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Polarization properties of high harmonics generated on solid surfaces
1 November 2001
Optics Communications 198 (2001) 419±431
www.elsevier.com/locate/optcom
Polarization properties of high harmonics generated on solid surfaces K. G al a,b,*, S. Varr oc a c
KFKI-Research Institute for Particle and Nuclear Physics, P.O. Box 49, H-1525 Budapest, Hungary b Department of Experimental Physics, University of Szeged, 6720 Szeged, Hungary KFKI-Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary Received 1 June 2001; received in revised form 14 August 2001; accepted 21 August 2001
Abstract A detailed study of the electromagnetic ®eld's con®guration inside plasmas generated on solid surfaces by ultrashort laser pulses is given in the present work. The radiation due to the nonrelativistic motion of the electrons in the overdense region is considered the main source of high harmonics, whose polarization dependence on the fundamental beam is treated to explain the experimentally observed polarization properties of the harmonics. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 42.65.Ky; 52.25.Qt; 52.50.Jm Keywords: Harmonic generation; Plasma production
1. Introduction Several laboratories developed lasers capable of delivering very short pulses t < 1 ps with high intensities I > 1016 W/cm2 . High order harmonics were observed, resulting from these interactions of laser beams with solid targets. It was recently observed by using ultrashort laser pulses that even and odd harmonics occur for both laser polariza-
* Corresponding author. Address: KFKI-Research Institute for Particle and Nuclear Physics, P.O. Box 49, H-1525 Budapest, Hungary. Tel.: +36-1-3922222/1467; fax: +36-13959151. E-mail addresses: [email protected] (K. Gal), [email protected] (S. Varr o).
tion [1±6]. Some theories [6±9] were developed to explain the polarization dependence of harmonic generation. One of the theories referring to the polarization dependence of high harmonics generated on solid surfaces have been developed by von der Linde [7] using a perturbative treatment. This theory consider as the source of the high harmonics the coupling of dierent order velocity perturbations to the dierent order electron densities induced by the electric ®eld of the incident laser light. Our earlier model [6] also consider the source of harmonics being the above mentioned coupling, but in this case the velocity and density components were calculated using the Fourier analysis of the Maxwell and continuity equation and of the equation of motion. A qualitative description was given by Refs. [8,10] using the simple
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 5 0 3 - 6
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K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
physical model of an oscillating plasma mirror and simulations with PIC codes were also carried out. The re¯ecting layer is considered to be the supercritical plasma, formed by the leading edge of the incident laser pulse, when the details of the density distribution are neglected. The phase modulation of this mirror gives rise to harmonic frequencies. All these theories predict that the even harmonics will be p-polarized no matter what the incident laser polarization is. The theories are in agreement with the experiments in case of odd harmonics which conserve the polarization of the fundamental laser beam. In the case of s-polarized incident beam there is a contradiction between the experiments and theory. The observation in this respect are also ambiguous. As we discussed in Ref. [6] the experiments carried out in this ®eld can be divided into two categories: the experiments where the pulse length of the incident laser beam was shorter than 100 fs showed the dominance of p-polarized harmonics [7,10±13]. The second category, where the experiments were performed with pulse length longer than 500 fs both s- and ppolarized harmonics were observed, keeping the polarization of the original laser beam [2,4±6]. In the present work we employ the theory of re¯ection and refraction, to understand how the electric and magnetic ®elds behave in plasmas where the index of refraction is complex. The role of the magnetic ®eld can be of importance in such a medium, which means that the term ~ v~ B=c in the equation of motion cannot be neglected. The nonrelativistic equation of motion of the electrons in the plasma governed by the complete Lorentz force due to the penetrating electromagnetic ®eld has been solved exactly, and high harmonic components of the electron's velocity were obtained. Hence, these are the source of the radiation containing high harmonics. In Section 2 the in¯uence of dierent plasma regions on the propagation of the electromagnetic ®eld is discussed. A subsection is dedicated to explain how the electromagnetic ®eld could penetrate into the overdense region. In Section 3 the equation of motion is solved and the nonlinearities of the electron's motion are presented. In Section 4 the radiation ®eld produced by the nonlinear motion of the electrons is depicted.
2. In¯uence of dierent plasma regions on the propagation of an electromagnetic wave In laser produced plasmas it is worth to distinguish two separate regions from the point of view of the optical density. There is an optically thin (underdense) and an optically dense (overdense) region, which will be de®ned in the followings. A detailed discussion of light propagation in these two regions is presented here. There are presented separates discussions for sand p-polarized incident beams. In Fig. 1 is shown the geometry of re¯ection and refraction for ppolarized incident beam and for s-polarization in Fig. 2. In these ®gures ~ k denotes the wave vector and h denotes the angle. ~ E is the electric ®eld vector and ~ B is the magnetic induction vector. In both cases the index i refers to the quantities that characterize the incident, the index r refers to the re¯ected light and the index t characterize the transmitted laser light, respectively. We mention here that for the clarity of the ®gures we used the geometry characteristic for the overdense region. 2.1. The propagation of a light wave in underdense plasmas In this subsection we discuss the behavior of the electromagnetic ®eld in the underdense plasma.
Fig. 1. The geometry of re¯ection and refraction for a ppolarized incident beam.
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
Fig. 2. The geometry of re¯ection and refraction for an s-polarized incident beam.
Our treatment is based on the Fresnel formulae and on the Snell±Descartes law [21]. The monochromatic ®eld associated with the laser±plasma interactions can be described by the electric and magnetic ®eld vector. Let us consider an obliquely incident monochromatic electromagnetic wave on a monotonically decreasing plasma pro®le: ~ ~ Ei E0 ei k~r
xt
:
1
The Fresnel relations, which are direct consequences of the Maxwell equations and appropriate boundary conditions, split automatically to two separate sets of equation for s- and for p-polarized monochromatic light (see Fig. 1 for p-polarized incident beam and Fig. 2 for s-polarized incident beam). In both cases the light is considered to impinge towards the surface from the left to the right. To obtain the con®guration of the electric and magnetic ®eld we have to know the angle between the wave vector ~ k and the z axis inside the medium. At an arbitrarily chosen z coordinate inside the plasma, in a layer of in®nitesimal thickness dz around this z coordinate, the electron density can be assumed constant. This layer having the index of refraction constant gl is bounded by surfaces l 1, l and l, l 1, which separates it from the layers having the index
421
Fig. 3. The structure of the plasma layers.
of refraction gl 1 (situated at z (situated at z dz): p gl z;
dz) and gl1 2
p z dz;
3
gl1
p z dz:
4
gl
1
In the followings indices l 1, l, l 1 will be attributed to the in®nitesimal layers around z, z dz and z dz, respectively. Fig. 3 shows the refraction and re¯ection of the light in the underdense region, between the above mentioned layers. In the case of laser±plasmas the dielectric function is de®ned as: z 1
x2p z ; x x im z
5
where x2p z 4pn ze2 =m denotes the local plasma frequency, x the light frequency and m z the collision frequency. The electron density is denoted by n z (cm 3 ). Propagation of an electromagnetic wave in a strati®ed medium, where the index of refraction p g z z depends only on one coordinate z, can be described by the equation: p p z sin h z sin h z z0 z z0 6 which is a consequence of the Snell±Descartes law. This way it is obtained for sin and cosine of h z:
422
sin h0 sin h z p z
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
7
and
s z sin2 h0 cos h z : z
8
As is shown below, collision do not play a role in the underdense region, hence the imaginary part of the dielectric function can be neglected. On the other hand in the overdense region the imaginary part should be taken into account. Re¯ection and refraction of a light wave can be studied in the case of a monotonically increasing density pro®le n z pncr eqz , where p and q are the parameters of the pro®le, n z denotes the electron density and ncr mx2 =4pe2 is the critical density. The surface where the electron density is equal to the critical density is called critical surface and we consider it being at the coordinate z2 as it can be seen in Fig. 4, where the quantities are measured in units of k. The physical meaning of the parameters p and q is the following: p ns =ncr , where ns represents the solid state density and q characterize the steepness of the pro®le. The scale length of the underdense region is d jz2 z0 j, where z0 denotes the coordinate where the density is 1% of the critical density, which we will consider approximately 0. The plasma scale length is de®ned as: L jz3 z0 j, where z3 denotes the coordinate where the density reaches the solid state density. It can be noted, that in the case of solid state media, where p
ns =ncr 10±100 in the underdense region there are considerably fewer particles than in the whole plasma: R z2 n z dz 1 Rzz03 9 : n z dz p z0 Therefore the processes which take place in the underdense region can be neglected. A very steep density pro®le p 40 and q 5 is chosen, similarly to Ref. [6]. As we are intending to develop a single electron model, we must investigate the conditions under which the collisional eects can be neglected. The mean free path can be calculated as la hvT z= 1:5 m zi, where m z 2:91 10 5 n z T z (s 1 ) is the electron collision frequency and vT z 4:19 0:5 107 T z (cm/s) is the electron thermal velocity. The average mean free path is calculated for the underdense and overdense regions. T z (eV) is the electron temperature and n z (cm 3 ) is the electron density. In the case of the interaction of high intensity lasers with matter the temperature for the laser±plasma interface may reach T0 200±300 eV [14]. As the light propagates towards higher densities, the temperature of the plasma decreases from T0 to the temperature of the solid state matter 0 as T z p0 eq z . The parameters p0 and q0 can be determined assuming the temperature of the electrons of the solid state matter having room temperature. Thus p0 8:3 10 5 and q0 6:8. The mean free path in the underdense region is greater than the wavelength so, than the whole underdense region. In the overdense region the mean free path is shorter than the wavelength, so there the collisions cannot be neglected. Neglecting collisions in the underdense region (5) becomes: z 1
Fig. 4. The density pro®le normalized to ncr depending on the normalized coordinate z/k.
