Physics of ultra-short laser interaction with matter: From phonon excitation to ultimate transformations

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Physics of ultra-short laser interaction with matter: From phonon excitation to ultimate transformations E.G. Gamaly, A.V. Rode

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Cite this article as: E.G. Gamaly, A.V. Rode, Physics of ultra-short laser interaction with matter: From phonon excitation to ultimate transformations, Progress in Quantum Electronics, http://dx.doi.org/10.1016/j.pquantelec.2013.05.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

PHYSICS OF ULTRA-SHORT LASER INTERACTION WITH MATTER: FROM PHONON EXCITATION TO ULTIMATE TRANSFORMATIONS E. G. Gamaly, A. V. Rode1 Laser Physics Centre, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 0200, Australia

ABSTRACT This review encompasses ultrafast laser interaction with matter in a broad range of intensities ~1010 W/cm2 – 1015 W/cm2. We consider the material transformation processes successively with increase of the absorbed laser intensity, beginning with the subtle atomic displacements and excitation of phonons, and further up the phase transitions, ablation, transformation into plasma, and interaction with plasma up to the relativistic limit. The laser pulse is considered as of ultra-short duration if it is shorter the time scale of major energy relaxation processes such as the electron-to-lattice energy transfer, heat diffusion, and hydrodynamic motion. We describe the material response from the first principles, aiming to establish analytical scaling relations, which link the laser pulse characteristics with the properties of the material. Special section is dedicated to the possibility of creating superhigh pressure and temperature with an ultrashort tabletop laser. The influence of the laser polarisation on the material ionisation is discussed. We consider theoretical and experimental aspects of a newly emerging topic of interaction the ultrashort vortex beams and sculptured beams possessing complicated spatial and temporal distribution of intensity, polarisation, and the geometrical Berryphase with matter. In conclusion, we discuss future directions related to the lasers and diagnostic tools on the attosecond time scale and with the photons energy in the x-ray range.

Keywords: Ultra-short laser interaction with matter; electron-phonon coupling; laser-induced melting; pulsed laser ablation; laser-induced micro-explosion; PACS codes: 52.38.–r; 78.70–g; 79.20Ds; 71.38.–k; 72.10Di; 78.47.J–; 64.70.–p

1

E-mails: [email protected]; [email protected]

2

1. INTRODUCTION The arrival of sub-picosecond lasers in the 1980’s boosted the intensity by several orders of magnitude and offered the possibility to study atomic and molecular systems in very non-equilibrium conditions in tabletop experiments. Several definitions are needed to clarify the boundaries of the research field of material transformations induced by ultra-fast laser-matter interaction. First, the laser pulse is considered as of ultra-short duration, typically well in the sub-picosecond domain, if it is shorter than all the major relaxation processes, including the electron-to-lattice energy transfer, heat diffusion and hydrodynamic motion. Therefore, during the pulse an atomic movement is insignificant in the laser-affected solid and the atomic structure remains intact. To create such conditions the laser intensity should be in the range from ~1010 W/cm2 up to 1016 W/cm2, the upper intensity being well below the value when the electron motion becomes relativistic. A µJ laser pulse focused down to a micron-sized focal spot delivers to the surface the energy density, or fluence, from mJ/cm2 up to hundreds Joules/cm2. The associated absorbed energy density per unit volume delivered into a sample transforms material from subtle excitations to phase transformations, such as transitions from one crystalline structure to another, from solid to liquid, conversion to plasma, ablation and, in the case of the laser tightly focused in the bulk of a transparent dielectric – creation of plasma at extreme pressure and temperature, termed as Warm Dense Matter (WDM), confined inside the solid. Second, the salient feature of the ultra-short pulse laser-matter interaction is that ultra-short pulses excite only electrons, leaving the lattice cold during the pulse, and for the time required for the transfer of the absorbed energy from the hot electrons to the lattice after the pulse. Therefore, phase transformations in a laser-affected solid occur in non-equilibrium conditions, creating phase states drastically different from their equilibrium counterparts. The statistical distributions in the electron and lattice sub-systems are time-dependent, and the time required for attaining the equilibrium state depends heavily on the parameters of the laser pulse and the characteristics of the material. Another important feature of the ultra-short laser-matter interaction is that the affected area is of only a few hundred atomic layers, which relaxes within picoseconds after the excitation. Understanding the phenomena occurring on a time-scale of picoseconds and on a space-scale of nanometers is essential for ultra-short laser-matter interaction, bringing these studies closer to research of nanostructures. Ultra-short laser-matter interaction studies are multidisciplinary by their nature. The areas of physics include the interaction of electromagnetic field with solids and plasma, elements of atomic physics, plasma physics, optics, solid-state physics, and statistical physics. The phenomena described in this review are arranged successively in accordance with increasing energy density deposited in the laser-affected material, and therefore with increasing laser intensity. The basics of laser-matter interactions are described in Section 2. Section 3 is devoted to the excitation of subtle atomic motion, followed by coherent atomic vibrations in various solids at various energy densities below and well above the threshold for phase transformation. The non-equilibrium processes preceding phase transitions allow deeper insight into the nature of well-known transformations such as melting. Ultra-fast phase transitions induced by ultra-short lasers are the subject of Section 4. Ultra-fast evaporation processes driven by short laser pulses, laser ablation, are the subjects of Section 5. Processes of laser-matter interaction when the laser focal zone is confined inside the bulk of a transparent solid are discussed in Section 6. In Section 7 we briefly discuss numerous applications of ultra-short laser-solid interaction. In conclusion, we address some unresolved fundamental and applied problems that may be encountered in the near future.

3 We are using CGSE (Gaussian) system of units in the description of electromagnetic fields interaction with matter in this review. The reasons are the following. The electric and magnetic fields enter in equivalent manner into the Maxwell equations, into the electro-magnetic tensor and into the forces imposed on charges, and thus have the same dimensions in CGSE directly reflecting their nature. On the contrary, the dimensions of electric and magnetic fields in SI units are different, leading to confusion and misunderstanding of their nature. Many prominent physicists such as Richard Feynman in his lectures used CGSE units for description of electromagnetic fields, indicating this inconsistency in the SI system of units (see also the text books of Il’inskii and Keldysh [1] and Kittel [2]). We inserted the Appendices at the end of the text where the links are presented between the CGSE/MKS (esu) and SI units. The field of research covered in this review is very broad and embraces many thousands of references. The list of presented references is by no means exhaustive and reflects the authors’ views. 2. FUNDAMENTALS OF ULTRA-SHORT LASER ABSORPTION IN SOLIDS 2.1. Absorption of laser energy In order to create conditions for the ultra-short laser pulse mode interaction with matter, the laser pulse should comply with several strict conditions. First, the high-energy contrast should be achieved. That means that there should be no pre-pulse or after-pulse accompanying the main pulse, as these may contain an amount of energy sufficient to produce phase changes before or after the interaction. In practical terms the contrast ratio, which is the ratio of the average intensity during the main pulse to that in the pre-pulse should constitute ~109 or higher. We assume for simplicity in the theoretical considerations here that the intensity is constant over the focal spot and drops to zero at the focal spot boundaries. The laser pulse in this case has a so-called “top-hat” or “flat-top” spatial intensity distribution over the focal spot. The set of initial laser parameters sufficient for the description of laser-matter interaction includes the laser wavelength λ, energy per pulse Elas, pulse duration tp, and the focal spot area Sfoc, assuming that the focal area is a circle. Useful characteristics are also the average intensity I during the pulse time, defined as I = Elas S foc t p and measured in [W/cm2], and the fluence F as the surface energy density, F = Elas S foc [J/cm2].

These laser

characteristics allow one to describe many details of laser-matter interaction. We mainly consider interactions with strongly absorbing media. The absorption coefficient is defined as the ratio of the absorbed fluence (or intensity) to the incident one. The laser energy is absorbed in a thin layer, the skin layer ls, with a typical thickness of several tens of nanometers, which is much smaller than the diameter of the focal spot. For this reason the description of the interaction is presented in one-dimensional approximation in most cases. The laser-affected material in the skin layer remains at a density close to the initial solid density, as the hydrodynamic motion is negligible during the laser pulse. However, the material properties, and particularly the optical properties, rapidly change from the very beginning of the laser pulse. Below we consider how the ultra-short pulse affects the properties of the material, which remains at the initial solid density during the interaction time. As the electric field of the laser pulse penetrates into a solid, the absorbed part of the incident energy swiftly changes the state of the solid and its properties. field is coupled via the Maxwell equations to the material equations: r r The electromagnetic j = σ E ; D = ε0 E [3]. The electric field, E, current density, j, conductivity, σ, and induction, D, all

4 are the fu unctions of thhree space coo ordinates andd time. The transient t dieleectric function is defined in i general ass:

()

ε = ε0 ω + i

4 πσ

ω

;

(2.11)

where ω = 2 π c λ is the t laser fieldd frequency. The optical properties p off a solid undeer the action of o powerful laser pulse arre time-depen ndent functionns changing in i the course of interactionn. For most of o e meann free path, lmmfp, and the diistance travellled by electrrons during thhe considered cases the electron , both b are less than the skiin depth ls soo that the norrmal skin app proximation is wave periiod, valid: lmppp << ls; ve ω << ls , here ve is the electtron velocity. Then the laaser electric field f inside thhe medium is described byy the well-knnown solutionn for the norm mal skin-effectt [3]: ⎛ ω ε ⎞ ⎛ ωn x ⎞ E = E0 exp⎜⎜ i x ⎟⎟ = E0 exp⎜ i x− ⎟; labs ⎠ ⎝ c ⎝ c ⎠

where ε = ε r + iε i ; ε r = n2 + k 2 ; ε i = 2nk .

(2.22)

Thee absorption labs, or the skin depth ls, is inverselly

proportionnal to the imaaginary part of o the refractivve index, N ≡ ε = n + ik :

labs =

c . ωk

(2.33)

Usually the t absorption length is of the order of several tens t of nanoometers (1-6))×10-6 cm that comprisess hundreds off atomic layerrs.

Figure 2.11. Laser electric field E penettration into thee solid in norm mal skin effect and the absorb bed laser energgy

Eabs ∝ E 2 .

We inntroduce the incident laserr intensity, I 0 = c E0

2

8 π , and expresss the imaginary part of thhe

dielectric function thrrough the reeal and imagginary parts of refractivve index. coefficiennt A is defined d by the Fresnnel formula [33]: 4n 4n . A = 1− R = ≡ 2 2 2 n +1 + k 1+ ε

T The absorptioon

( )

Then the absorbed a enerrgy density raate, the energyy per unit vollume per unitt time, is [4]:

(2.44)

5

2A I r, z, t ; labs

(

Qabs =

)

(2.55)

where labss is the electriic field absorp ption depth, the t skin depthh from Eq.(2.3). It is insttructive to note that the abbsorbed energgy density ratte in Eq.(2.5)) is expressedd through the parameters which w all coulld be experim mentally meaasured withouut any ad hocc assumption ns. Integration of Eq.(2.5)) over the tim me 3 gives the energy densitty W(tp) in [J//cm ] absorbeed during the pulse p t p:

()

W tp =

2 A Fp labs

tp

; Fp =

t dt ; ∫ I ()

(2.66)

0

Fp is the fluence, f the energy per uniit area deliverred to the surrface by a laseer pulse, and Fa =AFp is thhe absorbed laser fluence.. Analyysis of Eq.(2.6) allows onee making sevveral importannt conclusionns. The enthaalpy of meltinng in metals is of the ordeer of several kJ/cm3, whilee the absorptiion depth com mprises 10-5-1 10-6 cm. Thuus, the necesssary absorbedd laser fluencee threshold foor melting of metal is in a range of 1-100 mJ/cm2. Onne could estimate the ablaation threshold in a similarr way. The ennthalpy of vapporization is conventionallly 15-20 tim mes larger than that of fusiion. Thus, thhe ablation thhreshold for m metals is in a range 0.1–0.5 J/cm2. Thhese estimatess are in a quallitative agreem ment with thee experimentaal figures. 2.2. Materrial response during the ulltrashort pulsse The absorbed a laserr energy is coontained mosstly in the electron sub-sysstem at the en nd of the ultraashort pulsse (Fig. 2.2). Maximum electron e tempperature Te, which w is reachhed at the ennd of the laseer pulse, cann be calculated from the energy conseervation law.. The electroons excited by b laser to thhe temperatuure Te << εF constitute c the degenerated electron gas with the num mber density , the Ferm mi

()

energy ε F ne , and wiith the heat capacity, c Ce = π 2 k B2 Te 2ε F . The energgy density off such electroon gas reads:: .

(2.77)

By equalizing Eqs.(2.66;2.7) and solvving for the eelectron tempeerature one obbtains:

()

k B2 Te,2m t p =

4 ε F AFp

π 2 ne labs

.

(2.88)

For alumiinium, for exaample (ne = 1..8×1023cm-3, εF = 11.63 eV V), the enthalppy of melting in equilibrium m of ~1 kJ/ccm3. This lev vel of energy density in thee skin layer of o ls ~10-6cm iis reached forr λ = 800 nm at 2 the absorb bed laser flueence of 0.5 mJ/cm m . The maximum m eneergy per electtron increasess to 4.8×10-200 J at the endd of the pulsse, which corrresponds to the temperatture 0.3 eV ((3,520 K). The T maximum m electron temperature t in i ultrafast laaser interactioon is almost nine times hhigher than thhe equilibrium m melting teemperature off aluminium. The electronn temperaturee at the depossited energy density d arounnd the energyy for vaporizaation is in a range r of severral electron volts, v which iss sufficient foor ionisation of o the materiial. This is an a indication that t the assum mption of connstant optical properties duuring the pulsse fails. Thuus the maximuum electron temperature (iin Kelvin deggrees) in the ioonised solid is: i Te, max =

4 AFp 3k B ne labs

.

(2.99)

6 Maximum lattice temperature can be estimated under the assumption that the energy losses for ionisation, and for the electron and lattice heat conduction are all negligible. The maximum lattice temperature is reached at the moment when the electron-to-lattice energy transfer is completed, Te = TL = T, it can be calculated from the energy conservation:

()

Ce neT + CL na T =W t p .

(2.10)

At low excitation the electron heat capacity is taken as that for the degenerate electrons, while the lattice heat capacity has the value close to that of the Dulong-Petit law, CL = 3k B as for the unperturbed solid. In the conditions close to the ablation threshold CL ≈ Ce ≈ 3k B 2 , and if the electron and atomic densities are comparable, the maximum lattice temperature reads TL,max:

TL, max =

4 AFp

(

)

3k B ne + na labs

.

(2.11)

Figure 2.2. Qualitative time dependence of electron and lattice temperature in the skin layer. The dotted line is a Gaussian shape of fs-laser pulse; te-ph indicates the energy equilibration time.

2.3. Relaxation processes in laser-excited solids The salient feature of femtosecond laser-matter interaction is that only electrons are heated up by the laser energy during the pulse, while the lattice remains cold. The electron-lattice energy exchange through electron-lattice collisions (or, electron-phonon momentum exchange) governs transient optical and transport properties of excited metal while the energy exchange rate controls relaxation to equilibrium where the electron and lattice temperatures coincide. Thus, the knowledge of electronlattice momentum exchange rate and energy relaxation rate are of utmost importance for prediction and interpretation of the experimental results. The problem of energy exchange between electrons and crystalline lattice has been first formulated by Kaganov et al., [5]. The developed in [5] approach was simplified in [6], and twotemperature (electron/phonon) approximation was introduced with a constant energy exchange rate. The energy exchange rate for the calculation of energy transfer from electrons to the lattice was unjustifiably considered to be constant [7-11] or linearly dependent on lattice temperature [1] in the last decades. The electron diffusion (the energy transport) depends on the electron-phonon mom momentum exchange rate ν e− ph , the electron velocity ve, and on the electrons heat capacity. Therefore the heat conduction coefficient and characteristic cooling time both are the functions of electron and lattice temperature. Detailed examination of electron-electron, phonon-phonon, electronphonon collision rates and the electron-lattice/ions energy exchange rates, as well as the dependencies

7 of these rates on the electron temperature and density is presented in [12]. A general formula for the dependence of the electron collision rate on the electron energy ranging from a small excitation over the Fermi level (low temperature) up to the hot plasma state has not developed yet, to the best of our knowledge. However, the processes of electron-phonon interaction are well understood qualitatively and quantitatively in a low energy range k B Te << ε F ; as well as in the high-temperature ideal plasma state at k B Te > ε F . Moreover, the qualitative understanding also exists for transition from solid to ideal plasma through the non-ideal plasma state. The electron-phonon momentum exchange rate (the collision rate) and momentum exchange time are [12]:

( )

3 2

mom ν e− ph TL = Cω

k B TL ; h

( );

mom mom te− ph = ν e− ph

−1

(2.12)

where Cω is a dimensionless proportionality coefficient which depends on the material properties and should be determined from experiments. The momentum exchange rate is independent on temperature below the Debye temperature TL=TD, and is proportional to the lattice temperature in the range from the Debye temperature up to the melting point [1,2,13]. The electron-phonon momentum exchange time at the Debye temperature is of the order of 10 fs – 100 fs, it decreases with the increase of the lattice temperature. The above scaling gives a reasonable semi-quantitative estimate for the momentum relaxation time. The electron-phonon energy exchange rate and time are:

(k T ) = 2k T 2

ν

en e− ph

(T )≅ 3C L

ω

B L

εF h

B L

εF

mom ν e− ph ;

( ).

en en te− ph = ν e− ph

−1

(2.13)

The electron-phonon momentum and the energy exchange rates coincide at approximately k B TL ≈ ε F 2 when a metal approaches the ablation threshold, and solid is converted into non-ideal plasma. The approximation used for description of electron-phonon interaction fails at these conditions. However, in a non-ideal plasma the electron-electron collision rate increases in direct proportion to the increase in the electron energy, ν ee ∝ ω pe ε e ε F , when approaching the Fermi level from the low energy range, qualitatively supporting the above estimate. The growth rate slows down with increase of temperature when the solid becomes partly ionised and the Coulomb collisions become significant. The dependence of the collision rates on temperature becomes weaker due to the effects in non-ideal plasma discussed above. When the energy per atom approaches the ablation limit at k B TL ≈ ε b , the electron-phonon collision rate becomes close to the atomic frequency, max ν e− ph ∝ ε b h ~ ω at ~ ω pe apparently approaching the maximum value. In plasma, where the Coulomb

collisions dominate, the electron-ion momentum exchange rate decreases with the increase of

(

mom ∝ ne Te−3 2 ∝ ε F Te temperature, ν e−i

)

32

[13,14]. Thus, the existence of a maximum around the

electron plasma frequency seems qualitatively obvious. Eidmann et al. [15] suggested that the mom collision rate reaches its maximum value close to the electron plasma frequency, ν e−i ≈ ω pe ≈ 1016 s -1 . However the exact value of the maximum is still unresolved, to the best of our knowledge. We present the collision rate dependence on temperature in the whole range from electron-phonon interaction to electron-ion interaction in plasma in Fig.2.3 for Aluminium. For the electron-phonon rate the linear dependence is taken while the electron-ion collision rates in solid density Al plasma

()

(

mom approximated by the following form: ν e−i T ≈ 1.4 ×1016 ε F k B Te

)

32

s-1 [16].

8 At high excitation level k B Te ≈ ε F and above, when a solid becomes fully ionised, the energy transfer rate from electrons to ions with charge eZ is as the following [16-18]:

( )

en ν e−i = 12 2 π

12

ni Z 2 e 4 m1e 2 ln Λ ei . Te3 2 M i

(2.14)

Electron transfers only a small part of energy in a single collision with ion, ε e me M i . Hence, the energy transfer time is longer than the momentum transfer by the ratio of ion to electron mass: en ν e−i =

me mom en M i mom ν e−i ; te−i = te−i . Mi me

(2.15)

Therefore, the energy transfer time by the Coulomb collisions, in the conditions when the electron-ion momentum transfer is at its maximum ν eimom ≈ ω pe , resides in a range of several picoseconds. Note that these conditions are close to the optical breakdown and the ablation threshold. For example, in Copper (Mi = 63.54 a.u.) at εe ~ εF, the electron-ion momentum transfer rate is ν eimom ≈ 0.5ω pe ~1016 s-1, while the energy exchange rate in accordance to Eq.(2.14) equals to 8.6×1010 s-1. Thus, the electronto-ion energy transfer occurs in 11.6 ps.

mom Figure 2.3. Temperature dependence of the electron-phonon collision rate ν e− ph at low temperature (left solid

mom line) and electron-ion rate ν e− i at high temperature (right solid line) for Aluminium. Solid lines are the known

functions for electron–phonon interaction from this paper and for electron-ion collision rate from [16]. The dashed line is conjectured qualitative behaviour during solid-to-plasma transition.

The momentum exchange rate in metals can be extracted from the optical properties, which are well documented, see [19] for example. The electron-phonon momentum exchange rate directly relates to the real, ε r , and imaginary, εi , parts of the dielectric permittivity if the former obeys the

(

)

mom Drude-like form, which is applicable to metals ν e− ph ωl = ε i 1− ε r ; ωl is the laser frequency.

Therefore the Eqs.(2.12;2.13) allow one to express the energy exchange time through the single mom experimentally determined parameter, momentum relaxation time at room temperature, te− ph Troom , in whole temperature range without ad hoc assumptions. Taking Al, for example, ν

( )

mom e− ph

( ) (T ) ~10 room

15

12 -1 en s-1 at room temperature, and ν e− ph Troom = 4.29×10 s , so that the energy relaxation time is 233 fs.

One can present relaxation time ratios in the dimensionless form applicable to any metal characterized by its Fermi energy:

9

( ) =T (T ) T

mom te− ph TL mom te− ph

room

room

, and

L

( ) = ⎛⎜ ε (T ) ⎝ k T

en te− ph TL mom te− ph

room

⎞⎛ T ⎞2 room ⎟⎜ ⎟. room ⎠⎝ TL ⎠

F

B

(2.16)

These time ratios are plotted for Be (εF = 14.3 eV), Al (εF = 11.7 eV), Bi (εF = 9.9 eV), Cu (εF = 7.7 eV), and Au (εF = 5.53 eV), as an example, in Fig.2.4. The optical properties of Bismuth at 800 nm were measured at equilibrium from room temperature, 293 K, [20] up to 773 K well over the melting point of 544.7 K [21-24]. Under suggestion that electron has a free electron mass value, the momentum transfer rate for Bi follows the linear dependence on the temperature with good accuracy [25]. The momentum transfer rate extracted from equilibrium experiments at room temperature mom comprises ν e− = 2×1015 s-1 [21,22]. The corresponding energy exchange rate equals to ph Troom

( )

13

-1

1.17×10 s giving much shorter relaxation times than it would be estimated without considering the temperature dependence of the energy transfer rate.

Figure 2.4. Temperature dependences of the electron-phonon momentum relaxation time at elevated lattice temperature TL relative to that time at room temperature of Troom = 293 K, and the ratio of energy transfer to the momentum transfer times for beryllium, aluminium, bismuth, copper, and gold. The lowest curve is universal ration of momentum rate to the energy exchange rate from the theory.

2.4. Importance of the high-energy tail in the Maxwell distribution It is known that atoms from the high-energy tail of the Maxwell distribution play important role in such equilibrium processes as melting [26] and evaporation [27]. The proper concentration of thermal point defects (7–10%) is crucial for the initiation of the equilibrium melting at the melting point, Tmelt. The energy for formation of such defect is approximately 10×Tmelt. Therefore, in order to create such defects at the melting point the atoms with higher than the average energy should be generated. Similarly, in order to evaporate an atom in equilibrium, i.e. to remove it from a solid, the energy in excess of the binding energy εb should be supplied to the atom. Conventionally, the binding energy per atom is 10-20 times larger than the vaporisation energy. Thus, melting and vaporisation at the melting point and at the vaporisation temperature could proceed only if the high-energy tail in the atomic distribution is well established. The primary process of electron excitation by the ultra-short laser pulse occurs in totally nonequilibrium conditions. The following processes take place in succession for some time after the end

10 of the laser pulse while a solid attains the full equilibrium state. First, electron-electron and phononphonon collisions lead to the establishment of local quasi-equilibrium distributions in both subsystems. The electrons establish a conventional Fermi-Dirac distribution with Te < TF. However, the energy distribution of the atoms in the lattice has the form of the incomplete Maxwell distribution at the end of the laser pulse. The high-energy tail is absent because the creation of particles with the energy larger than the average energy needs much more collisions and therefore the longer time to be filled in. The main part of the Maxwell distribution establishes fast, it comprises 80-90% of the atoms. The time for building up the high-energy tail appears to be the longest of all the relaxation times. The electron-phonon momentum exchange rate, being responsible for light absorption and transport processes, plays an important role during the interaction. The electron-phonon energy exchange leads to equilibration of electron and lattice temperatures before the complete Maxwell distribution is established [28].

Figure 2.5.

Growth of the high-energy tail in Maxwell distribution of energy in the lattice with time:

t 1 < t 2 < … < t 5.

The time for building up the high-energy tail in the Maxwell distribution is [14]: −1

⎛ ε ⎞ ⎛ ε ⎞ ttail ≈ 0.85tmain ⎜ ⎟ exp⎜ ⎟. ⎝ kBT ⎠ ⎝ kB T ⎠

(2.17)

The time for building up the tail at the energy ε ~10kBT takes ~103 longer than that for establishing the main part of the equilibrium distribution. This is the longest of the relaxation times; it comprises from tens to hundred of picoseconds. It is necessary to note that in the processes of melting and ablation the role of the tail in phase transformation is not only in reaching a particular energy threshold. It is necessary to fill up the tail to the extent when the number of energetic atoms constitutes several percents of the total number of atoms. Therefore, time is required to fill up the high-energy tail in the energy distribution to create the necessary number of particles with the energy exceeding the threshold. This time is much longer due to the exponential dependence on the energy. 2.5. Two-temperature approximation

In many cases of ultra-short laser matter interaction a laser-excited solid remains intact, i.e. the macroscopic atomic motion could be neglected. Let us find how long and at what absorbed energy level this assumption holds. An atom could be removed from a solid (ablated) if it receives the

11 energy in excess of the binding energy. In the conditions close to the ablation threshold the kinetic velocity of a freed atom is of the same order of magnitude as the sound velocity vsound ~ 105 cm/s. Thus the expansion time could be estimated as the time when a surface atom moves on a distance comparable to the absorption depth, thydro ≈ labs vsound ~(2÷3)×10-11 s. This means that a solid remains practically intact during several tens of picoseconds after ~ 100 fs pulse even when the deposited energy density exceeds the ablation threshold. Thus, the hydrodynamic motion is not important during the laser pulse and long after of the electron-to-lattice energy transfer time. A legitimate description of the state of laser-excited medium during the time when the mass and momentum are unchanged, is given by two coupled equations, which express the energy conservation law [13]:

(

)

(

)

⎧∂ C n T e e e ⎪ = Qabs − ∇Qe−th − Qe− ph ⎪ ∂t ⎨ ⎪ ∂ CL na Te = Qe− ph ⎪⎩ ∂t

(2.18)

where Ce(Te) and CL(TL) are correspondingly heat capacities for the electrons and the lattice, both are functions of electron and lattice temperature; Qabs is the absorbed laser energy rate given by Eq.(2.5), where the spatial dependence of absorbed energy in the skin layer is taken into account:

Qabs =

⎛ 2x ⎞ 2A I t exp⎜ − ⎟; ls ⎝ ls ⎠

()

(2.19)

Qe-th is the electronic heat conduction flux:

Qe − th = κ e ∇Te ;

(2.20)

and Qe-ph is the energy transfer from electrons to the lattice (ions):

( ) (

)

Qe − ph = ν een− ph TL ne Te − TL .

(2.21)

The electron heat conduction coefficient κe is expressed through the electron’s heat capacity Ce, the electron number density ne, and diffusion coefficient [17]:

()

κ e = Ce Te ne Ddiff

(2.22)

Diffusion coefficient reads:

Ddiff =

le ve ve2 = mom 3 3ν e− ph

(2.23)

mom The electron mean-free-path, le = ve ve− ph , is conventionally expressed through the electron velocity

and electron-phonon momentum exchange rate. vF2

In the low-excitation regime Te << εF the electron velocity is close to the Fermi velocity = 2ε F me∗ where me∗ is the electron effective mass, so that the heat conduction coefficient in

metals is:

κe =

⎛T ⎞ 2 π 2 hk B e n ⎟. ⎜ e 9 Cω me∗ ⎝ TL ⎠

(2.24)

12 The heat conduction coefficient is temperature-independent after the electron-lattice temperature equilibration [29]. The characteristic time for cooling the absorption layer swiftly heated by laser follows directly from the heat conduction equation: l2 tcool ≅ s . (2.25) Ddiff The diffusion approximation for description the heat conduction in metals in a skin layer with thickness in a range 10–30 nm is a legitimate description of the transport processes at both low and high excitation levels. The only difference is that the electron-phonon collision rate at low excitation is replaced with electron-ion collision rate at high excitation near and above the ablation threshold. 2.6. Diffusion versus ballistic transport There are a number of papers where, while describing ultra-fast laser-matter interaction, the electronic energy transfer in laser-excited solid is treated as ballistic transport [30,31]. We should consider and compare the relevance of ballistic transport and electrons’ diffusion from the previous section to the processes in the fast heated solid. Transport of electrons is defined as ballistic when (i) – the collisions of externally injected electrons with material electrons could be neglected because the majority of electron states are filled up, and (ii) – the number density of the injected electrons is much lower than the density of material electrons, and thus the energy/momentum losses due to scattering are negligible. The motion of electrons then obeys the second Newton’s law. Thus, the ballistic, collisionless transport occurs when all electron collisions are negligible. To justify the ballistic transport mechanism, the influence of electron collisions should be negligible, thus the physical processes for collisionless transport has to be clearly elucidated. The examples justifying the application of the ballistic transport concept to the laser heating relate to the interaction of a single electron (or a weak electron beam) having energy slightly over the Fermi level with the degenerated electrons – see Quinn and Ferrel [32] and references therein. They stated explicitly that in ballistic transport “… the damping [of single electron states] is greatly retarded by the collective screening of the Coulomb field of the electron by the [degenerate] gas, acting as a dielectric medium. Without the screening the effect of the collisions would be so drastic that the mean life would be vanishingly small, and the concept of the single electron excitations […] would have no validity” [32]. In the degenerate gas all the states are filled up. Therefore this gas might be considered as a kind of vacuum in quantum electrodynamics. The above problem of electron’s damping is drastically different from the ultra-fast heating of metals by fs lasers. Indeed, at the intensity well below the damage threshold only small parts of electrons near the Fermi level is heated to kBTe << εF. As we show above, the electron-phonon collisions are the major reason for the electrons’ damping. At the intensity close to the ablation threshold the ultra-fast laser heats the whole sea of electrons in metal, raising the average electrons energy well over the Fermi level and inducing the consequent transformation of the electron distribution from the degenerate state to the Maxwell. Along with the transformation of electron distribution, the metal transforms to the solid density plasma. The electron transport during this transformation occurs in solid density plasma by diffusion where the main processes are electronelectron and electron-ion Coulomb collisions described in the previous section. Thus, in all ultra-fast laser-matter interaction processes the electron collisions is the dominant process defining the momentum/energy exchange and transport processes. Fig.2.6 illustrates qualitatively the transformation of the electron distribution function near the ablation threshold.

13

Figure 2.6. Evolution of energy distribution of electrons with increase of temperature in the material at the electron temperature kBTe = 0 (a); at kBTe << εF (b); kBTe ≅ εF (c); and kBTe >> εF (d).

3. EXCITATION AND DECAY OF COHERENT PHONONS The advent of femtosecond lasers in the early 1980s was accompanied by the development of a diagnostic technique for fast probing of swiftly excited solids by x-ray, optical and electronic probes with time resolution in the range 10 fs – 100 fs. These achievements generated a broad variety of experiments, which for the first time made possible the observation of new phenomena in swiftly excited solids on the femtosecond time scale and on a space-scale of tens of nanometers. The most spectacular observations are the oscillations in the optical probe reflection from a laser-excited solid with a frequency close to that of cold phonons in a solid [33,34]. It was also found that the intensity of an x-ray probe beam diffracted from the laser-excited solid oscillates with the cold-phonon frequency [35]. The excitation and detection of coherent lattice vibrations has been produced in many transparent and opaque materials, such as semi-metals [33-38], transition metals [39], cuprates [40], insulators [41], and semi-conductors [42]. It was then realised that the generation of a phonon is preceded and followed by a subtle laser-induced atomic motion that is imprinted into transient material properties of the excited solid [43]. The various stages of laser-induced atomic motion were identified on the basis of analysis of the experimental data [44]. First, the electronic excitation during the pulse, which is shorter than the electron-phonon energy transfer time, produces fast coherent atomic displacement followed by the harmonic vibrations of the lattice with the unperturbed phonon frequency because the lattice remains cold. Heating of the lattice leads to decay of harmonic modes due to non-linear phonon-phonon interaction, and the coherent vibrations cease to exist. Thus, observing the timedependent history of the reflected probe beam allows one to retrieve the atomic motion on the subpicosecond and picosecond time scale. These processes gradually transform the material phase state during the time period up to several tens of picoseconds. The onset of thermal expansion and instability manifests the onset of the transformation of the materials, eventually leading to the disorder and melting of the lattice. The studies of atomic motion on a femtosecond time scale and on a nanometer spatial scale at the pre-melting stage allow a better understanding of the microscopic processes preceding the transformation and disordering of the material. Studies of all stages of the laser-induced atomic motion allow us to obtain a deep insight into the microscopic nature of the material transformations produced by laser. The ability to drive and control transient atomic motion via an external photon flux opens a number of interesting applications, such as the possibility to induce phase transitions: transient phase state [43,45], paraelectric-to-ferroelectric [46] or insulator-to metal transitions [47], the selective

14 opening of the “caps” of nano-tubes in non-equilibrium conditions [48]. They also provide a basis for SASER (Sound Amplification by Stimulated Emission of Radiation) experiments [49]. Below we consider the interconnected processes of laser interaction with matter at a level of absorbed energy close to and exceeding the equilibrium enthalpy of melting. The laser-exerted forces start displacing atoms early in the pulse time. The atomic motion in turn affects the optical and material properties. Thus, observing the time-dependent history of the reflected probe beam allows one to retrieve the atomic motion on the sub-picosecond time scale. These processes gradually transform the material phase state during the time period up to several tens of picoseconds. For quantitative characterisation of ultra-fast processes in laser-excited solid we take the experimental and theoretical results for Bismuth as an example, which is one of the most studied solids in respect to the ultra-fast excitation. The enthalpy of fusion in Bi is 0.5 kJ/cm3 [20], just in the middle of the whole range of the melting enthalpy values for solids. It is essential that the temperature-dependent optical properties of Bi introduced below were retrieved from the experiments in equilibrium (see [25,50], and references therein) in agreement with the theory presented in Section 2 above. The experimental data indicate that Bismuth can be treated as a good metal obeying the Drudelike dielectric function at elevated temperature range from room temperature and up to 200 K over the melting point. Absorption coefficient and the skin depth for 800-nm laser light in Bismuth are respectively A = 0.26; ls = 2.98×10-6 cm [44]. Thus, 800-nm laser pulses with the fluence, F(tp) = 2.7 mJ/cm2 – 6.7 mJ/cm2, deposit the energy density in Bi from the enthalpy of melting in equilibrium to twice higher value (0.48 – 1.19)×103 J/cm3 [44]. The maximum electron temperature for the above

[

( ) π n l]

range of fluences was found to be k B Temax = 4ε F AF t p

12

2

e s

, Temax = 2,825 K – 4,450 K (0.24

– 0.38 eV). The maximum lattice temperature is, correspondingly, 701.5 K and 1,273 K (0.06 – 0.11 eV). Thus, the first distinctive feature of the ultra-fast excitation is apparent: the maximum electron temperature is at least three times larger than the lattice temperature at the deposited energy density comparable to the equilibrium enthalpy of melting. Therefore, one may expect a strong impact of the electron excitation on the transient state and on the atomic displacements in the excited material. It is worth noting that the melting point for Bismuth is Tmelt = 544.7 K. Hence, another question arises: when and how the solid transforms into a different phase state in non-equilibrium conditions under fslaser excitation? 3.1. Temperature dependence of optical properties in equilibrium The measurements of the reflectivity and the dielectric function by the ellipsometry technique give simultaneously the real and imaginary parts of dielectric function. Indeed, the reflectivity, R, is directly related to the real εr and imaginary εi parts of the dielectric function, ε = ε r + iεi through the Fresnel formula:

R=

ε −1 ε +1

2

.

