Thermodynamics of laser systems

PHYSICS REPORTS (Review Section of Physics Letters) 122. No. 1 (1985) 1—56. North-Holland, Amsterdam

THERMODYNAMICS OF LASER SYSTEMS Xavier de HEMPTINNE Catholic University of Leuven, Department of Chemistry, Celeshjnenlaan 201) F, B-3030 Heverlee, Belgium

Received October 1984

Contents: Preface 1. Irreversible thermodynamics 2. Rabi’s solution 3. Statistics 4. Equations of the motion 5. Integration—parallel 5.1. Regular regime 5.2. Chaos 5.3. Bistability 6. Integration — transverse 7. Open systems 8. Driven systems 8.1. Spectroscopy 8.2. Bistability 8.3. Bistable open systems

3 4 6 8 13 16 17 19 20 2I 24 27 27 30 31

9. Multilevel systems 9.1. Statistics 9.2. Coherent multiphoton absorption 9.3. Inverted systems 10. Radiation induced osmosis 11. Lamb dip Conclusion Appendix A. Master equations contra thermodynamics in quantum optics Appendix B. Lagrange’s undetermined multipliers Appendix C. Numerical integration Appendix D. Mode selective chemistry References General bibliography

32 33 36 38 38 40 44 45 50 50 53 54 55

Abstract: The laser is treated as a thermodynamic machine where production of radiation is the result of entropy creating spontaneous flows of energy and polarization through the system. One of the laws of thermodynamics governing flows of extensive properties is the symmetry principle of Curie. It is a fundamental law relying on the symmetry properties of Hamiltonian mechanics. Curie’s law determines how much flows are coupled. Going back to the first principles of statistical mechanics and using aforementioned symmetry requirements, a set of alternative Bloch equations is derived. The impact of the new formulation on laser physics is developed. It leads to a set of new conclusions for spontaneous pulsating regimes, for the spectral line shape of coherent radiation, for saturation effects, etc.

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THERMODYNAMICS OF LASER SYSTEMS

Xavier de HEMPTINNE Catholic University ofLeuven, Department of Chemist,y, Celestijnenlaan 200F, B-3030 Heverlee, Belgium

I

NORTH-HOLLAND-AMSTERDAM

X de Hemplinne, Thermodynamics of laser systems

3

Preface On December 14, 1900, Max Planck proposed to the German Physical Society a statistical interpretation for black body radiation based on the idea of quantization of action. This led to reconciliation of the experimental data gathered at both ends of the spectrum. Planck’s suggestion triggered a stream of consciousness that opened the way to the quantum description of the motion of individual particles. Light itself, which Maxwell had described as the manifestation of a wave-like perturbation, gained a second nature: it became a collection of photons, the signatures of individual atomic motions. Light’s intriguing double nature would remain a subject of discussion for years among leading scientists [1,2]. The invention of the ammonia maser in 1954 by J.P. Gordon, H.J. Zeiger and C.H. Townes [3] and the development of its optical analogue by T.H. Maiman in 1960 [4] after A.L. Schawlow and C.H. Townes’ (1958) suggestions [5],provided the expected experimental confirmation of collective action by radiators belonging to a given macroscopic system. Light, which is now coherent, has grown to a macroscopic property of radiating systems. Light carries energy. This must be supplied to the radiating systems by some kind of transport mechanism. Many recipes have been invented to force energy into radiating systems: electric glow or arc discharges in cooled gaseous environments, absorption of incoherent light in a cold medium, irreversible cooling of preheated gases by isentropic expansion (gasdynamic lasers), etc. Whatever is the detailed procedure chosen to this effect, the energy is stored somewhere as heat. Considering that, after its etymology, thermodynamics is the study of transport of heat in macroscopic systems and of the possible consequences thereof, it may be asserted that a laser is a thermodynamic machine comparable in some aspects to turbines (Carnot cycle) or Peltier diodes: heat is transformed into some kind of ordered alternative form of energy. It is sometimes alleged that thermodynamics describes only equilibrium or near equilibrium properties. In appendix A it is demonstrated that this allegation must be rejected. Flows of observables are indeed the direct consequences of the second law of thermodynamics: the urge for entropy creation. In the present monograph laser physics is discussed on the basis of thermodynamic principles. First principles are restated whenever it is felt to be necessary, thereby helping the reader in tracing the argumentation behind some of the more controversial statements. No matter how inventive one’s imagination may be in proposing new theories, nature remains an experimental fact to which our lucubrations must ultimately be confronted. Laser physics is a vast domain where a wealth of experimental information has been collected during a quarter of a century. Most experimental results have been described and interpreted using the currently dominating formalisms (Master equations, Fokker—Planck equations, rate equations, Bloch’s equations, .). By contrast, the thermodynamic approach has been carefully avoided [6]. Some phenomena are more sensitive than other ones to the exact formulation of the basic equations. With lasers, the spectral line shape (phase fluctuations), that depends decisively on transverse properties of radiating material, is probably the most critical data. It is unfortunately very difficult to measure accurately. There are other phenomena which have been rationalized using the traditional formalism and to which thermodynamics gives a new dimension. Worthwhile mentioning are the transition of laser regimes from C.W. to periodic or chaotic pulsing, saturation phenomena (bistability, lamb dip), multiphoton chemistry with its surprizing selectivity, geometrical phenomena resulting from the finite beam size (osmosis) etc. Critical experimental evaluation of the conclusions proposed by the thermodynamic approach against those of the more familiar theories is often perplexing because differences . .

4

X. de Hemptinne, Thermodynamics of laser systems

are usually quantitative and seldom qualitative. However the great number of adjustable parameters called upon by the supporters of the more traditional approaches contrasts with the tight requirements of thermodynamics, the laws of which restrict severely the system’s degrees of freedom. This monograph’s aim is to stimulate full and frank discussion. The author thinks that this goal is best reached by gathering in a single document the many facets of this considerably diverging approach to an old problem. This monograph is not a review paper. Its bibliographic section is therefore very skinny. In fact, the principles referred to and applied here have reached the textbook level. Some textbooks are quoted. Of course not all! Furthermore, considering the number of research papers, both experimental and theoretical, published monthly or even weekly on laser physics and related topics, trying to draft a picture reflecting present endeavour accurately enough is Utopia. The author has chosen the opposite way, thereby keeping his monographs’s length within reasonable limits. The author is indebted to Louis Bouckaert for reading the manuscript and making useful suggestions.

1. Irreversible thermodynamics The theory of laser action has been the object of many investigations to date [7—9]and present research literature remains very abundant. Most of this published material relates in some way or another to applications of Bloch’s original equations to optical systems [10]. The equations proposed by Bloch were meant to describe the simultaneous action of i) an electromagnetic field driving polarizable atoms and ii) exchange of heat and polarization between these atoms and the surroundings (parallel and transverse relaxation). The former process is a reversible Hamiltonian phenomenon, which is fully defined by the time dependent Schrodinger equation. By contrast, relaxations are irreversible. They rely on dissipative forces and hence do not fit in strict Hamiltonian mechanics. In the set of equations developed by Bloch, the driving forces acting on the flows of the extensive properties (observables) “energy” and “polarization” were taken to be a linear function of the remoteness from an assumed equilibrium value of the relevant observable. Irreversible or spontaneous processes belong to the scope of thermodynamics, the second law of which states that relaxation of non-equilibrium systems is driven by the urge for entropy creation. Let us clarify this by an example. We consider two physical systems which are separately in equilibrium. The macrostate of each is defined by a set of extensive properties (observables) Xk (energy, electric polarization, number of particles,. .). Their entropy is a function of the value of the observables: S = S(Xk). If the two systems are considered as parts of a single composite system, the total entropy equals the sum of that of the parts. Now we allow exchange of extensive properties between the subsystems to occur. The conservation law implies for every extensive property that its outflow Jk from one subsystem equals its inflow into the other one. The rate of entropy creation due to the irreversible flows is therefore: .

~

(1.1)

X. de Hemptinne, Thermodynamics of laser systems

5

Entropy creation by an irreversible process appears to be the sum of products of flows and conjugate “affinities” or “generalized thermodynamic forces”. Affinities are differences of the intensive variable (3S/3Xk) conjugate to the relevant observable (Xk). Equilibrium is characterized by vanishing affinities. After Onsager (1931) (see appendix A) we shall link the flows to the affinities by linear phenomenological equations. Coupling of flows generates off-diagonal coefficients [11], Jk~LkrAr.

(1.2)

Considering microscopic reversibility of Hamiltonian mechanics (time reversal invariance), Onsager demonstrated in his famous papers that the matrix of the phenomenological coefficients is symmetric (Lkr = Lrk). These are Onsager’s reciprocal relations. They have been verified experimentally in many systems. Another symmetry requirement, this one discovered by Curie (1908) [12] reduces the matrix of the phenomenological coefficients: “Macroscopic causes have always fewer elements of symmetry than the effects they produce”. Curie’s theorem has been reformulated by Prigogine for irreversible systems: “In isotropic systems flows and forces of different tensorial order are not coupled”. In the present monograph we shall be using this principle in several instances (flows of energy, polarization, matter, impulse). The conditions of applicability and particularly the implications of the concept “isotropic systems” as mentioned in Prigogine’s statement must therefore be carefully specified [13—15]. Suppose that our system was strictly isolated from its surroundings (conservative system). Its properties would then be defined uniquely by its (many particles) Hamiltonian. This Hamiltonian commutes with a certain number of symmetry operators representing observables. Let us clarify by an example. We consider a cell, enclosed between a pair of parallel windows and moving freely along the optical axis (z) of a cavity. If the radiation in the cavity is a standing wave (Fabry—Pérot cavity), the operation = z” commutes with the Hamiltonian. This operation switches the sign of the system’s impulse without changing its other macroscopic variables. By contrast, if the system was placed in a unidirectional ring cavity where light travels as running waves, ~ would not commute with the Hamiltonian, unless the cavity is void of radiation. The entropy being a measure of the system’s degeneracy, it is by definition invariant with respect to any unitary operation. Considering this invariance of the entropy it is easy to show (appendix A) that affinities and their conjugate observables are contravariant. (Their product is invariant.) We let now observables flow between the system and its surroundings (or thermostat). This irreversible flow creates entropy (eq. (1.1)). Entropy production is equally invariant with respect to any unitary operation, —~ —

~S)=S0.

(1.3)

Introducing the phenomenological equations in eq. (1.1) gives: =

~

LkrAkAr.

(1.4)

Let us consider two properties (1 and 2) which are allowed to flow between a system and its thermostat:

6

X de Hemptinne, Thermodynamics of laser Systems =

LUA~+ (L12 + L21)A1A2 + L22A~.

(1.5)

The response of the affinities to a unitary symmetry operator depends on the choice of the set of observables used to describe the system and of its relation to the symmetry properties of the Hamiltonian. Several cases may occur. If ~3~(A1) = A1, and ~(A2) A~or vice-versa, then, considering eq. (1.3) and Onsager’s reciprocal relations, we must have L12 = L21 = 0. By contrast, if the variables chosen to describe the system are “oblique” with respect to the Hamiltonian’s symmetry:

Ph(A2)

=

a11A1 + a12A2

(1.6)

=

a21A1 + a22A2

(1.7)

with a12, a21 0, the off-diagonal phenomenological coefficients are necessarily different from zero. Neglecting them would lead to violation of the system’s symmetry properties. Analysis of the symmetry of the Hamiltonian and of the impact of this symmetry on the intensive variables is therefore compulsory. In order to diagonalize the matrix of the phenomenological coefficients two necessary conditions are required: i) the generalized forces used in the kinetic expressions must be affinities and ii) the set of observables must be compatible with the symmetry properties of the Hamiltonian. Further considerations about Curie’s symmetry rule are treated in details in de Groot and Mazur’s “Non-equilibrium thermodynamics” (1969) [15]. There are schools where the generality of thermodynamic principles is questioned [6]. They suggest that irreversible phenomena be described by means of Master equations. No reference is made to intensive variables or affinities that are typical of thermodynamics. The predominance of thermodynamic principles versus Master equations is of course a matter for dispute. This question is discussed in appendix A. Thermodynamics yields no prediction about the numerical value of the kinetic constants nor about the precise mathematical expression of rate equations (higher order terms). Curie’s symmetry rule is however fundamental and it establishes special links between the mathematical expressions for the different flows, thereby introducing non-linearity in the overall process in a very natural way.

2. Rabi’s solution There are two ways to tackle the problem of the action of an electromagnetic field on radiating particles. In the semiclassical approach, the field is introduced in the Hamiltonian on the basis of Maxwell’s equations. By contrast, the full quantum treatment uses photon creation and annihilation operators in describing the field. It has been recognized by many authors that most results obtained with the semiclassical approach are validated by the experiment. Hence this approach will be used throughout the present monograph. Contrasting with the generally accepted belief that semiclassical mechanics fails to predict the quantum aspects of the interaction between light and matter, it will become evident that Maxwell’s equations are very general indeed, provided they are being used in connection with thermodynamic principles. In this chapter, the radiators will be taken to be two-level systems (spin systems). Let ~(t) be the electric field felt by a given radiator, the Schrodinger equation reads:

X de Hemptinne, Thermodynamics of laser systems [~‘~—

er~’(t)]çfr(t) = ih

t/i(t).

7 (2.1)

The eigenfunctions of the unperturbed Hamiltonian (~) will be symbolized by a) (energy: (Ia) and — U~ = /1w). In the presence of an electromagnetic field the wavefunction is a superposition of the stationary states: /3) (energy: U~)(Ui.

çli(t) = Ca(t) a) exp(—i Uatlh) + C5(t) I /3) exp(—iU5t/h).

4’ being

(2.2)

normalized:

CaC~+C5C~ 1,

(2.3)

the coefficients define a point (coordinates 0, 4,) on a unit diameter sphere: this is a spherical phase surface (fl. If one considers interaction with harmonic radiation with an angular frequency (WL) that differs from the transition frequency (w), the phase surface should then be made to rotate at the angular frequency v = w — WL. The angles 0 and 4, representing a given wavefunction are then defined as follows: CaC~C5C~=cos0, CaC~= ~sin 0 exp[i(vt

(2.4)

4,)].

+

(2.5)

The angles are related to the expectation values of the dipole moment (~u= ~aIerj/3)):

(4’IerI4’)

=

~ sin 0 cos(wLt 4,)

(2.6)

—

and of that of the Hamiltonian: —

er~’Iç1i)= ~(Ua+ U,~)+ (hw/2) cos 0— ~asin 0

C05(WLt —

4,) ~(t).

(2.7)

Unless specified (discussion of open systems), the energy scale will be zeroed at ~( U~.+ U~)without loss of generality. This simplifies the equations. An electromagnetic field acting on a particle causes the picture of 4’ on F to slide on the surface of the phase sphere. The differential equations governing this motion are directly related to the Schrödinger equation, ~h~a =

—~

~(t)

C~ei~ut,

(2.8) (2.9)

t. =

—~

~‘(t)

Ca e_i~0

The transformation suggested in eqs. (2.4) and (2.5) applied to eqs. (2.8) and (2.9) yields: 3, cos 0 = —2(pjh) ~(t) sin 0 sin(wLt

—

4,),

3,(sin 0 cos 4,)— v sin 0 sin 4, = 2(~a/h)~‘(t) cos 0 sin WLt,

(2.10) (2.11)

8

X. deHemptinne, Thermodynamics of laser systems

3,(sin 0 sin 4,)+ v sin 0 cos 4,

= 2(~/h)

~‘(t)cos 0 cos WLt.

(2.12)

In lasers, the main electromagnetic field acting on the radiators is a slowly fluctuating quasimonochromatic wave: =

E cos(wLt

—

4,) = E~cos WLt + E~sin WLt.

(2.13)

(It is sometimes convenient to represent the phase components of ~‘(t)as a “phasor” E.) Introducing this field in the set of eqs. (2.10), (2.11), (2.12) gives, on averaging over one cycle, the coarse grained motion of the wavefunction’s representation on the phase surface F: 3, cos 0 = (suE//i) sin(4,

—

4’) sin 0,

(2.14)

3,(sin Ocos4,)— ~sin Osin 4, = (1.tE/h)cos Osin

4’,

3,(sin 0 sin 4,) + v sin 0 cos 4, = —(1aEIh)cos 0 cos

4’.