n z : ncr
10
As the light propagates from the vacuum towards solid density, we can distinguish two dierent plasma regions: the underdense region, where the local plasma frequency xp is smaller than the light frequency x and the overdense region where it is larger than the light frequency. The underdense region can be further divided into two regions. The ®rst region is between z0 and z1 , where
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
z1 denotes the classical re¯ection point where z1 sin2 h0 and a second between z1 and z2 , where z2 represents the critical surface. Following the geometry in Fig. 3 at a given coordinate z, the electric (and the magnetic ®eld also) is composed by the refracted wave at the interface l 1, l (called primary wave) and the re¯ected wave at the interface l, l 1 (called secondary wave). The Fresnel formulae [15] gives us the transmission tl 1;l and the re¯ection coecients rl;l1 , which can be used to determine the amplitude of the electric and magnetic ®elds. We consider an unmagnetized medium, therefore we assume the magnetic permeability being 1 in the followings. In case of s-polarized laser light tl 1;l and rl;l1 reads as: tl
1;l
and
rl;l1
2 v u z 2 u t z dz sin h z dz 1 cos2 h z dz v u z dz u sin2 h z t z 1 cos2 h z v : u z dz u sin2 h z t z 1 cos2 h z
11
12
The electric ®eld in the layer l can be calculated by summing up the refracted wave at the l 1, l interface Elprim tl 1;l El 1 and the re¯ected wave at the l, l 1 interface Elsec rl;l1 Elprim tl 1;l rl;l1 El 1 . The local transmission t z and re¯ection r z coecients were calculated solving the dierential equation obtained by taking the Taylor expansion of z dz and z dz and E z dz and E z dz (the terms of order higher than 2 were neglected). t z
z sin2 h0 cos2 h0
E z z sin2 h0 cos2 h0
423
q x z sin2 h0 z exp i c 1=4 x z sin2 h0 sin h0 y xt E0 c cos2 h0 q x exp i z sin2 h0 z c x 15 sin h0 y xt E0 : c 1=4
The amplitude of the magnetic ®eld can be calculated using the relation B z
p zE z:
16
We are using the well known de®nition of the dielectric function z of the plasma (see Eq. (10)) to obtain the dependence of the electric and magnetic ®eld amplitude on the coordinate normalized to the wavelengths. The electric and magnetic ®elds normalized to the original electric ®eld E0 are shown in Fig. 5 for p 40, q 5, and sin h0 45°. It can be immediately seen that both E and B will be larger than the original E0 when the light approaches the classical turning point. The rise of the electric ®eld can be explained as a direct consequence of the ®rst Maxwell equation ~ 0, where D ~ ~ rD E denotes the electric displacement vector. The decrease of the dielectric
1=4
;
and for the local re¯ection coecient: 1=4 z sin2 h0 r z : cos2 h0 This way the electric ®eld reads:
13
14
Fig. 5. The dependence of the electric and magnetic ®eld amplitude normalized to E0 on the normalized coordinate z/k in the underdense region.
424
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
function leads to the increase of the electric ®eld. The propagation of p-polarized laser light can be derived in the same way when the scale length of the plasma is comparable to the wavelength. It is important to note here, that in the case of small scale length plasmas the penetration of the laser light into the medium is independent from the polarization of the incoming beam. 2.2. Frustrated total re¯ection in plasmas As was mentioned before total re¯ection takes place at the plane z z1 . An intriguing phenomenon is the penetration of the wave (evanescent wave) [16] into the denser medium (here the overdense region). The phenomenon, that the light could penetrate into an optically denser medium even if the angle of incidence is larger than the Brewster angle, is called frustrated total re¯ection (FTR). The phenomenon was studied already by Newton in the 18th century [17] and demonstrated experimentally by Bose [18] and Hall [19] in the case of two dielectric media. To investigate the phenomenon a most common solution was given by introducing a third medium [18,19], close to the ®rst one. All three media were assumed transparent at the wavelength of the operation. In case of three dielectric media a detailed description of the phenomenon has been given by Zhu [16]. In the present subsection we study how this phenomenon takes place in the case when the third media is conducting and the index of refraction is changing. In our case the ®rst medium is a thin layer near the coordinate z1 (the classical re¯ection point) and the third one is a layer near z2 (the critical density layer). The second layer is considered to be situated between z1 and z2 . It is shown here, that the variation of the dielectric function inside the second layer will not in¯uence the FTR. Following the notation used in Fig. 3 the ®rst medium is considered to have the index of refraction gl 1 , the second gl and the third gl1 . In the present analysis the layer l 1 is the last in®nitesimal layer of the underdense region, the second is situated between z1 and z2 , and the third one is considered being the ®rst in®nitesimal layer of the underdense region. The thickness of the second layer equals z1 z2 k=q ln 2. It can be seen that in the case when
q > 1 the thickness of the layer is considerably smaller than the wavelength, so it can be denoted by dz, as an in®nitesimal quantity. The transmission and re¯ection coecients of the second medium having the index of refraction gl are: t
1 tl 1 rl
id 1;l tl;l1 e 2id 1;l rl;l1 e
17
and r
rl 1;l rl;l1 e2id ; 1 rl 1;l rl;l1 e2id
18
where d is the phase dierence of consecutive re¯ected beams: q 2p dz g2l 1 g2l : 19 d k If the thickness of the second medium is almost in®nitesimally small, then e2id 1. Comparing the terms describing the primary and secondary ®elds in Eqs. (17) and (18) one can remark that the transmission coecient is larger than the re¯ection one, at an arbitrary z coordinate. Hence we conclude that the transmission coecient tends to in®nity, which means that this region will not reduce the amplitude of the electric and magnetic ®elds and no phase dierence will be introduced by this part of the plasma. It is also important that the direction of the wave vector is not in¯uenced by this second layer. In accordance with the theory presented by Ginzburg [20] our theory takes into account the increase of the electric ®eld near the classical re¯ection point. Here the coupling of the plasma wave to the electromagnetic wave near the critical surface is neglected even in the case of p-polarized incident laser beam. The bigness of the electric ®eld near the critical density is attributed to its increase at the classical turning point and its tunneling towards the critical density. The tunneling is attributed to the FTR. 2.3. The propagation of the inhomogeneous electromagnetic wave in the overdense region In this subsection we study the behavior of the electromagnetic ®eld in the overdense medium, where the electric and magnetic ®elds decrease exponentially. In high density media we cannot
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
neglect collisions. Eq. (5) can be expanded obtaining for the dielectric function: z 1
n z m n z i ; ncr x ncr
20
where m denotes the electron collision frequency, which is considered constant here. In the following we use the notations: c z 1
qz
pe ;
21
d z fp eqz ;
22
where f m=x. Thus the dielectric function can be given as z c z id z. Another unsolved problem of the laser±plasma interaction is the determination of the trajectory of the wave in the overdense material. Supposing, that no attenuation occurs in the underdense matter, the trajectory will be determined as in the general case of a refraction between a dielectric and a conducting media [21]. The same method is used as in Section 2.1. In this case the Snell± Descartes law can be written as: p p l 1 sin hl 1 l1 sin hl1 : 23 Comparing Eq. (23) with Eq. (6) it can be seen that in the overdense region the angle of refraction will have the same form as given by Eq. (7), but in this case sin h z will be complex. The formal identity will be valid for the transmission coecient and for the phase of the electric ®eld (see Eq. (15)), too. Substituting the complex dielectric function z in Eq. (15), it can be written in the form Eod z ei/ , where the amplitude of the electric ®eld in the overdense region is denoted by: Eod z
/ z
x c z c
425
sin2 h0 2 d z2 1=4
d z z cos 0:5 arctan c z sin h0 x 1 d z sin h0 y xt arctan : c 4 c z sin h0 25
Since z is complex, sin h z is complex too and this quantity has no longer the meaning of an angle of refraction [21]. The real angle of refraction can be determined from the real part of the phase of the electric ®eld. The surfaces where the real part of the phase is constant are given by the: 2
2 1=4
sin2 h0 d z d z cos 0:5 arctan z c z sin h0
c z
sin h0 y const:
26 relation, which de®nes planes whose normals make an angle hr z with the normal to the boundary. The tangent of the real angle of refraction is y=z: tan hr z
sin h0 sin2 h0 2 d z2 1=4 1 : d z cos 0:5 arctan c z sin h0 c z
27 In the case of q 40, p 5 and f 0:8 the angle dependence from z=k is shown in Fig. 6. It can be
cos1=2 h0 c z
2
2 1=8
sin h0 d z x c z sin h0 2 d z2 1=4 exp c 1 d z arctan sin z E0 2 c z sin h0
24 and / z is the phase of the electric ®eld in the same region:
Fig. 6. The dependence of the angle of refraction on the normalized coordinate z/k.
