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T HE U N I VE RS I T Y OF M I C H I G A N COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report INTENSITY FLUCTUATIONS OF THE RADIATION FROM A DISPERSIVE BLACK BODY R. K. Osborn A. Z. Akcasu ORA Project 07599 sponsored by: Advanced Research Projects Agency Project DEFENDER ARPA Order No. 675 under contract with: U. S. ARMY RESEARCH{ OFFICE-DURHAM CONTRACT NO,. DA- 51-124I-ARO(D) -403 DURHAM, NORTH CAROLINA.ad. d i.-nis te r'e d..ithro:ugh: OFFICE OF RESEARCH" ADMINISTRATION ANN ARBOR Apr 1 1966

This report was prepa.red as a, paper for submission to Physical Review.

TABLE OF CONTENTS Page ABSTRACT 1 1. INTRODUCTION 2 2. CORRELATIONS IN PHOTON DENSITY FLUCTUATIONS 4 3. FLUCTUATIONS IN THE COUNT RATE 11 4. SLAB GEOMETRY 14 5. DISCUSSION 15 REFERENCES 20 APPENDIX 21 FIGURE CAPTIONS 23 iii

INTENSITY FLUCTUATIONS OF THE RADIATION FROM A DISPERSIVE BLACK BODY R. K. Osborn and A. Z. Akcasu The University of Michigan Department of Nuclear Engineering Ann Arbor, Michigan ABSTRACT Statistical properties of the output fluctuations of a. photon detector measuring the intensity of radiation at a given frequency emitted by a dispersive black body are investigated using La.ngevin' s technique. The va.rianceto-mean ratio of the accumulated counts, and the power spectral density of the count rate are obtained in terms of properties of the emitting medium. The possibility and limitations of obtaining information about the emitter by measuring these quantities are discussed. It is found in particular that the temperature of the emitting medium can be determined in principle by observing the intensity fluctuations of the radiation at a single frequency. The photon fluctuations in a microwave cavity are also discussed in the framework of the present formulism and compared to previous work.

1. INTRODUCTION The purpose of this paper is to investigate the intensity fluctuations in radiation emitted by a dispersive black body. The experiment which we intend to analyze is illustrated in Fig. 1. The photons from the emitter are detected by a photon detector, e.g., photo-cathode, whose output Z(t) is assumed to be in the form of an electric voltage, and proportional to the number of photons absorbed per second in its active volume. Hence, Z(t) can be identified as the "instantaneous count rate." The detector is visualized as a uniform absorbing medium described by the microscopic absorption probability per photon rD(k) for photons with wave vector k. The emitter is assumed to be a, homogeneous:finite medium in which the atoms are in thermal equilibrium at a, temperature 9 (units of energy). The medium is characterized by a(k) and (k_) which are the absorption and emission rates, respectively, of photons with wave vector k. The scattering of photons in the medium is neglected. The medium is allowed to be dispersive with a photon speed v(k) which is different than the speed c in vacuum. Let N(x,k,t) denote the instantaneous number of randomly polarized photons at time t per unit volume about x in configuration space, and per unit volume about k in wave-vector space. We shall denote a point in the six-dimensional space by r = (x,k) to compress the notation. The instantaneous value of the photon density N(r,t) is a fluctuating function of time with a stationary mean value <N(r)>. These fluctuations are due to the statistical nature of the absorption and emission processes. We shall denote the fluctuating part of

N(r,t) by n(r,t), i.e., n(r,t) N(r,t) - <N(r)>. The aim of this paper will be to investigate some of the statistical properties of n(r,t) in terms of the observed fluctuations z(t) in the count rate (or absorption rate) of the detector, i.e., z(t) = Z(t) - <Z(t)>. In particular we shall consider the autocorrelation functions and the power spectral density associated with z(t) which are defined by Dz(T) - < z(t) z(t + T) > (1) and +00 GZ(w) e - iWT (DT)d (2) -00 we shall also discuss the variance-to-mean ratio of the accumulated counts, T a(T) -= z(t) dt, (3) 0 as a function of the gate time T, which is defined as* T <Z> < > T) 0 The interest in these particular statistical quantities stems from the fact that they a.re the quantities which are usually measured in fluctuation experiments. The fluctuations in the count rate can be expressed as z(t) = d6r z(r,t) (5) D use T T dt dt %(t-t') = (T-Tj) z(T) dT. o o -T

