ERRATA Page Line Read s Should Read iii 8 Resoponse Response 8aA aA. 8 1 at at 8 Eqo (4) r and r r and r 10 13 op (r,t), jO P(t) Oo op. 10 15 J tr,t) X\(r) J(rt)o X ( r) 10 17; 11 6 delete be 14 4 ttT t +T 23 19 Eqs. (25) Eq's. (.8)'S 28 21 U(t-t' ), U (t; ) 29 4 D(t) D 44 6 chapter section 44 12 i(... 44 16 numerator should read 2F+ M2M1 47 2 don do 50 16 posses possess 51 9 kk_.X 51 13 kh k15 52 11 subscrip subscript 53 4 and is and k is

ERRATA (Concluded) Page Line Read s Should Read 53 9 see Eq. (63b) see Eq. (65b) 54 5 Eq. (79) Eq. (79a) 54 7 X 2cXN 55 4 2TN2XhdA 42N2Xhd 55 19 1" " 58 4 " " 222+ 70 -7 ryh'Nh22aX2d+ N d4N 22X + 71 2 apposed opposed 2 2 22 n N2Xd\ Ti N2 d 73 1 2r+ ro O0 73 16 T II I? ti 74 7?" " " 87 7 Schalow Schawlow 95 15 complets completes 112 19 Schalow Schawlow

THE UNI VERSI TY OF M I C H I GAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report QUANTUM THEORY OF MULTIMODE CAVITIES AND APPLICATION TO THE STEADY-STATE SPECTRUM OF GAS OPTICAL MASERS P. Lambropoulos R. K. Osborn ORA Project 04836 under contract witha NATIONAL SCIENCE FOUNDATION GRANT NO. G-20037 WASHINGTON, D. C. administered through: OFFICE OF RESEARCH ADMINISTRATIO AN ANN ARBOR November 1964

This report was also a dissertation submitted by the first author in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1964.

TABLE OF CONTENTS Page A.BSTRA.CT V CHAPTER I. INTRODUCTION 1 II. FORMULATION OF THE PROBLEM 6 III. THE OUTPUT SPECTRUM 13 IV. METHOD OF SOLUTION 20 1. Time Evolution of Gi 20 2. The Response Function 26 3. The Susceptibility Function of A Two-Level System 34 4. Power Spectral Density of Spontaneous Transitions 42 5. Statistical Approximation 46 V. APPLICATION TO A. GAS OPTICAL MASER 50 1. The Spectrum of Spontaneous Emission 50 2. The Spectrum of Spontaneous Emission in the Statistical Approximation 55 30 The Susceptibility Function in the Statistical Approximation 59 4. The Susceptibility Function for the Case of Negligible Statistical Broadening 62 5. The Steady-State Output Spectrum 70 6. On the Loss Mechanism 77 VI. COMPARISON WITH OTHER THEORIES 82 VII. CONCLUSIONS 88 APPENDIX A. DAMPING THEORY 90 APPENDIX B. ON THE MATRIX ELEMENTS OF Gh 96 APPENDIX C. ON THE CA.LCULATION OF THE SUSCEPTIBILITY 100 APPENDIX Do CALCULATION OF THE DOPPLER WIDTH 105 iii

TABLE OF CONTENTS (Concluded) Page APPENDIX E. CALCULATION OF 2().107 REFERENCES 111 iv

ABSTRACT A quantum theoretical treatment of the interaction between the electromagnetic field in a multimode cavity with loss, and an assembly of particles, is presented. The theory is developed in the linear approximation, and the linear susceptibility, which is shown to account for absorption and stimulated emission, is defined. The field is shown to be driven by spontaneous emission from both the active maser material and the loss mechanism, in the absence of any other driving force. The steady-state power spectrum of an optical maser oscillator is studied. The spectrum is expressed in terms of the spectrum of spontaneous emission from the upper level of the active maser material, and in terms of the susceptibility. Homogeneous broadening, such as natural and collision, and inhomogeneous broadening, such as Doppler and statistical, are taken into consideration. Their effect on the spectrum of spontaneous emission and on the susceptibility is studied. Explicit forms of the susceptibility, valid under various conditions, are obtained. Consideration is also given to a model of a gas optical maser operating in a single mode. Spectrum narrowing is discussed, and a frequency pulling equation, which is found to contain a new term, is obtained.

Comparison to other treatments of the problem is made. It is found that our results are more general in several aspects, and that previous results are obtained as special cases of ours. Possible extensions of the work are also discussed. v'i

CHAPTER I INTRODUCTION The extensive literature on the subject eloquently attests to the interest stimulated by the discovery of the maser. As a source of electromagnetic radiation, the maser possesses an unprecedented spectral purity, directionality and intensity, to mention only some of its properties. At the same time, a fairly complex theoretical problem is posed; that is, the description of the electromagnetic field inside a cavity, and its interaction with material systems therein. For certain applications, the properties of the device are much better than needed and the accurate mathematical description appears to be rather unnecessary. It is in part for this reason that severely simplified models seem adequate for the description of the phenomenon. There are certain questions, on the other hand, which make the necessity for a refined and consistent analysis imperative. For example, the coherence time 12 of a maser beam is intimately related to the spectral width which requires a fairly accurate mathematical description. In addition, the phenomenon as such has an intrinsic interest independent of the applications. The present treatment is an attempt to present a deductive derivation of the steady-state spectrum of the power output of a maser oscillator~ It is generally agreed that the maser action is a quantum 1

2 mechanical phenomenon. The mere fact that one deals with emission of radiation by excited atomic systems ought to be fairly convincing in this respect. It is only natural, therefore, that we employ the quantum theoretical formalism. The basic axiom is the choice of the hamiltonian of the system. We take the non-relativistic hamiltonian of an assembly of particles interacting with the radiation field, and with each other. Spin is ignored. The radiation field is described by the quantum mechanical Max3,4 well's equations, in which the fields and sources are the expectation values of appropriate operators. This enables one to fully account for the quantum effects of the material system to any desired approximation. In Maxwell's theory however, the energy of the field is expressed in terms of the field vectors which are already averages of the respective operators. Quantum mechanically, the energy is given by the expectation values of the squares of those operatorso.5 An approximation inherent in Maxwell's theory therefore, is the replacement of averages of products by products of averages. It is presumed that this approximation is justified because of the high photon densities involved in the output of maser oscillators. The theory has been developed in the linear approximation which leads to the susceptibility functiono Mode coupling has been neglected. Among our main concerns has been to avoid introducing phenomenological parameters, especially in studying the interaction of the active maser

material with the radiation field. In this effort, damping theory has been used. Although most of the discussion has been devoted to the maser oscillator, the method is applicable to many problems involving the interaction of a material system with the radiation field. In particular, the method illuminates several aspects in the study of the 6 electric susceptibility; namely, the effect of broadening mechanisms. Chapter II is devoted to the formulation of the problem. The output spectrum is defined Chapter III, in terms of Fourier transforms of truncated functions. Equations for the field operators are developed, and their Fourier transforms are expressed in terms of the Fourier transform of the current operator, thereby reducing the problem to the calculation of the Fourier transform of the current operator Ga(t). The time evolution of G\(t) is taken up in Chapter IV section 1. An integral equation for the current operator is developed and iterated. In IV-2 the linear approximation is introduced. This linearization generates the response function, for the calculation of which damping theory is used. The result is specialized to a two-level system, in IV-3. The susceptibility, representing the effect of the material system on the field, is defined. In IV-4, the spectrum of spontaneous emission from the upper to the lower level of the two-level system is derived. It is shown that the spontaneous emission drives the field, in the absence of any other driving force. The output spectrum is then expressed in terms of the spectrum of spontaneous emission and

the susceptibility. The approximation made in replacing the average of a function by the function of the averages is discussed in IV-5. This approximation is used in Chapter V which is devoted to the application of the results obtained thus far to a model for the gas optical maser. As compared to an actual commercial device, the model is rather idealized. However, devices satisfying several of the conditions imposed by the model can be and have been constructed.7 The spectrum of spontaneous emission and various forms of the susceptibility are calculated. The effect of various broadening mechanisms is discussed. Moreover, the output spectrum for operation in a single mode is calculated and a frequency pulling equation is derived. In section 6 of this chapter we elaborate somewhat on a quantum mechanical description of the loss along lines similar to those of Ref. 8. Chapter VI is devoted to the comparison of the present theory to other treatments. Finally, the main conclusions are summarized in Chapter VII where possible extensions of the work are also discussed in brief. The use of damping theory brings this work to close relationship 9 10 with the work of A. Z. Akcasu on the applications of damping theory to the study of line shape. We have developed the formalism in parallel to that of the above references, especially Ref. 9. Expressions for shifts and widths, not dwelt upon here, can be obtained with little or no change therefrom.

The present treatment, being a linearized theory of a nonlinear system, cannot answer questions concerning the behavior of the oscillator beyond threshold. For example, it would predict that, when the gain exceeds the losses, the power increases without limit. It is known however, that existing non-linearities stabilize the system. The general formulation nonetheless contains this non-linear behavior. A question of terminology arises in connection with the frequency pulling equation in section 5 of Chapter V. According to the accepted electrical engineering terminology, the frequency shift due to the detuning between cavity resonance and active material resonance should be called frequency pulling; the shift which depends on the population inversion, and hence on external action (pump action), should be called frequency pushing. Throughout this study, the term maser will be used in a broad sense covering all maser oscillators irrespective of frequency. Whenever we refer to optical frequencies, we shall use the term optical maser."

CHAPTER II FORMULATION OF THE PROBLEM The system we propose to study consists of an assembly of material particles placed inside a cavity with highly reflecting wall.s The cavity may be a more or less closed structure, of the type encountered in microwave applications, or an open structure of the Fabry-Perot type~. 1 3' In either case, the electromagnetic field inside the cavity is describable in terms of normal modes. A. normal mode is a certain spatial distribution of the field vectors which is determined by the geometry of the cavity and the nature of the walls, through the eigenfunctions of a boundary value problem. In the absence of any dissipative interaction, the modes exhibit a harmonic time dependence with characteristic frequencies determined by the eigenvalues of the above boundary value problem. Dissipation (or loss) may result from imperfect reflection at the walls and/or from the geometry of the cavityo The diffraction loss around the edges of the plates of a Fabry-Perot resonator, for example, is an inherently geometric losso Even in a totally enclosed microwave cavity with perfect walls an opening is necessary for providing coupling with the external world. The effect of this opening is to introduce loss. Thus, in any realistic cavity some loss will be present. As a consequence of the loss, the time dependence of the modes is not purely harmonic but a more 6

7 complicated function of time. For sufficiently small loss however the Fourier spectra of the modes are peaked functions of frequency, centered at the characteristic frequencies of the cavity.15 If (r,t) and._(r,t) are the electric and magnetic fields, respectively, inside the cavity, their time evolution is governed by Maxwell's equations which in Gaussian units read5: V x _~ 1 _ - 43r J (r t), (la) c at c V X ~ l+ 1 X= O, (lb) c at v 4, (lc) Vh ~~ *0~ F -.(ld) where J and p are the macroscopic current and charge densities, respectively, which shall be referred to as the source terms, and c is the speed of light. If the macroscopic charge density is zero inside the cavity, one can dispense with the longitudinal part of the electric field and the current. Since this will be the case in all problems to be considered here,,6(r,t) and J (r,t) shall indicate the transverse parts of the electric field and the current, respectively, in the remainder of this treatment. Then, the fields can be derived from the vector potential A (r,t) through the equations.._ - VX A, (2a)

1 A (2b) c at It follows from Maxwell's equations that A. is governed by the equation 72 A. 1 -2A 4r J (2c) C2 at2 c The field vectors, A (r,t), and J (r,t) are interpreted as the expectation values of appropriate quantum mechanical operators that is, ~ (r,t) = Tr D LP' (t), (3a) J (r,t) Tr D 3P' (t), (3b) J (r,t) = Tr D JOP~ (t), (3c) A. (r,t) = Tr D A0p' (t), (3d) 16 where D is the density operator of the system, and the superscript op. indicates that the quantity it qualifies is an operator. The current operator is given by18 Jop (r,t) = e e A p.( (r ) + e o (rO) c + b (r-r ) Po _ e A~Po (r )) (4) c where eo and m0 are the charge and mass, respectively, of the oth particle. The summation extends over all particles present inside the cavity and the walls. r and p0 are the position and momentum operators, respectively, of the oth particle. The time evolution of the operators is governed by Heisenberg's equation of motionO For gpo (t), for example, we have

a' (t) = i j H, - o_(t), (5) at...h... where H is the total hamiltonian of the system. The non-relativistic hamiltonian of the system considered here is H =H +R 1 PC e o A.P (r + HI (6 2ml c where HR is the hamiltonian of the free radiation field, and H is a term containing particle-particle interactions, interaction between particles and external fields, as well as the hamiltonian of the external world. External fields are to be distinguished from the cavity field For example, pumping fields are to be considered as external fields as long as they do not have frequency components close to the cavity frequencies of interest, which we assume to be the case. Let x (r) A = 1,2,3, oo be a set of eigenvectors satisfying the equations v2 x~(r) + k2 X~ (r) = (7a) X1 ~ X>, (r) ~ (7b) and boundary conditions appropriate to the cavity under consideration. The domain of definition of these eigenvectors is the interior of the cavity. The k!s are the eigenvalues. For a certain class of boundary conditions of interest to us here, the set (XQh is orthogonal, real and complete.1 It will be assumed normalized to unityo Moreover,

10 the set {1k V._X\ (r) is orthonormal if _[XN is orthonormal. Expanding the vector potential, the current, and the field operators in terms of (X)A we have A p'(rt) = > i P (t) X (r), (8a) k J0P (rt) = XjiP (t) X (r) (8b) eoP (r,t) = > 4fi~T Q\ (t) _ (r), (8c) jP (rt) = ~ - PA (t) (VX (r)) (8d) where 5 - c k\' (8e) and P\, Q are hermitian operators which obey the commutation relations [Q,,P-A] = i 5\,h, (9a) [Q&A,Q,] = [Ph,PAt] = O. (9b) op0 The operator j\ (r,t) results from the modal decomposition of J (r,t), and is given by jx (t) J P (rtt) XX (r) d3 r eo (ro ) ~ po a me _- P, o _Xkea (r2 ).k' (r~). (lOa) A k, a me c

11 The second term in the right side of the above equation gives rise to mode coupling since it contains a summation over all modes. Since we wish to confine the present treatment to the case in which mode coupling can be neglected, we shall neglect all terms for which'.o The remaining term (for A = A\) represents a small correction to be the frequency o\ of the passive cavity, and we shall assume that \ is so redefined as to incorporate this correction. Thus, we shall take op. _ GA, (10b) Jh A where, in order to compress notation, we have introduced the operator au>\eoX (rO). p0 e (10c) Combining now Eqs. (2), which are also true in operator form, and Eqs. (8) we obtain a Ph (t) + 2 PA (t) = Cu G-(t), (11a) at2 and a PA (t) + Q2 (t) Oa. (lLb) at 18 We shall adopt the Heisenberg picture and hence all operators will be time dependent, their time dependence being determined by the time evolution operator U(tto) - e tto)H (12a)

12 Then, for example, GA (t) = U (t,to) G, (to) U (t,to). (l2b) In the calculations we shall choose to = 0 and shall omit the time argument whenever it is zero.