(2.15) (2.16)

These are the Rabi equations. The electric field which intervenes in the equations is the superposition of the amplitudes of all the fields felt by the particle. There is first of all the coherent field generated by the system if this is a laser (or driving the system if it is a target). Incoherent black body radiation issued by the surroundings (thermostat) is also part of the field acting on the particles. Considering the neighbourhood of the frequency WL, the spectral energy density per unit of volume is given by Planck’s equation: J.

3

flWL

U(s)

Urad(WL)dW~

(2.17)

.

rrc exp(hw1]k~T)—1

Finally, the theory of dipole oscillations learns that generation of radiation by an oscillator causes damping of the motion. This self-action may be expressed as an effective field acting on the oscillator. For this effective field, the theory of dipolar radiation yields: =

341T0c dt

((4’jer~4’))=

~

34ir0c

3sin 0 sin(wLt -4,).

(2.18)

It will be seen later that accounting explicitly for the latter two components of the field (black body radiation and self-effect) is compulsory only with systems that are very loosely coupled to their environment (thermostat), i.e. very diluted laser driven systems. In all other cases they may be neglected versus exchange of energy and polarization with the surroundings through thermodynamic transport processes.

3. Statistics Lasers and laser driven systems are composed of large numbers of identical radiators interacting with a common field by absorption or emission of radiation. They exchange also heat and polarization with

X de Hemptinne, Thermodynamics of laser systems

9

the surroundings (thermostat). Considering the large number of particles, statistical thermodynamic techniques are compulsory and particularly well suited in describing the relevant macroscopic properties. In a cavity the field is a function of the position. We consider a region with constant intensity of the field. In this region there are N identical radiating atoms. We assume a closed system. That means: N is a constant. Picturing every atom’s wavefunction on the phase sphere described above yields a cloud of points. Its surface density is n(0, 4,). This is the “vector representation” of the density matrix. In order to describe the set of radiators the density function is expanded in spherical harmonics: n(0, ‘b)= a

+

b cos O+Pc sin0 cos 4, +PsSlfl 0 sin 4, + Z(O, 4,).

(3.1)

Z(0, 4,) contains the contributions to n(0, 4,) which are higher than dipolar. This function does not intervene in the theory of laser action because it does not couple with the laser action driving properties of the heat sources (energy, polarization, number of radiators). The extensive properties of the set of radiators are related to the harmonic comments of n(0, 4,). (do- = sin 0 dO We have: dqS.)

N=4rra=Jn(0,cb)do-.

(3.2)

The coefficients Pc and Ps (the components of phasor j3) measure the phase components of the net polarization of the system [i3~(t)= N(4iIerI4i) = (4~-J3)~ (Pc cos WLt + P. sin wLt)I. (The bar on top of the matrix element represents an ensemble average.) Considering eq. (2.6) we have indeed: (4irI3)p~=

J

sin0 cos 4, n(0, çb)du,

(3.3)

(4ir13)p

J

sin 0 sin 4, n(0, 4,) do-.

(3.4)

5 Using eq.

=

(2.7),

U = (hw/2) =

the total energy of the system turns out to be:

J

cos 0 n(0, 4,) do-

—

13b(t) ~‘(t)

(4ir/3) (hw/2) b ~(4~/3)~.~(j3E). —

.

(3.5)

The “dot product” (~E) stands for (j3 E) pcEc + p~E5. Let a collection of states (i), each with a degeneracy 11l~ be populated with n identical particles. The result of Maxwell—Boltzmann’s statistics for the entropy of the distribution reads: .

~ 11~’i S=k8log~{fl—~-j. i

ft.

.

(3.6)

10

X de Hemptinne, Thermodynamics of laser systems

Transposing this result to the distribution in our phase surface requires the following substitutions: 111

—*

n

du/2ir,

=

n(0, 4,)do-.

The entropy of a particular distribution becomes then: SkmJdo-[n_nlOge2irn].

(3.7)

The question is now whether thermodynamics may provide predictions for the distribution n(0, 4,). The argumentation which follows suggests that the distribution is and remains “ergodic”, or in other words that its entropy is maximized, while respecting the external constraints (see appendix A). The Rabi equations (2.14, 2.15, 2.16) show that when a coherent electromagnetic field is acting on the system, the wavefunction’s images describe rigorously parallel motions. The shape of n(O, 4,) is not affected by the radiation. Only its orientation changes. By contrast eq. (3.7) indicates that the entropy of the set of radiators depends only on the shape of the distribution. Hence coherent light interacting with the system does not change its entropy. This is a “reversible” process in thermodynamics’ sense. Starting from an ergodic distribution, light does not perturb the ergodicity. In laser systems the radiators interact also with their surroundings (thermostat), exchanging energy and polarization. The thermostat is defined here as being the whole of particles, not resonant with the electromagnetic field, that may act on the radiators by collisions or any other exchange process (black body radiation). Transport phenomena occurring between the thermostat and the radiators are “irreversible”: their driving force is entropy creation. Flows of energy and polarization between the thermostat and the radiators may be thought of as being the superposition of two elementary processes: i) pure exchange of extensive properties and ii) rearrangement of the distribution function n(O, 4,) towards its maximum entropy configuration, with conservation of the set’s energy and polarization. One of the processes is rate determining. If step i) is fast, the set of radiators remains equilibrated with the thermostat. This being by definition not resonant to the radiation, and therefore unpolarized (at the optical frequency), straight equilibrium means that the radiators remain equally unpolarized. Laser action is then impossible. Hence step i) must be taken to be rate determining, while the wavefunction’s distribution remains perfectly ergodic. Using the method of Lagrangian multipliers (see appendix B), the statistical solution for an ergodic distribution compatible with a given energy (U) and polarization (~(t)) is found to be: n(0, 4,) = (N/f) exp[/3(hw/2) cos 0 + (A~ /3E~/2)~asin 0 cos 4, + (A. f3E~/2)~.t sin 0 sin 4,]. —

—

(3.8)

The partition function f(13, A~,A5) normalizes the distribution: f(/3, Ac, A~)=

J

do- exp[f3(hw/2) cos 0+ (A~ f3Ej2) ~ssin 0 cos 4, —

+

(A. f3E5/2) ~i sin 0 sin 4,]. —

(3.9)

The Lagrange multipliers /3, A~and A. that were introduced in the process of maximalization are intensive variables defining the state of the system. They are related to the extensive properties by the following logarithmic derivatives:

X. deHemptinne, Thermodynamics of laser systems

11

U=Ns9log~fI3$,

(3.10)

(4ir13) /.tPc = Na log~f/aA~,

(3.11)

(4ir/3) PPs = Na loge fl3As.

(3.12)

It is convenient to introduce a “vectorial” intensive variable (/3’, 0,.,, 4,~)by using the following transformation:

/3’ cos 0,., = (/lw/2) /3,

(3.13)

/3’ sin 0~cos 4,,., = ~(A~—/3Ec/2),

(3.14)

/3’sin 0~sin4,,,= ,a(A,.—/3E,./2).

(3.15)

It is easily verified that: (!hW/3)2

(/35)2

+ /.L2(Ac

—

/3 E~/2)2+ p~2(A.

— /3

E,./2)2.

(3.16)

Using this substitution, the partition function integrates as:

f

4ir sh(13’)//3’.

(3.17)

The entropy of the set of radiators, calculated on the basis of eq. (3.7), for the ergodic distribution (3.8) is: =

kB [loge (f~’2n)” N/3’ x(/3’)]. —

(3.18)

In this equation, X(/3’) is a generalized thermodynamic free energy: X(j3) =

3

logef(f3’)/s9f3’

=

{/3’ ch(/3’) sh(/3’)}//3’ sh(/3’). —

(3.19)

The partitial derivatives of the entropy versus the extensive properties are the so-called conjugate intensive variables. If one considers the energy and the phase components of the polarization as the system’s observables, the conjugate intensive variables are /3, A~and A,.: (1/kB) (aSI3U)~~.~, = /3

(3.20)

(4ir13) ~t A~,

(3.21)

(1/kB) (3SIop,.)~,~, = —(4ir/3) ~ A~.

(3.22)

(1/kB) (3S/ôpc) u, p,. =

—

By contrast, if the harmonic components of the distribution function are envisaged, the conjugate intensive variables are the components of the “vectorial” intensive variable,

12

X de Hemptinne, Thermodynamics of laser systems

(1/k8) (s9S/ t9b) Pc,p, = —(4ir/3) (hw/2) /3

(3.23)

(1/k8) (aS/ape) b,p~

(4ir/3)~u(A~—/3E~I2)

(3.24)

(1/kB) (3S/ôp,.) b,p.

(4ir/3) ,ii (As

=

—

/3E,./2).

(3.25)

In lasers, the heat bath that interacts with the radiators is characterized by an inverted population. This is usually obtained by the action of two independent sources at different temperatures, exchanging energy with different efficiencies with higher or lower energetic radiator particles. The heat bath is therefore characterized by a high energy at low entropy. In two-level systems it is easily described by means of a negative temperature. 3th For=the homogeneity the notations we shall define the reciprocal —(kBT,h)’. The ofintensive parameters conjugate to the phase temperature of the heat bath as / components of the polarization of the thermostat will be equally indexed with “th”. As the heat bath is by definition unpolarized, we have: ASh

—

f3thE/2 =

0.

(3.26)

Flow of extensive properties between the heat bath and the set of radiators is associated with creation of entropy. The extensive properties that will be considered here are the harmonic components of the distribution function: b, Pc and Ps. Taking account of eq. (3.26), the entropy creation reads: =

(4ir/3) k

8 [(hw/2)Jb( f3th

—

/3)

—

/.L

J~~(A~ /3E~/2) —

— p.

JJ,,(AS

—

/3E,./2)].

(3.27)

According to the principles of irreversible thermodynamics, the affinities or differences of the intensive variables in the set of radiators and in the thermostat act as generalized thermodynamic forces driving the flows. Though the precise relationship between the thermodynamic forces and the flows is model dependent, it surely contains a linear term. We shall consider the relationship to be linear. As the flows are related to properties which behave as orthogonal components of a three-dimensional vector, the dipolar components of n(0, 4,), application of Curie—Prigogine’s principle implies the absence of cross terms. Hence the flows read: Jb =

Jp~

=

(3/41T)

Ny11 (hw/2) (13th /3), —

(3.28)

—(3/4i~)Ny1 p. (A~ /3E~/2),

(3.29)

(3/4i~)Ny1 p. (A,. /3E~/2).

(3.30)

—

—

—

It should be stressed that the matrix of Onsager’s phenomenological coefficients would not have been diagonal if flows of U and had been considered as functions of their own conjugate affinities: Though the former one is scalar and the latter one behaves as a vector in a plane, these variables are indeed not linearly independent and neither are the conjugate affinities. The system’s Hamiltonian (eq. (2.1)) remains unchanged if both the signs of the elementary charge “e” and of the field’s amplitude are reversed simultaneously. Switching the sign of “e” is equivalent to sign reversal of “p.” (see eq. (2.6)). j3

X. de Hemptinne, Thermodynamics of laser systems

13

Working out eqs. (3.10), (3.11) and (3.12), using eq. (3.16) for the interrelation between the intensive variables, gives: (4ir/3) b

=

N X(f3 ) (hw/2) /3,

(

(3.31)

(A’—/3E/2).

4~/3)NX~)p.

(3.32)

From there we have: 2[E.(,~ /3E/2)]}.

(3.33)

—

U = N X(/3’) {(hw/2)2 /3 ~p. —

If the system’s extensive properties U and j3 are to remain unchanged on applying aforementioned symmetry operation (p. —p.; E -÷—E), eqs. (3.32) and (3.33) indicate that the conjugate intensive variables A and /3 do not change into ±Aand ±13unless E = 0 or /3 = A = 0. Variables U and j3 and their conjugate intensities appear to be oblique with respect to the system’s symmetry elements. This fact introduces off-diagonal elements in the corresponding phenomenological rate equations. By contrast, the transformation of coordinates defined by eqs. (3.13), (3.14) and (3.15) diagonalizes the matrix by introducing intensive variables (/3 and A /3E/2) which share the symmetry of Hamilton’s operator. Equations (3.28), (3.29) and (3.30) are the rate equations for parallel and transverse relaxation. They define two phenomenological coefficients (y 11 and y~).The equations are clearly at variance with the traditional rate equations, where flows are linearly related to the values of the relaxing properties themselves. By inspecting eqs. (3.31) and (3.32) it is clear that non-linearity and virtual coupling of the flows is introduced in the formalism by the function X(/3’)/f3’ that intervenes as a factor in the two equations. —~

—

4.

Equations of the motion

The continuity equation for the distribution n(0, 4,) combines the Rabi circulation of the wavefunctions on the phase sphere (velocity v) and a source J(0, 4,) that describes exchange of extensive properties between the set of radiators and the thermostat, ñ(O, 4,) =

—

div(nv) + J(0, 4,).

(4.1)

The spherical harmonic expansion of the Rabi term is obtained by multiplying the set of eqs. (2.14), (2.15) and (2.16) by n(0, 4,) and integrating on the phase surface. The components of J(0, 4,) are the thermodynamic flows (eqs. (3.28), (3.29) and (3.30)). By combining the results, the equations read: =

(p.1/1) (p,.E~ p~E~) + (3/4ir) Ny11 (hw/2) (13th

p~ vp~= —

—

(p.131)

—

bE. (3/4ir) Ny1 p. (A~ /3Ej2), —

—

/3),

(4.2)

(4.3)

14

X de Hemptinne, Thermodynamics of laser systems

+

~Pc =

(p./h)

bE~ (3/4ir) Ny1 p. (A,. —

—

/3E5/2).

(4.4)

These are the thermodynamic Bloch equations. In a laser the intensity of the field is position dependent. It follows that the harmonic components of the distribution function are also functions of the position. They should therefore be treated as densities (per unit of volume) and N should be replaced by the density of radiators. In laser systems, the set of Bloch equations must be supplemented by a suitable field equation, the whole forming the equations of the motion. The field equation accounts for the propagational properties of light in the cavity. It is based on Maxwell’s equations: 2~(z,t) 1 32~’(z,t) 82~(z,t) 45 3 3z2 c2 8t2 0~2 ~2 ( ) —

—

1

.

In a laser cavity, the amplification of light (bulk) and the load (windows) are located in different places. This makes solutions of the wave equation in the form of progressive waves or standing waves to be only approximations. Their validity requires that the round trip gain and loss be small versus the inverse round trip time. In the opposite case the system “superradiates”, an interesting phenomenon by itself that will however not be developed here. So we take the validity of the standing wave or the progressive wave approximations for granted. Cavities which are usually dealt with are Fabry—Pérot resonators where standing waves build up and the unidirectional ring cavities where light circulates as progressive waves. The advantage of considering the latter case is that all parts of the amplifying medium are addressed to successively by a field of approximately equal amplitude. Provided the “fine grained” time coordinate is given a suitable origin, the “mean field” treatment is correct. (The “fine grained” time coordinate measures time evolutions to the scale of the wave’s inverse frequency. This is opposed to the “coarse grained” scale that is compatible with the slower amplitude and phase modulation of the radiation.) The “mean field” treatment has been adopted by many authors. It will be used here too. The standing waves growing in a Fabry—Pérot cavity cause the field to have a different impact on radiators, depending on their position in the resonator. If the different regions of the cavity are strictly isolated from each other (solid state device), making transfer of matter, energy and polarization between adjacent regions of the amplifying medium impossible, then the heavy load due to production of electromagnetic energy where the amplitude of the field is highest results in a geometrical modulation of the inversion (spacial hole burning). In the discussion about “open systems” it will be demonstrated that radiators that are free to move (gas lasers) tend to accumulate in high field regions by translational relaxation (radiation induced osmosis). If this is the case, the amplifying power of the medium concentrates by itself where the load is heaviest, this is in regions where the gradient of the field is smallest (high amplitude regions of the standing waves). The impact of differential loading of the radiator’s inversion is thereby greatly reduced. This reduces the deleterious effect of hole burning on the laser output intensity and makes the “mean field” treatment a fair approximation in Fabry—Pérot cavities as well. In optically thin media, where ~(t)I2 = c2 I~(t)I2,the energy in the field averaged over one cycle is (V = volume of the cavity): Urad =

~

V/2) (E~+ E~).

