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 fluctuations
in the photon flux, over and above the fluctuations 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.Comment: 4 pages, 2 figures, revised version according to referee's comment

B. B. Kadomtsev

C. H. Henry

C. W. J. Beenakker

D. Wiersma

E. G. Mishchenko

E. V. Sukhorukov

H. Cao

K. E. Nagaev

L. Mandel

M. C. W. van Rossum

M. J. M. de Jong

M. Patra

S. Chandrasekhar

S. V. Gantsevich

Sh. Kogan

Sh. M. Kogan

V. S. Letokhov

V. V. Sobolev

English

arXiv

Crossref

Radiative Transfer Theory for Vacuum Fluctuations

Journal ArticleA 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 fluctuations in the photon flux, over and above the fluctuations 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

Mishchenko, Eugene

Beenakker, C. W. J.

The University of Utah: J. Willard Marriott Digital Library

Radiative transfer theory for vacuum fluctuations

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Beenakker, C.W.J.

Mishchenko, E.G.

Leiden University Scholary Publications

VOLUME 83, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER 1999Radiative Transfer Theory for Vacuum FluctuationsE G Mishchenko1 2 and C W J Beenakker11]Instituut-Lorentz, Umversiteit Leiden, PO Box 9506 2300 RA Leiden, The Netherlands2L D Landau Institute for Theoretical Physics, Russian Academy of Sciences, Kosygin 2, Moscow 117334, Russia(Received 6 July 1999)A semiclassical kinetic theoiy is presented for the fluctuating photon flux emitted by a disorderedmedmm in thermal equilibnum The kinetic equation is the optical analog of the Boltzmann-Langevmequation foi electrons Vacuum fluctuations of the electromagneüc field provide a new source of fluc-tuations in the photon flux, over and above the fluctuations due to scattermg The kinetic theory inthe diffusion approximation is applied to the supei-Poissoman noise due to photon bunchmg and to theexcess noise due to beatmg of incident radiation with the vacuum fluctuationsPACS numbers 42 50 Ar, 05 40 -a, 42 68 Ay, 78 45 + hThe theoiy of radiative transfer was developed by Chan-drasekhai [1] and Sobolev [2] to descnbe the scatteimgand absoiption of electromagnetic radiation by interstel-lar mattei It has become widely used in the study of wavepropagation m random media, with apphcations m medicalimagmg and seisrruc exploration [3] The basic equation ofladiative transfei theory is a kinetic equation of the Boltz-mann type that is denved from the Maxwell equations byneglecting mterference effects [4] It is a lehable approxi-mation whenevei the scattermg and absorption lengths aielaige compaied to the wavelength, which apphes to all butthe most stiongly disordeied mediaRadiative transfer theory has so far been restncted toclassical waves, excludmg purely quantum mechanicaleffects of vacuum fluctuations This lirmtation is feitstrongly in connection with the recent activity on landomlasers [5] These are amplifymg Systems in which thefeedback is piovided by multiple scatteimg fiom disorderrather than by mirrois, so that ladiative transfei theoryis an appiopuate level of descnption However, whilestimulated emission has been mcoiporated mto thisapproach a long time ago by Letokhov [6], spontaneousemission has not It is the puipose of our woik to removethis lirmtation, by presentmg an extension of the ladiativetiansfer equation that mcludes vacuum fluctuations andthe associated spontaneous emission of radiationOur Inspiration came from the field of electromcconduction in disoidered metals, where the notion of afluctuating Boltzmann equation (or Boltzmann-Langevmequation) has been developed extensively [7-9], followmgthe ongmal pioposal by Kadomtsev [10] In that contextthe fluctuations originale from landom scatteimg andthey conseive the paiticle number This same class offluctuations exists also in the optical context consideredheie, but with a different corielatoi because of the diffei-ence between boson and feimion statistics In addition,the photons have a new class of fluctuations, withoutparticle conseivation, oiigmating from landom absoiptionand emission events Vacuum fluctuations are of thesecond class We will extend the radiative tiansfei theoryto include both classes of fluctuations To demonstratethe vahdity of our "Boltzmann-Langevm equation forphotons," we solve the problem of the excess noise fromvacuum fluctuations m a waveguide geometry, foi whichan mdependent solution is known [11] We then apply itto the unsolved problem of the thermal radiation