(3.1)

The real part of dielectric function of solid Bi is εr = –16.25 at 800 nm at room temperature, indicating Bi has metallic property at this wavelength [20,22], so we can safely assume it can be described by the Drude form. It was experimentally proved that liquid Bi obeys the Drude-like form [21,23,24,51],

115 which is directly d linkeed to the plasma frequencyy ω pe and the electron-phhonon momenntum exchangge m mom rate, ν e− ph :

εr = 1−

ω 2pe

( )

ω +ν 2

; εi = 2

mom e− ph

ω 2pe

( )

ω +ν 2

mom e − ph

2

m ν emo− ph p ; ω

(3.22)

here ω is the laser fielld frequency, and ω 2pe = 4 π e2 ne me∗ iss the squaredd plasma frequ uency; e is thhe electron charge c and me∗ is the effective electron mass.

Thus the ellectron-phonoon momentum m

exchange rate and plaasma frequency are directtly connected d to the real and imaginarry parts of thhe dielectric function: mom ν e− ω 2pe ε ph h = i ; = 1− ε r ω 1− ε r ω2

(

⎡ ⎛ mom ⎞2 ⎤ ⎢ ⎜ ν e − ph ⎟ ⎥ ⎢1 + ⎜ ω ⎟ ⎥. ⎠ ⎥⎦ ⎢⎣ ⎝

)

(3.33)

The number n density of the coonductivity eelectrons can be directly retrieved froom the plasm ma frequencyy assuming th he electron mass m is knownn. We illustrrate this takinng bismuth as a an examplle. The follow wing bismuth h properties were w recovered from the elllipsometry measurements m in equilibrium m conditionss at the room temperature with the helpp of Eq. (3.3),, for the laserr wavelength of o 800 nm ( = 2.356×11015 s-1): ω 2pe ω 2 = 31; R = 0.74; = 2.1×1015 s-1; ne = 5.34×1022 cm-3; thee Fermi energgy εF = 5.4 eV (the Ferm mi velocity iss vF = 1.38×108 cm/s) under assumption that electtron has a freee m Cominss [21] found that t liquid Bi at 773 K has the followingg parameters:: reflectivity is electron mass. 23 R = 0.67; ω 2pe ω 2 = 81.58 that corrresponds to thhe number den nsity of free carriers c ne = 1.42×10 1 cm-3 . Thereforee all 5 valencce electrons are a transferreed into condu uction band iin liquid Bi. Comins [21] suggestedd that in liquiid Bi electronns have a freee mass valuee, then εF = 99.92 eV, and vF = 1.87×1008 cm/s. Thhe electron-ph honon momenntum exchangge rate reads ν emom 7×1015 s-1. Innterpolation oof − ph = 5.67 the data extracted fro om all availabble experimeents [21-23], and keepingg in mind thhe theory from m mom Section 2 shows that ν e− ph grows upp in direct prooportion to teemperature (F Fig.3.1): 15 ν emom − ph = 2 ×10

T Troom

.

(3.44)

The linearr dependencee of the momeentum exchannge rate on teemperature fitts the optical measuremennts for bismuuth in equilibrrium with suffficient accurracy; the prop portionality too the temperaature holds foor Bi well beefore and longg after the meelting point. Takinng the theoretiically establisshed link betw ween the temp perature depeendent energyy exchange annd the mom mentum exchaange frequenncy (see Secction 2) andd using the experimentaal temperaturre dependence Eq.(3.4) one obtainss the electroon-phonon energy e exchaange rate dependence d o on temperatuure: en 15 ν e− ph ≅ 10

2 ⎛ T ⎞2 -1 k B Troomm ⎛ T ⎞ 1 ×1013 ⎜ ⎜ ⎟ = 1.17 ⎟ s . ε b ⎝ Troom ⎠ ⎝ Trooom ⎠

(3.55)

The resullts of theoretiical calculations of the ennergy exchannge rate are ccompared in Fig.3.2 to thhe experimenntal data.

116

Figure 3.11. Temperature dependencies of the momenntum rate in equ uilibrium (left)), and energy exchange e (rightt). Solid liness correspond to o theoretical dependencies d E Eq.(3.4, 3.5). The T circles are the results froom ellipsometrry measuremeents from [22],, triangles – froom [21], squarees – from [23]..

Now it is possiblee to calculate the dielectricc function annd reflectivityy changes witth temperaturre, the resultss are presenteed in Fig.3.2. The dielectriic function at room temperrature was meeasured to be εr = –16.25,, εi = 15.4 [22,44] at 775 nm; the valuues are very close c to the literature dataa [20,51]. Thhe electron-pphonon collisiion rate and plasma p frequeency at room temperature w were found with w the help of o mom 2 2 ω 1.0. The theo Eqs. (3.3)) as ν e− = 0.893, and = 3 ory predicts w well the exper rimental resul lts ω ω ph pe of dielectrric function measurements m s for liquid bissmuth at 610 K [23] and 773 K [21].

Figure 3.22. Dependencee of real εr annd imaginary εi parts of the dielectric d functtion (a) and reeflectivity (b) of bismuth att 775 nm on temperature t in equilibrium. The solid linees were calcullated using Eqqs.(3.1, 3.2); thhe circles are results of elliipsometry meaasurements froom [22], trianggles –from [21], squares –froom [23], arrow ws nt Tm = 544.7K.. indicate thee melting poin

The electron e numb ber density in the conductioon band is rettrieved from tthe plasma frrequency undeer assumptioon that the ellectron mass is equal to thhe free electrron mass: m* = me. The result at room m temperatuure gives ne = 5.34×1022 cm m-3 (1.89 electtrons out of total 5 are in the t conductio on band) and εF = 5.4 eV V. At 773 K in liquid bismuth alll five electroons are in thhe conductioon band: ne =

17 1.42×1023 cm-3, εF = 9.92 eV, and the dielectric function obeys the Drude-like form [21,44]. Thus the experiments had shown that from 40 to 100% of the valence electrons are transferred to the conduction band at the temperature increase from the room temperature to the melting point. Thermal diffusivity (coefficient of thermal diffusion) relates to both the Fermi energy and the electron-phonon momentum exchange rate – see Eq.(2.23). The diffusivity values recovered from the temperature dependency of the momentum exchange rate in equilibrium from Fig.3.1, are Ddiff = (2.0 – 2.89) cm2/s for the temperature range 293 K – 793 K. These results are in good agreement with the recent non-equilibrium measurements from the x-ray reflectivity data of fs-laser excited bismuth giving D = 2.3 cm2/s [38]. Note the sharp difference of this value from 0.067–0.075 cm2/s given for the equilibrium conditions in the reference book [20,52]. Accordingly, the electron mean free path, mom lmfp = vF ν e− ph = 0.67 nm << ls, is much less than the skin layer depth and the film thickness, assuring the legitimacy of the diffusion approximation for the electron heat transfer. 3.2. Atomic motion in fs-laser excited solid A general scenario for the swift excitation of the atomic motion by fs-laser is as the following [12,14]. In the unperturbed solid at initial temperature the atoms oscillate around their equilibrium positions. The fast excitation heats electrons, while the atoms continue their vibrations as in the unperturbed solid. The electrons reach maximum temperature and create the electron temperature gradient in the skin layer at the end of the laser pulse. The force, proportional to the electronic pressure (temperature) gradient, acts on atoms inducing the coherent atomic motion. This force acts during the time shorter than the phonon’s period ( t ph = 2 π ω ph ~ few hundreds fs) and produces fast atomic displacement. The magnitude of this force is proportional to the absorbed energy density. Even at the energy density of the order of the enthalpy of melting in equilibrium the elastic force responsible for the atomic vibrations in the unperturbed solid is still larger than the electronic force. Therefore, the electronic force acts as a short blow inducing the atomic vibrations with the “cold” phonon frequency. The following stage of atomic motion in time corresponds to harmonic vibrations with “cold” phonon frequency, which lasts during the period for the energy transfer from the electrons to the lattice. The period of harmonic vibrations continues until the lattice acquires temperature close to the melting point. During this period atomic vibrations gradually lose the harmonic character. Non-linear interaction between different phonon modes becomes dominant and that eventually transforms a solid into a different phase or into a disordered state. One can also describe qualitatively the fast atomic motion in a swiftly excited solid using the language of the inter-atomic interaction (Fig.3.3). Indeed, the inter-atomic potential consists of the attractive electronic part and that of repulsion of ionic cores. Fast electronic excitation decreases attraction and therefore reduces the binding energy. The inter-atomic distance increases as a result of electronic excitation. Taking the inter-atomic potential in the Morse-like form allows one expressing the atomic displacement as a function of the electron temperature, binding energy, equilibrium interatomic distance and the gradients of attractive and repulsive parts of the potential. It is instructive first to re-visit the atomic vibrations in a “cold”, unperturbed solid. As a first approximation the spatial dispersion can be neglected and the excited phonon is considered as a standing wave.

18

Figure 3.3. Qualitative picture for the change of the potential energy due to electronic excitation.

The elastic force driving harmonic vibrations in a solid expresses through the quadratic perturbation term in the inter-atomic potential, which has the form:

ΔU el =

1 2

⎛ ∂ 2U ⎞ 1 2 2 2 ⎜⎜ 2 ⎟⎟ q ≈ 2 Mω0 q . ⎝ ∂q ⎠0

(

One estimates ∂ 2U ∂q 2

) ≈ε 0

b

(3.6)

d 2 ; here q is the cold phonon amplitude, εb is the binding (cohesive)

energy, and d is inter-atomic distance in equilibrium. The cold phonon frequency in Eq.(3.6) is ω02 ≈ εb Md 2 . The elastic force driving harmonic vibrations immediately follows from Eq.(3.6):

Fel =

∂ ΔU el ≈ M ω02 q . ∂q

(3.7)

The cold phonon amplitude at a temperature lower than the Debye temperature is estimated as [1]: 12

⎛ 2h ⎞ q0 ≈ ⎜ ⎟ ⎝ Mω0 ⎠

(3.8)

Now the elastic force explicitly expresses through the basic characteristics of a solid:

(

).

Fel ≈ Mω03 h

12

(3.9)

For example, in the unexcited Bi (atomic mass, MBi = 3.47×10-22 g, inter-atomic distance in cdirection d =3.3 Å, and εb = 2.16 eV) the longitudinal A1g optical mode (vibration in c-direction) has a frequency of 3×1012 s-1, which qualitatively complies with the estimate by the above formula for the cold phonon frequency. The amplitude of the cold phonon from Eq. (3.8) equals to q0 = 1.4×10-9 cm. Now the elastic force driving vibrations in cold Bi is Fel = M Bi ωo2 q0 = 4.4×10-6 erg/cm. The laser electric field at the moderate intensity induces internal atomic motion (deformations) in solids and liquids, while the total volume can be considered being constant. The atomic motion brings changes to the initially homogeneous dielectric function. The stress tensor for initially isotropic medium reads [3]:

19 ⎛ E 2δ

σ ik = −P ⋅ δik − ⎜⎜

⎝ 8π

ij

+

2⎡ ⎛ Ei E j ⎞ ∂ε ⎟ ⋅ ε jk + E ⎢na ⎜⎜ jk ⎟ 4π ⎠ 8 π ⎢⎣ ⎝ ∂na

⎞ ⎤ ED ⎟⎟ ⎥⋅ δij + i k ; 4π ⎠T ⎥⎦

(3.10)

where the displacement vector has a form Dk = εkj E j . We assume that the dielectric tensor modified

(p)

by the laser effect consists of two terms, the Drude–like term, ε D , and polarisation term, ε jk :

(p) ε jk = ε D ⋅ δ jk + ε jk .

(3.11)

In the transient state electron number density, dielectric function, electron and lattice temperature are all time-dependent. The volume forces induced by the laser field express through the stress tensor modified by the field action, σ ik , taken from Eq.(3.10) as follows [49]: fi =

(

)

(p) ∂σ ik ∂P ∂ε ik E2 ε D − 1 ∂E2 (p) =− + + = f i th + f i + f i pond . ∂x k ∂x k ∂x k 8 π 8 π ∂x i

(3.12)

Here we took into account that for the Drude-like part of the dielectric function the following relation holds, na ∂ε D ∂na = ε D − 1. The first term in Eq.(3.12) is the thermal force of electronic pressure

(

)

T

introduced earlier. The second term is the polarisation force with the polarizability inverse proportional to the atomic displacement (the Plazcek effect, see [14]), and third term is the ponderomotive force. Note that polarization and ponderomotive forces are effective only during the pulse, while thermal force drives atomic motion until spatial smoothing of temperature gradients. It is also worth noting that polarisation force in Eq. (3.12) is similar but not identical to the force driving phonons excitation in the Raman effect [54]. The difference relates to the fact that laser pulse duration is much shorter than the phonon’s period. Therefore, there is no interaction between laser electric field and vibrational field of atomic motion; the phonon frequency does not enter into polarisation force explicitly. Let us compare the laser-exerted forces to each other. We take the polarizability and dielectric tensor in the Placzek form [14,54], and estimate the changes as, ∂χ ik ∂x k ≈ χ 0 d , d being the

(

)

0

inter-atomic distance. The unperturbed polarizability estimates with the Lorentz-Lorenz formula, χ 0 = 3 ε − 1 4 π ε + 2 , [2]. Then the polarisation force in Eq. (3.12) reduces to the following:

( )

fi

( )

(p) (p) ∂ε ik E2 ⎛ ∂χ ik ⎞ E2 χ 0 E2 = =⎜ ≈ ⎟ ∂x k 8 π ⎝ ∂x k ⎠0 8 π d 8 π

(3.13)

Thermal force reaches its maximum at the end of the pulse. The total pressure is proportional to the absorbed energy density. Therefore the thermal force expressed through the fluence, F t ≈ I t ,

()

reads: f th ≈

Pe + PL 2 AIt ≈ 2 ; labs labs

(3.13)

where I = cE 2 / 8 π is the average laser intensity during the pulse. magnitudes of all forces during the laser pulse:

f th ≈

2 AIt 2 labs

;

f

(p)

≈

χ0 I d c

;

f

pond

≈

(ε

)

−1 I . labs c

D

Now one can compare the

(3.14)

220 At the beg ginning of thhe pulse the polarisation p annd ponderom motive forces ddominate the thermal forcce. After seveeral femtosecconds the therrmal force com mpares with the polarisatiion force in thhe heated layeer of tens off nanometers thickness. t Att the end of thhe pulse the thhermal force is significantly larger of thhe d forrces. intensity dependent Let us u consider, for example,, Bismuth inn conditions of the experriments of [444]. In thesse experimennts 50 fs, 800 0 nm, laser puulses excited Bi layer at fluuence of ~ 7 mJ/cm2 (labs = 28.3 nm; ε = -8 22.39; d = 3.3×10 cm m; na = 2.82×1 1022cm-3; χ0 = 0.2; 6.34×106 cm-1). Thhe magnitudees of the forcees acting on Bismuth atom ms at the end of the pulse aare the follow wing:

f

(p)

≈ 3.67 ×1014

erg e

c cm

4

~ f

pond

≈ 4.38 × 1014

eerg c cm

4

<< f th ≈ 4 × 1015

erg cm 4

.

The forcees above are volume forcees (force perr unit volumee). Let us coonvert the thermal force of o th 2 Eq.(3.13) into the forcee acting on a single atom, F ≈ 2 AIt na labs , and coompare it to thhe elastic forcce 9). One can eeasily see thatt the thermal fforce in Bism muth, excited to t driving attomic vibratioons of Eq.(3.9 the energyy density twiice of the equ uilibrium enthhalpy of meltting, is by thee order of maagnitude loweer then the force f driving “cold” harmoonic vibrationns. Another feature of thhe thermal forrce is the shoort period of action equal to t the pulse duration, d whicch is much sh horter the phonnon’s period. The laaser-exerted force can be considered aas a perturbation imposed on the harmo onic vibrationns during thee period when n lattice remaains cold. Then the oscillaations of an inndividual atom occur undeer the combiined action off elastic and laser-imposedd forces as folllows:

d 2 qk

M

d dt

2

= Fklas + Fkel ;

l where Fklas = Fth + F

(p)

(3.155)

+ F pond is thhe sum of the laser-exerted d forces; qk iss atomic disp placement in kk

direction. The above equation e holdds for the desccription of in nitial purely harmonic motiion. Howeveer, i heated by the energy ttransferred from fr electronns, and interaaction betweeen gradually the lattice is des becomess significant. Therefore,, the phenom menological damping witth different phonon mod mic vibrationns in a solid where w electronns coefficiennt γ should bee introduced. Now the equuation for atom are excited but lattice remains r cold reads: r

d 2 qk

+ 2γ

dt2

dqk F las + ω02 qk = k . dt M

(3.166)

The elastiic force is takken in the fo orm Eq. (3.9). The relativve effects of the elastic foorce and of thhe − laser-impoosed forces arre different inn a short-timee scale t < ω0−1 and in a lonng-time scale t >> ω0−1. Thhe elastic forrce is a slow one; it is not effective durring the period, which is much m shorter than t the periood of the atoomic vibration n. Correspon ndingly, the first f term in the t left-hand--side in equattion Eq. (3.166) dominatess, thus this eqquation reducees at t < ω0−1 tto the simplesst form of a Newton N equatiion:

d 2q dt

2

≈

()

Fklas t M

(3.177)

Approxim mate solution of o this equation is straightfforward:

())

qk t ≈

las 2 Fmax tp

2M

(3.188)

21 Thus, on a short-time scale, the laser-imposed forces produce a coherent displacement of atoms. It is

(

instructive to compare fast atomic displacement to the amplitude of cold phonon q0 ≈ 2h Mω0 th

Eq.(3.8). The electronic force, F ≈ 2 AIt p

2 na labs ,

)

12

of

is a dominant contribution into the sum of laser-

imposed forces to the end of the pulse. Let us consider the case when the absorbed energy density compares to the enthalpy of melting, 2 AIt p labs ≈ H fusion ~103 J/cm3 and calculate the coherent displacement of Bismuth atom to the moment when harmonic vibrations commence at about 300 fs after the pulse. One obtains from Eq.(3.18) that in these conditions Bismuth atoms are displaced on the distance less than 10-10 cm that is approximately one tenth of the cold phonon’s amplitude. One can see that the maximum value of the sum of the laser imposed forces at the absorbed energy density in excess of enthalpy of melting equals to approximately one tenth of the elastic force driving cold phonons, ω02 q0 >> Fklas M . Thus on a long-time scale t >> ω0−1 the laser-imposed forces act as a small perturbation. This perturbation results in a small change in the amplitude of the cold atomic vibration, Δqk(t). Therefore, one can search for solution of equation Eq.(3.16) in a form:

()

()

()

qk t =q k 0 t + Δqk t .

(3.19)

where q0 is the unperturbed solution:

⎧ qk 0 t = qk0 exp⎨−i ω02 − γ 2 ⎩

(

()

)

1/ 2

⎫ t⎬ . ⎭

(3.20)

Now Eq.(3.16) transforms into equation for the perturbation is as follows:

d 2 Δqk dt 2

+ ω02 Δqk + 2γ

dΔqk Fklas = . dt M

(3.21)

We assume as the first order approximation that the perturbation oscillates with the same frequency as unperturbed vibration and therefore that the perturbation affects only the vibration amplitude. Later on we account for the change in the vibration frequency. The change in the vibrational amplitude at t >> ω0−1 takes the form: Δq k =

⎧ Fth, k exp⎨ i ω02 − γ 2 2ω0 γ M ⎩

(

)

1/ 2

⎫ t −ϕ⎬ . ⎭

(3.22)

Note that only thermal force is effective after the pulse end. The phase and some pre-exponential constant should be introduced in order to stitch the above solution to the initial atomic displacement. 3.3. Non-linear effects due to the lattice overheating a). Thermal expansion The approximation of damped harmonic oscillations of atoms under the laser excitation is a valid description of atomic motion only during the period when the lattice remains cold and electronic force drives the atomic vibrations. After the end of the pulse the lattice temperature increases due to the energy transfer from the electrons. Phonon’s energy is then distributed in accord with the Boltzmann function φ ph q = exp{− ΔU el T } where the perturbation in the inter-atomic potential due to harmonic

()

vibrations reads as the following, ΔU el ≈ Mω2ph q 2 << T . Phonons do not interact in harmonic approximation; respectively the average position of vibrating atoms does not depend on temperature; moreover, the displacement of oscillating atom from the equilibrium position averaged over the

22 Boltzmann distribution is zero q = 0 [2]. However, the mean square displacement (or average energy of phonon treated as one-dimensional oscillator) is proportional to temperature,

q 2 = T Mω 2ph . The phonon-phonon interactions become important when increasing lattice temperature approaches the equilibrium melting point. The atomic vibrations loose their harmonic character; the non-linearity in the interaction potential should be taken into account. The perturbation in the interatomic potential with a 3-order term included reads [2,55]:

ΔU nl = C q 2 − gq 3 ; C ≈

εb 2d

; 2

1 ⎛ ∂ 3U ⎞ ε g = ⎜⎜ 3 ⎟⎟ ≈ b3 . 6 ⎝ ∂ x ⎠ 6d 0

(3.23)

Now the atomic displacement averaged over the distribution function with potential of Eq. (3.23) increases as the lattice temperature grows up [2,55]: 3gT T ≈ d. q nl = (3.24) 2 2ε b 4C This is the thermal expansion of heated solid. The atomic motion looses its oscillatory character and becomes chaotic (randomised) when the change in the average atomic position by Eq. (3.24) constitutes significant part of the “cold” oscillation amplitude. The displacement from the equilibrium atomic position due to non-linear interaction of phonons calculated by Eq.(3.24) at temperature equal to that of the equilibrium melting point coincides with 10% accuracy with the average displacement following from the Lindemann criterion of melting [56,57]. The non-linear processes of multi-phonon interaction at temperature close to the equilibrium melting point gradually result first in the mode softening (that is a decrease of the oscillation frequency). Later on the instability develops and as a result the squared frequency of atomic vibrations may turn negative. Let us now define the time scale when non-linear interaction becomes dominant. b). Three-phonon interaction: phonon’s decay time The probability of multi-phonon processes per unit time defines the characteristic phonon decay time. Probability of decay of an optical phonon in two acoustic phonons per unit time can be calculated with the help of quantum perturbation theory as the third order term of expansion of interatomic potential into series relative to the powers of atomic displacement [1]. We present here a simplified version of similar derivation estimating the perturbation Hamiltonian, H’, as the third term in the series of potential expansion in powers of atomic displacement similar to that in Eq. (3.23):

1 ⎛ ∂ 2U U ≈ U 0 + ⎜⎜ 2 2 ⎝ ∂x

⎞ 1 ⎛ ∂ 3U ⎟⎟ q 2 − ⎜⎜ 3 6 ⎝ ∂x ⎠0

⎞ ⎟⎟ q 3 + ... = U 0 + ΔU 2 + ΔU 3 + ... . ⎠0

(3.25)

Then the scaling for the probability for the phonon decay is the following:

w∝

2πΔU 32 ; hk B T

(3.26)

Let us take the average phonon energy as, MωD2 q 2 ≈ hω ph ≈ k B T , and U0 ≈ εb . Respectively, the third order perturbation term expresses in the following form:

23 3/2

ε ⎛k T ⎞ ΔU 3 ≈ b ⎜ B ⎟ 6 ⎝ εb ⎠

(3.27)

For the temperature in excess of the Debye temperature, kT >> hωD the single phonon decay rate using Eq.(3.24) and Eq.(3.26) reads:

wdecay ≈

π

(k T )

2

18hε b

(3.28)

B

For Bi at melting point of 544.7 K (εb = 2.16 eV) the decay rate equals to several picoseconds, which is in qualitative agreement with the experimental observations [44,58]. Non-linear phonon-phonon interactions also result in a dependence of phonon frequency on the lattice temperature considered in the next section. c). Decrease in the phonon frequency and increase in the inter-atomic separation The excitation of electrons and lattice heating lead to the decrease in the binding energy, in the phonon’s frequency, while the inter-atomic distance grows compared to that in a cold state. One can describe the property of slightly excited solid using a simplified form for the empirical chemical pseudo-potential [59,60]:

()

( )

( )

V r = VR exp −θ r −VA exp −λ r ;

(3.29)

where VR, VA, θ, λ (or s = θ λ and λ) are respectively the repulsive and attractive potentials along with their gradients. It follows from Eq.(3.29) that the inter-atomic distance in equilibrium (where the potential is a minimum, V d0 = −ε b ) can be expressed via the binding energy: ⎡ ⎤ 1 ⎢ s −1 V A ⎥ (3.30) d = ln λ ⎢⎣ s ε b ⎥⎦

( )

()

The excitation of electrons reduces the attraction part of the inter-atomic potential and the binding energy. Thus in the excited state at ε b, exc ≈ ε b − k B Te ; k B T << ε b0 the inter-atomic separation Eq.(3.30) can be presented as an expansion in series in respect to powers of a small parameter, k B T λε b0 . The increase in the inter-atomic spacing reads, d ≈ d0 + k B T λε b0 . This increase depends on the gradient of the attraction part of the potential: dq ≈

1 k B Te

λ ε b,0

.

(3.31)

One can see that the above relation qualitatively complies with the atomic displacement calculated as the thermal expansion from Eq.(3.24). However, in Eq.(3.31) the asymmetry in the inter-atomic potential (the difference in the gradients of attractive and repulsive parts) is taken into account. Similarly, the phonon frequency is calculated through the second derivative of the potential:

ω 2ph =

1 M

⎛ ∂ 2V ⎜⎜ 2 ⎝ ∂r

⎞ λ2 s ε b . ⎟⎟ = M ⎠r = d

(3.32)

Thus, the frequency of phonons in excited solid linearly decreases with the temperature growth as the first approximation when the inter-atomic distance change with the temperature is ignored:

24 ⎛

ω 2ph ≈ ω02⎜1− ⎜ ⎝

⎞ k B Te ⎟ . ε b0 ⎟⎠

(3.33)

The above estimate implicitly assumes that electrons are excited while the lattice remains cold. For example in Bi excited to the maximum electron temperature of 0.3 eV (binding energy 2.16 eV) the phonon frequency decrease predicted by Eq.(3.33) is ~7%. Thus for A1g mode the frequency of 3 THz in accord to Eq.(3.33) should decrease to 2.78 THz. The experimental result from [22,44] is 2.82 THz. One can see that when the temperature approaches to the equilibrium melting point (that is transition to the intermediate potential minimum corresponding to the liquid state) the second derivative of the potential passes through zero value. That is a manifestation of the onset of the vibrational instability when phonon’s frequency temporarily may turn negative. In fact at the stage when the non-linear inter-atomic interactions became essential, the transformations of the material should be considered with the statistical (thermodynamic) methods, – see in Section 4 below. 3.4. Laser-excited atomic motion imprinted into dielectric function and reflectivity In order to trace experimentally the subtle atomic motion following laser excitation one should determine how the atomic motion affects transient properties of the excited solid. An obvious way is to use the ultra-short x-ray and electronic beam probes diffracted from the excited sample. Analysis of time-dependent diffracted intensity in principle would allow directly trace the changing atomic positions. However, as we discuss later the interpretation of x-ray diffraction experiments should be carried on with caution and leaves some experimental features unclear. Another way is to use optical single and double simultaneous probes in order to measure the transient optical properties of excited solid. Both methods are complementary and allow obtaining a general picture of the ultra-fast transformation of the material. Our goal in this section is to trace the atomic motion and subsequent phase changes by time-dependent dielectric properties of laser-affected material. We assumed that the dielectric tensor modified by the laser effect consists of two terms, the Drude-like term, ε D , and

(p)

(p)

polarisation term, ε jk , ε jk = ε D ⋅ δ jk + ε jk . The polarisation term is a real number. Thus, the total dielectric function can be presented as a sum:

(p) ε jk = ε D ⋅ δ jk + ε jk ≡ ε r + iε i (p) ε r = ε jk + ε r, D ; ε i ≡ ε i, D

(3.34)

The Drude-like term has its conventional form of Eq (3.2). This function depends on the number density of the conductivity electrons, ne, and on the electron effective mass through the plasma frequency, ω 2p = 4 πe2 ne t me∗ . Number density of conductivity electrons may increase when a solid

()

is heated up to the melting point. For example it is known that dielectric properties of semi-metal Bismuth at the melting point and above can be well fitted by the Drude-like dielectric function under assumption that all five valence electrons are transferred into the conduction band. It is unknown to the best of our knowledge how the electron’s effective mass changes (if any) during a solid heating and melting. For example, it was established that in liquid Bismuth and in solid at room temperature and above room temperature the effective mass is equal to that of a free electron [21,22]. However, in simple metals such as aluminium, iron, copper, lead, the electron effective masses at room temperature are well above the free electron mass [2]. There is no data on how the ultra-fast excitation might change the effective electron mass. Therefore in the future analysis we ignore the

225 changes of o the electronn effective maass. The dieleectric functioon depends onn the temperatture-dependennt phonon am mplitude thro ough the electrron-phonon momentum m exxchange rate in i the explicitt form:

(0) q 2 ν e− ph ≈ ν e− ph 2 .

(3.355)

q0

Eq.(3.36) was obtainedd under assum mption that eleectron-phonoon scattering cross-section c is proportional n’s amplitude while the eleectron velocity equals to tthe Fermi vellocity and it is to the squuared phonon unchanged during the interaction. At A the initial stage of atom mic motion thhe electrons are a excited buut t Therefore one o can negllect change in the phonoon lattice still remains att the initial temperature. nt for the chaanges in the phonon’s am mplitude only thus taking Eq.(3.36) at frequencyy and accoun unperturbbed frequency y. The initial phonon ampplitude has a conventional c form of Eq. (3.8). Now thhe small pertturbation in th he electron-pphonon rate exxpresses throuugh the change in the phoonon amplitudde in the folllowing way:

()

(0) 2Δqq t .

Δν e− ph p ≈ ν e− ph

(3.366)

q0

The polarizability p y part from Eq.(3.34) exxpresses direectly throughh the laser-innduced atom mic displacem ment: ⎛ ∂χ ⎞ (p) ε ik = 4 π ⎜ ik ⎟ Δ Δq l t . ⎝ ∂x l ⎠0

()

(3.377)

()

Δq t duringg the pulse is expressed byy Eq. (3.18); laser-perturbe The coherrent atomic displacement d l ed atomic vibbrations on thhe later time presented p by E Eq.(3.22). The reeal and imagiinary parts off slightly pertuurbed dielectrric function arre the followiing:

⎞ Δν ⎞ Δn ⎛ ∂ε ∂ (p) (p) ⎛ ∂ε e− ph e + ⎜⎜ D, r ⎟⎟ Δε r = Δε jk + Δε r, D = Δε jk − ⎜ D, r ⎟ ⎝ ∂ ln ne ⎠0 ne ⎝ ∂ν e− ph ⎠0 ν 0,0 e− ph ⎞ Δν ⎛ ∂ε ⎞ Δn ⎛ ∂ε e− − ph e Δε i ≡ Δε i, D = ⎜ i,, D ⎟ + ⎜⎜ i, D ⎟⎟ ∂ e− ph ⎠0 ν 0, e−− ph ⎝ ∂ ln ne ⎠0 ne ⎝ ∂ν

(3.388)

Subscript “0” denotess that a derivvative is taken from the unperturbed function [444]. Thus, thhe c due to increase in the numbeer dielectric function in a swiftly exccited metal-liike solid is changing a due to vaariation of thee electron-phhonon momenntum exchangge density off conductivityy electrons and rate. Both h are explicit functions of electron and llattice temperrature. In Bissmuth the reaal part of the dielectric d funnction in equillibrium liquidd is slightly higher h than that in a solid d while the imaginary part p increases more thann two times. The numb ber density oof conductivvity electrons,, and electron n-phonon rate in liquid are almost 3 tim mes higher of those t in a soliid [21]. Onn the other hand the logaarithmic derivvatives of reaal and imaginnary parts off the dielectrric function on o electron number n densitty and derivaatives of real parts on the electron-phon non rate are of o the same sign in bothh states. Ho owever, the llogarithmic derivative d of the imaginarry part on thhe i a solid is slightly posiitive (0.86) while w in a liqquid it is stroongly negativve electron-pphonon rate in (–11.95) [43,44]. Thhus, the sign n of this derivative changges somewheere during thhe solid-liquiid wo conclusionns from this analysis. First, one cannnot present thhe transition time. One can make tw d fun nction in Bi as expansion iinto series du uring solid-liqquid transitionn. Second, thhe transient dielectric

26 transient phase state may have peculiar optical properties due to strong changes in the electronphonon collision rate directly related to the laser-excited atomic motion. Now it is instructive to present the time-dependent reflectivity of the probe laser beam from laserexcited solid through the solid internal properties. Such reflectivity was directly measured in numerous experiments from different solids. First, we express the Fresnel reflection coefficient R through the real and imaginary parts of the dielectric function ε = ε r + iεi – see Eq.(3.1). Small variation of reflectivity then reads: ⎛ ∂R ⎞ ⎛ (p) ⎞ ⎛ ∂R ⎞ ΔR = ⎜ ⎟ ⎜ Δε r, D + Δε jk ⎟ + ⎜ ⎟ Δε . ⎠ ⎝ ∂ε i ⎠0 i, D ⎝ ∂ε r ⎠0 ⎝

(3.39)

Substituting variations in the dielectric function from Eq.(3.40) one can present the small first order reflectivity variation expressed through the changes in polarisation, in the number of the conductivity electrons and in the electron-phonon rate with the coefficients, which are combinations of the unperturbed solid parameters: ⎛ ∂R ⎞ Δν e− ph Δne (p) + Cν e− ph ΔR = ⎜ ⎟ ⋅ Δε jk + Cne ne,0 ν e− ph,0 ⎝ ∂ε r ⎠0

(3.40)

The coefficients are the combinations of the derivatives taken from the unperturbed functions [44]. It is known from the numerous experiments that the relative amplitude of the reflectivity changes in observed oscillations is of the order of 10-3–10-4. One can use the expansion Eq. (3.40) for description of experiments with the deposited energy density larger than the enthalpy of melting if both coefficients in expansion have the same signs in solid and liquid states. This condition holds for experiments with Bismuth (see [43,44]) where Cne is positive and Cν e− ph is negative and they are of the same order of magnitude in solid and liquid states. Each term in Eq.(3.40) has clear physical meaning. Indeed, the first term describes the reflectivity decrease due to polarisation changes in the dielectric function during the pulse, which is positive, while the reflectivity derivative is negative in Bismuth. The second term describes increase in the reflectivity during the laser pulse due to the increase in the number of conductivity electrons. This contribution after the end of the pulse gradually decreases (remaining positive) due to the recombination of excited carriers. The third contribution into the reflectivity changes is the result of the effect of atomic vibrations on the optical properties of excited solid. This contribution is of the same physical nature as the first term and it is negative. All variations in optical properties are explicit functions of electron and lattice temperature, which are functions of laser and solid parameters. Summing up, one can conclude that both optical and x-ray probing of bismuth excited in almost identical conditions give similar qualitative picture in accord to the theoretical scenario [25,61]. First, the coherent atomic displacement is produced by the polarisation force and electron pressure force during the laser pulse that is recorded by the negative drop in the reflectivity. Then oscillations with the frequency corresponding to that of phonon in a sample at the initial temperature and red-shifted due to electron excitation were observed. The phonon’s amplitude gradually decreases while electrons transfer the energy to the lattice. Heating of the lattice and thermal expansion transformed initially harmonic vibrations of atoms into strongly non-linear motion that manifests the onset of solid-to-liquid phase transformation. This process is identified by the measurement and interpretation of the damping rate of the reflectivity oscillations. The observed dependence of the damping rate on

27 temperature is close to the dependence of the rate of the optical phonon decay into two acoustic phonons that confirms interpretation of the inverse damping rate as the phonon lifetime. The measurements and analysis of the reflectivity after the decay of oscillations gave the evidence that the solid experiences transition to some transient state; its reflectivity being the intermediate between those for solid and liquid. Following the Born criterion [65], we define the molten state as a phase state where the shear modulus is equal to zero. A disordered state is a state where short-range order dominates. Sometimes, but not always, both definitions coincide [55,65]. The difficulties in interpretation of fast laser-induced transformations in Bi relate to the fact that liquid bismuth in equilibrium conditions is denser than solid. In the typical experiment of the ultrafast excitation the temperature is a maximum at free vacuum-sample boundary. The thermal expansion during picoseconds after the end of the pulse may reduce the density of disordered sample. Therefore we should not exclude the possibility that Bi swiftly disordered by the ultra-fast laser and transformed into unusual transient state has density lower than a disordered liquid state in equilibrium. 4. ULTRAFAST PHASE TRANSFORMATIONS 4.1. Introduction Numerous experiments have demonstrated that phase transformations induced by the ultra-short laser action need more energy than the same transformations in equilibrium conditions. In this section we consider the processes occurring in a solid where the laser energy well in excess of the equilibrium enthalpy of melting is deposited by ultrafast pulses. Melting in thermodynamic equilibrium has been intensively studied for more than 100 years. Nevertheless this phenomenon is considered as a mystery [62], because it is still impossible to establish unequivocally the common properties of melting for different solids. However, important steps in developing a general theory for crystal-liquid transition were made by clarifying many common features of melting in different crystals. It was established that a succession of instabilities precedes the disordering in crystals, and the ultimate stability limit of a superheated crystal is determined by the equality of the crystal and liquid entropies [63,64]. Moreover, it was found by Fecht [26] that conditions for disordering depend heavily on the concentration of non-equilibrium point defects. In turn, the formation of defects depends on the presence of atoms from the high-energy tail of equilibrium distribution. The advent of fs-lasers and novel diagnostic techniques (ultra-short x-ray, optical and electronic probes) during recent decades has provided an opportunity to study the transient states of a material excited by ultra-short powerful lasers. The time-resolved observations of solid-liquid transition at fs and ps time scale allows one to uncover the effects of the non-equilibrium processes on the ultra-fast transformation of the material, and better understanding of the much slower process of equilibrium melting. The ultra-short pulse swiftly excites electrons to the average-per-electron energy several times higher than the equilibrium lattice temperature. The lattice temperature is established later, after the electrons had already shared their energy with the lattice, and the electron and lattice temperatures had been equilibrated. Therefore, it is quite reasonable to ask whether the energetic electrons able to disorder or, to melt, the lattice well before the energy is transferred to the lattice by the electron-phonon collisions. In order to answer this question one needs to examine the processes of interaction between the electrons and the lattice on a very short time-scale, and to find what would be the signatures of the disordered lattice that can be diagnosed with time resolution higher than the electron-to-lattice energy transfer time.