CHAPTER III THE OUTPUT SPECTRUM In interpreting energy transfer experiments, one would like to calculate the quantity 1 Tr D (JP' + PP? ). (13a) 8i This, in general, requires the solution of transport equations whose very formulation is not an easy task.17 For relatively high field densities however, (13a) can be approximated by 1 (I 2 ) (13b) 8~ I and this we assume to be the case in this study. Despite this approximation, the Maxwell's equations do provide a quantum mechanical description of the system if the source terms are interpreted correctly, that is as the expectation values of the appropriate operators. Under the assumption that the above approximation is satisfactory, the energy output per unit time, that is the power output, is given by h Pe (t), (14a) where pA (t) - Tr D PA (t) (14b) and Ad is a constant having the dimensions of inverse time, and relating the power output of the Ath mode to the energy in that mode 13

14 stored inside the cavity~ In general, the power output will be a function. of time. The system shall be said to be in steady state if W, defined by ttT W -I_ 1X pX (t') dt' (15a) T t is independent of t for T large enough. The lower limit of T is determined by the characteristic times of the physical processes taking place inside the cavityo Ultimately we shall approximate the power output by W - lim WT, (15b) T-rPoo It should be emphasized that WT has a physical content, while W is a mathematical quantity by which we approximate WrO Confining the present treatment to the steady-state we take t = 0 in Eqo (15a)o Thus W becomes W = lim 1 j p (t'P ) dt'K (15c) T-+oo T0 We now define PXT (W) /PX (t) e dt, (16a) I which is the Fourier transform of a truncated function equal to px (t) for 0 < t < T and zero otherwise. In terms of Fourier transforms WT readso WT X 2i i AT PhT(w) (l6b) A2%t T u~x

15 Since ph (t) is assumed to be a real valued function, we shall have (-" P*T (W)~ (16c) where * denotes the complex conjugate. Thus? Eq. (16b) becomes WT 1T IP2T (W)12 dw. (16d) Introducing RAT (w) - i_ PAT (w) | (17a) qT and RH (c) lim RAT (oW), (17b) T-Coo we obtain WT A J RT (w) dw, (-7c) and W J lhA 0 RA (ws) do. (17d) RA (w) is identified with the steady-state power spectral density of the Ath mode. The steady-state power spectral density of the output is Taking the Fourier transform of Eq. (1L4b) we obtain PAT (s) Tr D PA (t) e dt - Tr D PAT{), (19)

16 where we have interchanged trace with integration, and we have introduced the operator P T(0U) defined as the Fourier transform of P,(t). Combining Eqs. (16c) and (19) we have IPXT())I = (Tr D PGT()< (Tr D P\T(-)) Assuming that we are dealing with field densities high enough to justify the replacement of products of averages by averages of products, and vice versa, we take IPAT(c) = Tr D PXT(C) PXT(-w) (20) Then, from Eqs. (17b) and (18) we obtain R(wo) = lim 1 Tr D PXT(w) P\T(-w)' (21) T+o tT thereby reducing the problem to the calculation of PXT(w). The motion of Pj(t) is governed by Eq.(lla). The operator G\(t) appearing in the right hand side of this equation contains the coupling between the cavity field and the particle system. Note that GA, as defined by Eq. (10c), involves a summation over all particles present. It is convenient at this point to separate Gh in two parts as follows: G GA + (22) a,~ hL2: (22) L where T is an operator involving a summation over the particles of the wall or any other passive material that may exist inside the cavity, A while a involves a summation over the particles of what we shall call the active material. The latter is the material whose presence gives rise to the maser action and whose quantum effects on the field we wish

17 to study. L A Both G\(t) and GA(t) can be expressed in the form of a perturbation expansion in ascending powers of the field operators. If one retains the first two terms of the expansion and then take the Fourier transform of Eq. (11a), after some mathematical manipulations, the following equation is obtained: (0+2 P(o3) 2) LS 2 AS (2 + 2) PA(X0) 2 GT(c) + G\ G\T(W) -iWYY\ P\T (t) + (DA Y\(0\) PAT () (23) The first two terms on the right hand side are independent of the field and account for spontaneous emission from the passive and active material, respectively. The third term, involving the constant 7X\ accounts for the dissipative effect of the passive material (usually referred to as the loss mechanism) on the field. The constant 7\ depends on parameters such as the conductivity of the passive material. The fourth term accounts for the effect of the active material on the field. The function Y(0)) is the linear susceptibility and contains the effect of induced emission and absorption. Once the functions on the right side of Eq. (23) are known, one can solve for PXT(t) substitute into Eq. (21) and obtain an expression for the output spectrum. Thus, the main task in the remainder of this study will be the determination of the foregoing functions in terms of the dynamical parameters of the system.

18 Since the main objective of this treatment is the optical maser, it is presumed that spontaneous emission from a material in thermodynamic equilibrium at room termperature will be rather inconsequential at optical frequencies. For this reason, and in order to avoid LS mathematical complexity we shall neglect G. The term involving 7y shall be kept however, since it has an important effect on the output LS spectrum. Both GA and yZ have been discussed in Ref. 8, although not in Fourier domain and in a somewhat less general context. A brief discussion is presented in Chapter V, section 6. It should be mentioned that two assumptions inherent in describing the loss in terms of y\ are: the loss is small, that is ~h <1; (24) and mode coupling due to the loss mechanism can be neglected. It is also assumned that there is no interaction between loss mechanism and active material. At optical frequencies, and with gaseous materials in Fabry'Perot cavities, the foregoing conditions are satisfied. For solid state materials, the,loss mechanism and the active material might be correlated. This case is not taken up here. For lower frequencies, spontaneous emission from the passive material (usually referred to as thermal noise) may also be of importance. Then, one will LS ~ave to retain the term ~T (G, ) as well (see also Chapter V-6).

19 The problem is now effectively reduced to studying the motion A AS L of Gh(t), and determining G~T(w) and Yh( o)o Since G\ has been dispensed with, we shall omit the superscript A. Thus, the equation we now have is T -iwt +(- iw+ i +) T() = \ (t) e dt, (25) where Gr\(t) corresponds to the active material only. In conventional classical electrodynamics, one usually assumes that Y(:(w) has a real and an imaginary part. The dependence on X is then -taken to be of the form (see for example Ref. 6) 1 U-COO-ir The constant r is introduced in order to account for losses associated with the susceptibility, and 0o is a transition frequency characteristic of the material in question. Of course, this form is valid for values of X near wo. The question arises however, as to when the above form is valid, and how one can determine r in terms of the dynamical parameters of the system, The question presents itself also in connection with the spectrum of spontaneous emissiono It is precisely these questions that we attempt to answer herein.

CHAPTER IV METHOD OF SOLUTION 1. TIME EVOLUTION OF GI The time evolution of Gh(t) is governed by the equation %a (t) i [H, %G-(t)], (26) at where H is as given by Eq. (6). In order to proceed further a more detailed specification of the system, and hence the hamiltonian is necessary. Thus, assume that the cavity contains two kinds of materials: The active material whose atoms and/or molecules are capable of making radiative transitions, and a second material which shall be termed "the perturber." Both are assumed to be in the gaseous state. The perturber does not interact with the cavity field but it does interact with the active material through collisions. Let HP be the hamilPA tonian of the perturber, and V the energy of interaction between the active material and the perturber~ Note that if collisions with the walls are of importance in perturbing the active material, the hamiltonian of the wall should be included in HPo Using the expansion of the vector potential operator in terms of (XQi, and neglecting the term ec A (r ) as representing effects o 2moc2 of higher order, we can write the hamiiltonian in the form H = HE H gR + A + R + AX (27) 20

21 where HE is that part of the hamiltonian of the external world which is coupled to the active material, that is the hamiltonian of the pumping mechanism; HR is the hamiltonian of the free cavity field and is given by HR = \X (P2 + a Q); (28a) 2 HA is the hamiltonian of the active material; VRA is the energy of interaction between HR and HA and is given by P GA; (28b) EA. V represents the coupling between pumping mechanism and active material. Recall that GA is expressed as a sum over all particles of the active material (see Eq. (10c)), that is over electrons as well as nuclei. Since the materials actually used in cavities consist of atoms and/or molecules, the sum will have to be regrouped into partial sums each of which will represent the particles of one atom or molecule. This however, does not have to be done until a later stage (see also Appendix B). Combining now Eqs. (26)and (27) we obtain EA a,(t i [H(t) + (t) Gi [HE(t)] + (29) + i [H(t) G(t)] + i [H4t) + H, (t), G ],

22 where we have introduced A RA. PA (30) H H + V +V. ( The first commutator represents the effect of the pumping mechanism. Since we are interested only in the steady-state, we shall ignore this commutator and account for its effect by assuming that the populations of the states of the active material are kept at certain constant values by means of sufficient pumping. The approximation involved is the neglect of the details of the pumping mechanism, and is useful as long as one is interested in the steady-state spectrum only. At this point, motivated by Senitzky's work, we consider the equation of motion of Hx(t), namely; HX(t) = i [H, Hr(t)] at E P EA - i [H (t) + H (t) + V (t), HW(t)] + + i [H Rt) H(t)] (31) Neglecting the first commutator and calculating the second using Eqs. (28) and (30), we obtain a Hg(t) = i [HR(t), vt)]. (32) at 4 Integrating formally we have H'(t) = Hl(o) + i dtl[Htl) VW1)] (

Finally, in Eq. (29), we replace H (t) and HP(t) in the third commutator by HR(o) and HP(o). Then, using also Eq. (33), we obtain a G(t) i [Ho)+ o ) + Ho H(o, (t)] att P re t - R RA. - 1 1o dt, H(tl), V (t),' G(t). (3 4a) We now introduce the operator H defined by S Tr. R P H H +H +H H =HA R + +HPHR VH + PA (34b) For future use we also define o A R P H - H + H + H (34c) and v VRA + VPA (3 4d) S Then, H reads H = HO + V (3 4e) An integral equation equivalent to the integrodifferential equation (34a) is S S 2Ht iHt G (t) e ( t e i-. t SAG i (t-t ) HH _ e i (t-t5a where operators without t-ime ue argument are to be understood at t 0 From Eqs (25)~ and the commutation relations for Pk and Qk we have

24 [H(t2), "'Vkt2)] = - i- Qi (t2) Gg (t2). (3b) Combining Eqs. (35) we obtain Gx (t ) += G c (t) dtl dt2 U (t-tl)~ [Qx,(t2) Gx,(t2), Ga(t1)] US(t-tl), (36) where we have introduced SH t U (t) e t (37a) and GS(t) - uS Gt) uS(t) (37b) Equation (36) represents a set of, in principle, infinitely many coupled integral equationso Since this expression is to be substituted into Eq. (25), it is obvious that we shall have infinitely many, coupled equations for PAT(w) (. - 1,2,3..o). In any actual maser oscillator 20 21 only a finite number of modes oscillate simultaneously. This reduczes the set of equations to a finite set which still are coupled. UInder certain conditions furthermore, the modes may oscillate independently of each othero Limiting this treatment to the case in which the coupling terms can be neglected, we drop from the sum in Eq. (36) all except the Ath terms. Then, we hatve'n,(t) Gh(t)'i +2 > dt1 dt2 1, (t-t) 8)

25 Q[Q\(t2) G\(t2), Gx(tl)] U(t-tl). (38) Relatively little can be done with this integral equation without further approximations. Thus we resort to an iteration procedure, iterating the equation once and retaining terms up to and including the second order in Ga(t). Then, we obtain t ti - G (t) = Gj(t) + 2/ dtj dt 2 S(t-t ) S S S'[Q\(t2) G\(t2), G~\(tl)] U (t-t) ~ (39) Substituting into Eq. (25)we have (_2 + IX + e) P?\T(W) = \ ~T( ) +!T -iw~t ot ~tl S4+ ) i Jdt e Jdt1 dt2 U (t-tl)> whr o o -[Qx(t2) G~(t2), S (tz)] US(t-tl), (40a) where we have introduced G T S - iwt S~h~U Jt G(t) e I (40b) The crux of the problem is the handling of the right hand side of Eqo (40a). As will be shown subsequently, the first term corresponds to spontaneous transitions from the excited levels of the active material. The second term represents the response of the material system to the field, in the linear approximation. That is after the operator QA(t2) is taken out of the commutator, as discussed in the following section.