from asphencal landom medmmThe basic quantity of the kinetic theory is the fluctuatingdistnbution function /k (r, t) of the number of photons perumt cell (27r)~3dkdr m phase space (For simphcity, weignoie the polanzation dependence ) Conventional radia-tive tiansfer theoiy deals with the mean /k (r), which weassume to be time mdependent It satisfies the Boltzmannequationck - yk/k(/)]- /k~(/) (i)[For ease of notation, we wnte Xk mstead of (2ττ)"3 / dk,and <5kq mstead of (27r)3<5(k — q) ] The left-hand sideis the convection term (with c the velocity of hght mthe medmm and k a umt vector in the direction of thewave numbei k) The nght-hand_side contams gamjmdloss terms due to scattermg, Jkk'(f) = Wk_k '/k/(l + /k)'due to amphfication, Ik(f) — Wk~(l + /k), and due toabsorption /k~(/) = w^f^ The scattermg täte wkk' =Wk'k is elastic and Symmetrie The absorption and am-phfication rates wjf are isotropic (dependent only on k =|k|) and lelated to each other by the requirement that theBöse-Ernstem functionis the equihbiium solution of Eq (1) (at fiequency ω =ck and temperatuie T) This requirement fixes the latiow k " / w k = βχρ(/ζω/Α:βΓ) The temperature T is positivefoi an absoibmg medmm and negative foi an amplifymgmedmm such äs a lasei [12]We now extend the ladiative transfei equation (1) to m-clude the fluctuations <5/ = / — / Followmg the hne ofargument that leads to the Boltzmann-Langevm equation0031-9007/99/83(26)/5475(4)$1500 © 1999 The American Physical Society 5475VOLUME 83, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER 1999for electrons [7-10], we propose the kinetic equation(3)4+(/) - /k (/) +The argument is that the fluctuating / is propagated,scattered, absorbed, and amplified in the same way ästhe mean /, hence the same convection term and thesame kernels Jkk', 1^ appear in Eqs. (1) and (3). Inaddition, Eq. (3) contains a stochastic source of photons,- δ/k-k) + <5/k+ - S/k (4)consisting of separate contributions from scattering,amplification,_and absorption. This Langevin term haszero mean, L k = 0, and a correlator that follows fromthe assumption that the elementary stochastic processes<5/|f have independent Poisson distributions:, t)SJqq,(r', t') (5a)(5b)SJkk'(r,t)SI^(r',t') = 0, 5/k+(r,i)5/-(r',f) = 0,(5c)where we have abbreviated Δ = δ (r — r')S(t — t').Substitution into Eq. (4) gives the correlator_£k(r,f)i:q(r',r') = Δ_ _ _ _+ A<k(/)] - -W/) - -/qk(/) + <Μ4 (/) + (6)Equations (3) and (6) constitute the Boltzmann-Langevinequation for photons.To gain more insight into this kinetic equation we makethe diffusion approximation valid if the mean free pathis the shortest length scale in the System (but still largecompared to the wavelength). The diffusion approxima-tion consists in an expansion with respect to k in spheri-cal harmonics, keeping only the first two terms: /k =/o + k · (i, £k = £0 + k · L i, where /0, f j , £0, andL i do not depend on the direction k of the wave vec-tor, but on its magnitude k = ω/c only. The two terms/o and f i determine, respectively, the photon number den-sity n = p/o and flux density j = jcpf i, where ρ (ω) =4ττω2(2ττ€)~3 is the density of states. Integration ofEq. (3) gives two relations between n and j ,dr- n) + p£Q,(7)(8)where the diffusion constant D = $c2T and mean freepath / = er are determined by the transport scattering rater"1 = Xk/ wkk '(l — k · k')· The absorption length ξα isdefmed by Όξ~2 = w~~ — w + . (An amplifying mediumhas an imaginary ξα and a negative /eq.) In Eq. (7) wehave neglected terms of order (//^a)2, which are assumedto be -«1.Both Eqs. (7) and (8) contain a fluctuating source term.These two terms LQ and L \ have zero mean and correla-tors that follow from Eq. (6),£0(ω,ΐ,ί)£0(ω',τ',ί') =D(2/eq/Q + /eq + /θ) ,(9a)6c^p/+f0),£0(<»,r,D',r',i') = A'—^(2/e,/, + Λ ) ,(9c)where we have abbreviated Δ' = δ (ω — ω') δ (t —t') δ (r — r')· The correlator (9b) differs from the elec-tronic case [13-15] by the factor l + /0 instead ofl — /o· This is the expected difference between bosonand fermion statistics. The correlators (9a) and (9c) haveno electronic counterpart. They describe the statistics ofthe vacuum fluctuations.To demonstrate how the kinetic theory presented aboveworks in a specific Situation we consider the propaga-tion through an absorbing or amplifying disordered wave-guide (length L). The incident radiation is isotropic. Alltransmitted radiation is absorbed by a photodetector (seeFig. 