28 The primary effects of the excited electrons on a cold lattice are twofold. The first effect relates to the spatially inhomogeneous heating of electrons in the skin-layer. It is directly connected to the action of the electrostatic field of charge separation on the ion cores. The second effect appears in conditions of homogeneous excitation. It causes a deformation of the inter-atomic potential, a decrease in the attraction force due to swift electron excitation, and a corresponding increase in the inter-atomic separation. However, these effects produce atomic displacements similar to thermal expansion, which should not be confused with disordering. Statistical methods should be employed for the description of transformation of material in order to account for the disordering. The change from the long-range order of the crystal to the short-range order of a liquid is characterized by the correlation functions that could be extracted from the electron and x-ray diffraction experiments. The main parts of the local equilibrium distribution functions separately in the electronic and in the lattice subsystems are established very fast by the ultra-short pulse action, in a few femtoseconds. The main non-equilibrium processes, which require much longer time, are the electron-to-lattice energy transfer, and the process of building up he high-energy tail in the Maxwell distribution function [14]. It appears that statistical thermodynamics can be applied to describe the processes in both subsystems. Therefore, the time-dependent entropy, being the most fundamental measure of disorder, should be applied for the description of the degree of disordering induced by ultra-fast energy deposition. 4.2. Melting in equilibrium The major characteristics of melting in equilibrium are the melting point, the enthalpy and entropy of melting. The melting point in equilibrium at a pressure Pm is conventionally defined as the temperature at which the Gibbs free energies of the two phases are equal, and beyond which a thermodynamic instability occurs indicating commencement of the transformation process. The temperature increase during the melting process also indicates that the binding energy between the atoms is decreased. Thus the melting process leads to changes in the inter-atomic potential. The standard enthalpy of fusion (specific heat of melting), ΔH f , is the amount of thermal energy in a unit volume, which must be absorbed to change a solid phase to a liquid at the melting point. The corresponding entropy of fusion is the increase in entropy entirely due to heating a solid to convert it into liquid at the melting point, ΔSm = ΔH f Tm . There are several major differences between the solid and liquid states. First, the long-range order characteristics of the crystalline state change to the short-range order characteristics or to disorder in a liquid state. Second, a remarkable feature is that the shear modulus is equal zero in liquids. This feature directly relates to the change in the inter-atomic potential during the melting process. Third, the number of conductivity electrons in the conduction band may significantly increase in the molten phase, as happens in bismuth or in silicon. The entropy difference between the crystal and liquid phases, the entropy of fusion, reaches its maximum at the melting point. We briefly recollect below the numerous attempts to establish theoretically the criteria for melting in equilibrium. The commencement of the solid-to-liquid transformation is preceded by a succession of instabilities (or catastrophes), which defines the stability limit for the crystalline state. The Lindemann vibrational instability criterion [56,57] states that any solid melts when the amplitude of the vibrations of the atoms about their equilibrium positions exceeds some threshold value, qm . He found this threshold value from the condition that the average phonon energy at the melting point equals to the melting temperature in the energy units while the average phonon frequency is close to

29 the Debye frequency, MωD2 q 2 2 ≈ 3k B Tm 2 . By expressing the Debye frequency via the binding energy as MωD2 ≈ ε b d02 , the squared ratio of the average amplitude of atomic vibrations qm2 to the inter-atomic spacing d0 , reads: q m2 d02

≈

kTm

εb

≅ const .

(4.1)

This ratio is fairly constant for the alkali metals, it changes only ~15% from Li to Au [57]. One can see that the melting temperature constitutes approximately 3% from the binding energy over the whole periodic chart of elements with rare deviations to 15% for Gallium, 2% for Uranium and Lithium, and 4% for Beryllium. As temperature approaches the melting point, the atomic vibrations become anharmonic and the equilibrium positions of atoms are shifted due to thermal expansion. This shift is larger than the amplitude of vibrations calculated by Eq.(4.1) in harmonic approximation. The phonon vibrations become nonlinear, the vibration frequency decreases with the increase of temperature, and the moment when the squared phonon goes to zero, ω 2ph = 0, manifests the onset of the instability at which the lattice starts to reconstruct. As a consequence, the Lindemann criterion indicates just the onset of the vibration instability, and only qualitatively. It does not establish the melting threshold. Liquid state differentiates from a solid state by zero shear modulus. The Born’s criterion on melting [65] states that the temperature point at which the shear stress turns to zero signifies the transition to a liquid state. The condition of zero elastic stiffness, c44 = 0, allows one to find the ratio between the melting temperature and the binding k B Tm / ε b energy to be equal 0.35, which is about an order of magnitude higher than the measured values [57]. Qualitatively, the condition of zero shear modulus coincides with the onset of vibration instability when ω 2ph = 0 , because both the modulus and the squared phonon frequency are proportional to the second space derivative of the potential: c44 ∝ ω2ph ∝ ∂ 2U ∂x 2 . It should be noted that none of the above criteria predicts precisely melting temperature or gives quantitative characterization of the melting process. The entropy catastrophe of Fecht and Johnson [63] sets the ultimate limit for the crystalline stability at T >Tm. Fecht and Johnson [63] and Tallon [64] established that transition from solid to liquid occurs in a succession of steps with progressively elevated temperature, where every step is characterized by changes in entropy. The critical temperature corresponds to the state where the entropies of the solid and liquid are equal. Beyond this point a superheated solid cannot exist because the entropy of solid at T > Tcr becomes larger than in liquid, which is impossible from the point of view of thermodynamic. The catastrophic solid-toliquid transition commences in a state characterized by the values of critical entropy and critical temperature, which are interrelated. These critical values can be considered as the ultimate criteria defining disordering.

( )

The entropy of fusion defined as ΔS f Tm = ΔH f Tm and measured in the units of the Boltzmann constant kB. ΔS fusion = 1.5kB for binary inter-metallic compounds; for good metals it is (1.1–1.4)kB; while for the semi-metals and dielectrics it equals to (2–4)kB (see table 4.1 in [14]). The entropy of an overheated metal ΔS crit at the melting point is a sum of contributions from isothermal expansion ΔSexp, from heating the electrons ΔSe and the lattice ΔSth, and from the thermal point defects (vacancies) ΔSvac [26,63,64]:

30 ΔS crit = ΔSexp + ΔS e + ΔS th + ΔS vac .

(4.2)

(

)

The first term represents the isothermal changes ΔS exp ≅ k B ΔV V0 [64]. The isochoric contribution from electrons is calculated using the heat capacity of the degenerated electrons, Ce = π 2 k B2 Te 2εF , so

()

(

)

ΔS e Te = π 2 k B2 Te − T0 2ε F , where T0 is the initial temperature. Both isothermal and isochoric contributions are small in comparison to entropy of fusion because the melting temperature is small compared to the Fermi and binding energies. For example, in Al ΔSexp ≅0.2 kB and ΔSe ≅0.023 kB. The thermal disordering due to lattice heating makes a major contribution to the entropy: ΔS th TL = CL lnTL T0 . Taking Dulong-Petit values for the lattice heat capacity, contribution from

( )

( )

thermal disordering is in the range ΔSth = (3.5–5.5)kB. The last term in Eq.(4.2) is contribution from thermal point defects such as vacancies, anti-site defects, and interstitials, which are intrinsic features of a solid in equilibrium. The increase in vacancy concentration at melting point from ~10-3 to 0.1 leads to solid-liquid transformation. The change in the entropy arising from the presence of vacancies was found to be ΔS vac ≈ 2 k B [26]. For example, critical entropy for Al is ΔS crit = 6.41kB; this value is reached in the superheated state at the temperature Tms ≈ 1.38Tm [63]. Without vacancy contribution the critical temperature corresponding to the entropy catastrophe in Al increases dramatically to Tms ≈ 3Tm [63]. The critical entropy is in the range ΔS crit ~(5–7)kB for majority of metals. The value can be considered as a threshold criterion for disordering in non-equilibrium conditions. 4.3. Melting due to ultrafast overheating Succession of processes during the ultra-fast laser induced solid transformation is the following: the laser energy absorption, heating of the electrons, transfer of the absorbed energy to the lattice, and finally establishment of the complete equilibrium energy distribution. A key process, which allows statistical thermodynamics to be applicable for the quantitative characterisation of the disordering process, is fast set up of local quasi-equilibrium energy distribution in the electron and the lattice subsystems. The main part of the Maxwell distribution establishes in order of a few femtoseconds. Thus the time-dependent entropy is a rightful measure of the fast order-disorder transformation. Direct estimate of the maximum electron and lattice temperature can be made on the basis of energy conservation (see Section 2 above). The maximum electron temperature is reached at the end of the laser pulse as all the absorbed energy is confined in the electrons:

[

( ) π nl]

k B Temax = 4ε F AF t p

2

1/ 2

e s

, while the maximum lattice temperature is reached after the electron-

()

phonon temperature equilibration, k B TLmax ≈ 2 AF t p CL na ls . Consider as an example Aluminium (εF =11.63 eV; ne = 1.806×10 cm ; Tm = 933.47K) at the ultra-fast deposited energy density equal to 23

-3

[

the enthalpy of melting. Then the maximum electron temperature k B Temax = 2ε F ΔH f π 2 ne

]

1/ 2

= 0.23

eV = 2.9×kBTm is well over the melting point. Note that ultra-fast electron’s overheating is typical for all metals, where the enthalpy of melting is in the order of kJ/cm3 and higher. Now one easily finds that electrons transfer their energy to the lattice in Al during 25 femtoseconds raising it to the melting temperature, kBTm = 0.08 eV. During this time the main part of the Boltzmann distribution is also established. Now one should determine the time required for generation of the sufficient number of the thermal defects necessary for triggering a phase transformation. Thermal point defects play essential role in disordering of solids in the equilibrium. It is known that pure defect-free crystal could be significantly overheated above the melting point. In non-

31 equilibrium conditions fast melting may occur in strongly overheated solid without the defect contribution if the electron-to-lattice energy transfer is faster than the defect generation [14]. Timedependent generation of the thermal point defect proceeds in two interconnected steps with the different time scales. At the first step the atoms diffuse along the energy axis and reach the energy level necessary for defect formation, ε at ≥ ΔH vac . This is the time for the high-energy tail of the distribution to be generated – see Eq.(2.17):

(

⎛ ΔH ⎞ vac exp ⎜ ⎟. ⎝ kBT ⎠ vac

( )ΔkHT

)

ttail ΔH vac ≈ 0.85 tmain k B T

B

(4.3)

However, in order to fill the high-energy tail, atoms should increase their energy further to the point ε at ≈ 2 ΔH vac . Thus, the tail filling time ttft could be rightfully defined as the following: ⎛ 2 ΔH ⎞ kB T vac ttft ≈ 0.85 tmain k B T exp⎜ (4.4) ⎟. 2 ΔH vac ⎝ kBT ⎠

( )

The time for establishing the main part of the distribution function in a lattice can be presented with sufficient accuracy as tmain k B T ≈ h k B T . Now the defects (vacancies) formation time of Eq.

( )

(4.4) as a function of overheating θ = T Tm takes a simple form:

tdef ≡ ttft ≈ 4.58 × 10 −2

⎛ 18.56 ⎞ h exp⎜ ⎟. k B Tm ⎝ θ ⎠

(4.5)

A comparison of tail growing time of Eq.(4.5) with the electron-to-lattice energy transfer time shows that lattice heating occurs much faster than the filling up of the high-energy tail. Thus in nonequilibrium conditions a laser-excited solid is strongly overheated before the defect contribution becomes significant. Therefore the lattice overheating is the major reason for the commencement of lattice transformations. The electrons are heated first and it is suggested that electrons overheating might be a reason for the ultra-fast melting. The electron contribution into the entropy change is threefold. First contribution comes from the electron temperature rise. Two others relate to modification of electronic potential and atomic displacement, both due to the strong electronic excitation. The rapid electron temperature rise results in an isochoric increase in the entropy, ΔS e Te = π 2 k B2 Te − T0 2ε F . However, this change is small even at strong electrons’ overheating.

()

(

)

Indeed, even strong overheating of electrons in Al ten times over the melting point results in the entropy increase due to electronic contribution equals to ΔSe = 0.34 kB, that is ~ 20 times lower than the critical entropy value defined in the previous section [14, p.141]. The physical reason for this effect is obvious: even at very strong overheating the electron’s energy is much lower than the Fermi energy in Al, 10 k B Tm ε F = 0.06. Swift electronic excitation modifies the inter-atomic potential and induces an atomic displacement while the atoms still remain cold. The reduced atomic attraction effectively reduces the binding energy and increases the inter-atomic distance. For the majority of simple solids the binding energy is in the range of 2-5 eV. The electron temperature is much lower than the binding energy, k B Te << ε b , even for a strong superheated solid. Hence, the ratio of electronic temperature to the binding energy can be used as a small parameter for estimations of the modification in the interatomic potential and increase in the inter-atomic distance, volume, ΔV V ≈ 3 Δd d = 0.03, and entropy, ΔSexp ≈ k B ΔV V (at deposited energy density of ~1 kJ/cm3).

32 The swift laser heating of electrons is essentially a non-homogeneous effect: the space scale of electron temperature/pressure gradient equals to the half of the skin depth, ls/2. Therefore the displacement caused by the electronic gradient force, Fel = ∇k B Te ≈ 2 k B Te ls , should be added to the local effect of the increase in the inter-atomic spacing due to modification of the inter-atomic potential. The non-homogeneous displacement near the melting point approximately doubles the homogeneous displacement as a result of potential modification, giving the relative expansion of ΔV V ~0.1. The characteristic time scale of the process is the time required for the electron temperature to smooth across the skin depth by heat conduction, which is around a picoseconds. The change in the lattice entropy due to the isothermal volume change: ΔSexp ≈ k B ΔV V ~ (0.2kB–0.4kB). The increase of total entropy induced solely by the excited electrons is the sum of the contributions from the electron heating and the atomic displacement induced by excited electrons, it is in a range 0.7kB–0.8kB for strong overheating conditions. This constitutes one tenth of the critical entropy value required for melting at the deposited energy density several times exceeding the enthalpy of melting. Thus, the entropy approach clearly indicates that electronic excitation alone does not produce a specific disordering; it leads to the isothermal volume change similar to that due to thermal expansion [12]. Therefore, the sole electron contribution into the total entropy rise is only a small fraction of the catastrophic level even for a very large energy deposition several times higher than the enthalpy of melting. Ultra-fast disordering of the lattice, so called ultra-fast melting, can only occur either due to a strong overheating, or within a time period much longer than electron-lattice equilibration time. Ultra-fast melting always occur at the thermal stage when the statistical distribution and thermodynamic parameters, namely temperature, entropy, and pressure, are all fully established, and fully characterize the transient state of matter undergoing phase transition swiftly induced by ultrashort laser pulses. The specific contributions from the lattice heating and thermal point defect generation could be as essential for ultra-fast disordering as in equilibrium. A comparison of the growing time of the highenergy tail in the Maxwellian distribution Eqs.(4.3–4.5) with the electron-to-lattice energy transfer

[ ( )] = [ν (T )] (T

en en time, te− ph ≡ ν e− ph T

−1

en e− ph

−1

room

room

)

2

TL , shows that lattice heating occurs much faster

than the filling up of the high-energy tail of the distribution. Therefore, the thermal defects and associated disordering are absent before the electron-phonon temperature equilibration. For this reason fast melting should commence in non-equilibrium conditions of strongly overheated solid before the defect contribution becomes significant. The level of superheating can be evaluated from the energy considerations and from the comparison of the high-energy tail filling up time and energy transfer time [12,14]. For example, for Aluminium swiftly superheated to TL ≈ 3Tm the entropy increase due to the lattice heating equals to 6.98kB, above the catastrophe value of 6.3kB at which the entropy reaches the value for the liquid state [63]. Let us now summarise the main processes leading to the onset of transformation to liquid state. Electrons are heated to the maximum temperature at the end of the laser pulse. Statistical distributions within the electron and lattice sub-systems are established in 10-20 fs, this allows the use of averaged thermodynamics parameters such as temperature, pressure and entropy, the ultimate characteristic of disorder in a system, for description the state of each sub-system. Nevertheless, the crystal remains in the non-equilibrium state: the electron temperature still remains much higher than that for the lattice. Thus disordering and temperature equilibration are going along simultaneously. The entropy change contributed solely by strongly excited electrons is insufficient for the lattice disordering. The excited electrons produce

33 significant atomic displacement comparable to that of thermal expansion, and, as the thermal expansion does, the entropy changes are small in comparison to the catastrophic limit. The time for setting up the high-energy tail of the atomic distribution, containing approximately 10% of the atoms, appears to be longest of all relaxation times in ultrafast laser induced transformation. The high-energy tail is responsible for the generation of thermal point defects, which contribute half of the entropy change necessary for disordering. In the absence of the point defects the main contribution to the disordering comes from the lattice heating. The critical (catastrophic) value of the entropy is achieved if the lattice is superheated to the temperature approximately tree times the equilibrium melting point. At the same time the density of thermal point defects reaches several percents, necessary for triggering a catastrophic disordering. We note here that the onset of disordering by the lattice superheating could be reached if the energy losses from the skin layer due to heat conduction are negligible during the lattice heating by the energy transferred from electrons. The energy-lattice transfer time, teen− ph ≈ ε F 2ν emom − ph k B T ,

(

should be shorter than the cooling time,

ν

tcool = 3ls2 emom − ph

vF2

)

, to complete the phase transformation.

4.4. Heterogeneous and homogeneous nucleation in phase transformation The transformations of the material induced by a laser pulse in less than 100-fs time takes place in a surface skin layer with a thickness ls that comprises approximately hundred of atomic layers. Therefore, the implications of the space and time constraints on the kinetics of the phase transformation in such conditions are essential. A crystal lattice heated above its melting point up to the entropy catastrophe limit passes into explosively unstable state. Thermal point defects, or vacancies, are microscopic seeds for formation of the macroscopic nuclei of new phase in the bulk of a skin layer and at the skin-vacuum interface. Vacancies could condense at free surfaces, interfaces, and grain boundaries forming small clusters with the surface energy being proportional to the heat of formation of a vacancy [26]. The kinetics of melting depends strongly on the development of these small liquid-like clusters evolving in the crystalline phase and breaking the symmetry of the crystalline lattice. There are two processes that can drive the thermal phase transition in a skin layer. The temperature has maximum near the free surface; therefore the transformation into the new phase there is energetically favoured driving then the process of heterogeneous nucleation. Alternatively, the formation of liquid “seeds” by condensation of the thermal defects within an internally heated bulk crystal could also be a driving force for the transformation, if the inside temperature exceeds the melting point [27]. This is the process of homogeneous nucleation. Small seeds of the new molten phase are created in the overheated layer due to the lattice fluctuations generated by the thermal defects. For a closed system near the melting point the probability of such a fluctuation is [27]: ⎛ ΔS w ~ exp ⎜⎜ − f ⎝ kB

⎞ ⎟⎟; ⎠

(4.6)

where ΔS f denotes the entropy difference between solid and liquid. This probability is strongly enhanced by increase in the vacancy concentration because while ΔS f > 0 , the vacancy concentration increases to the critical value of 7.7 percent at the temperature close to the melting point [26].

34 These seeds, however, are unstable structures because the formation of an interface between the two phases requires extra energy to overcome the surface tension at that interface. Therefore homogeneous nucleation needs additional energy to proceed when compared to the heterogeneous nucleation. There is a critical radius rcr for a seed of the molten phase defined from the condition that the surface energy is balanced by the internal thermal energy of the seed. The critical radius of the seed depends on the surface tension, α, at the solid-liquid interface, melting point and the temperature of the overheated layer, kBT [27]. Analysis of the heating the Aluminium (α ~ 1 J/m2, melting point Tm = 933.5 K; na = 6.02×1022cm-3) skin layer of 13.1 nm (for 800 nm) shows that critical seeds of diameter comparable to the skin depth are formed at the temperature ≥ 2Tm . Thus when the critical seed radius compares to the thickness of the heated layer the phase transition should be strongly inhibited or terminated. One can conclude that the homogeneous nucleation in Aluminium layer terminates at the temperature ≥ 2Tm . Superheating experiments indicate that melting is usually a process driven by heterogeneous nucleation that is energetically more favourable. The phase transition starts from the target-vacuum interface due to the energy reason: there is no energetic barrier for conversion solid to liquid phase (no energy should be spent for the solid-liquid surface tension compensation) at the solid-vacuum interface. Thus, the melting wave starts at the outer surface at the moment when the entropy catastrophe conditions for the onset of melting were achieved. It is reasonable to suggest that the thermal defects condensation occurs with a rate proportional to the local speed of sound. Then the “phase transition wave” propagates inside the crystal with the speed of sound. Hence, the heterogeneous melting time for the layer with thickness of ls estimates as:

thetero ≈ ls vsound .

(4.8)

However, the temperature in the skin layer exceeds the melting point by ~3 times in the condition when the entropy catastrophe limit is already achieved. Therefore the three processes of heterogeneous melting wave propagating inside the layer from the outer surface, the homogeneous melting inside the crystal, and the cooling by the heat conduction, all are going simultaneously. Interplay between these processes defines the transition time and the life-time of a new phase, and after that – the time of reverse transformation due to cooling to the ambient temperature. We consider again the ultra-fast melting of Aluminium skin layer (ls = 13.1 nm; heat diffusion coefficient, D = 0.979 cm2/s; vsound = 5×105 cm/s – all parameters are at room temperature). The wave of the heterogeneous melting propagates through the skin layer in 2.6 ps under the assumption that the catastrophe condition holds everywhere along the skin during the propagation, while the cooling time for the layer is 1.75 ps. During this time the temperature drops to 0.7 of the maximum temperature. Thus, it is unlikely at the superheating to Tmax = 3Tmelt the fast Al melting would be completed during the first two picoseconds. It is clear that competition between the homogeneous and heterogeneous nucleation and the cooling, which defines the time and degree of the transformation of the material, is a common feature of the ultra-fast melting in different materials. Theoretical calculations [66] and molecular dynamic simulations [67] stressed the dominant contribution of the homogeneous contribution to the ultrafast melting process. Clearly, more theoretical and experimental studies are needed to obtain quantitative evaluation of relative contributions of heterogeneous and homogeneous processes in the laser-induced melting. 4.5. Transient state of matter created by ultrafast laser excitation

35 The ultra-short laser-excited solid can be temporarily transformed into the new transient phase state under the action of laser fluence, which is different from the mixture of liquid and solid states [45,68,69]. The heated material can exist in this new state for some time, of the order of nanoseconds; then cooling brings the laser-affected solid back to the ambient conditions. A transformation can bring short-lived changes in optical and material properties. These changes include the modification of the phonon spectrum, variation in the number of free carriers in the conduction band, vanishing of the shear modulus, coherent and chaotic atomic displacements, and finally the loss of the long-range order. The probing of the excited layer by single and double simultaneous optical beams allows measuring the time-dependent reflectivity of the probe beam with high temporal resolution comparable to the pulse duration of the probe beam (40-100 fs). Changes in the free carriers number, average phonon’s frequency and phonon’s lifetime along with the rate of transformation into liquid are all imprinted into the dielectric function and, can be recovered from the reflectivity measurements (see Section 3). The time-dependent diffraction pattern and diffracted intensity of short x-ray and electron beam probes, in principle, allows one to directly observe and identify the transient structural modifications, change in the inter-atomic spacing and disordering in the laser-excited layer. However, powerful laser-matter interaction induces phase transformation in non-equilibrium conditions and with significant spatial non-homogeneity. Spatial distribution of the absorbed laser energy across the skin layer is the primary source of non-homogeneity that affects the x-ray and electron beam probe diffraction. Therefore, the interpretation of the experiments with the ultra-fast x-ray and electron beam probes should take into account non-equilibrium conditions and spatial non-homogeneity within the penetration depth of the probes. For these reasons application the classical (equilibrium) DebyeWaller factor for the interpretation of fall-off the electrons and x-ray diffraction intensity in nonequilibrium conditions is questionable – see [25,12,14] for details. There is no obvious way of distinguishing the decrease in the diffraction beam intensity caused by the non-homogeneity from the disordering during melting. The Bragg conditions should be different at the different spots across the focus and along the depth in the skin layer due to non-homogeneity. One may expect the increase of the diffusive background in the diffracted beam intensity similar to that of the inelastic scattering. 4.6. Ultra-fast melting of metals and dielectrics First, let us recollect the differences and similarities of melting characteristics for materials in equilibrium. For the majority of simple solids the density in a liquid state is lower than that for the solid. For these group the melting entropy lies in a range 1.1-1.4 kB while for the materials, which have a liquid density higher than that for a solid state (H2O, gallium, bismuth, silicon, InSb), the melting entropy ranges in 2.3–3 kB. It is worth noting that the difference is small in comparison to the catastrophic value for disordering of (6–7)kB. On the other hand, the energy characteristics of melting, the ratio of melting temperature (in energy units) to the binding energy for the majority of elements from the periodic chart constitutes k B Tm ε b ≈ 3% . The enthalpy of fusion for broad range of metals and dielectrics varies in a range from 0.335 kJ/cm3 for ice and 0.5 kJ/cm3 for Bi at low end, up to 0.975 kJ/cm3 for InSb and 1.86 kJ/cm3 for Copper at the high enthalpy limit. Therefore in equilibrium, when the heating conditions are comparable, the process of melting of the majority of solids proceeds in a similar way. Consequently, one would not expect dramatic differences in the

36 ultra-fast melting by the swift laser action between metals and dielectrics, if the same absorbed energy density deposited in the material in the same period of time. Apparently, this is not the case. The laser interaction with metals and dielectrics is drastically different, and this difference results in the dissimilarities of the ultra-fast melting of diverse solids. Indeed, the metals are well absorbing even at the low laser intensity while most of the dielectrics, especially the wide band gap dielectrics, are transparent up to the optical breakdown threshold. Optical breakdown for dielectrics lies in the intensity range of a few tens of TW/cm2 [70]) that corresponds to fluence of ~ 1 J/cm2 for 100-fs laser pulse. At the threshold of optical breakdown, up to 10% of valence electrons are transferred to the conduction band, which leads to a strong increase of the intra-band absorption with minor effect on other parameters. This difference can be clearly seen when the dependence of the absorbed energy density on laser fluence is compared for metal and for dielectric. For illustration, such a difference is shown for aluminium and fused silica [71] in Fig.4.1. The absorbed energy density for Al plotted in Fig.4.1 following formula Al Eabs kJ cm3 ≈ 91.4 × F J cm2 , which gives the correct value for the experimentally observed

[

]

[

]

ablation threshold of (0.35-0.42) J/cm2 [4,72,73]. The corresponding formula for silica is chosen in

[

]

([

SiO2 kJ cm3 ≈ 17.5× F J cm2 the form Eabs

]) closely reproduces the ablation threshold predicted for 7

silica at 800 nm, 1.84 J/cm2 [72] and 1.5 J/cm2 damage threshold, as well as the experimentally observed dependence of the electron number density on fluence from [71]. One can easily see that the ablation thresholds for Al and for silica differ by about five times, while the absorbed energy density at that threshold in silica is ten times larger than in Al, thus signifying the dramatically different interaction modes in metals and dielectrics.

Figure 4.1. Dependence of the absorbed energy density on the absorbed laser fluence for aluminium and fused silica at 800 nm. The ablation threshold point ~0.35 J/cm2 is marked on the Al curve and 2 ~1.85 J/cm on SiO2 curve [4,72]. The solid line part of the fused silica dependence compiled from [68].

In conclusion of this section we stress that comparison of the experimental results with theory provided in [74] shows that optical, spectroscopic, x-ray and electron diffraction measurements of different ultra-fast laser excited solids clearly indicate that disordering (e.g. melting) commences and continues at the thermal stage, when electron and lattice temperatures are equilibrated. Even in the

37 case of strong overheating of Al over the ablation threshold melting occurs before ablation in picoseconds after the end of the pulse [73]. The high intensity necessary to achieve the melting conditions in dielectrics with the short laser pulse results first in the breakdown of dielectrics making then the interaction and the following melting close to that with metals. The material disordering caused by ultra-short laser excitation is characterized by time-dependent entropy at non-thermal and equilibrium stages. The entropy increase caused solely by the electronic excitation is less than one tenth of the critical value for the major lattice disordering associated with melting. Structural stability of a swiftly excited solid also enhanced due to the fact that the time for the thermal defect formation appears to be much longer than that for all other relaxation processes Therefore, the superheating of the lattice over the equilibrium melting point can drive the disordering. On the other hand the superheating results in the ultra-fast electron-to-lattice temperature equilibration ensuring a purely thermal character for melting of a solid. The catastrophic disordering leading to the liquid state is a consequence of the superheating of atoms in the crystal, and/or the formation (or presence) of characteristic point defects while solely electrons contribution is minor. 5. ULTRAFAST LASER ABLATION 5.1. Introduction Atoms escape from the surface if they gain the energy from the laser higher then the atomic binding energy. The binding energy of the material thus determines the ablation threshold; the absorbed energy in access of the binding energy constitutes the kinetic energy of atoms removed from the surface. Removal of material from the surface by laser irradiation termed as laser ablation to distinguish it from evaporation in equilibrium conditions [75-78]. Here we discuss the main characteristics of laser ablation, which distinguish it from the familiar evaporation in equilibrium, we describe mechanisms of ablation by ultra-short laser pulses, ablation rates and ablation thresholds, and other ablation characteristics as a function of laser irradiation and material properties. The absorbed energy density per atom in a laser-heated skin layer should be comparable to or above the heat of vaporisation in equilibrium. The heat of vaporisation in equilibrium for metals and dielectric is fairly comparable. Indeed, it equals to 324 kJ/mole (3.375 eV/atom) for Gold and 359 kJ/mole (3.74 eV/atom) for Silicon. However, the typical ablation threshold fluence by femtosecond lasers for metals is less than 1 J/cm2, while for dielectrics it is of the order of 2 J/cm2 – 4 J/cm2. Therefore laser ablation with a 100-fs pulse requires the intensity ≥ 1013 W/cm2 [79-83]. On the other hand it has been shown that optical breakdown threshold for dielectrics resides at intensities of the order of 1013 W/cm2 [70,82,84-87]. Thus at the very early stage of the 100-fs pulse a solid converts to plasma. The major part of the ultra-short laser pulse interacts with plasma in non-equilibrium conditions. The experimentally observed difference in ablation thresholds for metals and dielectric is explained by the additional energy expenses for ionisation of dielectrics before the ablation. By contrast, the material removal with long (nanoseconds) low intensity (108 W/cm2 – 109 W/cm2) pulse proceeds in equilibrium, where heating, melting, and evaporation of the target proceeds in succession, without plasma formation. A distinctive feature of the ultra-short interaction mode is that the laser pulse is shorter than the energy equilibration processes in the irradiated material. Thus, the non-equilibrium phenomena play an important role, modifying significantly the mode of material removal. One of these phenomena, which we discuss in detail later in this Section, is the fast formation of lattice distribution function

38 with a truncated high-energy tail. The consequence of this effect is existence of different ablation modes. There are three modes of ablation, depending on the laser and target parameters and on the ambient conditions, if the target in vacuum or in gaseous environment. The electrostatic ablation, the extreme ablation mode, is completely non-equilibrium and nonthermal. This mode is realised when a short powerful pulse elevates the average electron energy during the pulse in excess of the sum of the binding energy of the ions plus the energy necessary for the electron to escape from a solid [4]. The lattice remains cold during the pulse. The energetic electrons escaping from the solid create a strong electrostatic field of charge separation, which pulls ions out of the solid. We should stress that the electrostatic ablation should not be confused with the Coulomb explosion. In electrostatic ablation fast electrons are escaping the solid and pull out the cold ions by the force of electrostatic attraction between negatively charged electron and positive parent ion. The Coulomb explosion is a repulsion of ions of the same charge in specific conditions when electrons are swiftly, faster than the Coulomb force action, are removed from the parent ions [88,89]. The second, transitional, mode is realised when electrons have enough time to transfer the energy to the lattice and the average energy of the ions (ion temperature) exceeds the binding energy but the distribution function is far from the equilibrium Maxwell distribution [4,72]. In these conditions, the majority of ions escape the solid before the equilibrium distribution is established. The third mode, thermal equilibrium ablation, coincides with evaporation in equilibrium conditions. This mode may occur after the pulse and after the time when full Maxwell distribution is established, including the high-energy tail with energy in excess of the binding energy. In this mode the ablation is the same as evaporation in equilibrium: only particles with energy in excess of the binding energy from the high-energy tail can escape the solid. An ambient gas next to the ablation surface also helps to establish the equilibrium distribution. This effect leads to a decrease in the ablation threshold [4,72]. It is noteworthy that absorbed energy density at the ablation threshold in extreme non-equilibrium conditions is 2-3 times higher than the equilibrium enthalpy of evaporation. High repetition-rate lasers (up to 100 MHz) are used in practical applications of the laser ablation process [90,91]. In this case even at the maximum available scanning speed a big number of pulses hit the same focal area. This leads to the coupling between successive laser pulses and thereby to incubation of the heating and evaporation processes. At repetition rate above a certain threshold (depending on the heat conduction in a target) the multiple-pulse action may result in a reduction of the laser-ablation threshold thus introducing another variety of the ablation mode. Below we present practical recipes for efficient ablation and control the phase-state of the vapour as a self-consistent function linking laser parameters, namely, the pulse duration, energy per pulse, and wavelength, to the parameters of the material to be ablated. 5.2. Thermal evaporation Let us recollect the evaporation rate in equilibrium conditions. The equilibrium evaporation rate for a solid in vacuum at temperature T follows from conventional thermodynamics [27]:

(n v ) i i

equilibrium

=

P 2π mi k B T

(1− R);

(5.1)

39 where P is pressure of saturated vapour at the temperature T, the same in solid and in vapour, R is the average coefficient for vapour atoms reflection from a solid-vapour interface, and mi is atomic mass. In order to find the saturated pressure one needs to know relation between pressure, density and temperature in solid and liquid, so called equation of state, which is unknown in the whole range of density and temperature. Qualitatively correct equation of state for such an estimate can be obtained under the assumption that at the temperature above the Debye temperature (T > TD) the atoms in a solid state are still undergoing the harmonic vibrations [27]:

() ()

()

⎧ε = ε V + c k T N ≡ ε V + ε c s B c th ⎪ ⎨ ε ⎪ P = th Γ V + Pc ⎩ V

(5.2)

The total number of particles, N, is constant. The energy at T = 0, so called “cold” energy or specific energy, εc, is temperature independent and it defines a “cold” pressure, Pc = − dε c dV . Γ V is the

()

Gruneisen coefficient, Γ >0, which defines the ratio of the thermal pressure to thermal energy. The Gruneisen coefficient changes during the evaporation process from ~ 2 at the solid state to 2/3, which corresponds to the ideal gas in vapour. The temperature dependent term, εth, in general includes the ionic and electronic parts. Solid and vapour parameters near the solid-vapour interface are explicitly linked at thermodynamic equilibrium. The temperature, pressure and chemical potential at the solid-vapour interface are equal for both phases. The pressure at the solid-vapour interface in equilibrium follows from the condition of equality of the chemical potential for the solid and for liquid μ s = μ g [53]: c −cs

⎛T ⎞p P = C⎜ ⎟ ⎝ TD ⎠

⎛ε −ε ⎞ exp⎜⎜ c g ⎟⎟; ⎝ kB T ⎠

(5.3)

where c p , c s are respectively the specific heats for the vapour and the solid (both enter into the exponent in the dimensionless form in the previous equations and up to Eq.5.6); ε c , ε g , are the cold energy in the solid and the energy per atom in the gas. The Debye temperature, TD , is used for normalisation, C is a constant. The heat of phase transition equals to the difference between the enthalpy in the gas and in solid state: wg − ws = c p − cs k B T + ε g − ε c . One can neglect the interaction

(

)

of atoms in vapour in comparison to that in solid, εg<< εc. Thus, the energy required for separation an atom from the bulk at zero temperature equals to the binding energy in solid, because ε c V → 0 = ε b . Then Eq. (5.4) converts into familiar form [53]:

(

)

c −cs

⎛T ⎞p P = C⎜ ⎟ ⎝ TD ⎠

⎛ ε ⎞ exp⎜ − b ⎟ ⎝ kB T ⎠

(5.4)

( )

One can express the specific heat for the vapour c p = γ γ −1 through the adiabatic constant,

γ = c p cV . For γ = 5/3 as in ideal gas it makes c p = 5/2, while cs changes from cs = 3 for a solid up to cv = 3/2 in a vapour state. Inserting Eq.(5.4) into Eq.(5.1) one obtains the explicit expression for the evaporation rate in equilibrium:

(n v ) i i

equilibrium

( )

∝ 1− R T

c p −cs −1/ 2

⎛ ε ⎞ exp⎜ − b ⎟ ⎝ kB T ⎠

(5.5)

40 It is reasonable to take cs ~3/2 and c p = 5/2 and denote a characteristic density of the saturated vapours near solid-vapour interface as nvap (that density can be obtained only from experiments). Then the evaporation rate in equilibrium can be expressed in the form with clear physical meaning: 1/ 2

⎛k T ⎞ ∝ nvap ⎜ B ⎟ equilibrium ⎝ M ⎠

( ) ni vi

⎛ ε ⎞ exp⎜ − b ⎟ ⎝ kB T ⎠

(5.6)

This is a flow of evaporated atoms through the unit area. Thus, only an exponentially small amount of particles with energy ε b from the high-energy tail in the Maxwell distribution evaporates from a solid in equilibrium [53]. The rate of evaporation in equilibrium is low due to condition, k B Tevap << ε b , however velocity of evaporating particles equals to a local speed of sound. Let us show that the number density of evaporating atoms is proportional to those in the high-

(

energy tail of Maxwell distribution, dN v = n M 2 πk B T

(

from ε b k B T nε ≥ε b ≈

)

1/ 2

2

π

)

3/2

(

)

exp − Mv2 2k B T 4 π v2 dv . Integrating

to infinity one gets the number of atoms at the energy εb >> kBT:

(

n εb k B T

)

1/ 2

(

)

exp − ε b k B T .