E~,E,. are phase components of the amplitude in the wave’s maxima.

(4.6)

X. de Hemptinne, Thermodynamics of laser systems

15

On performing their Rabi circulation, the radiators feed energy into the electromagnetic field, but in the mean time energy escapes through the optics with a rate constant (~V/2) (a~+ 2K) (E~+ E~)= (pcEs p,.E~). —

2K.

The energy balance reads: (4.7)

Separation of the equation into its phase components defines two equations which are to be added to the set (4.2), (4.3) and (4.4) to form the equations of the motion for laser action, E~+KE~~(41r/3)

E~+KE,.(41r/3)

/1

EoV /1

(4.8)

Ps,

(4.9)

Pc.

In a tuned cavity K is a function of w. In monomode arrangements it is highly peaked at wi,. Spontaneous emission of radiation has not been mentioned explicitly in the equations of the motion. In fact it is part of the energy flow prevailing between the set of radiators and its thermostat. The transport mechanism of this part of the energy flow is exchange of black body radiation. Let us consider a set of radiators in equilibrium with its thermostat: /3 = /3th = —(k 8T)~A~= A,. 0. In terms of the stationary state population we have: NaNse$~~~0,

(4.10)

U = (hw/2) Na (1— e~$*~~) = (4ir/3) (hw/2) b.

(4.11)

Planck’s formula for the spectral density of the black body radiation in the neighbourhood of the resonant frequency states:

Urad(W)

dw =

dw exp(—/3hw)

—~--~

IT c

—

1

(4.12)

.

This spectral density is related to the autocorrelation function of the field according to the Wiener— Kintchine theorem: 2/IT)

(1I6~)Urad(W) =

(

J

= (1/2ir)

ds cos(ws) (~(t) ~(t+ s))

(E(t).

f

E(r) dT).

(4.13)

The hooked brackets represent time averages. In this equation the factor 1/6 accounts for the propagation of radiation along three directions, the waves being uncorrelated, and for two states of polarization. Introducing this result in the integration of eqs. (4.2), (4.3) and (4.4) yields:

16

X. de Hemptinne, Thermodynamics of laser systems

=

J E(r)

(p./h)~bKE(t)

dr).

(4.14)

The rate of transfer of photons into the field is 1J7(hw) = (2mr/3)/i By replacing in eq. (4.14) b using eq. (4.11), the known Einstein coefficient for spontaneous decay is obtained: —(p./h)2

U/(hw)

by

Na

/13 ~‘

6ir0c

3Na.

(4.15)

This effect of black body radiation is balanced by the energy loss due to the self-effect of oscillating dipoles (eq. (2.19)).

5. Integration parallel —

The equations of the motion for laser action are the equations (4.2), (4.3) and (4.4), and (4.8) and (4.9). In view of their integration it is useful to perform the substitutions proposed by eqs. (3.13), (3.14) and (3.15). Considering eqs. (3.31) and (3.32), the harmonic components of the distribution function (b and j3) are easily expressed in terms of the generalized potential X(j3’) and of the angles 0,., and 4,~.(The suffix of 0,., and 4,,., will be omitted.) The cavity will be supposed to be untuned or broad banded and tuned to resonance with the radiators. For homogeneity of the notations we introduce a dimensionless variable /3~h to characterize the inversion of the thermostat, 13~h=

(hw/2) /3th =

—

hw/2knT,h.

(5.1)

The transformed equations read:

XO =

X sin(4i —44— ~fIIf3~sin 0 + (y11 3th cos 0— ~ cos2 0 + 71 sin2 0)13’,

(p.E/h)

—

y±)/3’ sin 0 cos 0,

(5.2) (5.3)

X = y11 f

E + KE (p.1/1) (hw/2) NX sin 0 sin(4, -44, 0V

E(çb ~) —

= ~ [(/10)/2)

/1 Ecu = (p.1/1)

2 cot 0] cos(4’ —44,

(5.4) (5.5)

NX sin 0- E

e~V

NX sin 0 cos(çli —44.

(5.6)

The last equation (5.6) is separable and will be documented in chapter 6. It determines the phase fluctuations of the output radiation.

X. de Hemptinne, Thermodynamics of laser systems

17

The mutual phase angle (i/i 4,) relaxes according to eq. (5.5). The sign of the function between the brackets decides whether the angle goes to 17/2 or to ir/2. In normal laser operation the sign is positive (sin 0 1; cot 0 0), yielding a relaxed value of (4’— 4,) = 17/2. After relaxation of the mutual phase angle (cos(4i = 0) we are left over with three first order non-linear differential equations. —

—

—

4,)

5.1. Regular regime The coefficients in the equation define a characteristic time for the evolution of the laser variables: =

(p.1/1) [(/1w/2)]1/2

(5.7)

The first step in solving the equations is the search for steady state solutions. In view of the high non-linearity of the equations this must be done numerically. When the steady state values of the laser variables /3’, 0 and E have been calculated for a given set of relaxation constants (yli, 7’ and K) and a given inversion of the thermostat (/3~h),the stability of this solution is estimated by linearizing the equations in its neighbourhood. This yields a characteristic equation of order 3, the roots of which are easily computed (appendix C). The results are mapped in fig. 1 for the value I9th = 10 given to the inversion of the thermostat.

I

.03 A~j~ ~

B

\\

SUBTHRESHOLD DOMAIN

—~

3

__

-2

-

1

10910 ~i

~ t

1

)

Fig. 1. Map indicating the laser regimes as a function of the relaxation parameters y~,y~and ic. The resultsare based on the stability criterion of the steady state focus. Calculations assume (hu/2) $th = 10. The relaxation parameters are scaled versus the characteristic time defined by eq. (5.7). Analogous results calculated for (hw/2) $th = 5 are published in J. Chem. Phys. 79 (1983) 727—735. The scaled values of y~are indicated on the relevant curves. Region A: exponential relaxation. Region B: damped oscillating relaxation. Region C: limit cycles = spontaneous pulsing, periodic or chaotic.

18

X. de Hemptinne, Thermodynamics of laser systems

Threshold for laser action is dictated by the condition: [X(~i3~)]2//3~1,.

(5.8)

In subthreshold conditions (> sign in the latter inequality) the trivial solution (0 = 0, E = 0, /3’ = yields three negative roots to the characteristic equation, indicating stability. Above threshold at least one of the roots is positive: the trivial solution becomes unstable. Beyond threshold three cases may occur, depending on the values of the relaxation constants. In region A (fig. 1) the three roots of the characteristic equation are real and negative: the focus is stable. Fluctuations relax exponentially. In region B, one root is real and negative while the two other ones are complex with a negative real part. The focus is again stable but fluctuations relax by performing damped oscillations. In region C the real part of the complex roots is positive: the steady state is unstable and any perturbation drives the laser variables into a limit cycle where the output electromagnetic field is pulsed. The traditional Bloch equations do predict also the possibility of occurrence of limit cycles. However the conditions required to this end (inversion) are extreme and it has been recognized in the literature that they are unattainable in real systems [16].By contrast with thermodynamics limit cycles come very naturally, even in mild conditions of inversion, because the equations introduce a coupling between flows of energy (b) and polarization (j3). The periodicity and the shape of the pulses in the limit cycle domain can only be determined by numerical integration (Runge—Kutta). An example of the result of such a computation is given in fig. 2. -

point

b”

200

400

600

800

timelt •

point

•‘c~’

200

400

600

800

time It Fig. 2. Examples of limit cycle trajectories. The time scale is indicated in units of ~ (eq. (5.7)). Radiated intensity: arbitrary scale. The values of the relaxation constants (Vu. yi and K) and the inversion ((has/2)/3th = 10) are those indicated on fig. 3: points b and c (K/y1 = 10).

X. de Hemptinne, Thermodynamics of laser systems

19

Pulsing behaviour of lasers is quite fascinating and it has triggered intense research in recent years, especially after Casperson’s (1978) observations with the Xe laser [17].The very systematic experiments performed by N.B. Abraham and coworkers [18]indicate that the tendency of the Xe laser to deliver its radiation as pulses increases on decreasing the pressure of He used as a buffer gas. This observation is compatible with the result on display in fig. 1. In this figure, it is shown indeed that limit cycle behaviour is favored by a low value of Yii (energy transfer to the radiators). This is probably what N.B. Abraham has reached on lowering the buffer pressure. It may indeed be anticipated that a buffer increases the efficiency of energy transport in the system while the small polarizability of He should not contribute much to the transverse relaxation (y,). 5.2. Chaos There is a broad range of values of the relaxation constants y~, y, and K where the strong non-linearity of the three “parallel” equations ((4.2), (4.3) and (4.4)) create spontaneously a pronounced chaotic behaviour. Results of the corresponding Runge—Kutta simulations are mapped in fig. 3. This figure displays the region enclosed inside the “limit cycle” domain, where pulsing of the laser variables turns systematically chaotic. Careful inspection of the calculated trajectories indicates that chaos originates from beating of two oscillations generating phenomena. These are: i) slow recovery after a pulse (dominating in the upper periodic region) of fig. 3 and ii) a long cavity lifetime (dominating in the lower periodic region). Discussion of the various forms of chaos is beyond the scope of this monograph, though the question is interesting indeed. Previously mentioned measurements of N. Abraham’s group on the Xe laser indicate that the laser regime tends to change from stable over periodic to chaotic on tuning the cavity towards the line centre. Let us therefore examine the role of detuning (i’) on the equations of the motion. Applying the vectorial transformation to eqs. (4.3) and (4.4) in the non-resonant case results in the

0

S UBTH REI S HO LD DOMAIN

I

~

2

STEADY REGIME —2

I

-2

-1 10910

I

Fig. 3. Map indicating the laser regimes as a function of the relaxation parameters with (hwI2) $th

=

10 and ely

1 = 10. The dotted lines a, b and c are explained in the text. Conditions represented by pointsb and c have been taken for calculating the trajectories in fig. 2. T—T’: threshold conditions.

20

X de Hemptinne, Thermodynamics of laser systems

modification of eq. (5.5) as follows: E(4’ -4,

Now

(4’

—

-

ii)

(p./h)

NX sin 0 + E2 cot

o] cos(4’ -44.

(5.9)

4,) does not relax to a stable value. It oscillates about

(cos(4’—

44)= —((p.E/h) v[Y2X

sin 0+ (p.E/h)2 cot

(5.10)

0]_1).

This lowers the average value of sin(4’ 4,) to be used in eqs. (5.2) and (5.4), thereby modifying the time scale (eq. (5.7)) associated to the equations of the motion. Y is increased by a factor (sin(4’ We take a laser characterized by a given set of relaxation rate constants ~ 7’, K). If the cavity is detuned versus the radiator’s resonance, all the scaled values to be plotted on the relevant maps (fig. 1 or fig. 3) increase by the same factor. In particular this makes the working point plotted in fig. 3 to move along a 45 degree line up towards the system’s threshold. Depending on the set of rate constants valid at the line centre, detuning should keep the system in a C.W. regime (line a), or it should bring it from periodic pulsing over a C.W. regime to the system’s threshold (line b) or it should be chaotic near the line centre, periodic when detuning is low, steady at moderate detuning and dead far from resonance. This is exactly what N. Abraham’s group has observed [18]. —

—

5.3. Bistability We finally turn back to subthreshold conditions, showing that the equations of the motion predict bistability for a limited range of values of the laser parameters. The steady state form of eq. (5.3) gives a relation between /3’ and 0. The potential X(/3’) may then be computed using eq. (3.19). This enables calculation of the function (X2 cos 01/3’), which is plotted against 0 in fig. 4. 15

1t14

Tt/2

e Fig. 4. Function [X(fl’)]2cos 9/fl’, obtained by eliminating E between the steady state forms of eqs. (5.2) and (5.4), plotted versus the angle 9. The steady state focus is the intersection of the relevant curve with the horizontal X2 cos 9113’ = e y~r2. If y~>y~jand if e VI is greater than its threshold value, there are two intersections (besides the stable trivial solution). The right-hand side one is stable.

X. de Hemptinne, Thermodynamics of laser systems

21

By eliminating E between the steady state forms of eqs. (5.2) and (5.4) the following relation is obtained: X2CoSO//3’=Ky.L9~.

(5.11)

The steady state solutions of the equations of the motion are obtained by intersecting the relevant curve on fig. 4 by the horizontal defined by eq. (5.11). The normal threshold is given by the point where the curves on fig. 4 meet on the vertical axis (eq. (5.8)). If y~~ y~the existence of an intersection requires Ky 1 to be smaller than its threshold value. By contrast, if yi> Yli there is a range of values of Ky1 greater than the normal threshold where the horizontal intersects the curve in two points. This indicates that two other foci exist besides the stable trivial solution. The right-hand intersection is found to yield a stable steady state (negative roots of the characteristic equation) while the left-hand side one is unstable. The system is bistable. In order to observe this bistability, a cavity with a variable loss factor (K) is required. The laser is then brought to radiate by lowering K under its threshold value. If K is then increased, thethesteady state 2) until maximum solution shifts along the relevant curve and the laser remains in action (0 < 0 < IT! of the curve is reached. At this point the laser stops radiating and the variables return to their trivial value.

6. Integration transverse —

Equation (5.6) predicts that the average rate of change of the output radiation’s phase angle vanishes as soon as the mutual phase angle 4i 4, has relaxed (cos(4i 4,) = 0). It will be shown here that the average square rate of change of i/i does not vanish. We consider conditions where the laser variables relax to a stable focus (regions A or B in fig. 1). Let i/i = 0. We have then: —

E~=E;

E=0.

E~=0;

—

(6.1)

Equations (4.8) and (4.9) yield: (hw/2) KE~=(4IT/3)(/2/h)

~, 0 V

Ps,

(6.2) (6.3)

and eq. (3.32) indicates: A~—f3E~/2=O.

(6.4)

On squaring eq. (4.9) and averaging afterwards we get ~

4i~

/1w

2_

p~.

(6.5)

22

X. de Hemptinne, Thermodynamics of laser systems

Here we assume Gibb’s ergodic hypothesis, which is one of the foundations of statistical thermodynamics and has never been disavowed by the experiment. It states that with ergodic distributions a time average on one system equals an ensemble average over systems ergodically spread over the phase space. The left-hand side of expression (6.5) represents the average square transverse velocity of the phasor 2~/J2.The right-hand side is proportional to the average square fluctuation of the relevant phase E or E component of The statistical properties of the complete set of N radiators are described by means of the canonical partition function fN!N!. Let dF represent the volume element of the N2-dimensional phase space (df’ = H~do~).We have: ~.

JNIN! =

J

~

dF exp[f3U

+

(4ir/3) ~ ()~j3)1.

(6.6)

.

The relevant average phase components of the set’s polarization are obtained by deriving the canonical partition function with respect to the conjugate intensive variable (compared with eq. (3.11)), 41T

~

~~ipc

Ia!TM

1

4ir

~fdr

‘lIT

Pcexp[13U+~IL (A .~3)].

(6.7)

The thermodynamic average square of this phase component is the second derivative of the canonical partition function: ~1~~

2 ~~2p2~JdF()

1

—

a2f N

1

flIT

2

4IT ,i2p~exp[f3U+_[L (A .j3)].

(6.8)

The individual partition function f(f3’) (sum over states) is taken from eq. (3.17). The relationship between the intensive variables is fixed by eq. (3.16): (8/3’/8A~)~, ~, = ~ (A~— /3E~/2)/,8’.

(6.9)

Considering condition (6.4) we have: N

2X(13’)/f3’.

(6.10)

1a

This result is to be transferred in eq. (6.5). The average value of the field intensity E2 is taken from eq. (6.2) where the relevant phase component of 13(Ps) is given by eq. (3.32) (sin 4, = 0) and eq. (3.15): (4IT/3)~p~=

N~tX(f3’) sin 0.