where z(r,t) is the fluctuating part of the "count rate density" at r in the phase space. The autocorrelation function Oz(T) can be expressed in terms of z(r,t) as z(T) = d6 r dr' < z(r,t) z(r',t + T) >. (6) D D Both, in (5) a,nd (6) the integrations are performed in the active volume of the detector in the phase space. It follows from (6) that the statistical quantities Dz(T), GZ(cu) a~nd n(T) can readily be obtained once the correlation function -pz(r,r', T ) < z(r,t) z(r',t + T) > (7) associated with the count rate density is determined. Thus, our main task will be to calculate cpz(r,r', T) in terms of the parameters {a(k), F(k) a.nd rD(kY describing the optical source and the emitter. In order to achieve this we must first determine the correlation function. CPn(ryr' T) - < n(r,t) n(r',t + T) > (8) appropriate to the photon density flucutations and then use the relations between (Pn a,nd (pz as discussed in section 3, by Eq. 31. 2. CORRELATIONS IN PHOTON DENSITY FLUCTUATIONS The mean photon density <N(r,t)> in a,n inhomogeneous, dispersive medium satisfies, when the photon scattering is neglected, the following approximate transport equation

a+ V v(r) + a(r] < N(r,t) > = E(r) p(k) (9) where (_(r) = a(r) - ~(r), p(k) = density of photon states = k2/4n3, Q = /k We consider the optical source, vacuum and the detector as a single inhomogeneous medium. Then, both a(r) and E(r) are zero in the vacuum, and a(r) = rD(k) within the detector. Equation (9) is a simple statement of photon balance. The right hand side represents the spontaneous emission, the term E(r) < N(r,t) > denotes the rate of stimulated emission, and the terms ~ V v(r) < N(r,t) > and a(r) < N(r,t) > a.ccount for the rates of loss of photons by streaming and absorption. A careful derivation of (9) can be found elsewhere.l In a stationary system, as we assume here, the mea.n photon density is independent of time. Therefore, (9) reduces to Q ~ V F(r) + t(r) F(r) So(r) (10) where we have defined F(r).= v(r) < N(r) >, (lla) Z(r) = a(r)/v(r), (Ub) SO(r) = 8(r) p(k). (I1c) clearly, F(r) is the mean photon flux. Equation (O10) can readily be solved by the method of characteristics to obtain

F(r) = du S(x - Qu, k) p(u,r) (12a) 0o where p(u,r) exp L E(x' - U'~_, k) du (12b) The neutron density < N(r) > in various regions can be obtained from (12) as: a) In the emitter 1 - exp [-Z(k) ~S(fs))] < N(S) > = So(k) (k) (13) __S 0 c G(k) b) In the de'tector v(k) 1 - exp [-Z(k)Ls(,)] e)] -,D(k)D( (14) < N(r) > So(k) e (14) - c - (k) where the distances YS(2SQ-),- LS(x,Q) and ID(x,_), as well as LD(x,Q) which will be used later, are indicated in Fig. 1. We shall now investigate the fluctuations n(r,t) in the photon density by means of Langevin's technique. The discussion of this technique as a method for investigating fluctuation phenomena, in other physical systems can be found elsewhere, and will not be reproduced here in detail. This approa.ch, which is somewhat phenomenological, is adopted in this work for its simplicity as opposed to the more detuctive kinetic approach'23 The Langevin technique starts with the "stochastic" equation t+ _ ~ V v(r) + c(r)] n(rt) = s(r,t) (15) where s(rTt) is a random source introduced to account for the fluctuations about the mean photon density. This random driving force is called the "noise