26 It is perhaps in place to note the difference between the time dependence of operators bearing the superscript S and those which do iSt not. The former's time dependence is determined by Ui(t) e-t HSt while -the latter's is by the time evolution operator for the whole i Ht system, namely U(t) = e - 2. THE RESPONSE FUN]CTION Since we wish to confine this treatment to the linear approximation, we write the integrand of the right hand side of Eq. (40a) as follows ~ Q7(t2) UiSt-tl) [Gr(t2) G~(t,)] US(t-tl) ~ (41) In doirg so we have neglected the commutator of Q (t2) and US(t-t-) This comnmutator can be neglected in the zeroeth order approximation, i.e. in the absence of any interaction between field and material systemb In the next order approximation, the commutator yields terms linear in G\ and consequently the corresponding term in (41) will be of third order in G?. The approximation invoived in (41) therefore, is to neglect terms of order higher than. the second in G\ consistently with our previous assumptions. in view of the fact that ultimately we shall take the trace with e A d ityt opt2) -Tr wD U (t-tu) [f t ( t 1.tz] 1'd(t<t,)f (42)

27 and we replace (41) by Q\(t2) A(t Ytlt2) (43) Formally, the approximation involved in replacing (41) by (43), is replacing the average of a product by the product of the averages. This is done in a way such that the field operator is separated from the particle operators. The resulting function A(ttlt2) is a response function representing the effect of the material system on the field. Again quantum effects of order higher than the second in G\ are neglected. The equations for the field operators thus obtained are linear and as will be seen subsequently they account for spontaneous emission, induced emission, and absorption. This is essentially the dielectric approximation.22 As long as more than one photon processes do not play an important role, the formalism is expected to be adequate for the study of the power spectrum. If photon scattering, for example, becomes important, which may be the case with some solid state materials, a major revision will be necessary. The approximation is somewhat similar to the irreversibility approximation frequently used in the description of systems in contact with a thermostat, where the interaction is assumed to be small enough (or the thermostat large enough) for the effect of the system of interest on the thermostat to be negligible. Here, the situation is different in several aspects. The active material, which corresponds to the thermostat, is neither in thermal equilibriim nor is

28 it a large system. In fact, for maser action to take place, it is necessary that the level populations be inverted. However, if we consider as thermostat the active material plus the pumping mechanism, then we do have a large system. Moreover, we may assume that the level populations of the active material are kept at a desired value through sufficient pumping. Thus, the approximation underlying (43) and the subsequent calculations essentially involves the deletion of information concerning the pumping mechanism and the build up of the oscillations, thereby restricting the present treatment to the steady-state. Incidentally, it is important to note that an additional difference from the thermodynamic problem is that here we have a steady but not an equilibrium state. Substituting (43) into Eq. (40a) we obtain (_2 + inur + 2~) P2T(m) = cc GA\T() + T i t to i dt e dt dt2 QJ(t2) A (tt1,t2), (44) h o o o from which one recognizes that A(t,t:,t2) is a functional relating the effect of the material system on the fie~ld at all times previous to t, to the field at time t. Turning now to the calculation of A(t,tl,t2), and using Eqs. (37) and the identity u(t-t') = US(t) US (t'),

29 we obtain A(ttlt2) = (t,tl-t,2) Tr U (t) D U (t) U (tl-t2) G, U (tl-t2) G - - Tr UJS(t) D(t) (t) GA US(tl-t2) GX US(tl-t2), (45) where we have used the identity Tr A B = Tr B A, which is valid for any two operators A, B. Noting that the second term in the right hand side of Eq. (45) is the complex conjugate of the first, because the corresponding operators are the hermitian adjoints of each other, and setting u8.(t) D U ) D(t), (46a) we obtain 3t A(t,tl-t2) = 2i Im Tr D (t) U (tl-t2) G U (tl-t2) GA, (46b) where Im indicates the imaginary part. As indicated above, A depends on the difference tl-t2 and not on the specific values of tl and t2. Let now kU > be a representation diagonalizing HA, I > a representation diagonalizing HP, and EC, Ep the corresponding energy eigenvalues. Then HAO > - E|I >, (47a)

30 and HP lp > = E p >. (47b) The spectrum of HP is assumed to be continuous. In addition, let jr> be the representation diagonalizing H R and let lo> be the vacuum photon field state.5 We introduce the representation IM > - la >Ip>Io>, (48a) and we proceed to calculate the trace in Eq. (46b) using this representation. Thus, we have A(t,ti-t2) = 2i ImX D [t ) U. (tl-t2) MM1 MM1M2M3 s 1G,,MM2 UM Ms(tl-t2e) G (48b) where we have neglected the off-diagonal matrix elements of D, and have introduced the simpler notation G MM instead of < M IGjdM1 >. It is assumed that the eigenvalue problems (47a) and (47b) can be solved and that the corresponding eigenfunctions are known to us. Then, the matrix elements of G\ can be calculated. We shall have the occasion to elaborate on this point in considerable detail at a later stage (see also Appendix B). The remaining problem is the calculation of the matrix elements of US(t). By definition US(t) =, where tH = HHA,HP HR As is readily verified, H is diagonal in the representation IM >.

The calculation of the matrix elements of US(t) in this representation is precisely the problem solved by damping theory. As shown in Appendix A, the matrix elements of U (t), for t > 0, are given by i (EM + sM -irM) t UM(t) = e -i, (49a) and S S UMM' (t) = VMM (t-T) UMt (T) dT, (49b) where EM = Ec + Ep, (49c) SM() = MM + PP X M' (49d) )M,#M -EM, M'tM =X7 IV MtMI2 6 b(%EM,), (49e) M'AM s = SM(EM)' (49f) 7YM = ~ -~M (EM~)* (49g) In Eq. (49d), PP indicates that the Cauchy principal part is to be taken whenever an integration overX is performed. It is important to note that 7M is non-negative. The quantities sM and YM represent the shift and width of the energy of the state IM > caused by the interaction with the perturber and the vacuum fieldo A comprehensive discussion of these quantities has been presented by Akcasuo'0

32 As is seen from Eq. (49b), the off-diagonal matrix elements of U (t) are linear in V. If these matrix elements were substituted into Eq. (48b), they would yield terms of at least second order in VA. However, Eq. (48b) is already of second order in Gg.o Assuming that both of these coupling constants are small quantities, we shall neglect the off-diagonal matrix elements of U S(t). Then, Eq. (48b) becomes A(t,tl-t2) = 2i Im D (t) U (tl-t2).__ M~ MM MM sGA,MM1 UM1Ml (t -t2) G\ M1M = = 2iIm D(t) UMM(tl-t2) U t MM 12. (50).JN1 M1M1 MM 1 Introducing the symbols -, rM -, S (51) and using Eq. (49a) we obtain A (t,tl-t2) - 2i Im D IG (52) MM1 -i(iM + SM -irM) (tl-t2) i(0L, + SM1 + irMa) (t1-t2) Note that for Eq. (49a) to apply we must have t > 0. The difference (tl-t2) appearing in the above equation must, therefore, be non-negativre. That this is indeed the case can be readily verified if it is recalled that in Eqo (44) A(t,t1-t2) appears in the integrand of a double integral whose limits of integration are such that ti > t2.

33 It is perhaps desirable at this point to iterate some of the physical ideas underlying the calculations presented in this chapter. In calculating the trace, we have neglected the excited states |1 > of the photon field and have retained only the vacuum state. The necessity for considering the vacuum field stems from the fact that its coupling to a particle system does give rise to a shift and width of the energy levels of the system. The excited states, on the other hand, have been neglected from Eq. (48b) since we wish to confine this treatment to the linear approximation. Indeed, the excited states would yield terms proportional to < ctpnIDIctpi > that is, proportional to the number of photons present. These terms give rise to terms non-linear in Q, when substituted into Eq. (44). Thus, what we essentially do is to consider the vacuum and excited fields as two separate dynamical systems, up to a certain point. The excited field is described in terms of p\(t) = Tr D PA(t) and q,(t) = Tr D QA(t) which are expectation values of operators The vacuum field cannot be described in terms of expectation values of the field operators. It is taken into consideration in so far as it affects the material system. As will be seen subsequently, its effect appears as a shift and width in the spectrum of spontaneous emission and in the susceptibility. Analogous effects are caused by the perturber. In subsequent chapters, we study these effects in considerable detail for a material system with two internal energy states, usually referred to as a two-level system.

34 3. THE SUSCEPTIBILITY FUNCTION OF A. TWO-LEVEL SYSTEM The results of the preceding section are now applied to the case of a two-level system which in fact is the central objective of this study. Physically, a two-level system corresponds to an atomic or molecular system, whose transition frequency between two particular levels is close to the frequency of interest, the other transition frequencies being much different. By frequency of interest, we mean the frequency of that mode of the relevant cavity which has the lowest loss. Then, we may disregard the other states of the system and treat it as a two-level system. It is important to note however, that both levels are excited levels, in general, and have finite lifetimes. Ideally, we would desire a fourlevel system with energies Eo< El < E2 < E3. Eo would be the ground state energy. The pumping would take place from the ground state to 13>. The transition 3 + 2 should be very fast compared to 2 + 1 (typically by one order of magnitude), and presumably non-radiative. This scheme would reduce the possibility of saturation of the pumping mechanism, as well as of interference between pumping and cavity fields. The maser action would take place between 12> and!l>~ Here, we consider the simpler case of a two-level system, these two levels referring to internal degrees of freedom of the atom. The atom as a whole is subject to thermal motion. To account for the effect A of this motion we separate the hamiltonian H into two parts as follows: HA HAI +HAe (53)

35 HA refers to the internal and H to the external (center of mass) degrees of freedom. Let Im > and IK > be defined by AI H Im > - Em Im >, (54a) and Ae (54b H IK > - EK K > (4b) The eigenvector laG > is now written I1 > = Im > IK >, (55a) where E Em + EK. (55b) We now assume that eAI possesses only two eigenstates represented by il > and 12 >. Their energies will be denoted by El and E2, where E2 > El. The eigenvector IK > is left unspecified for the moment. For the sake of mathematical simplicity we shall assume that the energy eigenstates 11 > and 12 > are non-degenerate. The presence of degeneracy does not affect the qualitative conclusions and can be handled without difficulty as discussed in Ref. 9, for example. The vector IM > is now written IM> = Im > JK > Ip > 10 >, (56) where m = 1,2. Invoking the steady-state assumption we replace DS(t) by D and assume that the latter can be written as follows: -D = Ir0 > Im > IK > IP > Drm D DPP < Pl rnKp (57a) <KI < ml < 01.

