Excess noise for coherent radiation propagating through amplifying random media

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Excess noise for coherent radiation propagating through amplifying random media

M. Patra and C. W. J. Beenakker

Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands 共Received 27 January 1999; revised manuscript received 18 May 1999兲

A general theory is presented for the photodetection statistics of coherent radiation that has been amplified by a disordered medium. The beating of the coherent radiation with the spontaneous emission increases the noise above the shot-noise level. The excess noise is expressed in terms of the transmission and reflection matrices of the medium, and evaluated using the methods of random-matrix theory. Intermode scattering between N propagating modes increases the noise figure by up to a factor of N, as one approaches the laser threshold. Results are contrasted with those for an absorbing medium.关S1050-2947共99兲02411-7兴

PACS number共s兲: 42.50.Ar, 42.25.Bs, 42.25.Kb, 42.50.Lc

I. INTRODUCTION

The coherent radiation emitted by a laser has a noise spec-tral density P equal to the time-averaged photocurrent I¯. This noise is called photon shot noise, by analogy with electronic shot noise in vacuum tubes. If the radiation is passed through an amplifying medium, P increases more than I¯ because of the excess noise due to spontaneous emission 关1兴. For an ideal linear amplifier, the共squared兲 signal-to-noise ratio I¯2/ P drops by a factor of 2 as one increases the gain. One says that the amplifier has a noise figure of 2. This is a lower bound on the excess noise for a linear amplifier关2兴.

Most calculations of the excess noise assume that the am-plification occurs in a single propagating mode. 共Recent ex-amples include work by Loudon and his group 关3,4兴.兲 The minimal noise figure of 2 refers to this case. Generalization to amplification in a multimode waveguide is straightforward if there is no scattering between the modes. The recent inter-est in amplifying random media关5兴 calls for an extension of the theory of excess noise to include intermode scattering. Here we present such an extension.

Our central result is an expression for the probability dis-tribution of the photocount in terms of the transmission and reflection matrices t and r of the multimode waveguide.共The noise power P is determined by the variance of this distribu-tion.兲 Single-mode results in the literature are recovered for scalar t and r. In the absence of any incident radiation, our expression reduces to the known photocount distribution for amplified spontaneous emission 关6兴. We find that intermode scattering strongly increases the excess noise, resulting in a noise figure that is much larger than 2.

We present explicit calculations for two types of geom-etries, waveguide and cavity, distinguishing between photo-detection in transmission and in reflection. We also discuss the parallel with absorbing media. We use the method of random-matrix theory关7兴 to obtain the required information on the statistical properties of the transmission and reflection matrices of an ensemble of random media. Simple analytical results follow if the number of modes N is large 共i.e., for high-dimensional matrices兲. Close to the laser threshold, the noise figure F exhibits large sample-to-sample fluctuations, such that the ensemble average diverges. We compute for arbitrary N⭓2 the distribution p(F) of F in the ensemble of

disordered cavities, and show thatF⫽N is the most probable value. This is the generalization to multimode random media of the single-mode resultF⫽2 in the literature.

II. FORMULATION OF THE PROBLEM

We consider an amplifying disordered medium embedded in a waveguide that supports N(␻) propagating modes at frequency␻ 共see Fig. 1兲. The amplification could be due to stimulated emission by an inverted atomic population or to stimulated Raman scattering 关1兴. A negative temperature T ⬍0 describes the degree of population inversion in the first case or the density of the material excitation in the second case关3兴. A complete population inversion or vanishing den-sity corresponds to the limit T→0 from below. The minimal noise figure mentioned in the Introduction is reached in this limit. The amplification rate 1/␶ais obtained from the

共nega-tive兲 imaginary part ⑀

⬙

of the 共relative兲 dielectric constant, 1/␶a⫽␻兩⑀

⬙

兩. Disorder causes multiple scattering with rate

1/␶sand共transport兲 mean free path l⫽c␶s 共with c the

veloc-ity of light in the medium兲. We assume that␶s and ␶a are

bothⰇ1/␻, so that scattering as well as amplification occur on length scales large compared to the wavelength. The waveguide is illuminated from one end by monochromatic radiation共frequency␻0, mean photocurrent I0) in a coherent

state. For simplicity, we assume that the illumination is in a single propagating mode 共labeled m0). At the other end of

the waveguide, a photodetector detects the outcoming radia-tion. We assume, again for simplicity, that all N outgoing modes are detected with equal efficiency ␣. The case of single-mode detection is considered in Appendix A.

