Dynamic effect of phase conjugation on wave localization
Bemmel, K.J.H. van; Titov, M.L.; Beenakker, C.W.J.
Citation
Bemmel, K. J. H. van, Titov, M. L., & Beenakker, C. W. J. (2002). Dynamic effect of phase
conjugation on wave localization. Physical Review B, 65(17), 174203.
doi:10.1103/PhysRevB.65.174203
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Dynamic effect of phase conjugation on wave localization
K. J. H. van Bemmel, M. Titov, and C. W. J. BeenakkerInstituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
共Received 22 October 2001; published 26 April 2002兲
We investigate what would happen to the time dependence of a pulse reflected by a disordered single-mode waveguide if it is closed at one end, not by an ordinary mirror, but by a phase-conjugating mirror. We find that the waveguide acts like a virtual cavity with resonance frequency equal to the working frequency0of the phase-conjugating mirror. The decay in time of the average power spectrum of the reflected pulse is delayed for frequencies near0. In the presence of localization the resonance width iss⫺1exp(⫺L/l), with L the length of the waveguide, l the mean free path, andsthe scattering time. Inside this frequency range the decay of the average power spectrum is delayed up to times t⯝sexp(L/l).
DOI: 10.1103/PhysRevB.65.174203 PACS number共s兲: 42.65.Hw, 42.25.Dd, 72.15.Rn
I. INTRODUCTION
The reflection of a wave pulse by a random medium pro-vides insight into the dynamics of localization.1– 4 The re-flected amplitude contains rapid fluctuations over a broad range of frequencies, with a slowly decaying envelope. The power spectrum a(,t) characterizes the decay in time t of the envelope at frequency. In an infinitely long waveguide
共with N propagating modes兲, the signature of localization,5,6
具
a共,t兲典
⬀t⫺2 for tⰇN2s, 共1兲is a quadratic decay of the disorder-averaged power spectrum
具
a典
, which sets in after N2 scattering timess.
The decay共1兲 still holds over a broad range of times if the length L of the waveguide is finite, but much greater than the localization length ⫽(N⫹1)l 共with l⫽cs the mean free
path兲. What changes is that for exponentially large times t
Ⰷsexp(L/l) the quadratic decay becomes more rapid ⬀exp(⫺const⫻ln2t). This is the celebrated log-normal
tail.7–11 We may assume that the finite length of the wave-guide is realized by terminating one end by a perfectly re-flecting mirror, so that the total reflected power is unchanged. In this paper we ask the question what happens if instead of such a normal mirror one would use a phase-conjugating mirror.12,13The interplay of multiple scattering by disorder and optical phase conjugation is a rich problem even in the static case.14 –16Here we show that the dynamical aspects are particularly striking. Basically, the disordered waveguide is turned into a virtual cavity with a resonance frequency0set
by the phase-conjugating mirror.
We present a detailed analytical and numerical calculation for the single-mode case (N⫽1). For times tⰇs we find
that a(,t) has decayed almost completely except in a nar-row frequency range⬀s⫺1exp(⫺L/l) around 0. In this
fre-quency range the decay is delayed up to times t
⯝sexp(L/l), after which a log-normal decay sets in. The
exponentially large difference in time scales for the decay near0 and away from 0 is a signature of localization.
II. FORMULATION OF THE PROBLEM A. Scattering theory
A scattering matrix formulation of the problem of com-bined elastic scattering by disorder and inelastic scattering
by a phase-conjugating mirror was developed by Paasschens
et al.15 We summarize the basic equations for the case of a single propagating mode in the geometry shown in Fig. 1. A single-mode waveguide is closed at one end (x⫽0) by either a normal mirror or by a phase-conjugating mirror. Elastic scattering in the waveguide is due to random disorder in the region 0⬍x⬍L. For simplicity we consider a single polar-ization, so that we can use a scalar wave equation.