36 That is, we assume that the populations of the particle states are kept constant through external means. The effect of the pumping mechanism is accounted for by assuming certain values for D in steady-state. The mm above assumption about D will be used in the calculation of A(t,t1-t2) and the calculation of the spectrum of spontaneous emission, because in both cases only the vacuum field is considered. When we write Tr D QA(t), we shall mean the complete density operator of the system. However, its knowledge is not necessary for our purposes since we have equations for the quantities Tr D QA(t) and Tr D PA(t) themselves. To simplify notation, we denote D22 and D11 by D2 and D1 respective.ly. These quantities represent the expected values of the populations of the respective levels in steady-state. Using now Eqs. (56) and (57a) we obtain DMM = Dmm DKK D pP (57b) Noting that GA is diagonal in IP > since it does not contain any perturber operators, and that EM =Em + EK + Ep, Eq. (52) becomes A(t,tl-t2) = 2i Im Dmm DKK DpplGA;mKmmKL 12 KKlp -i(m + + K p + SM -iFM) (tl-t2) e ei((wmi + (J) + CUP + SM1 + i rM) (tl-t2), (58)

37 where M and M1 are abbreviations for mKpO and mlKlpO respectively. Note that p is the same in both. The symbols wm' aco and up are defined by Em EX m _ m, ( EK = (59) consistently with Eq. (51). As shown in Appendix B, Ga can be written as follows: G; = f Zd x_(Rj), (60) where d is the dynamic electric dipole moment operator and operates only on the internal degrees of freedom; Rj is the position operator of the center of mass of the jth atom, and the summation extends over all atoms of the active material. In deriving Eq. (60) it has been assumed that 2(r) does not vary appreciably over the dimensions of the atom, which is essentially the dipole approximation. If we denote the polarization vector of the \th mode by ~A, then X2\ (Rj) - c X> ( Rj), (61a) and Eq. (60) becomes G?\ - 4 (d *- X \X (Rj). (61b) Assuming that the atoms of the active material are uncorrelated, one can show (see Appendix B) that Gh;mKmlKI = 4_ _ I<m d.~ ml>I ~T<KyXm(Fj)1K>l. (62)

38 To compress writing we introduce dAmmi -2 =m; | An |d 2 |ml |,j (63a) and x X, 4l J<jIX(Rj)lK,> * (63b) Xj,KK1 -eb Using Eqs. (62) and (63), and noting that w cancels in Eq. (58), the p latter becomes 2 A (t,tl-t2) = 2i Im Dmmn d 2 mm=l,m DKK Dpp X-jKKz e i(mn% 1 + + 8i1 -rl) t jKKlpei(lmm +K + S -i MM rl) t2, (64) where we have defined Wmmml = m-%,ml, (65a) Kljl _ K -t OK1, (65b) SM1 = SM-SM1, (65c) rMM, - rM + r-M (65d) Substituting Eq. (64) into the second term in the right hand side of Eq. (44), and after a lengthy calculation we obtain 4 T -ict t tt iwIx. dt e dt1 dt2 Q(t2)A(t,tI-t2) = 2'hg Q ) d2, mmlr DKK Dpp.jKK 1 mm1=l jlKK.p

59 [(mm + KK1 + SM -i ) ( + + + SM -i 1(66) -[(U + WKK1 + SMj + iPr1)(- + Xm1 + e1 + S~M1 +irM 1)] }. From Eq. (lib), taking Fourier transforms, we have Q4T ( ) = - P (T ) (67) Using this equation, and defining 2 Y(u) = Dmm D mml DEK2 Dpp Xhj,Kz mml=l jKKlP {[(mm + ( K1 + S1M i ( + MM + + K1) + SM1 -irMM )]- + (68a) -[( + (+m + + S) (- + + KK1 + S +ir )]-l, Eq. (66) becomes 4 T t t2 ia(D? dt e dti Q (t2) A(t,t,t ) = 4f o o o (68b) =aYh(\)PhT() (.) Y.\(w) is the susceptibility function corresponding to the )\th mode. Its value at u represents the effect of the material system on the wth Fourier component of the field. The basic steps of the calculation of the triple integral, leading to Eq. (66), are presented in Appendix C. Two approximations have been made: Terms containing the damping factor e M1T have been neglected, in view of the fact that ultimately we shall take the limit for T + 0. Moreover, we have neglected terms of the order

-1 of Wmml as compared to terms of the order of (wo-mml). This is justified by the fact that we are dealing with frequencies of the order of 10 -1015 cps, and narrow spectra (aCK, and SM are small shifts). Although the summation over m,ml in Eq. (68a) extends from 1 to 2, the equation can be applied to the case of more that two internal states as well. The spacing of the levels however, would have to be small compared to %mlj because otherwise the second of the above assumptions would not be justified. Recall that d is the dynamic electric dipole moment operator. Assuming that the atoms of the active material do not exhibit a permanent electric dipole moment, in either of the two states, the diagonal matrix elements of dA will vanish. Introducing the simpler notation 12 2 (69a d~, d?"12 = d,21 (69a) and eo = 21 =- 12, (69b) recalling that M stands for mKpO, and Ml for mKlpO, and performing the summation over m, ml in Eq. (68a), we obtain Y(1)) =' d DK pp Xh j' mc jKKlp (W o4iKK1+ SKpo, zKpo + i r Kpo,Klpo) D2 (~-~o-~KK1 -S;Kpo,LKipo -i r2Kpo,Kipo)

41 (Wo-C4K1- S 2KiKrlKpo, 2Kl )po D1,' (70) ( o-*oItKl+ SIlKpo,2Klpo -i rlKpo,2Kpo) Again, we have neglected terms of the order of ((w*o)-1 as compared to terms of the order of (-c~)-),. The above equation can be simplified somewhat if one notes that the shift S, as well as qfK1, are small quantities as compared to co. Thus we may replace Wo+UkK1 + SapolKlpo + i r by w + i r+ and i 2K~po K PO 0 2Kpo,lKlpo' ano -Kl Kpo,2K ipo 1Kpo,2K po 1Dby O+ i r lKpoipo in the denominators of Eq. (70). Note that the same approximation cannot be made in the remaining factors because there, the quantities S and WKK1 are compared to (w-c 0o) which is of the same order. Upon making the above approximations, and noting that S S S] po, Kpo 2K po,l po' ) and.i':71~ = F+ (7 lb ) iKpo,'K &po.[ - 1po, -lK.po' ( Eq. (70) simplifies to _= Y dai e pp XdjK jKKLp (Wo iP'po,,K)po) WOkKI -S2KpolK pO- i2Kpol-Klp) (lom1 r 2Kpo, 1K lP~ +i ( + t~c ) -KK l - S. 2KpoK 1K Zpo ) (72) D1 (wotir2K lpo, lKpo) ( Wwo~lK1 SK ipo s lKpo- ir' polKpo )

42 The remaining task is to average over the states of the center of mass and the perturber. 4. POWER SPECTRAL DENSITY OF SPONTANEOUS TRANSITIONS We have found that PAT(w) obeys Eq. (44) which in terms of the susceptibility function reads (- _+iy) PAT(() = GT(Uc))^ (75) The power output spectrum is R(w) = R ((W) where R (j) = lim eOT Tr D PAT(^) P\T( w). (b) T + oo From Eqs. (73) and (74b) we obtain S S ~ lim 1 Tr D GT((u) GET(-a) T+ oo T Rh (c=) * 22 (75) it-~ + ~ -~~ (~) -W The numerator represents the power spectral density of spontaneous transitions, as will be shown in this section. It provides the force that drives the field, since no other driving force has been assumed, which is the case in actual maser oscillators. If an additional driving force, such as an external field, is present, its power spectral density should be added to the numerator.

43 Using Eq. (40b), we obtain, S S T T TrDGhT() T (-W) = TrD dtdt' e G(t)G\ \(t'). (76) TrDG -i)0(t ) 00 To insure that the right hand side remains real after the transformation to follow we write it as follows: T T SS 1 TrD (t-t') S S TrDGT (w)G(j-) = Tr 00 dtdt'[e ( )G\ (t)G (t') + + e G(tt) (tI )G (t). Introducing a new variable T. defined by t - +T, and after some manipulations, we obtain T TrDGT (~C)GT (-W) = Re dT (T-T) e [ TrDG (ST)G + + TrDGGh (-T)], where we have interchanged trace with integration and Re indicates the real part. Introducing I (w) = lim -TrDGrT ()77a) in order to compress writing, and making use of the steady-state assumption, we obtain rT - iC)T S I?\() = 21im Re dTe TrDG (T)G-. (77b) T+oo o

44 From the definition of Gr(T) we have, Tr D G(T) GA = S = DM UMMl(T) G\,M1M2 U (T) G\,M3M (78a) M3 M MM1M2M3 where we have neglected the off-diagonal matrix elements of the density operator, and IM> is as defined by Eq. (56). The subsequent calculations and approximations are parallel to those of the preceding chapter; namely, we retain only the diagonal matrix elements of U (T) and use the results of damping theory; we assume that GA is off-diagonal in the representation Im>, and that D can be written as in Eq. (57a). Then, for a two-leve.l system, we obtain 2 Tr D G'(T) GA: Dmm,mml DKK Dpp X KK mml-1l jKK P 0e'(n~1l + SmKpo,mlKlpo + mKpo,mKlpo) Substituting into Eq. (77b) and neglecting the term which is proportional to (wcu-+o)-1 we obtain I(X) = D2 d)X DKK Dpp Xhj,KK, jKK1p?MM (79a);. W.. ( * (@-@o~KlS1AlMm M1

45 where 2> = K2>IK>IP >Io>, (79b) and M>l = Ij1>JlK.>p>jo>. (79c) I:(a) is the spectrum of spontaneous transitions 12> -+ 1> into the \th mode. As can be seen from Eqs (715) and (75), the spontaneous emission is the force driving the field, At low values of population inversion, that is of the quantity (D2-Di), the spontaneous emission spectrum has a dominant effect on the spectrum of the outputo As -the degree of inversion increases, the induced emission takes over~ For a quantitative discussion see section 5 of Chapter Vo The spontaneous emission spectrum is represented by Eq. (79a) as a formal average over the states of the center of mass of the active material, and the states of the perturber~ A similar average appears in Eqo (72) which represents the susceptibility function of a twolevel system~ In most practical problems. one has to resort to nrumerical calculation in order to perform the averagesO For a gaseous active material and a Fabry-Perot cavity ho wever, the calculation is simplified considerably. As a consequence, one is able to obtain useful results in a more or less closed form, as shown subsequentlyo From Eqso (75) and (77a) we have 5,(L) = I\ GX)) (80) A-,:o +,i(;,J-,~ -'e, Y~,() 1J

46 Thus the output spectrum is expressed, through Eqs. (74a) and (80), in terms of the spectrum of spontaneous emission and the susceptibility function. We shall now use these results to study the spectrum of a gas optical maser, in steady-state. 5- STATISTICAL APPROXIMATION Before embarking on the calculation of the averages in Eqs. (72) and (79a) we discuss briefly the approximation involved in replacing the average of a function by the function of the averages. This approximation shall be referred to as the statistical approximation. Following Akcasu9 we consider a function Z = f(A,B) where A and B are functions of some set T of stochastic variables. It is assumed that we have a probability distribution P(T) defined on T. The mean value of Z is then defined by -z f(A,B) P (T) dT. (81) The mean values of A. and B, which are denoted by A and B, are defined in a similar fashion. If c and a are the deviations of A and B from their mean values we shall have A = A + c, (82a) and B =B +. (82b) Expanding Z in a Taylor series we obtain Z_ f(AB) f _ _

47 (+A8B A B 2Al iAB. (83a) + 2c13 ~ ~A~B,'~,~ ~2(~B'~'.i~','t+''' Terms linear in aC and f don not appear since the mean values of (X and P vanish by definition. Here, we shall assume that the mean value of a also vanishes. This is the case for example, when a and P depend on different sets of stochastic variables. Thus we take z = (AB) + 2 7 (a ~,A,B + 2 (B 2 K (83b) We apply now this result to two functions which will be of interest to us in connection with the spontaneous emission spectrum and the susceptibility function. First we consider the function Z1 defined by Z1 A (84) B2+A2 Using Eq. (83b) we obtain - r ~ ~ 2 - Z 2_A 2 1 -(2 A-3B (85a) B2+A 2 (A2+B2)2 Assuming that the second term inside the square brackets is small compared to 1, and using the approximation (l-x)'- (l+x), the above equation becomes

48 = + 2 _'(85b) L A+B2 A 2+B2D If it can be assumed that the correction terms are small we have Z = - A, (86a) A2+B2 which is replacing the average of the function Z1 by the function of the averages. If however, we replace the average of Z1 by the ratio of the average, that is if we take Z1 A, we obtain B2+A 2 ZL......, (86b) T2 + T2 + F where we have assumed that 2 << A2. As discussed in Ref. 9, Eq. (86b) is a better approximation than Eq. (86a), and it is the former that we shall use in this treatment. In any event, one can go back and use Eq. (85b) if greater precision is desired. We now consider the function Z2 = B (87) Using Eq. (83b) we find Z2 - _ + _B. (88) BiB-~iA) (fBiA)2 (If the correction terms inside the square brackets can be assumed to be small compared to unity then what we obtain is the function of the averages. If this approximation is not satisfactory, which will be

49 the case in the calculation of the susceptibility where we shall -have a function of the form Z2, but we can nevertheless assume that ca ~2 2 then we obtain z2 - -+ _ ___)1. (89) In this study we shall use Eqs. (86b) and (89). For further discussion of the statistical approximation in connection with the function Z1, Ref. 9 should be consulted.

