VOLUME 73, NUMBER 6 PH YS ICAL REVIE% LETTERS 8 AUrUs~ 1994 Effect of Absorption on the Wave Transport in the Strong Localization Regime V. Freilikher, M. Pustilnik, and I. Yurkevich The Jack and Pearl Resnick Institute of Advanced Technology, Department of Physics, Bar lian -University, Ramat-Gan 52900, Israel (Received 5 April 1994) The asymptotically exact expressions for all statistical moments of the transmission coefficient are obtained for a 1D disordered lossy system by mapping the Fokker-Planck problem onto a Schrodinger equation with imaginary time. It is shown that even arbitrary small absorption leads to the quantization of the spectrum of the initial Fokker-Planck equation. The quantization reveals itself as the appearance of a new scale: the disorder-induced absorption length. PACS numbers: 42.25.8s, 71. 55. Jv, 73.20.Dx where the complex dielectric permittivity e(x) = e'(x) + ie" consists of the real random function e'(x) with zero mean (elastic scattering) and the imaginary constant is" (absorption). The scattering problem is characterized by reflection and transmission coefficients r and t defined by the form of wave function outside the disordered lossy region 0 & x & L: Ik02 + reIk0x + ) L e — ik0 x &0. (2) Both r and t are the functions of length L of the disordered region. This makes it possible to accept as a Electron transport in one-dimensional elastic disordered systems and classical wave propagation in randomly layered lossless media have been well studied (see, for example, [1 — 3]). Much recent attention has been paid to electron transport in disordered structures with dissipation caused by inelastic scattering processes [4,5]. However, the problem of wave transport through a disordered system, that absorbs energy due to finite conductivity, still remains unresolved. The first results for power reflection coefficient from a lossy one-dimensional media were obtained almost twenty years ago [6]. Nevertheless, up to now there have been no methods devised to calculate analytically the statistical moments of transmitted energy through a slab where both scattering and absorption are present. This problem is of particular interest for many reasons. First, all real media have inelastic scattering channels, and many of these channels can be described as a slight absorption. Second, some two- and three-dimensional problems may be mapped onto the 1D Schrodinger equation with an effective complex-valued potential [7]. Third, the 1D disordered system, where the effect of absorption can be studied rigorously, is the only model expected to be exactly solvable. Let us consider a wave field U satisfying the Helmholtz equation ( d2 + ko[l + s)x)]jU(x) = O. ( dx2 starting point the so-called invariant imbedding equations (see, for example, Refs. [8, 9]), which are exact nonlinear equations for r and t with respect to the length L: dr(L) i = — k, . (L.)[. — '' ~ . (L)e'" ], - dL 2 dt(L) i dL 2 = — koe(L)t(L) [1 + r(L)e "' I . - It is necessary to notice that Eq. (3) being closed for r allows us to obtain the distribution function of the phase and the squared modulus R = ~r2~ of the reliection coefficient. In the absence of absorption knowledge of the distribution function of R gives us complete information on T due to energy conservation, R + T = 1 (T = ~t2~). This is not the case for lossy media, because in the presence of absorption the conservation law connects not only R and T but also involves a random amount of absorbed intensity. It demands Eqs. (3) and (4) to be solved simultaneously in order to get statistical information about T. Let us introduce averages of the form Z (Tttt Rt I ) The advantage of this choice is that these quantities turn out to satisfy the closed set of equations: d Zpg g: (m ) ( pg n~) Z~ p~) dv — n '(Z„„, — Z „-)) + m(m — 1)Z, „„ — P(m + 2n)Z with the evident initial conditions Z „(r = 0) = 6„)). Here we have introduced the dimensionless length v- = L/l and a parameter characterizing the absorption strength p = l/l„where l is the localization length with no absorption and l, is the absorption length in an ordered (clean) system. Equations (6) are derived by the standard averaging procedure (see, for example, [9] and references therein) and while neglecting the rapidly oscillating terms (under the assumption kol » 1. k))l„» 1. 810 0031-9007/94/73(6)/810(4) $06.00 1994 The American Physical Society