equivalent source" in the theory of neutron fluctuations.4,2 Since < n(r,t) > 0 by definition, one requires that < s(r,t) = O. (16) Physically, s(r,t) represents the natural fluctuations in the rate of absorption, stimulated emission and spontaneous emission processes. In addition to (16-), one attributes the following statistical property to s(r,t) as a, postulate: < s(r,t) s(r',t) > Qs(r).(r' - r) 8(t' t), (17) where Qs(r) = [ a(r) + t(r)] < N(r) > + So(r). (18) Equation (17) implies that the fluctuations in the noise-equivalent source are uncorrelated in phase space and in time. Equation (18) is based on the assumptions that a) absorption, stimulated emission, and spontaneous emission of photons are statistically independent, and b) each of these processes is described by a Poisson distribution. The foregoing statistical properties, i.e., (16) and (17), are sufficient to investigate the space, time and energy correlations of the photon density fluctuations, viz., cp(r,r',T) = < n(r,t) n(r',t + T) >. In order to obtain n, we must first solve (15) for n(r,t). Defining f(r,t) - v(r) n(r,t) (19) and taking the Fourier transform of (15), one~finds

~ V f(r,io) + [ i + Z(r)] f(r,ico) = (r, i) (20) v( r) where f and s denote the Fourier transforms of the respective time functions. This equation is readily solved in a similar fashion to (10) to obtain r(ri w) = du s(x-uQ_,k, iW) exp [- i v/ du' o v(x-uo.k) (x) p(u,r) (21) The inverse Fourier transform yields v1t) Lkt du' n(r,t): (r) du s -uQh,t - v-r) v(x-u',,) 0 0 (x) p(u,r) ( 22)) which is the desired solution of (15). We are now in a position to evaluate cp (r,r',t) in terms of the assumed statistical properties of s(r,t). Substituting (22) into the definition of CPn in (8), one finds 00 00 cp_(r,r',T) = du jdu' p(u,r) p(u',r') Qs(X - uQ, k) 0 0 -1 (x) b(k'-k)' 6[x'- x-u'Q' + uQ] [v(r)v(r' )] u' u.dq.....,:, dq (X) 6LT _ vj x d,~ + x v 1(25) v(x -qQ- k' k)' v(x-qQ,k) where we have used (17) and (18) to evaluate < s(r,t) s(r',t') >. This equation can be simplified by making use of k' = k, Q' = Q and x' = x + (u'-u)Q in ap

propriate terms: 00 pn(r,r' T) = (k' -k)[v(r) v(r')] du QS(x - uQ,k) p (u,r) ~00 u'-u~0 (x) du' exp T (+qQk dqH [x'-x-(u'-u)Q] o o 5 T - dq/v(x+qQPk) j (24) 0 we shall now perform the integration on u' assuming that T > 0. The case of T < 0 need not be considered separately because the following relation holds cPn(r,r', - ITI) = (Pn(r', r, -I,) (25) which can be verified easily using the definition of Pn(r,r',T) in (8). The last delta function in (24) can be replaced by* ( u' -u-ro) v[x+ro0,%k] (26) where ro = ro(r) is a number obtained by o T = dq/v(x+qQ,k). (27) o In the case of a constant speed, (27) yields ro = VT. The integration on u' can now be performed easily yielding O0 (P (rnr' T) =B(k'-k) 6['-x-Or,] exp 7- O(x+ 1q.k)dq] (x) v (r) du p2(ur) QS(x-ua2k) (28) Use 6[(x)] = Z 6(x-xi) where the xi are the roots of (x) = 0 which are IfT (xi) l assumed to be simple, and ~' = d~/dx.

where Qs(x-u:Q,k) and p(u~r) were defined before in (17) and (12b) respectively. We shall now specialize (28) to the case of a uniform emitter, vacuum, and detector as indicated in Fig.l. We shall be interested in the values of cPn(r,r',T) within the detector. One can see that Qs(x-uQ,k) = rD(k) < N(x-uQ,k) > for O < ul - [ol(k) + E(k)] < N(x-u_,k) >+ So(k) for u2 < u < uK 0 elsewhere Hence the integration on u will be performed only in the intervals (0, ul) and (ul, 2). On the other hand (27) gives rO = Tc because v(x + q2,k) =c in the detector and in vacuum. After some lengthy but elementary calculations one finds cpn(rr',T) = (k' -k) 6(xT -x-QTc) exp [-rDi] < N(r) > (x) [1 + Q(r)],9 for > 0 (29a) where k(~) -D(k) iD(-xn0 -,(k)~ )(x~0 Q(r),) (.k)( e - ), (29b) c(k) and where the distances 1D and LS are already. indicated in Fig. 1. When T < 0, one obtains from (25) and (29) cpn(r,r',T) = 6(k_-k) 6(x -x + _Q TIc) exp [-rDITI] (x) < (r' ) > [1 + Q(r')], for T < O. (3O) 10