The phase-conjugating mirror consists of a four-wave mixing cell:12,13Two counterpropagating beams at frequency
0mix with an incident beam at frequency0⫹to yield a
retroreflected beam at frequency 0⫺ 共for Ⰶ0). The mixing is due to the presence in the cell of a medium with a large third-order nonlinear susceptibility 共e.g., BaTiO3 or
CS2).
For xⰇL the wave amplitude at frequencies ⫾⫽0 ⫾ is an incoming or outgoing plane wave,
u⫾in共rជ,t兲⫽Re⫾inexp关⫺ik⫾共x⫺L兲⫺i⫾t兴⫾共y,z兲,
共2a兲
u⫾out共rជ,t兲⫽Re⫾outexp关ik⫾共x⫺L兲⫺i⫾t兴⫾共y,z兲.
共2b兲
Here k⫾⫽k0⫾/c is the wave number at frequency ⫾,
with k0 the wave number at 0 and c⫽d/dk the group velocity. The transverse wave profile⫾( y ,z) is normalized such that the wave carries unit flux.
The reflection matrix relates the incoming and outgoing wave amplitudes, according to
冉
⫹ ⫺*冊
out ⫽冉
r⫹⫹ r⫹⫺ r⫺⫹ r⫺⫺冊
冉
⫹ ⫺*冊
in . 共3兲The reflection coefficients are complex numbers that depend on. They satisfy the symmetry relations
r⫺⫺* 共兲⫽r⫹⫹共⫺兲, r⫺⫹* 共兲⫽r⫹⫺共⫺兲. 共4兲 If there is only reflection at the mirror and no disorder, then one has simply
冉
r⫹⫹ r⫹⫺r⫺⫹ r⫺⫺
冊
⫽冉
⫺e2ik⫹L 0
0 ⫺e⫺2ik⫺L
冊
共5兲for a normal mirror and
冉
r⫹⫹ r⫹⫺r⫺⫹ r⫺⫺
冊
⫽冉
0 ⫺ie2iL/c
ie2iL/c 0
冊
共6兲for a phase-conjugating mirror operating in the regime of ideal retroreflection. 共We will assume this regime in what follows.兲
We wish to determine how the reflection coefficients are modified by the elastic scattering by the disorder. For this we need the elastic scattering matrix
S⫽
冉
r t
⬘
t r
⬘
冊
. 共7兲The reflection coefficients r,r
⬘
and transmission coefficientst,t
⬘
describe reflection and transmission from the left or from the right of a segment of a disordered waveguide of length L. The matrix S is unitary and symmetric 共hence t⫽t⬘
). The basis for S is chosen such that r⫽r⬘
⫽0, t(⫾)⫽eik⫾L inthe absence of disorder. The relationship between the coeffi-cients in Eqs.共3兲 and 共7兲 is15
r⫹⫹共兲⫽r
⬘
共兲⫹t共兲关1⫺r*共⫺兲r共兲兴⫺1r*共⫺兲t共兲,共8a兲
r⫹⫺共兲⫽⫺it共兲关1⫺r*共⫺兲r共兲兴⫺1t*共⫺兲, 共8b兲 for a phase-conjugating mirror. For a normal mirror there is no mixing of frequencies and one has simply
r⫹⫹共兲⫽r
⬘
共兲⫺t共兲关1⫹r共兲兴⫺1t共兲, 共9a兲r⫹⫺共兲⫽0. 共9b兲
In each case the matrix of reflection coefficients is unitary, so
兩r⫹⫹共兲兩2⫹兩r⫹⫺共兲兩2⫽1. 共10兲
B. Power spectrum
We assume that a pulse⬀␦(t) is incident at x⫽L 关corre-sponding to⫾in⫽1 for all in Eq.共2兲兴. The reflected wave at x⫽L has amplitude
uout共t兲⫽Re e⫺i0t
冕
0⬁d
2 兵关r⫹⫹共兲⫹r⫹⫺共兲兴e
⫺it
⫹关r⫺⫺* 共兲⫹r⫺⫹* 共兲兴eit其. 共11兲
共We have suppressed the transverse coordinates y,z for
sim-plicity of notation.兲 Using the symmetry relations 共4兲, we can rewrite this as
uout共t兲⫽Re e⫺i0t
冕
⫺⬁ ⬁ d
2关r⫹⫹共兲⫹r⫹⫺共兲兴e
⫺it.