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K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
Fig. 7. The variation of the normalized electric and magnetic ®elds in the overdense region on the normalized coordinate z/k.
seen that together with the decrease of the angle the wave vector becomes more and more parallel to the z axes, according to the fact that the overdense region is an optically dense medium. If the density pro®le is steeper, so the real angle of refraction will decrease faster (see Eq. (27)). Fig. 7 illustrates the decrease of the electric and magnetic ®elds in the overdense region. It shows that the decrease of the magnetic ®eld is slower than that of the electric ®elds. The slow decay of the magnetic ®eld is in accordance with the fast increase of the amplitude of the dielectric function as it can be seen in Fig. 8. The local maximum of the magnetic ®eld can be attributed to the faster increase of the amplitude of the dielectric function than the decrease of the exponential factor, in the near critical region. It
can be seen, that after the local maximum the magnetic ®eld decreases exponentially. It can be seen, that in the case of thin plasma layers (l k) the decay of the electromagnetic ®elds takes place in the overdense region. Practically no attenuation occurs in the underdense medium, if one neglects nonlinear processes. Hence, the basic excitation processes of the electrons take place in the overdense medium. Even in this case however both the electric and the magnetic ®elds decrease within the skin depth. Therefore it is necessary to determine it and compare its value with the scale length of the interaction. In case of a medium having the dielectric function of the form (20) the skin depth was determined solving the following implicit equation: cos1=2 h0 1=8 2 2 c dskin sin h0 d dskin x c dskin sin h0 2 d dskin 2 1=4 exp c 1 d dskin arctan sin z 1 2 c dskin sin h0 28 and it was found for the parameters mentioned above being: dskin 0:7k. To avoid the appearance of the anomalous skin eect the amplitude of the electron's must be smaller than the skin depth. This amplitude amax can be determined from the equation of motion. It is considered p being amax 0:13lk, where l 10 9 k (lm) I (W/cm2 ) is the intensity parameter (I is the intensity of the laser light). Even in the case when the intensity is approximately relativistic, i.e. l 1 the amplitude is smaller than the wavelength.
3. The source of high harmonics
Fig. 8. The ratio of the magnetic and electric ®eld in the overdense region.
The nonlinear motion of electrons is considered to be the source of nonlinear properties of the scattered laser light. In this subsection we study this motion. The motion of the electrons in the ®eld of an electromagnetic wave is studied. For
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
427
will have two velocity components vx0 and vz0 . The equation of motion reads: 8 dvx0 > > m > > > dt > > > > > > <
0 0z eE exp ix t n i/1 c v z0 z0 e B exp ix t n0 i/0 c c
> > > > > dvz0 > > m > > > dt > :
vx0 e B exp ix t c
z0 n c
0
i/0 : 30
Fig. 9. The rotation of the coordinate system.
simplicity, we assume a coordinate system where the x0 y 0 z0 axes are parallel with the electric x0 k~ E, magnetic y 0 k~ B and wave z0 k~ k vectors, respectively. The equation of motion is solved in this coordinate system, which is the same for both polarizations. The quantities from the original coordinate system are transformed to x0 y 0 z0 system, and after solving of the equation of motion, the resulting trajectories are transformed back to the original coordinate system (see Figs. 1 and 2). The rotation of the coordinate system is shown in Fig. 9 for s-polarized incident beam. The transformation is made for an arbitrary z coordinate, therefore the dependence of the velocity on sin hr z is taken into account. Even in the nonrelativistic case v=c 1, the smallness of the term ~ v~ B=c can be compensated by the smallness of ~ E. This way the electron's motion will be governed by the complete Lorentz force:
The aim of our treatment is to obtain the higher harmonics of the fundamental laser frequency. Therefore we make the transformation of variable t0 t n0 z=c, where n0 depends on the properties of the material and on the angle of incidence in the case of propagation of inhomogeneous waves in conducting media [21]. The local real index of refraction can be determined comparing Eqs. (7) and (27): n0 c z
2
2 1=4
sin2 h0 d z
d z cos 0:5 arctan : c z sin h0
31
29
After performing the transformation of variables mentioned above, Eq. (30) can be written: 8 dv 0 vz 0 x0 > 1 n eE exp ixt0 i/1 m > c > < dt v z0 e B exp ixt0 i/0 c > > > : m dvz0 1 n0 vz0 e vx0 B exp ixt0 i/ : 0 dt c c 32
where the electric ®eld is de®ned as ~ E E exp i xt ~ k 0 ~ r0 /1 and the magnetic ®eld as: ~ B B exp i xt ~ k 0 ~ r0 /0 . The quantities E and B are the amplitudes of the electric and magnetic ®elds and ~ v denotes the velocity of the electron. By solving the equation of motion we get high harmonic velocity components of the fundamental laser frequency. Taking the projections of the velocity to the coordinate axes we found that the electron will oscillate in the x0 ±z0 plane and it
On the left hand side of the above equations the terms containing the velocity component vz0 are small, because vz0 stems from the coupling of the vx0 component and of the magnetic ®eld. This way it can be considered as a second order correction to the velocity. In order to ®nd the solution of Eq. (32) in an elegant way we will not neglect the small term vz0 B=c for symmetry reasons. The consistency of this process has been numerically demonstrated. This way Eq. (32) simpli®es to:
m
d~ v dt
e~ E
~ v e ~ B; c
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K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
du dt0
e E exp ixt0 i/1 m
i
e B exp ixt0 i/0 u; m 33
where we introduced the complex velocity u vx0 ivz0 . Solving the dierential equation and separating the variables it is obtained after the back transformation that: vx 0
1 e 1 e E sin /1 /0 E xL m xm x x X 1n x n 1 L L exp z0 /0 cos c x n! x h x i xL z0 /0 n xt /0 sin sin /1 /0 c x
34
and vz0
1 e 1 e E cos /1 /0 E xL m xm x x X 1n x n 1 L L z 0 /0 cos exp c n! x h x i xL x 0 sin z /0 n xt /0 : cos /1 /0 x c
35
In Eqs. (34) and (35) xL z denotes the local Larmor frequency de®ned as: xL z
eB z : mc
36
As it is shown in Fig. 9 the velocity components in the laboratory frame are: 8 < vx vx0 ; v vz0 sin h z; 37 : y vz vz0 cos h z: Similarly we obtained the velocity components in case of a p-polarized laser light with the values of vx0 and vz0 given by Eqs. (34) and (35), as in the case of s-polarized incident beam: 8 < vx vx0 cos h z vz0 sin h z; v 0; 38 : y vz vx0 sin h z vz0 cos h z: The above velocity components contain high harmonic components, therefore they will serve as the source terms of high harmonics for a single electron.