1). Because of the one dimensionality of the ge-ometry we need to consider only the χ dependence of jand n (we assume a unit cross-sectional area). The trans-mitted photon flux / = /Q d ω j ( a ) , L, t) fluctuates aroundits time-averaged value, I(t) = 7 + 8I(t). The (zero-fre-quency) noise power P = /Ü^ dt 8Ι(ί)δΙ(0) is the corre-lator of the fluctuating flux. We will compute P by solvingthe differential equations (7) and (8) with boundary con-ditions η(ω,Ο,ί) — η·ιη(ω,ί), n(a>,L,t) = 0, dictated bythe incident radiation at one end of the waveguide and bythe absorbing photodetector at the other end.D(9b)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 aphotodetector.5476VOLUME 83, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER 1999Combining Eqs. (7) and (8) we find equations for themean and the fluctuations of the photon number,at χ = L,d2nί/2δ« δηdx2 ~ T2p_d£ic dx D (H)](ω,1) =8j(a>,L,t) =Sa• tanh(s/ +Dnmξa smhs(15)The homogeneous differential equation has Green function£a) smh(.s - χ > / ξ α )ξα smhsX+ DPsmhsfJo= -ξα smhs(12)where we have defmed s = L/'ξα and x< (x>) is the small-est (largest) of χ and x'. (In the amphfymg System ξα isimagmary so the hyperbohc functions become tngonomet-nc functions.) The mhomogeneous equations (10) and (11)have the solutionη(ω,χ) = eq [smhs —,smhiη,η(ω)smh(j -smhs(13)δη(ω,χ,ή =£oD+ δη ιη(ω,ί)- */£,)smhs(14)The flux density at the photodetector follows from Eq. (7)(16)[Notice that the extra term <* £\ m Eq. (7) is canceled bythe delta function m d2G/dxdx'.]The time-averaged flux 7 = 7m + 7 th is the sum oftwo contnbutions, the tiansrrutted mcident flux 7m =/ο άω Dnm/(£a smhs), and the thermal flux 7th =/ο άω (Z)p/eq/^a)tanh(5/2). The transmitted m-cident flux per frequency mterval is a fractionT = 4D/(c£a smlu) of the mcident flux density 70 ~4C«m . A fraction R = l — 4£>/(c£a tanhs) of themcident flux is reflected The thermal flux per frequencymterval is a fraction l - T - R = (4D/c^a)tanh(s/2)of the blackbody flux density jo = jcp/eq. This isKirchhoff's law of thermal radiation.The noise power P follows from the autocorrelatorsof £Q and L\ [given by Eq. (9), with /0 = n0/p fromEq. (13)]. The autocorrelator of 8nm and the cross corie-lator of £Q and L\ contnbute only to order (l/ξα)2 andcan therefore be neglected. The noise power P = Pm +PÜ\ + P&K is found to consist of three terms, given by= /ΛPex =r co j-.—/^m = /m + / άω - —JO οΡξαΓ ΰρ/eq smh2(s/2)2scosh(2j·) + smh(2s) - 4s(17a)[8s + 4s coshs - 7smhs - 4smh(2s) + smh(3s)], (17b)άωin», smh2(s/2)ra smh4s[-6s — 4s coshs + 4smhs + 3smh(2s)]. (17c)The two terms Pm and Pth describe separately the fluctu-ations m the tiansmitted mcident flux and in the thermalflux. Both terms are greater than the Poisson noise (themean photon flux 7 th, 7m) äs a consequence of photonbunchmg. The third term Pex is the excess noise whichin a quantum optical formulation originales from the beat-mg of the mcident radiation with vacuum fluctuations mthe medmm [16]. Here we find this excess noise fiom theserruclassical radiative transfer theory. The expressions forPth and Pex in Eq. (17) are the same äs those that followfrom the fully quantum optical treatment [l 1,17]. This is aciucial test of the validity of the semiclassical theory. Theexpression for Pm agrees with the quantum optical theoryfor the case that the mcident radiation ongmates fiom athermal source [18]. The case of coherent mcident radia-tion is beyond the reach of radiative transfer theory.We envisage a vanety of applications foi the Boltz-mann-Langevin equation foi photons obtamed in this pa-per. Although we have concentrated here on the waveguidegeometry, in oider to be able to compare with results in thehterature, the calculation of the noise power m the diffu-sion approximation can be readily generahzed to arbitrarygeometry. As an example, we give the noise power ofthe thermal radiation emitted by a sphere (per umt surfacearea),Pth = / thXdu)2/W,2 s2eq^4ξ a sinn,Γ / smh z\2I dz coshzJo V z l(18)where s = R/'ξα is the ratio of the radius R of the sphereand the absorption length ξα. The mean thermal flux isgiven by 7th = /ö άω Dpf&^~l(coths - l/s). The le-sult for 7th could have been obtamed from the conventionalradiative transfer theory usmg Kirchhoff's law, but the re-sult for Pth could not.5477VOLUME 83, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER 1999A dimensionless measure of the magnitude of thephoton flux^fluctuations is the Mandel paiameter [19],g = (P — /)// In a photocount expenment, countingn photons in a time i with umt quantum efficiency, theMandel paiameter is obtamed fiom the mean photo-count n