3. FLUCTUATIONS IN THE COUNT RATE This section is devoted to the calculation of the count rate correlation function defined previously in (7). It is shown in reference 2 that Pz(r,r',T) cain be expressed in terms of cpn(r,r',T) a.s zcp(r,r',T) < N(r) > rD(k) b(r'-r) 6(T) 2 + rD(k) n (r,r' T -< N(r) > g0(rr') (31) where gl(r,r',T) is the Green's function which satisfies the photon transport equation in the detector, i.e., + c Q Vx + rD(k) 1(r,r',T) = 5(r'.r) 6(T) (32) Here we have used explicitly the fact that the detector medium is non-dispersive and a pure photon absorber. The solution of (32) is readily found to be gl(r,r',T) = 6(k' -k) 5(x' -x-Q TC) exp [-rD(k) T]. (33) Substituting (29a) and (33) into (31), one obtains pZ( r, rD(k) ( k' -k) 5(x'-x) 6(T) + rD(k) Q(r) 5(x'-x-QTC) exp [-rDT] T >. (4) The Autocorrelation Function The autocorrelation of the detector output, defined by (6), is now obtained by integrating (34) on r and r'. The result is 11

z( D d r rD(k) < N(r) > b(T) + rD(k) exp LITI rD] (x) Q(r) U [Q (xQ) - ITIC]} (35) where U[x] is the unit step function arising from* d3x 65(x' -x- T I C) = dq 6(q-lTlc) = U[iD(x, ) - Tic] V V Observe also that we have replaced T by ITI since after the integration over r and r', (34) becomes an even function of the time lag T. The Power Spectral Density Fourier transforming (35) with respect to T yields the power spectral density associated with the count rates: Gc(W) = d6r rD(k) < N(r.) > [1 + Q(r) I (r,)] (36) D where +00 _r)(k) ITl _,1 iWUT I(r,) dT erD(k)TI U['(x) - Tic] e -00 exp[-DQD] [o)/rD) sin(cO /c) - cos (w./c)] + 1 2 (37) 1 + (c/rD) The Variance-to-Mean Ratio The variance-to-mean ratio, r(T), was defined in (4). We first note that Use 5(x'-x-x jTic) = b(q-jTic) 5( __)/q where q = x'1-x 12

the mean count ra~te is < Z > dr rD(k) < N(r) >. (38) D We then perform the integral indicated in the right hand side of (4). The fina,l result is found to be I(T) = 1 +rD d6r rD(k) Q(r) <N(r) > j(kT)/ <Z > (39) where T (_kT) - rD(k) (1 T) exp [-rDu] U [~V(x,2) - uc ] 0 (1 - e D) 1 [1-(1+ ID)e D D], Tc> I,(X-),:rDT D(x,) (40b) The limit of la.rge gate time, i.e., Tc >> LD(xQ) and T rD >> 1, is of pa~rticular interest because large gate times are easier to work with experimentally. In this case, the second term in (40a) can be ignored as compared to the first one, and (39) reduces to ~(T + ~) = 1 + d r rD(k) Q(r) < N(r) > (1 - e /< Z > D (41) We shall now apply these general results to a particular geometry in order to gain more insight into the intensity fluctuation phenomenon. 13