4. The scattered laser light It is well known in plasma physics, that longitudinal current density perturbation (parallel to the surface's normal) may give rise to an electron-plasma wave [22]. The other component, the transverse one gives the source term for the re¯ected light, with high harmonic components. We are interested in the study of this latter component. Comparing Eqs. (37) and (38) it was observed that the velocity components, which give contribution to the scattered light are dierent for s- and ppolarized incident laser beam. The solutions of the Maxwell equations in the Coulomb gauge are the retarded potentials [22]. In nonrelativistic case the retarded vector potential for a point charge can be written as: Z ~ 1 vt s ~ A ~ R; t e 4p j~ R ~ r sj 1~ jR ~ r sj s ds; 39 d t c where ~ vt is the transverse part of the electron's velocity, normal to the z axes, containing high harmonic components (see Eqs. (34) and (35)). The vector ~ R and ~ r s are the position vectors of the point of observation and that of the source, respectively. Assuming the source dimension being smaller than the wavelength, we can use the approximations which are valid for the so called radiation zone [22]. In this case we can make the approximation ~ r s ~ r0 and the vector potential will get the following form: 1 1 1~ ~ ~ ~ jR ~ r0 j : A R; t e 40 vt t 4p j~ c R ~ r0 j It can be seen that the vector potential (40) characterize a spherical harmonic wave. In order to obtain the time and space dependence of the vector potential we make the following replacements in Eqs. (34) and (35): t!t /1
1~ jR c /0 !
~ r0 j;
1 d r0 arctan ; 2 c r0
41 42
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
Fig. 10. The fundamental scattered component of the vector potential.
/0 !
1 arctan 4
d z c z sin h0
429
Fig. 11. The intensity distribution of the harmonics for a ppolarized incident beam.
1 d r0 arctan ; 2 c r0 43
i1=4 x 0 x xh 2 2 z ! sin h0 y c r0 sin2 h0 d r0 c c c d r0 r0 : cos 0:5 arctan 44 c r0 sin h0
The fundamental scattered harmonic can be seen in case of an s-polarized incident beam in Fig. 10. We considered the radiating electron being at coordinate r 0; 0; 0 and supposed, that the radiation which induced its motion has the intensity parameter l 1. Its shape is characteristic for a spherical harmonic wave. The variation of the higher harmonics will have the same form for both laser polarizations. In order to characterize the eciency of the harmonic production we calculated the time averaged intensity of the harmonics Z t 1 ~ Sn E ~ B ds; 45 2t t where t is assumed to be considerably larger than the period of the harmonics t > 2p=x. In Figs. 11 and 12 the intensity distribution of the harmonics are plotted as a function of the order for dierent polarizations. The intensity decreases with the order. The eciency of high harmonics generated by p-polarized laser beam is higher than for s-polarized laser beams.
Fig. 12. The intensity distribution of the harmonics for an spolarized incident beam.
In the case of an s-polarized incident laser beam the harmonics will be mainly s-polarized, but all this harmonics will have a p-polarized component, which is proportional to sin hr z. The p components gets important, if the number of the electrons in the underdense region is not negligible compared to the overdense pregion. The intensity parameter l 10 9 k (lm) I (W/cm2 ) determines the magnitude of dierent harmonics. The intensity of a given harmonic decreases with the decrease of the intensity parameter l proportional to the term el ln 1 , where n represents the order of the harmonic.
5. Conclusions In the present work the con®guration of the electromagnetic ®eld inside a plasma created on a
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solid surface by a high intensity laser radiation, whose scale length is in the range of the wavelength was studied. It was found that in the underdense region the electric ®eld increases due to the decrease of the dielectric function keeping the divergence of the electric displacement vector zero. Between the classical re¯ection point and the overdense region the electric ®eld shows the phenomenon called FTR and penetrates in the overdense region. In the overdense region the collision frequency is not negligible, so the plasma will behave as a conducting medium and will absorb the inhomogeneous light wave emerging from the underdense region. In the case of a very steep density pro®le the underdense region is thinner than the overdense region. In such cases the electric ®eld will be high enough around the critical density to expel very hot electrons from the plasma [23]. In the overdense region the electromagnetic ®eld decrease exponentially. In the present study the processes which take place in the overdense region are considered to be the main source of harmonic generation. In such a dense medium the magnetic ®eld of an electromagnetic wave becomes high enough to in¯uence the motion of the electrons. By taking into account the complete Lorentz force we obtain nonlinearities already in the ®rst step of our calculations. The electrons whose motion is governed by the complete Lorentz force are the source terms of the harmonics. The intensity of a given harmonic component is always larger at least by an order of magnitude for p-polarized incident beam than for an s-polarized incident beam. In the case of an s-polarized incident beam there appears a p-polarized component for all harmonics, but this term will be negligible compared to the s component. There is no well de®ned intensity threshold for the appearance of the harmonics. This model might explain the polarization dependence of the experiments, which was made with laser pulses longer than 500 fs [2,4±6]. In this case, if the intensity of the incident laser light was high enough to produce a plasma layer comparable to the wavelength, the light penetrated into the overdense region and produced high harmonics. The eciency of harmonic generation is determined by the parameters p and q, which characterize the density pro®le. The parameter p is
connected to the atomic number of the target material. The atomic number of the target must be high enough to neglect the processes which take place in the underdense region. In the case of targets having low atomic number the steepness of the density pro®le is small, therefore there are too much electrons in the underdense region, so the processes which take place here cannot be neglected. If the atomic number of the target is too high there are practically no electrons in the underdense region and the surface will coincide to the critical layer and will behave as the oscillating mirror as is suggested in Refs. [8,10]. The parameter q is mainly determined by the pulse length of the incident laser beam, which in¯uence the scale length of the plasma. This way, if the pulse length is shorter than 100 fs [7,10±13] the same phenomenon takes place as in the case of high atomic number targets. The experimental results reported in Ref. [6] are in agreement with the present model. The experimentally found eciency ratio of the second and third harmonic was of the order of 10 2 , and the same results was obtained here. Theoretically and experimentally it was obtained that the high harmonics preserve the polarization of the fundamental beam.
Acknowledgements The authors gratefully acknowledge I.B. F oldes, G. Kocsis, J.S. Bakos and S. Szatmari for fruitful discussions. This work was supported by the Hungarian OTKA Foundation under contract numbers T029376, T032375, T023526 and T029179 and by NATO Science for Peace Program (Sfp-971989) and by Title of project: KFKI-Condensed Matter Research Centre (KFKI-CMRC) Contract No: ICA 1-CT-2000-70029.
References [1] P.A. Norreys, M. Zepf, S. Moustaizis, et al., Phys. Rev. Lett. 76 (1996) 1832±1835. [2] I.B. F oldes, J.S. Bakos, G. Veres, et al., IEEE J. Sel. Top. Quant. Electron. 2 (1996) 776±781.
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431 [3] I.B. F oldes, J.S. Bakos, Z. Bakonyi, et al., Phys. Lett. A 258 (1999) 312±316. [4] D.M. Chambers, P.A. Norreys, A.E. Dangor, et al., Opt. Commun. 148 (1998) 289±294. [5] A. Ishizawa, K. Inaba, T. Kanai, et al., IEEE J. Quant. Electron. 35 (1999) 60±65. [6] G. Veres, J. Bakos, I.B. F oldes, et al., Europhys. Lett. 48 (1999) 390±396. [7] D. von der Linde, in: R.M. More (Ed.), Laser Interactions with Atoms, Solids, and Plasmas, Plenum Press, New York, 1994, pp. 207±237. [8] R. Lichters, J. Meyer-ter-Vehn, A. Pukhov, Phys. Plasmas 3 (1996) 3425±3437. [9] P. Gibbon, IEEE J. Quant. Electron. 33 (1997) 1915±1924. [10] D. von der Linde, H. Schulz, T. Engers, et al., IEEE J. Quant. Electron. 28 (1992) 2388±2397. [11] S. Kohlweyer, G.D. Tsakiris, C.G. Wahlstr om, et al., Opt. Commun. 117 (1995) 431±438. [12] L.A. Gizzi, D. Giulietti, A. Giulietti, et al., Phys. Rev. Lett. 76 (1996) 2278±2281.