CHAPTER V APPLICATION TO A GAS OPTICAL MASER 1. THE SPECTRUM OF SPONTANEOUS EMISSION The model for the gas optical maser we shall consider consists of a tube of length L (typically 100 cm), containing a gaseous active ma21 terial. The side walls of the tube are transparent to light, while the end plates are highly reflecting, with reflectivity of the order of 99% or better. This structure forms a Fabry-Perot cavity with a high quality factor. It has been shown that, the modes of this cavity which have the lowest loss are the even symmetric modes whose frequencies are c-the t (90) where c is the velocity of light, and A a large integer of the order 6 of 10. For a typical He-Ne optical maser, the separation of these frequencies is of the order of 160 Me/sec. There are also modes of next lowest loss which posses odd radial symmetry, and their frequencies differ from the frequencies of the previous modes by, typically, 1 Mc/ sec. Here, we shall neglect these modes. Moreover, it has been shown that, the transverse field of the even symmetric modes does not vary appreciably over the diameter of the tube, usually of the order of 2-5 cmo These modes correspond to propagation of light along the axis of 50

the tube, and inside the tube one has a standing wave pattern. In order to simplify the analysis, we shall ignore the variation of the mode vectors over the diameter, and shall take X\(r) = cx Srink)z, (91a) where XX is a constant normalization factor, and kx is the wave number related to the frequency as follows: N = c k-. (9lb) The z-axis is taken along the axis of the tube and the x,y-axes on a plane perpendicular to the axis. If we introduce a vector kh defined by kA = (ook), (91c) where the numbers inside the parenthesis are the cartesian components of kg, we shall have %X(r) = X> ( eik, ). (91d) 2i For each mode, only one polarization is present the other being eliminated by using windows of the Brewster's'21 angle type at the ends of the tube. The active material inside the cavity is assumed to consist of an assembly of uncorrelated atoms (or molecules) whose center of mass is

52 subject to thermal motion. Thus, the states of the center of mass shall be taken to be free particle states given by IK> = (2wr)-3/2 e -iK R (92) where, R is the position operator of the center of mass, and K the wave vector. The energy of the state IK> is,n 2K2 ~~~~~~EK ~~~(93) 2m where m is the mass of the atom. Recalling the definition of XAjMKK1 as given by Eq. (63b), and using Eqs. (92) and (9ld) we obtain AXJ El -'_ b (KK1_ ) 2ich_ - (K- K j (-K (94a) Note that the subscrip j in the left hand side, which refers to the jth atom, now becomes redundant and will be deleted. Whenever a summation over j occurs, as in Eqs. (72) and (79), it will be replaced by multiplication by N, the number of atoms of the active material. From Eq. (94a) we now have 2 - 2 ) xh2 _~ na- =x (K-x+k) - - X,KK1 -I (94b -- 6 (K-Ki-k)ij o (94b)

55 Moreover, we shall assume that DKK is a Maxwellian distribution with temperature T, or mean energy 3- E, where 2 $ =k T, (95a) and is the Boltzmann's constant. The summation over K is then replaced by integration, according to DKK b3 s/ 2 -dKe-K (95b) K where b2= 2__ (95c) 2m,.% Recalling now the definiton of'CKI (see Eq. (63b)), and using Eq. (93) we obtain =i (K2-K ). (96a) Because of the presence of the delta functions (or Kronecker deltas, if a discrete spectrum of K's is assumed) in Eq. (94b), the only terms that will survive in a summation over K and KL are those for which K K + k ~ Using this relationship Eq. (96a) yields K2mc2 m Neglecting ho4/2mc2 as small compared to Cal and introducing Sd- (Kk ), (96b) m -A

54 we have'KK1 = + Sd~ (96c) In Eq. (96b) we have changed K1 to K. The same will be done in the summation X from which K has now disappeared. KK 1 Introducing the foregoing simplifications into Eq. (79), we obtain IA(@) = D 2 Td ( X Dpp. Kp +tr MM1 (97a) rMMi + ((o-wo-Sd- M?)2i)2 + r(+i' + - 2+ 2 (w'-O+Sd-SMI)2 + (r ) where IMp = IK>12>lp>1o> (97b) IM,> = IK>l>Ip>lo> (97c) An additional approximation has been made in Eq. (97a). It has been assumed that the significant effect of the center of mass motion is contained in Sd, and K1 has been replaced by K in M Xand SC!,. Since DK depends only on the magnitude of the vector K, and since we integrate over all K-space, both terms inside the square

55 brackets in Eq. (97a) yield the same quantity, when averaged. It suffices therefore to retain one of the two multiplied by two, thus obtaining 2NN2Xdf X KK Dp Kp (97d) (W-o-S -S I)2 + (r+ )2 2 where N2 = ND2 is the expected number of atoms of the active material, in the upper state 12>, in steady-state. The above equation gives the spectrum of spontaneous emission as a superposition of Lorentzians. Subsequently, we shall use the statistical approximation to replace the right hand side of (Eq. 97d) by a single Lorentzian. 2. THE SPECTRUM OF SPONTANEOUS EMISSION IN THE STATISTICAL APPROXIM.ATION The spectrum Ik(c) as represented by Eq. (97d) has the form of zl as defined by Eq. (84). Setting A' (98a) and 3B ~ O~ —S~d- 3M, (98b) we have ~((> W 2 X iDKKDPP EA (99) we~~~~J~A

We now introduce X2 KD M'= X DDppK2po (looa) Kp Kp DKKDppMl = DKKDpprKlpo' (100b) Kp Kp and ro 2 r2 + rl = A rn and r2 are the widths of the lower and upper states of the active material, in vacuum, and averaged over the states of the center of mass and the perturber. Similarly, we introduce S2 — XDDPPSM2 = >Dc DppSK2po (lOla) Kp Kp S1 DJKDPPSMI = D}DppSKlpoD (101b) Kp Kp and So - S2-S! (101c) The subscript o in For and So should not be confused with the vacuum state appearing in the right hand sides of the above equations. Again, SI and S2 are the shifts of the lower and upper states of the active material, in vacuum, and averaged over the states of the center of mass and the perturber. Akcasu9 discusses the averaged widths and

57 shifts of the states in considerable detail. Here, we simply note that both shift and width can be separated in two parts: One due to the vacuum field, and another due to the perturber. For further information see Ref. 9. Observing that Sd, being the average over all K-space of an odd function of K, vanishes, and using Eqs. (101), we obtain B C= A-m%-So. (102) Moreover, we have = XD~KKDpp (B.T)2 (103a) Kp Combining Eqs. (98b) and (102) we obtain = KKSd KKpp( (b) K Kp where again use of the facts that Sd =, and that Sd does not depend on p has been made. We now define 2 )740 S 2 C4 ~i~m n(104a) d -KK S = X KK 2 K K 2 2 2 Fr - ) I DK~pp(SaMVS) -O s (o104b) Kp and

58 2 - r+2 + r2 + r2 (104c) e o d s With the foregoing definitions, and combining equations (103b), (102), (100c), (99) and (86b), we obtain 22 (u) = 2'N2XAd\ _ __ (105) 2 (_0o- So)l + r2 In obtaining this result we have assumed that the statistical fluctuations of the widths can be neglected. This assumption is inherent in the condition 22<<A2 under which Eq. (86b) has been derived. Thus, the spectrum of spontaneous emission into the Ath mode is shown to be a Lorentzian centered at wo+So and having an effective width Fe* This effective width consists of three terms. The first term +2 is the su of the widths from the interaction with the 0o is the sum of the widths arising from the interaction with the vacuum field (natural width), and the interaction with the perturber (collision broadening). Also the third term F2 is due to the same interaction but it is different in nature. It appears as a width, while actually is due to the statistical fluctuations of the shifts. It is usually referred to as statistical broadening. The second term?d is due to the recoil of the center of mass of the atom when it emits a photon. This is essentially the Doppler broadening. For a Maxwellian distribution of center of mass velocities, rI can be readily calculated. The calculation is carried out in Appendix D with the result

59 2 2 kT (106) rd M c~h — Ic2 In the limit of zero temperature or infinite mass it vanishes as it should. Up to this point, wo has denoted the frequency of the transition 2 - 1. Then, SO is the shift due to the interaction with the vacuum field and the perturber, averaged over the states of the perturber and the motion of the center of mass. In interpreting experiments however, it may be preferable to include the vacuum shift in cO. Then, So should be reinterpreted as due to the interaction with the perturber only and averaged as before. 3. THE SUSCEPTIBILITY FUNCTION IN THE STATISTICAL APPROXIMATION We now proceed to calculate the susceptibility function for a gas optical maser, in the statistical approximation. The starting point is Eq. (72). Since DKK depends on the magnitude of K only, and since aKK1 = Sd, we may choose one of the signs and then multiply by two because we average over the whole K-space. Moreover, assuming that the recoil effect is adequately accounted for by Sdi we replace S2Ko and S by S 2Kpoand Kp and ~2Kp o, -LK lp 2K pol.Kpo 2Kpo,lKpo 2Kpo, anpo r+ by. Then, using also Eqs. (97b) and (97c), Eq. 2K zpo. 1Kpo by 2KpoL1Kpo' (72) becomes = 2irNL )Q=a, (D2-D1)

6o KK Dpp (107a) W"o -S0 d -irt where we have used Eq. (94b) and have renamed K1 to K. Since we have already assumed (see V-2) that the statistical fluctuations of the widths can be neglected, we replace the two factors in the right hand side of Eq. (lO7a) by their averages. After some mathematical manipulations, and using Eqs. (100) we obtain Y (ca) = N2X [ (D2-D) ( o%-i ro) ] <(_)X (Do DKK Dpp E It....pp + ( 107b) i -uo-Sd-SMi -i rM+Mb Kp 2 + 2 2 where we have replaced o+ (Fo) by 0o2 in the denominator, since r <<o-' We shall also introduce Z, defined by z - (D2-D1) (%O-irO+) (1O7c) because it will appear in several of the following equations. The problem is now reduced to the calculation oft), defined by ( ) X DE D'P (108) X Kp (lSd-SR~4t-i rM>' Kp The statistical fluctuations of SMU will be neglected, as was done in previous instances. On the contrary, we shall retain the statistical fluctuations of the shift and as will be shown subsequently, two different methods of averaging suggest themselves, depending on whether

F2 is negligible as compared to C2 or not. Consider first the more 5 d general case in which both r2 and d2 are to be retained. Identifying "I o-Sd-Si;4 -with B, and +M~M, with A, the right hand side of Eq. (108) assumes the form of the function z2 difined by Eq. (84). As shown in IV-5, the average value of z2 can be approximated as in Eq. (89), provided one assumes that a2<<A2. Thus, combining Eqs. (89), (100), (101), (102), (103) and (104), we obtain () 1 1 1 + 2 (c))(w-o-SO)-i rf L (109) r2 + r2 + d s If the correction term inside the square brackets can be neglected as compared to unity, one has W) a. ) 1i (110) cl)-Wo-So-i rO which defines O(c(). In phenomenological treatments of the problem, one obtains an expression for the susceptibility function resembling O() + && (o). Actually, So is entirely ignored, and ro is replaced by the effective width Fe, appearing in the spectrum of spontaneous emission (see Eq. (105)) which is again assumed on phenomenological grounds. However, Eq. (109) shows that the form w(co) involves at least two assumptions; namely that both Doppler and statistical broadening are zero (or very small). In addition, even if this is so and even if

62 SO is zero (or negligible), the imaginary part of the denominator is not the effective width re. In the limit of zero temperature or infinite mass of the emitting atom, r2 vanishes as Eq. (106) shows. Then, one still has a correction due to statistical broadening. It appears therefore, that Eq. (109) is useful when s2 is either comparable to rd or much larger In the s d or much larger~ In the second case in fact, one may neglect rd entirely. In order to obtain higher order corrections, if necessary, one can consider the Taylor series given in Eq. (83a) and supplement it with additional terms. In conventional line shape experiments and interpretations, the second order corrections seem to be adequate. The spectra of optical masers however, are extremely narrow and one should be prepared to go to higher order corrections when relevant experiments with well stabilized masers become feasible. There is a third case, not discussed thus far, namely the case in which 2 >>F. Then, r2 can be neglected from Eq. (109), and the red s S sulting expression gives the susceptibility function with a correction, due to the motion of the center of mass, of the first order in the temperature. For this case, in which r2 is negligible, we shall now proceed to obtain higher order corrections. 4. THE SUSCEPTIBILITY FUNCTION FOR THE CASE OF NEGLIGIBLE STATISTICAL BROADENING In this section we calculate(cu) for the case in which the statistical fluctuations of both SMj 4 and FM can be neglected. Then, we

replace the above quantities by their average values in Eq. (108), which now becomes () KK + g(ill) K WDo-Sd-S -i r0 The only unaveraged quantity in this expression is Sd which does not depend on p. Note that Sd cannot be replaced by its average since Sd = o and one would lose all information about its effect. Using Eqs. (95b) and (96b) we obtain 3 -b2K2 b3 -3/2 dKe (112) /~'~ (~-co-So-i r+ ~ (K'k ) The direction of k is fixed and has been chosen as the z-axis. Let /j be the angle between K and k_ the azimuthal angle, and p. - cos Then H) becomes 2 2 +1 ~~ -b K, ) = 2b-mr do dK e - (115) j_) -: O m (u-%o-So)_-Kk-p.-i m r+ where we have performed the integration over ~. Observing now that the denominator has the integral representation m (wcow-So)-Kk^\[,-i r+ nd s t-i[ (uio-So)-Kkiq-wrobti_]xn1 and substituting into Eq. (113) we obtain

64 00 -[ r+ +i (u-o )So1) x ) -= 2ib3m dx ed 00 + -btK2 iKkktx Ke dK e dK e d (115) o -1 where we have interchanged the order of integrations. The calculation of the integral is relatively straightforward, albeit somewhat lengthy, and is presented in Appendix E. The result is () = i bNT bm gb +i (C-(o-So S (116a) where the function ~(z), for any complex number z, is defined by 2 ~23 cj(z) = e Erfc(z), (116b) and the complementary error function is defined as follows: 00 _t2 Erfc(z) = e dt. (116c) Combining now Eqs. (116a), (107) and (108), we obtain the following expression for the susceptibility: Y~(e ) = 2 NwXkd\ Z i m) 2 2 bm _=2 1 (2i(um -11hk u (117k)

65 The argument of the function s in Eq. (117a) is therefore, inversely proportional to the square root of the temperature, and directly proportional to the square root of the mass, provided the width r+ and shift So are slowly varying functions of the temperature and the mass. In order to investigate the behavior of Yh(W) for | k-s (O+ i(-o-So) >> 1, (118) we note that the function ~(z) has the asymptotic expansion23 00 (z) 1 + (-1)m 1.3.(+l) (119) 2m=l (2Z2)m which is valid for Izl - 0o, and largzj < -t. In Eq. (117a) we have 4 Rez>o and consequently the condition for the argz is satisfied under all circumstances. Retaining the first two terms in Eq. (119), and after substituting into Eq. (117a) we find ~ 3/2 Na)X\d. Z Yo\J (o-%O-So-ir+) + 2 (120a) + 3 rd (w-o-So-iro) 2J where we have used the fact that = 2 r (120b)