(6.11)

Putting all this together yields the average rate of change of the output radiation’s phase angle j2_ ‘~‘

K2

N,8’X(,8’)sin20~

612

23

X. de Hemptinne, Thermodynamics of laser systems

Close to saturation, this is in the “good cavity” limit, we have sin 0 limiting value of 4,2 becomes then:

lim (4, )=

K 3YVf3th

8-s,r/2

,

2(hw/2) (p.1/1) 0V

—~

1 and

/3’/X(13’)

—~

3. The

6.13

.

For this result, the limiting value of /3’ has been determined by eliminating the field’s amplitude (E) between eqs. (5.2) and (5.4). By applying the procedure developed above to higher order fluctuations of the relevant phase component of j3, the corresponding higher order derivatives of the canonical partition function are involved: 4,2n

K = ~1a~f” aA~ [a log~(fN)/aA5]

(6.14)

.

In macroscopic systems (N—~x),functions multiplying lower powers of N may be neglected versus the higher power ones. This simplifies drastically the process of successive derivations. As a result, the following recursion shows off: 4,2n

...x(2n_1)(~2)hi

1X3X5X

(6.15)

Fluctuating variables are conveniently described by means of an autocorrelation function. For the amplitude of the electric field in the cavity in steady state conditions this reads: 2 K(s) = ~E(t)E(t + s)) = E2 cos[4,(s) 4i(0)]. (6.16) E The hooked brackets refer to time averages on a given system while the bar on top of the function is a statistical average over an ensemble. Once again the equality is found on Gibbs’ ergodic hypothesis. In order to perform a Maclaurin expansion of the correlation function, its successive derivative must be determined for s = 0. Considering the absence of correlation between the successive derivatives of 4, we have: .

—

K(0)= 1

(6.17)

K”(0)=—~

(6.18) (619)

K’~(0)

—

+ 45

~

~2

—

10

~2

The odd derivatives vanish. Using the result presented as eq. (6.15) the correlation function reads:

(6.20)

24

X. de Hemptinne, Thermodynamics of laser systems

K(s)= 1_

2s2+~(~ii2)2s4_S(c~2)3s6+

+F(s)

(6.21)

where F(s) =

~

—

s’~+

-~

(45 ~

—

10 ~2)

~6

~

(6.22)

The latter function depends on the rate of creation of fluctuations in thermodynamic systems. Though a given system jumps in the course of time from one configuration in the phase space to another one on every integration with the surroundings or with black body radiation, adopting one after the other all the positions that are statistically accessible, averaging over an ensemble is a static (time independent) process. Every element of the ensemble is characterized by a given distribution function in the phase space. On averaging over the properties of the ensemble the kinetics of the jumps do not intervene. Hence the ensemble averages of powers of second and higher derivatives of 4, vanish, and so does F(s). Equation (6.21) sums up as follows: / 2 \1/2( K(s) = (—=~=) J di~exp(—~j2/24,2)cos(~s).

(6.23)

IT 4,2

The correlation function is a Gaussian and its spectrum is also a Gaussian. This result is opposed to the generally accepted belief that the spectral line shape of laser radiation should be Lorentzian. If the resonator presents a sharp 0-factor or if it is slightly off-resonant, the spectrum is expected to be deformed. Its width and shape should however still be the expression of the thermodynamic properties of the radiating system.

7. Open systems Atoms in a beam of light may present a number of nearly degenerate states (rotational manifold) of which only one is resonant with the field. The non-resonant states belong to the heat bath. Besides exchange of energy and polarization, there may be exchange of matter between the set of resonant atoms and the thermostat. In this case the thermodynamic system representing the set of radiators is said to be an “open system”. In thermodynamics this concept has indeed been reserved to systems where the number of atoms is not fixed and where it depends on the conditions of the environment. “Chemical” equilibrium between the system and its surroundings requires equality of the “chemical potential” or of the “absolute activity” of the system and of the reservoir with which exchange of matter is allowed to occur. Neglecting a factor /3 the chemical potential is the derivative of the entropy versus the number of atoms at constant energy and polarization. For adequate comparison of the statistics of the resonant particles and of their heat bath analogues, a suitable common origin must be chosen for the energy scale. The best choice is to zero the scale at the heaviest populated heat bath level. If the system is inverted and used as a laser we take: Ua = 0, U~= —/1w. For non-inverted (driven) systems the convention gives: U, 9 = 0, U,. = hw. In all cases the ratio of the minority versus the majority populations in the heat bath is: exp(—P/3~hIhw).

X. de Hemptinne, Thermodynamics of laser systems

25

Introducing the new origin in eq. (3.5) yields: U__(hw/2)(~b÷N)—~~ii(P~E).

(7.1)

(The upper (negative) sign refers to inverted systems (/3 > 0), the lower (positive) sign to non-inverted (driven: /3 <0) systems.) The conservation laws are related to exchange of energy, polarization and number of atoms. Considering eq. (7.1), it is clear that conservation of U and j3 implies a relationship between the observable “inversion” (b) and N. We have indeed: (4ir/3) (~b)u,,9= ~ Hence the set of linearly independent observables is N, ((4ir/3)b ±N) and j5. Using this set of observables and eq. (3.18) for the entropy, the intensive variable conjugate to the number of atoms becomes: 1

/as\

rf(/3’)l

—i—i

kB

\aNJU,P

=logi—i—/3 eL2ITNj

~hw i— \ 2

.

(7.2

Exchange of particles with the thermostat is a thermodynamic flow which intervenes in the definition of the general source term J(0, 4,) in eq. (4.1). Its kinetic expression must be discussed in terms of the relevant intensive variable. To the first order we consider a linear relationship. The Curie—Prigogine principle is also applicable to this flow because the observables are linearly independent; there are no cross terms,

7N1[fOs\ ~lR~)]

JN

1

11as~ ~

L\.~)~,~JJ

(7.3)

Flow of matter between the thermostat and the resonant state requires transitions between nearly resonant states (rotational manifold). This may be supposed to be fast when scaled versus the rate of exchange of energy and polarization. If this is indeed the case, the relevant rate of flow may be eliminated adiabatically. We have dynamical equilibrium between the set of radiators and its surroundings (steady state: JN = 0). The number of resonant atoms is then given by the following equation: = Nth

~

1(13’)

exp[(hw12)

(/9th

—

/3)1].

(7.4)

Dthf(/3th)

t2th is the rotational partition function or rotational degeneracy of the heat bath states. (As usual, functions or variables referring to the heat bath are indexed with “th”.) Equation (7.4) is the law of mass action. This new value of N is now to be entered in the equations of the motion (eqs. (4.2), (4.3) and (4.4), and eqs. (4.8) and (4.9)), and the discussion goes along the same lines as above. As for closed systems (see chapter 5), analysis of the laser regimes begins with the calculation of the steady state laser variables. The equations of the motion are then linearized in the neighbourhood of this focus and the roots of the resulting characteristic equation yield the necessary information about the corresponding regime’s stability. The results are mapped in fig. 5. In this example again the inversion of the heat bath is assumed to define a reciprocal negative temperature with f3~h= 10. If there is no light in the cavity, eq. (7.4) reduces to:

26

X. de Hemptinne, Thermodynamics of laser systems

d’,,=.01 2

DI

\1

\1

\ \

\ \

\\

\

SUBTHRESHOLD

____DOMAIN_

S TA B LE -2

-3

FOCUS

-2

-~

0 10 (~1t)

Log

1

DI 2 ‘~~‘3

Fig. 5. Map indicating laser regimes with “open” amplifying media, as a function of the kinetic parameters. DT: “dead threshold”, LT: “live threshold”, (hw/2)f3th = 10. Scaled values of y~as indicated on the relevant curve.

N = Nth/Q~h. Then, referring to the discussion for the threshold conditions in closed systems (eq. “normal” threshold for open systems reads: 2//3~h Ky1Y~ = [X(/3~h)] where the time scale is defined by:

= (1/1) [(/iw/2)JVh]1/2

(7.5) (5.8)),

the (7.6)

(7.7)

This will be coined as a “dead” threshold. It is pictured on fig. 5 by the line DT—DT. It correlates with the conditions where the trivial solution changes from stable to unstable. In addition to this “dead” threshold, the enhanced non-linear amplification caused by the equation fixing the number of active atoms (7.4) results in the appearance of a second “live” threshold (line LT—LT). While the laser is running we have indeed /3 NSh/t2~h).The resulting higher amplification requires a stronger damping (K) to choke production of coherent radiation. Between the “dead” and the “live” thresholds the system is bistable. Bistability of laser systems is spectacular and very easy to observe. This is usually done by modulating the resonator’s loss factor by means of an absorber placed inside the cavity. Unfortunately a possible side-effect of this procedure is the appearance of passive 0-switching caused by radiative

X. de Hemptinne, Thermodynamics of laser systems

27

exchange of energy and polarization between the inverted system and the absorber. Active bistability has been ascribed to exchange of matter with the surroundings also by other authors [19].Though their formulation is different, relying on rate equations where the present approach prefers a thermodynamic treatment, the principles put forward are alike. “Limit cycle” behaviour does not show up in the realm of constants on display in fig. 5. Relaxation towards a stable focus proceeds by damped oscillations in all cases except for the lowest value of Vu proposed on the map (102 in units Y~),in a tiny triangular region in the lowermost right-hand part of the display (y~ 30, K 10~),where relaxation is exponential. The present model’s main restriction is its assumption that there is an unlimited supply of potential radiators in the heat bath (Nib is a constant). More realistically one should take:

NIh— N

=

~

exp[(hw/2)

(13th —

/3)j].

(7.8)

Far from threshold the chemical equilibrium condition between the set of radiators and the thermostat would then yield almost, N = NIh making the system behave as if it was “closed”. A general solution for this model would however require an additional assumption for 12th~ 8. Driven systems Driven systems may be enclosed in a resonator in which the field of an external source is injected or they may be irradiated in open field. The former case is investigated presently by many groups in view of obtaining reliable and fast passive optical bistable devices to be used as optical switches. The latter case is that of spectroscopy. Analyis of driven systems in a resonator requires the solution of the full set of five equations of the motion, the field in the cavity being the superposition of the injected field and that created by the radiators. By contrast, in spectroscopy the absence of a resonator reduces the set of equations required for the description of absorbers to the thermodynamic Bloch equations, the field being completely determined by the source. We shall consider this latter case first. 8.1. Spectroscopy The system which is irradiated is in thermal contact with its environment (thermostat) to which it delivers the energy that has been absorbed from the field. Absorbers are usually not placed in an inverted environment: we shall therefore take: /9th = —(kBT,h)1 <0. In vector notation, the thermodynamic Bloch equations read (compare eqs. (4.2), (4.3) and (4.4)):

6 = —(p.1/1) (j3 A E) + (3/4IT) Nyju (hw/2) + ii(i

13) =

—

(p.1/1) b (iE)

—

y±

(/9th

[/3’IX(f3’)]j3.

—

/3),

(8.1) (8.2)

v = w WL is the detuning of the field versus the transition’s resonance. The cross product (j3 A E) stands for (j3 A E) = pOE. p5Ev. The imaginary number i multiplying a phasor rotates this phasor anticlockwise over 1712: if ~3= (Pc, Ps), then (i13) = (Ps, Pc). —

—

28

X. de Hemptinne, Thermodynamics of laser systems

We assume a constant incident light intensity (E2) and we investigate the steady state absorption of energy by the target. In steady conditions the “parallel” thermodynamic parameters (b, PI~/9’, X(/9’), 0) reach constant values. By contrast, the polarization’s phase angle keeps varying, as it is slaved to the field’s phase fluctuations, which themselves depend on the thermodynamics of the source. Integration of eq. (8.2) yields:

j3(t) =

-

b exp{-(iv + yj3’/X)t}

f

exp{(iv + y±/3’/X)r}(i E(r)) dr.

(8.3)

When squared and averaged this result becomes:

=

b~(p.E/h)2

J J d~

1

exp[(y1/9’/X)

dT2

(T1

+

r2)] cos[v(Ti

—

72)]

K(ri

—

72),

(8.4)

where K(Ti 72) is the field’s phase angle correlation function. We adopt the solution proposed in eq. (6.23). This leads to the following result: —

2 (2IT4i2)”2 —

= b~ (p.EIh)

J

exp(_,~2/2~?)d~ (y±/3’/X)2 + (~+ ~)2~

(8.5)

The same procedure applied to eq. (8.1) gives:

Jb

= b (p.E//1)2 Re{

or

2

Jb

= b (p.E/h)

J

J

exp[(iP

+

y 1/9’/X)r] K(T) dr}

exp[(y±/9’IX)7] cos(vT) K(T) dT,

(8.6)

(8.7)

where the energy transferred by the target to its surroundings is —Jo: Ju

(4rr/3)(/1w!2) J,, = N

2 (/9th /3). (hw/2) Introducing the correlation function in eq. (8.7) leads to the following result: —

(8.8)

711

J

2 (2IT4,2)~’2

Jb =

b (p.EIh)

(y

1/9’/X) exp(_~2/2~) (y 2 + (ii + ,~ d~ 1/3’/X) 1.

Equations (8.5), (8.8) and (8.9) fix all the thermodynamic parameters of the target.

(8.9)

X de Hemptinne, Thermodynamics of laser systems

29

The form of the integrals in eqs. (8.5) and (8.9) suggests two different regimes for the interaction of a target with laser radiation, depending oi~..therelative values of the transverse rate constant (yj3’/X(/3’)) and of the radiation’s spectral width ((4,2)1/2). If (71f3’/X)2 ~ 4i~,the function which is integrated over behaves as a t5-function multiplying the radiation’s spectrum:

(y

+00

1f3’/X) exp(_n2/2c&2)

lim

I I

,

2

2

(yj3/X) +(v+~)

yi-cOLJ

1

—

2/2ç1i2).

dfl

I = ~rexp(—v

(8.10)

i

The system feels then faithfullyjhe Gaussian shape of the radiation’s spectrum. By contrast, if (y 2 ~ 4,~,the target’s phase memory is shorter than the average stability of the 1f3’/X) radiation’s phase angle and the radiation is felt as if it was virtually monochromatic:

(y

~ [I

1$’/X) exp(_i~2/2~) 2 + (~+ ~)2 d~]= (2~2)h/2(y (y1/3’/X) 2 + ~2 (y1/9’/X) 1$’IX) —

(8.11)

The interaction is pressure broadened (y±depends on collisions and is therefore a linear function of pressure) and Lorentzian shaped. Very low pressure regime. In low pressure regime (y~—~O), eq. (8.5) becomes —

P12 = b2 (p.E//1 )2

(17)1/2

24,

X(/3’) exp(- p2/2~2) Vu3

(8.12)

The target’s vectorial intensive variable orients itself close to the equatorial plane of its phase surface (lim 7~.0(Itg 01) = cx). The average density of the wave function’s representations is half-way up: the atomic populations are saturated (N,. = Na). Equation (8.1) multiplied by b, eq. (8.2) multiplied by (ij3)Nand the latter equation’s complex 2+ L,312 = [(3/417) X(/3’)]2, we obtain the analogue of conjugate are added together. Considering that b eq. (5.3). Having assumed that the transverse rate constant is very small, the parallel one (Yuu) must be too. Energy transfer to the surroundings is almost inhibited. However energy may be stored in the atoms themselves as a change of their thermodynamic potential X(/9’). Introducing the conditions sin 0 = 1 and cos 0 = 0 in eq. (5.3), we get indeed:

X=—y

1/3’.

(8.13)

This indicates that transverse relaxation makes the generalized potential and its conjugate intensive variable to relax slowly to zero. Considering the relationship between the density function and the intensive variables (eq. (3.8)), the wavefunctions tend to spread out uniformly over the phase space, reaching thereby phase saturation.

30

X. de Hemptinne, Thermodynamics of laser systems

Higher pressure region. In the higher pressure regime (y±f3’/X)2~

—

1P12

b2(p.E//1)2[(y

—

4,2

2+ v2]~

eq. (8.11) is to be used. Equation (8.5) becomes then: (8.14)

1/9’/X)

and eq. (8.9):

(y±/3’/X) 2+ v2 2E//1) (y1f3’/X) 2

Jb—b(/

(815)

.

