Radiative transfer theory for vacuum fluctuations

VOLUME83, NUMBER26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER1999

Radiative Transfer Theory for Vacuum Fluctuations

E. G. Mishchenko1,2 _{and C. W. J. Beenakker}1

1_{Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands}

2_{L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Kosygin 2, Moscow 117334, Russia} (Received 6 July 1999)

A semiclassical kinetic theory is presented for the fluctuating photon flux emitted by a disordered medium in thermal equilibrium. The kinetic equation is the optical analog of the Boltzmann-Langevin equation for electrons. Vacuum fluctuations of the electromagnetic field provide a new source of fluc-tuations in the photon flux, over and above the flucfluc-tuations due to scattering. The kinetic theory in the diffusion approximation is applied to the super-Poissonian noise due to photon bunching and to the excess noise due to beating of incident radiation with the vacuum fluctuations.

PACS numbers: 42.50.Ar, 05.40. – a, 42.68.Ay, 78.45. + h

The theory of radiative transfer was developed by Chan-drasekhar [1] and Sobolev [2] to describe the scattering and absorption of electromagnetic radiation by interstel-lar matter. It has become widely used in the study of wave propagation in random media, with applications in medical imaging and seismic exploration [3]. The basic equation of radiative transfer theory is a kinetic equation of the Boltz-mann type that is derived from the Maxwell equations by neglecting interference effects [4]. It is a reliable approxi-mation whenever the scattering and absorption lengths are large compared to the wavelength, which applies to all but the most strongly disordered media.

Radiative transfer theory has so far been restricted to classical waves, excluding purely quantum mechanical effects of vacuum fluctuations. This limitation is felt strongly in connection with the recent activity on random lasers [5]. These are amplifying systems in which the feedback is provided by multiple scattering from disorder rather than by mirrors, so that radiative transfer theory is an appropriate level of description. However, while

stimulated emission has been incorporated into this

approach a long time ago by Letokhov [6], spontaneous emission has not. It is the purpose of our work to remove this limitation, by presenting an extension of the radiative transfer equation that includes vacuum fluctuations and the associated spontaneous emission of radiation.

Our inspiration came from the field of electronic conduction in disordered metals, where the notion of a fluctuating Boltzmann equation (or Boltzmann-Langevin equation) has been developed extensively [7 – 9], following the original proposal by Kadomtsev [10]. In that context the fluctuations originate from random scattering and they conserve the particle number. This same class of fluctuations exists also in the optical context considered here, but with a different correlator because of the differ-ence between boson and fermion statistics. In addition, the photons have a new class of fluctuations, without particle conservation, originating from random absorption and emission events. Vacuum fluctuations are of the second class. We will extend the radiative transfer theory

to include both classes of fluctuations. To demonstrate the validity of our “Boltzmann-Langevin equation for photons,” we solve the problem of the excess noise from vacuum fluctuations in a waveguide geometry, for which an independent solution is known [11]. We then apply it to the unsolved problem of the thermal radiation from a spherical random medium.

The basic quantity of the kinetic theory is the fluctuating distribution function fk共r, t兲 of the number of photons per unit cell共2p兲23dkdr in phase space. (For simplicity, we

ignore the polarization dependence.) Conventional radia-tive transfer theory deals with the mean f_k共r兲, which we assume to be time independent. It satisfies the Boltzmann equation c ˆk ? ≠fk ≠r 苷 X k0 关Jkk0共 f兲 2 Jk0_k共 f兲兴 1 I1 k共 f兲 2 I_k2共 f兲 . (1)

[For ease of notation, we writeP_kinstead of共2p兲23R_dk, and dkq instead of 共2p兲3d共k 2 q兲.] The left-hand side is the convection term (with c the velocity of light in the medium and ˆk a unit vector in the direction of the wave number k). The right-hand side contains gain and loss terms due to scattering, Jkk0共 f兲 苷 wkk0fk0共1 1 f_k兲,

due to amplification, Ik1共 f兲 苷 wk1共1 1 fk兲, and due to absorption Ik2共 f兲 苷 wk2fk. The scattering rate wkk0 苷

wk0_k is elastic and symmetric. The absorption and

am-plification rates wk6are isotropic (dependent only on k 苷

jkj) and related to each other by the requirement that the

Bose-Einstein function

feq共v, T兲 苷 关exp共 ¯hv兾kBT兲 2 1兴21 (2)

is the equilibrium solution of Eq. (1) (at frequency v 苷

ck and temperature T ). This requirement fixes the ratio w2k兾w1k 苷 exp共 ¯hv兾kBT兲. The temperature T is positive

for an absorbing medium and negative for an amplifying medium such as a laser [12].