共12兲
The time correlator of the reflected wave becomes
uout共t兲uout共t⫹t
⬘
兲 ⫽1 2 Re e i0t⬘冕
⫺⬁ ⬁ d 2冕
⫺⬁ ⬁ d⬘
2 e i(⬘⫺)tei⬘t⬘ ⫻关r⫹⫹共兲⫹r⫹⫺共兲兴关r⫹⫹* 共⬘
兲⫹r⫹⫺* 共⬘
兲兴, 共13兲plus terms that oscillate on a time scale 1/0. We make the
rotating wave approximation and neglect these rapidly oscil-lating terms. The power spectrum a of the reflected wave is obtained by a Fourier transform
a共,t兲⫽
冕
⫺⬁ ⬁
dt
⬘
cos关共0⫹兲t⬘
兴uout共t兲uout共t⫹t⬘
兲⫽Re
冕
⫺⬁ ⬁ d␦
2 e
⫺i␦ta共,␦兲, 共14兲
where we have introduced the correlator in the frequency domain
a共,␦兲⫽1
4关r⫹⫹共⫹␦兲⫹r⫹⫺共⫹␦兲兴关r⫹⫹* 共兲
⫹r⫹⫺* 共兲兴. 共15兲
Integration of the power spectrum over time yields, using also Eq. 共10兲,
冕
⫺⬁ ⬁ dt a共,t兲⫽Re a共,␦⫽0兲 ⫽1 4⫹ 1 2 Re r⫹⫺共兲r⫹⫹* 共兲. 共16兲For a normal mirror r⫹⫺()⫽0 and a(,␦⫽0)⫽1 4,
ex-pressing flux conservation. For the phase-conjugating mirror there is inelastic scattering, which mixes the frequency com-ponents and⫺. The constraint of flux conservation then takes the form
a共,␦⫽0兲⫹a共⫺,␦⫽0兲⫽12. 共17兲
This follows from the symmetry relations共4兲 and the unitar-ity of the reflection matrix. Equation 共17兲 implies that a(
⫽0,␦⫽0)⫽1 4.
K. J. H. van BEMMEL, M. TITOV, AND C. W. J. BEENAKKER PHYSICAL REVIEW B 65 174203
III. RANDOM SCATTERERS
We assume weak disorder, meaning that the mean free path l is much larger than the wavelength 2/k0. The
mul-tiple scattering by disorder localizes the wave with localiza-tion length equal to 2l. We consider separately the case of a phase-conjugating mirror and of a normal mirror.
A. Phase-conjugating mirror
We first concentrate on the degenerate regime of small frequency shift and will simplify the expressions by put-ting ⫽0 from the start. We note that r⫹⫹(0)⫽0, r⫹⫺(0)
⫽⫺i, as follows from Eq. 共8兲 and unitarity of the scattering
matrix 共7兲. Using Eqs. 共8兲 and 共15兲, we arrive at the power spectrum in the frequency domain
a共0,␦兲⫽ i
4兵r
⬘
共␦兲⫹关1⫺r*共⫺␦兲r共␦兲兴⫺1
⫻关t2共␦兲r*共⫺␦兲⫺it共␦兲t*共⫺␦兲兴其. 共18兲
The scattering amplitudes have the polar decomposition r
⫽
冑
R exp(i), r⬘
⫽冑
Rexp(i⬘
), and t⫽i冑
1⫺Rexp关12i(⫹
⬘
)兴, with R,,⬘
real functions of frequency. The phase
⬘
may be assumed to be statistically independent of R (⫾␦),(⫾␦), and uniformly distributed in (0,2).共This is the Wigner conjecture, proven for chaotic scattering in Ref. 17.兲 In this way only the last term in Eq. 共18兲 survives the disorder average具
•••典
,4
具
a共0,␦兲典
⫽冓
t共␦兲t*共⫺␦兲1⫺r*共⫺␦兲r共␦兲
冔
⫽m兺
⫽0⬁
Zm, 共19兲
where we have defined Zm⫽
具
t(␦)t*(⫺␦)⫻关r*(⫺␦)r(␦)兴m
典
.The moments Zm satisfy the Berezinskii recursion relation18,19 ldZm dL ⫽m 2共Z m⫹1⫹Zm⫺1⫺2Zm兲⫹共2m⫹1兲共Zm⫹1⫺Zm兲 ⫹2is␦共2m⫹1兲Zm, 共20兲
with s⫽l/c the scattering time. 共The mean free path l