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[13] M. Zepf, G.D. Tsakiris, G. Pretzler, et al., Phys. Rev. E 58 (1998) R5253±R5256. [14] I.B. F oldes, J.S. Bakos, K. Gal, et al., Laser Phys. 10 (2000) 264±269. [15] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. [16] S. Zhu, A.W. Yu, D. Hawley, R. Roy, Am. J. Phys. 54 (1986) 601±607. [17] I. Newton, Optics, Dover, New York, 1952. [18] J.C. Bose, Proc. R. Soc. London A 62 (1897) 300. [19] E.E. Hall, Phys. Rev. 15 (1902) 73. [20] V.L. Ginzburg, Propagation of Electromagnetic Waves in Plasmas, Gordon and Beach, New York, 1960. [21] M. Born, E. Wolf, Principles of Optics, Pergamon Press, Oxford, 1980. [22] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1963. [23] C.E. Max, Second harmonic production from solid targets, in: R. Balian, J.C. Adam (Eds.), Laser Plasma Interactions, North-Holland, Amsterdam, 1982, pp. 304±410.
Optics Communications 198 (2001) 419±431
www.elsevier.com/locate/optcom
Polarization properties of high harmonics generated on solid surfaces K. G al a,b,*, S. Varr oc a c
KFKI-Research Institute for Particle and Nuclear Physics, P.O. Box 49, H-1525 Budapest, Hungary b Department of Experimental Physics, University of Szeged, 6720 Szeged, Hungary KFKI-Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary Received 1 June 2001; received in revised form 14 August 2001; accepted 21 August 2001
Abstract A detailed study of the electromagnetic ®eld's con®guration inside plasmas generated on solid surfaces by ultrashort laser pulses is given in the present work. The radiation due to the nonrelativistic motion of the electrons in the overdense region is considered the main source of high harmonics, whose polarization dependence on the fundamental beam is treated to explain the experimentally observed polarization properties of the harmonics. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 42.65.Ky; 52.25.Qt; 52.50.Jm Keywords: Harmonic generation; Plasma production
1. Introduction Several laboratories developed lasers capable of delivering very short pulses t < 1 ps with high intensities I > 1016 W/cm2 . High order harmonics were observed, resulting from these interactions of laser beams with solid targets. It was recently observed by using ultrashort laser pulses that even and odd harmonics occur for both laser polariza-
* Corresponding author. Address: KFKI-Research Institute for Particle and Nuclear Physics, P.O. Box 49, H-1525 Budapest, Hungary. Tel.: +36-1-3922222/1467; fax: +36-13959151. E-mail addresses: [email protected] (K. Gal), [email protected] (S. Varr o).
tion [1±6]. Some theories [6±9] were developed to explain the polarization dependence of harmonic generation. One of the theories referring to the polarization dependence of high harmonics generated on solid surfaces have been developed by von der Linde [7] using a perturbative treatment. This theory consider as the source of the high harmonics the coupling of dierent order velocity perturbations to the dierent order electron densities induced by the electric ®eld of the incident laser light. Our earlier model [6] also consider the source of harmonics being the above mentioned coupling, but in this case the velocity and density components were calculated using the Fourier analysis of the Maxwell and continuity equation and of the equation of motion. A qualitative description was given by Refs. [8,10] using the simple
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 5 0 3 - 6
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K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
physical model of an oscillating plasma mirror and simulations with PIC codes were also carried out. The re¯ecting layer is considered to be the supercritical plasma, formed by the leading edge of the incident laser pulse, when the details of the density distribution are neglected. The phase modulation of this mirror gives rise to harmonic frequencies. All these theories predict that the even harmonics will be p-polarized no matter what the incident laser polarization is. The theories are in agreement with the experiments in case of odd harmonics which conserve the polarization of the fundamental laser beam. In the case of s-polarized incident beam there is a contradiction between the experiments and theory. The observation in this respect are also ambiguous. As we discussed in Ref. [6] the experiments carried out in this ®eld can be divided into two categories: the experiments where the pulse length of the incident laser beam was shorter than 100 fs showed the dominance of p-polarized harmonics [7,10±13]. The second category, where the experiments were performed with pulse length longer than 500 fs both s- and ppolarized harmonics were observed, keeping the polarization of the original laser beam [2,4±6]. In the present work we employ the theory of re¯ection and refraction, to understand how the electric and magnetic ®elds behave in plasmas where the index of refraction is complex. The role of the magnetic ®eld can be of importance in such a medium, which means that the term ~ v~ B=c in the equation of motion cannot be neglected. The nonrelativistic equation of motion of the electrons in the plasma governed by the complete Lorentz force due to the penetrating electromagnetic ®eld has been solved exactly, and high harmonic components of the electron's velocity were obtained. Hence, these are the source of the radiation containing high harmonics. In Section 2 the in¯uence of dierent plasma regions on the propagation of the electromagnetic ®eld is discussed. A subsection is dedicated to explain how the electromagnetic ®eld could penetrate into the overdense region. In Section 3 the equation of motion is solved and the nonlinearities of the electron's motion are presented. In Section 4 the radiation ®eld produced by the nonlinear motion of the electrons is depicted.
2. In¯uence of dierent plasma regions on the propagation of an electromagnetic wave In laser produced plasmas it is worth to distinguish two separate regions from the point of view of the optical density. There is an optically thin (underdense) and an optically dense (overdense) region, which will be de®ned in the followings. A detailed discussion of light propagation in these two regions is presented here. There are presented separates discussions for sand p-polarized incident beams. In Fig. 1 is shown the geometry of re¯ection and refraction for ppolarized incident beam and for s-polarization in Fig. 2. In these ®gures ~ k denotes the wave vector and h denotes the angle. ~ E is the electric ®eld vector and ~ B is the magnetic induction vector. In both cases the index i refers to the quantities that characterize the incident, the index r refers to the re¯ected light and the index t characterize the transmitted laser light, respectively. We mention here that for the clarity of the ®gures we used the geometry characteristic for the overdense region. 2.1. The propagation of a light wave in underdense plasmas In this subsection we discuss the behavior of the electromagnetic ®eld in the underdense plasma.
Fig. 1. The geometry of re¯ection and refraction for a ppolarized incident beam.
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
Fig. 2. The geometry of re¯ection and refraction for an s-polarized incident beam.
Our treatment is based on the Fresnel formulae and on the Snell±Descartes law [21]. The monochromatic ®eld associated with the laser±plasma interactions can be described by the electric and magnetic ®eld vector. Let us consider an obliquely incident monochromatic electromagnetic wave on a monotonically decreasing plasma pro®le: ~ ~ Ei E0 ei k~r
xt
:
1
The Fresnel relations, which are direct consequences of the Maxwell equations and appropriate boundary conditions, split automatically to two separate sets of equation for s- and for p-polarized monochromatic light (see Fig. 1 for p-polarized incident beam and Fig. 2 for s-polarized incident beam). In both cases the light is considered to impinge towards the surface from the left to the right. To obtain the con®guration of the electric and magnetic ®eld we have to know the angle between the wave vector ~ k and the z axis inside the medium. At an arbitrarily chosen z coordinate inside the plasma, in a layer of in®nitesimal thickness dz around this z coordinate, the electron density can be assumed constant. This layer having the index of refraction constant gl is bounded by surfaces l 1, l and l, l 1, which separates it from the layers having the index
421
Fig. 3. The structure of the plasma layers.