We now extend the radiative transfer equation (1) to in-clude the fluctuations df 苷 f 2 f. Following the line of argument that leads to the Boltzmann-Langevin equation

VOLUME83, NUMBER26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER1999 for electrons [7 – 10], we propose the kinetic equation

c ˆk ? ≠fk ≠r 苷 X k0 关Jkk0共 f兲 2 J_k0_k共 f兲兴 1 I_k1共 f兲 2 I_k2共 f兲 1 Lk. (3) The argument is that the fluctuating f is propagated, scattered, absorbed, and amplified in the same way as the mean f, hence the same convection term and the same kernels Jkk0, Ik6 appear in Eqs. (1) and (3). In addition, Eq. (3) contains a stochastic source of photons,

Lk 苷 X k0 共dJkk0 2 dJ_k0_k兲 1 dI1 k 2 dI 2 k , (4)

consisting of separate contributions from scattering, amplification, and absorption. This Langevin term has zero mean, Lk 苷 0, and a correlator that follows from the assumption that the elementary stochastic processes

dJkk0, dIk6 have independent Poisson distributions:

dJkk0共r, t兲dJ_qq0共r0, t0兲 苷 Dd_kqd_k0_q0J_kk0共 f兲 , (5a) dIk6共r, t兲dIq6共r0, t0兲 苷 DdkqIk6共 f兲 , (5b) dJkk0共r, t兲dIq6共r0, t0兲 苷 0, dI 1 k共r, t兲dIq2共r0, t0兲 苷 0 , (5c) where we have abbreviated D苷 d共r 2 r0兲d共t 2 t0兲. Substitution into Eq. (4) gives the correlator

Lk共r, t兲Lq共r0, t0兲 苷 D " dkq X k0 关Jkk0共 f兲 1 J_k0_k共 f兲兴 2 J_kq共 f兲 2 J_qk共 f兲 1 d_kq关I1 k共 f兲 1 I 2 k共 f兲兴 # . (6) Equations (3) and (6) constitute the Boltzmann-Langevin

equation for photons.

To gain more insight into this kinetic equation we make the diffusion approximation valid if the mean free path is the shortest length scale in the system (but still large compared to the wavelength). The diffusion approxima-tion consists in an expansion with respect to ˆk in spheri-cal harmonics, keeping only the first two terms: fk 苷

f0 1 ˆk ? f1, Lk 苷 L0 1 ˆk ? LLL1, where f0, f1, L0, and

L

LL1 do not depend on the direction ˆk of the wave vec-tor, but on its magnitude k 苷 v兾c only. The two terms

f0and f1determine, respectively, the photon number den-sity n 苷 rf0and flux density j 苷

1

3crf1, where r共v兲 苷

4pv2共2pc兲23 is the density of states. Integration of Eq. (3) gives two relations between n and j,

j 苷 2D≠n ≠r 1 1 3 lrLLL1, (7) ≠ ≠r ? j苷 Dj 22 a 共rfeq2 n兲 1 rL0, (8) where the diffusion constant D 苷 1₃c2t and mean free

path l 苷 ct are determined by the transport scattering rate

t21 苷Pk0w_kk0共1 2 ˆk ? ˆk0兲. The absorption length j_ais

defined by Dj_a22 苷 w2 2 w1_{. (An amplifying medium} has an imaginary ja and a negative feq.) In Eq. (7) we

have neglected terms of order共l兾ja兲2, which are assumed

to be ø1.

Both Eqs. (7) and (8) contain a fluctuating source term. These two terms L0and LLL1have zero mean and correla-tors that follow from Eq. (6),

L0共v, r, t兲L0共v0, r0, t0兲 苷 D0 D rj2 a 共2feqf0 1 feq1 f0兲 , (9a) LLL1共v, r, t兲LLL1共v0, r0, t0兲 苷 'D0 6c rlf0共1 1 f0兲 , (9b) L0共v, r, t兲L1共v0, r0, t0兲 苷 D0 D rj2 a 共2feqf1 1 f1兲 , (9c) where we have abbreviated D0 苷 d共v 2 v0兲d共t 2

t0兲d共r 2 r0兲. The correlator (9b) differs from the

elec-tronic case [13 – 15] by the factor 1 1 f₀ instead of

1 2 f₀. This is the expected difference between boson and fermion statistics. The correlators (9a) and (9c) have no electronic counterpart. They describe the statistics of the vacuum fluctuations.