4. SLAB GEOMETRY We consider an experiment indicated in Fig. 2. The source is an infinite slab of thickness a, and the detector is a cylinder of length I alnd cross section A. We look at the photons with wave vector kio which is parallel to the axis of the detector. The relative positions of the detector and the source are shown in Fig. 2. The following quantities are needed to calculate the autocorrelation function, power spectral density and variance-to-mean ratio: LD= Q, D = I -x, lD = x = a LS - Q(x) = ~eZDx(l e-Zx) < N(x) > = _ p (1 - e a' )e -Dx <Z> = v Z(1- e~ )( - e )A We shall discuss only the power spectral density and the variance-to-mean ratio, because the autocorrelation function is usually used only to obtain the power spectral density. Equations (36) and (41) reduce in the present case to Z.a 1+ e2ZD~_2 eZD~ L1 Gz(wU):<z > + (e os c (42) 1 + 2- 1-eD 1 + — - ~'DrD and r~(T + oo)= 1 + - (1 - e (43) Two limiting cases are of particular interest: 14

a) Za >> 1 ED >> 1 These conditions imply that the emitter is optically thick and that all the photons entering the detector are detected. One finds V13. 1 -(w) c a 1 + (w/rD)2' and j(T +oo) = 1+. (45) b) Za, >>l, n <<1 This case differs from the previous one in that the detector is now optically thin: G (w) = v V [1 + a( (s) in)-] (46) 2c ] and G(T + a) = 1 +a (DI) 2 (47) where V = AI, i.e., is the detector volume. 5. DISCUSSION It is observed from (44) and (46) that the break frequency in the power spectral density, i.e., rD or (c/i) depending on the value of ZDI, is related to the characteristics of the detector only, and hence does not yield any information about the optical properties of the source medium. This implies that 15

one does not gain more information by measuring Gz(w) at different frequencies. However, one can measure ( E/c), which is related to the source, by determining Gz(o) for very small (w << Min (c/i, rD)) and very large (c >> rD) values of wC. Indeed, one finds from (42) that Gz ()-ZD = 1-e ) (1 - e ) ) Gz(") - (T +t) - i (48) As already indicated in (48) this quantity (a) is also obtainable from the variance-to-mean experiment. The choice between these two experimental techniques may depend on the mode of operation of the detector. If the detector is operated in the current mode, i.e., the output Z(t) is a continuous fluctuating voltage, the power spectral density can be determined by conventional techniques. If the detector is operated in the pulse-mode, i.e., Z(t) is a train of pulses, then the variance-to-mean experiment is the natural choice. Consider now the information contained in &/o, assuming that it is measured. If we are looking at a black-body system in which the atoms are in thermal equilibrium at a temperature T, one can show that a = exp [ woo/KT] (49) where K is Boltzmann's constant and hcoo is the energy of the observed photons. Recalling that a - al-, one finds - = [exp (+.o/Kr)- 1] -1 16

h \, L "-'/ ~- 1 h(0 = exp[-hwo/KT], (Io/KT) >> 1.(50) Thus, one can obtain the temperature of a black-body source by observing the fluctuation in the intensity of radiation at a single frequency. According to equation (50), it appears that it would be preferable to measure the fluctuations of the low energy photons (hwo/KT << 1). However, it must be borne in mind that both the power spectral density and the variance are proportional to the average photon density which is, in turn, proportional to the ratio of the photon speed in the emitting medium and the speed of light in a vacuum. In at least some circumstances this fact presents a significant limdtation on the feasibility of such measurements at very low frequencies. -To illuminate this point a little, consider the case in which the emitter is a plasma —at least in sofar as its dispersive properties are concerned. According to reference 1, the speed, v, which appears in Eq. (9) is defined to be v = c (51) co/k where B is the index of refraction and cw as a function of k is given by the dispersion relation for the propagation of transverse electromagnetic waves in the medium. In the plasma case, at a sufficiently low order of approximation, this dispersive relation is 2 2 2 U C= Co+ W (52) 0 e where ce is the plasma frequency and co is the vacuum frequency of the photons 17