We denote by p(n) the probability to count n photons within a time ␶. Its first two moments determine the mean photocurrent I¯ and the noise power P, according to

I ¯_⫽1

␶¯ ,n P⫽ lim_␶→⬁ 1

␶共n2⫺n¯2兲. 共2.1兲

FIG. 1. Coherent light共thick arrow兲 is incident on an amplifying medium共shaded兲, embedded in a waveguide. The transmitted radia-tion is measured by a photodetector.

PRA 60

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关The definition of P is equivalent to P⫽兰_⫺⬁⬁ dt␦I(0)␦I(t), with␦I⫽I⫺I¯ the fluctuating part of the photocurrent.兴 It is convenient to compute the generating function F(␰) for the factorial cumulants ␬j, defined by

F共␰兲⫽

兺

j⫽1 ⬁ _␬ j␰ j j ! ⫽ln

冉

n

兺

⫽0 ⬁ 共1⫹␰兲n_p_共n兲

冊

_. _共2.2兲 One has n¯⫽␬1, n2⫽␬2⫹␬1(1⫹␬1).

The outgoing radiation in mode n is described by an an-nihilation operator a_nout(␻), using the convention that modes 1,2, . . . ,N are on the left-hand side of the medium and modes N⫹1, . . . ,2N are on the right-hand side. The vector aout consists of the operators a1

out ,a2 out , . . . ,a2N out . Similarly, we define a vector ainfor incoming radiation. These two sets of operators each satisfy the bosonic commutation relations

t

⬘

t r

冊

. 共2.5兲

Reciprocity imposes the conditions t

⬘

⫽tT, r⫽rT, and r

⬘

⫽r

⬘

T_.

The operators c account for spontaneous emission in the amplifying medium. They satisfy the bosonic commutations relation共2.3兲, which implies that

VV†⫽SS†⫺1. 共2.6兲

Their expectation values are

具

cn共␻兲cm

†_共_␻

_⬘

_兲

_典

_⫽⫺_␦

nm␦共␻⫺␻

⬘

兲f 共␻,T兲, 共2.7兲

with the Bose-Einstein function

f共␻,T兲⫽关exp共ប␻/kT兲⫺1兴⫺1 共2.8兲 evaluated at negative temperature T (⬍0).

III. CALCULATION OF THE GENERATING FUNCTION The probability p(n) that n photons are counted in a time ␶ is given by关10,11兴

p共n兲⫽ 1 n!

具

:W

n_e⫺W_:

_典

_, _共3.1兲

where the colons denote normal ordering with respect to aout, and W⫽␣

冕

0 ␶ dt

兺

n⫽N⫹1 2N

a_nout†共t兲a_nout共t兲, 共3.2兲 a_nout共t兲⫽共2␲兲⫺1/2

冕

0 ⬁

d␻e⫺i␻ta_nout共␻兲. 共3.3兲 The generating function 共2.2兲 becomes

F共␰兲⫽ln

具

:e␰W:

典

. 共3.4兲 Expectation values of a normally ordered expression are readily computed using the optical equivalence theorem关12兴. Application of this theorem to our problem consists in dis-cretizing the frequency in infinitesimally small steps of⌬ 共so that ␻p⫽p⌬) and then replacing the annihilation operators

a_nin(␻p),cn(␻p) by complex numbers an p in _{, c}

n p共or their

com-plex conjugates for the corresponding creation operators兲. The coherent state of the incident radiation corresponds to a nonfluctuating value of a_{n p}in with 兩a_{n p}in兩2⫽␦_nm

0␦p p02␲I0/⌬

共with␻0⫽p0⌬). The thermal state of the spontaneous

emis-sion corresponds to uncorrelated Gaussian distributions of the real and imaginary parts of the numbers cn p, with zero

mean and variance

具

(Re c_{n p})2

_典

_⫽

_具

_{(Im c}

n p)2

典

⫽⫺

1

2f (␻p,T).