66 as can be seen by comparing Eqs. (117b) and (106). For the sake of comparison, we give the expression for Y)(wo) resulting from the considerations in V-3. It is obtained by combining Eqs. (107), (108) and (109), and has the form 2 2:W(Su2 (+-a0-SO-ir) + (12oc) 2 + r2 +,d s (wa no-So ir+)2J If r2 can be neglected as compared to F2, the two expressions assume s d' the same form except for two differences. The coefficient in Eq. (120a) is slightly smaller than that of Eq. (120c), their ratio being approximately 0.9. The correction term in Eq. (120a) on the other hand, is three times larger than the corresponding term in Eq. (120c). In view of the drastic approximations made in calculating Y,(cu), the above differences are not too surprising. It is presumed that in the extreme case in which Fs is entirely ignorable, Eq. (117a) (from which the asymptotic expansion has been derived) gives a better approximation. In addition, it has the advantage of expressing YZ(w) in effectively closed form. In the case in which Fs cannot be ignored, it is Eq. (120c) that must be used. As mentioned above, Eq. (117a) is in effectively closed form. This enables one to obtain correction terms up to any desired order in the quantity o P+i(u-.o- 0So)j, for large values of this quantity, /)

67 through the asymptotic expansion of &(z). A further advantage of Eq. (117a) is that one can obtain approximate expressions for Yh(cn) in the case in which we have b mk (+O+i(WD.O-SO)) << 1. (121a) This we now proceed to discuss. The function ~ (z) has the following series representation 2 00 6-(z) = L p (. +i) (l2lb) n-o 2 where P(x) is the gamma function. Retaining the first three terms of the series and substituting into Eq. (117a), we obtain Y() = a X\d\ z i 1i ro++i(o-0o —So) + + [ r++i (L-O _SO ) 1] 2 (122) 2 \Fd where we have used Eq. (120b). It should be emphasized again that this equation contains the inherent assumption that the statistical broadening is ignorable. Presumably, in the range of validity of this expansion, the above assumption is likely to be satisfied, since inequality (121a) also implies relatively large Doppler broadening. As an attempt to decide about the form of Y (m) that should be used in the analysis of an actual system, we consider briefly the first He-Ne gas optical maser developed at the Bell Telephone Laboratories.

68 20 According to Bennetts paper, the maser consists of a discharge tube 100 cm long and with an inside diameter of 1.5 cm, filled with He at 1mm Hg pressure and Ne at O.lmm Hgo The transition used in the maser 24 action is the 2s -+ 2p4 (Paschen notation) transition of Ne, The associated frequency is approximately 1.64xLO 1 cpso The Doppler width is estimated to be of the order of 800 Mc/sec, while the natural width of the order of 50 Mc/sec. The lowest loss cavity modes have a width of the order of 0.5 Mc/sec and the modes are separated by 160 Mc/sec. At room temperature and for m - 20amu, Eq. (117b) yields bm A 3.56 x 10 sec(123) Under these circumstances, the whole power output will be practically within at most 10 cps about A = 1.64 x 1015 cps. Interpreting r+ as the natural width and noting that in the present case it is much smaller than 109 cps, we conclude that, for those values of a for which we have a substantial amount of power, we shall have i(w-)o-.So) + o+I<l09 cps. Combining this with Eq. (123) we find bm (++i (c-o-So) I< 0.356 (124) In any event therefore, the argument of t in Eqo (117a) is smaller than unity and it is the series expanslon rather than the asymptotic

69 expansion that one should use. Moreover, since the right hand side of Eq. (124) is not much smaller than unity, for a different system the inequality might be reversed. Attempting to analyse the foregoing optical maser in terms of,Y(w) would have several weak points. The most serious difficulty arises from the fact that the maser exhibited strong mode cou.pl.ing. Also the frequency stability of the system was not particularly satisfactory becuase of fluctuations of the mechanical construction, Our analysis is aimed particularly, albeit not inevitably, at the single mode operation of a well stabilized masero According to a recent report, such systems have been constructed, and the hope that one will be able to perform measurements on such systems, in the immediate future, can be hardly considered as optimistic. A third difficulty comes from the fact that most existing treatments, dealing with the interpretation of actual experiments, are phenomenological. And it is not always clear what the parameters quoted really represent. In the foregoing discussion on the dependence of YA(G) on the mass and temperature, we have ignored the dependence of So and +o on these quantities, by assuming that their variation is slowo The dependence nevertheless exists and it may be imperative to take it into consideration in actual situations. A fairly extensive study of this problem is presented in Ref, 9 whose formulation we h.ave followed closely~ Here, we simply note that one is ultimately faced with the necessity of nu

70 merical calculations, if comparison with experimental results is contemplated. 5. THE STEADY-STATE OUTPUT SPECTRUM The steady-state power spectrum of the Ath mode (Rh(j)) is given by Eq. (80). Combining this equation with Eqs. (22) and (105), we obtain the following expression for the output spectrum R(w)> R(T) eo f s eu eiso (125)'....... 2+ 2]1~, j'"2... 2 2' 2 The expression for the spontaneous emission spectrum, appearing above, has been derived under the assumption that the mode involved represents photons travelling along the axis of the tube, Photons travelling at any angle with respect to the axis are lost after a small number of reflections, and consequently no appreciable amount of energy in those modes can build up inside the cavity. Moreover, from the modes with longitudinal propagation, only the ones lying within one or two widths of the spontaneous emission line will oscillate. Thus, although the summation in Eq. (125) was initially understood over all modes which in principle are infinite, for the gas maser this summation is effectively reduced to a small number of terms. A further reduction comes from the assumption, made at an earlier stage, that only the lowest loss modes oscillate Under these circumstances, the num - ber of modes that need be considered in a typical He-Ne gag optical

71 maser may be as low as three. Incidentally, this is another aspect that greatly simplifies the analysis of a gas maser as apposed to a solid-state maser25 where more modes must be considered. Eq. (125) gives the steady-spectrum in a general form, in terms of the susceptibility Y-(w) and the shifts and widths of the relevant states. To proceed further one will have to decide about the form of Y-(w) that is appropriate to the system under consideration, and the values of the parameters involved. Here we shall discuss two special cases. Assume that we have a well stabilized maser operating in a single mode. That is, most of the energy is concentrated in one mode, the other modes having practically no energy at all. Then, the summation in Eq. (125) reduces to one term and although the index A is now unnecessary, we shall retain it for notational convenience. The transition frequency wo and the cavity mode frequency a~ are assumed to be of the same order, typically 1015 cps. Due to the anticipated narrow spectra, we introduce the following approximations: Xc c) LU o 1, (126a) and ~3t)~\ CWt ~ (11\ +W *a-D 2w (126b) In addition, we asstume that 2 and r2 can be neglected. We then have a(d s e2,o+2 and Eq. (120c) yields

72 2 2 z (12)a) MA\YA (a) - 2 g9\ + (1-27a) w-o0-S -iro where we have introduced 2 irNX\dX g2= (127b) Observing that (W-_-So) +(ro ) ~! _-S o-iro 12 we have I-o2+iao)W2YS(a))I2 [(-no os )2+ (rO+) 2 = 4 (_-\ ) (- Loiro+) + i Y (W-% -iro ) - (128a) - g\ (D2-D1) %+ig\(D2-D1) ro+a, where we have introduced no -- o+S~o (128b) and have used the expression for Z given by Eq. (107d). Note that the approximation w\A - Wo - 2o is valid since So is much smaller than aco. Substituting into Eq. (125) and dividing numerator and denominator by (rFo) we obtain RX (a) =

73 rN2Xdg (129) 2ro [(w-w))+ + (w-%o)+g (D2-D1)] +[2 A)(D2-DI) o]2 2I O+ 2 2 r+ 2D-Dl)%I O0 The width yh of a good Fabry-Perot cavity is of the order of 0.5 Mcps, while,according to Bennet, the natural width of the upper level of the maser transition is of the order of 50 Mcps. Here, Fo+ is the sum of the widths of both levels and contains both natural and collision widths. We may assume therefore, that Yh7 << rF. Moreover, we assume that the maser is stabilized well enough to have lQo-0o < Y7/2. Although usual gas optical masers are not so stable, stabilities of the order of 10 over periods of several hours have been reported recently. Under the foregoing conditions, most of the power output is expected to be concentrated within a few yks about a~ and %o. That is, the power is essentially contained in a frequency range such that Iu-.cI << Fo+. This implies that I (o)(h-0o) i < (w-Qo). Since (O-2o) is of the order of yk we may neglect (0O)( ) from the ro+ denominator of Eq. (129) which now simplifies to 2 2 ~kN2Xhdx R(w) =.2 o+. (130) [(X<\u)+ (-_~0o)+g\(D2-D1)] 2+ - (D-D)(Do] r+ 2 r+ 0 0 If wy is the vralue of cD at which R(CD) attains its maximum value, we shall have

74 2r+ gA2 (D2-Dg) = O. Solvring this equation for cm and retaining only terms linear in Y7\/2+ we obtain (tin = (D Qt(-lo) jF: gr(D2-D2 ). (131) Also, if we neglect terms of order higher than the first in X 0 R(a) becomes 2 2 AN 21X2d 2r R(w) = 2o (132) (W)-) 2 + ( i (D2-D1) o)2 To the extent that the conditions under which the above equation has been derived are satisfied, the power output spectrum of a well stabilized optical gas maser operating in a single mode has a Lorentzian shape. The line is centered at cum and has a full-width at half-maximum (6w)1/2 = D)2%j. () (133) This width is larger or smaller than the cavity mode width y,, depending on whether (D2-D1) is negative or positive respectively. (Note that (D2-D1) varies from +1 to -1). The width decreases as the degree of inversion, that is (D2-D1), increases. Results similar to Eq. (132) have been derived (through different arguments) and discussed elsewhere.25'26 Thus, we shall not dwell on it any further. However,

75 Eq. (13.1) deserves further attention since it contains a term which does not appear in previous treatments. This equation is usually referred to as the "linear frequency pulling" equation. If the term g\(D2-D1) is neglected, the equation agrees with the well-known result obtained for the first time in Ref. 26. Here, we obtain an additional correction term. This term is presumably a consequence of the more refined model we have used. In Ref. 26, as well as in other treatments, one introduces a phenomenological width which masks the fact that this width is due to many effects which give rise to separate widths and shifts for each level, as shown in earlier chapters. One cannot expect therefore, such models to predict effects associated with this fine structure, so to speak, of the effective width. Since (D2-D1), for most masers, will vary between 0.5 and 1, the order of magnitude of g- (D2-Dl) will be determined by g?\. It is not aprioriobvious therefore, that this term is ignorable under all circumstances. It is perhaps illuminating to compare this term to the term 2g<(D2-Dj) wo/rF which accounts for the 26 spectrum narrowing. For an ammonia maser for example, we have wo 6 20 8 10 For an optical gas maser, we have 10. Thus the frer+ quency shift term is, typically, seven orders of magnitude smaller than the narrowing term. In usual devices, one would not expect this shift term to be of importance. In a well stabilized maser operating in a single mode however, it might represent a significant effect.

76 As a second special case, let us consider an optical maser operating in a single mode and assume that the spontaneous emission spectrum is very broad compared to ye. The previous example suggests that, for population inversion high enough, the output spectrum will be narrower than Y7. One may assume therefore, that the spectrum of spontaneous emission is constant over the frequency range of interest, and replace it by its value at the center of the line, that is Q,. Thus in Eq. (125) we replace (W- O-S )2 + r2 by F2e Again, this assumption is valid if the maser is stabilized well enough for Qo and OX to differ by an amount of the order of yA at most. Let furthermore, Y1j(c) and Ya~(w) be the real and imaginary parts, respectively, of Y-(a). That is Y\(W) = Y1\(W) + i y (W). (134) Then, using also Eqs. (126), we obtain N2x\d\rFo 2 2 R(cu) 2 e (135) + Y())2 2 22\()) This, in general, is not a Lorentzian since yj1 and yg\ depend on a. If these quantities are slowly varying functions of wc or perhaps constant, then the spectrum does become a Lorentzian centered at aI If YZl, (136a) and having a full-width at half-maximum

77 Yh - aY2\h. (136b) It is seen therefore, that the real part of the susceptibility appears as a shift of the frequency of oscillation with respect to the cavity mode frequency, while the imaginary part appears as a width. The detailed structure of the spectrum is contained in Eq. (135). This equation is likely to correspond more closely to the spectrumn of an actual gas optical maser than Eq. (129) does, with the additional complication of considering two or three more modes. For example, according to Bennet's estimates, the Bell Telephone Laboratories He-Ne gas maser would have Fe> 800 Mcps and y\ X 0.5 Mcps. The distance between modes was 160 Mcps. For single mode operation therefore, all conditions under which Eq. (135) was derived are satisfied. Moreover, it is conjectured that even for two-mode operation, in which Qo lies between two cavity modes, Eq. (135) will approximate the actual spectrum adequately, when summed over the modes in question. It must be emphasized that the comparison of the present theory, as well as of other theories, to experimental results is hindered mainly by the inadequate frequency stability of usual devices. 6. ON THE LOSS MECHANISM A.s pointed out in Chapter III, the coupling of the cavity field to the loss mechanism gives rise to the damping constant Vh and a driving term representing the fluctuations of the loss mechanism. The problem has been discussed by Senitzky and in this section we