observed. Thus, for this example, one finds that [1+ (-) ]2 -1/2 C ado Therefore, if one were to attempt to measure the fluctuations in the emergent photon distribution at frequencies for which wo/e << 1, one must expect that both the variance and the power spectral density will be proportional to this small quantity and that hence the feasibility of a statistically significant measurement is called into question. Thus the desirability of measuring the variance at frequencies such that o/KT << 1 is in competition with the feasibility of any measurement at all if the corresponding index of refraction is small. Finally, it must be pointed out that the present theory is predicated upon the assumption that the principal mechanisms for the emission and absorption of radiation involve free-free praticle transitions only, i.e., that particle states of short life time do not play a significant role in the observations discussed above. The point is that Eq. (15), which purported by describes the fluctuating photon density, has been derived with the explict use of a coarsegraining, or averaging, in time which implies a loss of information regarding the fluctuations in time intervals of the order of, or less than, the life time of atomic bound states. Consequently the effects of emission and absorption on Eq. (15) cannot be represented simply by a term of the form, on. Instead there should be resonant elastic and inelastic scattering terms, as well as fission-like terms which describe the multiple production of photons following the absorption of a single one. In such an event, the equation 18

describing the fluctuating photon density is not simply soluable as is the case in the present example. Furthermore, because of the possibility that two or more photons may have common ancestors due to the fission-like processes, an entirely new mechanism for correlation is introduced into the system (quite analogous to the correlations observed in neutron counting experiments in multiplying media). The contribution to the variance, or spectral density due to this latter mechanism may be as important —or even more so —than the one investigated in this work. The present approach based on the stochastic equation (15) has been applied also to the study of photon fluctuations in a microwave cavity as a special case in which the transport effects can be neglected. The description of the physical system and the results are given in the appendix. (5) The results agree exactly with those obtained by McCombie using the master equation. 19

REFERENCES 1. E. H. Klevans, A Theory of Photon Transport in Dispersive Media, Ph.D. Thesis, University of Michigan (1962). 2. A. Z. Akcasu and R. K. Obsorn Nucl. Sci, Eng., (1966) (in print). 3. R. K. Osborn and M. Nateson, J. Nucl. Eng., Part A/B, 19, 916 (1965). 4. C. E. Cohn, Nucl. Sci. Eng., 8_, 472 (1960). 5. W. McCombie, Fluctuation, Relaxation and Resonance in Magnetic Systems p. 193, Scottish Universities Summer School, 1961, edit. by D. Ter Haar, New York, Plenum (1962). 20

APPENDIX FLUCTUATIONS IN A CAVITY The system under consideration consists of a microwave cavity, a source and two detectors. The source and the detectors are located a.t the center of the ca.vity, which supports a radiation mode of frequency wo. The dimensions of the source and the detectors are small a.s compared to the wave length of the radia.tion. This assumption eliminates transport effects which play an important role in the general problem treated in the text. The source is described by kfo) and aS(ko) which are the emission and absorption rates. The detectors consist of atoms which absorb photons. The re-emission of photons by the detector atoms is neglected. Thus, the detectors are described only by the absorption rate aD(ko). Dropping the arguments in (k ), aS(ko) and ~D(ko), and defining a = 2% + aS one obtains the stoclhastic equation (15) in present case as ( + a) n(t) = s(t) c6t The correlation function associated with the photon density n(t) is obtained by the technique described in the texts as < n(t) n(t + T) > = (a/c) N exp[ -CaT ] where N = (~p/C) and denotes the mean number photons in the mode under consideration. Recall that p is the density of photon states which is unity if there is only one mode with frequency wo. if there are degenerate modes with 21

the same frequency, then p will denote the number of such modes. The crosscorrelation between the accumulated counts of the detectors in a collection time T is found to be = 2 c l+aT - exp[-pT] <C1(T) C2(T) > 2 TC(T)2 p where C(T) is the common value of the mean number of counts of the detectors in T. For large gate times, i.e., oT >> 1, this reduces to < C1(T) C2(T) > = 2 C(T) /Tap which is identical to (4.14) of reference (5) apart from differences in notation. 22

FIGURE CAPTIONS Fig. 1. Intensity fluctuation experiment. Fig. 2. Slab geometry. 23

LD(x,fl)uu,(,) - IO1= k/k x' Z(t) output VACUUM EM ITTER DETECTOR Fig. 1

"-1~~~~~~~~ x=O CYLINDRICAL SLAB EMITTER DETECTOR Fig. 2

N9ER0i10N6 O95 79G~'RN 3 t15Olx 5 79