and the vanance vam m the long-time limit<2 = hmr_oo(vam — ~n)/~n We assume a frequency-ie-solved measurement, so that the Integrals over frequencym Eqs (17) and (18) can be omitted The Mandelparameter for thermal radiation from a waveguide and asphere is plotted m Fig 2, äs a function of s (s = L/ξαfoi the waveguide and s = Κ/ξα foi the sphere) Both thesmall- and laige-s behavior of Q is geometry independentQ = j§s2feq for s «. l and Q = ^feq for s » l TheBöse-Ernstem function /eq(w, T) is to be evaluated at thedetection frequency ω and temperature T of the mediumThe plot in Fig 2 is for /eq = 10~3, typical for opticalfrequencies at 3000 KMuch larger Mandel parameters can be obtamed in am-phfymg Systems, such äs a random laser Since com-plete population Inversion corresponds to T —> 0~, one has/eq = -1 in that case [12] Equaüons (17) and (18) applyto amphfied spontaneous emission below the lasei thresh-old if one uses an imagmary ξα The absolute value | ξα \ isthe amphfication length, and we denote s = L/\£a\ for thewaveguide geometry and s = Κ/\ξα\ for the sphere Thelaser threshold occurs at s = π in both geometnes Wehave mcluded in Fig 2 the Mandel parameter for thesetwo amphfying Systems for the case of complete popu-lation Inversion Agam the result is geometry indepen-dent for small s, Q = -rj.y2|/eql for s <5C l At the laserthreshold (s = π) the Mandel paiametei diverges in thetheory considered here An important extension for futurework is to mclude the nonhneanties that become of crucialQ0FIG 2 Mandel parametei β = (P - /)// for the thermalradiation from an absorbmg medium and for the amphfied spon-taneous emission from a medium with a complete populationInversion 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 latio of the ladms of thesphere or of the length of the waveguide to the absorpüon oramphfication length The laser threshold in the amphfyingcase is at s = π Το show both cases in one figure, the Q forthe absorbmg medium has been rescaled by a factor of 104(corresponding to /eq = 10~3)impoitance above the laser threshold The simplicity of theladiative tiansfei theory developed here makes it a promis-ing tool for the exploiation of the nonlmeai regime m arandom lasetSince radiative transfer theory was ongmally developedfoi apphcations in astiophysics, we imagine that the exten-sion to fluctuations piesented here could be useful in thatcontext äs wellWe acknowledge discussions with M Patia Thiswoik was suppoited by the Dutch Science FoundationNWO/FOM[1] S Chandrasekhai, Radiative Transfer (Dover, New York1960)[2] V V Sobolev, A Treatise on Radiative Transfer (Van Nostiand, Punceton, 1963)[3] Diffuse Waves m Complex Media, edited by J P Fouque,NATO ASI Ser C531 (Kluwer, Doidrecht, 1999)[4] M C W van Rossum and T M Nieuwenhuizen, Rev ModPhys 71, 313 (1999)[5] H Cao, Υ G Zhao, S T Ho, E W Seelig, Q H Wang, andR P H Chang, Phys Rev Lett 82, 2278 (1999) Earlierwork on random lasers has been reviewed by D Wieismaand A Lagendijk, Phys World l, 33 (1997)[6] V S Letokhov, Zh Eksp Teor Fiz 53, 1442 (1967) [SovPhys JETP 26, 835 (1968)][7] Sh M Kogan and A Ya Shul'man, Zh Eksp Teor Fiz56, 862 (1969) [Sov Phys JETP 29, 467 (1969)][8] S V Gantsevich, V L Guievich, and R Kaühus, RivNuovo Cimento 2(5), l (1979)[9] Sh Kogan, Electronic Noise and Fluctuations m Sohds(Cambridge Umversity Press, Cambridge, 1996)[10] B B Kadomtsev, Zh Eksp Teor Fiz 32, 943 (1957) [SovPhys JETP 5, 771 (1957)][11] M Patra and C W J Beenakker, Phys Rev A 60, 4059(1999)[12] The negative temperature of an amphfying medium isdetermmed by the relative population of the uppei andlower levels of the atormc transition, Niower/Nuppa =e\p(nü,/kBT), where Ω is the tiansition frequency Acomplete population Inversion corresponds to T —* 0~[13] K E Nagaev, Phys Lett A 169, 103 (1992)[14] M J M de Jong and C W J Beenakker, Phys Rev B 51,16867 (1995)[15] E V Sukhorukov and D Loss, Phys Rev B 59, 13054(1999)[16] C H Henry and R F Kazarmov, Rev Mod Phys 68, 801(1996)[17] C W J Beenakker, in Diffuse Waves m Complex Media(Ref [3])[18] According to the quantum optical theory of Ref [11], theintegial over frequency m the expression for P,n contamsan additional factoi g = (a^ a^ aa) {a^ a)'2 — l, where αis the annihilation operatoi of the incident radiation Equa-tion (17) corresponds to thermal radiation, when g = l,while g = 0 for coherent radiation The noise terms PH,and Pex aie independent of g[19] L Mandel and E Wolf, Optical Coherence and QuantumOptics (Cambudge Umversity Press, New Yoik, 1995)5478