of refraction gl 1 (situated at z (situated at z dz): p gl z;
dz) and gl1 2
p z dz;
3
gl1
p z dz:
4
gl
1
In the followings indices l 1, l, l 1 will be attributed to the in®nitesimal layers around z, z dz and z dz, respectively. Fig. 3 shows the refraction and re¯ection of the light in the underdense region, between the above mentioned layers. In the case of laser±plasmas the dielectric function is de®ned as: z 1
x2p z ; x x im z
5
where x2p z 4pn ze2 =m denotes the local plasma frequency, x the light frequency and m z the collision frequency. The electron density is denoted by n z (cm 3 ). Propagation of an electromagnetic wave in a strati®ed medium, where the index of refraction p g z z depends only on one coordinate z, can be described by the equation: p p z sin h z sin h z z0 z z0 6 which is a consequence of the Snell±Descartes law. This way it is obtained for sin and cosine of h z:
422
sin h0 sin h z p z
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
7
and
s z sin2 h0 cos h z : z
8
As is shown below, collision do not play a role in the underdense region, hence the imaginary part of the dielectric function can be neglected. On the other hand in the overdense region the imaginary part should be taken into account. Re¯ection and refraction of a light wave can be studied in the case of a monotonically increasing density pro®le n z pncr eqz , where p and q are the parameters of the pro®le, n z denotes the electron density and ncr mx2 =4pe2 is the critical density. The surface where the electron density is equal to the critical density is called critical surface and we consider it being at the coordinate z2 as it can be seen in Fig. 4, where the quantities are measured in units of k. The physical meaning of the parameters p and q is the following: p ns =ncr , where ns represents the solid state density and q characterize the steepness of the pro®le. The scale length of the underdense region is d jz2 z0 j, where z0 denotes the coordinate where the density is 1% of the critical density, which we will consider approximately 0. The plasma scale length is de®ned as: L jz3 z0 j, where z3 denotes the coordinate where the density reaches the solid state density. It can be noted, that in the case of solid state media, where p
ns =ncr 10±100 in the underdense region there are considerably fewer particles than in the whole plasma: R z2 n z dz 1 Rzz03 9 : n z dz p z0 Therefore the processes which take place in the underdense region can be neglected. A very steep density pro®le p 40 and q 5 is chosen, similarly to Ref. [6]. As we are intending to develop a single electron model, we must investigate the conditions under which the collisional eects can be neglected. The mean free path can be calculated as la hvT z= 1:5 m zi, where m z 2:91 10 5 n z T z (s 1 ) is the electron collision frequency and vT z 4:19 0:5 107 T z (cm/s) is the electron thermal velocity. The average mean free path is calculated for the underdense and overdense regions. T z (eV) is the electron temperature and n z (cm 3 ) is the electron density. In the case of the interaction of high intensity lasers with matter the temperature for the laser±plasma interface may reach T0 200±300 eV [14]. As the light propagates towards higher densities, the temperature of the plasma decreases from T0 to the temperature of the solid state matter 0 as T z p0 eq z . The parameters p0 and q0 can be determined assuming the temperature of the electrons of the solid state matter having room temperature. Thus p0 8:3 10 5 and q0 6:8. The mean free path in the underdense region is greater than the wavelength so, than the whole underdense region. In the overdense region the mean free path is shorter than the wavelength, so there the collisions cannot be neglected. Neglecting collisions in the underdense region (5) becomes: z 1
Fig. 4. The density pro®le normalized to ncr depending on the normalized coordinate z/k.
n z : ncr
10
As the light propagates from the vacuum towards solid density, we can distinguish two dierent plasma regions: the underdense region, where the local plasma frequency xp is smaller than the light frequency x and the overdense region where it is larger than the light frequency. The underdense region can be further divided into two regions. The ®rst region is between z0 and z1 , where
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
z1 denotes the classical re¯ection point where z1 sin2 h0 and a second between z1 and z2 , where z2 represents the critical surface. Following the geometry in Fig. 3 at a given coordinate z, the electric (and the magnetic ®eld also) is composed by the refracted wave at the interface l 1, l (called primary wave) and the re¯ected wave at the interface l, l 1 (called secondary wave). The Fresnel formulae [15] gives us the transmission tl 1;l and the re¯ection coecients rl;l1 , which can be used to determine the amplitude of the electric and magnetic ®elds. We consider an unmagnetized medium, therefore we assume the magnetic permeability being 1 in the followings. In case of s-polarized laser light tl 1;l and rl;l1 reads as: tl
1;l
and
rl;l1
2 v u z 2 u t z dz sin h z dz 1 cos2 h z dz v u z dz u sin2 h z t z 1 cos2 h z v : u z dz u sin2 h z t z 1 cos2 h z
11
12
The electric ®eld in the layer l can be calculated by summing up the refracted wave at the l 1, l interface Elprim tl 1;l El 1 and the re¯ected wave at the l, l 1 interface Elsec rl;l1 Elprim tl 1;l rl;l1 El 1 . The local transmission t z and re¯ection r z coecients were calculated solving the dierential equation obtained by taking the Taylor expansion of z dz and z dz and E z dz and E z dz (the terms of order higher than 2 were neglected). t z
z sin2 h0 cos2 h0
E z z sin2 h0 cos2 h0
423
q x z sin2 h0 z exp i c 1=4 x z sin2 h0 sin h0 y xt E0 c cos2 h0 q x exp i z sin2 h0 z c x 15 sin h0 y xt E0 : c 1=4
The amplitude of the magnetic ®eld can be calculated using the relation B z
p zE z:
16
We are using the well known de®nition of the dielectric function z of the plasma (see Eq. (10)) to obtain the dependence of the electric and magnetic ®eld amplitude on the coordinate normalized to the wavelengths. The electric and magnetic ®elds normalized to the original electric ®eld E0 are shown in Fig. 5 for p 40, q 5, and sin h0 45°. It can be immediately seen that both E and B will be larger than the original E0 when the light approaches the classical turning point. The rise of the electric ®eld can be explained as a direct consequence of the ®rst Maxwell equation ~ 0, where D ~ ~ rD E denotes the electric displacement vector. The decrease of the dielectric
1=4
;
and for the local re¯ection coecient: 1=4 z sin2 h0 r z : cos2 h0 This way the electric ®eld reads:
13
14
Fig. 5. The dependence of the electric and magnetic ®eld amplitude normalized to E0 on the normalized coordinate z/k in the underdense region.
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K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
function leads to the increase of the electric ®eld. The propagation of p-polarized laser light can be derived in the same way when the scale length of the plasma is comparable to the wavelength. It is important to note here, that in the case of small scale length plasmas the penetration of the laser light into the medium is independent from the polarization of the incoming beam. 2.2. Frustrated total re¯ection in plasmas As was mentioned before total re¯ection takes place at the plane z z1 . An intriguing phenomenon is the penetration of the wave (evanescent wave) [16] into the denser medium (here the overdense region). The phenomenon, that the light could penetrate into an optically denser medium even if the angle of incidence is larger than the Brewster angle, is called frustrated total re¯ection (FTR). The phenomenon was studied already by Newton in the 18th century [17] and demonstrated experimentally by Bose [18] and Hall [19] in the case of two dielectric media. To investigate the phenomenon a most common solution was given by introducing a third medium [18,19], close to the ®rst one. All three media were assumed transparent at the wavelength of the operation. In case of three dielectric media a detailed description of the phenomenon has been given by Zhu [16]. In the present subsection we study how this phenomenon takes place in the case when the third media is conducting and the index of refraction is changing. In our case the ®rst medium is a thin layer near the coordinate z1 (the classical re¯ection point) and the third one is a layer near z2 (the critical density layer). The second layer is considered to be situated between z1 and z2 . It is shown here, that the variation of the dielectric function inside the second layer will not in¯uence the FTR. Following the notation used in Fig. 3 the ®rst medium is considered to have the index of refraction gl 1 , the second gl and the third gl1 . In the present analysis the layer l 1 is the last in®nitesimal layer of the underdense region, the second is situated between z1 and z2 , and the third one is considered being the ®rst in®nitesimal layer of the underdense region. The thickness of the second layer equals z1 z2 k=q ln 2. It can be seen that in the case when
q > 1 the thickness of the layer is considerably smaller than the wavelength, so it can be denoted by dz, as an in®nitesimal quantity. The transmission and re¯ection coecients of the second medium having the index of refraction gl are: t
1 tl 1 rl
id 1;l tl;l1 e 2id 1;l rl;l1 e
17
and r
rl 1;l rl;l1 e2id ; 1 rl 1;l rl;l1 e2id
18