To demonstrate how the kinetic theory presented above works in a specific situation we consider the propaga-tion through an absorbing or amplifying disordered wave-guide (length L). The incident radiation is isotropic. All transmitted radiation is absorbed by a photodetector (see Fig. 1). Because of the one dimensionality of the ge-ometry we need to consider only the x dependence of j and n (we assume a unit cross-sectional area). The trans-mitted photon flux I 苷R`₀dv j共v, L, t兲 fluctuates around

its time-averaged value, I共t兲 苷 I 1 dI共t兲. The (zero-fre-quency) noise power P 苷R`<sub>2`</sub>dtdI共t兲dI共0兲 is the

corre-lator of the fluctuating flux. We will compute P by solving the differential equations (7) and (8) with boundary con-ditions n共v, 0, t兲 苷 nin共v, t兲, n共v, L, t兲 苷 0, dictated by the incident radiation at one end of the waveguide and by the absorbing photodetector at the other end.

FIG. 1. Isotropic radiation (solid arrows) is incident on a wave-guide containing an absorbing or amplifying random medium. The transmitted radiation (dashed arrows) is absorbed by a photodetector.

VOLUME83, NUMBER26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER1999 Combining Eqs. (7) and (8) we find equations for the

mean and the fluctuations of the photon number,

d2_n dx2 2 n j2 a 苷 2rfeq j2 a , (10) d2dn dx2 2 dn j2 a 苷 r c dL1 dx 2 rL0 D . (11)

The homogeneous differential equation has Green function

G共x, x0兲 苷 2ja

sinh共x,兾ja兲 sinh共s 2 x.兾ja兲

sinhs , (12)

where we have defined s 苷 L兾jaand x,共x.兲 is the small-est (largsmall-est) of x and x0. (In the amplifying system ja is

imaginary so the hyperbolic functions become trigonomet-ric functions.) The inhomogeneous equations (10) and (11) have the solution

n共v, x兲 苷 rfeq

sinhs关sinhs 2 sinh共x兾ja兲 2 sinh共s 2 x兾ja兲兴

1 nin共v兲 sinh共s 2 x兾ja兲 sinhs , (13) dn共v, x, t兲 苷 rZ L 0 dx0G共x, x0兲 µ 1 c dL1 dx0 2 L0 D ∂ 1 dnin共v, t兲 sinh共s 2 x兾ja兲 sinhs . (14)

The flux density at the photodetector follows from Eq. (7)

at x苷 L, j共v, L兲 苷 Drfeq ja tanh共s兾2兲 1 Dnin jasinhs , (15) dj共v, L, t兲 苷 Ddnin jasinhs 1 Dr sinhs Z L 0 dx 3 µ sinh共x兾ja兲 L0 D 1 cosh共x兾ja兲 L1 cja ∂ . (16) [Notice that the extra term ~ L1in Eq. (7) is canceled by the delta function in ≠2G兾≠x≠x0.]

The time-averaged flux I 苷 Iin 1 Ith is the sum of two contributions, the transmitted incident flux Iin 苷

R`

0dv Dnin兾共jasinhs兲, and the thermal flux Ith 苷

R`

0dv共Drfeq兾ja兲 tanh共s兾2兲. The transmitted

in-cident flux per frequency interval is a fraction

T 苷 4D兾共cjasinhs兲 of the incident flux density j0苷 1

4cnin. A fraction R 苷 1 2 4D兾共cjatanhs兲 of the incident flux is reflected. The thermal flux per frequency interval is a fraction 1 2 T 2 R 苷 共4D兾cja兲 tanh共s兾2兲

of the blackbody flux density j0 苷 1

4crfeq. This is Kirchhoff’s law of thermal radiation.

The noise power P follows from the autocorrelators of L0 and L1 [given by Eq. (9), with f₀苷 n0兾r from Eq. (13)]. The autocorrelator of dninand the cross corre-lator of L0 and L1 contribute only to order 共l兾ja兲2 and

can therefore be neglected. The noise power P 苷 Pin 1

Pth 1 Pexis found to consist of three terms, given by

Pin 苷 Iin 1 Z ` 0 dv Dn 2 in 8rja 2s cosh共2s兲 1 sinh共2s兲 2 4s sinh4s , (17a) Pth 苷 Ith 1 Z ` 0 dvDrf 2 eq 4ja sinh2共s兾2兲

sinh4s 关8s 1 4s coshs 2 7 sinhs 2 4 sinh共2s兲 1 sinh共3s兲兴 , (17b) Pex 苷 Z ` 0 dvDfeqnin 2ja sinh2共s兾2兲

sinh4s 关26s 2 4s coshs 1 4 sinhs 1 3 sinh共2s兲兴 . (17c)

The two terms Pinand Pthdescribe separately the fluctu-ations in the transmitted incident flux and in the thermal flux. Both terms are greater than the Poisson noise (the mean photon flux Ith, Iin) as a consequence of photon bunching. The third term Pex is the excess noise which in a quantum optical formulation originates from the beat-ing of the incident radiation with vacuum fluctuations in the medium [16]. Here we find this excess noise from the semiclassical radiative transfer theory. The expressions for

Pth and Pexin Eq. (17) are the same as those that follow from the fully quantum optical treatment [11,17]. This is a crucial test of the validity of the semiclassical theory. The expression for Pinagrees with the quantum optical theory for the case that the incident radiation originates from a thermal source [18]. The case of coherent incident radia-tion is beyond the reach of radiative transfer theory.