共Note that f ⬍0 for T⬍0.兲 To evaluate the characteristic function 共3.4兲 we need to perform Gaussian averages. The calculation is described in Appendix B.

The result takes a simple form in the long-time regime ␻c␶Ⰷ1, where␻cis the frequency within which S(␻) does

not vary appreciably. We find

F共␰兲⫽Fexc共␰兲⫺ ␶ 2␲

冕

0 ⬁ ln储1⫺␣␰f共1⫺rr†⫺tt†兲储d␻, 共3.5兲 Fexc共␰兲⫽␣␰␶I0兵t†关1⫺␣␰f共1⫺rr†⫺tt†兲兴⫺1t其m₀m₀, 共3.6兲 where 储•••储 denotes the determinant and 兵•••其m₀m₀ the

m0,m0 element of a matrix. In Eq. 共3.6兲 the functions f, t,

and r are to be evaluated at␻⫽␻0. The integral in Eq.共3.5兲

is the generating function for the photocount due to amplified spontaneous emission obtained in Ref. 关6兴. It is independent of the incident radiation and can be eliminated in a measure-ment by filtering the output through a narrow frequency win-dow around ␻0. The function Fexc(␰) describes the excess

noise due to the beating of the coherent radiation with the spontaneous emission关1兴. The expression 共3.6兲 is the central result of this paper.

By expanding F(␰) in powers of␰we obtain the factorial cumulants, in view of Eq. 共2.2兲. In what follows we will consider only the contribution from Fexc(␰), assuming that

the contribution from the integral over ␻ has been filtered out as mentioned above. We find

␬k⫽k!␣ k_␶_fk_⫺1_I 0关t † 共1⫺rr†_⫺tt†_兲k_⫺1_t_兴 m₀m₀, 共3.7兲

where again ␻⫽␻0 is implied. The mean photocurrent I¯

⫽␬1/␶ and the noise power P⫽(␬2⫹␬1)/␶ become

I ¯_⫽_␣_I

0共t†t兲m₀m₀, P⫽I¯⫹Pexc,

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P_exc⫽2␣2f I₀关t†共1⫺rr†⫺tt†兲t兴_m

0m0.

The noise power P exceeds the shot noise I¯ by the amount Pexc.

The formulas above are easily adapted to a measurement in reflection by making the exchange r→t⬘, t→r⬘. For ex-ample, the mean reflected photocurrent is ¯I ⫽␣I0(r

⬘

†r

⬘

)m₀m₀, while the excess noise is

兴m₀m₀. 共3.9兲

IV. NOISE FIGURE

The noise figure F is defined as the 共squared兲 signal-to-noise ratio at the input I₀2/ P0, divided by the signal-to-noise

ratio at the output, I¯2/ P. Since P₀⫽I₀ for coherent radiation at the input, one hasF⫽(Pexc⫹I¯)I0/ I¯2, hence

F⫽⫺2 f共t †_rr†_t_⫹t†_tt†_t_兲 m₀m₀ 共t†_t_兲 m₀m₀ 2 ⫹ 1⫹2␣f ␣共t†_t_兲 m₀m₀ . 共4.1兲 The noise figure is independent of I0. For large amplification

the second term on the right-hand side can be neglected rela-tive to the first, and the noise figure becomes also indepen-dent of the detection efficiency␣. The minimal noise figure for given r and t is reached for an ideal detector (␣⫽1) and at complete population inversion ( f⫽⫺1).

Since (t†rr†t⫹t†tt†t)m₀m₀⫽兺k兩(t†r)m₀k兩2

⫹兺k兩(t†t)m₀k兩2⭓(t†t)m₀m₀ 2

, one has F⭓⫺2 f for large am-plification 关when the second term on the right-hand side of Eq. 共4.1兲 can be neglected兴. The minimal noise figure F⫽2 at complete population inversion is reached in the absence of reflection 关(t†_r)

m₀k⫽0兴 and in the absence of intermode

scattering 关(t†_t)

m₀k⫽0 if k⫽m0兴. This is realized in the

2

, which is possible only in the absence of intermode scattering.