(5.7)

Note that in equilibrium the establishment of the Maxwell distribution goes much faster than evaporation, tevap ≈ ω D−1 exp ε b T >> teq ≈ ωD−1. By contrast, under non-equilibrium conditions,

(

)

particle removal and the establishment of distribution function goes together, teq ≥ tevap . This implies that the majority of atoms in non-equilibrium ablation should have enough energy to break bonds, and have almost mono-energetic distribution. We consider these processes in the next Section. 5.3. Fast transformation from solid to plasma at absorbed energy close to ablation threshold Let us consider the succession of processes occurring in a solid hit by the ultra-short (≤100 fs) laser. First, it is clear that the action of fs pulse on metals and dielectrics differs by the amount of absorbed energy and magnitude of laser intensity. Metals absorb well even at low laser intensity. By contrast, absorption in dielectrics, especially in wide band gap dielectrics, is low at low intensity for wavelength as short as 200 nm. Therefore, in order to achieve significant absorption in visible range of 400-700 nm in dielectrics, the solid should be ionised. The required intensity is up to several tens of TW. The state of the solid heated to the enthalpy of vaporisation in equilibrium and the state of the same solid heated by the ultra-short pulse at a similar energy density is drastically different. The absorbed ultrashort laser energy is contained solely in the electrons during the relatively long period required for transferring the energy to the ions. The electrons are reaching maximum temperature at the end of the pulse while the ions remain cold. The maximum electron temperature in energy units is comparable to the ionisation potential (or Fermi energy for metals), kBTe ~ Ji ~ εF ~ 10 eV. Thus during the pulse the electrons and atoms distribution functions, heat capacities, the electron collisions rates and the interrelated optical and transport parameters are all changing from those in a solid to those in plasma (see Section 2). 5.3.1. Transformation of metals

41 Electrons in the conduction band of metal are heated mainly due to processes of collisional absorption (Inverse Bremsstrahlung). In the course of heating, the heat capacity of electron component transforms to that for ideal gas C = 3kB/2. Zel’dovich and Raizer [92] indicated that this transition occurs at kBTe ~ εb/3 (εb is the binding energy). Accordingly, the Fermi distribution of electrons gradually changes to the Maxwell distribution of plasma. Indeed, the heat capacity of the degenerated gas at such electron temperature, Ce ≈

π 2 εb kB , 6 εF

is close to that of ideal gas. The initial Fermi distribution changes and after the electron-electron equilibration, which takes tee ~(ωpe)-1 ~0.1 fs, it transforms into the Maxwell distribution in plasma (see Fig.5.1). The electron-phonon collisions are gradually overwhelmed by electron-ion collisions (see Fig.2.3) when all laser-affected material in the skin-layer becomes fully transformed into plasma. In metals near the ablation threshold the number density of electrons in the conduction band is practically the same as in the initial state. Therefore, change in the optical and transport properties of excited metal occurs mainly due to increase in the momentum transfer rate (see Section 2). The electrons transfer the absorbed energy to the lattice (ions). Maximum of the ions temperature is achieved by the time of the electron-ion temperature equilibration. Note that maximum electron temperature is at least two times larger than the maximum lattice temperature. However the ion distribution function remains different from the full Maxwell form due to the absence of high-energy tail long after the end of the laser pulse (see Ch.2 and Fig 2.5). Hence several modes of ablation are contributing into the material removal depending on the interplay between many competing processes, which we discuss below. The major difference between the ablation of metals and dielectrics relates to the fact that dielectrics should be ionised first before the ablation can start. The ionisation (the optical breakdown) leads to the dramatic increase in the density of electrons in the conduction band and related change of the all properties of excited dielectric similar to those in metals heated to the absorbed energy density comparable to the enthalpy of vaporisation. 5.3.2. Ionisation of dielectrics At low light intensity the absorption in transparent dielectrics is low. The complex dielectric function has large real and very small imaginary parts. The increase in the laser intensity leads to transfer of valence electrons into conduction band. Conductivity electrons absorb the laser energy via inverse Bremsstrahlung and resonance absorption as it occurs in metals. The skin layer of dielectric converts into plasma by ultra-short pulses of ~100 fs at the average intensity around 1-10 TW/cm2 [80,81]. It is conventionally suggested that the ionisation threshold is achieved when the electron number density in conduction band reaches the critical density nc = me ω 2 4 π e2 , corresponding to the incident laser wavelength [16,93]: nc =

t ion

∫ W (I (r, z, t))dt ;

(5.8)

0

( ( ))

where W I r, z, t is the sum of ionisation rates from electron impact and multi-photon mechanisms, which is presented later. The energy spent for ionisation should be accounted for in the energy balance. The moment when critical density is achieved defines the ionisation time. It follows from (5.8) that the breakdown threshold depends on the total energy density per surface area (fluence)

42 absorbed during the ionisation time. The relative roles of impact ionisation and multi-photon ionisation depend dramatically on the relation between the electron oscillation energy in the laser field, ε osc = e2 E 2 4meω 2 (averaged over the light period and for linear polarisation), and the ionisation potential, Ji, or band gap Δg [4,79]. The ratio of ionisation potential (or band gap) to the energy of oscillations is the adiabaticity parameter, γ 2 = Δ g / ε osc . It is useful for separating different ionisation regimes. At γ > 1 the multi-photon ionisation prevails. For example γ = 1.5 for silica at λ = 0.8 μm and I = 1014 W/cm2 indicating that multi-photon process dominates. The electron impact ionisation is the main ionisation mechanism in the long (nanosecond) pulse regime. Near the ablation threshold good approximation for experimentally measured ionisation threshold

(

[71,79-81] is formula for multi-photon ionisation rate in the limit γ >>1, wmpi = Bω n3ph/ 2 ε osc Δ g

(

)

)

n ph

.

Here n ph = Δ g + ε osc hω ≈ Δ g hω is the number of photons that should be absorbed in order to move electron through the band gap Δ g . In practice it is taken in the form wmpi = β n ph I

n ph

. For

example, for fused silica at 800 nm where five photons are necessary for transferring electron from valence to conduction band through the band gap of Δ g = 7.6 eV (six photons for Δ g = 9.0 eV) it

(

[

converts to the following form: wmpi = 1.3×1010 I 1013 W/cm2

])s 5

−1

.

The energy of electron oscillations in laser field is transformed into electron kinetic energy via collisions. The ionisation by the electron impact develops into the avalanche process when the removed electron kinetic energy exceeds the band gap. Generally, the Keldysh impact formula

(

)

2

should be applied for the probability of the avalanche process per unit time, wav = wimp ε el Δ g −1 . Therefore the electron impact ionisation rate is proportional to laser intensity. For practical

[ ]

applications it is used in the form, wimp = α I s −1 . Using fused silica as an example at 800 nm, one

[ ]

(

[

])

gets, wimp s −1 ≈ 1014 I 1013 W/cm2 . One can see that at intensity 1015 W/cm2 both mechanism

equally contribute, and the ionisation time is less than a femtosecond. 5.3.3. Laser-modified properties near the ablation threshold The material properties of metal and dielectric are strongly modified at the absorbed energy density approaching ablation threshold. The changes in optical and transport properties occur due to momentum transfer rate approaching to its physical maximum. In metals the number of electrons in the conduction zone changes only slightly, thus the changes of optical properties in the skin depth are minor. In contrast, the electron number density in dielectrics increases dramatically, raising the absorption and reducing the absorption length down to the level of the skin depth of a few tens of nanometers similar to that in metals. The characteristic values for the electrons number density ne, electron-ion momentum exchange rate νei, absorption coefficient A, and the skin-depth ls, can be estimated with reasonable accuracy allowing calculation of the ablation threshold in the skin-effect approximation, i.e. keeping the above quantities constant during the pulse time and calculating spatial distribution of electric field in the form, E x = E 0 exp − x ls , and accordingly intensity and temperature distributions. It has been

( ) () (

)

shown [4] that such an approach allows determining the ablation thresholds in good agreement with experiments. The expansion time for laser-excited material estimates as a skin depth divided by the

43 sound velocity: texp ≈ ls vsound . It is conventionally of the order of a few tens of picoseconds. Thus, a femtosecond pulse interacts with a solid target whose density remains nearly constant during the laser pulse. 5.3.4. Electron-to-ion momentum and energy transfer time. The electrons’ collision rate changes from the electron-phonon interactions at low-temperature to the electron-ion interaction in solid density plasma (see Section 2). The effective collision frequency

( )

mom has a broad maximum ν e−i

max

≈ ω pe ~ ve d ~ 1016 s-1 near the ablation threshold, and as the first

order approximation does not depend on the temperature [4,15,79]. Thus, the electron-ion momentum transfer rate is much larger the laser frequency in the visible range, ν emom − i ≈ ω pe >> ω . Therefore the electron mean free path, lmfp = ve ν emom − i is of the order of a few angstroms, lmfp << lskin , justifying the validity for the normal skin effect approach. The electron-ion energy transfer time in plasma expresses through the momentum exchange rate

(

mom Mi as teien ≈ meν e−i

) ∝T −1

3/2 e

M i me (see Section 2). In accord with the previous paragraph the

minimum energy transfer time at the maximum of momentum rate reads

(t ) ei

min

≈ M i ω pe me

[15,79,88,94,95]. Note that in plasma state this time increases as temperature grows up. Taking Cu as an example (MCu = 63.54 a.u., ne = 8.47×1022 cm-3, ωpe = 1.64×1016 s-1), we estimate the ion heating time as tei = 7 ps, which is in agreement with experimental values in the literature [15,96]. Similar estimate for silver gives 15 ps, which qualitatively correlates with the measured 71 ps [96], and for gold 27.4 ps, which is comparable with the measured 70-120 ps [98,99]. These estimates for many other materials demonstrate that the ions remain cold during the sub-picosecond laser pulse and long after the end of the pulse for both metals and dielectrics. 5.3.5. Heat diffusion time The electron thermal diffusion coefficient D is expressed through the electron-ion momentum transfer rate in accord with general kinetic expression as follows [3]: v ve2 ∝T5 2 D = le, mfp e = mom (5.9) 3 3ν e−i The electron heat conduction time theat, is the time for the electron diffusion across the skin-depth ls, reads: l2 theat ≈ s . (5.10) D For copper close to the ablation conditions with 780 nm laser (ls = 67.4 nm), the heat diffusion coefficient estimates as D ≈ ve2 3ω pe ~1 cm2/s. Therefore the maximum electron heat conduction time equals to theat~ 45 ps following Eq.(5.10). Note that in plasma state the electron-ion temperature equilibration time, teien ∝ M iTe3 / 2 me , is especially large for heavy ions and high temperature, while the heat conduction time, independent on the ions mass, decreases as lheat ∝ T −5 2 . Therefore in Gold, for example, at the deposited energy density of 0.375 J/cm2, slightly in excess of ablation threshold for 400 nm [30] the cooling time for a 30 nm skin layer is 2-3 ps; this estimate is in agreement with the experimentally observed drop in the reflectivity on the rear side of free standing gold foil. It

44 makes the introduction ad hoc of ballistic transport in the hot dense plasma, where the Coulomb collisions dominate, unsubstantiated. 5.3.6. Dielectric function and absorbed energy density below ionisation threshold In metals the modification of the dielectric permittivity and other optical properties mainly relates to the change in the collision rate. The permittivity of ionising dielectric can be presented as a sum of the dielectric (polarisation) contribution and the Drude-like part due to excitation of electrons into the conduction band, ε = ε0 + Δε D . The electron-phonon collision rate ν e − ph ≈ 2 πk B T h < ω is less than the laser frequency. Therefore below the breakdown threshold the number density of electrons in the conduction band is less than the critical density and therefore the laser frequency is larger than the plasma frequency ( ω2p − e < ω 2 → ne < ncr ). The contribution of excited electrons to the real part of

( )

permittivity, Δε D

re

(

)

2

≈ 1− ω pe ω < 1, is positive and small in these conditions. Hence, we can take

the real part for the permittivity and for the refractive index equal to that of the unperturbed state. The imaginary part in these conditions takes the following form:

( ) Δε D

im

≈

ω 2pe ν e− ph ne ν e− ph . = ncr ω ω2 ω

One can see that the imaginary part of the refractive index, which is responsible for the absorption length, grows up with the increase of electron density, but remains much lower than the real part. The relations between the permittivity and the refractive index in these conditions are n02 ≈ ε re ; 2n0 k ≈ Δε D . Thus the absorption coefficient can be taken in the unperturbed form and

( )

im

the imaginary part can be neglected: A0 ≈ 4n0

(n +1) . 2

0

The main dependence of the absorbed

energy density on the number density of excited electrons comes from the absorption length through the imaginary part of the refractive index:

labs =

λ 2 πκ

=

n0 λ π Δε D

( )

im

=

2cn0 ncr

ν e− ph ne

.

Now the absorbed energy density E abs in a dielectric approaching the optical breakdown reads: Eabs = A0

ν e − ph ne F. cn0 ncr

(5.11)

The number density of excited electrons is a strong non-linear function of laser intensity when multiphoton absorption mechanism dominates, ne ∝ I n , n > 1. Temnov et al. [71] measured the dependence of the number of excited electrons as a function of the laser intensity in fused silica irradiated by 50 fs pulses at 800 nm (ω = 2.356×1015 s-1). The measured dependence of the electron

( [ ]).

density on fluence can be approximated as ne = 1019 F J cm2

6

At higher intensity the growth

saturates approaching to the critical density of ncr = ω2 me 4πe2 = 1.745×1021cm-3 at 800 nm. The dependencies of the absorbed energy density on the laser fluence for Aluminium (as example of a good metal) and for fused silica (as a dielectric) are presented at Fig. 4.1 for comparison. One can clearly see the dramatic difference in the interaction and therefore in the ablation thresholds for metals

45 and dielectrics: the dielectrics require much more energy to be deposited, and thus much higher laser fluence for ionisation. As soon as the electron critical density is achieved, the absorbed energy density increases sharply at the ablation threshold. 5.3.7. Dielectric function and absorbed energy density above the breakdown threshold At the intensities above 1014 W/cm2 the ionisation time for a dielectric is just a few femtoseconds, shorter than a typical pulse duration of ~100 fs. The electrons produced by ionisation then dominate the absorption in the same way as the free carriers do in metals, and the characteristics of the lasermatter interaction become independent of the initial state of the solid target. As a result, the inverse Bremsstrahlung absorption, together with the resonance absorption for p-polarized light at oblique incidence becomes the major absorption mechanisms for both metals and dielectrics. For this reason one can use the Drude-like dielectric function for description of the optical properties of laser-excited metals as well as the ionised dielectric. The condition ν emom − i ≈ ω pe >> ω is fulfilled if the first ionisation is completed at the early stage of the pulse. The link between the real and imaginary parts of the Drude dielectric function and refractive index can be then simplified:

(Δε )

≈

ω2 ; ω 2pe

(Δε )

≈

ω pe ; ω

D

D

re

im

( )

⎡ Δε D n ≈ k =⎢ ⎢ 2 ⎣ 4π c λ pe = ;

1/ 2

⎤

im ⎥

⎥ ⎦

⎛ ω ⎞1/ 2 ⎛ λ ⎞1/ 2 = ⎜⎜ pe ⎟⎟ = ⎜⎜ ⎟⎟ ; ⎝ 2ω ⎠ ⎝ λ pe ⎠

ω pe

here λpe is the characteristic spatial scale inverse proportional to the square root of the electron density. For good metals, λpe ≈190 nm. The Fresnel absorption coefficient and the absorption depth are the explicit functions of the electron number density and laser wavelength: A = 1− R ≈ ls ≈

c

ωκ

4κ

( )

2

κ +1 + κ 2

;

(5.12)

.

Thus, the ratio of the absorption coefficient to the skin depth appears to be a function of electron density and laser wavelength [4,88,95,99-101]:

Eabs =

8π 2 AF ≈ Cκ ne , λ F λ ls

( )

(5.13)

( )

The function Cκ ne , λ is: −1

1/ 2 ⎞ −1 ⎛ ⎛λ ⎞ ⎛ 1 λ pe ⎟ 1 ⎞ pe ⎜ . Cκ ne ; λ = ⎜1+ + 2 ⎟ = 1+ ⎜⎜ ⎟ + ⎜ ⎝ λ ⎟⎠ 2λ ⎟ ⎝ κ 2κ ⎠ ⎝ ⎠

( )

46

( )

One can see that in visible range (500-1000 nm) Cκ ne ; λ is weak function of the material and laser parameters. For example, for copper ablation at 780 nm (ω = 2.415×1015 s-1; ωpe = 1.64×1016 s-1; λpe = 230 nm) it has the value 0.56, while for gold ablation at 1064 nm (ω = 1.77×1015 s-1; ωpe = 1.876×1016 s-1; λpe = 201 nm) the value is 0.654. Thus, we assume for the further estimates that the absorbed energy density near the ablation threshold is directly proportional to the incident fluence and inverse proportional to the laser wavelength. The numerical coefficient Ck ~ 0.6; it could be corrected in Eq.(5.13) for any particular material. 5.3.8. Distribution functions for electrons and ions at ablation threshold At low intensity at the short time in the beginning of the pulse the phonon-phonon collision rate is responsible for the time of establishing the average energy density of the lattice, the lattice

(

temperature, teq ~ ν ph − ph

)

−1

≈ h TL . This time comprises a few femtosecond, however ionisation

occurs faster. The electron-electron collision time is proportional to inverse electron plasma -1 frequency, i.e. tee ~ (ωpe) ~ 5×10-2 fs for electrons in metal or for electrons swiftly transferred into conduction band in dielectrics. Therefore the electrons equilibrium energy distribution (the Maxwell distribution) establishes early in the pulse time and then follows the laser intensity evolution adiabatically adjusting to any changes. Therefore the electrons heat capacity equals to that for monoatomic ideal gas. 5.3.9. Maximum electron and ion temperatures The specific heat of the atoms and electrons becomes comparable when the electron and lattice temperature in the energy units approaches the binding energy, Catom ≈ Ce = 3k B 2 at Te = TL ≥ Tb (see Section 2). The electron temperature, Te, in a skin layer can be presented in a simple form assuming that all the interaction parameters apart from laser intensity are time-independent during the pulse. The energy conservation law for the conditions of normal skin effect and under the assumption that all the parameters but laser intensity are time independent during the pulse takes the simple form of an equation [4,88,95,99-101]:

⎧ 2x ⎫ 2E k B Te = abs exp⎨− ⎬; 3ne ⎩ ls ⎭

2 A F0 Eabs = ; ls

tp

F0 =

t dt ; ∫ I () las

(5.14)

0

where Eabs, F0 are respectively the absorbed energy density and the incident fluence in the skin layer, A = I/I0 is the absorption coefficient, I0 = cE2/8π is the incident laser intensity, ne and Ce ~3/2kB are the number density and the specific heat of the conducting electrons. The ion temperature reaches its maximum at the electron-ion temperature equilibration. The maximum ion surface temperature can be found from the energy conservation if the heat conduction losses are neglected:

(

)

k B Tmax x = 0 =

2Eabs . 3 ne + na

(

)

(5.15)

Note that in ablation of heavy metals such as Gold heat conduction becomes effective well before electron-ion temperature equilibration as was indicated before. The maximum electron temperature is always higher than the ion temperature:

47

Te, max Ti, max

= x =0

(n + n ) . e

a

ne

(5.16)

The electron temperature is two times higher the ion temperature for one conductivity electron per atom. If the maximum ion temperature is below the binding energy then the contribution of thermal ablation may only occur after the time necessary for the building up the high-energy tail in the Maxwell distribution for ions. 5.4. Mechanisms of ablation by ultra-short laser pulses An atom can be removed from a solid if its total energy exceeds the binding energy, which is the energy of vaporization per particle, ε tot > ε b . The kinetic energy of a free particle should allow the atom to leave the solid ε kin = ε tot − ε b > 0 . On the basis of the previous analysis we outline three possible modes of ablation, depending on the relation between the absorbed energy density at the end of the pulse, the pulse duration, and the electron-to-ion energy transfer time [4]. The extreme nonequilibrium and non-thermal mechanism of material removal, the electrostatic ablation, takes place when electron temperature exceeds the energy threshold defined in the following Section. The second ablation mode takes place when electrons delivered part of their energy to ions in excess of the binding energy, kB Tion > ε b , faster than the time necessary to build up the high-energy tail in the Maxwell distribution. The ion’s energy distribution function in this period is still far from equilibrium. The third mode, a purely thermal evaporation, takes place after establishing the full Maxwell distribution of ions energy. In either case the removal of atoms requires the atom to acquire energy above the binding energy. These three cases are considered below. 5.4.1. Electrostatic ablation: energetic electrons pull the ions out of solid The electrons on the surface of the skin layer gain large energy to the end of the pulse while ions remain cold, k B Te >> k B Ti . The condition of quasi-neutrality (ne = Zni, Z is the ion charge) also holds. We are interested here in the processes during the period longer than characteristic time for the electron’s oscillations, t >> ω −pe1 ~0.1 fs. Then the electron inertia is negligible and can be ignored. The description of the plasma motion in these conditions falls in the frames of hydrodynamic description when the electrostatic field of the electron-ion charge separation adiabatically adjusts to the spatial distribution of electron and ion density. This field is a driving force for the ion motion with velocity ui << ue [17]. Thus the scenario of non-equilibrium ion removal by the laser pulse shorter then electron-ion energy exchange time is as the following. Electron, which gained energy from laser in excess of the sum of work function plus atomic binding energy, leaves a solid, creates the electrostatic field, which pulls ions off the solid with the force [17]:

Felst = eEelst = −k B Te

∇ne ; ne ≈ ni . ne

(5.17)

Thermal spreading of nearly mono-velocity ion motion can be neglected because the electron-ion energy exchange time is much longer then pulse duration. It is reasonable to assume that Z = 1 near the ablation threshold. During the pulse the ponderomotive force of the laser electric field and the polarization force in general should be included. It has been shown in [4] that the ponderomotive and polarisation forces are comparable to the electrostatic force only at the very beginning of the laser pulse when the laser intensity and electron temperature is low. Later during the pulse

48

(

)

Felst >> Fpond , Fpol , thus both forces can be neglected in comparison with the electrostatic force. Thus

the equation of motion for ions reads:

M

dui ≈ Felst . dt

(5.18)

It should be especially stressed that the electrostatic ion removal is of a different nature from the Coulomb explosion. Removal of ions by the force of the electrostatic field is a result of the Coulomb attraction of negative (electron) and positive (ion) charges trying to maintain quasi-neutrality (charge conservation). The Coulomb explosion occurs in conditions when electrons are swiftly (during the time shorter of the Coulomb force action) removed on the distance exceeding the Debye length. Then repulsion of a few positive charges of parent ions results in ions moving apart of each other (exploding). Locally charge conservation is violated. The electrostatic force, Felst = −k B Te ∇ne ne , can be evaluated under the assumption that the density gradient of escaping electrons is inverse proportional to the Debye length, lD = ve ω pe . It is also necessary to account for the electron’s energy losses for the work function by taking the electron velocity as:

(

⎡2 T − ε e esc ve = ⎢ ⎢ me ⎣

)⎤⎥

1/ 2

⎥ ⎦

.

Taking into account, 3ne k B Te 2 ≈ 2 AF0 ls , and ∇ ln ne ≈ 1 lD the electrostatic force near the solidvacuum boundary at the end of the pulse reads:

Felst ≈ −k B Te∇ ln ne ≈

4 AF0 . 3ne ls lD

(5.19)

This force depends on time explicitly through the electron temperature, thus it reaches its maximum at the end of the laser pulse. The electrostatic force of Eq. (5.19) pulls the ions out off the solid if the electron energy is larger than the sum of the binding energy plus the work function. The maximum energy of ions dragged from the target reaches: ε i t = Zε e = k B Te − ε esc − ε b (for Z = 1). One can see that the electron

()

temperature of k B Te = ε esc + ε b sets the threshold for the ablation of metals. Note that the maximum electron temperature in the energy units at the ablation threshold for metals is approximately twice the binding energy. An atom is separated from a solid if it is dragged away to the distance larger than the Debye length, lD ≈ ui t . The time necessary to move an ion to this distance could be estimated with the help of the equation of motion for ions Eq. (5.18) with the force from Eq.(5.19) corrected for the energy losses for electron (work function) and ion detachment (binding energy) from a solid. However, the simple and reasonable estimate can be made taking the maximum ion velocity as

[( ≈ [2(k T

ui ≈ 2 k B Te, max − ε b − ε esc ve

B e, max

− ε esc

)M]

1/ 2

i

)m]

1/ 2

e

and,

correspondingly,

the

electron

velocity

as

. Then the time required for the removal of an ion from the target, the

ion ablation time, reads: 1/ 2

tabl

1/ 2

l 2 ⎛ M i ⎞ ⎛ k B Te − ε esc ⎞ ≈ D ≈ 1/ 2 ⎟ ⎜ ⎟ . ⎜ ui 3 ω pe ⎝ me ⎠ ⎝ k B Te − ε esc − ε b ⎠

(5.20)

49 This qualitatively correct expression formally diverges at the ablation threshold, k B Te = ε esc + ε b , indicating the absence of the removal of the ions as the ablation time goes to infinity. However, when the laser fluence even slightly exceeds the ablation threshold, this time becomes comparable and even shorter then the pulse duration. For example, for Copper (Mi = 63.5 a.u.; ω pe = 1.64 ×1016 s −1 ) at

k B Te = ε esc + ε b + Δ (taking Δ = 0.2 ε b ) this time comprises 57.6 fs, while at the extreme case of k B Te >> ε b it equals to 23.5 fs. It should be noted that for high intensities well above the ablation threshold the energy conservation equations should include the energy losses for the ion heating, and for electron ionisation and emission. This effect of electrostatic acceleration of ions is well known from the studies of the plasma expansion [17] and ultra-short intense laser-matter interaction [88,95,102]. The electrostatic ablation is the only mechanism for atoms removal during the laser pulse in the non-equilibrium conditions close to and above the ablation threshold. Two processes responsible for terminating the electrostatic ablation are a space charge that builds up in the plasma plume and the two-dimensional effects associated with the plume expansion. The characteristic time for the electrostatic ablation comprises several tens of femtoseconds and occurs during the laser pulse. 5.4.2. Non-equilibrium, semi-thermal ablation ( k B Tion ≥ ε b ) Removal of ions by escaping electrons becomes impossible if the electron temperature does not reach the electrostatic ablation threshold. The maximum electron temperature is more than two times = ne + na ne , depending on the number larger than the maximum ion temperature, Te, max Ti, max x =0

(

)

of conduction electrons per atom. Therefore the removal of atoms is only possible after the energy transfer from electrons to the ions. This process usually takes several picoseconds. Therefore ablation in these conditions occurs well after the end of the pulse. The surface atoms can leave a solid when the average ion temperature in the surface layer becomes comparable to the binding energy. This process can contribute to the total ablation only if the maximum electron temperature is 2-3 times higher than the binding energy. Note that this ablation mode can be efficient even when only the main part of equilibrium distribution without the high-energy tail is established. Therefore, this mode can be termed as semi-thermal and non-equilibrium one. 5.4.3. Thermal evaporation ( k B Tion < ε b ) Finally, the ion-ion collisions fully establish the equilibrium distribution including the highenergy tail. This requires longer time due to the occurrence of many collisions. In plasma, where the

(

Coulomb collisions dominate, the tail building time is ε b k B Tion

)

3/2

times longer than that for

establishing the equilibrium temperature. Then the conditions in ablating target and in the vapour next to a solid surface are close to those in equilibrium; the ablation characteristics can be estimated by the results taken from conventional thermodynamics. The ablation rate can be estimated as for the equilibrium evaporation rate into vacuum in accord with the formulas presented in Section 5.2 above. However, caution should be exercised when applying thermodynamic relations to laser ablation with short pulses because conditions in the expanding plume are usually far from equilibrium. 5.5. Ablation thresholds The ablation threshold by a single laser pulse is defined as a minimum amount of energy that initiates the material removal process. The definition of the ablation threshold relates to the obvious

50 physical limit independently of any subjective views: at least one atom should be removed from the target surface to mark a beginning of ablation process. In real laser ablation experiments the focal spot diameter is much larger than the absorption depth, which is of just a few tens of nanometers. Therefore the description of the process as one-dimensional is a good approximation. We define the ablation threshold as the amount of energy necessary for removal of a single mono-atomic layer from the surface. This definition complies well with the classical theory of evaporation and with the experimental results obtained by extrapolation of the ablated depth/mass vs laser fluence dependence to the ‘zero’ thickness. Note that the thickness of the monolayer is comparable to the mean free path

( )~

of atomic collisions in a solid: lmfp = nσ

−1

(1023×10-15)-1 ~10-8 cm).

Therefore, only kinetic

approach should be used for the description of ablation phenomena in the outermost surface layer, even in the conditions close to thermal equilibrium. It is well known that the surface atoms are loosely bound to the bulk making part of bonds dangling or saturated with foreign atoms [103,104]. The removal of atoms from the surface layer at the ablation threshold depends on the energy absorbed in the outermost surface layer. The energy distribution is responsible for the relative contributions of the non-equilibrium and thermal evaporation. The establishment of quasi-equilibrium distribution in the outermost surface layer also strongly depends on the environment: either for the solid placed in vacuum or in the ambient gas of specific atomic content and pressure. Below we consider the processes responsible for the removal of swiftly heated atoms from the surface layer into vacuum, and consider relative contribution from thermal and non-thermal processes near the ablation threshold. First, the process of non-thermal ablation in vacuum is considered. Next, the processes of the energy transfer are accounted for. These processes are important for understanding the ablation thresholds with the pulse duration in the transitional range from the short pulse to the long pulse regimes. Then, it is demonstrated that the presence of an ambient gas (air) next to the ablating solid leads to a decrease of the ablation threshold due to significant contribution of thermal evaporation. 5.5.1. Ablation thresholds in vacuum Let us consider first the thresholds for the electrostatic ablation in extreme non-equilibrium conditions for metals and dielectrics [4]. a). Ablation thresholds for metals. The minimum energy required for electron to escape the solid equals to the work function. In order to drag a cold ion out of the target by the force of electrostatic field the electron must have an additional energy equal to or larger than the ion binding energy. Hence, the ablation threshold for metals can be defined as the condition for the electron energy to reach, in a mono-atomic surface layer d << ls, the value equal to the sum of the atomic binding energy and the work function by the end of the laser pulse. From the energy conservation one obtains the equation allowing to obtain the ablation threshold fluence:

εe = ε b + ε esc =

4 AF0 3 lsne

The threshold laser fluence for ablation of metals is then defined as:

Fthm =

3 ln ε b + ε esc s e . 4 A

(

)

(5.21)

51 It is assumed that the number density of the conductivity electrons is unchanged during the lasermatter interaction process and the electron heat capacity is equal to that of the ideal gas. In dense plasma for the laser wavelength in the visible range the approximate relation holds (see Eq.(5.15)) as: A ls ≈ 4 π Cn λ . Hence, the threshold fluence is proportional to the laser wavelength: Fthm ∝ λ . This relation agrees well with the experimental data. b). Ablation threshold for dielectrics. The ablation mechanism for the ionised dielectrics is similar to that for metals. However, there are several distinctive differences [4]. First, an additional energy is needed to create the free carriers, i.e. to transfer the electron from the valence band to the conductivity band in order to increase the absorbed energy density to the level needed for ablation. Therefore, the energy at least equal to the band gap (the ionisation potential Ji) should be delivered to the valence electrons. At this stage the absorbed laser fluence for the breakdown threshold should be delivered well before the end of the pulse. The number density of free electrons then saturates at the level ne ~ na, where na is the number density of atoms in the target. Then additional energy equal to the binding energy per atom should be absorbed. Hence the threshold fluence for ablation of dielectrics Fthd is defined as follows:

Fthd =

3 ln εb + J i s e . 4 A

(

)

(5.22)

Therefore, in general the ablation threshold for dielectric in the ultra short laser-matter interaction regime is higher than that for the metals. The absorption in the ionised dielectric also occurs in a skin layer; thus one can use the relation Eabs = 2 AF ls for estimates of absorbed energy density in dielectrics. The ablation thresholds of Eqs.(5.21) and (5.22) do not depend explicitly on the pulse duration and intensity. However it is just a first order approximation. Certain, though weak, dependence is hidden in the absorption coefficient and in the number density of free electrons. c). Link between the short pulse and the long pulse regimes of ablation. Many experimental and theoretical studies of the ablation threshold and the ablation rate for metals irradiated with laser pulses clearly demonstrate the presence of two different ablation regimes depending on the pulse duration [4,15,94,95,105,106-108]. The ablation threshold for ultra-short pulses is practically independent of the pulse duration. The ablation with pulses longer than 100 ps proceeds in equilibrium conditions because the electron-ion temperature equilibration establishes before the end of the pulse. The heat conduction and hydrodynamic motion dominate the long pulse ablation process, t pulse > te − i , ttherm .