Hence, the degree of saturation depends on the ratio of Rabi’s flopping frequency to the rates of the two phase perturbating parameters (transverse relaxation and detuning). In low intensity conditions we have approximately cos 0 = —1. The equations simplify then as follows: Jb

= (3/417) N/3 (hw/2) (p.E//1)2

(y~/3’/X)2+

~

(8.16)

Absorption is linear in the intensity of the incident radiation. Non-linearity of the system’s response occurs as soon as the approximation cos 0 —1 is invalidated.

8.2. Bistability For optical switches, the target is enclosed in a cavity as if it was a laser without an inversion mechanism. Coherent light is injected into the cavity. Some of it is absorbed and transferred to the heat bath by relaxation. The equations developed above are easily extended to the “non-inverted laser” case by considering a suitable field equation where the amplitude of the injected field is taken account of (JE 0J). We shall limit ourselves to the resonant case (i.’ = 0) and assume that the system’s transverse relaxation is fast when scaled against the injected radiation’s phase fluctuations. The relaxed equations of the motion read then:

(p.E//1) X = Vu 3th

cos 0

=

0

/9~h S~fl

(Yui — yu) ~

sifl20

(8.17)

(Vu, COS2 0 + Vu sin2 0) /9’

(8.18)

Yji /

K(E+Eo)

(8.19)

(p.I/1)(/1w/2)NXS~fl0.

The energy flux to the heat bath J~ YIu(/3h f3’ cos 0) which equals the energy absorbed from the injected radiation is plotted in fig. 6 against the energy of the incident beam. Conditions for bistable action (non-linearity) are easily recognized by the S-shape of the relevant curves. —

—

X. de Hemptinne, Thermodynamics of laser systems

10

—

31

________

8— —

~

IuuuOl

120

2 IARB.SC)

J-..E

Fig. 6. Passive bistability in a “closed” system. The energy absorbed and transferred to the thermostat (proportional to ~ injected intensity (arbitrary scale).

—

13th) is plotted versus the

8.3. Bistable open systems For open systems the same discussion is valid provided the variable N be taken from eq. (7.4). Results are exemplified in fig. 7 for a set of values of the relaxation constants. Some of the curves are rather unusual. The results labeled Yu/Yiu = 0.5 and Y’/Yuu = 1 show cases where the absorption of energy by the system [J~ Nyiu (/9~h /3’ cos 0)] increases almost linearly with the incident energy (linear domain) in the low intensity branch, reaching values which are eventually higher than the residual absorption at saturation. Above a critical incident intensity the system switches -~

—

4

r-10

2

1T/~10~5I~u

IogJ (ARB.SCALE) Fig. 7. Passive bistability in “open systems”. The energy absorbed and transferred to the thermostat (proportional to N(/3 — 13th)) is plotted versus the injected intensity (log—log scale).

32

X. de Hemptinne, Thermodynamics of laser systems

suddenly to saturation. The (lower) residual rate of transfer of energy from the radiation to the thermostat is then independent of the incident intensity. Turning the intensity down brings no modification to the saturation in a fairly large range of light fluxes. Below a new critical value the system switches suddenly back down to the low branch of the absorption curve.

9. Multilevel systems Harmonic oscillators are isochronous. In quantum mechanics their level spacing is constant and the transition moments increase as one goes up the energy ladder. Therefore, when the action of a perturbation involving transitions between adjacent levels is envisaged, it is necessary to consider its impact on all the stationary states at once. This is performing an infinite order perturbational treatment. If spacial coherent light interacts with a collection of oscillators, it is always possible to perform a linear combination of the coordinates, which results in a transcription of the Hamiltonian where only one collective mode interacts with the field. All the other collective modes are independent. If this single active mode absorbs one quantum of energy (/1w), this quantum is shared by the N oscillators of the system. The effective quantum for a single oscillator reduces then to /1w/N This vanishes for a macroscopic system. The condition is therefore fulfilled where classical mechanics are valid [20]. It will be shown later that the full quantum treatment of harmonic oscillators is exactly equivalent to a classical treatment, where due account is taken of the interaction with black body radiation and of the oscillating dipole’s self-action. This discussion will parallel that proposed for spontaneous emission in chapter 4 (eq. (4.15)). Comparison between classical and quantum dynamics for coherent driving of harmonic and anharmonic oscillators has been documented by several authors in the past [21]. Our conclusions parallel theirs. The classical amplitude and phase of every oscillator’s motion may be represented as a phasor I in a suitable phase space F. This phase space is a plane. If the phase space is considered to be rotating at an angular velocity w (the oscillator’s resonance) and if the oscillator is driven resonantly, the classical equation of the motion reads: =

(e/2mw) (iE),

(9.1)

where E represents the amplitude and the phase of the driving field. In off-resonant conditions (w w~= v 0) the oscillators experience the superposition of the fluctuating field and of a steadily rotating phase: —

E = El e~t)ei~~r.

(9.2)

It is then more convenient to consider F rotating at the radiation’s angular frequency w~.The driven motion itself may be separated in a fluctuating part and a steadily rotating part: (9.3) Using this substitution, the classical motion is described as follows: +

i~9= (el2mw) (iE).

(9.4)

X de Hemptinne, Thermodynamics of laser systems

33

If self-effect is considered, a supplementary damping term ~(e2w2/4IToc3m)9should be added to the left-hand side of eq. (9.4). The classic—quantum correspondence yields: e2/4m = (/1w12) (p.1/1)2.

(9.5)

9.1. Statistics Macroscopic targets are composed of many identical oscillators. Every one is to be pictured in the phase space F. As a result the macroscopic system defines a cloud of points with local density n(9). Before turning on the radiation, the system may be assumed to be in equilibrium with its surroundings. n(9) is then definitely ergodic. This is: the energy being given, the entropy is maximized. The system is unpolarized. Absorption and emission of radiation are physically reversible processes. They do not change the entropy. The evolution of n(9) due to the driving field is a parallel displacement of all the oscillators in F as it may be inferred from eq. (9.4). The continuity equation for the density in F combines the interaction with the field and a source J(9) describing the exchange of extensive properties with the surroundings (thermostat):

ñ(9) =

—

div[9

.

n(9)] + J(9).

(9.6)

J(9) is an entropy creating thermodynamic flow. It is impossible for a system, that is initially ergodic and that interacts with its surroundings through entropy creating processes, to grow at any time non-ergodic. Ergodicity remains therefore a permanent property of the system. The oscillator’s energy is composed of two parts: vibrational energy (U~,b)and potential energy (that of a dipole in an electromagnetic field (Uei)), U

~mw2J

912

n(9)do—~(E P).

(9.7)

.

The net polarization of the system is likewise:

P=eJ9n(9)du.

(9.8)

Using the same technique as in chapter 3 (eq. (3.8)) the ergodic density in F appears to be:

n(9) = (N/f) exp{~mw2/9I9I2+e[(A~ /9E/2). 9]}, —

(9.9)

where the partition function is defined:

J

do exp{~mw2/3 1912+ e[()’—E/2)

The partition function is easily integrated:

.

9]}.

(9.10)

X. de Hemptinne, Thermodynamics of laser systems

34

f= —(2ir/m w2$)exp[_e2!(A’~~/9E/2)!2/2mw2f3].

(9.11)

The intensive variables /3 and ,~are defined by the following logarithmic derivatives: U=

N8

log~f/a/3,

(9.12)

P=Nalogef/aic.

(9.13)

Using the integrated value of

f

the relationship between the extensive properties and the intensive

variables turns out to be: /3

2mw2 Ne2 I(A~_/9EI2)I2lpE /92

pNe2(AT/9E12) mw

(9.14)

(9.15)

/3

In the absence of a coherent field, the system is unpolarized. Then only the first term remains in the energy. This is the incoherent part of the energy. This classical result is also the high temperature limit of the energy known from thermodynamics. We have got this result here because we have neglected the interaction with black body radiation and the self-effect of oscillating dipoles. The second term is the additional coherent energy caused by slaving the oscillators to a common coherent driving field. Using eq. (3.7) which is equally applicable here, the entropy is found to be: S = kn{loge[~~’]

—

~

—

[(~~/3E/2) fi}. —

(9.16)

Solution of the continuity equation (9.6) requires an expansion of n(9) in dipolar and quadratic components. The quadratic component is directly related to the system’s vibrational energy; the dipolar component, to its polarization. We shall therefore envisage the set of extensive properties U~tband P (rather than the total energy and the polarization), because this set is linearly independent. The conjugate intensive variables are: (1/kB) (aS/aU~~b)~N =

—/9

(9.17)

(1/kB) (aS/aP)UV~b.N=

—(A~—/9E/2).

(9.18)

Exchange of matter between the resonant atoms and the rotational states belonging to the heat bath (non-resonant with the electromagnetic field) is also a thermodynamic flow. Its role has been fully documented in chapter 8, where it was suggested that high field conditions freeze all the particles in their resonant motion. If the system has collected all the potential radiators, it behaves as if it was closed (eq. (7.8)). Referring to the discusssion in chapter 3 (eqs. (3.28), (3.29) and (3.30)), the thermodynamic flows of energy and of polarization may be written:

X. de Hemptinne, Thermodynamics of laser systems

J u,~,= N ~ (/1w/2)2 (/9th .Ip =

Ny~p. ~(AT —

35

(9.19)

— /9),

f3E/2).

(9.20)

Let us recall that the unpolarized thermostat is characterized by: A~th /3thE/2 = 0. Using these expressions for the thermodynamics flows and using eq. (9.4) for the action of the driving field, the equations of the motion, which are the quadratic and the dipolar components of the continuity equation, turn out to be: —

Uvib

—~w(P A E)+Ny,,(/1w/2)2(f3th_f3),

ft = N (p.2//1) (iE)

—

i.’(iP)

—

Nyu p.2

~

—

(9.21)

/3E/2).

(9.22)

If the self-effect of the oscillating dipoles is considered, a supplementary damping term —2y. U~~b should be added to the right-hand side of eq. (9.21) and —y~Pshould be added to the right-hand side of eq. (9.22) with: Vs =

2.

w2

317

0c

(9.23)

3(/1w/2) (p.1/1)

We examine now the role of black body radiation on the classical harmonic oscillator. We assume that the set of oscillators is very loosely coupled to its surroundings (y~~ = 7-i- = 0) or else that it is in equilibrium with its thermostat, so that the thermodynamic flows vanish. We may consider here safely v = 0 because there is no external driving field which defines for F another angular velocity than the oscillator’s resonance. Equation (9.22) with the self-effect integrates as follows:

P = N~

f

exp{ys(r — t)}

E(r) dr.

(9.24)

This result entered in eq. (9.21) gives in a stationary regime (U~,b= 2 N (/1w/2) (p.1/1)

J

0):

exp{y

5(r— t)} [E(7)

. E(t)]

dT = 2 Vs U~Ib.

(9.25)

Black body radiation being incoherent, the only term which matters in the expansion of the exponential is the constant. Then, considering the correlation function applicable to black body radiation (eqs. (4.12) and (4.13)) we have: N/1w exp(—/9/1w) 1

U~Ib=

.

(9.26)

—

It obtaining this relation the radiation’s correlation function was taken to be twice that of eq. (4.13). The

36

X de Hemptinne, Thermodynamics of laser systems

field is indeed supposed not to escape: it is reflected onto itself. We must therefore consider the two opposite directions of propagation. The result for the steady energy of the set of vibrators is identical to the thermodynamic result for quantum oscillators. As’it was announced, quantum mechanics appears to be the result of the action of black body radiation on a classical motion. 9.2. Coherent multiphoton absorption We shall now examine the action of coherent light. In order to simplify the expressions, the action of black body radiation and the oscillator’s self-action will be considered as parts of the thermodynamic flows to the thermostat. They will not be mentioned otherwise in the equations. The most fascinating experiments involving infrared laser radiation and molecular oscillators, resulting in multiphoton excitation require “collisionless conditions”. This term usually denotes inefficiency of close encounters between oscillators and heat bath particles, in dissipating vibrational energyon interpenetration of the respective electron clouds (V V, V T relaxations). The cross-section of this (parallel) relaxation process is indeed small (molecular diameter squared). Its rate constant ~ is a linear function of the pressure. In low pressure conditions we may readily have Yuu (p.E//1). Energy that accumulates in the system by absorption from the incident radiation is then hardly allowed to flow away to the surrounding. High excitation of the set of oscillators may then readily be anticipated. Exchange of polarization with the heat bath requires also collisions. The transverse rate constant Vu is therefore equally linearly related to the pressure. However, this process involves interactions between oscillating dipoles and induced oscillating dipoles, that are much longer ranged. No matter how low the pressure may be, transverse relaxation may never be neglected. As a matter of fact, its lower limit would be the oscillating dipoles’ self-effect. Considering eq. (9.15) the flow of polarization may be expressed in terms of the actual value of the set’s polarization: —

—

‘~

-1i~=

V±(hw/2) J$~P.

(9.27)

This result is to be introduced in the relevant equation of the motion (9.22):

ft+[yu(/1w/2)!sl+iPIPN(p.2//1)iE,

(9.28)

which integrates then as follows:

P= N~-

J

exp[(yu~I/9!+i~)(7

t)] iE(r)dT.

(9.29)

This result is then introduced in eq. (9.21) where it is assumed that ~ is negligible: /1wp.E2

uVIb=N—~-(-~-)Re{

J

/1w exp[(7u-~-!$!+i~) T]K(T)dr}.

(9.30)

Using the correlation function for the driving field that has been elaborated in chapter 6 (eq. (6.23)),

37

X. de Hemptinne, Thermodynamics of laser systems

the rate of absorption of energy from the incident radiation integrates as follows:

(~)

/1w p.E U~~b = N~

2

—

(2~4,2)1/2

~

~

((/1/2)1/91)2 +

(~

+

~)2

dfl.

(9.31)

Discussion of this result with respect to the value of the transverse relaxation constant and of the fluctuations of the field goes along the same lines as for two-level systems (chapter 8). If Vu is almost vanishing, eq. (8.10) must be applied. The system responds then to the radiation only if its resonance lies within the radiation’s spectral line width:

Uvib = N (hw/2) (p.EI/1)2 (ir/24!?)1t’2 exp(— ,,2/2~).

(9.32)

The opposite hypothesis (y~(/1w/2)1/91 > (4~2)1/2) seems to be more realistic in most experimental circumstances. If then we may neglect the line width versus the rate of transverse relaxation, we have: /1w (p.E\2 (yj. (/10)/2)1/91) U~Ib N -i-- ~ (~,(/1w/2) 1/91)2 +

33

—

(9.

~

Considering that in a steady electromagnetic field (E2 = constant) the polarization is constant (eq. (9.29) squared), and therefore also the coherent contribution to the vibrational energy (eq. (9.14)), storage of energy results in the increase of the incoherent part of the energy (1/91 decreases). Saturation is reached when 1/91 = 0. Then U~ 11,vanishes. 2) is a function of the time. The amount of energy In pulsed operation, the intensity (p.E//1) eq. (9.23). accumulated during the pulse is found (Jg by integrating If 1/91 does not change too much during the process of absorption of energy, we have: -~

—

~ U.~

—

/1w (Vu (/1w/2) 1/91) N 2 (Vu (/1w/2) 1/91)2 +

1

2

J

/p.E\2

~

dt.

(9.34)

pulse

This equation clearly shows the “fluence” (integrated intensity) dependence of the energy deposition, a fact well known to experimentalists in the field of vibrational multiphoton chemistry. At high energy storage, decrease of 1/91 should reduce the absorption efficiency. It has been claimed recently that chaos generating terms in the equations of the motion were the origin of incoherent energy storage in driven oscillators [22]. As it is stressed in appendix A, chaos generating Hamiltonians lead to forces that do not sum up to zero. The resultant is then balanced by a force acting on or coming from the surroundings (thermostat). This is why chaos generating Hamiltonians may be called “non-conservative”. Considering that our conclusions link energy storage to the existence of an energy conserving (transverse) relaxation phenomenon, above-mentioned assessment is beautifully substantiated by the thermodynamic theory. The latter theory segregates indeed the conservative and the non-conservative parts of the Hamiltonian and treats the latter ones in terms of transverse relaxation. In fact, the wealth of information hidden in the detailed form of the non-conservative part of the

38

X. de Hemptinne, Thermodynamics of laser systems

Hamiltonian is unnecessary, as our experimental observations are by themselves integrations of all the interactions. Furthermore, experimental evidence for choosing particular expressions for this part of the Hamiltonian is usually lacking. This makes the thermodynamic approach still preferable.