To make analytical progress in the evaluation of F, we will consider an ensemble of random media, with different realizations of the disorder. For large N and away from the laser threshold, the sample-to-sample fluctuations in numera-tors and denominanumera-tors of Eqs. 共4.1兲 and 共4.2兲 are small, so we may average them separately. Furthermore, the

‘‘equiva-lent channel approximation’’ is accurate for random media 关13兴, which says that the ensemble averages are independent of the mode index m0. Summing over m0, we may therefore

writeF as the ratio of traces, so the noise figure for a mea-surement in transmission becomes

F⫽⫺2 f NⱮtr 共t †_rr†_t_⫹t†_tt†_t_兲Ɑ Ɱtr t†_t_Ɑ2 ⫹N 1⫹2␣f ␣Ɱtr t†_t_Ɑ, 共4.3兲 and similarly for a measurement in reflection. The brackets Ɱ•••Ɑ denote the ensemble average.

V. APPLICATIONS A. Amplifying disordered waveguide

As a first example, we consider a weakly amplifying, strongly disordered waveguide of length L 共see the inset of Fig. 2兲. Averages of the moments of rr† and tt† for this system have been computed by Brouwer 关14兴 as a function of the number of propagating modes N, the mean free path l, and the amplification length ␰a⫽

冑

D␶a, where 1/␶a is the

amplification rate and D⫽cl/3 is the diffusion constant. It is assumed that 1/NⰆl/␰_aⰆ1 but the ratio L/␰_a⬅s is arbitrary. In this regime, sample-to-sample fluctuations are small, so the ensemble average is representative of a single system.

The results for a measurement in transmission are

I ¯_⫽4␣l 3L I0 s sin s, 共5.1兲 Pexc⫽ 2␣2l 3L f I0s

冋

3 sin s⫺ 2s⫺cot s sin2_s ⫹ s cot s⫺1 sin3_s ⫺ s sin4_s

册

. 共5.2兲 For a measurement in reflection, one finds

I

¯_⫽_␣_I₀

冋

₁_⫺ 4l

3Ls cot s

册

, 共5.3兲 FIG. 2. Noise figure of an amplifying disordered waveguide

共length L, amplification length␰a兲 measured in transmission 共solid

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P_exc⫽2␣ 2_l 3L f I0s

冋

2cot s⫺ 1 sin s⫹ cot s sin2s ⫹s cot s⫺1 sin3s ⫺ s sin4s

册

. 共5.4兲 The noise figure F follows from F⫽(Pexc⫹I¯)I0/ I¯2. It is

plotted in Fig. 2. One notices a strong increase inF on ap-proaching the laser threshold at s⫽␲.

B. Amplifying disordered cavity

Our second example is an optical cavity filled with an amplifying random medium 共see the inset of Fig. 3兲. The radiation leaves the cavity through a waveguide supporting N modes. The formulas for a measurement in reflection apply with t⫽0 because there is no transmission. The distribution of the eigenvalues of r†r is known in the large-N limit关15兴 as a function of the dimensionless amplification rate ␥ ⫽2␲/N␶a⌬␻ 共with ⌬␻ the spacing of the cavity modes

near frequency␻₀). The first two moments of this distribu-tion are N⫺1Ɱtr r†rⱭ⫽ 1 1⫺␥, 共5.5兲 N⫺1Ɱtr r†rr†rⱭ⫽2␥ 2_⫺2_␥_⫹1 共1⫺␥兲4 . 共5.6兲

The resulting photocurrent has mean and variance

I ¯_⫽_␣_I₀ 1 1⫺␥, 共5.7兲 Pexc⫽2␣2f I0␥ ␥⫺␥2_⫺1 共1⫺␥兲4 . 共5.8兲

The resulting noise figure for␣⫽1 and f ⫽⫺1,

F⫽1⫺␥⫹␥

2_⫹_␥3

共1⫺␥兲2 , 共5.9兲

is plotted in Fig. 3. Again, we see a strong increase ofF on approaching the laser threshold at␥⫽1.