where d is the phase dierence of consecutive re¯ected beams: q 2p dz g2l 1 g2l : 19 d k If the thickness of the second medium is almost in®nitesimally small, then e2id 1. Comparing the terms describing the primary and secondary ®elds in Eqs. (17) and (18) one can remark that the transmission coecient is larger than the re¯ection one, at an arbitrary z coordinate. Hence we conclude that the transmission coecient tends to in®nity, which means that this region will not reduce the amplitude of the electric and magnetic ®elds and no phase dierence will be introduced by this part of the plasma. It is also important that the direction of the wave vector is not in¯uenced by this second layer. In accordance with the theory presented by Ginzburg [20] our theory takes into account the increase of the electric ®eld near the classical re¯ection point. Here the coupling of the plasma wave to the electromagnetic wave near the critical surface is neglected even in the case of p-polarized incident laser beam. The bigness of the electric ®eld near the critical density is attributed to its increase at the classical turning point and its tunneling towards the critical density. The tunneling is attributed to the FTR. 2.3. The propagation of the inhomogeneous electromagnetic wave in the overdense region In this subsection we study the behavior of the electromagnetic ®eld in the overdense medium, where the electric and magnetic ®elds decrease exponentially. In high density media we cannot
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
neglect collisions. Eq. (5) can be expanded obtaining for the dielectric function: z 1
n z m n z i ; ncr x ncr
20
where m denotes the electron collision frequency, which is considered constant here. In the following we use the notations: c z 1
qz
pe ;
21
d z fp eqz ;
22
where f m=x. Thus the dielectric function can be given as z c z id z. Another unsolved problem of the laser±plasma interaction is the determination of the trajectory of the wave in the overdense material. Supposing, that no attenuation occurs in the underdense matter, the trajectory will be determined as in the general case of a refraction between a dielectric and a conducting media [21]. The same method is used as in Section 2.1. In this case the Snell± Descartes law can be written as: p p l 1 sin hl 1 l1 sin hl1 : 23 Comparing Eq. (23) with Eq. (6) it can be seen that in the overdense region the angle of refraction will have the same form as given by Eq. (7), but in this case sin h z will be complex. The formal identity will be valid for the transmission coecient and for the phase of the electric ®eld (see Eq. (15)), too. Substituting the complex dielectric function z in Eq. (15), it can be written in the form Eod z ei/ , where the amplitude of the electric ®eld in the overdense region is denoted by: Eod z
/ z
x c z c
425
sin2 h0 2 d z2 1=4
d z z cos 0:5 arctan c z sin h0 x 1 d z sin h0 y xt arctan : c 4 c z sin h0 25
Since z is complex, sin h z is complex too and this quantity has no longer the meaning of an angle of refraction [21]. The real angle of refraction can be determined from the real part of the phase of the electric ®eld. The surfaces where the real part of the phase is constant are given by the: 2
2 1=4
sin2 h0 d z d z cos 0:5 arctan z c z sin h0
c z
sin h0 y const:
26 relation, which de®nes planes whose normals make an angle hr z with the normal to the boundary. The tangent of the real angle of refraction is y=z: tan hr z
sin h0 sin2 h0 2 d z2 1=4 1 : d z cos 0:5 arctan c z sin h0 c z
27 In the case of q 40, p 5 and f 0:8 the angle dependence from z=k is shown in Fig. 6. It can be
cos1=2 h0 c z
2
2 1=8
sin h0 d z x c z sin h0 2 d z2 1=4 exp c 1 d z arctan sin z E0 2 c z sin h0
24 and / z is the phase of the electric ®eld in the same region:
Fig. 6. The dependence of the angle of refraction on the normalized coordinate z/k.
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K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
Fig. 7. The variation of the normalized electric and magnetic ®elds in the overdense region on the normalized coordinate z/k.
seen that together with the decrease of the angle the wave vector becomes more and more parallel to the z axes, according to the fact that the overdense region is an optically dense medium. If the density pro®le is steeper, so the real angle of refraction will decrease faster (see Eq. (27)). Fig. 7 illustrates the decrease of the electric and magnetic ®elds in the overdense region. It shows that the decrease of the magnetic ®eld is slower than that of the electric ®elds. The slow decay of the magnetic ®eld is in accordance with the fast increase of the amplitude of the dielectric function as it can be seen in Fig. 8. The local maximum of the magnetic ®eld can be attributed to the faster increase of the amplitude of the dielectric function than the decrease of the exponential factor, in the near critical region. It
can be seen, that after the local maximum the magnetic ®eld decreases exponentially. It can be seen, that in the case of thin plasma layers (l k) the decay of the electromagnetic ®elds takes place in the overdense region. Practically no attenuation occurs in the underdense medium, if one neglects nonlinear processes. Hence, the basic excitation processes of the electrons take place in the overdense medium. Even in this case however both the electric and the magnetic ®elds decrease within the skin depth. Therefore it is necessary to determine it and compare its value with the scale length of the interaction. In case of a medium having the dielectric function of the form (20) the skin depth was determined solving the following implicit equation: cos1=2 h0 1=8 2 2 c dskin sin h0 d dskin x c dskin sin h0 2 d dskin 2 1=4 exp c 1 d dskin arctan sin z 1 2 c dskin sin h0 28 and it was found for the parameters mentioned above being: dskin 0:7k. To avoid the appearance of the anomalous skin eect the amplitude of the electron's must be smaller than the skin depth. This amplitude amax can be determined from the equation of motion. It is considered p being amax 0:13lk, where l 10 9 k (lm) I (W/cm2 ) is the intensity parameter (I is the intensity of the laser light). Even in the case when the intensity is approximately relativistic, i.e. l 1 the amplitude is smaller than the wavelength.
3. The source of high harmonics
Fig. 8. The ratio of the magnetic and electric ®eld in the overdense region.
The nonlinear motion of electrons is considered to be the source of nonlinear properties of the scattered laser light. In this subsection we study this motion. The motion of the electrons in the ®eld of an electromagnetic wave is studied. For
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
427
will have two velocity components vx0 and vz0 . The equation of motion reads: 8 dvx0 > > m > > > dt > > > > > > <
0 0z eE exp ix t n i/1 c v z0 z0 e B exp ix t n0 i/0 c c
> > > > > dvz0 > > m > > > dt > :
vx0 e B exp ix t c
z0 n c
0
i/0 : 30
Fig. 9. The rotation of the coordinate system.
simplicity, we assume a coordinate system where the x0 y 0 z0 axes are parallel with the electric x0 k~ E, magnetic y 0 k~ B and wave z0 k~ k vectors, respectively. The equation of motion is solved in this coordinate system, which is the same for both polarizations. The quantities from the original coordinate system are transformed to x0 y 0 z0 system, and after solving of the equation of motion, the resulting trajectories are transformed back to the original coordinate system (see Figs. 1 and 2). The rotation of the coordinate system is shown in Fig. 9 for s-polarized incident beam. The transformation is made for an arbitrary z coordinate, therefore the dependence of the velocity on sin hr z is taken into account. Even in the nonrelativistic case v=c 1, the smallness of the term ~ v~ B=c can be compensated by the smallness of ~ E. This way the electron's motion will be governed by the complete Lorentz force:
The aim of our treatment is to obtain the higher harmonics of the fundamental laser frequency. Therefore we make the transformation of variable t0 t n0 z=c, where n0 depends on the properties of the material and on the angle of incidence in the case of propagation of inhomogeneous waves in conducting media [21]. The local real index of refraction can be determined comparing Eqs. (7) and (27): n0 c z
2
2 1=4
sin2 h0 d z
d z cos 0:5 arctan : c z sin h0
31
29
After performing the transformation of variables mentioned above, Eq. (30) can be written: 8 dv 0 vz 0 x0 > 1 n eE exp ixt0 i/1 m > c > < dt v z0 e B exp ixt0 i/0 c > > > : m dvz0 1 n0 vz0 e vx0 B exp ixt0 i/ : 0 dt c c 32
where the electric ®eld is de®ned as ~ E E exp i xt ~ k 0 ~ r0 /1 and the magnetic ®eld as: ~ B B exp i xt ~ k 0 ~ r0 /0 . The quantities E and B are the amplitudes of the electric and magnetic ®elds and ~ v denotes the velocity of the electron. By solving the equation of motion we get high harmonic velocity components of the fundamental laser frequency. Taking the projections of the velocity to the coordinate axes we found that the electron will oscillate in the x0 ±z0 plane and it
On the left hand side of the above equations the terms containing the velocity component vz0 are small, because vz0 stems from the coupling of the vx0 component and of the magnetic ®eld. This way it can be considered as a second order correction to the velocity. In order to ®nd the solution of Eq. (32) in an elegant way we will not neglect the small term vz0 B=c for symmetry reasons. The consistency of this process has been numerically demonstrated. This way Eq. (32) simpli®es to:
m
d~ v dt
e~ E
~ v e ~ B; c
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du dt0
e E exp ixt0 i/1 m
i
e B exp ixt0 i/0 u; m 33
where we introduced the complex velocity u vx0 ivz0 . Solving the dierential equation and separating the variables it is obtained after the back transformation that: vx 0
1 e 1 e E sin /1 /0 E xL m xm x x X 1n x n 1 L L exp z0 /0 cos c x n! x h x i xL z0 /0 n xt /0 sin sin /1 /0 c x
34
and vz0
1 e 1 e E cos /1 /0 E xL m xm x x X 1n x n 1 L L z 0 /0 cos exp c n! x h x i xL x 0 sin z /0 n xt /0 : cos /1 /0 x c
35
In Eqs. (34) and (35) xL z denotes the local Larmor frequency de®ned as: xL z
eB z : mc
36
As it is shown in Fig. 9 the velocity components in the laboratory frame are: 8 < vx vx0 ; v vz0 sin h z; 37 : y vz vz0 cos h z: Similarly we obtained the velocity components in case of a p-polarized laser light with the values of vx0 and vz0 given by Eqs. (34) and (35), as in the case of s-polarized incident beam: 8 < vx vx0 cos h z vz0 sin h z; v 0; 38 : y vz vx0 sin h z vz0 cos h z: The above velocity components contain high harmonic components, therefore they will serve as the source terms of high harmonics for a single electron.