78 shall elaborate somewhat on his method. Recall that (see Eq. (27)) the total hamiltonian of the system was E R A P RA PA EA H =H +H +H +H +V +VA+VEA The loss is the result of the interaction of the radiation field inside the cavity with some other system (e.g. the walls of the cavity, or the host crystal in a solid-state maser). Let HL be the hamiltonian of this system which we refer to as the loss mechanism. Also, let V be the energy of interaction between the radiation field and the loss mechanism. Considering the interaction of each particle of the latter with the radiation field, as we did with the active material, we can write VRL as follows: VL = - P=j G?, (137a) L where GX is the current operator of the loss mechanism whose definition is analogous to the definition of CGi (see Eq. (10c)). The total hamiltonian now becomes H = +HP+V+VP +. ( 37b) It is assumed that HL is coupled only to H and not to X or HP. Following Chapters III and IV-l, and treating the loss mechanism as in Ref. 8, we obtain

79 (U2 Pbc \() 2 2 L (, -2s(W) PXT(@U 2T U+ G2 T(f) + 4 T -it t t Ii (t-t)HL + o- dt e dtl dt2e 0 0 0 i (t-tl)H L * [Qx(t2) GO(t2), G (tl)] e /, (138a) where L L._ Lt GA(t) = et' Ga e 1i. (138b) We now treat the last term in the right hand side of Eq. (138a) according to Chapter IV. That is, we pull QA(t2) out of the commutator and we replace the operator multiplying Q\(t2) by its expectation value. Moreover, we assume that H has an energy spectrum densly spaced, and that its density matrix is diagonal in the energy representation, its diagonal matrix elements being -En/kTL De n (139 nn - eEn/kTL ( n where H In> EnIn> In calculating the trace with the density matrix we have a summation over the states In>. Assuming that the spectrum is dense enough for this summation to be replaced by integration, and after a rather lengthy calculation which is presented in Ref. 8, one obtains

80 (2 2 Y2(W) PAT(T) = + 2) GT(L ) - (140) _icuyh P\T (X) The quantity y7 is a constant which arose from the expectation value of the operator that multiplied QA(t2). It is the susceptibility of the loss mechanism, and it turns out to be a constant, that is independent of w, because of the assumptions made about the properties of H. Additional assumptions introduced during the calculation are: Y) is small compared to w and no mode coupling exists. In fact the question of mode coupling does not arise at all in Ref. 8 because a single harmonic oscillator is considered there. Thus, one has a model for the quantum mechanical description of loss. Eq. (140) shows therefore that the equations we used in the present treatment are in agreement with the above model, except for one difference. In our equa2 L tions we did not have the term caG T(w) appearing in Eq. (140). It should be clear from the considerations in IV-4 that this term, when one calculates the output spectrum, will give rise to the quantity L 1 LL L I~(c) - lim - Tr D GT() G\T (-), (141a) T- oo L L where D is the density operator of the loss mechanism, and G T(D) is defined by L T.-it,, H t i 1L L~~~~~~~~~~~~~~~. -

81 IN(u) represents the spectrum of spontaneous emission from the loss mechanism, and in calculating the output spectrum it will be added to the spectrum of spontaneous emission from the upper level of the active material. When we consider a gas optical maser therefore, I(U(w) represents spontaneous emission from the walls of the tube at room temperature, and at frequencies of the order of.10 cps. In principle, this contribution is present. But it is extremely unlikely that its neglect could be of any importance, as far as the output spectrum is concerned. Even if one considers an optical maser amplifier, in which case IL(cw) would constitute noise, its effect can presumably be neglected since it will be masked by the much more important term of spontaneous emission from the upper maser level. For masers in the range of microwaves however, spontaneous emission from the cavity walls may not be ignorable in which case I.() would have to be taken into consideration.

CHAPTER VI COMPARISON WITH OTHER THEORIES The present study was motivated by the work of Wagner and Birnbaum.25 Although their work is particularly aimed at the solid-state maser, their formulation is rather general. They describe the electromagnetic field classically, in terms of the cavity modes, that is pand qi. The active material is treated as an assembly of fluctuating dipoles, with no permanent dipole moment. By "fluctuating" is meant that dipoles which are in the upper state can decay to the lower state spontaneously. If a field is present, dipoles in the upper state can decay and dipoles in the lower state can make transitions to the upper state at a rate which is proportional to: the number of photons present (that is the square of the field), the number of dipoles in the respective levels, and the square of the coupling constant which is the matrix element of the dipole moment. The fluctuation represents spontaneous emission. Taking the spontaneous emission as the driving force, and calculating the induced dipole moment by using second order perturbation theory, they are able to obtain an equation for the spectrum similar to Eq. (125). They assume that the spontaneous emission spectrum has the form ( )+ (142a) 82

83 where r is a phenomenological width including all broadening effects. This width is also used in their calculation of the susceptibility function which has the form g2(D2-Dl) (142b) C-Co- i r In both (142a) and (142b), we have omitted non-germane multiplicative factors. Structurally, (142a) resembles our expression for the spectrum of spontaneous emission, except for two differences. First, no shift appears in (l142a). Secondly, we have seen that the width Fe consists of several parts and we have exhibited explicit formulas for them, indicating their dependence on the dynamical parameters of the system. The issue however, becomes even more important when one considers the susceptibility function. As we saw in Chapter V, Y\(wu) takes the form (142b) only if the Doppler and statistical broadening can be neglected. Then, if in addition we neglect the shift SO or reinterpret O0, our results reduce to those of Wagner and Birnbaum. If the Doppler and the statistical broadening cannot be neglected, then (142b) does not coincide with our expression for the susceptibility. Nevertheless, under certain conditions we were able to express Y\(cm) in the form of a series whose first term was similar to (142b). However, one difference still remains. That is, the quantity r, appearing in the first term of the series for Y(cO), is not the effective width Ee'

84 appearing in the spontaneous emission spectrum. While, according to (142a) and (142b) the same r appears under all circumstances. Moreover, we have shown that, under certain conditions, Y\(w) may have a form entirely different than (142b) (see Eq. (122)). In addition to the above differences in the results obtained, the present treatment also differs in the derivation of the equations. In fact, Wagner and Birnbaum do not derive their equations. They rather construct them. Here, we construct the hamiltonian, and then derive the equations through Heisenberg's equations, making suitable approximations. This approach has the advantage of exhibiting the approximations involved, and lends itself to generalizations in order to account for phenomena such as mode coupling, non-linear effects etc. Also, it has the intellectually pleasing feature that one does not have to assume that the spontaneous emission is the driving force, since it inevitably follows from the formulation. Actually, it was shown that the field is, in principle, driven by spontaneous emission from both the active material and the loss mechanism. The foregoing differences stem mainly from the difference in the degree of refinement of the two models. Lumping all effects into a constant r, as in (142a), has the advantage of leading to simpler expressions. At the same time however, one loses considerable information about the relative importance of several aspects that may alter the results even qualitatively.

85 Part of this treatment is also related to Senitzky's work. In order to study the electromagnetic field inside a cavity, Senitzky has considered the problem of a single harmonic oscillator coupled to material systems. Indeed, each mode of the cavity corresponds to a harmonic-oscillator, and it seems reasonable to consider a single harmonic oscillator, if one wishes to study the single mode operation. This would undoubtedly be correct in an entirely enclosed, perfect cavity with only one mode excited. Of course, perfect cavity implies no coupling with the external world, and one would have to redefine the connection between theory and measurement. In any event, the problem treated here is not of this nature. The cavity is quite open and clearly, when an atom placed inside the cavity emits spontaneously a photon of wavelength 10 O cm, it does not know that it is inside the cavity. It emits as if it were in free space. When the emission is induced, the presence of the cavity is felt strongly because the induced emission is proportional to the number of photons present in the final state, and it is the cavity that selects the photons which stay in it for a relatively long time. If one considers a single harmonic oscillator and attempts to calculate the spectrum of spontaneous emission, as we did in IV-4, no natural broadening is found. In fact, if the collision and the Doppler broadening are neglected, the spectrum becomes a delta function. This is to be expected since, the natural broadening is intimately connected with the fact that the atom can de

86 cay into a continuous spectrum. We have avoided this difficulty by considering not a single oscillator but the whole field as represented by the vector potential A. In order to obtain equations for pi and qg we expand A(r,t) in terms of the cavity modes. However, when we develop and study the time evolution of G\(t), we retain the coupling of the particle system to the whole radiation field. The RA S coupling term V is contained in U (t). Thus, when we calculate the matrix elements of U (t) we expand the vector potential not in terms of the cavity modes but of plane waves, thereby being able to account for natural broadening (see also Appendix A). Natural broadening is of quantitative importance in some cases while it is not in other cases. Obtaining it or introducing it phenomenologically however, is a matter of consistency of the formulation. This, we regard as an essential difference between the present approach and Ref. 8. In addi tion, here we have considered not a single mode but a multimode cavity, we have formulated the problem as a many-body problem in terms of the density operator, and we have employed Heitler's damping theory which, to our knowledge, has not been applied to the maser problem thus far. The method used in the calculation of the spectrum of spontaneous emission is the generalization of a technique developed by Ekstein 27 and Rostoker 27 These authors have not considered broadening effects and their results are expressed in terms of delta functions. The introduction of broadening requires a different treatment of the auto

87 correlation function. Their result is recaptured by taking I' = o. Papers dealing with problems directly or indirectly connected with our work abound in the research journals. Refs. 28-31 and the references already cited constitute only a small sample. The first paper dealing with the maser is the paper by Gordon, Zeiger and 26 Townes. This paper was later extended to the optical maser by 28 Shallow and Townes. Some of the results of the first paper are special cases of ours. The second contains all the fundamental ideas that led to the construction of the first optical maser but the analysis is rather qualitative. More closely related to our work 30 22 are the papers by KemenyO and McCumber. In addition to the different techniques that they use, their emphasis is more on the mathematical than the physical aspects of the problem. Lastly, one cannot fail to mention Lamb's32 work differing from ours in intention and content considerably.

CHAPTER VII C ONC LUSIT ONS The present theory is a basically linear theory of a multimode cavity in which mode coupling can be neglected. In so far as the theory is valid, it has been shown that: The electromagnetic field inside the cavity, in the absence of any other driving force, is, in principle, driven by spontaneous emission from material systems existing inside the cavity as well as the loss mechanism. For a gas optical maser, the field is effectively driven by spontaneous emission from the active material only. The effect of the material system on the field is represented by the susceptibility. A model for a gas optical maser has been studied, and explicit expressions for the spectrum of spontaneous emission and the susceptibility have been derived. For operation in a single mode and adequate frequency stability, one finds that the output has a Lorentzian shape whose width decreases as the population inversion increases. Moreover, a new term is found in the equation determining the center frequency of the Lorentzian. Since line shape measurements on lines of the narrowness of the optical maser output are not available, the only test of the theory has been the comparison with other theories. This comparison suggests that we have a more refined model capable of accounting for several phenomena that other models do not account for. 88

89 Further work along the same lines could be directed toward calculating the spectrum of the output for more than one mode oscillating simultaneously. However, such a calculation would be more meaningful if comparison to experiment were feasible. Also, one might attempt to extend the theory to include mode coupling. It is quickly recognized though, that the mathematics will become very complex. Perhaps the only thorough treatment of mode coupling existing today is Lamnb's32 work. His equations are extremely cumbersome and the whole work leans heavily on numerical calculations. Thus, it appears that mode coupling inescapably leads to mathematical complexity independently of the underlying model.