VI. NEAR THE LASER THRESHOLD

In the preceding section we have taken the large-N limit. In that limit the noise figure diverges on approaching the laser threshold. In this section we consider the vicinity of the laser threshold for arbitrary N.

The scattering matrix S(␻) has poles in the lower half of the complex plane. With increasing amplification, the poles shift upwards. The laser threshold is reached when a pole reaches the real axis, say at resonance frequency␻th. For␻

near␻th the scattering matrix has the generic form

Snm⫽ ␴n␴m ␻⫺␻th⫹ 1 2i⌫⫺i/2␶a , 共6.1兲

where␴_n is the complex coupling constant of the resonance to the nth mode in the waveguide, ⌫ is the decay rate, and 1/␶a the amplification rate. The laser threshold is at ⌫␶a

⫽1.

We assume that the incident radiation has frequency ␻0

⫽␻th. Substitution of Eq.共6.1兲 into Eq. 共4.1兲 or 共4.2兲 gives

the simple result

F⫽⫺2 f ⌺ 兩␴m₀兩2 , ⌺⫽

兺

n⫽1 2N 兩␴n兩2, 共6.2兲

for the limiting value of the noise figure on approaching the laser threshold. The limit is the same for detection in trans-mission and in reflection. Since the coupling contant 兩␴_m

0兩

2

to the mode m₀of the incident radiation can be much smaller than the total coupling constant⌺, the noise figure 共6.2兲 has large fluctuations. We need to consider the statistical distri-bution p(F) in the ensemble of random media. The typical 共or modal兲 value of F is the value Ftyp at which p(F) is

maximal. We will see that this remains finite although the ensemble averageⱮFⱭ of F diverges.

A. Waveguide geometry

We first consider the case of an amplifying disordered waveguide. The total coupling constant ⌺⫽⌺l⫹⌺r is the

sum of the coupling constant ⌺_l⫽兺_nN_⫽1兩␴_n兩2 _{to the left end}

of the waveguide and the coupling constant ⌺_r ⫽兺n_⫽N⫹1

2N _兩_␴

n兩2 to the right. The assumption of equivalent

channels implies that

Ɱ1/FⱭ⫽⫺_{2 f N}1 Ɱ⌺l/⌺Ɑ⫽⫺

1

4 f N. 共6.3兲 Since the average of 1/F is finite, it is reasonable to as-sume that Ftyp⬇Ɱ1/FⱭ⫺1⫽⫺4 f N, or Ftyp⬇4N for

com-plete population inversion. The scaling with N explains why the large-N theory of the preceding section found a divergent FIG. 3. Noise figure of an amplifying disordered cavity,

con-nected to a photodetector via an N-mode waveguide. The curve is the result共5.9兲, as a function of the dimensionless amplification rate

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noise figure at the laser threshold. We conclude that the di-vergency ofF at L/␰_a⫽␲ in Fig. 2 is cut off at a value of order N, if F is identified with the typical value Ftyp.

B. Cavity geometry

In the case of an amplifying disordered cavity, we can make a more precise statement on p(F). Since there is only reflection, there is only one⌺⫽兺_nN_⫽1兩␴n兩2. The assumption

of equivalent channels now gives

Ɱ1/FⱭ⫽⫺ 1

2 f N. 共6.4兲

Following the same reasoning as in the case of the wave-guide, we would conclude that Ftyp⬇Ɱ1/FⱭ⫺1⫽⫺2 f N.

We will see that this is correct within a factor of 2.

To compute p(F) we need the distribution of the dimen-sionless coupling constants un⫽␴n/

冑

⌺. The N complex

numbers un form a vector uជ of length 1. According to

random-matrix theory 关7兴, the distribution p(S) of the scat-tering matrix is invariant under unitary transformations S →USUT _{共with U an N⫻N unitary matrix兲. It follows that}

the distribution p(uជ) of the vector uជ is invariant under rota-tions uជ→Uuជ, hence

p共u₁,u₂, . . . ,u_N兲⬀␦

冉

1⫺

兺

n 兩un兩

2

冊

_. _共6.5兲

In other words, the vector uជ has the same distribution as a column of a matrix that is uniformly distributed in the uni-tary group关16兴. By integrating out N⫺1 of the un’s we find

the marginal distribution of um₀,

p共um₀兲⫽

N⫺1

␲ 共1⫺兩um₀兩2兲N⫺2, 共6.6兲

for N⭓2 and 兩um₀兩2⭐1.