NARCIS 

VOLUME 83, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER 1999Radiative Transfer Theory for Vacuum FluctuationsE. G. Mishchenko1,2 and C. W. J. Beenakker11Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands2L. 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 disorderedmedium in thermal equilibrium. The kinetic equation is the optical analog of the Boltzmann-Langevinequation for electrons. Vacuum fluctuations of the electromagnetic field provide a new source of fluc-tuations in the photon flux, over and above the fluctuations due to scattering. The kinetic theory inthe diffusion approximation is applied to the super-Poissonian noise due to photon bunching and to theexcess noise due to beating of incident radiation with the vacuum fluctuations.PACS numbers: 42.50.Ar, 05.40.–a, 42.68.Ay, 78.45.+hThe theory of radiative transfer was developed by Chan-drasekhar [1] and Sobolev [2] to describe the scatteringand absorption of electromagnetic radiation by interstel-lar matter. It has become widely used in the study of wavepropagation in random media, with applications in medicalimaging and seismic exploration [3]. The basic equation ofradiative transfer theory is a kinetic equation of the Boltz-mann type that is derived from the Maxwell equations byneglecting interference effects [4]. It is a reliable approxi-mation whenever the scattering and absorption lengths arelarge compared to the wavelength, which applies to all butthe most strongly disordered media.Radiative transfer theory has so far been restricted toclassical waves, excluding purely quantum mechanicaleffects of vacuum fluctuations. This limitation is feltstrongly in connection with the recent activity on randomlasers [5]. These are amplifying systems in which thefeedback is provided by multiple scattering from disorderrather than by mirrors, so that radiative transfer theoryis an appropriate level of description. However, whilestimulated emission has been incorporated into thisapproach a long time ago by Letokhov [6], spontaneousemission has not. It is the purpose of our work to removethis limitation, by presenting an extension of the radiativetransfer equation that includes vacuum fluctuations andthe associated spontaneous emission of radiation.Our inspiration came from the field of electronicconduction in disordered metals, where the notion of afluctuating Boltzmann equation (or Boltzmann-Langevinequation) has been developed extensively [7–9], followingthe original proposal by Kadomtsev [10]. In that contextthe fluctuations originate from random scattering andthey conserve the particle number. This same class offluctuations exists also in the optical context consideredhere, 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, withoutparticle conservation, originating from random absorptionand emission events. Vacuum fluctuations are of thesecond class. We will extend the radiative transfer theory0031-90079983(26)5475(4)$15.00to include both classes of fluctuations. To demonstratethe validity of our “Boltzmann-Langevin equation forphotons,” we solve the problem of the excess noise fromvacuum fluctuations in a waveguide geometry, for whichan independent solution is known [11]. We then apply itto the unsolved problem of the thermal radiation from aspherical random medium.The basic quantity of the kinetic theory is the fluctuatingdistribution function fkr, t of the number of photons perunit cell 2p23dkdr in phase space. (For simplicity, weignore the polarization dependence.) Conventional radia-tive transfer theory deals with the mean fkr, which weassume to be time independent. It satisfies the Boltzmannequationck̂ ?