APPENDIX A DAMPING THEORY In this appendix we present a brief derivation of Eqs. (49) by using damping theory. The discussion follows that of Ref, 9. Let H be the hamiltonian of a system. It is assumed that H can be written H = H + V, (A.1) and that the eigenvalue problem H In> = En> (A2) can be solved. Then, the problem we wish to solve is to calculate the matrix elements of U(t), where i - Ht U(t) = e X (A3) in the representation In>). We introduce the resolvent operator R(z), defined by R(z) — 1 (A4) z-H where z is a complex number. The operator U(t) is the inverse Laplace transform of R(z), that is +oo+iE -it z/1 U(t) = dz R(z) e (AS) 2hi -oo+i e 9o

91 where e > o. The problem is now reduced to finding the matrix elements of R(z). Let N and Q be two new operators defined by R = N + N Q N, (A6) and the condition that N be diagonal and Q non-diagonal in the representation (In>). Then, NQN will be non-diagonal, and consequently N and NQN will be the diagonal and non-diagonal parts of R respectively in the representation (In>). Introducing the operator Ro defined by - z- (A7) R(z) can be expanded as follows: R(z) 11 _ 1 Ro = Ro z-H-V 1-ROV 1-VRo 00 00 (RoV)nRo = Ro(VRO) * (A8) n=o n=o To obtain integral equations for N one writes Eq. (A4) in the form (z-H -V) R = 1, (Ag) which by virture of Eq. (A6) becomes (z-H~-V) (N+NQN) = 1. (A10) Equating the diagonal and non-diagonal operators on both sides we obtain

92 Q = Vnd + [VndNQ]nd - [(VNQ)d NQ]nd, (All) and N = [z-H~ — r(z)], (A12) 2 where we have defined ~r(z) (V+VNQ).(A13) 2 d The subscripts d and nd denote the diagonal and non-diagonal parts, respectively, of an operator. From Eqs. (A6) and (A12) follows that the diagonal matrix elements of R are given by Rnn (z) = 1 (A14) z-En-rnn(z) To find the off-diagonal matrix elements of R., we iterate Eq. (All) treating N as independent of Q. Thus, we have Q = Vnd + VndNVnd +... (A15) Keeping the first term only and using Eq. (A6) we obtain Rmn(z) = Nmm Vmn Nnn = Vmn (A16) [zEm_ -; rm] [z-En- rn] 2 2 The matrix e~lements of U(t) therefore are:

93 +oo+i t -it z/T Umn (t) = 1 dz Vrmn e, (Al7) 2 2 for m f n. For m = n, we have +oo+ic 2 nn From these equations follows that the Laplace transform of Umn(t) is the product of the transforms of Umm(t) and Unn(t). The invrersion integral in Eq. (ALy7) therefore, can be expressed as the convolution of Umm(t) and Unn(t) as follows: Umn(t) = Vmn U(t-T) nn(T) dT t Vmn UM(T) Unn(t-T) d. (Al9) Thus, the problem reduces to calculating the inversion integral for Umr(t) only. One first investigates the analyticity of the integrand in Eq. (A18), and in particular the analyticity of i.(z). By substituting Eq. (A.15) into Eq. (A13) and replacing N by (z-H~)-s we obtain' r (z) = Vd + [V 1 Vndd + +(A20) Rtii2 teisz-Ho nd (l Retainin the first two terms only we have

94' r (z)j) Vnn + IvnI (l2 2 nn() Vnn z-E nfn This equation shows that the singularities of Pnn(z) lie on the portion x>Eo of the real axis, where Eo is the lowest eigenvalue of H. The singularities are simple poles when the spectrum of H~ is discrete. When part of the spectrum of H~ is continuous rnn(z) has a branch cut along that part of the real axis which corresponds to the continuous spectrum. It can also be shown that Imrnn(z) and Rez have always opposite signs. Hence, the denominator in Eq. (A18) can vanish only on the real axis. Noting that, lim 1 = PP lIi~r(x) E+o X~iEi x where PP denotes the Cauchy principal part, Eq. (A21) yields lim' (x+iE) = in(X) - i yn(x), (A22) c*o 2 2 where Yn(x) 2jtj IVn12 b(x-EA), (A25) Yn (X > VA n n n7n and n(X) = Vnn + PP X (A24) Ahn n It follows then that the integrand in Eq. (A18) is analytic in the complex plane cut by Imz = o and Rez>Eo. Expressing the complex in

95 tegral as a real integral by shifting the path of integration properly, we obtain - ixt/~ E [X-En-A(x ) ]2 + ~~ y(x) 2 E n-"~-n~x)nYn o 2 For a continuous spectrum and since the quantities en(x) and yn(x) are small quantities the integrand attains its maximum value near x = En. The main contribution to the integral thus comes from the vicinity of En* Assuming that {n(x) and yn(x) are slowly varying functions of x, we replace them by their values at En. Moreover, for En>>Eo we can extend the integration to -mo. Then, Unn(t) is approximated by _ i (En+in-iYn)t Unn(t) = e /A (A26a) for t>o, where sn - n Yn(E (A26b) and 1n m ffiAn'(En)* (A.26c) This complets the derivation of Eqs. (49) which we have used in the present treatment. The presentation has been rather sketchy with several subtle questions passed over. More elaborate discussions of the theory can be found in Refs. 9, 10, 18, and 19.

APPENDIX B ON THE MATRIX ELEMENTS OF GC The operator GA is defined by G J o X ma EC(r).p0 (B1) where the summation is over all particles that is, electrons as well as nuclei. Assuming that the particles are grouped into atoms, we separate the summation in two parts as follows: G?= -X (L r )-Eoj (B2) j oj j where j is an index referring to the atoms and oj refers to the oth particle of the jth atom. Assuming now that the mode vector X(r) does not vary appreciably over the dimensions of the atom, we replace 25j(r.o) by X\(Rj), where Rj is the position operator of the center of mass of the jth atom. This is essentially the dipole approximation. Then, introducing the operator d defined by d = me oj (B3) oj J Eq. (B2) becomes Gi Ixf(i)* (B4) 96

97 Note that d operates only on the internal degrees of freedom of the atom. The hamiltonian H (see Eqs. (30)) can be written H = H Hj. (B5) j Assuming that the atoms of the active material are uncorrelated and that they do not interact between each other, we shall have [H j, H Ij'] = o. (B6) S RPt Then, we introduce the operators H. = H +H +H. and J j i S Uj(t) = e (B7) S which defines Uj(t). Introducing furthermore, Ghj defined by GA, j _ d * X (Rj), (B8) we have G =E G j. (B9) Combining now Eqs. (B5)-(B9) we obtain a (t) = uS(t) Gh uS(t) = - Uj(t) Gh Uj(t) = Gj(t), (B10) j J

98 which defines G\j(t). From this equation it is also seen that [GC (t), Cj (t ) ] o, (B11) for all j,j'. From Eq. (42) we have (Bl2a) A (t,tl,t2) = Tr D A (t,tlt2), where A P(t,tlt) = U -tl) [ (t2), (t)] (t-t). (2b) In the linear approximation and for uncorrelated atoms we have A p'(ttl,t2) = A~P (t,t l,t) = j? S'~t - S S S = Uj (t-tz) [Gh\j(t2), Gj(tl)] Uj(t-tl) (B13 op which defines A j. This equation justifies Egs. (62) and (64). Also, in calculating the spectrum of spontaneous emission we have to calculate the quantity Tr D G7(T)Gx. By virtue of Eq. (B10) we have Tr D G (T) G\ = Tr D G\j(T) GAj+ J + Tr D G;j(T) G7jt. (B14) jj' (ifi t)

99 For uncorrelated atoms, and if the off-diagonal matrix elements of D can be neglected, the second term vanishes. The first term gives rise to the right hand side of Eq. (78b). To summarize: the results of this Appendix show that, in calculating the spectrum of spontaneous emission, as well as the susceptibility, we may take IG\,mK,mlKl 2 1kznI9.A\ImJ~I J X (Rj) IK1>12 (B15) where X (r) = c X(r), (B16) provided the atoms of the active material interact with the electromagnetic field independently of each other.

APPENDIX C ON THE CALCULATION OF THE SUSCEPTIBILITY For the purposes of this Appendix, we introduce 0kT(W) defined by T ict t t 1 (DT() = dt dt 2 Q(t2) A(t,tl-t2), (c1) 0 0 where 2 A (t,tl-t2) = 2i Im D d2 D D X mm dDml XKK pp AjKK I mmn=l jKKl -i(~~mm!1~Kjl++SrMMMj iM )tl i(omml+l+Ml -ir+M + t e e (C2) We also introduce the symbols =pM1 E (cDmnlt9(K1+SM 4 (C3a) and 2 2 Bn - Dmm dmml DKK Dpp X\j,KK1' (C3b) where all subscripts are lumped into n and B is real. Then, we have -i(swI-iFM )tl A (t,tl-t2) = 2i Im I Bn e n n e, (C4a) which, if written as the difference of the right side and its complex conjugate, becomes 100

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103 + Q\T(w) Q\T(u) (C8) (Xl ire) (X~nl irMW(Q 1-ir )'k The first term in the right hand side of the above equation shall be + T neglected as containing the damping factor e MM1 in view of the fact that ultimately we shall let T + co. Moreover, recall that = mml +fK 1 + SMM1 where cmml shall be a frequency of the order of 10 4aK1K+ M1 where Um, shall be a frequency of the order of 10 10 5 cps, and o1lK SM~ 1 MM small quantities as compared to L. rMM1 mi ~~~mml For the purpose of investigating orders of magnitude therefore, we may neglect these small quantities. If we denote by wo the maser transition frequency, the term (w+S4M1) will give rise to resonance terms of the form(u-wo). In view of the narrow spectral lines involved, we shall be interested in w's of approximately the same order as 0O which means that u-Luo << co. Thus, we shall have ( uo) -l<< ( (- o ) and consequently the third term in the right side of Eq. (C8) will be much smaller than the second. Retaining therefore only the second term we have 4NT1(I() = QAT(L) (C9a) 1o (+nw irch) (u+dmle-iro 1) Note that this term contains also antiresonance terms of the order of jO(Luuco) which should be neglected if the approximation is to be

104 consistent. This is done at a later stage in Chapter VIo. The calculation of T(Qc) proceeds along the same lines and the 2 same approximations are made. The result is Bh/A~T2((QT) = QT)(C9b) (~ +irM l) (~+I +lir+ +iPMml MM1 Combining now Eqs. (C3), (C5) and (C9) we obtain 2 TT(c) = QAT() X Dm d2,ml D D X2 j mml-l jKK p ([(+mW x 1+sE +irj) (- m +lslir+Sf +irjl)]- (C o10) fro[((ml,++ M (whichlEq.6l+6);Mflol o from which Eq. (66) follows,

APPENDIX D CALCULATION OF THE DOPPLER WIDTH We have, by definition, 2 C 2 2 rd = (Kk (D1) K m Assuming a Maxwellian distribution of velocities for the center of mass motion, DK becomes K 3 -3/2 3 -b K 2 D ~ - Ir d K e, (D2) K where 2 b = m (D3 ) 2m-g and Q is the mean energy. Since for the integration over K the vector is fixed, we transform to spherical coordinates taking kA as the zaxis. Callings the angle between k_ and K and setting v = cos, we obtain 2t +1 2ab3 2 0 22 r k77 k dJ V dv K e dK. (D4) I/ m 0 0 Using the relation 00 xn e x dx = 1.3...(2n-1) o 2 ao105

106 carrying out the integrations, and using (D3) we obtain F2 2 k d A Noting that g = kT where k is the Boltzmann's constant and T the temperature, and that k2 2 /C2, we have 2 2 k T (D4) mc2

APPENDIX E CALCULATION OF 4) From Eq. (115) we have 2ib m -(d+i)x 2 -b2K 2ibdm ) dx e ( K e dK. +1 _iKk tx' e d), (El) -.1 where we have introduced m r+ (E2) and m (c-noo-So). (E3) Integrating with respect to [ we obtain +1 -iKkIx 2( e dp = 2 Sin (Kkhx). (E4) Kk\x -1 Kkx Then, Eq. (El) becomes 00 -(o+irI)x 00 b2t'?(CD) = 2ib3m d e Sin(k hxJ)dt, (E5)'' h hjo khx o where we have made the transformation K2 = t. (E6) 107

108 From Ref. 33, Vol. II, p. 57, we have e -St Sin 7fSdt = S, (E 2s ( which is valid for s > o. Using this formula we obtain 00 2 2 -b2 t k s-x k e Sin (kx 4t) dt =e 4b2 (E8) o 2b Substituting into Eq. (E5) it becomes 00 -(o+iq)x kh 2 () i Im, dx e 4b2 (E9) This can also be written oo 22 co 2 2 =W - dx e Cos(qx) dx + 00 -22 OX + m dx e -a x -oX + -J dx e Sin (ax) dx, (E10) where we have introduced c2 kh2 (Ell) 4b2 The abovre integrals are Fourier Cosine and Sine transforms and from Ref. 34 pages 15 and 74 we have 00 22 -OC x -ox e Sin(rx) dx = 0 - -1 i e 4e2 (i)Erfce[ (c-in)] - 4 o-

109 -e 4a (o+i)2 Erfc [A1 (o+ij]I, (E12) 2a and 00 2 2 -a: x -ox e Cos (rx) dx = o = l< { e 4 (0ir) Erfc [ (o-iq)] + (E13) + e - (o+i )2 Erfc [ 1 (o+ir)], 26 where Erfc(z) is defined by 00 Erfc(z) = e dt. (E14) z Eqs. (E12) and (E13) are valid for ReoC > o and ~ > o. The first condition is always satisfied since a 2 is real and positive. But q assumes positive as well as negative values. Consider first the case of positive v. Then j = I and therefore, Cos (~x) = Cos (| lx) and Sin (rx) = Sin (|ljx). Using these relations in Eq. (ElO) and then Eqs. (E12) and (E13), which are now applicable since hI|>o, after some straightforward manipulations we obtain I —(1) = i m e~ (oilrl) Erfc (O+ilI), (E) 6' 2aKo 2i which is valid for r > o. Let now i < o. Then r = -rIJ and

110 Cos (rx) = Cos (Hlix) and Sin (rx) = -Sin (Nl|x). Again, using these relations in Eq. (E10) and proceeding as before we obtain 7(0) = i m ee (o-ilil) Erfc( ), (E16) which is valid for ~ < o. But for r > o we have Jlj = r, and for < o we have -Irj = -. Consequently Eqs. (E15) and (E16) can be combined to the single equation ( o+i2r)2 ( i m e Erfc ( C+i) (E17) 2C6i 2cz which is valid for all values of n, including zero. That the case r = o is included in Eq. (El7) can be readily verified by calculating the integral in Eq. (E9) for T = o and comparing the result to what Eq. (17) gives for T = o. Introducing now the function L_(z) defined by z2 (z) = e Erfc(z), (E18) and using Eqs. (E2), (E3), (Ell) and (E17) we obtain ) ) = i bm;7 bm- ro+ + i(-O)0o-So) ]. (El9) lhlk- - klkh

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