The distribution ofF⫽⫺2 f兩um₀兩⫺2 becomes

p共F兲⫽⫺2 f 共N⫺1兲

冉

1⫹2 f F

冊

N⫺2

F⫺2_, _共6.7兲

for N⭓2 and F⭓⫺2 f . We have plotted p(F) in Fig. 4 for complete population inversion ( f⫽⫺1) and several choices of N. It is a broad distribution, all its moments are divergent. The typical value of the noise figure is the value at which p(F) becomes maximal, hence

Ftyp⫽⫺ f N, N⭓2. 共6.8兲

In the single-mode case, in contrast, F⫽⫺2 f for every member of the ensemble关hence p(F)⫽␦(F⫹2 f )兴. We con-clude that the typical value of the noise figure near the laser threshold of a disordered cavity is larger than in the single-mode case by a factor N/2.

VII. ABSORBING MEDIA

The general theory of Sec. II can also be applied to an absorbing medium, in equilibrium at temperature T⬎0.

Equation 共2.4兲 then has to be replaced with

aout共␻兲⫽S共␻兲ain共␻兲⫹Q共␻兲b共␻兲, 共7.1兲 where the bosonic operator b has the expectation value

具

b_n†共␻兲bm共␻

⬘

兲

典

⫽␦nm␦共␻⫺␻

⬘

兲f 共␻,T兲, 共7.2兲

and the matrix Q is related to S by

QQ†_⫽1⫺SS†_. _共7.3兲

The formulas for F(␰) of Sec. III remain unchanged. Ensemble averages for absorbing systems follow from the corresponding results for amplifying systems by substitution ␶a→⫺␶a. The results for an absorbing disordered

wave-guide with detection in transmission are

I ¯_⫽4␣l 3L I0 s sinh s, 共7.4兲 Pexc⫽ 2␣2l 3L f I0s

冋

3 sinh s⫺ 2s⫹coth s sinh2s ⫺s coth s⫺1 sinh3s ⫹ s sinh4s

册

, 共7.5兲 where s⫽L/␰_a with␰_a the absorption length. Similarly, for detection in reflection one has

I ¯_⫽_␣_I₀

冋

₁_⫺ 4l 3Ls coth s

册

, 共7.6兲 Pexc⫽ 2␣2_l 3L f I0s

冋

2 coth s⫺ 1 sinh s⫺ coth s sinh2s ⫺s coth s⫺1 sinh3s ⫹ s sinh4s

册

. 共7.7兲 These formulas follow from Eqs. 共5.1兲–共5.4兲 upon substitu-tion of s→is.

For an absorbing disordered cavity, we find 关substituting ␥→⫺␥ in Eqs.共5.7兲 and 共5.8兲兴

FIG. 4. Probability distribution of the noise figure near the laser threshold for an amplifying disordered cavity, computed from Eq.

共6.7兲 for f ⫽⫺1. The most probable value is F⫽N, while the

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I ¯_⫽_␣_I 0 1 1⫹␥, 共7.8兲 Pexc⫽2␣2f I0␥ ␥2_⫹_␥_⫹1 共1⫹␥兲4 , 共7.9兲

with␥ the dimensionless absorption rate.

Since typically fⰆ1 in absorbing systems, the noise fig-ureF is dominated by shot noise, F⬇I₀/ I¯. Instead ofF we therefore plot the excess noise power Pexcin Figs. 5 and 6. In

contrast to the monotonic increase of P_excwith 1/␶_a in am-plifying systems, the absorbing systems show a maximum in Pexc for certain geometries. The maximum occurs near

L/␰a⫽2 for the disordered waveguide with detection in

transmission, and near ␥⫽1 for the disordered cavity. For

larger absorption rates the excess noise power decreases be-cause I¯ becomes too small for appreciable beating with the spontaneous emission.