≠fk≠rXk0Jkk0 f  2 Jk0k f  1 I1k  f 2 I2k  f  . (1)[For ease of notation, we writePk instead of 2p23Rdk,and dkq instead of 2p3dk 2 q.] The left-hand sideis the convection term (with c the velocity of light inthe medium and k̂ a unit vector in the direction of thewave number k). The right-hand side contains gain andloss terms due to scattering, Jkk0 f   wkk0fk01 1 fk,due to amplification, I1k  f   w1k 1 1 fk, and due toabsorption I2k  f   w2k fk. The scattering rate wkk0 wk0k is elastic and symmetric. The absorption and am-plification rates w6k are isotropic (dependent only on k jkj) and related to each other by the requirement that theBose-Einstein functionfeqv, T   exph̄vkBT  2 121 (2)is the equilibrium solution of Eq. (1) (at frequency v ck and temperature T ). This requirement fixes the ratiow2k w1k  exph̄vkBT . The temperature T is positivefor an absorbing medium and negative for an amplifyingmedium such as a laser [12].We now extend the radiative transfer equation (1) to in-clude the fluctuations df  f 2 f. Following the line ofargument that leads to the Boltzmann-Langevin equation© 1999 The American Physical Society 5475VOLUME 83, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER 1999for electrons [7–10], we propose the kinetic equationck̂ ?≠fk≠rXk0Jkk0 f 2 Jk0k f1 I1k  f 2 I2k  f 1 Lk . (3)The argument is that the fluctuating f is propagated,scattered, absorbed, and amplified in the same way asthe mean f, hence the same convection term and thesame kernels Jkk0 , I6k appear in Eqs. (1) and (3). Inaddition, Eq. (3) contains a stochastic source of photons,Lk Xk0dJkk0 2 dJk0k 1 dI1k 2 dI2k , (4)5476consisting of separate contributions from scattering,amplification, and absorption. This Langevin term haszero mean, L k  0, and a correlator that follows fromthe assumption that the elementary stochastic processesdJkk0, dI6k have independent Poisson distributions:dJkk0r, tdJqq0r0, t0  Ddkqdk0q0Jkk0 f  , (5a)dI6k r, tdI6q r0, t0  DdkqI6k  f  , (5b)dJkk0r, tdI6q r0, t0  0, dI1k r, tdI2q r0, t0  0 ,(5c)where we have abbreviated D  dr 2 r0dt 2 t0.Substitution into Eq. (4) gives the correlatorLkr, tLqr0, t0  D"dkqXk0Jkk0 f  1 Jk0k f  2 Jkq f  2 Jqk f  1 dkqI1k  f  1 I2k  f #. (6)Equations (3) and (6) constitute the Boltzmann-Langevinequation for photons.To gain more insight into this kinetic equation we makethe diffusion approximation valid if the mean free pathis the shortest length scale in the system (but still largecompared to the wavelength). The diffusion approxima-tion consists in an expansion with respect to k̂ in spheri-cal harmonics, keeping only the first two terms: fk f0 1 k̂ ? f1, Lk  L0 1 k̂ ? L 1, where f0, f1, L0, andL 1 do not depend on the direction k̂ of the wave vec-tor, but on its magnitude k  vc only. The two termsf0 and f1 determine, respectively, the photon number den-sity n  rf0 and flux density j 13crf1, where rv 4pv22pc23 is the density of states. Integration ofEq. (3) gives two relations between n and j,j  2D≠n≠r113lrL 1 , (7)≠≠r? j  Dj22a rfeq 2 n 1 rL0 , (8)where the diffusion constant D  13c2t and mean freepath l  ct are determined by the transport scattering ratet21 Pk0 wkk01 2 k̂ ? k̂0. The absorption length ja isdefined by Dj22a  w2 2 w1. (An amplifying mediumhas an imaginary ja and a negative feq.) In Eq. (7) wehave neglected terms of order lja2, which are assumedto be ø1.Both Eqs. (7) and (8) contain a fluctuating source term.These two terms L0 and L 1 have zero mean and correla-tors that follow from Eq. (6),L0v, r, tL0v0, r0, t0  D0Drj2a2feqf0 1 feq 1 f0 ,(9a)L 1v, r, tL 1v0, r0, t0  'D06crlf01 1 f0 ,(9b)L0v, r, tL1v0, r0, t0  D0Drj2a2feqf1 1 f1 ,(9c)where we have abbreviated D0  dv 2 v0dt 2t0dr 2 r0. The correlator (9b) differs from the elec-tronic case [13–15] by the factor 1 1 f0 instead of1 2 f0. This is the expected difference between bosonand fermion statistics. The correlators (9a) and (9c) haveno electronic counterpart. They describe the statistics ofthe vacuum fluctuations.To demonstrate how the kinetic theory presented aboveworks in a specific situation we consider the propaga-tion through an absorbing or amplifying disordered wave-guide (length L). The incident radiation is isotropic. Alltransmitted radiation is absorbed by a photodetector (seeFig. 1). Because of the one dimensionality of the ge-ometry we need to consider only the x dependence of jand n (we assume a unit cross-sectional area). The trans-mitted photon flux I R`0 dv jv, L, t fluctuates aroundits time-averaged value, It  I 1 dIt. The (zero-fre-quency) noise power P R`2` dt dItdI0 is the corre-lator of the fluctuating flux. We will compute P by solvingthe differential equations (7) and (8) with boundary con-ditions nv, 0, t  ninv, t, nv, L, t  0, dictated bythe incident radiation at one end of the waveguide and bythe 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 aphotodetector.VOLUME 83, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER 1999Combining Eqs. (7) and (8) we find equations for themean and the fluctuations of the photon number,d2ndx22nj2a 2rfeqj2a, (10)d2dndx22dnj2arcdL1dx2rL0D. (11)The homogeneous differential equation has Green functionGx, x0  2jasinhx,ja sinhs 2 x.jasinhs, (12)where we have defined s  Lja and x, x. is the small-est (largest) of x and x0. (In the amplifying system ja isimaginary so the hyperbolic functions become trigonomet-ric functions.) The inhomogeneous equations (10) and (11)have the solutionnv, x rfeqsinhssinhs 2 sinhxja 2 sinhs 2 xja1 ninvsinhs 2 xjasinhs, (13)dnv, x, t  rZ L0dx0 Gx, x0µ1cdL1dx02L0D∂1 dninv, tsinhs 2 xjasinhs. (14)The flux density at the photodetector follows from Eq. (7)at x  L,jv, L Drfeqjatanhs2 1Dninja sinhs, (15)djv, L, t Ddninja sinhs1DrsinhsZ L0dx3µsinhxjaL0D1 coshxjaL1cja∂.(16)[Notice that the extra term ~ L1 in Eq. (7) is canceled bythe delta function in ≠2G≠x≠x0.]The time-averaged flux I  Iin 1 I th is the sum oftwo contributions, the transmitted incident flux Iin R`0 dv Dninja sinhs, and the thermal flux I th R`0 dv Drfeqja tanhs2. The transmitted in-cident flux per frequency interval is a fractionT  4Dcja sinhs of the incident flux density j0 14cnin. A fraction R  1 2 4Dcja tanhs of theincident flux is reflected. The thermal flux per frequencyinterval is a fraction 1 2 T 2 R  4Dcja tanhs2of the blackbody flux density j0 14crfeq. This isKirchhoff’s law of thermal radiation.The noise power P follows from the autocorrelatorsof L0 and L1 [given by Eq. (9), with f0  n0r fromEq. (13)]. The autocorrelator of dnin and the cross corre-lator of L0 and L1 contribute only to order lja2 andcan therefore be neglected. The noise power P  Pin 1Pth 1 Pex is found to consist of three terms, given byPin  Iin 1Z `0dvDn2in8rja2s cosh2s 1 sinh2s 2 4ssinh4s, (17a)Pth  I th 1Z `0dvDrf2eq4jasinh2s2sinh4s8s 1 4s coshs 2 7 sinhs 2 4 sinh2s 1 sinh3s , (17b)Pex Z `0dvDfeqnin2jasinh2s2sinh4s26s 2 4s coshs 1 4 sinhs 1 3 sinh2s . (17c)The two terms Pin and Pth describe separately the fluctu-ations in the transmitted incident flux and in the thermalflux. Both terms are greater than the Poisson noise (themean photon flux I th, Iin) as a consequence of photonbunching. The third term Pex is the excess noise whichin a quantum optical formulation originates from the beat-ing of the incident radiation with vacuum fluctuations inthe medium [16]. Here we find this excess noise from thesemiclassical radiative transfer theory. The expressions forPth and Pex in Eq. (17) are the same as those that followfrom the fully quantum optical treatment [11,17]. This is acrucial test of the validity of the semiclassical theory. Theexpression for Pin agrees with the quantum optical theoryfor the case that the incident radiation originates from athermal 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 waveguidegeometry, in order to be able to compare with results in theliterature, the calculation of the noise power in the diffu-sion approximation can be readily generalized to arbitrarygeometry. As an example, we give the noise power ofthe thermal radiation emitted by a sphere (per unit surfacearea),Pth  Ith 1Z `0dv2Drf2eqs2ja sinh4sZ s0dzµcoshz 2sinhzz∂23sinh2zz2, (18)where s  Rja is the ratio of the radius R of the sphereand the absorption length ja. The mean thermal flux isgiven by Ith R`0 dv Drfeqj21a coths 2 1s. The re-sult for I th could have been obtained from the conventionalradiative transfer theory using Kirchhoff’s law, but the re-sult for Pth could not.5477VOLUME 83, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER 1999A dimensionless measure of the magnitude of thephoton flux fluctuations is the Mandel parameter [19],Q  P 2 I I . In a photocount experiment, countingn photons in a time t with unit quantum efficiency, theMandel parameter is obtained from the mean photo-count n and the variance varn in the long-time limit:Q  limt!