ac-counts only for backscattering, so that the scattering time in a kinetic equation would equal 12s.) The initial condition is
Zm(L⫽0)⫽␦m,0. In Appendix A we derive an analytical
re-sult for
具
a(0,␦)典
in the small frequency range ln(1/s␦)ⲏL/lⰇ1. It reads
具
a共0,␦兲典
⫽1 2冕
⫺⬁ ⬁
dkik共⫺2is␦兲ik⫺1/22⫺3ik⫺1/2 ⫻⌫2共1
2⫹ik兲⌫共 1
2⫺ik兲⌫⫺1共1⫹ik兲⌫⫺1共ik兲
⫻exp关⫺共1 4⫹k
2兲L/l兴. 共21兲
The initial decay is determined by the contributions of the poles at k⫽⫺1 2i, ⫺ 3 2i, ⫺ 5 2i,
具
a共0,␦兲典
⫽14⫹ 1 4is␦exp共2L/l兲⫺ 1 18s 2␦2exp共6L/l兲 ⫹O共␦3兲. 共22兲The result 共21兲 is plotted in Fig. 2 for L/l⫽12.3. We compare with the data from a numerical solution of the wave equation on a two-dimensional lattice, using the method of recursive Green functions.20 共The method of simulation is the same as in Ref. 15, and we refer to that paper for a more detailed description.兲 The agreement with the analytical curves is quite good, without any adjustable parameter. The
␦ dependence of
具
a(0,␦)典
for large L/l occurs on an exponentially small scale, within the range of validity of Eq.共21兲.
A Fourier transform of Eq.共21兲 yields the average power spectrum in the time domain for ln(t/s)ⰇL/lⰇ1, with the
result
具
a共0,t兲典
⫽18 3/2共L/l兲⫺3/2exp共⫺L/4l兲 s ⫺1/2t⫺1/2 ⫻ln共4t/s兲exp关⫺共l/4L兲ln2共4t/s兲兴. 共23兲The leading logarithmic asymptote of the decay is log-normal,⬀exp关⫺(l/4L)ln2t兴, characteristic of anomalously
lo-calized states.7–11
These results are calculated for⫽0 and remain valid as long asⰆs⫺1exp(⫺L/l). This can be checked by
perform-ing a Taylor expansion in of Eq. 共8兲, using the polar de-composition for r,r
⬘
,t. We still have r⫹⫹()⬇0 andr⫹⫺()⬇⫺i as long as d/dⰆ1⫺R. In order of mag-nitude this corresponds to sⰆexp(⫺L/l). This is the
de-generate regime. For sⰇexp(⫺L/l) the power spectrum
FIG. 2. Average power spectrum for reflection by a disordered waveguide (L/l⫽12.3) connected to a phase-conjugating mirror
关solid curves, from Eq. 共21兲兴 or a normal mirror 关dashed curves,
from Eq.共28兲兴. The data points follow from a numerical simulation. There is no adjustable parameter in the comparison. Notice the much faster frequency dependence for the phase-conjugating mirror
a(,␦) is dominated by the term r
⬘
(⫹␦)r⬘
*(). The decay of具
a(,␦)典
then occurs in the ranges␦ⱗ1. Thesame is true for the normal mirror, which we consider in the next subsection. The presence of the mirror is now only of importance for very small␦ 关ln(1/s␦)ⲏL/lⰇ1兴, when
a(,␦)⬇1
4. For sⰇ1 the average power spectrum
具
a(,␦)典
in the range ln(1/s␦)ⰇL/l is the same as thatfor a normal mirror, leading to exactly the same log-normal decay in the time domain. This is proven in Appendix B.