VIII. CONCLUSION

In summary, we have studied the photodetection statistics of coherent radiation that has been transmitted or reflected by an amplifying or absorbing random medium. The cumulant generating function F(␰) is the sum of two terms. The first term is the contribution from spontaneous emission obtained in Ref. 关6兴. The second term Fexcis the excess noise due to

beating of the coherent radiation with the spontaneous emis-sion. Equation 共3.6兲 relates Fexcto the transmission and

re-flection matrices of the medium.

In the applications of our general result for the cumulant generating function, we have concentrated on the second cu-mulant, which gives the spectral density Pexcof the excess

noise. We have found that P_excincreases monotonically with increasing amplification rate, while it has a maximum as a function of absorption rate in certain geometries.

In amplifying systems we studied how the noise figureF increases on approaching the laser threshold. Near the laser threshold the noise figure shows large sample-to-sample fluctuations, such that its statistical distribution in an en-semble of random media has divergent first and higher mo-ments. The most probable value of F is of the order of the number N of propagating modes in the medium, independent of material parameters such as the mean free path. It would be of interest to observe this universal limit in random lasers.

ACKNOWLEDGMENTS

We thank P. W. Brouwer for helpful comments. This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek共NWO兲 and the Stichting voor Fundamenteel Onderzoek der Materie 共FOM兲.

APPENDIX A: SINGLE-MODE DETECTION We have assumed throughout this paper that all N modes propagating through the waveguide are detected at either the left or the right end. At the opposite extreme one can con-sider the case of single-mode detection. This is particularly relevant in a slab geometry, where the cross-sectional area of the photodetector is much less than the area of the random medium共see Fig. 7兲. The number of detected modes is then much smaller than the number of modes N propagating through the medium. The limit of single-mode detection is FIG. 5. Excess noise power Pexcfor an absorbing共solid line, left

axis兲 and amplifying disordered waveguide 共dashed line, right axis兲, respectively, in units of␣2_l_{兩 f 兩I}

0/L. The top panel is for detection

in transmission, the bottom panel for detection in reflection.

FIG. 6. Excess noise power Pexcfor an absorbing共solid line, left

axis兲 and amplifying disordered cavity 共dashed line, right axis兲, in units of␣2_{兩 f 兩I}

0.

(7)

reached when the photodetector covers an area comparable to the area of one speckle or smaller.

Single-mode detection of thermal radiation was consid-ered in Ref.关6兴. Denoting the detected mode by the index n0,

the mean photocurrent was found to be

I ¯ thermal⫽

冕

0 ⬁d␻ 2␲jthermal共␻兲, 共A1兲 jthermal共␻兲⫽␣f共1⫺rr†⫺tt†兲n₀n₀, 共A2兲

and the noise power

Pthermal⫽

冕

0 ⬁d␻ 2␲jthermal 2 共␻兲. 共A3兲

In this case of single-mode detection the noise power con-tains no information beyond what is contained in the photo-current.

The same holds for the excess noise considered in this paper. The mean transmitted photocurrent in a narrow fre-quency interval around ␻0 is given by

I

¯_⫽_␣_I₀_兩t_n

0m0兩

2_, _共A4兲

and the excess noise

Pexc⫽2I¯ jthermal共␻0兲共A5兲

is simply the product of the mean transmitted photocurrent and thermal current density. Noise measurements in single-mode detection are thus not nearly as interesting as in multi-mode detection, since the latter give information on the scat-tering properties that is not contained in the mean photocurrent.

APPENDIX B: DERIVATION OF EQ.„3.6…

To evaluate the Gaussian averages that lead to Eq.共3.6兲, it is convenient to use a matrix notation. We replace the sum-mation in Eq.共3.2兲 by a multiplication of the vector aout_with

the projectionPaout, where the projection matrixP has zero elements except P_nn⫽1, N⫹1⭐n⭐2N. We thus write

f共␻_p兲 . 共B7兲 We eliminate the cross terms of ainand c in Eq.共B6兲 by the substitution

c

⬘

*⫽c*⫺M⫺1Cain, 共B8兲 leading to

具

:e␰W:

典

⫽exp关ain*_共A⫹C†_M_⫺1_C_兲ain_兴

⫻

冕

d兵c_{n p}

⬘

其exp共⫺c

⬘

M c

⬘

*兲. 共B9兲 The integral is proportional to the determinant of M⫺1, giv-ing the generatgiv-ing function

F共␰兲⫽const⫺ln储M储⫹ain*_共A⫹C†_M⫺1_C_兲ain

⫽const⫺ln储M储⫹2␲I0

⌬ 共A⫹C†M⫺1C兲m₀p₀,m₀p₀.