9.3. Inverted systems For strictly harmonic oscillators, when the system is inverted (/3 > 0) the partition function (eq. (9.10)) diverges. The problem is unphysical. In fact molecular oscillators are anharmonic. Their potential energy surface is quadratic only in the lowermost energy region. The relationship between the motion’s amplitude and the vibrational energy exhibits usually an inverted bell-like shape, with the dissociation energy (Ud) as the upper bound U(9) = Lid V(9). This function must replace ~mw2l9j2in the exponential of eq. (9.10) and exp(f3Ud) may then be factorized. The wavefunction’s representations in F in the inverted case is the negative picture of the non-inverted density. n(9) is smaller in the regions about the origin (191 = 0) and highest in the external ones (close to dissociation threshold). If 1/91 is big, the only part of F which really matters is the domain close to the origin where the gradients of n(9) are highest. There V(9) is almost harmonic. By contrast, with small values of 1/91 the anharmonic part of F bears an increased importance in the system’s response to electromagnetic radiation. If we use eq. (9.4) as the equation of the motion for anharmonic oscillators, detuning i-’ is itself a function of 1912. In the regions of F where the Rabi frequency (p.E//1) exceeds translation of the representations of the particles in F due to the field exceeds their anharmonic dephasing. There the harmonic oscillator’s results are valid. By contrast, in the regions where the anharmonic dephasing has the lead, rotation of the representations in F washes out any angular anisotropic distribution of the density function n(9) and the relevant regions of F bring no net contribution to the interaction with the field (the sum of all interactions coming from those regions cancel). Exact integration of the partition function for anharmonic oscillators cannot be done in general because it depends on the amount and of the type of anharmonicity that is involved. The arguments presented above for “open systems” (chapter 7) applied to the density in F suggest however that particles will tend here again to maximize their response to the field by accumulating in the most favourable regions of the phase space. —

~(92),

10. Radiation induced osmosis Osmosis will be discussed in the context of harmonic oscillators though it is quite general and could have been developed for two-level systems also. Equation (9.14) indicates that the total energy splits into a potential energy contribution (oscillating electric dipoles in an oscillating electric field: ~P E) and genuine vibrational energy. In the latter contribution there is an incoherent part (—N//3) and a part directly linked to the coherent driving field. Let us call this latter contribution the coherent contribution: U~ 0= Nu. By comparison of eqs. (9.14) and (9.15) it comes that the coherent contribution is proportional to the average amplitude squared of the system’s polarization. Using eq. (9.29) squared we have: .

X. de Hemplinne, Thermodynamics of laser systems 0

N/1w p.E2 U,~

0=

Nu =

-~---~—

(-i-)

39

0

J dr~J dr2 exp[yu

/1w -~—

1/91 (ri + 72)] cos[v(1-1

—

72)]

K(r1

—

72).

(10.1)

We assume the system’s phase is shorter than the instability 1/3J)2~ th~t 4,~).This means that the memory system feels the radiation as if field’s it wasphase monochromatic. ((y~.(/1w/2) Equation (10.1) integrates as follows: u

=

~(hw/2)(p.E//1)2 [(Vu (/1w/2) 1/91)2 +

~2]_1

(10.2)

(compare with eq. (8.14)). The partition function for the set of oscillators (eq. (9.11)) is an exponential function of this coherent energy: f

=

—

(2ir/mw2/9) exp(—/3u).

(10.3)

We have assumed a non-inverted driven system: /3<0. The partition function is therefore an exponentially increasing function of the incident intensity. It has been assumed hitherto that the number of particles in the driven system could be modified only by relaxation of nearly degenerate states (rotational relaxation). This process does not imply physical displacement of matter. In gaseous systems particles are also allowed to move physically. This allows them to generate density gradients. Transport of matter is governed by a diffusion equation quite analogous to eq. (7.3). It is driven by gradients of the chemical potential: JN =

—ND grad(3S/t9N) = —ND grad[log~(f/N)].

(10.4)

Laser beams have finite cross-sections. Their wings are characterized by a gradient of the intensity. If then a mechanical dynamic equilibrium (steady state: JN = 0) is to be reached, a density gradient must develop across the beam which parallels the gradient of the intensity. This simplification is valid only in the low pressure range where absorption of energy is negligible (eq. (9.33) indicates that absorbance is proportional to Vu and vanishes at low pressure). If p represents the local density of resonant particles, we have at dynamic equilibrium: grad[loge(p)]

—

grad(/3u).

(10.5)

Osmosis or physical displacement of resonant particles towards the core of the driving beam is easy to observe with any absorbing gas using an acoustically non-resonant spectrophone. The author has used a cylindrical cell (diameter: 2 cm) along the axis of which a mechanically chopped CO 2 laser beam of about 30 watt of power was sent. The beam was reduced in diameter by means of a telescope to a width of about 2 mm. The experiments were performed with pure ethylene or with pure SF6. Identical results are obtained for all the absorbed laser lines. The results presented in fig. 8 show how the pressure transients obtained by chopping the incident beam change as a function of the sample’s pressure. At low pressure, light sucks the target into the beam causing transient depressions during the illumination phases. By contrast, at higher pressure, heating of the sample along the beam by V—T

40

X. de Hemptinne.

Thermodynamics of laser systems

4ltorr

48torr

b9torr

L

0

L

ID

I

L

I

Fig. 8. Pressure transients in chopped C0

2-laser irradiated SF6 (any line: here P26 (10.6 sm)) for different pressures of the target. Time scale: 2 ms/div. Pressure in the cell as indicated. L: irradiation phase. D: dark phase.

relaxation of the absorbed energy causes the pressure to increase during the illumination phases. Compensation of these opposite actions was observed for C2H4 at 3.5 torr and for SF6 at 5 torr. The pressure transients reduce then to spikes as the beam switches from on to off and from off to on. For accurate observation, chopping of the beam must be rather slow (±150 hertz) to allow the establishment of mechanical dynamic equilibrium in the cell (translational relaxation).

11. Lamb dip Spectroscopists like to classify systems interacting with light as “homogeneously” and “inhomogeneously” broadened [23].The latter ones are characterized by an absorption spectrum presenting a shallow minimum at the very top of the line (at resonance) when the system is irradiated in low pressure (Doppler limited) conditions. This phenomenon is known as the “Lamb dip”. Doppler shifts of atoms moving at various velocities along the beam’s direction of propagation is claimed to be a typical cause for inhomogeneous broadening of the relevant transition.

X. de Hemplinne,

Thermodynamics of laser systems

41

In the literature, the velocity groups of atoms experiencing different Doppler shifts are treated as independent systems. The only link that is recognized between them is the common electromagnetic field to which they are coupled. This total independence of the velocity groups, besides whatever does the field, will be reconsidered here. There is indeed no reason to deny possible exchange of energy and polarization between the velocity groups themselves. It seems indeed to be a contradiction to assume that exchange of extensive properties would proceed more swiftly between the quasi-unpolarizable thermostat and radiators belonging to any velocity group than between dipoles themselves which are nearly resonant to each other and therefore very much alike. Hence we shall adopt the opposite hypothesis: that of a dynamical equilibrium between the velocity groups. Description of the velocity distribution requires a suitable partition function (sum over states) for translation. Interaction with light depends only on the velocity component which parallels the beam’s direction of propagation. Velocity components along the two other directions of space may therefore safely be omitted in the treatment. We assume here that the system is a set of harmonic oscillators. It is easy to transpose the theory to two-level systems. Individual motions are defined by three variables: the particle’s velocity (v) and the amplitude and phase of its vibration (or resonant motion: 9). Four extensive properties describe the system: its total impulse Ni3=Jvn(v,9)dvdo-,

(11.1)

its kinetic energy (UK), the energy associated with the optical transition and the corresponding polarization (eqs. (9.7) and (9.8)). The most favourable statistical distribution (ergodic distribution) is found by maximizing the system’s entropy, considering above-mentioned extensive properties as constraints (appendix B). To this purpose two new Lagrangian multipliers are required: ~ for the impulse and /9K for the kinetic energy. The ergodic distribution n(v, 9) now reads:

~

Mv2

N

~/9E (11.2)

F is the partition function for translation and vibration. We integrate the vibrational part of the partition function and get:

/9)J

F = —(217/rn w~

dv exp[~v+ I3KMv2/2 /3 u(v, J)], —

(11.3)

where u(v, .~) is the coherent contribution to the vibrational energy of velocity group v (see eqs. (10.1) and (10.2)). For ease of the notation we shall represent the Rabi frequency squared, that is proportional to the intensity of the radiation by the symbol J: [Jg = (p.EI/1)2]. We shall also write: 7~= 7j (/1w/2) 1/91. Detuning of the zero-velocity group is v 0: vvo—wv/c.

(11.4)

42

X. deHemptinne, Thermodynamics of laser systems

With this notation the coherent contribution to the driven energy is: u(v, 5) =

~(hw/2)~

~

(,i~—

wv/c)2]1.

(11.5)

The problem is now to determine the value of Using eq. (3.7) with the distribution function (11.2), the entropy becomes: C.

S = kB{ loge(F”/N!)

—

CN5 /9KNMv2/2 + /3 —

J

u(v, 5)

n(v) dv }.

(11.6)

Given the intensity of the radiation, the entropy is a function of the system’s extensive properties: its impulse (average particle velocity) and its kinetic energy (average square velocity). The conjugate intensive variables are the negative partial derivatives of the entropy versus the relevant extensive property. In order to derive correctly eq. (11.6), the function u(v, .9~)must be expanded in powers of v. The integrals of the successive terms are the average values of the successive powers of the velocity:

J

u(v, 5) n(v) dv

=

u(0, 5) N + u’(O, 5) Nv

+ U(0, ~

Nv2+

U(0, ~

Nv3 ~

(11.7)

Introducing this result in eq. (11.6) gives for the intensive variable conjugate to the average velocity (impulse):

~_

(o~i~)r

—[C—

/9

u’(O, J)1.

(11.8)

Likewise the intensive variable conjugate to the average kinetic energy becomes: ~

(

3fl

kB a(Nv2)

=

-

~[/9KM

-/9

u”(O, ~)1~

(11.9)

;

Neglecting the effect of radiation pressure, the system traversed by the beam is characterized by a reflexion symmetry: it does not matter which is the inlet and which is the outlet window. With respect to this symmetry element, the impulse is antisymmetric while the kinetic energy is symmetric. This makes the two extensive properties to be independent as far as the principle of Curie and Prigogine is concerned. Onsager’s phenomenological equations relating flows of these properties in the cell to gradients of the conjugate intensive variables are diagonal (uncoupled). We have therefore: Jimpulse

-~—

Jkinetic energy

grad[~ j3 u’(O, 5)]

(11.10)

—

—

grad[/9KM

—

/3 u”(O, 5)]

.

(11.11)

We suppose the cell to be immobile: NIY=J vn(v)dv =0.

(11.12)

X

de Hemptinne, Thermodynamics of laser systems

43

Collisions with the windows make this condition to be valid everywhere in the cell. It may hence be claimed that transport of impulse throughout the cell is a strong mechanism. This fast transport of impulse ensures dynamical equilibrium of the conjugate intensive variable: Jimpuise = 0. Considering that in the dark zones of the cell the velocity distribution is symmetric about v = 0, so ~ = 0 where S = 0, the requirement that eq. (11.10) vanishes yields at any place throughout the cell the value of the Lagrangian multiplier as a function of the local intensity:

C(S)

=

/9

u’(O, 5)

=

/3(hw/2) [F212~

(11.13)

By contrast there are no strong mechanisms for transport of kinetic energy across the cell. We shall take therefore /9K to be approximately equal to its initial value (thermostat value, prior to irradiation). This value may in fact be slightly modified by translational heating as a result of V—T relaxation of the driven motion. The resulting modification does however hardly alter the conclusions. The absorption coefficient observed by the spectroscopists is the sum of the absorptions by all the velocity groups: =

±00

U(v) n(v) dv.

(11.14)

Absorption of energy by a single velocity group (L1(v)) is given by eq. (9.33). Using this result, the integrated rate of energy abstraction from the incident field becomes:

i

1.52

2~~2 V

0

1.3”

Fig. 9. Absorption coefficient for coherent light (log2/units) 2. The versus curves detuning are computed of the radiation. for YI2C2M.BK Detuning = —40w2. is scaledModification in units of of Vi this = yL(hw/2) value produces J3~.The field minor only intensity quantitative is represented changes. by .~ — (hw/2) j$I(jsE/h) 71

44

X. de Hemptinne, Thermodynamics of laser systems ±00

U=

/1w -~--5

~J N

dv

_______________

(~i)2+ (Po- wv/c)2

exp[/9 u’(O, 5) v +

Mv2 /9K—~——/9

u(v, 5)].

(11.15)

where u’(O, 5) is given by eq. (11.13) and u(v, 5) by eq. (11.5). The absorption coefficient is obtained by dividing eq. (11.15) by the density of particles (N/F Nth/Fth = density), and by the incident radiation’s intensity. Given the molecular parameters m, Al, w, p., the absorption spectrum depends on two experimental parameters: the ratio of the Lorentzian to the Doppler components of the spectral broadening (yi~2c2M/3K/2w2)and the intensity of the radiation. Figure 9 displays a set of results calculated on the basis of eq. (11.15). The curves reproduce faithfully the well-known experimental Lamb dip. The general argumentation supporting this treatment of the Lamb dip is that dynamic equilibria are often shifted with respect to true equilibria, because they are coupled to irreversible processes. This coupling yields typically non-linear responses of the system to its driving perturbation: the incident radiation. Laser induced osmosis was one example of this general phenomenon; the Lamb dip is another one. There are probably many more (laser cooling etc.).

Conclusion The present monograph’s backbone is the suggestion that relaxations intervening in laser action be described according to the principles of thermodynamics. This idea contrasts with the widespread trend of discussing relaxations using Master equations. The latter technique describes indeed individual steps of non-equilibrium processes, but when flows are coupled, the necessary approximations may lead to questionable conclusions. The result may be for instance the violation of essential symmetry properties of the Hamiltonian. In the discussion proposed here at least this consideration is taken care of. The field’s part of Bloch’s equations has been the object of many discussions during the past decennium. It does however not seem to be so deficient: Schrödinger’s equation or even classical mechanics render indeed correctly the known quantum effects, in the latter case at least as a counterpart of the interaction with black body radiation. Other approaches do exist that describe incoherent phenomena of radiation in more elegant terms. Their main drawback when laser systems are considered is that here incoherence is a minor side-effect, to be considered as a perturbation on a generally classical phenomenon, where the opposite is true in non-laser physics. Therefore it is believed that other field equations and other descriptions of field interactions would yield identical conclusions. The thermodynamic counterpart of Bloch’s equations leads to conclusions which are easily validated by the experiment requiring only a small number of adjustable parameters (three rate constants and the thermostat’s inversion). There are no additional assumptions, besides perhaps the linearity of the phenomenological equations. Some conclusions are novel (line shape, osmosis, . others are more or less diverging reformulations of previously described phenomena (pulsing regimes, bistability, multiphoton excitation, saturation phenomena, . .). Many other phenomena could have been discussed along the same lines (lens forming and propagational phenomena, cooling,.. .). All result from interaction between light and matter. Coherent emission and absorption are collective phenomena: they are a macroscopic property of matter. This makes them specially suited for a thermodynamic description. Why has confrontation with thermodynamic principles been so carefully avoided in laser physics to date? This remains an open question. . .),

.