(

)

Therefore long pulse ablation threshold becomes directly dependent on the pulse duration. The heat conduction depth exceeds the field penetration depth ltherm >> ls 2 . The ablation threshold for this case is defined by condition that the absorbed laser energy, A F , is fully converted into the energy of

(

broken bonds in a layer with the thickness of the heat diffusion depth ltherm = Dth t p

)

1/ 2

during the

laser pulse [105,106]:

(

AF ≅ Dth t p

)

1/ 2

ε b na ;

where Dth is the temperature-independent thermal diffusion coefficient (or, thermal diffusivity). The ablation threshold fluence for the long pulses (>100ps) immediately follows from this equation:

(D t ) ≈

0.5

Fth

th p

A

ε b na

.

(5.23)

52 This is a well-known ‘square-root’ time dependence of ablation threshold on the pulse duration [109]. The difference between the ablation mechanisms for the long-pulse thermal regime and the short-pulse non-equilibrium mode is two-fold. Firstly, the laser energy absorption mechanisms are different. The intensity for the long pulse interaction is in the range 108-109 W/cm2 for the pulse duration range from nanoseconds to hundred of picoseconds. The ionisation is negligible, and the dielectrics are almost transparent up to the UV-range. The absorption is weak, and it occurs due to the inter-band transitions, defects and excitations. In the short-pulse regime the intensity is in excess of 1013 W/cm2 and any dielectric is almost fully ionised in the interaction zone. Therefore, the absorption due to the inverse Bremsstrahlung and resonance absorption on free carriers dominates the interaction, and the absorption coefficient amounts to several tens percent. Secondly, the electron-tolattice energy exchange time in a long-pulse ablation mode is shorter than the pulse duration. By these reasons, the electrons and ions are in equilibrium, and ablation has a conventional character of thermal evaporation. An intermediate regime occurs for the conditions when the pulse duration is longer than electronion energy transfer time and heat conduction time, t p > theat , te − i . This regime holds for the laser

(

)

11

pulse durations tp > 100 ps and at the intensities less than 10 W/cm2, where the heat conduction should be taken into account during the pulse. The thickness of a target layer heated during a pulse longer than a picosecond becomes ltherm + ls 2 . The ablation threshold for this case can be obtained with the help of Eq. (5.23), by replacing ls/2 with ltherm + ls 2 :

Fthm ≈

⎞ 3 n ⎛l ε b + ε esc e ⎜ s + ltherm ⎟ . 2 A⎝2 ⎠

(

)

(5.24)

The long-pulse threshold fluence of Eq.(5.23) immediately follows from this equation in the limit ltherm >> ls 2 . Thus, formulas for the ablation thresholds for the ultra-short and for the long laser pulses are naturally linked through a continuous function of the pulse duration. However, the experimentally observed transition from the ablation threshold for the non-equilibrium regime to that for the thermal regime occurs at much longer pulse durations, for example, at up to ~100 ps in gold [4,81]. This indicates that the thermal mechanism for some reason does not contribute to the ablation rate at fluences near the threshold, as might be expected, even when the pulse-width is up to ten times longer than the electron-lattice equilibration time. It was found [72] that the time to establish the high-energy tail of the Maxwell energy distribution of atoms at the surface must be considered along with the time for equilibration of the electron and lattice ‘temperatures’. Specifically in vacuum, the time needed to transfer energy from the highenergy Maxwell tail from atoms in the bulk to the atomic layer at the surface (bulk-to-surface energy transfer time tb-s) becomes the crucial parameter that determines the relative contribution of equilibrium (thermal) evaporation and non-thermal ablation to the material removal rate, especially near the ablation threshold. The bulk-to-surface energy transfer time thus reads [12,72]:

[

tb − s = na v σ b − s

]

−1

⎛ ε ⎞ ≈ tmain exp ⎜ b b ⎟; ⎝ kB T ⎠

(5.25)

where tmain is the time to establish the main part of Maxwell distribution and b is the ratio of close neighbour atoms in the bulk to those on the surface – see [14]. A conservative estimate can be made by taking the maximum surface temperature at the non-thermal ablation threshold Tm ~ εb, tmain = 0.2 ps. According to Eq.(5.25), the bulk-to-surface energy transfer time increases dramatically with decreasing temperature: the bulk-to-surface energy transfer time appears to be much larger the heat

53 conduction time: tb-s ~ 80 ps >> tth ~30 ps [72]. One can see that the bulk-to-surface energy transfer time exceeds markedly the electron-to-lattice energy transfer time and the heat conduction time at fluences that are below the threshold for non-thermal ablation. Thus, non-equilibrium ablation in vacuum completely dominates thermal evaporation for pulse durations shorter than bulk-to-surface energy transfer time. In other words, in vacuum, thermal evaporation at the ablation threshold and below is completely negligible for tpulse < tb-s. This result explains the experimentally observed fact that the ablation threshold independence on the pulse duration holds for up to 100 ps in nonequilibrium ablation [4]. This implies that thermal ablation will only dominate when the pulse duration is comparable to or longer than the bulk-to-surface energy transfer time. 5.5.2. Ablation thresholds in ambient gas The experiments in gases revealed that the ablation thresholds for several metals are less than half of those measured in vacuum [72,110]. The analysis shows that this difference is caused by the differences in establishing the atomic energy distribution in surface layer at the solid-vacuum and the solid-air interface [72]. In the case of ablation in vacuum the energy distribution of atoms at the surface is Maxwell-like but with its high-energy tail truncated at the binding energy. Therefore the energy per atom at the ablation threshold should be higher then for the full Maxwell because the contribution of thermal evaporation is prohibited. In gas media, however, collisions between the gas atoms and the surface markedly reduce the time for establishing full equilibrium distribution allowing thermal evaporation to proceed before the surface cools down. Let us consider the physical mechanism responsible for ‘turning on’ the thermal evaporation in the presence of gas that results in reduced ablation threshold in air. After the beginning of the laser pulse, the air next to the heated surface layer gains energy through collisions with the solid target. This results in the establishment of the Maxwell distribution in air near the air-solid interface. Hence the air plays the same role as the saturated vapour in classical thermal evaporation. The presence of air introduces a new pathway allowing the creation of the highenergy tail of the Maxwell distribution in the surface layer augmenting the bulk-to-surface energy transfer process discussed earlier. Thus, there are now three processes acting at the same time which determine the ablation conditions at the solid-air interface: i)

Evolution of the Maxwell distribution in air near the surface due to air-solid collisions;

ii) Evolution of the Maxwell distribution at the surface due to bulk-to-surface energy transfer, and iii) Cooling of the surface layer by heat conduction. We concluded that mechanism (ii) was too slow to result in thermal evaporation when kBT < εb. Therefore the presence of air significantly increases thermalization rate reducing the time for building up the high-energy tail at the surface. The ablation rate then can be calculated using the equilibrium relations. Let’s consider all these processes in sequence. The air-solid equilibrium energy distribution is established by collisions of air molecules with the solid. The gas-kinetic mean free path in air in standard conditions is lg-k = 6×10-6 cm [92]. The equilibration time teq needed to establish the Maxwell distribution in the gas can be estimated as teq ≈ tg −k ≈ lg −k vth , where vth is the average thermal velocity in air. We estimate this time at room temperature (vth = 3.3×104 cm/s) as teq ~1.8×10-10 s. The bulk-to-surface energy transfer time calculated by the Eq. (5.27) at the maximum temperature (kB Tmax ≈ ε b 2 ) for the threshold fluence

54 conditions in air constitutes tb−s ≈ tmain e12 ~30 ns >> teq for Cu, Al, and Fe after the pulse. Thus, only the air-surface collisions could lead to the formation of the high-energy Maxwell tail, and therefore to thermal evaporation from the surface. The evaporation rate can be calculated in the following way. During the time when the solid-air temperature equilibration is completed the surface temperature has dropped due to thermal conduction

(

to the level Teq ≈ Tm t p teq

)

1/ 2

. Here Tmax ∝ Fabs (absorbed fluence) is the maximum temperature at

the end of the laser pulse at the experimentally determined threshold fluence for ablation in air. These values were presented in Table 5.4. Thermal evaporation starts after the equilibration time t > teq and the temperature at the solid-air surface continue to decrease in accordance to the linear heat conduction law. We suggest that thermal evaporation proceeds at a vapour density corresponding to the temperature at the solid-air interface. The number of atoms ablated per unit area after establishing the Maxwell equilibrium can be estimated as follows: ∞

nvt

therm

=

∫ (nv)

therm

t eq

dt .

(5.26)

A reliable estimate of the evaporation rate can be obtained with the numerical coefficients extracted from the known experimental data in equilibrium at the temperature close to the experimental conditions. We also assume that the vapour-air mixture with a predominance of air plays the role of the saturated vapour over the ablated solid. Thus, we take:

(n v) a

therm

≈

nair k B Teq

(2π M

a k B Teq

)

1/ 2

Then, the equation Eq. (5.26) takes the following form: ∞

nvt

therm

=

⎛ k T nv dt ≈ nair ⎜⎜ B eq therm ⎝ 2π M a

∫( ) t eq

1/ 2

⎞ ⎡ atoms ⎤ ⎟⎟ teq ⎢ ⎥. ⎣ cm2 ⎦ ⎠

(5.27)

The ablation threshold is achieved when the above number of thermally evaporated atoms equals to the number of atoms in the mono-atomic surface layer na d mono : nvt

therm

= na d mono .

(5.28)

The above expression combines air, laser and material parameters at conditions of ablation threshold. The values predicted by Equation Eq.(5.27) are presented in Table 5.3 for comparison with the areal density of a monolayer. It is clear that the predicted number of the thermally ablated atoms is, in fact, close to the number of atoms in a mono-atomic layer. Table 5.1. The predicted numbers of atoms thermally evaporated after the pulse once the Maxwell distribution has been established compared with the number of atoms in a monolayer [72].

nvt

15

therm

-2

, 10 cm ; Eq.(5.29)

na×dmono, 1015 cm-2 (number of atoms in a monolayer)

Al

Cu

Fe

Pb

2.4

5.28

1.67

0.45

1.72

2.16

2.0

1.15

55 Table 5.1 suggests that thermal evaporation well after the end of the laser pulse can be responsible for the removal of a mono-atomic layer for Al, Cu, and Fe at the fluences corresponding to the threshold measured in air [72]. This is in a good agreement with the experiments, as the threshold fluence was introduced as the fluence needed to remove a single atomic layer. Therefore, we can conclude that the presence of air decreases the single pulse ablation threshold by approximately a factor of two relative to the vacuum case due to the contribution of thermal ablation assisted by the presence of the air well after the end of the pulse. The presented theoretical predictions compares favourably with a large number of experimental results [4,12,72,111]. Numerical studies of this effect were recently presented by Bulgakova et al. [112]. 5.6. Ablation rate, mass, and depth by a single pulse The dependence of over-threshold ablation on laser and material parameters can be obtained on the basis of the major factors of the ultra-short pulse energy absorption. The electron temperature grows up linearly in time and its magnitude decreases exponentially with depth into material: Te x, t = (2I o t Cene )exp −2x ls . Thus, the electron temperature at the outermost surface layer

( )

(

)

reaches its maximum at the end of the pulse. At the threshold fluence, Fthr, this outermost atomic layer is removed from the bulk by the electrostatic field of the electrons escaping from a solid after the pulse. The total absorbed energy should be increased in order to remove the next layer. Therefore, ablation at Fa>Fthr starts before the end of the pulse at t = Fthr Fa t p . Note that the

(

)

temperature calculation presented above does not take into account the energy losses on the ablation during the pulse. Optimistic but rather exaggerating estimate can be based on the assumption that all the absorbed energy above the threshold is spent on ablation, i.e. on breaking bonds plus kinetic energy of ablated atoms. Then the ablation depth can be estimated on the basis of energy conservation as follows: F − Fthr max labl ≈ lmono + ; lmono ≈ na−1 3 . (5.29) n ε b + ε esc

(

)

The losses for kinetic energy of ablating atoms and for the heating of the rest of non-evaporating material should be added into parenthesis in the denominator in Eq.(5.29). Therefore the above estimate gives linear dependence on the absorbed fluence setting an upper limit for the ablated depth of material set by the energy conservation for any ablation mode. Conservative minimizing estimate for the ablation depth may be obtained from the spatial temperature dependence under the assumption that ablation wave propagating inside the solid stops at the depth where the electron temperature corresponds to that at the threshold fluence. The maximum electron temperature is proportional to the total absorbed fluence, Fa. Therefore the ablation depth can be estimated as: l ⎛F ⎞ min ≈ s ln ⎜ a ⎟ + lmono . labl (5.30) 2 ⎝ Fthr ⎠ The ablated plume in this estimate is overheated at Fa>Fthr, which is reflected in the weak ablation depth dependence on the absorbed fluence. The ablated mass, which could be measured in experiments, is connected to the ablation length, the laser focal spot area, Sfoc, the atomic number density, and the atomic mass, M, by an obvious relation: mabl = S foc na M labl , (5.31) min max < labl < labl given by Eqs.(5.29,5.30). where labl is in the range labl

56 5.7. Phase state of laser-ablated plume One of the areas of laser ablation applications is the use of the laser-ablated plume for deposition of thin films, nanoclusters and nanocomposite materials, and to create a medium for chemical reactions [72,111-114]. The ability to control the composition of the laser-produced plume in these applications is a significant physical and technological challenge. It is well known from numerous experiments that laser plumes contain macroscopic particles and liquid droplets. The presence of mixture of phases in the plume is a clear indication that a broad variety of processes take place during the ablation and the plume expansion. There are two major reasons for complex phases coexistence in the plume. The first relates to the spatial intensity variations across the laser focal spot. The processes that appear progressively with increase of laser intensity are structural phase transitions, destruction of the target integrity (appearance of cracks, flaking of the surface), melting, ionisation and ablation. Another source of non-homogeneity in the plume, particularly the formation of liquid droplets, is condensation of vapour during the expansion. As a result, an expanding plume consists of a mixture of gas, vapour and liquid phase. Therefore, in order to ensure a homogeneity of a plume one needs two conditions to be fulfilled: i) the local fluence at every point across the focal area should exceed the ablation threshold, and ii) the conditions which maintain the expanding plume in the gas phase should be satisfied. Below we formulate both conditions and demonstrate their implementation in an experiment. 5.7.1. Local energy thresholds for phase transitions In order to define the local thresholds one needs to take into account the spatial distribution of the absorbed energy in two dimensions: across the focal spot and inside the target surface:

Ee (r, x, t) =

⎧ 2x ⎫ 2F(r, t) exp⎨− ⎬ ; nels ⎩ ls ⎭

where F(r,t) is the spatial distribution of the absorbed fluence: F(r, t) = φ (r)

(5.32) t

∫ A( t′)I ( t′)dt′.

We

0

assume that the focal spot is a circle. The r-coordinate corresponds to the distance from the centre of a circular focal spot on the target surface, while the x-coordinate is normal to the surface; φ(r) is a dimensionless function (for example, Gaussian-like) describing the axially symmetric spatial distribution. The absorbed energy in a mono-atomic surface layer with thickness da = na−1/ 3 << ls at the end of the laser pulse becomes: Ee (r, x, t) ≈

2F(r, t p ) ne ls

.

(5.33)

A local threshold by definition depends on position within the focal spot. The condition that the absorbed energy defined by Eq.(5.33) equals the energy, ε transf , required for the particular phase transformation to occur defines the local threshold for any laser-induced transformation of the material:

ε transf =

( ).

2F r, t p ne ls

(5.34)

557 It is obvious that the threshold energy density (fluence), Fthr ∝ ε transf sccales with thee characteristtic quired for a particular p typee of phase traansition. Forr example, thee ratio of ablaation thresholld energy req to the mellting thresholld equals the ratio r of the heeat of vaporissation H vap too the heat of melting H melt . This ratio varies for most materials within a rangge ~ 5 – 30. For examplee, for a siliconn target used in i the experiiments below H vap = 10.6 kJ/g; H melt = 1.66 kJ/g and H vap H meltt = 6.4. One can c see that in i order to coontrol the phaase-state of thhe plume, eneergy in excesss of the phasse transition threshold shoould be deposited. This condition empphasizes that tthe threshold energy shoulld be signifiicantly higherr than that reequired to just break the inter-atomic bonds. Suffficient kinettic energy sh hould be addittionally deliv vered to an unnbound atom in order to reemove it from m the solid annd maintain it i in the desireed vapour staate for a certaiin time. Thee production oof a droplet-frree laser plum me imposes th he additional condition thaat the gas statte of the vapour should be effectively co ollisionless. 5.7.2. Criiterion for tottal atomizatioon of ablated pplume The energy e threshoold for total atomisation a oof the ablated plume can bbe calculated on the basis of o thermodynnamic argum ments similar to those usedd to derive thhe criterion foor complete vaporization v o of material by b strong shoock waves [922]. The depposited laser energy e is useed to break thhe inter-atom mic bonds and d also providdes the kinetic energy to the t expandingg plume. Thhe magnitudee of the kinettic energy shhould be sufficcient to keep the expandinng plume in a gaseous statee containing non-interactin n ng atoms. This T condition n determines the absorbedd energy threeshold Fatom ffor total atom misation in thhe plume. The equation e of staate of the ablated materiall in conditionss close to thaat for the solidd-vapour phasse transition can be preseented as a suum of the elaastic pressure (related excllusively to thhe inter-atom mic n) and the theermal pressurre in the form m of Eq.(5.2).. The importtant practical issue of target interaction ablation and a further exxpansion of vapour relatees to the defiinition of thee phase state of the ablateed matter in the different areas of phaase space in ppressure - den nsity, P, n, orr temperature, density, T, n, he phase states of the exxpanding abllated materiaal lie along curves c for thhe parameterr planes. Th adiabatic expansion (enntropy is consserved duringg expansion), as shown in F Fig.5.1.

Figure 5.11. P–V diagram m of vapour sttates in the pluume at variouss levels of absoorbed laser eneergy and thus at different innitial normalized pressures. The total atom mization (gas phase) p is achievved at the condditions when thhe

58 P–V curve goes above the critical point and therefore above the curve of phase equilibrium (following from the Clapeyron–Clausius equation, black curve) separating the states with the mixture of phases and the gas phase.

If the initial temperature of the expanding plume is lower than a specific value Tcrit defined below, the expansion curves cross the phase equilibrium curve that separates the states of a single phase from the states with a mixture of phases. The maximum of the phase equilibrium curve in T , n (or T,V = n −1 ) plane is the critical point, Tcrit . The phase equilibrium curve in PT-plane ends at a critical point, at Pcrit , Tcrit . The difference between the solid, the melt and the vapour ceases to exist in a −1 critical state, i.e. the density of all phases has the same value, Vcrit = ncrit . The states above the first

adiabatic curve touching the critical point represent the atomic state of a homogeneous phase at V > Vcrit n < ncrit . Although the critical values are poorly known for many materials, it has been

(

)

established, however, that Tcrit constitutes a small fraction of the binding energy, usually Tcrit ≅ (0.1–0.2)εb [92]. The adiabatic curve, which separates the single-phase area from the mixture of phases, and the phase equilibrium curve have only one common point, e.g. the critical point. One can find the initial temperature for a material at the initial density that begins to expand along this adiabatic curve by applying the above condition with the help of equation of state Eq.(5.2):

( ) ( )

Pcrit = cs ncrit Tcrit Γ ncrit + Pc ncrit

(5.35)

Note that the subscript ‘c’ marks the cold pressure. The critical and initial parameters are linked through the equation of adiabatic expansion: Tcrit ⎛ ncrit =⎜ T0 ⎝ n0

⎞ ( crit ) . ⎟ ⎠ Γ n

(5.36)

The density-dependent Gruneisen coefficient Γ plays the same role as the adiabatic constant, Γ ∝ γ −1 . At the critical point, vapour is described as an ideal gas, Γ ncrit ≈ 2 3 . Now, the initial temperature

( )

T0 of the laser-excited solid that should pass through the critical point during adiabatic expansion is easily expressed through the critical parameters from Eqs. (5.35,5.36):

( )

2/3 3 Pcrit + Pc ncrit ⎛ n0 ⎞ T0 = ⎜ ⎟ 2 csncrit ⎝ ncrit ⎠

(5.37)

The critical parameters can be related to the binding energy and the initial density [92,53]. For example, using the experimental data for aluminium [115] and interpolation for the ‘cold’ pressure one obtains: Pcrit = 1.84 kBar 5.55×10-3εbn0, n0 ncrit ≈ 26 , and Pc ncrit ≈ 0.015ε b n0 . Thus, the

( )

adiabatic expansion curve for aluminium touching the phase equilibrium curve at the critical point must start at k B T0 ≈ 4.9 ε b , where T0 is the temperature at the end of the laser pulse [116]. The practical conclusion from this result is that the aluminium skin layer heated homogeneously to k B T0 ≥ 5ε b at its initial density then expands adiabatically in a homogeneous state of the atomic gas. Accordingly, all states of expansion starting at lower temperature shall cross the phase equilibrium curve inevitably entering into states containing a mixture of phases. A similar qualitative conclusion for atomisation of the ablated plume can be made for all solids because the binding energy for most solids lies in the range of 2-5 eV. Thus, the energy per atom approximately 3-5 times larger than the binding energy, Eabs > (3–5)εb, should be deposited into a solid in order to transform the target into a fully atomised gas. We should note here that the exact value of the above numerical coefficient

59 depends strongly on the equation of state in a range n0 > n > ncrit for a particular material. Therefore, the laser energy density necessary for transformation of the ablated substance into the atomised gas phase should comply with the following condition: Fatomise ≥ 4Fthr . The qualitative dependence of normalised pressure, P ε b n0 , on normalised volume n0 n during the adiabatic expansion starting at

( )

the same initial density n0 and with the different initial normalised pressure is plotted in Fig.5.1. The black curve corresponds to the states of phase equilibrium (Clapeiron-Clausius curve). This curve separates the states representing the mixture of phases (at P < Pcrit ) from the states of a homogeneous phase or atomised vapours (at P > Pcrit ). We should stress that this function is poorly known for most of materials. One clearly sees from Fig.5.1 that the expansion from the initial state with energy Eabs > 3− 5 ε b keeps the expanding vapour in a gas state whilst at a lower initial energy the vapour

( )

condensates into liquid droplets and thus a mixture of states is unavoidable. We should keep in mind that the thresholds above were introduced locally, i.e. different regions of the beam can create different phase states depending on the local beam intensity. High contrast ratio is required between the energy of the short pulse and the pre-pulse in order to achieve the short pulse interaction mode. A detailed discussion of the methods for providing a high contrast ratio can be found in [88,95,117]. Efficient methods to suppress the pre-pulse include gain narrowing, the use of a saturable absorber, and frequency conversion into harmonics. On a much shorter, fs-range time scale, temporal shaping of the ultra-short laser radiation tightly focused by a high-NA microscope objective inside transparent material provides an opportunity to control the ionisation rate and to reduce the absorbing energy volume below the diffraction limit [118]. It was shown that it was possible to control filaments propagation in non-linear aqueous solution [119], to optimize the efficiency of white continuum generation in air [120], and to compensate spherical aberrations by changing temporal shape of ultra-short laser pulses [121,122]. The adaptive control is achieved with a help of spatial phase modulator placed into the fs-laser dispersive system before the amplification stage. Temporal shaping is modified by phase-only filtering in liquid crystal array without loss of total energy. The control over the ionisation rate and deposited energy density offer potential benefits for laser-based 3D optical memory procession and 3D microstructuring. Temporal shaping of femtosecond pulses provides the ability to control the plume composition. It was demonstrated that temporal shaping offers the ability to control the efficiency of nanoparticle generation through variation of the spatial temperature and density profiles in the laser ablated plume [123]. The optimal tailoring of the laser pulses allows one to enhance the atomization of the supercritical fluid in the plume inducing a more efficient transition to the gas phase. Turning to a spatial distribution of the laser intensity over the target surface, it is very difficult to obtain complete atomisation using a Gaussian beam and it is obvious that it would be preferable to use a ‘top-hat’ intensity distribution where the absorbed fluence everywhere exceeds about four times the ablation threshold. A simple way to move towards the top-hat profile is to truncate the low energy wings in the spatial distribution with an aperture and employ a relay-imaging focusing scheme to image the top-hat beam onto the target. This idea has been implemented in fs-laser ablation of silicon and resulted in deposition of high surface quality silicon films [116]. Total atomisation in the plume was proved in a number of works on fs-laser ablation and deposition of high surface quality diamondlike films, where the laser intensity was several times above the ablation threshold – see, for example, [124].

60 5.7.3. Nanocluster formation in the plume A mixture of phases in a plume ablated at the intensity level below the total atomisation level leads to efficient formation of nanoclusters and nanoparticles through inelastic atomic collisions. Atoms re-ensemble into clusters in an expanding plume in a process that controlled by the density and temperature in the plume directly steered by the laser parameters. Laser ablation has proven to be an efficient method for producing nanoclusters of different atomic content, shape and internal structure. Initially long (10-30 ns) low repetition rate (10-30 Hz) lasers with the average intensities (2–4)×109 W/cm2 have been employed. Metal and metal oxide clusters [125,126], fullerenes [127], carbon and boron nitride nanotubes [128-131], silicon clusters [132,133] and many other structures have been produced with laser ablation in similar conditions. Special conditions are often required complementing laser ablation to promote a particular type of structure to grow. For example, in case of carbon nanotube growth in a low repetition rate regime, the laser plume cools down after the pulse below the minimal temperature required for the nanotube formation due to the long gap between the successive pulses. Therefore, additional heating of the ambient gas was necessary in order to maintain the proper conditions for cluster formation. Cluster formation by long pulse low repetition rate lasers conventionally occurs during the laser plume interaction with an ambient gas in a chamber placed into a furnace with controlled temperature and under a continuous flow of a noble gas. The advent of powerful femtosecond lasers with the repetition rate up to 100 MHz along with a better understanding of physics of laser-matter interaction and cluster formation prompted a new approach to formation of nanoclusters by short single pulses. It has been shown that control over the cluster size is possible by choosing the optimum combination of laser and target parameters, thus eliminating additional heating and even removing the necessity for ambient gas in the chamber [134,135]. The size of a cluster has a significant effect upon various material properties and, therefore, provides a relatively simple experimental avenue to control those properties [131]. There is a critical cluster size of several nanometers that separates atomic ensembles composed from the same atoms but possessing different material properties. Atoms re-ensemble into clusters in an expanding plume in a process that controlled by the density and temperature in the plume directly steered by the laser parameters. Thus it is possible to control the properties of nanoclusters, which are the building blocks for any nanomaterial, via control over their size [134]. Temperature and density of atoms in the adiabatically expanding plume remain appropriate for formation of clusters through the atom-to-atom and atom-to-cluster attachments over a certain time period after the laser pulse. The longer the period of appropriate atomic density and temperature in the plume, the more successive inelastic collisions occur, and hence the larger the clusters are formed. Thus the plume expansion time in vacuum is a major factor in determining the cluster size. This time depends on the combination of laser intensity and material parameters. Similarly, in the case of expansion in ambient gas the diffusion time of the ablated material through the gas determines the cluster formation time and hence the cluster size. Clusters are formed in the dense area near the laser focal spot close to the ablation surface. By applying a simple kinetic model it was shown that formation of clusters in a plume created by a single ultra-short laser pulse is possible [134]. Formation of carbon nanoclusters in a single-laser-pulse created ablation plume was studied both in vacuum and in a noble gas environment at various pressures. The developed theory provides cluster radius dependence on combination of laser parameters, properties of ablated material, and type

661 and pressu ure of an ambbient gas in agreement a witth experimen nts. The expeeriments weree performed on o carbon naanoclusters foormed by laseer ablation of graphite tarrgets with 122 ps 532 nm laser pulses at MHz-rangge repetition rate r in a broaad range of am mbient He, Ar, A Kr, and Xee gas pressurees from 2×100−2 to 1500 Torr T (Fig. 5.2aa,b) [134].

(aa)

(b)

Figure 5.2 2. Dependencee of average cluuster size on A Ar gas pressuree (a) and on thee type of gas (b b); the error baars represent thhe accuracy off the cluster sizze measuremennts ±0.5 nm. (aa) – The laser fluence f is 8 J/ccm2; the curve is

rcl ∝ n1/Ar5 ; (b) – the bufffer gas pressuree 50 Torr; the ddashed curve iss a scaling baseed on rcl ∝ rat1/ 2 (from [134]). Theree are a numbber of imporrtant conclusiions from th hese studies. First, nanoccluster growtth process taakes place in a relatively short time beelow ~10 ns in a laser-abblated plume produced by a single laseer pulse. Seccond, the bufffer gas pressuure has two distinguished rregions affectting the clusteer size. At thhe pressure raange below ~50 ~ Torr the nnumber densitty of atoms inn the plume iss higher than in i the backgground gas an nd the effect of confinemeent of the lasser plume is negligible n (Fig. 5.2a). Thhe cluster grrowth processs in the pllume, which was termed d as expansiion-limited aggregation a o of nanoclusters [134], is determined by b the collisioons between the t ablated attoms. This conclusion c waas m conditions, and by a verry supportedd by experimeental observaation of nanocluster growtth in vacuum weak, if any, a cluster siize dependencce on buffer ggas pressure at a the pressurre range below w 50 Torr. As A soon as it has formed d in the expaanding plumee, there is noo further sizee increase wh hile the clusteer t the buffer b gas. Third, in thee pressure ran nge above ~50 ~ Torr the kinetic theorry diffuses through provides a semi-quan ntitative agreeement betw ween the predicted and tthe measuredd cluster sizze p of groowth is usuallyy referred to aas dependence on gas pressure. In thiss high pressurre range the process noclusters [1336]. diffusion-limited aggreegation of nan c formaation occurs by b a monom mer addition, i.e. by atom--to-atom or atom-to-cluste a er The cluster collisions, in a space cllose to the abblation surfacee in a few nannoseconds tim me after the laaser pulse. Thhe p expanssion time in vacuum, or by atomic diiffusion in thhe cluster sizze is controlled by the plume ambient gas g in the preessure range above the abblated plume density. This time in turrn is explicitlly connectedd to the laser,, target materrial, and fillinng gas parameeters. Experiimental studiees and analyssis of the ressults show thaat a single ulltra-short pulsse of ~100 fss duration cann produce a few f nanometeer small clussters with a very v weak deppendence on the t laser fluen nce and the aambient gas tyype or pressurre [124,137]. The developed d theeoretical anallysis shows that t the ultra--short laser ppulses could be applied foor producingg nanoclusters of differentt materials w with cluster sizze below thee critical valuue of 3 nm – 5 nm, depennding on the material, whhich separatess the nano-prroperties from m those of thee material in a

62 bulk [134]. Nanoclusters with predictable size in a range of 3-10 nm by single femtosecond pulses with the fluence of 5-10 J/cm2. The cluster size can be controlled mainly by the change in the ambient gas pressure in combination with the laser fluence. 5.8. Accumulation effects in ablation with high-repetition-rate short-pulse lasers The use of ultra-short pulses with the top-hat intensity distribution offers good control over the phase state of ablated plume. Each low-energy high-intensity pulse evaporates relatively a few (~1011 – 1012) atoms per pulse [79,83,84]. However, high average ablation rate per second is desirable for industrial applications of laser ablation. To compensate for the reduced ablated mass per pulse, high pulse repetition rates are then used to achieve a high average ablation rate. The high repetition-rate up to 100 MHz maintains the average atomic flow in a plume at a high level of 1019 – 1020 atoms/sec appropriate for laser deposition and micromachining applications. On the other hand, the interaction of pulse trains from MHz repetition rate lasers with matter appears to be significantly different from single pulse interaction due to accumulation of the effects of successive pulses. The number of pulses on the target surface at high repetition rate may reach thousands per spot because the scanning speed of the laser over the target surface is normally too slow to physically separate successive pulses as the time between the successive pulses for MHz lasers is only of the order of hundred of nanoseconds. The coupling between the successive pulses with single pulse energy below ablation threshold may have a positive effect by increasing the overall ablation rate. While single pulse energy may be insufficient to produce ablation, the accumulation of heat between the pulses gradually transforms the laser-matter interaction into a regime where much stronger laser-target coupling occurs overcoming the ablation threshold. The mechanisms for coupling the successive pulses in ablation of metals and transparent dielectrics with low heat conduction such as silica and glasses are different. The build-up of temperature and density leads to a change in the laser-matter interaction mode from absorption in the skin-layer to absorption on a plasma density gradient [90]. The cumulative effect in high-repetition-rate-laser interaction with matter has been observed during ablation of carbon by a laser with 76 MHz repetition rate [83,84]. The effects of highrepetition-rate ablation that include smoothing of the spatial energy distribution and cumulative heating, have been observed in ablation and deposition of chalcogenide glasses [138]. Cumulative heating in the bulk of a transparent glass was also reported in [139]. The cumulative effect of increasing the electron density above the plasma critical density in a silica substrate heated by a train of 20-30 femtosecond pulses with a time gap of 500 fs was demonstrated my molecular dynamic simulations in [140]. The coupling between consecutive pulses occurs in three stages. First, the temperature accumulation takes place when absorbed energy of many pulses simply sums up, while the heat losses are negligible, until the thermal evaporation threshold is achieved. Second, the density of vapours begins builds up after ablation near the ablating surface due to short time gap between the consecutive pulses, this is a direct consequence of high repetition rate of the laser pulses. Finally, the vapour became ionised due to increase of temperature and density, and laser-matter interaction changes to the interaction with plasma. The energy absorption occurs near the critical density defined by the incident laser wavelength.

63 The cumulative heating increases the evaporation rate to some saturation level at which the energy deposition is balanced by the dissipation through heat conduction from the absorption volume. Further increase in the number of pulses heating the target at the same spot on the surface tends to produce deep crater, leading to a decrease in the ablation rate due to absorption of laser energy in the dense ablated vapours [141]. Cumulative ablation by high repetition rate lasers allows efficient ablation of any transparent solids in the conditions when single pulse energy is insufficient for evaporation. The high quality thin films of different materials, including low absorbing dielectrics, can be deposited by using very high repetition rate lasers (1-100 MHz) by an appropriate self consistent choice of laser wavelength, repetition rate, and energy per pulse. The appropriate combination of laser and material parameters determines the optimum conditions for ablation and deposition. Control over the phase state of a plume, the elimination of droplets, and high ablation rate can all be simultaneously achieved as it has already been demonstrated using silicon and chalcogenide glasses [90,116]. 6. ULTRAFAST LASER INTERACTION IN CONFINED CONDITIONS 6.1. Introduction Intense ultrafast laser pulses tightly focused in the bulk well below the surface of transparent material interact with matter in the condition where the conservation of mass is fulfilled. All the matter affected by the electromagnetic field of laser radiation remains in the interaction zone and preserved inside the dense solid. It was shown two decades ago that the plasma produced by laser pulses in confined geometry generates strong shock waves with the pressure at the shock wave front exceeding several times the pressure produced by the same pulses focused on the surface of the material [142]. The laser induced structural changes depend on the laser intensity in the interaction zone. They could be separated into two distinct categories as non-destructive and reversible, which occur below the optical breakdown threshold, and irreversible structural changes at the above breakdown intensity. The examples of non-destructive phase transitions are the photo-refractive effect in LiNbO3 [143-145], photodarkening in chalcogenide glasses [146,147], and photopolymerisation of hybrid materials [148,149]. The structural changes produced by a single pulse in a small micron-size volume can be used as a memory bit for high-density 3D optical storage [150-152], for formation of optical waveguides [153] and in other photonics and material-processing applications [154]. Irreversible structural changes, often considered as damage, occur at an intensity level above the optical breakdown threshold. It was first indicated by Glezer and Mazur in 1997 [150], and later experimentally demonstrated by [152-154] that unique conditions of extremely high pressure and temperature with record high heating and cooling rates are created in the energy deposition region. A strong shock wave generated in the interaction region expands into the surrounding cold material and compresses it, which may result in the formation of new states of matter. Shock wave propagation is accompanied by a decompressing rarefaction wave behind the shock front, leading to the formation of a void inside the material. The pressure at the shock wave front reaches the magnitude of several TPa, triggering dramatic changes in the compressed material in the form of a densified shell surrounding the void. The essential distinctive feature of laser-driven microexplosion is that the modified material remains compressed and confined in a strongly localized region inside a bulk, and can be investigated later by Raman spectroscopy, electron beam, x-ray, and other structural diagnostics techniques.