B. Normal mirror
For comparison we discuss the known results for a disor-dered waveguide connected to a normal mirror instead of a phase-conjugating mirror. Since r⫹⫺⫽0, one has from Eq.
共15兲
4
具
a共,␦兲典
⫽具
r⫹⫹共⫹␦兲r⫹⫹* 共兲典
⬅R1. 共24兲 The quantities Rm⫽具
关r⫹⫹(⫹␦)r⫹⫹* ()兴m典
satisfy the Berezinskii recursion relation18,19ldRm
dL ⫽m
2共R
m⫹1⫹Rm⫺1⫺2Rm兲⫹2is␦mRm. 共25兲
The initial condition is Rm(L⫽0)⫽1 for all m. The solution
for ln(1/s␦)ⲏL/l is known21and gives the average power
spectrum
具
a共,␦兲典
⫽1 2冑
⫺2is␦冉
K1关2冑
⫺2is␦兴 ⫹1冕
⫺⬁ ⬁ dk k sinh共k兲共14⫹k 2兲⫺1⫻K2ik关2
冑
⫺2is␦兴exp关⫺共1 4⫹k
2兲L/l兴
冊
, 共26兲with K a Bessel function. 关The result 共26兲 does not require
L/lⰇ1, in contrast to Eq. 共21兲.兴 The initial decay is
domi-nated by the contributions of the poles at k⫽⫺1 2i, ⫺ 3 2i, ⫺5 2i,
具
a共,␦兲典
⫽1 4⫹ 1 2is␦L/l⫺14s 2␦2exp共2L/l兲⫹O共␦3兲. 共27兲Comparison of Eqs.共26兲 and 共27兲 with Eqs. 共21兲 and 共22兲 shows that the decay is much slower for a normal mirror than for a phase-conjugating mirror. The characteristic frequency scale is larger by a factor exp(2L/l). So Eq. 共26兲 is not suf-ficient to describe the entire decay of
具
a(,␦)典
, which oc-curs in the range s␦ⱗ1. The decay in this range isob-tained by putting the left-hand side of Eq.共25兲 equal to zero, leading to5,22
具
a共,␦兲典
⫽14⫺ 12is␦exp共⫺2is␦兲Ei共2is␦兲, 共28兲
where Ei is the exponential integral function. The range of validity of Eq.共28兲 is ln(1/s␦)ⰆL/l and L/lⰇ1. The
re-sult 共28兲 is plotted in Fig. 2 and is seen to agree well with data from the numerical simulation.
For ln(t/s)ⰆL/l 共and L/lⰇ1) one can perform the Fourier
transform of Eq. 共28兲 to get the average power spectrum in the time domain5
具
a共,t兲典
⫽12s共t⫹2s兲⫺2, t⬎0. 共29兲It decays quadratically ⬀t⫺2 for t/sⰇ1. For exponentially long times tⰇsexp(L/l), one should instead perform the
Fourier transform of Eq. 共26兲. One finds that the quadratic decay crosses over to a log-normal decay ⬀exp
关⫺(l/4L)ln2t兴,7
the same as for the phase-conjugating mirror.
IV. CONCLUSION
We have shown that the interplay of phase conjugation and multiple scattering by disorder leads to a drastic slowing down of the decay in time t of the average power spectrum
具
a(,t)典
of frequency components of a reflected pulse. The slowing down exists in a narrow frequency range around the characteristic frequency0of the phase-conjugatingmir-ror 共degenerate regime兲. If is outside this frequency range
共nondegenerate regime兲, the power spectrum decays as
rap-idly as for a normal mirror.