X. de Hemptinne, Thermodynamics of laser systems

45

Master equations are simplifications of Liouville’s equation, itself based on the system’s Hamiltonian Our continuity equation (eqs. (4.1) and (9.6)) is nothing else than Liouville’s equation to which a source (J) is added. In strictly isolated conditions the source would vanish. Hamilton operators being conservative, the system would then keep for ever the memory of its initial conditions. In a system composed of many interacting particles, an initial perturbation may of course seem to dissipate by exchange with the many degrees of freedom but, the system being finite (no matter how big it is), its information (entropy) is doomed to remain in the system and eventually to recreate periodically its initial conditions. Strictly isolated systems are however inconceivable. Ours is certainly not: the source in Liouville’s equation makes it to interact with the whole universe, which it is part of. The universe being immense, recursion of any fluctuation is delayed to astronomical times. It may therefore be claimed that Liouville’s equation is incomplete if not supplemented by an adequate source. [24].

Dedication To the memory of my grandfather Alexandre who was a respected chemical physicist (chemical effects of electric discharges in gases) and in honor of my father Marc whomade himself a name in molecular spectroscopy. Appendix A. Master equations contra thermodynamics in quantum optics Defining the conditions of dissipativity is a fundamental problem of non-equilibrium statistical mechanics. Dissipativity is the property of many-particles systems to forget in the course of time the information about their initial conditions. The thermodynamic equivalent is the trend of the entropy to grow until it reaches a maximum value compatible with the constraints imposed to the system. The system is then said to be “ergodic”. Though dissipativity has been discussed by several authors in the past [24], some controversial points remain. There is in particular the debate about the validity of the laws of thermodynamics in non-equilibrium systems [6]. The literature on non-equilibrium statistical mechanics is often limited to “conservative” systems: this is when all the forces are derivable from a potential and sum up to zero. Such systems are indeed easily studied using the Hamiltonian formalism. However it is clear physically that systems cannot be conservative if friction or similar forces are present [25]. This excludes in fact the discussion of Brownian motion and similar problems. It has sometimes been suggested that dissipativity could occur in certain conservative systems as a consequence of the distortion of the Hamiltonian during collisions [24]. This allegation clearly opens a question of semantics. Dissipation as it is meant here is indeed not the alteration of trajectories in the phase space due to higher order perturbations of the Hamiltonian. It is instead a phenomenon whereby the detailed information about the state of the system is lost. In conservative systems, no matter how complicated a Hamiltonian may be, its eigenvalues are constants of the motion. Transitions between eigenstates require external intervention, be it for instance exchange of radiation of collisions with the walls with transfer of impulse. The latter statement seems to be conflicting with recent published material on the appearance of dynamical chaos [26, 27]. It is not. It should be stressed indeed that chaotic and dissipative behaviour requires Hamiltonians presenting odd powers of the coordinates (e.g. gravitation, the Henon—Heiles

46

X. deHemptinne, Thermodynamics of laser systems

system, etc.) or collisions with the boundaries (hard sphere gas). In neither cases do the forces sum up to zero (the potential is not invariant with respect to the inversion). The forces must therefore be caused by an external source: such systems are non-conservative. The entropy is by definition S = kB loge 11, where (1 is the number of states the system may be in. If our knowledge about the system is complete, as it may be with conservative systems, we have .f2 1 and the entropy vanishes and remains zero at all times. In fact, with conservative systems, the concept “entropy” is academic. Strictly conservative systems are however hardly conceivable, excepting perhaps the universe as a whole. For radiating atoms, spontaneous emission backed by absorption of black body radiation issued from the surroundings couples the system to its neighbourhood. For non-radiating systems, information is transferred by gravitational interaction. Finally, considering that the constraints are fixed by the boundaries enclosing the system, exchange of extensive variables (energy, impulse, momentum, etc.) occurs whenever the system “feels” its boundaries. This is the case even for highly diluted “collisionless” atomic beams, the properties of which are determined by their initial strong interaction with the beam forming device. Dissipative interactions cause incoherent transitions between the eigenstates thereby increasing the uncertainty about the system. When all the eigenstates that respect the constraints have become equally probable, the system’s entropy is maximized. Subsequent Hamiltonian interactions cannot destroy the ergodicity of the system’s probability distribution. It is common knowledge that solving Liouville’s equation and integrating the canonical equations are equivalent problems. In systems with many degrees of freedom, Liouville’s equation has the distinct advantage over the canonical equations to summarize the behaviour of the whole system in a single equation. The treatment followed next is classical but extension to quantum mechanics is simple [28]. Let n be the density of the points representing individual motions in the phase space F. This phase space is dimensioned according to the degrees of freedom of a single particle. All the particles are taken to be identical. Liouville’s equation is easily extended to non-conservative systems by adding a suitable source to usual continuity equation. Using Poisson’s brackets, Liouville’s equation reads: ñ=[H,n]+J~.

(A.1)

Description of irreversible processes requires a correct expression for the source (J~)or of its effect on the system’s macroscopic properties. There are in principle two routes for solving Liouville’s equation. One treats individual jumps in F by means of kinetic equations and predicts in a second step changes of the macroscopic properties by integrating the latter result over the phase space. This is the “Master equation” approach. By contrast, thermodynamics starts by expanding loge(n) as a linear superposition of a complete set of orthogonal functions representing the set of observables. This procedure mimics the expansion of wavefunctions in a Hilbert space. In a second step observables are allowed to flow, driven by generalized forces (affinities). The nature of the generalized forces is defined by considering the symmetry properties of the Hamiltonian. We shall start with the latter approach. The physics of ergodic many-particles systems is defined by a limited number of constraints or observables (Xk). In quantum mechanics the word “observables” denotes operators that commute with the Hamiltonian. Their expectation values are the above-mentioned extensive properties. The number of independent observables equals the number of irreducible representations of the symmetry elements of the Hamiltonian. The classical counterpart of quantum operators are functions 4,k (p, q) defined in F, that give the

X. de Hemptinne, Thermodynamics of laser systems

47

corresponding expectation value on weighted integration in the phase space: Xk =

J

n(p, q) 4k(P, q)dF.

Let us expand loge(n) as a superposition of aforementioned functions 4k(p, loge(n) = AN + ~

(A.2)

q):

A,. 4’k(P, q).

(A.3)

k~~N

If the partition function is defined as follows: f(Ak) =

J

exp[ ~ A,. çb,.(p, q)] dF,

(A.4)

kN

it is easy to verify that in ergodic systems the average value of the extensive properties Xk are linked to their conjugate intensive variable by the following general equation: Xk =

N~loge(f)/3A,..

(A.5)

From first principles it is known (eq. (3.7)) that the entropy of the system is given by:

-

SknJ(n—nlogen)dF.

(A.6)

Using eq. (A.3) the entropy becomes: S= kB[(1—AN)N— ~ A~X~],

(A.7)

kN

and its time derivative is: .‘=

[—~isr— ~

~

(A.8)

kN

The entropy measures the system’s degeneracy. It is therefore invariant with respect to all the the Hamiltonian. In quantum mechanics it is known that every state is completely determined by its set of eigenvalues in response to a complete set of mutually commuting observables (operators). The relevant set of eigenvalues may therefore be considered as the components of a vector. Vectors referring to orthogonal states are mutually orthogonal. We consider the set of degenerate (w.r.t. the Hamiltonian) states with energy U The observables being mutually commuting, the number of orthogonal eigenstates equals that of the independent observables and that of the symmetry operations acting on the coordinates, which leave the Hamilsymmetry elements of

48

X. de Hemptinne.

Thermodynamics of laser systems

tonian unchanged. Any of the latter symmetry elements (~,), excepting however the identity operation, results in the interchange between orthogonal states. Their action on the vectors of eigenvalues is a transformation from one vector into another. We have: (A.9) where w,., is the relevant member of the symmetry group’s character table (wkl = ±1) (e.g. sign reversal of the coordinates reverses the sign of the impulse but keeps the internal energy unchanged). Considering the invariance of the entropy with respect to any ~,, eqs. (A.7) and (A.8) indicate that the extensive properties (Xk) and their conjugate intensive variables (Ak) are contravariant (their products are invariant). The same argumentation holds for AN and N versus the other variables, because N is an integer while the vector of Xk is scalable. Their characters are clearly different. Interaction of the system with its surroundings (thermostat) causes extensive properties to be exchanged. Considering the conservation law of observables (first law of thermodynamics), effiux from the thermostat equals influx to the system (Xk)the~o,t. = (X~)system= Exchange causes the thermostat’s entropy to change according to an equation analogous to (A.8). The second law of thermodynamics says that the total entropy change (sum of that of the system and that of the thermostat) is positive. It vanishes when equilibrium is reached between the system and its thermostat. Entropy production reads therefore: —

=

kB

~

[(Ak)th

—

(Ak)5Y5]Jk

= ~

AkJk

0

(A.10)

where Ak is the “affinity” conjugaie to flow Jk. This result is well known in irreversible thermodynamics [13, 14, 15]. Description of irreversible processes in terms of entropy production is not very useful unless it is supplemented by suitable kinetic equations relating the flows to the affinities. Onsager has suggested to link the flows and the affinities by linear phenomenological equations. Linearity is only a convenient approximation, but it is usually verified under wide ranges of experimental conditions. Coupling of flows generates off-diagonal coefficients [11]: Jk~Lk,Al.

(A.11)

Onsager has demonstrated in his famous paper that the matrix of the phenomenological coefficients is symmetric (LkI = L,k). Introducing eq. (A.11) in eq. (A.10) yields: S=

~ Lk,AkA,.

(A.12)

If the set of observables used to describe the system and its irreversible flows is an independent (orthogonal) set where eq. (A.9) is valid, invariance of the entropy production for all the symmetry operations requires that the matrix of the phenomenological coefficients be diagonal.

X de Hemptinne, Thermodynamics of laser systems

49

As a conclusion we may state that for kinetic equations to be diagonal it is necessary that affinities (differences of intensive variables in the system and in the thermostat) be used as the generalized forces and that the flows be related to independent extensive properties or observables. This is a generalization of Curie’s well-known symmetry rule [12, 15]. It has been stressed above that the Master equation’s approach is based on the kinetic treatment of individual jumps in the phase space. It should lead of course to the same result as the thermodynamic treatment. Unfortunately the mathematical tool for accurate determination of the individual kinetic constants is so difficult [24] that approximations are required (Fokker—Planck equations, Rate equations, etc.). Our final interest is to determrne flows of macroscopic observables or macroscopic properties in our system (coherent emission etc.). This necessitates integration of the individual contributions in F. Integration may however lead to a summation of the approximations and finally to incorrect results. In the study of Brownian motion, the only exchange with the thermostat that is considered is exchange of “impulse” (friction). Exchange of impulse is hardly conceivable without simultaneous exchange of kinetic energy. The latter process is however assumed to be so fast that it remains equilibrated with the heat bath. Exchange of kinetic energy is eliminated adiabatically and the process is controlled by its rate determining step: friction. Furthermore in this particular case the observables are clearly orthogonal (see comment after eq. (A.9)), so flows of impulse and of kinetic energy are not coupled. The non-equilibrium (steady state = dynamical equilibrium) value of the system’s temperature equals therefore its true equilibrium value (statistical equilibrium = temperature of the thermostat). This is the ideal case to be solved by Master equations. In quantum optics several observables are allowed to be exchanged between the set of radiators and their thermostat (surroundings). The two main observables to be considered here are the inversion (energy) and the polarization (at the optical frequency). They may be shown to be orthogonal in the sense quoted above. If the rates of exchange are very different, one flow may be eliminated adiabatically versus the other one and the process behaves as if it was controlled by a single rate determining process. This leads to the “rate equations” approximation of quantum optics that has been very useful. Unfortunately even in this case the published rate equations are only rough approximations, as the driving force for the irreversible flow between the pump and the radiators is generally expressed as a linear function of the value of the observable itself, which is very far from being comparable to an “affinity” (intensive variable). Sometimes the rates of exchange may be comparable. Then, considering the heavy feedback in lasers, it may be shown that oscillating instabilities occur (limit cycles: see chapter 5). The system’s variables are then driven into high amplitude variations. In such cases the approximation consisting in replacing affinities by observables in the kinetic equations (Bloch’s equations) is disastrous. In quantum optics other irreversible flows between the set of radiators and the thermostat must be considered too: rotational redistribution (see chapter 7) and translational redistribution, flow of impulse (see chapter 11), flow of matter (see chapter 10), etc. Some of them are orthogonal when the electromagnetic field is off and become oblique when the electromagnetic field is on. This is particularly the case with the observable “impulse” (due to Doppler effect in non-resonant light: see chapter 11). Coupling of flows (off-diagonal phenomenological coefficients in eq. (A.11)) results then in a shift of dynamical equilibria with respect to static values causing non-linear phenomena unpredicted by the conventional approximations to Master equations. The conclusion is therefore clearly in favour of thermodynamics.

50

X de Hemptinne, Thermodynamics of laser systems

Appendix B. Lagrange’s undetermined multipliers The method of Lagrange’s undetermined multipliers is described in details in many textbooks on statistical thermodynamics. It works as follows. Let n(0, 4,) be any distribution of the variables 0 and 4,. The entropy associated to this distribution is given by eq. (3.7) (we drop here the factor 2irN which may be considered as an integration constant) S

=

kB

f

do- [n

—

n loge(n)].

(B.1)

We want to define the distribution which maximizes the entropy while subject to a number of constraints. The constraints are a given energy (eq. (3.5)), a given number of particles (eq. (3.2)), a given amplitude and phase of the system’s polarization (eqs. (3.3) and (3.4)). There could be also a given impulse (eq. (11.1)), etc. Every equation defining a constraint is multiplied by an undetermined multiplier: a for N, /9 for U, A for p.j3 and the result is added to the function S (eq. (B.1)). This creates a new function F of n(0, 4,), the coordinates of the absolute maximum of which equal those of the entropy’s bounded maximum: F

kB

J

do- [n

—

n loge(n)+ an

+

/3n

(~

cos

~

—

~p.

sin 04, .

E) + n A .4, p. sin o].

(B.2)

Maximizing F is obtained by nullifying its derivatives with respect to n for any value of 0 and 4,. We have therefore: log(n)= a +/9(~cos 0—~p.sin oç~.E)+p. sin 0A~.ç~.

(B.3)

Using eq. (3.2) exp(a) is easily identified as N/f (or 217N/f).

Appendix C. Numerical integration Applying the “vectorial” (polar) transformation defined by eqs. (3.13), (3.14) and (3.15) to eqs. (3.31) and (3.32) gives for the harmonic components of the distribution function:

NX cos 0

(C.1)

NX sin 0 cos 4,

(C.2)

5 = NX sin 0 sin 4,

(C.3)

(417/3) b

=

(417/3) Pc =

(41.r13)p with (eq. (3.19)): /9’ ch(/9’) — —

sh(/9’)

/9’sh(/3’)

C4

X. de Hemptinne, Thermodynamics of laser systems

51

Introducing this transformation in the set of eqs. (4.2), (4.3), (4.4), (4.8) and (4.9) yields readily the set (5.2) to (5.6). (We take here p = 0.) We assume that the phase angles have reached their relaxed value (sin(4, 4,) = 1). This leaves us with three equations ((5.2), (5.3) and (5.4)) which may be written in compact form as follows: —

O=F(0,X,E),

(C.5)

J~=G(O,X),

(C.6)

E=H(o,x,E).

(C.7)

The steady form of the equations is obtained by nullifying their left-hand side (time derivatives). Our first aim is indeed to determine the steady values of the laser variables (On, X0 and E0) (which we shall call the “focus”) for given sets of rate-constants (yi,

)‘u

and K) and a given inversion of the thermostat

Equations (5.3) ((C.6)) appears to be independent of E. The field’s amplitude may therefore be eliminated between eqs. (5.2) and (5.4). The intersection between the resulting equation and the steady form of eq. (5.3) is then computed by successive approximations. Thermodynamic stability of the focus must now be determined. To this end, the equations of the motion (eqs. (5.2), (5.3) and (5.4)) are expanded in the neighbourhood of the focus (to the first order):

.aF

aF

aF

3G aG X=-~-(0—00)+~(X—X0)

(C.8)

(C.9)

aH oH OH E=—~-(0—00)+~(X—X0)+—~(E—E0).