64 The extreme conditions produced in the ultrafast laser driven micro-explosions can serve as a novel microscopic laboratory for high pressure and temperature studies and materials processing, well beyond the pressure levels achieved in diamond anvil cell. First ultrafast laser-induced microexplosion experiments were performed with sapphire, silica, polystyrene, and germanium oxide [155-158]. 6.2. Limitations for laser power concentration There are a number of restrains for delivering high energy density to the focal area of transparent material by intense ultrashort laser pulses [14]. In order to create the high energy density in a bulk of transparent material one should minimize the energy losses along the transport path and to reduce the focal volume to the smallest possible. The first obstacle for a powerful laser beam propagating long distance in a transparent media is self-focusing caused by the intrinsic non-linearity of the medium, which may result in optical breakdown above certain threshold power of the beam. The critical value for the laser beam power, Wcr , reads [159]: Wcr =

λ2 2 π n0 n2

;

(6.1)

where n2 is the nonlinear part of the intensity dependent refractive index, n = n0 + n2 I , and λ = λ0/n is the wavelength of light in the medium with the refractive index n, λ0 is the wavelength in vacuum. Self-focusing of the laser beam occurs at a distance Ls-f [159]: Ls− f

⎞−1/ 2 2 π n0 r02 ⎛ W0 = −1⎟ ⎜ λ ⎝ Wcr ⎠

(6.2)

Here r0 is the minimum waist radius of the Gaussian beam. For example, in fused silica (n0 = 1.45; n2 = 3.54×10-16 cm2/W) for λ = 800 nm, the critical power comprises 2 MW. The self-focusing distance for a laser pulse with W0 = 2Wcr and for a minimum waist r0 ~λ equals to ~9λ. The second limitation comes from the diffraction limit. For a Gaussian beam the focus take the radius of a circle of the first Airy minimum, rmin, as a minimum beam waist radius at the focus that expresses as follows: f 0.61 λ0 rmin = 0.61 λ = ; (6.3) a NA

[

( )]≅ na

where NA = n sin arctan a f

f is numerical aperture and a is the beam radius at the aperture

of the focusing optics. The focal volume at the FWHM-intensity level thus comes to [155]: VFWHM = 0.95

n λ30

(NA)

4

.

(6.4)

For example, focusing a low-intensity ideal Gaussian-shape 800-nm laser pulse with NA = 1.35 in fused silica should concentrate the energy to a volume of ~0.2 µm3. However, interaction with the media at high intensity significantly changes the absorption volume, which we will consider further down in this Section. Further down, there is a limitation imposed by spherical aberrations, which is more pronounced with the increase of numerical aperture. As a result, the focal area in the transparent media becomes elongated in the direction of the beam propagation, and thus the energy density is reduced. The

65 function Ψ(θ,n,Lf) describing spherical aberrations depends on the converging angle θ of a ray, refractive indices n, which in turn depends on the laser wavelength, and on the geometrical depth Lf of focusing into the refractive media [160,161]:

(

)

Ψ(θ , n, L f ) = −k L f n1 cos θ1 − n2 cos θ2 ;

(6.5)

here k = 2π/λ is the wave vector in vacuum, and θ1 and θ2 are related through Snell’s law. The point spread function If, the intensity distribution in the focal area, can be presented in spherical coordinates (r, z), in the frames of scalar Debye theory as [161]:

(

)

I f r, z,θ , L f = ⎧⎪ α ⎫⎪ = ⎨ P θ1 sin θ1 ts + t p cos θ2 J0 k1, r1 ,sin θ1 exp −i Ψ θ , n, L f + k2 z2 n2 cos θ2 dθ1⎬ ⎪⎩ 0 ⎪⎭

∫ ()

[

] (

(

where ts = 2 sin θ2 cos θ1 sin θ1 + θ2

) and

)

[((

)]

)

[ (

) (

t p = 2sinθ2 cos θ1 sin θ1 + θ2 cos θ1 − θ2

()

2

)] are

(6.6)

the Fresnel

coefficients for s- and p-polarisations, correspondingly; P θ1 = cos θ1 is the apodization function obeying the sine condition (the commercial objective lenses are designed to satisfy the sine condition); J0 k1 , r1 , n1 ,sin θ1 is the zero-order Bessel function of the first kind; and θ1 is varied fro

(

( )

)

zero to α = arctan a f . And finally, there is a limitation relates to non-linear ionisation when the pulse intensity is reaching and exceeding the optical breakdown threshold. The non-linear ionisation is directly related to plasma formation with electron density equal to and above the critical density determined by the laser frequency ncr = ω 2 me 4 πe2 , for 800 nm laser pulse ncr = 1.7×1021 cm-3. Ionisation non-linearity of the solid material influences the beam structure and its spectrum at lower intensity before the breakdown. After the breakdown light is absorbed in the skin layer of plasma and energy absorption volume shrinks well below the focal volume, thus increasing the energy density. For example, a 100-fs laser pulse with ~100 nJ energy tightly focused to a micron-size focus below the surface produces the energy density of several MJ/cm3 in a sub-micron volume. The motion of the ionisation front in the direction opposite to the pulse propagation during the pulse increases the absorption volume and reduces the absorbed energy density. The ionisation threshold in fact is function of absorbed fluence because the time is necessary for generation of critical density of electrons. Therefore, the statement that there is a clamping intensity of 1013 W/cm2 [162,163] does not correspond to experimental results [81,108,172-174]. Proper account for intensity-dependent ionisation time supported by experiments on laser-induced confined microexplosion demonstrate that laser intensity up to 5×1015 W/cm2 can efficiently be absorbed in the focal area in the bulk of transparent material [164]. 6.3. Non-destructive interaction: Formation of diffractive structures in photo-refractive materials In accordance with the increase of intensity in the focal region, material modification takes a number of distinctive steps. First, changes in optical properties occur due to moderate modification of electronic and structural properties; this followed by phase transitions due to re-arranged bonding and crystal structure. With further increase in intensity, material decomposition begins, accompanied by breaking of inter-atomic bonds, and ionisation. There are numerous effects induced by fs lasers in different materials at the intensity below the damage threshold: the photorefractive effect, the formation of colour centres, photodarkening in silica and chalcogenide glasses, etc. The analysis

66 below is restricted to the phenomena in fs-laser excited photo-refractive crystals. This particular material is chosen because the fs-laser interaction with a photorefractive crystal is attractive for applications in 3-D optical memories with additional ability to design of read-write-erase devices. The major difference between the fs-laser interaction with photo-refractive crystals and that with the long pulse lasers relates to the fact that the laser field of high intensity is applied during the period shorter of all major relaxation times. Particular important is the fact that the field of spontaneous polarisation has no effect during the pulse. Indeed, as it follows from the experiments [165], the lowest intensity at which changes of the refraction index produced by a single 800 nm, 150 fs laser pulse could still be detected is of the order of 4×1011 W/cm2 (6 nJ per pulse). This intensity is just 2-3 times lower than the ionisation threshold. The main processes contributing to formation of a diffractive structure in the photo-refractive materials under light illumination at low intensity are presented in [166]. The photo-excitation of free carriers into the conduction band occurs during the interaction time. The excited free carriers are subject for the following processes: recombination, drift in a local field of charge separation, in photovoltaic field and in diffusion field related to the carriers’ density and temperature gradients, while ions remain fixed. The carriers recombine in a different location from where they were created because the recombination time is longer than the pulse. Thus, the gradients in space distribution of charge carriers and correspondent electric field are created. This field then induces a refractive index modulation via the electro-optical effect. It is also important to define the time for the transition to quasi-steady charge distribution and the total life-time of the charge distribution that defines the reliability of this process for 3D optical memory applications. 6.3.1. Electron excitation by the low intensity laser field The excitation rate of electrons from valence to conduction band by the laser beam at low intensity, I [W/cm2], in accordance with [166] reads:

dne αI =Φ ; dt hω

(6.7)

it is proportional to the number density of photons with the energy hω , arriving to the unit volume per unit time, α I hω ; here Φ is the quantum efficiency of a single photon, Φ = 5×10-4, and α is inverse of the absorption length (for Lithium Niobate α ~0.0033 cm-1). It is shown in [145] that Eq.(6.7) underestimates the number of excited electrons in lithium niobate affected by ultra-short laser (150 fs, 800 nm; hω = 1.55eV; intensity of 1012 W/cm2) by nine orders of magnitude. As we show later the excitation at high intensity should be treated in a quite different way. Another issue relates to the relative role of laser field and inherent photovoltaic field on the modification of the refractive index by the laser action. Buse presents the excitation of electrons as a stationary low intensity process when excitation of electrons from Fe2+ centres (number density nFe 2+ ) is balanced by recombination on Fe3+ ( nFe 3+ ) centres [143]:

dne = qSInFe2+ − γnFe3+ ne . dt

(6.8)

The source terms in Eqs.(6.8) coincides with that of Eq.(6.7) assuming that absorption occurs on Fe2+ centres, α = l −1 = nFe 2+ S (S is the absorption cross section), and q = Φ hω . In a stationary case one obtains the number density as the following:

67

ne =

qα I . γnFe 3+

(6.9)

The photovoltaic current has a conventional form: j phv = ene ve .

(6.10)

The electron acceleration by all the electric fields and stationary velocity follows from the Newton equation: me

dve = eElas + eEint − ν ef me ve ; dt

(6.11)

here ν ef is the effective collision frequency responsible for resistance, me is the electron mass, and Eint represents all fields (including photovoltaic Ephv) in a medium except for the incident laser field in the first term. In conventional photo-refractive effect the electron motion is stationary and laser field is neglected. Thus the velocity of electron accelerated by the photovoltaic field reads:

ve =

eE phv

ν ef me

.

(6.12)

On the other hand the photovoltaic current, jph, excited in non-centrosymmetric crystal by the laser excitation with intensity I, depends on the material properties through the form [164]: j ph = α G I ;

(6.13)

where G is the glass constant, which is a characteristic of particular material, and α (cm-1) is the inverse of the absorption length as above. Now inserting Eq.(6.9) and Eq.(6.12) into Eq.(6.10) and comparing with Eq.(6.13) one obtains the expression for the photovoltaic field in the form: E phv =

G γ nFe 3+ ν ef me q

e2

.

(6.14)

One can see that the photovoltaic field does not depend on the laser intensity, and it actually represents the field of spontaneous polarisation, the inherent property of ferroelectric photo-refractive crystals. 6.3.2. Electron excitation by the high-intensity ultra-short pulse Three major mechanisms contribute to the light absorption in solids [14]: (i) – the inter-band transitions, for example, single and multi-photon absorption, (ii) – the intra-band transitions, which is absorption on electrons in the conduction band, and (iii) – absorption on the donor (acceptor) level that locates inside the band gap. It was found that the diffractive structure formation by 150 fs pulses can be observed at intensity ~ 4×1011 W/cm2 at 800 nm (6 nJ per pulse). This intensity is just 2-3 times lower than the ionisation threshold for the dielectrics [152]. It is instructive to note that the critical electron density, which signifies breakdown threshold, for 800 nm (ω = 2.35×1015 s-1) is nc = meω 4 πe2 = 1.735×1021 cm-3. The electric field amplitude at the

(

intensity of 1012 W/cm2 equals to E = 8 πI c

)

1/ 2

= 9.15×104 CGS = 27.46 MV/cm, which is 2 orders

of magnitude larger than the photovoltaic field of ~105 V/cm in Lithium Niobate [167]. The qualitative indication of the electric field strength is the oscillation energy of free electron and oscillation amplitude in this field. The oscillation energy for linear polarised light reads [88,95]:

68

ε osc = 9.3×10 −14 I λ2 ;

(6.15)

Here I is intensity in W/cm2 and εosc is in eV, and λ is in µm. At 1012 W/cm2, 800 nm, this energy equals to 0.06 eV, while the oscillation amplitude of free electron is around of an Angstrom. One may expect that the oscillation amplitude of bonded electron is comparable to the displacements of Li of 0.9 Å and Nb of 0.5 Å in LiNbO3 responsible for the spontaneous polarisation. Therefore it might be expected that during the laser pulse of such intensity the oscillating electrons affect the intrinsic crystal field and ferroelectric properties. In these conditions, consideration must be given to the major mechanisms responsible for excitation of electrons to the conduction band from all constituent atoms in the crystal. It is known that ionisation by the electron impact (avalanche ionisation) and the ionisation produced by simultaneous absorption of multiple photons are the two most important mechanisms for electron excitation in close to the ionisation threshold conditions. a). Avalanche ionisation. A few seed electrons in the conduction band oscillate in the electromagnetic field of the laser in dielectrics Eq. (6.15). Electron can gain net energy by multiple electron-phonon (lattice) collisions and eventually be accelerated to the energy in excess of the bandgap ε >Δgap. Energetic electrons create an avalanche of ionisation events. The probability of such event per unit time (ionisation rate) estimates as follows [93]: wimp ≈

ε osc

ω 2 ν e − ph

(

Δ gap ν e2− ph + ω 2

)

;

(6.16)

e, m*, νe-ph, and ω are respectively the electron charge, effective mass, electron-phonon momentum exchange rate and the laser frequency. The electron-phonon momentum exchange rate approximately 2 expresses as ν emom − ph ≈ 2 π I 0 ε F ε b TL h ∝TL h at the crystal temperature being larger than the Debye

(

)( )

temperature TL > TD [12,25,50]. This rate at the room temperature of 293 K constitutes 3.83×1013 s-1. For Lithium Niobate (Δgap = 3.8 eV) under the action of 800 nm laser (ω = 2.35×1015 s-1 is mach higher than the electron-phonon collision rate) the avalanche excitation rate as function of laser intensity then expresses as:

wimp ≈ 4.46 ×1013

I 14

10

s −1 ,

here the intensity I is in W/cm2 units. Note that ionisation rate of metal dopants is approximately twice higher because the doping introduces an additional energy level in the band gap. b). Multi-photon ionisation. It is reasonable to take the multi-photon ionisation rate (probability of ionisation per atom per second) in the form [93]:

wmpi ≈ ω

⎛ ε ⎞n ph osc ⎟⎟ ; ⎝ 2Δ gap ⎠

n3ph/ 2 ⎜⎜

(6.17)

where n ph = Δ gap hω is the number of photons, which an electron should absorb in order to be transferred from valence to the conduction band. Again taking as an example Lithium Niobate under the action of 800 nm laser one gets the intensity-dependent multi-photon ionisation rate in the form: 2.58

⎛ I ⎞ wmpi ≈ 4.55×1015 ⎜ 14 ⎟ ⎝ 10 ⎠

[s ]. −1

669 c). Numbber of excited d electrons. The T number density d of eleectrons ne creaated to the ennd of the pulsse jointly byy the avalanch he and multi--photon proceesses can be obtained withh the help off the simplifieed rate equattion [4]:

dne = ne wimp + na wmpi . dt

(6.188)

Let’s assume for simplicity thhat laser intennsity is consttant during thhe laser pulsse (flat top-hhat d As we show w later the reccombination during d the puulse time is neegligible. Theen intensity distribution). solution to t Eq.(6.18), with the in nitial conditioon ne (t = 0) = ne0 and wimmp and wmpi are both tim me independeent, is straighttforward:

⎧⎪ nw ne I , λ , t = ⎨ ne0 + a mpi 1 − exxp −wimp t wimp ⎩⎪

(

)

[

(

⎫⎪

)]⎬⎭⎪ exxp(w t). imp

(6.199)

Let uss again considder Lithium Niobate N underr the action of 800 nm, 150 fs laser pulse and find thhe intensity level for the optical breaakdown to occcur. It is coommonly accepted that brreakdown of a o when the t plasma frequency of exxcited electroons equals to the frequencyy of impinginng material occurs laser light. Thus onee can concludde that the siimple model presented heere predicts the t breakdow wn 12 2 12 1 2 d 1.5×10 W/cm W in sem mi-quantitativee agreement with 1.0×10 W/cm from m threshold to be around experimennts of [145] (ssee Fig.6.5).

Figure 6.11. (a) – Avalannche (solid linne) and multi--photon (dasheed line) ionizattion rates for 150 fs, 800 nm n pulses in LiNbO L nization at intennsities above 5.5 3. Note that multi-photon ionization takes over the avalanche ion TW/cm2. (b) ( – Normalizzed electron deensity vs laser intensity at 1550 fs, 800 nm;; the horizontaal line marks thhe critical den nsity. The opttically detectabble changes off refractive ind dex in Fe: LiN NbO3 and undo oped LiTaO3 are a marked by y arrows at 1 TW W/cm2 and 1.37 TW/cm2, resspectively (from m [145]).

d). Recom mbination ratte. Recombinnation in threee body collisions may occuur with electroon, an ion, orr a neutral attom, each off those can act a as a thirdd body. Onee can take ann atomic cro oss section foor estimationn of e-i collission rate in thhe case when ionisation deegree is below w the breakdoown threshold. The probaability of recoombination peer unit volum me per unit tim me (recombinnation rate) then is a product of probab bility of e-i collision, c ν coll = ne n+ σve , aand the probability for a third body presence p in thhe vicinity off colliding paarticles, p3b . The last one can be approx ximated by thhe ratio of atoomic volume to t the averag ge volume peer atom, p3b ≈ 4πrat3 na 3 , if an atom can be consideered as a third body. In thhis

70

(

case this probability is close to unity. Thus recombination time estimates as trec ≈ neσ at ve

)~ −1

(1021×10-15×106)-1 and it is around 1 ps at under-threshold condition (ne <1.7×1021 cm-3). At the electron and ion density close to the breakdown threshold the recombination in the triple Coulomb collisions when the electron acts as a third body can be of importance. In this case the

( ) 2

Coulomb collisions are characterized by the cross section πr02 ≈ π e2 ε el ; here ε el is the electron energy. Excited electrons in the conduction band can be approximately treated as the degenerated Fermi gas with the energy

(

εF = 3π 2 ne

)

2/3

h2 2m = 5.83×10 −27 ne2 / 3 erg , and heat capacity

Ce = π 2Te 2ε F . At ne = 1021 cm-3 εel ≈ ε F = 5.83×10-13 erg = 0.36 eV, and vF = 2.5×107 cm/s. One

can easily see that the Coulomb cross section is of two orders of magnitude higher than the atomic cross section. The electron mean free path lmfp = vF ν e − ph ~10-6 cm is much shorter than the lasermodified absorption length as it is shown below. Therefore the recombination time in the triple Coulomb collisions can be in the order of femtoseconds and hence the recombination during the pulse time might be significant and fast. On the other hand the recombination time is inverse proportional to the electron density. Taking recombination cross-section and electron velocity time independent one obtains, trec ≈ t0 n0 / ne , and very slow decrease in time for the electron number density, ne t = ne t p t0 t ; here t0 is recombination time at the end of the pulse. For ne t p = 1021 cm-3 t0 = 1

() ( )

()

()

ps. On the other hand if the electron number density comparable to that for metal dopants, i.e. ne t p 18

-3

= 10 cm , then the recombination time is t0 ~ 1 ns as it suggested at low intensity studies [166]. 6.3.3. Modification of the properties of the laser-excited solid Now let us consider the change in the optical properties under the assumption that intensity during the pulse time is lower than the breakdown threshold and therefore the laser-induced modification could be reversible. The total dielectric function for a dielectric modified by electrons excitation at high intensity is the sum of unperturbed function and contributions from excited electrons: 4πσ 4πσ i 4 πσ r ε = ε0 ω + i ≡ ε0 ω − +i ≡ ε0 ω − Δε r + iΔε i ; me me ω here σ re, im are the real and imaginary parts of conductivity, respectively. It is reasonable to take the

()

()

()

dielectric function for excited electrons in the Drude form: e2 neν e − ph e2 ne e2 neω σ= = +i m ν eff − iω me ν e2− ph + ω 2 me ν e2− ph + ω 2

(

(

)

)

(

)

(6.20)

The effective collision rate, νeff, includes all collision processes. Then one can find the contributions to the real and imaginary parts above at ω >> ν e − ph as follows:

Δε r ≈

ω 2pe ω

2

; Δε i ≈

ω2pe ω

3

; σr =

ω Δε i . 4π

(6.21)

Note that in the considered here conditions the inequality holds, Δε i << Δε r . The modified refractive index, N ≡ ε = n + ik , then reads:

(

n ≈ ε0 − Δε r

)

1/ 2

; k≈

Δεi . 2n

The change in the refractive index due to electron excitation is negative:

(6.22)

71 −Δn Δε r ≈ 2 . n0 2n0

(6.23)

The conditions in photo-refractive crystal created by the action of intense femtosecond pulse are in sharp contrast to those produced by long pulse or cw lasers. All processes during the pulse are transient. The quasi-stationary state is achieved long after the pulse. We shall consider now all relevant physical processes contributing toward the changes in the optical properties of the material during the pulse and after the pulse. a). Processes during the pulse. Let us estimate the electron current, j phν = e neν e created during the short (150 fs) and intense (1012–1013 W/cm2) laser pulse. The pulse generates the number of excited electrons of the order of 1021 cm-3. The electrons oscillate during the pulse in the high frequency electric field with the amplitude (2.7–8.7)×107 V/cm, which is several orders of magnitude higher than the inherent field of spontaneous polarisation in photo-refractive crystals. Therefore it is impossible to establish a stationary distribution of electric field during the pulse. Indeed, the photoconductivity from Eq.(6.20) in the condition that the collision rate can be neglected in comparison to the laser frequency, reads:

σ ph ≈

e2 neν e − ph meω 2

(6.24)

It is a strong function of the carrier number density. For 800 nm pulse, collision rate of 3.8×1013 s-1, and ne = 1021 cm-3 it yields σph ~2 ohm-1cm-1 (1.74×1012 s-1 in the Gaussian units). If the recombination in triple Coulomb collision is significant during the pulse time the photoconductivity might be lower and photovoltaic field higher. The above conductivity value defines the time for establishing quasi-stationary distribution of electric field, tstat ≅ ε st 4 πσ ph ~1 ps. Thus, stationary distribution could be established only after the end of the pulse. b). Processes after the pulse. After the pulse the electrons are subject for recombination, for a drift in a local field of spontaneous polarisation, which is inherent to a photo-refractive crystal, and for diffusion under the electrons temperature and density gradients, while the ions remain fixed in their positions. The diffusion field is much smaller than other fields, so it can be neglected. The essential difference in comparison with conventional long pulse low intensity case is that the field of charge separation, which is finally responsible for a quasi-permanent change in the refractive index, establishes due to the spontaneous polarisation field in the absence of the external field of laser irradiation. Because the recombination time is longer than the pulse duration, the carriers recombine in a different location from where they were created, most probably in the iron sites. Thus there are strong grounds to assume that the charge separation field, Ecs, equals to Es - the internal field of spontaneous polarisation in the ferroelectric crystal, is unaffected by the electron excitation. It is conventionally assumed that quasi-stationary distribution establishes during the so-called Maxwell time [168]:

tstat ≈ ε st 4 πσ d

εst is the static dielectric function in the absence of the external field. The conductivity in the absence of photo excitation, the so-called dark conductivity, reads:

σd ≈

e2 ne

meν e − ph

(6.25)

72 13 -1 (mom) Note that σ d >> σ ph because ω >> ν e − ph . Taking εst = 29; ne ~1018 cm-3, ν e− ph ≈ TL h = 3.83×10 s ,

and the electron mass as for a free electron, the time for establishing the stationary distribution is less than a picosecond. Spontaneous polarisation Ps in LiNbO3 has a maximum of Ps = 7.1×10-5 Coulomb/cm2 at the Curie point of 1480 K [2]. Respective electric field equals to Es = Ps 3ε0 = 2.67×106 V/cm (

[

]

ε0 = 8.854 ×10 −12 A 2s 4 kg m3 ). Spontaneous polarisation strongly depends on temperature. Glass et al., (1974) gave the value for the photovoltaic field in iron-doped LiNbO3 at room temperature Es Tr ~100 kV/cm, which we use for the following estimates.

()

6.3.6. Possible mechanisms for changes in the refractive index after the pulse Generation of large amount of excited electrons immediately results in the decrease in the refractive index – see Eq. (6.14):

ω Δn Δε re ≈ 2 ≈ 2pe 2 . n0 2n0 2n0ω 2

−

(6.26)

At the electron number density of ne = 1021 cm-3 at the end of the pulse the change constitutes Δn n0 = –4.8×10-2. This index modification decreases in proportion to the decreasing number density of free carriers and after several nanoseconds reaches undetectable level of Δn n0 ~10-5. However, we should note that the real recombination time is unknown. This might be the case of perfect undoped crystal with the absence of defects serving as trapping centres. Most probably, the field of spontaneous polarisation dominates the process of charge separation in a metal-doped crystal. Then the change in the refractive index due to the electro-optic effect reads [168]:

Δn ≈

n3 r Ecs 2

(6.27)

We take the known value of n3 r ≈ 3×10 −8 cm V for Lithium Niobate. Now, the refractive index changes can be estimated as Δn ≈ 1.5×10−8 (cm /V ) × Ecs . For Ecs ~102 kV/cm these changes are expected to be around Δn ≈ (1.5) ×10 −3 . This is close to the values observed in experiments. The interaction of single intense femtosecond laser pulses tightly focused in the bulk of LiNbO3 and LiTaO3 crystals, both with pure crystals and with crystals doped with Fe, was recently studied in [145,152,169-171]. Single 150 fs laser pulses at 800 nm with energy per pulse in a range of 3–50 nJ, were focused inside a crystal to the depth of 50 microns and to the focal spot with diameter of 1.8 μm (Sfoc = 2.54×10-8 cm2), so that intensity in the interaction region varied from 1.0 TW/cm2 to 16.7 TW/cm2. The optically detectable change of refractive index were observed in Fe: LiNbO3 at the energy per pulse 3.8±0.5 nJ and at 5.2±0.5 nJ in pure LiNbO3 crystal. These figures correspond to intensity of 1.0 TW/cm2 and 1.37 TW/cm2 respectively. The permanent modification of LiTaO3 was observed at 32±5 nJ (~10 TW/cm2) that might be considered as a result of breakdown. Therefore the breakdown threshold locates at the average intensity ≤10 TW/cm2. It also has been found that the laser-induced material transformation is fully reversible at the energy per pulse of 14.5 nJ (3.8 TW/cm2). One can see that the theoretical estimates for a breakdown are in qualitative agreement with the measurements. Thus it is confirmed that generation of free carriers by intense fs-pulses occurs due to intertwined avalanche and multi-photon processes.

73 The changes of refractive index were recovered from the light transmission measurements. In zcut pure LiTaO3 crystal (cut is perpendicular to c-axis) the maximum measured relative change in refractive index was 2.5×10-4. In y-cut pure LiNbO3 (cut is parallel to c-axis) the change was 5×10-4. Maximum reversible change in Fe: LiNbO3 (400 ppm doping) constitutes ~10-3. The laser-affected area has characteristic index modulation pattern +Δn/-Δn/+Δn that corresponds to dark/bright/dark regions along the crystallographic c-axis. This occurs in both Fe-doped and undoped crystals. This feature is a qualitative evidence of the fact that the photovoltaic process was followed by the electrooptic effect, and both are responsible for the index modulation. There is however a substantial difference in the after-pulse behaviour of laser-affected regions in Fe-doped and undoped LiNbO3. Index modulation in Fe-doped crystal is long-lived while the effect in undoped crystal completely disappears in 0.25-0.3 seconds, most probably due to recombination of the free carriers. On the other hand the long life of the charge separation and resulted modulation of refractive index in Fe-doped crystal most probably occurs due to the presence of the metal trapping centres separated by distance −1/ 3 of nFe ≈ 10 −6 cm from each other, which is much larger than free carrier mean free path of ve ν e − ph . The last fact indicates that the non-local approach should be applied for the current calculation instead of the Ohm's law. Summing up, it has been demonstrated that the interaction of intense femtosecond pulse with photorefractive crystal at conditions close to the breakdown threshold has several distinctive features in comparison to that of the long pulse (or cw) lasers. First, the high number density of excited electrons modifies the dielectric function and leads to the negative change in refractive index, – 4.8×10-2, exceeding that due to the charge separation long after the pulse. The amount of change depends on laser intensity and independent on the beam polarization. However, this index change is transient, it disappears when the recombination is completed at the nanosecond time scale. Second, the dominance of high frequency laser field, which is two orders of magnitude higher than the field of spontaneous polarization, makes the stationary charge distribution impossible during the pulse. Third, the diffusion and recombination of the charge carriers continue long after the end of the pulse (on the nanosecond time scale). The main driving force responsible for the current is the field of spontaneous polarization: the current terminates when the field of charge separation balances this field. Quasistationary distribution of charges that results in change of the refractive index due to the Pockels effect occurs well after the pulse. Modification of refractive index derived from this theory is in a semi-quantitative agreement with experiment. In the model presented here the modification of refractive index should be independent on polarization that is also in agreement with observation of index modifications in dielectrics [152]. These findings suggest that the laser electric field with high amplitude modifies not only linear properties of the material. Most probably strong AC field also induces transient ferroelectric and non-linear properties of a crystal. Therefore a new avenue opens up for the studies of the intensity-dependent transient phase transformation induced by femtosecond lasers at intensity close but below the damage threshold. Pump-probe experiments with doubleprobes might provide the information of time-dependent dielectric function of the excited crystal with fs resolution, while harmonics generation may provide information on transient non-linear properties. 6.4. Laser-matter interactions confined in a bulk at high intensity Single short pulse tightly focused inside the bulk of a transparent solid can easily generate the energy density in excess of the Young modulus of any of existing solid within a focal volume less than a cubic micron [155,156]. The pressure of the order of several TPa inside a focal volume leads to formation of a cavity (void) surrounded by a shell of compressed material. These two features of

74 the phenomenon delineate two areas of studies and applications. The first area relates to formation of different 3D structures, photonic crystals, waveguides, gratings etc. making use of multiple voids (separated or interconnected) created in the different space points of a crystal. For these studies the most important part is the void formation. As it is shown later in this section, in order to produce a void one has to generate a pressure in excess of the strength (the modulus) of a material. The second area of research relates to the studies of material transformations under high pressure-temperature conditions, which are possible to create in tabletop laboratory experiments. The interaction of a laser with matter at intensity above the ionisation and ablation threshold proceeds in a way similar for all the materials [4]. The material converts into plasma in a few femtoseconds at the very beginning of the pulse, changing the interaction to the laser-plasma mode, increasing the absorption coefficient and reducing the absorption length, which ensures a fast energy release in a very small volume. A strong shock wave is generated in the interaction region and this propagates into the surrounding cold material. The shock wave propagation is accompanied by compression of the solid material at the wave front and decompression behind it, leading to the formation of a void inside the material. The laser and shock wave affected material is confined in the shell that surrounds the void and this shell is the major object for studies of new phases and new material formation. Single pulse action thereby allows a formation of various three-dimensional structures inside a transparent solid in a controllable and predictable way. Let us first to underline the differences between the intense laser-matter interaction at the surface of a solid and the case when laser-matter interaction is confined deep inside a solid by comparing the pressure created at the absorption region at the same intensity and total absorbed energy. At the intensity well over the ionisation and ablation thresholds any material converts into plasma in a few fs time. Therefore the interaction proceeds most of the time in laser-plasma interaction mode. In these conditions the pressure at the ablated plume-solid interface (in laser-surface interaction) constitutes from the sum of thermal pressure of plasma next to the boundary plus the pressure from the recoil momentum of expanding plasma. Significant part of absorbed energy is spent on the expansion and heating of the ablated part of a solid. Therefore the ablation pressure in the case of surface interaction m depends on the absorbed intensity by the power law Pabl ∝ I abs ; m < 1. Alternatively, there is no expansion loss in confined interaction. Hence the maximum pressure is proportional to the absorbed intensity Pconf ∝ I abs and it is almost twice larger than in the surface interaction mode. Full description of the laser-matter interaction process and laser-induced material modification from the first principles embraces the self-consistent set of equations that includes the Maxwell’s equations for the laser field coupling with matter, complemented with the equations describing the evolution of energy distribution functions for electrons and phonons (ions) and the ionisation state. A resolution of such a system of equations is a formidable task even for modern supercomputers. Therefore, the thorough analytical analysis is needed. We split below this complicated problem into a sequence of simpler interconnected problems: the absorption of laser light, the ionisation and energy transfer from electrons to ions, the heat conduction, and hydrodynamic expansion. 6.4.1. Absorbed energy density Let us first to define the range of laser and focussing parameters necessary for obtaining high pressure inside the interaction region. A 100 nJ laser pulse of duration tp ≤100 fs with the average intensity I > 1014 W/cm2 focussed into the area S foc ∝ λ2 delivers the energy density >10 J/cm2, well above the ionisation and ablation thresholds for any material [4]. The focal volume has a complicated three-dimensional structure. As a first approximation (that is also useful for scaling purposes) the

75 focal volume is the focal area multiplied by the absorption length. The absorbed laser energy per unit time and per unit volume during the pulse reads:

dEabs 2 A = I r, z, t dt labs

(

)

(6.28)

labs is the electric field absorption depth labs = c ω k . We assume that the electric field exponentially decays inside a focal volume, E = E0 exp{−x / labs} as it does in the skin layer; A is the absorption coefficient defined by the Fresnel formula [3] as the following:

A=

4n

( )

2

n +1 + k 2

2ε i

=

2

.

(6.29)

1+ ε1/ 2 k

The duration of a typical short pulse of ~100 femtosecond is shorter than the electron-phonon and electron-ion collision times. Therefore the electron energy distribution during the pulse has a delta function like shape peaked near the energy that can be estimated from the general formula of Joule heating Eq. (6.28) under the assumption that the spatial intensity distribution inside a solid, and material parameters are time independent. We denote the energy per single electron by εe (it should not be confused with the dielectric function that is always without a subscript here). Then the electron energy density change reads:

( )= 2 A I t . () l dt

d ne ε e

abs

We show later that the ionisation degree at I > 1014 W/cm2 is high, Z > 1, the number density of electrons is large, and electrons heat capacity can be taken as that for ideal gas. Thus from the above one can make a rough estimate of the electron temperature to the end of the pulse:

Te ≈

2A I t t. 1.5k B ne labs

()

The electron temperature rises to tens of electron volts at the very beginning of the pulse. Fast ionisation of a solid occurs that affects absorption coefficient and absorption length. Thus, the next step is to introduce the model where the optical properties are dependent on the changing electron density and electron energy. 6.4.2. Ionisation Optical breakdown of dielectrics and optical damage produced by an intense laser beam has been extensively studied over several decades. Analytical estimates of the breakdown threshold, ionisation rates and transient number density of electrons created in the absorption region allows one to obtain the general picture of the processes in qualitative and quantitative agreement with computer simulations. a). Ionisation thresholds. It is generally accepted that the breakdown occurs when the number density of electrons in the conduction band reaches the critical density expressed through the frequency of the incident light by the familiar relation, nc = me ω 2 4 π e2 . Thus, laser parameters, (intensity, wavelength, pulse duration) and material parameters (band-gap width and electron-phonon effective collision rate) at the breakdown threshold are combined by condition, ne = nc .