We envisage a variety of applications for the Boltz-mann-Langevin equation for photons obtained in this pa-per. Although we have concentrated here on the waveguide

geometry, in order to be able to compare with results in the literature, the calculation of the noise power in the diffu-sion approximation can be readily generalized to arbitrary geometry. As an example, we give the noise power of the thermal radiation emitted by a sphere (per unit surface area), Pth苷 Ith1 Z ` 0 dv2Drf 2 eqs2 jasinh4s Z s 0 dz µ coshz 2 sinhz z ∂2 3 sinh 2_z z2 , (18)

where s苷 R兾ja is the ratio of the radius R of the sphere

and the absorption length ja. The mean thermal flux is

given by Ith 苷

R`

0dv Drfeqj21a 共coths 2 1兾s兲. The

re-sult for Ithcould have been obtained from the conventional radiative transfer theory using Kirchhoff’s law, but the re-sult for Pth could not.

VOLUME83, NUMBER26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER1999 A dimensionless measure of the magnitude of the

photon flux fluctuations is the Mandel parameter [19],

Q 苷 共P 2 I兲兾I. In a photocount experiment, counting n photons in a time t with unit quantum efficiency, the

Mandel parameter is obtained from the mean photo-count n and the variance varn in the long-time limit:

Q 苷 limt!`共varn 2 n兲兾n. We assume a

frequency-re-solved measurement, so that the integrals over frequency in Eqs. (17) and (18) can be omitted. The Mandel parameter for thermal radiation from a waveguide and a sphere is plotted in Fig. 2, as a function of s (s苷 L兾ja

for the waveguide and s 苷 R兾jafor the sphere). Both the

small- and large-s behavior of Q is geometry independent:

Q 苷 ₁₅2s2_f

eq for s ø 1 and Q 苷 1

2feq for s ¿1. The Bose-Einstein function feq共v, T兲 is to be evaluated at the detection frequency v and temperature T of the medium. The plot in Fig. 2 is for feq苷 1023, typical for optical frequencies at 3000 K.

Much larger Mandel parameters can be obtained in am-plifying systems, such as a random laser. Since com-plete population inversion corresponds to T ! 02, one has

feq 苷 21 in that case [12]. Equations (17) and (18) apply to amplified spontaneous emission below the laser thresh-old if one uses an imaginary ja. The absolute value jjaj is

the amplification length, and we denote s 苷 L兾jjaj for the

waveguide geometry and s 苷 R兾jjaj for the sphere. The

laser threshold occurs at s苷 p in both geometries. We have included in Fig. 2 the Mandel parameter for these two amplifying systems for the case of complete popu-lation inversion. Again the result is geometry indepen-dent for small s, Q 苷 ₁₅2s2j feqj for s ø 1. At the laser threshold 共s 苷 p兲 the Mandel parameter diverges in the theory considered here. An important extension for future work is to include the nonlinearities that become of crucial

FIG. 2. Mandel parameter Q 苷 共P 2 I 兲兾I for the thermal radiation from an absorbing medium and for the amplified spon-taneous emission from a medium with a complete population inversion. The solid curves are for the sphere geometry [Eq. (18)]; the dashed curves are for the waveguide geometry [Eq. (17b)]. The parameter s is the ratio of the radius of the sphere or of the length of the waveguide to the absorption or amplification length. The laser threshold in the amplifying case is at s 苷 p. To show both cases in one figure, the Q for the absorbing medium has been rescaled by a factor of 104

(corresponding to feq 苷 1023).

importance above the laser threshold. The simplicity of the radiative transfer theory developed here makes it a promis-ing tool for the exploration of the nonlinear regime in a random laser.

Since radiative transfer theory was originally developed for applications in astrophysics, we imagine that the exten-sion to fluctuations presented here could be useful in that context as well.

We acknowledge discussions with M. Patra. This work was supported by the Dutch Science Foundation NWO兾FOM.

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exp共 ¯hV兾kBT兲, where V is the transition frequency. A complete population inversion corresponds to T ! 02_.

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[18] According to the quantum optical theory of Ref. [11], the integral over frequency in the expression for Pincontains

an additional factor g 苷 具ayayaa典 具aya典22_{2 1, where a}

is the annihilation operator of the incident radiation. Equa-tion (17) corresponds to thermal radiaEqua-tion, when g苷 1, while g苷 0 for coherent radiation. The noise terms Pth

and Pex are independent of g.

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