`varn 2 nn. We assume a frequency-re-solved measurement, so that the integrals over frequencyin Eqs. (17) and (18) can be omitted. The Mandelparameter for thermal radiation from a waveguide and asphere is plotted in Fig. 2, as a function of s (s  Ljafor the waveguide and s  Rja for the sphere). Both thesmall- and large-s behavior of Q is geometry independent:Q 215s2feq for s ø 1 and Q 12feq for s ¿ 1. TheBose-Einstein function feqv, T  is to be evaluated at thedetection frequency v and temperature T of the medium.The plot in Fig. 2 is for feq  1023, typical for opticalfrequencies 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 hasfeq  21 in that case [12]. Equations (17) and (18) applyto amplified spontaneous emission below the laser thresh-old if one uses an imaginary ja. The absolute value jjaj isthe amplification length, and we denote s  Ljjaj for thewaveguide geometry and s  Rjjaj for the sphere. Thelaser threshold occurs at s  p in both geometries. Wehave included in Fig. 2 the Mandel parameter for thesetwo amplifying systems for the case of complete popu-lation inversion. Again the result is geometry indepen-dent for small s, Q  215s2j feqj for s ø 1. At the laserthreshold s  p the Mandel parameter diverges in thetheory considered here. An important extension for futurework is to include the nonlinearities that become of crucialFIG. 2. Mandel parameter Q  P 2 I I for the thermalradiation from an absorbing medium and for the amplified spon-taneous emission from a medium with a complete populationinversion. 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 thesphere or of the length of the waveguide to the absorption oramplification length. The laser threshold in the amplifyingcase is at s  p. To show both cases in one figure, the Q forthe absorbing medium has been rescaled by a factor of 104(corresponding to feq  1023).5478importance above the laser threshold. The simplicity of theradiative transfer theory developed here makes it a promis-ing tool for the exploration of the nonlinear regime in arandom laser.Since radiative transfer theory was originally developedfor applications in astrophysics, we imagine that the exten-sion to fluctuations presented here could be useful in thatcontext as well.We acknowledge discussions with M. Patra. Thiswork was supported by the Dutch Science FoundationNWOFOM.[1] S. Chandrasekhar, Radiative Transfer (Dover, New York,1960).[2] V. V. Sobolev, A Treatise on Radiative Transfer (Van Nos-trand, Princeton, 1963).[3] Diffuse Waves in Complex Media, edited by J.-P. Fouque,NATO ASI Ser. C531 (Kluwer, Dordrecht, 1999).[4] M. C. W. van Rossum and T. M. Nieuwenhuizen, Rev. Mod.Phys. 71, 313 (1999).[5] H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, andR. P. H. Chang, Phys. Rev. Lett. 82, 2278 (1999). Earlierwork on random lasers has been reviewed by D. Wiersmaand A. Lagendijk, Phys. World 1, 33 (1997).[6] V. S. Letokhov, Zh. Eksp. Teor. Fiz. 53, 1442 (1967) [Sov.Phys. JETP 26, 835 (1968)].[7] Sh. M. Kogan and A. Ya. Shul’man, Zh. Eksp. Teor. Fiz.56, 862 (1969) [Sov. Phys. JETP 29, 467 (1969)].[8] S. V. Gantsevich, V. L. Gurevich, and R. Katilius, Riv.Nuovo Cimento 2(5), 1 (1979).[9] Sh. Kogan, Electronic Noise and Fluctuations in Solids(Cambridge University Press, Cambridge, 1996).[10] B. B. Kadomtsev, Zh. Eksp. Teor. Fiz. 32, 943 (1957) [Sov.Phys. JETP 5, 771 (1957)].[11] M. Patra and C. W. J. Beenakker, Phys. Rev. A 60, 4059(1999).[12] The negative temperature of an amplifying medium isdetermined by the relative population of the upper andlower levels of the atomic transition, NlowerNupper exph̄VkBT, where V is the transition frequency. Acomplete population inversion corresponds to T ! 02.[13] K. E. Nagaev, Phys. Lett. A 169, 103 (1992).[14] M. J. M. de Jong and C. W. J. Beenakker, Phys. Rev. B 51,16 867 (1995).[15] E. V. Sukhorukov and D. Loss, Phys. Rev. B 59, 13 054(1999).[16] C. H. Henry and R. F. Kazarinov, Rev. Mod. Phys. 68, 801(1996).[17] C. W. J. Beenakker, in Diffuse Waves in Complex Media(Ref. [3]).[18] According to the quantum optical theory of Ref. [11], theintegral over frequency in the expression for Pin containsan additional factor g  ayayaa aya22 2 1, where ais the annihilation operator of the incident radiation. Equa-tion (17) corresponds to thermal radiation, when g  1,while g  0 for coherent radiation. The noise terms Pthand Pex are independent of g.[19] L. Mandel and E. Wolf, Optical Coherence and QuantumOptics (Cambridge University Press, New York, 1995).

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Radiative transfer theory for vacuum fluctuations

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