(C.10)

This yields a set of coupled linear differential equations which is equivalent to a single differential equation of order three. The general solution reads: 51’ + A 2 + A 83’ (C.11) 0(t)= O~+ A1 e 2 es 3e

X(t) = X

51’ + B

0+

E(t)= E

52’ + B

53’

(C.12)

51+ C 52+ C 53’ 2e 3e

(C.13)

B1 e

2e

3c

0+ C1 e

where s~,s 2 and s3 are the roots of the following secular equation:

X

52

10F

OF

]

de Hemplinne, Thermodynamics of laser systems

OF

~-Sj

OG

rOG

1

LOX OH

i

l——sl

—

00

OH

0

=0.

(C.14)

IOH

-~

If a couple of roots is complex, it may be transformed into an amplitude and a frequency. If the roots are negative or if the real part of complex roots is negative, the laser variables relax to their focal value whenever they are perturbed: the focus is stable. In the opposite case, the slightest perturbation drives the system away from its focus: the focus is unstable. A particularly interesting case of unstable foci is the “limit cycle” domain discussed in fig. 1 and in fig. 3. Here integration must be performed numerically using a suitable Runge—Kutta procedure. This is a step-by-step integration. A time interval iXt is defined. Then, given a set of laser variables (0k, X1, E1) valid at time t1, a new set of variables (02, X2, E2) is computed, to be valid at time t2 = t1 + ~t. The procedure used for the trajectories presented in fig. 2 goes as follows. Given a set of laser variables (0k, X1, E1), the functions F, G and H (see eqs. (C.5), (C.6) and (C.7)) and their derivatives (the coefficients in eqs. (C.8), (C.9) and C.10)) are computed. The secular determinant is solved. Its roots are the rate constants for the evolution of the variables. (We assume here the roots to be real. The slight adaption required when roots are complex is trivial.) Assuming ~t to be small enough, the new set of variables is (see eqs. (C.11), (C.12) and (C.13)): 02 = 01 + A1 = X1 +

es1~t+

B1

A2 es2At + A3 es3~t

es1~t+ B2

e52~~t + B3 Cs3~t

E2=Ei+Cies1~t+C2e52~t+C3e53~.

(C.15)

(C.16) (C.17)

The matrix of the integration constants (A1, A2, A3, etc.) is calculated by fitting the successive derivatives of the laser variables at time t = t~: A1+A2+A3=0

(C.18)

A1S1 + A2S2 + A3S3 = F(01, X1, E1)

(C.19)

A1S~+ A2S~+ A3S~= FOF/O0 + G OF/OX + HOF/OX.

(C.20)

The accuracy of the method may be estimated by comparing trajectories obtained using different values of the elementary step t~t.The smaller is the step, the more reliable are the results (not considering unavoidable rounding off errors introduced at every step). The counterpart of enhanced accuracy is however an increased calculation time. Rounding off errors introduce stochastic perturbations at every step in the trajectory calculations. There are conditions (values of the rate constants y~,Vu, K) where the perturbation created in the preceding step tends to relax in the next ones. The result is a remarkably stable periodic trajectory. Any

X. de Hemptinne, Thermodynamics of laser systems

53

artificial disturbance is indeed soon washed out. This is the “periodic” domain on display in fig. 3. In the opposite case stochastic perturbations tend to be amplified and to accumulate, resulting in chaotic pulsating behaviour. The chaotic domain’s boundary in fig. 3 has been determined on the basis of the calculated trajectories’ stability.

Appendix D. Mode selective chemistry One of the laser chemist’s dreams is to succeed in promoting selective chemical processes by tuning incident radiation to resonance with particular molecular motions. There have been many attempts in this direction, unfortunately with poor results. This has led to the widespread opinion that multiphoton absorption of energy by polyatomic target molecules involves all the available vibrational degrees of freedom simultaneously. Energy absorbed by a particular motion would scramble into a structureless “quasi-continuum” representing all the molecule’s high energy vibrational levels. The fact that diatomics do not dissociate on absorption of resonant I.R. laser light is claimed to confirm the alleged requirement of a manifold of vibrational degrees of freedom. At least a couple of selective laser induced chemical reactions have been published in recent years, casting some doubt on the pertinency of aforementioned conclusions. These are Hall and Kaldor’s dissociation and dismutation of cyclopropane and the author’s time of flight experiments on CO2 laser excitation of ethylene, with conversion to the triplet state [29,30]. Multiphoton vibrational excitation of molecules is a process which involves the electronic ground state. If dissociation or any other chemical transformation is expected, the ground state potential energy surface must be intersecting potential energy surfaces leading to the required dissociation. Such intersections do not occur with diatomics. In chapter 9 it was shown that high vibrational excitation (multiphoton) of molecular oscillators by laser radiation requires transverse (phase) relaxation of the driven mode in order to divert the slaved motion’s energy into ergodic vibrational energy of the same mode. Recent literature ascribes this transformation of “quasi-periodic” into ergodic motion to chaos generating terms in the system’s Hamiltonian [22]. It has been stressed in chapter 9 (see also appendix A) that this suggestion is not at all conflicting with the thermodynamic theory of relaxation. Chaos generating terms in the Hamiltonian represent indeed interactions of the driven motion with its surrounding thermostat. The long ranged transverse relaxation makes energy absorption pressure dependent. However, parallel relaxation must be kept as low as possible (“collisionless” conditions or slow V—V relaxation) (eq. (9.33) or (9.35)) because it results in a loss of energy in the driven motion. Energy absorbed from the incident radiation is stored as a rise of the relevant motion’s temperature (~/3> 0). In polyatomics one mode becomes hot the other ones remaining cold. At this stage mode selectivity is preserved. The laser pulse will be assumed to be short. From now on intramolecular V—V transfer starts, feeding the other modes towards generalized ergodicity. This phenomenon has been observed very accurately with CO2-laser excited ethylene using a time of flight technique and analysing the evolution of the molecules’ vibrational state on the basis of the modification of the fragmentation spectrum following electron impact ionization [30]. Though high energy vibrations of polyatomics may involve complex motions, whereby several “normal modes” are coupled due to anharmonicity and/or Conolis interactions, the conservation laws of conservative Hamiltonian mechanics prevent randomization of the energy if the molecules are -

54

X

de Hemptinne, Thermodynamics of laser systems

assumed to be strictly isolated from each other. Distortion of the vibrational Hamiltonian during gaskinetic collisions (parallel relaxation) relieves temporarily the conservation laws allowing flow of energy to the other modes. The rate of intramolecular energy transfer between two modes depends on which modes are involved. This has been demonstrated clearly in the aforementioned experiment on ethylene. The rate of V—V relaxation was indeed shown to depend on the nature of the laser driven mode. When electronic excited potential surfaces cross the ground state surface, internal conversion may occur. With some molecules this may be an antibonding state leading to dissociation. In ethylene the lowest electronic state is the triplet form. It is known that this state intersects the ground state potential energy surface when the molecule is in a nearly perpendicular configuration (torsion angle) [31]. Several intramolecular flows may compete, one leading to conversion and other ones wasting the energy into the vibrational heat bath. One expects then that the kinetics of the observed chemical process is determined by the ratio of the kinetic flows leading to the product and to the waste. Mode selectivity may be obtained whenever this ratio is modified by changing the very nature of the driven mode on tuning the incident radiation to another resonance. This has clearly been shown by the author to be the case with ethylene on its conversion to the triplet state, this conversion competing with V—V relaxation. Generalized vibrational ergodicity and finally translational (and rotational) heating terminate the energy randomization.

References [1] Louis de Brogue, Certitudes et Incertitudes de Ia Science (Editions Albin Michel, Paris, 1966). [2] C. Cohen-Tannoudji, in: Laser Spectroscopy VI, eds. H.P. Weber and W. Luthy (Springer-Verlag, Berlin, 1983) p. 2. [3] J.P. Gordon, H.J. Zeiger and C.H. Townes, Phys. Rev. 95 (1955) 1264—1274. [4] T.H. Maiman, J. Opt. Soc. Am. 50 (1960) 1134. [5] A.L. Schwalow and C.H. Townes, Phys. Rev. 112 (1958) 1940—1949. [61 H. Haken, Synergetics, 2nd ed. (Springer.Verlag, Berlin, 1978). [7] M. Sargent III, M.O. Scully and W.E. Lamb, Laser Physics (Addison-Wesley PubI. Co., Reading, Mass., 1974). [8] F.T. Arecchi and E.D. Schultz-Dubois, editors, Laser Handbook, vol. I, part A (North-Holland Publ. Co., Amsterdam, 1972). [9] H. Haken, Laser Theory, in: Encyclopedia of Physics, vol. XXV/2C, ed. S. Flugge (Springer-Verlag, Berlin, 1970). [10] F. Bloch, Phys. Rev. 70 (1946) 460—474. [11] L. Onsager, Phys. Rev. 37 (1931) 405; 38 (1931) 2265. [12] P. Curie, Oeuvres (Gauthier-Villars, Paris, 1908) p. 129. [13] I. Prigogine, Introduction to thermodynamics of irreversible processes, 2nd ed. (Interscience Publishers, New York, 1961). [14] A. Katchalsky and P.F. Curran, Nonequiibrium Thermodynamics in Biophysics (Harvard University Press, Cambridge, Mass., 1965). [151S.R. De Groot and P. Mazur, Non Equilibrium Thermodynamics (North-Holland Pubi. Co., Amsterdam, 1969). [16] L.W. Casperson, in: Laser Physics, eds. J.D. Marvey and D.F. Walls (Springer-Verlag, Berlin, 1983) pp. 88—106. [17] LW. Casperson, IEEE J. Quarn. Elec. QE-14 (1978) 756—761. [18] N.B. Abraham, T. Chyba, M. Coleman, R.S. Gioggia, N.J. Halas, L.M. Moffer, S.N. Liu, M. Maeda and J.C. Wesson, in: Laser Physics (see ref. [16])pp. 107—131. [19] E. Arimondo, F. Cassagrande, L.A. Lugiato and P. Glorieux, Applied Phys. B-30 (1983) 57—77. [20] A. Messiah, Mécanique quantique (Dunod, Paris, 1960). [21] (a) W.E. Lamb, in: Laser spectroscopy IV, eds. H. Walter and K.W. Rothe (Springer-Verlag, Berlin, 1979) p. 296 and in: Laser spectroscopy III, eds., J.L. Hall and J.L. Carlsten (Springer-Verlag, Berlin, 1977) p. 116. (b) B. Walker and R.K. Preston, J. Chem. Phys. 67 (1977) 2017—2028. [22] E.V. Shuryak, Soy. Phys. JETP 44 (1976) 1070—1080; D.A. Jones and I.C. Percival, J. Phys. B At. Mol. Phys. 16 (1983) 2981—2996; H.W. Gaibraith, J.R. Ackerhalt and R.W. Milonni, J. Chem. Phys. 79 (1983) 5345—5350; J.R. Ackerhalt, H.W. Gaibraith and P.W. Milonni, Phys. Rev. Lett. 51(1983)1259—1261.

X. de Hemptinne, Thermodynamics of laser systems

55

[23] W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1981). [24] I. Prigogine, Non equilibrium statistical mechanics (Interscience Publishers, New York, 1962). [25] H. Goldstein, Classical Mechanics (Addison-Wesley, Cambridge, 1951) p. 3. [26] S.A. Rice, in: Photoselective Chemistry, eds. J. Jortner, R.D. Levine and S.A. Rice, Advances in Chemical Physics, vol. XLII (John Wiley, New York, 1981) pp. 117—200. [27] P.1. Belobroy, G.P. Berman, G.M. Zaslavskii and A.P. Slivinski, Soy. Phys. JETP 49 (1979) 993—997. [28] D. ter Haar, in: Physical Chemistry, an advanced treatise, vol. II (Statistical Mechanics) ed. H. Eyring (Academic Press. New York, 1967) pp. 53—108. [29] R.B. Hall and A. Kaldor, J. Chem. Phys. 70 (1979) 4027. [30] X. de Hemptinne and D. De Keuster, J. Chem. Phys. 73 (1980) 3170—3177. [311R.S. Mulliken and C.C.J. Roothaan, Chem. Rev. 41(1947) 219.

General bibliography It is not possible to give a full account of the very broad research literature covering the different topics on discussion in this monograph. This is left over to review papers which are many in this field. Some of the latest are: G. Singh, Statistical properties of single-mode and two-mode ring lasers, Physics Reports 108 (1984)

—

217—273.

— S. Mukamel, Collisional broadening of spectral line shapes in two-photon and multiphoton processes, Physics Reports 93 (1982) 1—60. L. Allen and C.R. Stroud Jr., Broadening and saturation in n-photon absorption, Physics Reports 91 (1982) 1—29. The present discussion refers to fundamentals in physics, quantum mechanics, statistical mechanics and thermodynamics. The relevant questions have grown mature since long and are available for study in major textbooks. Additional reading suggestions given hereafter are only exemplary.

—

Statistical thermodynamics: R.K. Pathna, Statistical Mechanics (Pergamon Press, 1972). G.H. Wannier, Statistical Physics (John Wiley, New York, 1966). —C. Kittel, Elements de physique statistique (Dunod, Paris, 1961) (translated from: Elementary statistical physics (John Wiley)). D. ter Haar, Elements of Thermodynamics (Addison-Wesley, Reading, Mass., 1966). D. ter Haar, Elements of Statistical Mechanics (Rinehart, New York, 1954).

-— —

— —

Maxwell’s equations and dipole radiation (self-effect) H. Bremmer, in: Encyclopedia of physics, vol. XVI, ed. S. Flugge (Springer-Verlag, Berlin, 1958) pp. —

423—640.

—R.P. Feynmann, R.B. Leighton and M. Sands, The Feynmann lectures on physics (Addison-Wesley PubI. Co., Reading Mass., 1965) vols. I and II. Limit cycles — P. Glansdorff and I. Prigogine, Thermodynamic theory of structure, stability and fluctuations (WileyInterscience, London, 1971). Non-linear phenomena bistability Laser Physics, eds. J.D. Marvey and D.F. Walls (Springer-Verlag, Berlin, 1983). —

—

56

—

—

X. de Hemptinne, Thermodynamics oflaser systems

R. Bonifacio, L.A. Lugiato and M. Gronchi, in: Laser Spectroscopy IV, eds. H. Walter and K.W. Rothe (Springer-Verlag, Berlin, 1979) p. 426. V.P. Chebotayev, in: Coherent Nonlinear Optics, eds. M.S. Feld and V.S. Letokhov (Springer-Verlag, Berlin, 1980).

Multiphoton excitation and reactions N. Bloembergen and E. Yablonovitch, in: Laser Spectroscopy III, eds. J.L. Hall and J.L. Carlsten (Springer-Verlag, Berlin, 1977) p. 86. CD. Cantrell, V.S. Lethokov and A.A. Makarov, in: Coherent Nonlinear Optics, eds. M.S. Feld and V.S. Lethokov (Springer-Verlag, Berlin, 1980) pp. 165—270. —C.D. Cantrell, S.M. Freund and J.L. Lyman, in: Laser Handbook vol. 3, ed. M.L. Stitch (NorthHolland PubI. Co., Amsterdam, 1979) pp. 485—576. J.I. Steinfeld, ed., Laser-induced chemical processes (Plenum Press, New York, 1981); contributions by H.W. Galbraith and J.R. Ackerhalt and by W.C. Danen and J.C. Jang. V.5. Lethokov, Nonlinear laser chemistry (Springer-Verlag, Berlin, 1983). D.S. King, in: Dynamics of the Excited State, ed. K.P. Lawley, Advances in Chemical Physics vol. L (John Wiley, Chichester, 1982). —

—

—

— —

Intra- and intermolecular relaxations Energy Storage and Redistribution in Molecules, ed. J. Hinze (Plenum Press, New York, 1983). Photoselective chemistry part I, eds. J. Jortner, R.D. Levine and S.A. Rice, Advances in Chemical Physics vol. XLVII (John Wiley, New York, 1981).

— —