76 The ionisation threshold for the majority of transparent solids lies at intensities in between (1013 – 1014) W/cm2 (λ ~1 μm) with a strong non-linear dependence on intensity. The conduction-band electrons gain energy in an intense short pulse much faster than they transfer energy to the lattice. Therefore the actual structural damage (breaking inter-atomic bonds) occurs after the electron-tolattice energy transfer, usually after the end of the pulse. It was determined that in fused silica the ionisation threshold was reached to the end of 100 fs pulse at 1064 nm at the intensity 1.2×1013 W/cm2 [70]. Similar breakdown thresholds in a range of (2.8 ± 1)×1013 W/cm2 were measured in interaction of 120 fs, 620 nm laser with glass, MgF2, sapphire, and the fused silica [172]. This behaviour is to be expected, since all transparent dielectrics share the same general properties of slow thermal diffusion, fast electron-phonon scattering and similar ionisation rates. The breakdown threshold fluence (J/cm2) is an appropriate parameter for characterization conditions at different pulse duration. It is found that the threshold fluence varies slowly if pulse duration is below 100 fs. For example, for the most studied case of fused silica the following threshold fluences were determined: ~2 J/cm2 at 1053 nm; ~ 300 fs and ~ 1 J/cm2 at 526 nm; ~ 200 fs [81]; 1.2 J/cm2 (620 nm; ~ 120 fs) [172]; 2.25 J/cm2 at 780 nm; ~ 220 fs [108,173]; 3 J/cm2 at 800 nm; 10 – 100 fs [174]. b). Ionisation rates: Avalanche ionisation. In interaction of lasers in a visible range with wide band gap dielectrics the direct photon absorption by electrons in a valence band is rather small. However, a few seed electrons can always be found in the conduction band. These electrons oscillate in the laser electromagnetic field and can be gradually accelerated to the energy in excess of the band-gap. Electrons with εe > Δgap collide with electrons in the valence band and can transfer them a sufficient energy to excite into the conduction band. Thus the number of free electrons increases, which provokes the effect of avalanche ionisation. The probability of such event per unit time (ionisation rate) can be estimated as follows:

1 dε e 2ε osc ω ν eff = . 2 Δ gap dt Δ gap ν eff + ω2 2

wimp ≈

(

)

(6.30)

Electron is accelerated continuously in this classical approach. The oscillation energy is proportional to the laser intensity and to the square of the laser wavelength. At relatively low temperature corresponding to low intensities below the ablation threshold the effective collision rate, νeff, equals to the electron-phonon momentum exchange rate νeff = νe-ph. The electron-phonon momentum exchange rate increases in proportion to the temperature. For example, the electron-phonon momentum exchange rate in SiO2 is of νe-ph = 5×1014 s-1 [70] and it is lower of the laser frequency for visible light, ω ≥ 1015 s-1. Then the ionisation rate from Eq. (6.30) grows in proportion to the square of the laser wavelength in correspondence with the Monte-Carlo solutions to the Boltzmann kinetic equation for electrons [70]. With further increase in temperature and due to the ionisation, the effective collision rate becomes equal to the electron-ion momentum exchange rate, and reaches maximum approximately at the plasma frequency (~1016 s-1) [15,4]. At this stage the wavelength dependence of the ionisation rate almost disappears due to ω < ν e− i ≈ ω pe , as it follows from Eq. (6.30) in agreement with rigorous calculations of [70]. It is instructive to estimate the ionisation rate for an extreme case of high intensity and for one of the largest band gap of sapphire, when the oscillation energy of a free electron in conduction band, εosc (see Eq.6.17), equals to the band gap, ε osc ≈ Δ gap = 9.9 eV. Then the intensity of conventional 800 nm laser ( ω = 2.356 × 1015 s −1) equals to 1.7×1014 W/cm2. In the beginning of the ionisation process, when ω > ν e − ph , ionisation rate is wimp ≈ 2ν e − ph ~1014–1015s-1.

77 When the collision rate reaches its maximum, ω < ν e − i ≈ ω pe , the ionisation rate equals to wimp ≈ 2ω 2 / ν e− i ≈ 2ω 2 / ω pe ~5×1014 s-1.

c) Ionisation rates: Multi-photon ionisation. Multiphoton ionisation has no intensity threshold and hence its contribution can be important even at relatively low intensity. Multi-photon ionisation creates the initial (seed) electron density, n0, which then grows by the avalanche process. It proved to be a reasonable estimate of the ionisation probability (per atom, per second) in the multi-photon form [93]:

wmpi ≈ ω

⎛ ε ⎞n ph osc ⎟⎟ ; ⎝ 2 Δ gap ⎠

n3ph/ 2⎜⎜

(6.31)

here n ph = Δ gap hω is the number of photons necessary for electron to be transferred from the valence to the conductivity band. The multi-photon ionisation process is important at low intensities as it generate the initial number of seed electrons, though small number, they are further multiplied by the avalanche process. The multi-photon ionisation rate dominates, wmpi > wimp, for any relationship between the frequency of the incident light and the effective collision frequency in conditions when εosc > Δgap. However, even at high intensity the contribution of avalanche process is crucially important: at wmpi ~ wimp the seed electrons are generated by multi-photon effect whilst final growth is due to the avalanche ionisation. Such an inter-play of two mechanisms has been demonstrated with the direct numerical solution of kinetic Fokker-Planck equation [93]. Under the condition εosc = Δgap, hω = 1.55 eV, n ph = Δ gap hω ~ 6.4, and ω = 2.356 × 1015 s −1, the multi-photon rate comprises wmpi −1 ~5.95×1015 s-1. The ionisation time estimates as tion ≈ wmpi . Thus, the critical density of electrons (the

ionisation threshold) is reached in a few femtoseconds in the beginning of a 100-fs pulse. After that the interaction proceeds in a laser-plasma interaction mode. d). Ionisation state during the laser pulse. In order to estimate the electron number density generated by the ionisation during the laser pulse the recombination processes should be taken into account. Electron recombination proceeds in dense plasma mainly by three-body Coulomb collisions with one of the electrons acting as a third body [92]. The cross section for the Coulomb collision reads

( ) 2

σ e − i ≈ π e2 ε el Z 2 , while the probability for a third body (electron) presence in the vicinity of

( ) 3

3 = e2 ε el . colliding particles is proportional to the cube of the Coulomb impact distance, p3b ∝ rCoul

The growth rate of the electron number density is the balance of ionisation and recombination terms [155]:

dne ≈ ne wion − β e ni ne2 . dt

(6.32)

Here the ionisation rate is wion = max{wmpi , wimp } ~1015s-1, and the recombination rate is β e ni ne , where the coefficient βe is expressed as the following [92]:

β e = 8.75×10 −27 ln Λ Z 2 ε el9 / 2 .

(6.33)

We assumed that ne = Zni , the electron energy εel is in eV; ln Λ is the Coulomb logarithm. One can

(

−1 see that ionisation time, tion ≈ wion , and recombination time trec ≈ β e ni ne

) , are of the same order of −1

78 magnitude, ~10-15 s, and both are much shorter then the pulse duration. This is a clear indication of the ionisation equilibrium, and that the multiple ionisations Z >1 take place. Therefore, the electron number density at the end of the pulse can be estimated in a stationary approximation as the follows: ne2 ≈ wion Zβ e . Taking w ion ~ 1015 s-1; εe ~ 30 eV; Z = 5; lnΛ ~ 2, one obtains, that number density of electrons at the end of the pulse becomes comparable to the atomic number density ne ~1023 cm-3. 6.4.3. Increase in the absorbed energy density due to modification of optical properties We demonstrated above that the swift ionisation during the first femtoseconds in the beginning of the pulse produces the electron number density comparable to the critical density for the incident laser light, ne = nc . The free-electron number density grows up and becomes comparable to the ion density to the end of the pulse. Respectively, the electron-ion collision rate reaches its maximum that equals approximately to the plasma frequency in the dense non-ideal plasma. With further increase of electron temperature the electron-ion exchange rates decrease due to domination of the Coulomb collisions. The optical properties of this transient plasma are described by the Drude-like dielectric function; they are changing in accord with the change in electron density and temperature. Let us estimate the absorption coefficient and absorption length in the beginning of the laser pulse and at the end of the pulse. The dielectric function and refractive index in conditions, ν e − i ≈ ω pe >> ω , are estimated as the following: 1/ 2

⎛ε ⎞ ω ω2 εr ≈ 2 ; ε i ≈ pe ; n ≈ k = ⎜ i ⎟ . ω ω pe ⎝2⎠

(6.34)

For example, after the optical breakdown of fused silica glass by 800 nm laser at high laser intensity (ω = 4.7×1015 s-1; ωpe =1.45×1016 s-1) the real and imaginary parts of refractive index are n ~ k = 1.18 thus giving the absorption length of ls = 54 nm, and the absorption coefficient A = 0.77 [155]. Therefore, the optical breakdown and further ionisation and heating converts silica into a metal-like plasma medium reducing the energy deposition volume by up to two orders of magnitude when compared with the focal volume, and correspondingly massively increasing the absorbed energy density and consequently the maximum pressure in the absorption region. For the interaction parameters presented above (I = 1014 W/cm2; A = 0.77; ls = 54 nm; tp = 150 fs) the pressure corresponding to the absorbed energy density equals to 4.4 TPa, ten times higher than the Young modulus of sapphire, one of the hardest of dielectrics. The general approach presented above is applicable for estimating parameters of any wide band gap dielectric affected by high intensity short pulse laser. 6.4.4. Energy transfer from electrons to ions: relaxation processes after the pulse The hydrodynamic motion starts after the electrons transfer the absorbed energy to the ions. The following processes are responsible for the energy transfer from electrons to ions: recombination; electron-to-ion energy transfer in the Coulomb collisions; ion acceleration in the field of charge separation; electronic heat conduction. Below we compare the characteristic times of different relaxation processes. a). Impact ionisation and recombination. The electron temperature in the energy units at the end of the pulse is much higher then the ionisation potential. Therefore, the ionisation by the electron impact continues after the pulse end. The evolution of the electron number density can be calculated in the frame of the familiar approach [92]:

79

dne ≈ α e ne na − β e ni ne2 ; dt

(

(6.35)

) (

here α e = σ e ve ne I Te + 2 exp − I Te

) [cm s] 3

is the impact ionisation rate and βe is the

recombination rate connected to αe by the principle of detailed balance. One can see that for parameters of experiments in question (σe ~2×10-16 cm2; εe ~ 30 eV) the time for establishing the

( )

ionisation equilibrium is very short τ eq ≈ α e ne

−1

~10-16 s. Thus the average charge of multiple

ionised ions can be estimated from the equilibrium conditions using Saha equations. Losses for ionisation lead to temporary decrease in the electron temperature and in the total pressure [155]. However the fast recombination results in the increase in the ionic pressure. b). Electron-to-ion energy transfer by the Coulomb collisions. The Coulomb forces dominate the interactions between the charged particles in the dense plasma created by the end of the pulse. The parameter that characterizes the plasma state is the number of particles in the Debye sphere,

(

N D = 1.7 ×109 Te3 ne

)

12

[16]. Plasma is in ideal state when ND >> 1. In the plasma with parameters

estimated above for the fused silica (Z = 5, lnΛ = 1.7; ne = 3×1023 cm-3; εe = 50 eV) ND is of the order of unity, that is a clear signature of the non-ideal conditions. The maximum value for the electron-ion momentum exchange rate in non-ideal plasma approximately equals to the plasma frequency, ν ei ≈ ω pe ~3×1016 s-1 [4,15]. Hence electrons in ionised fused silica transfer the energy to ions over a

( ) ≈ (M

time teien ≈ ν eien

−1

i

)

meν ei ~(1–2) ps [155].

c). Ion acceleration by the gradient of the electron pressure. Let us estimate the time for the energy transfer from electrons to ions under the action of electronic pressure gradient when ions are initially cold. The Newton equation for ions reads:

∂M i ni ui ∂P ≈− e ∂t ∂x The kinetic velocity of ions then estimates as follows:

ui ≈

Pe t M i ni Δx

(6.36)

The time for energy transfer from electrons to the ions is defined by condition that the ions kinetic energy compares to that of the electrons, M i ui2 2 ~ ε e . With the help of Eq. (6.36) one obtains the energy transfer time ( Zni ≈ ne ):

tel−st

−1/ 2 Δx ⎛ ε e ⎞ ~ ⎜ ⎟ ; Z ⎝ 2M i ⎠

(6.37)

here Δx ≈ labs = 54 nm is the characteristic space scale. Then taking the time for the electron-to-ion energy transfer by the action of the electrostatic field of charge separation equals to tel-st ~ 1 ps. d). Electronic heat conduction. Energy transfer by non-linear electronic heat conduction starts immediately after the energy absorption. Therefore heat wave propagates outside of the heated area before the shock wave emerges. The thermal diffusion coefficient is defined conventionally as the

80 following, D = le ve 3 = ve2 3ν ei , where le, ve and nei are the electron mean free path, the electron velocity and the momentum transfer rate respectively. The characteristic cooling time is conventionally defined as tcool = ls2 D . For the conditions of experiments [156] nei ~ ωpe ~ 3×1016s-1; 2 εe = 50 eV, and the cooling time is tcool = 3ω pe me labs 2ε e = 14.9 ps.

Summing up the results of the energy deposition in confined microexplosion we shall note that the major processes responsible for the electron-to-ion energy transfer in the dense plasma created by the tight focussing inside a bulk solid are different, and much shorter, when compared to those in the plume created by laser ablation. The ions acceleration by the gradient of the electronic pressure and the electron-to-ion energy transfer by the Coulomb collisions both comprise ~1 ps. The thermal ionization and recombination are in equilibrium, this permits the description of the plasma state by the Saha equations. The electronic non-linear heat conduction becomes important in the first ~15 ps after the pulse, and dominates the return to the ambient conditions. 6.4.5. Shock wave propagation, stopping, and void formation It was shown above that a focal volume as small as 0.2 μm3 can be illuminated by focusing 800 nm laser pulses in the bulk of fused silica glass (n = 1.453) with a microscope objective with NA = 1.35. The original focal volume shrinks to much smaller energy deposition region due to fast decrease in the absorption length, approximately five times less than the averaged focal radius. Modified shape of the absorption region has a complicated shape, which is practically unknown. Therefore, it is reasonable to assume for the further calculations that the absorption volume is a sphere of a smaller radius than the focal volume. One can see that 100 nJ of laser energy focussed in the volume of 0.2 μm3 create the energy density of 5×105 J/cm3 equivalent to the pressure of 0.5 TPa (5 Mbar). However this energy absorbs in much smaller volume thereby generating a pressure in excess of P = 10 TPa. All absorbed energy is confined in the electron component at the end of the 150-fs pulse. a). Shock wave generation and propagation. The hydrodynamic motion of atoms and ions starts when the electrons have transferred their energy to ions. This process is completed in a few picoseconds time, however one should note that the energy transfer time by the Coulomb collisions increases in proportion to the electron temperature. So, in solid-state density plasma formed in confined microexplosion the higher the absorbed energy, the longer the time for hydrodynamic motion to start. The pressure in a range of several TPa builds up after electron-ion energy equilibration; this pressure considerably exceeds the Young modulus for majority of materials. For example, the Young modulus for sapphire equals to 0.3 – 0.4 TPa, and that for silica is ~0.07 TPa. The high pressure generates the shock wave propagating from the energy absorption region into the surrounding cold material. The bulk modulus of the cold material equals to the cold pressure, Pc. This cold counter pressure finally decelerates and stops the shock wave. Because the shock driving pressure significantly exceeds the cold pressure, P>>Pc, the strong shock emerges compressing a solid to the limit allowed by the equation of state of a solid, which does not depend on the magnitude of the driving pressure. The maximum density of a perfect gas with the adiabatic constant γ is as the follows:

ρ=

γ +1 ρ0 . γ −1 e

(6.38)

81 The adiabatic constant for a cold solid is conventionally estimates as γ ~ 3 [92]. Therefore maximum density after the shock front is expected to be ρmax = 2 ρ0 . The compression ratio gradually decreases to unity along the shock propagation, deceleration and transformation into a sound wave. Note that the temperature in the compressed solid behind the shock front in the limit of P >>Pc grows in proportion to the driving pressure:

⎛ γ −1 ⎞ P T =⎜ ⎟ T0 ⎝ γ +1 ⎠ Pc

(6.39)

Thus material is compressed and heated behind the shock wave front. Hence, the conditions for transformation to another phase might be created and this phase might be preserved after unloading to the normal pressure. The final state of matter may possess different properties from those in the initial state. b). Shock wave stopping. The shock wave propagating in a cold material loses its energy due to dissipation, e.g. due to the work done against the internal pressure (Young modulus) that resists material compression. The distance at which the shock wave effectively stops defines the shockaffected area. At the stopping point the shock wave converts into a sound wave, which propagates further into the material without inducing any permanent changes to a solid. The distance where the shock wave stops can be estimated from the condition that the internal energy in the whole volume 3 enclosed by the shock front is comparable to the absorbed energy: 4 π P0 rstop 3 ≈ Eabs [92]. The stopping distance obtained from this condition reads: 1/ 3

rstop

⎛ 3E ⎞ ≈ ⎜ abs ⎟ . ⎝ 4 π P0 ⎠

(6.40)

In other words, at this point the pressure behind the shock front equals to the internal pressure of the cold material. One can reasonably suggest that the sharp boundary observed between the amorphous (laser-affected) and crystalline (pristine) sapphire in the experiments [155,156] corresponds to the distance where the shock wave effectively stopped. The sound wave continues to propagate at r > rstop apparently not affecting the properties of material. For 100 nJ of absorbed energy and sapphire, taking Pc = 0.4 TPa for sapphire, one gets from Eq.(6.40) rstop = 180 nm, which is in qualitative agreement with the experimental values. c). Rarefaction wave: formation of void. The experimentally observed formation of void, which is a hollow or low-density region, surrounded by a shell of the laser-affected material, can be qualitatively understood from the following simple reasoning. Let us consider for simplicity spherically symmetric explosion. The strong spherical shock wave starts to propagate outside the centre of symmetry of the absorbed energy region, compressing the material. At the same time, a rarefaction wave propagates to the centre of symmetry decreasing the density in the area of the energy deposition. This problem qualitatively resembles the familiar hydrodynamic phenomenon of strong point explosion (P >> P0) in homogeneous atmosphere with counter pressure taken into account. It is characteristic of a strong spherical explosion that material density decreases rapidly in space and time behind the shock front in direction to the centre of symmetry. Practically, the entire mass of material, initially uniformly distributed in the energy deposition region inside a sphere of radius r ~ labs, is concentrated within a thin shell near the shock front some time after the explosion. The temperature increases and density decreases towards the centre of symmetry, while the pressure is almost constant along the radius [92].

82 This picture is qualitatively similar to that observed in the experiments [155] as a result of fs-laser pulses tightly focussed inside sapphire, silica glass and polystyrene. A void surrounded by a shell of laser-modified material was formed at the focal spot. Hence, following the strong point explosion model, one can suggest that that the whole heated mass in the energy deposition region was expelled out of the centre of symmetry and was frozen after shock wave unloading in the form of a shell surrounding the void. One can apply the mass conservation law for estimate of the density of compressed material from the void size measured in the experiment. Indeed, the mass conservation relates the size of the void to compression of the surrounding shell. We assume that in conditions of confinement no mass losses could occur. One can use the void size and size of the laser-affected material from the experiments and deduce the compression of the surrounding material. The void formation inside a solid only possible if the mass initially contained in the volume of the void was pushed out and compressed. Thus after the micro-explosion the whole mass initially confined in a volume with of radius rstop resides in a layer in between rstop and rv, which has a density ρ = δ ρ0 ; δ > 1:

(

)

4π 3 4π 3 3 rstop ρ0 = rstop − rvoid ρ. 3 3

(6.41)

Now, the compression ratio can be expressed through the experimentally measured void radius, rvoid , and the radius of laser-affected zone, rstop , as follows:

(

rvoid = rstop 1 − δ −1

)

1/ 3

.

(6.42)

It was typically observed that rvoid ~0.5rstop in experiments of [155]. Applying Eq.(6.42) one obtains that compressed material in a shell has a density 1.14 times higher than that of crystalline sapphire. Note that the void size was measured at the room temperature long after the interaction. 6.4.7 Density and temperature in the shock- and heat-affected solid There are two distinctive regions in the area affected by laser action, which is confined inside a cold solid. First is the area where the laser energy is absorbed. Second area relates to the zone where shock wave propagates outside the absorption zone, compresses and heats the initially cold crystal and then decelerates into the sound wave, which apparently does not affect the pristine crystal. In the first area the crystal is heated to the temperature of tens of eV (~105 K). It is swiftly ionised at the density close to that of a cold solid. In picosecond time the energy is transferred to ions and shock wave emerges. Conservatively, the heating rate estimates approximately as 10 eV/ps ~ 1017 K/s. Then this material undergoes fast compression under the action of the micro-explosion and its density may increase to the maximum of ~2 times the solid density. The energy dissipation in the shock wave and by the heat conduction takes nanoseconds. In the second zone heated and compressed exclusively by the shock wave the maximum temperature reaches several thousands Kelvin, approximately 10 times less than in the first zone. Heating and cooling rates in this zone are of the order of ~1014 K/s. Phase transformations in quartz, silica and glasses induced by strong shock waves have been studied for decades, see [92,175] and references therein. However all these studies were performed in one-dimensional (plane) open geometry when unloading into air was always present. The pressure ranges for different phase transitions to occur under shock wave loading and unloading have been established experimentally and understood theoretically [175]. Quartz and silica converts to dense phase of stishovite (mass density 4.29 g/cm3) in the pressure range between 15 – 46 GPa. The

83 stishovite phase exists up to a pressure 77 - 110 GPa. Silica and stishovite melts at pressure > 110 GPa that is in excess of the shear modulus for liquid silica ~ 10 GPa. Dense phases usually transform into a low density phases (2.29 – 2.14 g/cm3) when the pressure releases back to the ambient level. Numerous observations exist of amorphisation upon compression and decompression. An amorphous phase of silica denser than the initial state sometimes forms when unloading occurs from 15 – 46 GPa. Analysis of experiments shows that the pressure release and the reverse phase transition follows an isentropic path. In the studies of shock compression and decompression under the action of shock waves induced by explosives or kJ-level ns lasers, the loading and release time scales are in the order of ~1 – 10 ns [176-179]. The heating rate in the shock wave experiments is 103 K/ns =1012 K/s, 5 orders of magnitude slower than in the confined micro-explosion. In contrast, the peak pressure at the front of shock wave, driven by the laser in confined geometry, reaches the level of TPa, that is, 100 times in excess of pressure value necessary to induce structural phase changes and melting. Therefore, the region where the melting may occur is located very close to that where the energy is deposited. The zones where structural changes and amorphisation might occur are located further away. Super-cooling of the transient dense phases may occur if the quenching time is sufficiently short. Short heating and cooling time along with the small size of the area where the phase transition takes place can affect the rate of the direct and reverse phase transitions. In fact, phase transitions in these space and time scales have been studied very little. The refractive index changes in a range of 0.05–0.45 along with protrusions surrounding the central void that were denser than silica were observed as a result of laser-induced micro-explosion in a bulk of silica [150]. This is the evidence of formation of a denser phase during the fast laser compression and quenching; however, little is known of the exact nature of the phase. Thus, we can conclude that a probable state of a laser-affected glass between the void and the shock stopping edge may contain amorphous or micro-crystallite material denser than the pristine structure and with a larger refractive index than the initial glass [150]. 6.4.8. Upper limit for the pressure achievable in confined interaction The micro-explosion can be considered as a confined one if the shock wave affected zone is separated from the outer boundary of a crystal by the layer of pristine crystal m-times thicker than the size of this zone. On the other side the thickness of this layer should be equal to the distance at which the laser beam propagates without self-focussing Ls− f W Wc – see Eq.(6.4); W is the laser power, Wc

(

)

is the critical power for self-focussing from Eq.(6.3). This condition expresses as the following: Ls− f = m rstop

(6.43)

Absorbed energy, Eabs = A Elas , expressed through the laser power W = Elas t p ( Elas , t p are respectively energy per pulse and pulse duration), and the radius of shock wave affected zone are connected by the equation: 1/ 3

rstop

⎛ 3 AW t ⎞ p ≈ ⎜⎜ ⎟⎟ 4 π P cold ⎠ ⎝

(6.44)

84 The condition of Eq. (6.43) with Eq.(6.4) for the self-focusing length inserted then turns to the equation for the maximum laser power at which micro-explosion remains confined and self-focussing does not affect the crystal between the laser affected zone and outer boundary: 1/ 3 ⎛ W ⎞1/ 3 ⎛ W ⎞1/ 2 2n0 πr02 ⎛ 4πPcold ⎞ ⎜⎜ ⎟⎟ = ⎜ ⎟ ⎜ −1⎟ . m λ ⎝ 3AWc t p ⎠ ⎝ Wc ⎠ ⎝ Wc ⎠

(6.45)

The left-hand side in Eq. (6.45) can be calculated if the laser pulse duration and focal spot area are both known. Taking, for example, sapphire (n0 = 1.75); Wc = 1.94 MW; λ = 800 nm; 4 πPcold 3 = 1.67 MJ; tp = 100 fs; πr02 = 0.2 μm2; m = 3) one gets 0.6 in the LHS of Eq. (6.45). Thus the maximum laser power allowed in these conditions equals to ~1.3Wc = 2.5 MW or 250 nJ of the energy in 100 fs laser pulse. For conditions considered above the maximum pressure that can be achieved in absorption volume confined inside the transparent crystal might be up to 27 TPa, approximately 3 times higher that was achieved in the experiments [156]. 6.4.9. Ionisation wave propagation towards the laser beam There is an additional effect in the focal zone which can influence the size of the volume absorbing the laser energy at the laser fluence above the optical breakdown threshold Fthr,, namely, the motion of ionisation front with critical density in the direction towards the laser beam propagation. It was first discovered in studies of breakdown in gases [93]. Indeed, the intense beam with the total energy well above the ionisation threshold reaches the threshold value at the beginning of the pulse. Laser energy increases and the beam cross-section where the laser fluence is equal to the threshold value, the ionisation front, starts to move in the opposite to the beam direction. The beam is focussed to the focal spot area, S f = π rf2 . The spatial shape of the beam path is a truncated cone, the intensity at any time to be independent of the transverse coordinates (see figure 3).

Figure 6.2. Scheme of the path of the converging focused beam as a truncated cone. The motion of the ionisation front toward the beam propagation is indicated by the arrows.

The ionisation time, i.e. the time required for generating the number of electrons to reach the critical density in the conduction band, is defined by Eq.(1). Therefore, the threshold fluence is achieved at the beam cross-section with a radius increasing as the following:

( )

()

r z, t = rf + z t tgα

(6.46)

85 where, z is the distance from the focal spot, which is usually a circle with radius rf. Then at any moment t during the pulse the relation holds:

E las (t ) = Fthr πr 2 (z,t )

(6.47)

( ) πr F

We introduce the dimensionless parameter, f = Elas t p

2 f thr

= Flas Fthr , as the ratio of the

maximum fluence to the threshold fluence. Then the ionisation front moved the distance z(tp):

()

z tp =

rf tgα

(f

1/ 2

)

−1

(6.48)

Correspondingly, the ionisation time can be evaluated as: 1/ 2 ⎤ ⎡ ⎛ 1⎞ tion = t p ⎢1− ⎜1− ⎟ ⎥ . ⎢ ⎝ f⎠ ⎥ ⎣ ⎦

(6.49)

One can see that if the total fluence equals to that for the threshold, f = 1, the ncr is reached only at the end of the pulse [71,81,180], the ionisation time equals to the pulse duration and thus there is no movement of the ionisation front. These simple geometrical considerations are in qualitative agreement with the experiments in sapphire and silica. Indeed, the voids measured in sapphire and in silica in the reference [155-158] are slightly elongated; the Eq.(6.48) gives zm = 0.67rf and zm = 0.45rf for sapphire and silica respectively. Summing up the results of this section, the effects of the blue shift of the pulse spectrum and the intensity dependence of the group velocity are small and rather positive for achieving high absorbed energy density. The negative effect of the ionisation front motion at the laser energy well above the ionisation threshold leads to a large decrease in the absorbed energy density. The negative effect of defocusing needs further careful studies with the solution of Maxwell equations. 6.4.10. Modelling of macroscopic explosions by micro-explosion in tabletop experiments The micro-explosion can be described solely in the frames of the ideal hydrodynamics if the heat conduction and other dissipative processes, all characterized by specific length scales, could be ignored. The hydrodynamic equations contain five variables: the pressure, P, the velocity, v, the density, ρ, the distance, r, and the time, t. The last three of them are independent parameters, and the other two can be expressed through the previous three. The micro-explosion can be fully characterized by the following independent parameters: the radius of the energy deposition zone, R0, the total absorbed energy, E0, and the initial density ρ0. Then the initial pressure, P0 = E0 R03 , and the

(

initial velocity, v0 = P0 ρ0

)

1/ 2

are combinations of the independent parameters. One can neglect the

energy deposition time and time for the energy transfer from electrons to ions (picosecond) in comparison to hydrodynamic time of a few nanoseconds. Then, the hydrodynamic equations can be reduced to the set of the ordinary equations with one variable [92], ξ = r v0 t , describing any hydrodynamic phenomena with the same initial pressure and density (velocity), but with the characteristic distance and time scales changed in the same proportion. When the energy of explosion

(

increases, the space, R0, and time scales are increased accordingly to R0 = E0 p0

)

1/ 3

; t0 = R0 v0 .

The similarity laws of hydrodynamics in micro- and macroscopic explosions suggest that microexplosion in sapphire (E0 = 10-7 J; ρ0 ~4 g/cm3; R0 = 1.5×10-5cm; t0 = 5.5×10-12s) is a reduced copy of

86 macroscopic explosion that produces the same pressure at the same initial density but with the energy deposition area size and time scales changed in accordance with the above formulae. For example, the energy of 1014 J (that is equivalent to 25,000 tons of high explosive or one nuclear bomb) released in a volume of 4 cubic meters (R0 = 1.59 m) during the time of 20μs exerts the same pressure of 12.5 TPa as the laser-induced micro-explosion in sapphire does. Thus, exactly the same physical phenomena occur at the scale 107 times different in space and in time, and 1021 times different in energy. Therefore, all major hydrodynamic aspects of powerful macroscopic explosion can be reproduced in the laboratory tabletop experiments with ultra-short laser pulses. 6.5. Summary Let us summarise the main conclusions of this Section on ultrafast laser-induced material modifications in confined geometry: •

In the conditions close but below the optical breakdown threshold the femtosecond laser pulse creates optically detectable changes in the refractive index. The modifications in refractive index are short-lived and transient. The short-lived modification occurs due to excitation of electrons of all constituent atoms. Permanent modification occurs in the doped sites due to the field of spontaneous polarisation.

•

Femtosecond laser pulse tightly focussed by high-NA optics leads to absorbed energy density in excess of the strength of any existing material. A void surrounded by a compressed shell is formed as a result of the confined micro-explosion in the focal volume.

•

Warm Dense Matter at the pressure exceeding TPa and the temperature more then 100,000 K is created in the table-top experiments, mimicking the conditions in the cores of stars and planets.

•

Confined micro-explosion studies open several broad avenues for research, such as formation of three-dimensional structures for applications in photonics, studies of new materials formation, and imitation the inter-planetary conditions at the laboratory tabletop.

7. INTERACTION OF ULTRAFAST NON-GAUSSIAN AND VECTOR PULSES WITH MATTER 7.1. Introduction In the previous sections we considered primarily ultrashort laser pulses with Gaussian intensity profiles. Perfect Gaussian beams have the lowest divergence when compared to any other cavity beam modes or profile shapes, and thus minimal spot size and largest Rayleigh length when focused. While Gaussian beams are most commonly used in laser-matter interaction studies, other beam intensity profiles like top-hat beams with spatially uniform intensity are widely used for material processing applications like photolithography and pulsed laser deposition – see Fig. 7.1 [181-185]. Non-Gaussian beam shapes have important benefits as they can be explicitly designed to meet the requirements of specific material configuration and illumination conditions, for example in drilling holes of precise diameter and with sharp edges, in pulsed laser deposition, or in laser-induced forward transfer technique (LIFT) [186-188]. It was demonstrated recently that ultrafast optical vortex beams are able to form highly reproducible sub-wavelength ring structures with the feature sizes of the order of 0.1λ on glass surface [189]. A considerable advance has been achieved in drilling high aspect ratio micro-channels with sub-micron diameter with a single shot Bessel beam [190]. These examples show the important and increasing role of ultrafast pulses with non-Gaussian intensity profiles in several laser micro-processing applications.

87 Hand in hand with designed intensity variation beams, a new class of cylindrical vector beams is emerging in the laser-matter interaction studies. Cylindrical vector beams are the class of light beams with spatially variant direction of polarisation, where polarisation states exhibit cylindrical symmetry (figure 7.2) [191,192]. They attracted a lot of attention recently, mainly due to the ability to shape focused laser spots in high-NA focusing optics. Studying vectorial structure of optical beams with axial symmetry in polarisation with sub-wavelength resolution will provide insights into the directionality of electric field and radiation pressure in the nanoscale domain. The complex polarisation structure is of interest because of the fast growing applications of such beams in optical trapping of particles [193-196], surface plasmon excitation for sensing [197], plasmonic nanolithography [198,192], micro-Raman spectroscopy, confocal and 3D imaging optical microscopy [199-202]. In this section we present a brief description of beams with non-Gaussian intensity distribution and spatially variant states of polarisation, demonstrate the ways to visualise complex polarisation structure in the focal spot of a tightly focused ultrashort laser pulse with sub-wavelength resolution, and discuss the problems to be resolved for understanding the interaction of femtosecond vector beams of high intensity with matter. 7.2. Non-Gaussian intensity profiles Laser pulses with non-Gaussian intensity profiles like flat top-hat, doughnut-shaped beams and Bessel beams have each distinctive propagation properties. An important characteristic of any beam propagation is the beam divergence θ, which determines the focal spot size W0: W0 =

2λ M 2

πθ

,

(7.1)

where M2 is a propagation constant characterising the difference to the ideal Gaussian beam, θ = M θ0 , W0 = M w0 , w0 and θ0 are respectively the minimum radius at e −2 level of intensity and total angle of divergence of the diffraction-limited Gaussian beam [203]. The beam diameter df at the focus of a lens with a numerical aperture NA, is: df =

⎛ ⎛θ ⎞ ⎛D 2λ M 2 ; NA = nsin ⎜ ⎟ = nsin⎜ arctan⎜ π NA ⎝2⎠ ⎝2 f ⎝

⎞⎞ nD ⎟⎟ ≅ ⎠⎠ 2 f

(7.2)

where n is a refractive index of the media (n = 1 for air; 1.453 for fused silica; 1.79 for sapphire; and 2.4 for diamond at 800 nm [204]), D is a lens aperture and f is a focal distance. The real nonGaussian beam will focus at the same location as the perfect Gaussian beam, but unlike Gaussian beams, the depth of focus, or Rayleigh length zR is shorter:

zR =

π W02 . λ M2

(7.3)

Beams with top-hat intensity distribution are widely used in material processing and in laser ablation as they maintain constant irradiation conditions across the beam profile, and thus even lasermatter interaction settings. Top-hat beams usually produced using various types of beam homogenisers or with specially designed refractive optical systems, so called ‘π-shaper’ [205,206]. They have M2 value much higher than 10, thus the focal spot is much larger. As a result, the energy concentration when focused with a lens and the depth of focus is much lower than with Gaussian beams. It should be noted that top-hat beams, unlike Gaussian ones, demonstrate their uniform intensity distribution only at the focus or at the imaging plane of the focusing system, the uniformity across the beam is not maintained as they propagate [185].

88

Figure 7.1. Intensity profiles (a) – of a Gaussian beam, (b) – top-hat beam, (c) – doughnut-shape vortex beam, and (d) – Bessel beam (the profile intensities are not in scale).

Bessel beams are the beams whose electric field distribution E(r,φ,z) is proportional to a zeroorder Bessel function J0 [207,208]:

(

)

( ) ( )

E r,ϕ , z = E0 exp ik z z J0 k r , r ;

(7.4)

where r and φ are transverse and polar coordinates, kr and kz are the wave vectors in longitudinal and transfer directions, and z is the coordinate in propagation direction. The beam shape has a central core with a series of concentric rings, like a ‘bull-eye’ pattern (figure 7.1d). A salient feature of the Bessel beams is that the intensity profile of the central core I(r,φ,z) is independent on z, propagating in a ‘non-diffracting’ manner:

(

) (

)

2

(

I r,ϕ , z ∝ E r,ϕ , z ; I r,ϕ , z

)

z≥0

( )

= I r,ϕ .