共B10兲 The additive constant follows from F(0)⫽0. The term ⫺ln储M储 is the contribution from amplified spontaneous emission calculated in Ref.关6兴. The term proportional to I0is

the excess noise of the coherent radiation, termed Fexc in

Sec. III.

(8)

冕

0 ␶

ei⌬(p⫺p⬘)tdt⫽␶␦p p⬘. 共B11兲

The matrices defined in Eq.共B5兲 thus become diagonal in the frequency index, An p,n⬘p⬘⫽ ␣⌬␶␰ 2␲ 关S †_共_␻ p兲PS共␻p兲兴nn⬘␦p p⬘, 共B12兲

and similarly for B and C. We then find 共A⫹C†_M⫺1_C_兲

n p,n⬘p⬘

⫽␣␰₂⌬_␲␶共S†_P关1⫹_␣␰_{f VV}†_P兴⫺1_S_兲

nn⬘␦p p⬘, 共B13兲

where f, S, and V are evaluated at␻⫽␻p. Substitution into

Eq. 共B10兲 gives the result 共3.6兲 for Fexc(␰).

Simplification of Eq. 共B10兲 is also possible in the short-time regime, when ⍀c␶Ⰶ1, with ⍀c the frequency range

over which SS†differs appreciably from the unit matrix. The generating function then is

Fexc共␰兲⫽␣␰␶I0

冋

t†共␻0兲

冉

1⫺ ␣␰␶ 2␲

冕

0 ⬁ d␻f共␻,T兲 ⫻关1⫺r共␻兲r†_共_␻_兲⫺t共_␻_兲t†_共_␻_兲兴

冊

⫺1 t共␻0兲

册

m₀m₀ . 共B14兲

关1兴 C. H. Henry and R. F. Kazarinov, Rev. Mod. Phys. 68, 801 共1996兲.

关2兴 C. M. Caves, Phys. Rev. D 26, 1817 共1982兲.

关3兴 J. R. Jeffers, N. Imoto, and R. Loudon, Phys. Rev. A 47, 3346 共1993兲.

关4兴 R. Matloob, R. Loudon, M. Artoni, S. M. Barnett, and J. Jef-fers, Phys. Rev. A 55, 1623共1997兲.

关5兴 D. Wiersma and A. Lagendijk, Phys. World 10 共1兲, 33 共1997兲. 关6兴 C. W. J. Beenakker, Phys. Rev. Lett. 81, 1829 共1998兲. 关7兴 C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 共1997兲. 关8兴 R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, Phys.

Rev. A 52, 4823共1995兲.

关9兴 T. Gruner and D.-G. Welsch, Phys. Rev. A 54, 1661 共1996兲. 关10兴 R. J. Glauber, Phys. Rev. Lett. 10, 84 共1963兲.

关11兴 P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 共1964兲. 关12兴 L. Mandel and E. Wolf, Optical Coherence and Quantum

Op-tics共Cambridge University Press, New York, 1995兲. 关13兴 P. A. Mello and S. Tomsovic, Phys. Rev. B 46, 15 963 共1992兲. 关14兴 P. W. Brouwer, Phys. Rev. B 57, 10 526 共1998兲. The formulas in this paper refer to an absorbing slab. For an amplifying slab one should replace␰aby i␰a. Note that Eq.共13c兲 contains a

misprint: The second and third term between brackets should have, respectively, signs minus and plus instead of plus and minus.

关15兴 C. W. J. Beenakker, in Diffuse Waves in Complex Media, NATO Science Series C531, edited by J.-P. Fouque共Kluwer, Dordrecht, 1999兲.