The slowing down can be interpreted in terms of a long-lived resonance at 0, which is induced in the random
me-dium by the phase-conjugating mirror. This resonance is known from investigations of the static scattering properties.15 The resonance is exponentially narrow,
⬀s⫺1exp(⫺L/l), in the presence of localization 共withs the
scattering time, L the length of the disordered region, and l the mean free path兲. The resonance leads to the exponentially large differences in time scales for the decay of the power spectrum in the degenerate regime and the nondegenerate regime.
We have restricted the calculation in this paper to the case of a single propagating mode, when a complete analytical theory could be provided. We expect that the N-mode case is qualitatively similar: An exponentially large difference in time scales ⬀exp(L/) for the decay in the degenerate and nondegenerate regimes provided the medium is localized关L large compared to the localization length ⫽(N⫹1)l]. In the diffusive regime we expect
具
a(,t)典
to decay on the time scale of the diffusion time s(L/l)2. The difference with thenondegenerate regime 共or a normal mirror兲 is then a factor (L/l)2 instead of exponentially large.
In final analysis we see that phase conjugation greatly magnifies the difference in the dynamics with and without localization. Indeed, if there is no phase-conjugating mirror the main difference is a decay⬀t⫺3/2in the diffusive regime versus t⫺2 in the localized regime,6 but the characteristic time scale remains the same 共set by the scattering times). We therefore suggest that phase conjugation might be a promising tool in the ongoing experimental search for dy-namical features of localization.23,24
K. J. H. van BEMMEL, M. TITOV, AND C. W. J. BEENAKKER PHYSICAL REVIEW B 65 174203
APPENDIX A: POWER SPECTRUM IN THE FREQUENCY DOMAIN
We show how to arrive at the result共21兲 starting from the recursion relation共20兲. We assume ln(1/s␦)ⲏL/lⰇ1. It is
convenient to work with the Laplace transform
Zm共兲⫽
冕
0
⬁dL
l exp共⫺L/l兲Zm共L兲 共A1兲
of the moments Zm. The recursion relation共20兲 transforms
into Zm共兲⫺␦m,0⫽m 2关Z m⫹1共兲⫹Zm⫺1共兲⫺2Zm共兲兴 ⫹共2m⫹1兲关Zm⫹1共兲⫺Zm共兲兴 ⫺共2m⫹1兲Zm共兲, 共A2兲 with⫽⫺2is␦.
For small兩兩 and large m this equation can be written as a differential equation m2 2Z共m,兲 m2 ⫹2m Z共m,兲 m ⫺共⫹2m兲Z共m,兲⫽0, 共A3兲
where m is now considered to be a continuous variable. The solution of Eq.共A3兲 is
Z共m,兲⫽C共,兲共m兲⫺1/2K冑1⫹4共2
冑
2m兲. 共A4兲The factor C(,) is determined by matching to the solution of Eq. 共A2兲 for m→0, m→⬁, which has been calculated
in Ref. 25. The result is
C共,兲⫽41/2⌫共12⫹ 1 2
冑
1⫹4兲 ⫻⌫⫺1共1⫹1 2冑
1⫹4兲⌫⫺1共 1 2冑
1⫹4兲 ⫻exp关1 2冑
1⫹4ln共/8兲兴. 共A5兲To obtain the power spectrum 共19兲 we replace the sum over m by an integration, with the result
兺
m⫽0 ⬁ Zm共兲⫽21/2⫺1/2⌫2共12⫹ 1 2冑
1⫹4兲 ⫻⌫共1 2⫺ 1 2冑
1⫹4兲⌫⫺1共1 ⫹1 2冑
1⫹4兲⌫⫺1共 1 2冑
1⫹4兲 ⫻exp关1 2冑
1⫹4ln共/8兲兴. 共A6兲There are poles at⫽n(n⫹1), n⫽0,1,2, . . . , and a branch cut starting at ⫽⫺1/4. When doing the inverse Laplace transform we put the integration path in between the poles and the branch cut. The final result is given by Eq.共21兲. The reason that we need the condition L/lⰇ1 is that Eqs. 共A4兲 and 共A5兲 are only correct for mⰇ1. The first terms in the sum兺m⬁⫽0Zmare important for L/lⱗ1, but can be neglected
for L/lⰇ1.