(7.5)

In addition to non-diffracting, Bessel beams have a remarkable ability to reconstruct itself if the central part of the beam is blocked. The outer rings of the Bessel beam ‘refill’ the central maximum, preventing it from spreading. If the central core is blocked by an object with radius rb, the outer rings replenish and restore the beam at some distance lsh, which is a length of a shadow of the block,

( )

(

lsh ≅ rb k k r , where k = k r2 + k z2

)

12

[209].

It has to be noted that finite apertures of optical elements form the Bessel beams with finite dimensions. The experimentally close approximation to the ideal Bessel beams, the quasi-Bessel or Bessel-Gauss beams formed in laboratory experiments, have the central non-diffracting maximum over a limited propagation distance. The intensity in the central axis decreases with distance and eventually goes to zero converting into a ring-shaped beam. The Rayleigh range for a Bessel beam can be approximated as [207]: z rBessel ≅

π D df 4λ

(7.6)

89 Comparison Eq.(7.6) with Eq.(7.3) shows that the Bessel beams exhibit much longer Rayleigh length as D >> df. When a Gaussian beam with intensity I0 and beam waist w0 illuminates an axicon with angle α, the resulted intensity distribution has a form [210,211,185]:

( )

⎛ 2z n −1 tan α ⎞ 2 ⎟ J02 k r n −1 tan α . I r, z = 2 π k z tan2 α n −1 I 0 exp⎜ − ⎜ ⎟ w0 ⎝ ⎠

( )

(

)( )

( ( ) )

(7.7)

With the help of Eq.(7.7), the central core expresses as: df =

2.405 . k(n −1) tan α

(7.8)

Substituting Eq.(7.8) into Eq.(7.6) one obtains the Rayleigh length in the form [208]: z rBessel ≅

D . 2 n −1 tan α

(7.9)

( )

In practice, the Bessel beams can be formed by a number of techniques generating a cone-like propagation of wave vectors: with a circular slit [207,208], with a conical-shaped axicon [209], or by computer-generated hologram techniques using spatial light modulators (SLMs) [213]. While the amplitude-modulating circular slit method is the simplest among all, the high energy loss due to blocking the central part of the initial Gaussian beam makes it inapplicable for material processing. Alternatively, the spatial phase shaping axicon and SLM methods are most suitable for femtosecond laser ablation applications. In particular, variable spatial shaping with SLM adds the flexibility of a constant or tuneable intensity distribution over the propagation of Bessel-Gaussian beam in the media with specific refractive index [214]. The non-diverging central maximum of Bessel-Gaussian beams is very attractive for applications that require large depth of constant beam parameters such as material processing of highly uneven surfaces for example. They are becoming increasingly important in 3-D imaging applications [202]. The conical geometry of energy delivery from the rings to the central lobe offers high resistance to filamentation due to nonlinear Kerr effect when compared with Gaussian beams [215]. The robustness of Bessel beams to the presence of obstacles and self-healing ability makes them very attractive for laser processing of dielectrics with powerful femtosecond pulses. The most spectacular demonstration of the unique ability of Bessel beams is drilling sub-wavelength 200-nm channels with >100 aspect ratio by a single shot of 230-fs 800 nm pulse in glass slides [190,216]. An annular beams or doughnut-shaped vortex beams are optical beams with helical shape phase fronts and ring-like intensity distribution with zero intensity on the axis due to phase singularity. To create optical vortex a phase shift of the form exp ilφ is applied to an incident laser beam where is

( )

the azimuthal coordinate about the beam centre in the transverse plane. The beam centre is a point in an optical field around which the phase of the field changes by an integer l multiple of 2π; this integer is called topological charge, it represents the number of rotations of the optical field while propagating the distance of one wavelength [217]. The Poynting vector in optical vortex has an azimuthal component, which results in angular momentum along the beam axis [218]. Spatial intensity distribution I in vortex beams is described as the follows [189]:

90

(

⎞ ⎟ 2 ⎟ ⎟ ⎠

( ) P W (7.10) [] (w ξ ) π w ), l = 0, ±1; ±2; ... is a topological charge, r is radial (transverse) coordinate,

⎡ W ⎤ 2 l +1 r 2 l I ⎢ 2 ⎥= l!π ⎣ cm ⎦ where ξ = 1+ i zλ

⎛ ⎜ 2r 2 exp⎜ − ⎜ w0 ξ ⎝

2( l +1)

tot

0

2 0

w0 is a waist of a Gaussian beam at l = 0 (both r and w0 in cm), and Ptot is a total power in the beam. The intensity profiles for the beams with l = 0, 1, and 2 are presented in Fig. 7.2. Transformation of the Gauss beam into the vortex beams with different topological charges leads to decrease in the maximum intensity while the total energy per pulse remains conserved: l =±2 l = ±1 l =0 (Gauss ) : I max : I max = I max

2 1 : :1 = 0.271: 0.368 :1 e2 e

Figure 7.2. Intensity distribution in a Gaussian beam (l = 0) and in vortices with topological charges l = 1 and l = 2. The r-scale is in the w0-units for the Gaussian beam; the total power is the same in all three beams.

Rapid development of singular optics in recent years offered several methods to produce vortex beams using holographic displays in SLMs and diffractive optical elements. The most effective for generating high-power femtosecond vortex pulses are spiral phase plates [219], spiral phase mirrors [220] as they all offer high efficiency when compared with fork-like holograms and diffractive optics, and uniaxial birefringent crystals [221], all with much higher damage thresholds. Birefringent crystals in particular produce high-intensity ultrafast vortex pulses without spectral broadening, thus they do not require correcting elements to compensate for topological charge dispersion as they are inherently polychromatic. The advantages of application of optical vortex pulses for material processing were recently demonstrated by producing regular arrays of micron-size rings with down to λ/10 structures on a glass surface using fs-pulses just above the ablation threshold fluence [171]. Another interesting application was demonstrated in revealing polarisation structure of tightly focused fs-pulses with nanoscale resolution [222]. Visualisation of the electric field in a spatially variant azimuthal and radial polarisation states, including its longitudinal z-component, was achieved by irradiation of glass surface by a single pulse with sub-threshold intensity. We are reviewing the complex polarisation structure of the cylindrical vector beams in the next section.

91 7.3. Cylindrical vector beams Cylindrical vector beams are the class of laser beams that are described by a solution to vector Helmholtz equation with cylindrical boundary conditions [191,192]. In contrast to paraxial solutions, polarisation direction in vector beams is spatially varies, while the amplitude of the field possess azimuthal symmetry. For the lowest azimuthal order modes the electric field makes a constant angle 0 with the beam radius, with the limiting cases 0 = 0° and 0 = 90° of correspondingly radial and azimuthal polarisation states (figure 7.3).

Figure 7.3. Cylindrical vector beams with spatially variant polarisations states; the arrows indicate the direction of oscillating electric field. (a) – radially polarised, 0 = 0°; (b) – azimuthally polarised, 0

= 90°; (c) – linear superposition of radially and azimuthally polarisation states, 0° <

0

< 90°.

The direction of polarisation affects intensity distribution in the focus of high-NA optics. Indeed, while the doughnut-shape intensity distribution is identical in the near-field of cylindrical vector beams, in the focus of high-NA optics the beams with different cylindrically symmetric polarisation states behave very differently. The azimuthally polarised beam preserves its doughnut intensity profile with zero intensity on the axis, whereas the radially polarised beam has intensity maximum due to a strong longitudinally polarised component, a z-component of the electric field [191]. The transverse radius of focal spot of the radially polarised beam is smaller than the focal spot of the Gaussian beam. It is termed as a ‘needle beam’, as it has sub-wavelength FWHM, and ~10 aspect ratio [223]. The axial component of the Poynting vector in such a focus is zero, this distinctive attribute makes such beams attractive for trapping metallic nanoparticles as the destabilising radiation pressure is absent in the azimuthal direction [224,225]. Moreover, it was shown that the polarisation state of cylindrical vector beams provides an extra degree of freedom as the optical strength and aspect ratio of the trap can be controlled by the polarisation angle only [225]. Beams with spatially variant polarisation states can be produced in a variety of ways [221,226,227]. One of the ways is to use birefringent crystals and focus the beam along the optical axis. The radial and azimuthal polarisations focus in different planes and can be selected by a spatial filter (see Fig. 7.4). An additional double half-wave plate offers a flexibility to form a cylindrical vector beam with the particular state of polarisation. The main advantage of using a birefringent crystal is a high damage threshold, which is important for studies of interaction of powerful fs-pulses with various polarisation states with matter and their applications.

92

Figure 7.4. Generation of cylindrical vector beams. Calcite birefringent crystal with a spatial filter acts as a radially symmetric analyser, while the double half-wave plates rotate linearly polarisation independent on the incident spatially variant polarisation angle [221,226,227]. Below are SEM images of the revealed replicas of longitudinal (z-component) and transverse (radial) components of polarisation obtained in the focus of NA=0.9 microscope objective irradiating a glass surface with 775-nm 200-fs single pulse at the intensity near the ablation threshold [222]. Note the size of ~0.1λ of the ablated crater produced by the beam possessing a longitudinal z-component of the field.

Comprehensive studies established several applications of the unique vectorial focal distribution using polarisation state as an extra degree of freedom, both with reflective and refractive optics. These include confocal and two-photon microscopy [228], dark-field imaging [229], molecular orientation [230,231], and micro-ellipsometry [226]. Space variant polarization can be used to shape focused laser spots. By tuning the angle of polarisation, the relative intensity contribution from transversal and longitudinal components can be adjusted continuously. It was shown that focal area engineering with a broad range of intensity profiles, both in transverse and longitudinal directions, can be performed by proper control of spatial polarisation orientation in vector beams [227]. The intensity pattern from doughnut-shape to top-hat to needle-shape can be obtained in the focus using generalized cylindrical vector beams by balancing the strong z-component and transverse rcomponent with a two half-wave-plate polarisation rotator. For example, flat-top intensity profile highly independent from numerical aperture can be achieved for NA = 0.8 at the angle 0 = 24° [227]. In the next session we will discuss the polarisation-related specific peculiarities in interaction of highintensity ultrafast vector pulses with matter. 7.4. Interaction of ultrafast sculptured pulses with matter Coupling of electromagnetic laser energy into any matter states – solid, liquid, or plasma, depends on intensity, wavelength, angle of incidence, and on the state of polarisation of the incident beam. The polarisation dependence of absorption is described by Fresnel equations which link s- and pcomponent of polarisation states for a particular angle of incidence with material optical property through the refractive index [280].

93 a). Interaction with metals. By definition, the short pulse interaction occurs when the hydrodynamic motion is negligible and the vacuum-solid boundary maintains a perfect density step. In this case the interaction falls in the frame of the normal skin effect at the non-relativistic intensity. The major mechanism of absorption is the inverse Bremsstrahlung absorption, which accounts for electron collisions with ions (see Section 5.3 above). On quantum mechanical language this process is described as absorption of a photon by an electron in the presence of the third body (an atom or an ion). Classically, this process is described as the energy gain by the electron oscillating in high frequency external field through collisions. Both approaches gave similar results. There is no apparent dependence of the absorption on the photon’s quantum state. In s-polarised beam the electric field is perpendicular to the plane of the incidence, while at p-polarisation it lies in the plane of incidence, thus at normal incidence absorption of both polarisation states are identical. The polarisation dependence reveals itself in the case of oblique incidence of the beam. The influence of polarisation might be two-fold: (i) due to angular dependence given by the Fresnel formulas, and (ii) by change of the field spatial distribution inside a solid. In s-polarisation the angular dependence induces change only trough the Fresnel formula for absorption, while the electric field component parallel to the target surface remains the same at any angle. In p-polarisation the ratio of field components parallel and perpendicular to the target surface changes with the angle. b). Interaction with dielectrics. The important effect directly related to the quantum properties of photon is strong dependence of multiphoton ionisation rate on polarisation ([232] and references therein). It was noted that the ionisation rate for circular polarisation is several orders of magnitude lower than those for linear polarisation. The explanation of this effect directly relates to the angular momentum. The selection rule for angular momentum dictates that for each absorbed photon the magnetic quantum number has to increase by one unit (in the Plank constant). The ground state usually has low angular momentum. This implies that after absorption of many photons the angular momentum may become very large and electron is spatially farther away from the nucleus. Thus the final state separates from the origin by a wider and stronger barrier. That causes poor overlap of the wave functions of electron in the final ionised and the ground state, which results in lower ionisation cross section. The experimental results in earlier studies were, however, more controversial – see a discussion in [71]. The first clear evidence for dominance of circular over linear polarisation was demonstrated in the experiments on multi-photon ionisation of Na in the 8.5 GHz microwave field [233]. The experiments of Temnov et al. [71] on multi-photon ionisation of sapphire and fused silica induced by 50 fs, 800 nm powerful laser pulses clearly demonstrated the difference between the circular and linear polarisation states, in qualitative agreement with the theoretical predictions. The six-photon ionization rate in both materials was found to be significantly higher for linear polarization of the pump pulses. The ratio of cross-sections for circular to linear polarization in fused silica is σ Sicirc σ Silin circ lin σ sapph = 0.15. = 0.27, while in sapphire it is σ sapph

However, the difference in the intensity

breakdown threshold for both polarisations is in a range of 15%-23%. The authors’ explanation for this difference relates to the fact that the breakdown threshold is associated with achieving the critical density of conductivity electrons. The intensity for achieving the given number density of electrons

( )

expresses through the six-photon cross section as I ∝ σ 6

(

estimated as I6lin / I6circ = σ 6lin σ 6circ

)

1/ 6

1/ 6

. Therefore, the ratio of thresholds

is in qualitative agreement with the experiments.

94 The dependence of the impact ionisation (avalanche) rate on polarisation is straightforward. Indeed, this rate is proportional to the electron oscillation energy and absorption. Dependence of absorption on polarisation is presented in the Fresnel formulae. The oscillation energy reads ε osc = 1+ α 2 e2 E2 4π ne ω 2 ; here α = 0 stands for the linear polarisation and α = 1 for the circular

( )

polarisation of light. c) Modification in high-NA focusing. It was demonstrated in several papers that in the focus of highNA objective, where paraxial approximation fails, the internal structure of the beam drastically changes. Neminen et al., [234] claimed that strong focussing even converts the circularly polarized Gaussian beam into the longitudinal vortex possessing the topological phase, phase singularity and axial component of the electric field. Bomzon et al., [235] showed that the energy distribution in the focal plane of a beam focused with high-NA lens possesses a shape, which depends on polarization. A focal spot from circularly polarized light has a circular symmetry in contrast to the asymmetric distribution observed when linearly polarized light is focused with the lens of NA = 0.95. Focusing a beam through a high-NA objective also causes space-varying polarization and depolarization. They found the threshold at which the increase in NA starts changing the amount of the geometrical phase in the beam. The ratio between the angular momentum and the energy of the entire beam might be considered as a portion of the beam possessing the Berry phase. This ratio is almost constant when the NA is smaller than 0.2, after which it monotonically increases to a value of about 1.4 at NA = 0.95. The authors explained this apparent paradox showing that the lens acts as a low pass filter. The lens only transmits spatial frequencies lower than a cut-off wave number. The high spatial frequencies carry more angular momentum, related to the geometrical phase, per unit energy than the low spatial frequencies. The cut-off wave number increases with NA. Therefore increasing the aperture and hence the NA inevitably increases the amount of optical angular momentum (OAM) per unit energy in the focal region. 7.5. Interaction of ultrafast vector pulse with matter In tightly focussed vector beam with spatially variant polarisation states the polarisation along the beam propagation normal to the target surface becomes essential. There is an additional absorption mechanism for ultra-short powerful pulse, so called vacuum absorption, or Brunel effect, that is important at and above the intensity level 1016 W/cm2. Efficient absorption was predicted and verified by experiments with intense beam having a field component perpendicular to the surface of a metal target [236]. The absorption occurs when an oscillating electron is dragged into the vacuum and then sent back by the oscillation force and loosing its energy in collisions with the target material. The electron can escape the target if the amplitude of oscillations, vosc/ω, is large enough making the absorption coefficient, A ≈ vosc/c, sufficiently large to contribute in the total absorption; here vosc = eE mω , e and m the charge and mass of electron, E, ω are the electric field and frequency of the laser field. For lasers in visible range the absorption by this mechanism dominates (A ~ 1) at the intensity approaching the relativistic limit. A remarkable feature of the vortex beam is that it possesses the spatial-dependent topological phase, the Berry phase. Let us discuss what are microscopic (or, quantum mechanical) and macroscopic implications of the topological phase on the laser-matter interaction.

95 In quantum language the topological phase is an addition to the conventional dynamic phase of the wave function of a photon, which arises when the photon performs a motion along the non-planar loop with helical or spiral trajectory. In the classical language the Berry phase of plane polarized photon beam is directly connected to the angle of plane of polarization rotation, similar to that occurring under the action of imposed magnetic field (Faraday and Cotton-Mutton effects) or polarization rotation in anisotropic medium. Therefore the interaction of such a photon with electron differs from that for photon not possessing the Berry phase simply by direction of polarization. Similarly, in the classical language, the vector potential for the photon beam possessing the Berry phase has an additional phase multiplier formally equivalent to the orbital momentum for a massive particle. Recently the problem of separation orbital angular momentum and spin momentum for the vortex beam are broadly discussed in literature. Let us discuss briefly the implication of the beam angular momentum on the laser-matter interactions. The Berry phase is gauge invariant. The gauge invariance leads to two conclusions: first, that the relativistic photon is a massless particle; and second, is the condition of the transversality of the wave k • A(k ) = 0, which forbids the separation of total momentum of photon for orbital angular momentum (OAM) and spin angular momentum (SAM) components. In other words, these conditions imply that it is impossible defining the orbital momentum of a massless relativistic particle as a local function of space coordinates because such a particle (photon) cannot be localized due to uncertainty principle. Thus, local separation of OAM and SAM for a single photon is impossible [237]. Many authors have made the heuristic separation of OAM and SAM for the photon beam in the form of coordinate-space integrals. Iwo Bialynicki-Birula and Zofia Bialynicka-Birula [238] have shown that the orbital angular momentum and the spin parts cannot be expressed as integrals of local densities: they are intrinsically macroscopic nonlocal objects and therefore they are characteristics for a beam and not for a single photon. Such separation appears to be useful for description of vortex beam propagation through different media and in particular through optical elements and lenses. In laser-matter interaction at the high intensity a conversion of the energy of the incident electric field into the internal energy of absorbing substance is the main process of the interaction resulting in various transformations of laser-affected material. The absorption process consists of energy and momentum transfer from the laser electric field to electrons at any spatial point of the focal spot. Therefore if the electric field (or more precisely, the vector potential) in any spatial point is defined then the initial conditions for quantum mechanical problem of laser-matter interaction are known. The major difference in the microscopic physics of the vortex beam-matter interaction is expected from the presence of the geometrical phase introduced as a new feature of the vector-potential in any spatial point. However as it was shown the topological phase results in the plane of polarization rotation. Therefore, if the changes in the spatial distributions of intensity and polarization during the focusing process can be accounted for, the problem of vortex beam-matter interaction reduces to the known scenario. The spatial distribution and time dependence of the intensity and polarization introduce only the quantitative changes in comparison to a well-studied problem of the Gaussian beam-matter interaction [14]. 8. CONCLUDING REMARKS AND FUTURE DIRECTIONS Summing up the major results of the studies covered by this review we conclude that the major concepts of laser-matter interaction at non-relativistic intensities with sub-picosecond pulse duration shorter than the main relaxation times are well established, and in full agreement with the previous

96 experimental and theoretical studies. Studies of ultra-fast laser excitation of coherent phonons, laser superheating without melting, and laser ablation rather than equilibrium evaporation conducted in the last two decades have all demonstrated significant progress in understanding and control of ultra-fast laser-induced processes, from a delicate atomic motion to complete ionisation and ablation of the material. The major concepts presented in this review describe the time sequence of the most important phenomena during and after the interaction of ultra-short laser pulses with matter. The prime process is the absorption of the laser light by electrons, while atoms in a lattice remain at their initial unperturbed state during the pulse. The distinctive characteristic of ultra-short laser-matter interaction is that the local statistical distributions in the electron and in the lattice sub-systems are established in two stages. In the first stage, both sub-systems have different temperatures, and then the exchange of momentum and energy via electron-phonon interaction with temperature-dependent rates gradually leads to equilibration of temperatures. The longest of all the relaxation phenomena is the process of building up the high-energy tail in the Maxwell-Boltzmann distribution. These concepts are successively embedded into the qualitative and quantitative description of the sequence of major phenomena in this review. They form a sound basis for future studies in search of a deeper understanding of phenomena in shorter time and smaller space scales. Ultra-fast lasers enable many new applications in medicine and biomedical applications (corrective eye surgery, two- and three-photon imaging), precision micro-processing of materials, and all-optical signal processing in telecommunications, which can only work with ultrafast pulses [154,239,240]. The general aim of future studies might be formulated as to achieve the ability to generate, observe and control material transformation phenomena at the atomic space and time scales, i.e. at Angstrom and sub-femtosecond resolution. Clearly, further progress towards this depends on progress in ultrafast laser development, sophisticated diagnostic tools, and in further theoretical development. We briefly outline some emerging problems and areas of research, which are expected to shape the field of ultra-short laser-matter interaction in the coming years. 8.1. Ultra-short lasers 8.1.1. Solid-state laser systems Rapid progress in ultra-fast lasers is expected in further shortening of the pulse length, broadening of the spectral range of available photon energy into deep infra-red, UV, and x-ray spectral ranges, and scaling the energy per pulse above the 100-µJ level with increased repetition rate into the GHz domain, subsequently increasing the average power to the kW level and above [239,241]. The invention of Chirped Pulse Amplification technology (CPA and OPCPA) [240], creation of Kerr-lens mode-locking [241] and development of the semiconductor saturable absorber mirror (SESAM) [242] made it possible to produce pulses of a few light cycles with intense Ti:sapphire systems [243,244] and with high average power and broad-range variable repetition rates from 150 kHz to 28 MHz [245,246]. The average output power has been pushed beyond the 100-W level for the first time with a SESAM modelocked Yb:Lu2O3 thin-disk laser oscillator generating 141 W, with 738 fs pulses, at 60 MHz repetition rate [247,248], and the 100-GHz repetition rate barrier was overcome with 1.5 µm 1.6 ps pulses [249]. The adaptive optics and spatial light modulators make the top-hat intensity distribution over the focal spot area achievable, and assures that the focal spot size is close to the diffraction limit.

97 8.1.2. Attosecond lasers While atomic and molecular vibrations occur on a time-scale of tens to hundreds of femtoseconds, the motion of individual electrons in molecular orbitals and the inner shells of atoms occurs in the sub-femtosecond time domain. The characteristic length scale is comparable to interatomic distance, and the period of plasma oscillations is of the order of 0.1 fs. In situ observation of a nanometer-scale excited area with spatial and temporal resolution is necessary for understanding and control over the atomic excitation and ionisation processes. Generation of intense EUV laser pulses of attosecond-range pulse duration (1 as = 10-18 s) approaching single optical cycle limit with high signal-to-noise ratio will undoubtedly be available as a pump-probe tool in the near future [248, 250-252]. Real-time observation of valence electron motion was already achieved with 24 as resolution using a streaking technique with XUV (~80 eV), 80-as pulses [253]. Attosecond pulse interferometry makes it possible to directly observe chemical reactions in real time [254]. Recent developments in powerful femtosecond fiber laser amplifier systems are expected to break the 1-kW power barrier [255], and the first MHz system generating high-order harmonics has already been demonstrated [256]. The development of high-average power, high repetition-rate, compact, tabletop ultrafast VUV/XUV sources will enable new measurements in photoelectron imaging spectroscopy, surface science, metrology, and biology. One may expect that attosecond VUV/XUV pulses will become viable diagnostic probes to study the excitation and post-excitation processes in solids. We refer our readers to an excellent review [257] for a comprehensive introduction to the exciting new area of research on attosecond physics, which emerged at the beginning of the millennium. 8.1.3. Free electron lasers Relativistic electron beams in free electron lasers (FEL) act as an active medium where free-tofree energy transition leads to self-amplified spontaneous emission. The beam is passed through a periodic array of magnets – the undulator – and the electrons, while wiggling in the magnetic field, start emitting photons. A resonant frequency occurs due to the difference between the geometric path length of wiggling relativistic electrons and a straight-line path of their x-ray radiation, the electrons are bunched at a radiation wavelength down to 1 Å. Bunching of the elctrons results in the increased coherency of their radiation. For this reason FEL does not need a resonator cavity, unlike a conventional laser, and is particularly well-suited to generating short-wavelength X-rays. Recent progress in the development of extreme ultraviolet and X-ray FEL has demonstrated the capability to produce high-brightness X-ray pulses of femtosecond duration, with the potential to achieve attosecond durations [258,259]. With femtosecond pulses and peak brightness that exceeds the brightest synchrotron sources by ten orders of magnitude, the XFEL naturally lends itself to timeresolved studies on the nanoscale [260-262]. Many scientific studies will benefit from this new capability, such as phase transitions in magnetic materials, electron dynamics in chemical systems, which occur on femto- and attosecond time-scales, and studies of correlated solid-state phenomena, which require angular-resolved photoemission with both spatial and temporal resolution. 8.1.4. “Sculptured” laser beams Intensity distribution across the conventional laser beam has a Gauss-like form. In 1974, Nye and Berry considered basic properties of dislocations in wavefronts and introduced a so-called vortex beam, where the light propagates with its phase twisted around the beam axis [263,264]. The phase

98 on the axis of such a twisted beam is indeterminable, which implies that the intensity at the axis is equal to zero. The intensity distribution across the beam has a doughnut-like shape with a dark hole in the centre. The spiral phase of vortex beams carries spin and orbital angular momentum, which introduce torque on an electric dipole and can be observed in the orbiting motion of trapped particles in optical tweezers [265-267]. Recent development of efficient beam converters generating powerful femtosecond optical vortex pulses [221] offers new opportunities for studies of laser-matter interaction with such beams and formation of specific micro-and nano-structures. 8.2. Theory and Computer Modelling The ultra-short laser-matter interaction with a solid surface generates a chain of interconnected non-equilibrium processes, which in turn change the transient optical properties of laser-affected material, thus influencing the interaction mode. Comprehensive computer modelling of these processes should include the set of Maxwell equations coupled with the material equations, as a compulsory element (such as kinetic equations, those for the density matrix, molecular dynamics simulation etc.). Detailed studies of time evolution in distribution functions from the non-equilibrium excited state to their equilibrium form are required for understanding and control over the interconnected phenomena in the ultra-fast laser-induced transformations of the material. It should be stressed that at high intensity the ultra-short pulse removes (ablates) a material in completely nonequilibrium conditions, where traditional hydrodynamics is inapplicable. A good example of selfconsistent formulation of complex multi-disciplinary problems for computer simulation is the code LASNEX developed for laser-fusion problems [268]. There are several physical problems related to light beams tightly focused deep within a transparent solid, which need further theoretical and computational studies. The most general one is on the propagation of intense ionizing laser beam through the initially transparent medium. It is still waiting for the full numerical solution. The ionising beam gradually transforms the transparent medium to absorbing and reflecting plasma, finally stopping the beam propagation. To resolve such a problem the set of coupled Maxwell and material equations, which includes the whole variety of physical, optical, and transport processes, should be computed in 3D space coordinate and in time. The action of the field on the medium and the respond of the medium on the laser field should be taken into account. It is still formidable problem even for modern supercomputers. The problem of shock-wave generation by swiftly excited electrons in hot, short-lived, and multicomponent solid-density plasma created by a confined micro-explosion inside a solid appears to be quite different from conventional hydrodynamic shocks. Experiments revealed that unexpected spatial separation of ions with different masses occurs during the short time of shock-wave generation and decay [269,270]. This effect is the evidence of the complicated structure of the shock-wave front smeared by the ion diffusion, which evolves during the shock propagation. Thus, it seems that kinetic calculations of electrons coupling with different ion species are necessary in order to understand the complicated structure of the emerging shock wave. 8.3. Controlled excitation of phonons and coherent sound waves The excitation of phonon modes in different solids and observation in time of their evolution until the decay of the phonons has been performed in many laboratories. However, selective and controlled excitation of the desirable phonon mode from 3n of all available modes (n is the number of atoms in the primitive cell) remains difficult. If it were possible to excite and control a specific

99 phonon mode, then a range of possibilities would look viable. We mention some suggestions based on experiments performed, and on theory. With the help of molecular dynamics simulations, Dumitrica et al. [271] found that selective excitation of specific phonon modes by femtosecond laser could lead to the opening of the carbon nanotube cap. Ultra-fast bond-weakening and simultaneous excitation of two coherent phonon modes of different frequencies localized in the spherical cap and cylindrical body of a carbon nanotube might be responsible for the opening of the cap in non-equilibrium conditions. Stimulated emission of terahertz phonons in a super-lattice under vertical electron transport has been observed by Kent et al. [272]. The authors suggested that such a super-lattice might form the basis of a SASER device (Sound Amplification by Stimulated Emission of Radiation) – creating a directed flow of coherent phonons and generating coherent sound waves, equivalent to a laser generating electromagnetic waves. One may conjecture that excitation of resonant phonons by short laser may also lead to a similar device. Electron-phonon interaction and electron and phonon spectra in general are responsible for the conductivity and optical properties of solids. There were experiments using two pump pulses, which excited the same phonon mode. It is possible, in principle, to excite different phonon modes simultaneously by multiple pumps thus creating a particular phonon spectrum. Hence, there remains the challenge of pursuing changes in conductivity and optical properties by selective excitation of a particular phonon spectrum. 8.4. Control over chemical reaction-rates and transient phase states Transient phases produced by ultra-short pulses were observed in different laser-excited materials and some were identified. For example, Collet et al. [273] show that a 300-fs laser pulse transforms charge-transfer molecular material from a paraelectric to a ferroelectric state, which was completed in 500 ps after the pulse. X-ray probing revealed macroscopic ferroelectric reorganisation with longrange structural order to be fully established. The recovery of the equilibrium state occurs in 1 ms. Ultra-short pulses produce transient phase transformations in Gallium [68,274] and in Bismuth [43,44,275] at the absorbed energy-density in excess of the equilibrium enthalpy of melting into a state different from the equilibrium liquid of a corresponding solid, or any other known phases of these materials. The phase transitions are reversible: solids recovered to their initial state in several nanoseconds after the excitation. However, the identification of these transient states is still elusive. There are some results, which suggest that control over the phonon modes and their coupling will allow one to control the chemical reaction rates [276]. Hillebrand et al. [277] found the strong enhancement of the near-field resonant coupling in the infrared by phonons in polar dielectrics (SiC). This coupling proved to be very sensitive to the chemical and structural composition of samples, permitting nanometre-scale analysis of minerals and semiconductors. When a femtosecond pulse shorter than any phonon period hits a crystal, non-equilibrium phonons and a lattice distortion are induced. Williams [278] suggested that the controlled distortion would then initiate metal-insulation transition and an accompanying charge-density wave. The challenge for future studies is to identify the new transient phases and find methods to preserve them in a meta-stable state. 8.5. Warm Dense Matter on the tabletop: mimicking conditions in stars and creating new materials

100 It was demonstrated that tight focusing of a conventional tabletop laser inside the bulk of a transparent solid creates pressure exceeding the strength of any material, and the shock wave compresses a solid, which afterwards remains confined inside a crystal. High pressure and temperature are necessary to produce super-dense, super-hard and super-strong phases or materials, which may possess other unusual properties, such as ionic conductivity. In nature, such conditions are created in the cores of planets and stars. Extreme pressure was recreated by strong explosions, by diamond anvil cell presses and with powerful ns-lasers. All these methods were cumbersome and expensive. By contrast, ultra-short lasers create extreme pressure and temperature along with record high heating and cooling rates by focusing 100 – 200 nanoJoules of conventional femtosecond laser pulse into a sub-micron volume confined inside a solid [279]. Recently, the crystalline phase of aluminium, bcc-Al has been discovered in ultra-fast laser-induced micro-explosions [269]. These results open the possibilities of formation of new high-pressure phases and prospects of modelling in the laboratory the conditions in the cores of planets and macro-explosions. The first results might be considered a proof-of-principles step. However, it is obvious that, with this simple and inexpensive method for creation of extreme pressure/temperature, the focus in research is shifted to post-mortem diagnosis of shock-compressed material. Micro-Raman, x-ray and electron diffraction, and AFM and STM studies with resolution on the sub-micron level are needed for identification of the new phases. Another challenge is to develop a pump-probe technique with time resolution capable if in situ observation of shock-front propagation inside a crystal. Summing up, the prospects for the fundamental study of ultra-fast laser-matter interaction and its technological applications look extremely encouraging. As the technology becomes smaller, less expensive, more robust, less power-hungry and more energy-efficient, it allows the increased exploitation of ultrafast phenomena, ultimately entering our everyday lives. 9. ACKNOWLEDGEMENTS We are indebted to my colleagues with whom we have been working together on several projects included into this review, and who helped us to shape understanding many phenomena involved. We would like to acknowledge Barry Luther-Davies, Saulius Juodkazis, Vladimir Tikhonchuk, Wieslaw Krolikowski and the late Lewis Chadderton for many enlightening discussions. This research was supported under Australian Research Council's Discovery Project funding scheme (project number DP120102980). Partial support to this work by Air Force Office of Scientific Research, USA (FA9550-12-1-0482) is gratefully acknowledged.

101

APPENDIX A: Maxwell’s equations

Si units

∇ ×E = −

Faraday’s law Ampere’s law

∇×H=−

Gaussian units

∂B ∂t

∇ ×E = −

∂D +J ∂t

∇×H=−

1 ∂B c ∂t

1 ∂ D 4π + J c ∂t c

Gauss’s law

∇ ⋅E = ρ

∇ ⋅ E = 4π ρ

Gauss’s law for magnetism

∇ ⋅B = 0 D =εE B=μH

∇ ⋅B = 0 D =εE B=μH

Material constitutive relations

APPENDIX B: Dimensions, units and conversion factors [281]

Physical parameter Length; l Time; t Velocity; v Mass; m Density; ρ Force Energy; E Power; W

Si units

Conversion factor

Gaussian units

meter (m) second (s)

102 1

centimeter (cm) second (s)

m s kilogram (kg) kg

102

cm s gram (g)

103

g

m3

10-3

newton (n)

105

Joule (J)

107

erg (

107

erg g cm2 = 3 s s g1/ 2 cm3 / 2 s

watt ( W =

J ) s

cm3 g cm dyne ( 2 ) s

g cm2 s2

Charge; q

coulomb

3×109

Current; i

q ampere ( i = ) t

3×109

volt m

1 ×10 −4 3

ampere m

4π×10-3

oersted (

tesla

104

gauss (

Electric field; E Magnetic field; H Magnetic induction; B

)

g1/ 2 cm3 / 2 s2 g1/ 2 cm1/ 2 s g1/ 2 cm1/ 2 s g1/ 2 cm1/ 2 s

)

)

102 APPENDIX C: Constants [281]

Physical parameter

Symbol

Value, Si units

Value, Gaussian units

Elementary charge

e

1.6022×10-19 C

4.8032×10-10

Electron mass Atomic mass unit

me ma h h c

9.1094×10-31 kg 1.6605×10-27 kg h = 6.6261×10-34 J s h = h/2π = 1.0546×10-34 J s 2.9979×108 m/s

ε0

8.8542×10-12 F/m

1

μ0

4π×10-7 H/m

1

a0

5.2918×10-11 m

5.2918×10-9 cm

π a02 kB

8.7974×10-21 m2

8.7974×10-17 cm2

1.3807×10-23 J/K

1.3807×10-16 erg/K

2.4180×1014 Hz

2.4180×1014 Hz

1.6022×10-19 J

1.6022×10-12 erg

1.1604×104 K

1.1604×104 K

1.2398×10-6 m

1.2398×10-4 cm

6.0221×1023 1/mol

6.0221×1023 1/mol

Planck constant Speed of light Permittivity of free space Permeability of free space Bohr radius Atomic cross section Boltzmann constant Frequency associated with 1 eV Energy associated with 1 eV Temperature associated with 1 eV Wavelength associated with 1 eV Avogadro number

NA

g1/ 2 cm3 / 2 s 9.1094×10-28 g 1.6605×10-24 g 6.6261×10-27 erg s 1.0546×10-27 erg s 2.9979×1010 cm/s

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