4. The scattered laser light It is well known in plasma physics, that longitudinal current density perturbation (parallel to the surface's normal) may give rise to an electron-plasma wave [22]. The other component, the transverse one gives the source term for the re¯ected light, with high harmonic components. We are interested in the study of this latter component. Comparing Eqs. (37) and (38) it was observed that the velocity components, which give contribution to the scattered light are dierent for s- and ppolarized incident laser beam. The solutions of the Maxwell equations in the Coulomb gauge are the retarded potentials [22]. In nonrelativistic case the retarded vector potential for a point charge can be written as: Z ~ 1 vt s ~ A ~ R; t e 4p j~ R ~ r sj 1~ jR ~ r sj s ds; 39 d t c where ~ vt is the transverse part of the electron's velocity, normal to the z axes, containing high harmonic components (see Eqs. (34) and (35)). The vector ~ R and ~ r s are the position vectors of the point of observation and that of the source, respectively. Assuming the source dimension being smaller than the wavelength, we can use the approximations which are valid for the so called radiation zone [22]. In this case we can make the approximation ~ r s ~ r0 and the vector potential will get the following form: 1 1 1~ ~ ~ ~ jR ~ r0 j : A R; t e 40 vt t 4p j~ c R ~ r0 j It can be seen that the vector potential (40) characterize a spherical harmonic wave. In order to obtain the time and space dependence of the vector potential we make the following replacements in Eqs. (34) and (35): t!t /1
1~ jR c /0 !
~ r0 j;
1 d r0 arctan ; 2 c r0
41 42
K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
Fig. 10. The fundamental scattered component of the vector potential.
/0 !
1 arctan 4
d z c z sin h0
429
Fig. 11. The intensity distribution of the harmonics for a ppolarized incident beam.
1 d r0 arctan ; 2 c r0 43
i1=4 x 0 x xh 2 2 z ! sin h0 y c r0 sin2 h0 d r0 c c c d r0 r0 : cos 0:5 arctan 44 c r0 sin h0
The fundamental scattered harmonic can be seen in case of an s-polarized incident beam in Fig. 10. We considered the radiating electron being at coordinate r 0; 0; 0 and supposed, that the radiation which induced its motion has the intensity parameter l 1. Its shape is characteristic for a spherical harmonic wave. The variation of the higher harmonics will have the same form for both laser polarizations. In order to characterize the eciency of the harmonic production we calculated the time averaged intensity of the harmonics Z t 1 ~ Sn E ~ B ds; 45 2t t where t is assumed to be considerably larger than the period of the harmonics t > 2p=x. In Figs. 11 and 12 the intensity distribution of the harmonics are plotted as a function of the order for dierent polarizations. The intensity decreases with the order. The eciency of high harmonics generated by p-polarized laser beam is higher than for s-polarized laser beams.
Fig. 12. The intensity distribution of the harmonics for an spolarized incident beam.
In the case of an s-polarized incident laser beam the harmonics will be mainly s-polarized, but all this harmonics will have a p-polarized component, which is proportional to sin hr z. The p components gets important, if the number of the electrons in the underdense region is not negligible compared to the overdense pregion. The intensity parameter l 10 9 k (lm) I (W/cm2 ) determines the magnitude of dierent harmonics. The intensity of a given harmonic decreases with the decrease of the intensity parameter l proportional to the term el ln 1 , where n represents the order of the harmonic.
5. Conclusions In the present work the con®guration of the electromagnetic ®eld inside a plasma created on a
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K. Gal, S. Varro / Optics Communications 198 (2001) 419±431
solid surface by a high intensity laser radiation, whose scale length is in the range of the wavelength was studied. It was found that in the underdense region the electric ®eld increases due to the decrease of the dielectric function keeping the divergence of the electric displacement vector zero. Between the classical re¯ection point and the overdense region the electric ®eld shows the phenomenon called FTR and penetrates in the overdense region. In the overdense region the collision frequency is not negligible, so the plasma will behave as a conducting medium and will absorb the inhomogeneous light wave emerging from the underdense region. In the case of a very steep density pro®le the underdense region is thinner than the overdense region. In such cases the electric ®eld will be high enough around the critical density to expel very hot electrons from the plasma [23]. In the overdense region the electromagnetic ®eld decrease exponentially. In the present study the processes which take place in the overdense region are considered to be the main source of harmonic generation. In such a dense medium the magnetic ®eld of an electromagnetic wave becomes high enough to in¯uence the motion of the electrons. By taking into account the complete Lorentz force we obtain nonlinearities already in the ®rst step of our calculations. The electrons whose motion is governed by the complete Lorentz force are the source terms of the harmonics. The intensity of a given harmonic component is always larger at least by an order of magnitude for p-polarized incident beam than for an s-polarized incident beam. In the case of an s-polarized incident beam there appears a p-polarized component for all harmonics, but this term will be negligible compared to the s component. There is no well de®ned intensity threshold for the appearance of the harmonics. This model might explain the polarization dependence of the experiments, which was made with laser pulses longer than 500 fs [2,4±6]. In this case, if the intensity of the incident laser light was high enough to produce a plasma layer comparable to the wavelength, the light penetrated into the overdense region and produced high harmonics. The eciency of harmonic generation is determined by the parameters p and q, which characterize the density pro®le. The parameter p is
connected to the atomic number of the target material. The atomic number of the target must be high enough to neglect the processes which take place in the underdense region. In the case of targets having low atomic number the steepness of the density pro®le is small, therefore there are too much electrons in the underdense region, so the processes which take place here cannot be neglected. If the atomic number of the target is too high there are practically no electrons in the underdense region and the surface will coincide to the critical layer and will behave as the oscillating mirror as is suggested in Refs. [8,10]. The parameter q is mainly determined by the pulse length of the incident laser beam, which in¯uence the scale length of the plasma. This way, if the pulse length is shorter than 100 fs [7,10±13] the same phenomenon takes place as in the case of high atomic number targets. The experimental results reported in Ref. [6] are in agreement with the present model. The experimentally found eciency ratio of the second and third harmonic was of the order of 10 2 , and the same results was obtained here. Theoretically and experimentally it was obtained that the high harmonics preserve the polarization of the fundamental beam.
Acknowledgements The authors gratefully acknowledge I.B. F oldes, G. Kocsis, J.S. Bakos and S. Szatmari for fruitful discussions. This work was supported by the Hungarian OTKA Foundation under contract numbers T029376, T032375, T023526 and T029179 and by NATO Science for Peace Program (Sfp-971989) and by Title of project: KFKI-Condensed Matter Research Centre (KFKI-CMRC) Contract No: ICA 1-CT-2000-70029.
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