APPENDIX B: EQUIVALENCE OF NORMAL AND PHASE-CONJUGATING MIRROR IN THE NONDEGENERATE REGIME
We show that the average power spectrum
具
a(,␦)典
in the range ln(1/s␦)ⰇL/l is the same for a normal mirror and aphase-conjugating mirror in the regime sⰇ1.
First we consider the normal mirror. One can write
具
a(,␦)典
in terms of R,,⬘
, using the polar decomposition and Eqs.共9兲 and 共15兲. Only two terms survive the average over,
4
具
a共,␦兲典
⫽具
r⬘
共⫹␦兲r⬘
*共兲典
⫹冓
t2共⫹␦兲t2*共兲
关1⫹r共⫹␦兲兴关1⫹r*共兲兴
冔
. 共B1兲 The first term is also present for the phase-conjugating mirror, so we only need to consider the second term. This term can be written as冓
t2共⫹␦兲t2*共兲 关1⫹r共⫹␦兲兴关1⫹r*共兲兴冔
⫽兺
n,m 共⫺1兲 n⫹m具
t2共⫹␦兲t2*共兲rn共⫹␦兲rm*共兲典
⫽兺
n具
t2共⫹␦兲t2*共兲rn共⫹␦兲rn*共兲典
, 共B2兲where we have averaged over in the last step.
Now we consider the phase-conjugating mirror in the regimesⰇ1. In that regime the phase() is independent of the
4
具
a共,␦兲典
⫽具
r⬘
共⫹␦兲r⬘
*共兲典
⫹冓
t2共⫹␦兲r*共⫺⫺␦兲t2*共兲r共⫺兲 关1⫺r*共⫺⫺␦兲r共⫹␦兲兴关1⫺r共⫺兲r*共兲兴
冔
⫹
冓
t共⫹␦兲t*共⫺⫺␦兲t*共兲t共⫺兲关1⫺r*共⫺⫺␦兲r共⫹␦兲兴关1⫺r共⫺兲r*共兲兴
冔
. 共B3兲 The first term is also present for the normal mirror. ForsⰇ1, t() is independent of t(⫺). The second term is then muchlarger than the third term because of the large fluctuations in the localized regime (LⰇl). The second term can also be written as
冓
t2共⫹␦兲r*共⫺⫺␦兲t2*共兲r共⫺兲 关1⫺r*共⫺⫺␦兲r共⫹␦兲兴关1⫺r共⫺兲r*共兲兴冔
⫽兺
n,m具
t2共⫹␦兲t2*共兲rn共⫹␦兲rm*共兲rm⫹1共⫺兲rn⫹1*共⫺⫺␦兲典
⫽兺
n具
t2共⫹␦兲t2*共兲rn共⫹␦兲rn*共兲典具
rn⫹1共⫺兲rn⫹1*共⫺⫺␦兲典
. 共B4兲Comparison with Eq. 共B2兲 for a normal mirror shows that the two expressions are the same as long as we can replace
具
rn⫹1(⫺)rn⫹1*(⫺⫺␦)典
by 1 for the relevant terms in the summation over n. It is now convenient to write rn(⫹␦)rn*()⫽Rn()关1⫺C(,␦)兴n. The average over兵r(),r(⫹␦),t(),t(⫹␦)其 is dominated by configurations where the transmittance T is large. For small␦this corresponds to configurations where 1⫺R() and兩C(,␦)兩 are much larger than typical values of these quantities. For these dominating configurations the number of relevant terms in the summation over n is relatively small and for these relatively small n we can replace
具
rn⫹1(⫺)rn⫹1*(⫺⫺␦)典
by 1. We therefore conclude that for small␦, the average power spectrum具
a(,␦)典
is the same as for a normal mirror. The above argument breaks down if具
rn⫹1(⫺)rn⫹1*(⫺⫺␦)典
starts to deviate from 1 for the relevant terms in the summation. This is the case for ln(1/s␦)ⱗL/l.1Scattering and Localization of Classical Waves in Random
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