Localization in a Disordered Multi-Mode Waveguide with Absorption or Amplification

An analytical and numerical study is presented of transmission of radiation through a multi-mode waveguide containing a random medium with a complex dielectric constant $\epsilon= \epsilon'+i\epsilon''$. Depending on the sign of $\epsilon''$, the medium is absorbing or amplifying. The transmitted intensity decays exponentially $\propto\exp(-L/\xi)$ as the waveguide length $L\to\infty$, regardless of the sign of $\epsilon''$. The localization length $\xi$ is computed as a function of the mean free path $l$, the absorption or amplification length $|\sigma|^{-1}$, and the number of modes in the waveguide $N$. The method used is an extension of the Fokker-Planck approach of Dorokhov, Mello, Pereyra, and Kumar to non-unitary scattering matrices. Asymptotically exact results are obtained for $N\gg1$ and $|\sigma|\gg1/N^2l$. An approximate interpolation formula for all $\sigma$ agrees reasonably well with numerical simulations.Comment: 13 pages, RevTeX, 1 postscript figur

Probability of Reflection by a Random Laser

A theory is presented (and supported by numerical simulations) for phase-coherent reflection of light by a disordered medium which either absorbs or amplifies radiation. The distribution of reflection eigenvalues is shown to be the Laguerre ensemble of random-matrix theory. The statistical fluctuations of the albedo (the ratio of reflected and incident power) are computed for arbitrary ratio of sample thickness, mean free path, and absorption or amplification length. On approaching the laser threshold all moments of the distribution of the albedo diverge. Its modal value remains finite, however, and acquires an anomalous dependence on the illuminated surface area.Comment: 8 pages (revtex), 3 figures, to appear in Phys.Rev.Let

Brightness of a phase-conjugating mirror behind a random medium

A random-matrix theory is presented for the reflection of light by a disordered medium backed by a phase-conjugating mirror. Two regimes are distinguished, depending on the relative magnitude of the inverse dwell time of a photon in the disordered medium and the frequency shift acquired at the mirror. The qualitatively different dependence of the reflectance on the degree of disorder in the two regimes suggests a distinctive experimental test for cancellation of phase shifts in a random medium.Comment: 4 pages LaTeX. 2 Postscript figures include

Reflection of light from a disordered medium backed by a phase-conjugating mirror

This is a theoretical study of the interplay of optical phase-conjugation and multiple scattering. We calculate the intensity of light reflected by a phase-conjugating mirror when it is placed behind a disordered medium. We compare the results of a fully phase-coherent theory with those from the theory of radiative transfer. Both methods are equivalent if the dwell time \tau_{dwell} of a photon in the disordered medium is much larger than the inverse of the frequency shift 2\Delta\omega acquired at the phase-conjugating mirror. When \tau_{dwell} \Delta\omega < 1, in contrast, phase coherence drastically affects the reflected intensity. In particular, a minimum in the dependence of the reflectance on the disorder strength disappears when \Delta\omega is reduced below 1/\tau_{dwell}. The analogies and differences with Andreev reflection of electrons at the interface between a normal metal and a superconductor are discussed.Comment: 27 pages RevTeX with 11 figures included with psfi

Field and intensity correlations in amplifying random media

We study local and nonlocal correlations of light transmitted through active random media. The conventional approach results in divergence of ensemble averaged correlation functions due to existence of lasing realizations. We introduce conditional average for correlation functions by omitting the divergent realizations. Our numerical simulation reveals that amplification does not affect local spatial correlation. The nonlocal intensity correlations are strongly magnified due to selective enhancement of the contributions from long propagation paths. We also show that by increasing gain, the average mode linewidth can be made smaller than the average mode spacing. This implies that light transport through a diffusive random system with gain could exhibit some similarities to that through a localized passive system, owing to dominant influence of the resonant modes with narrow width.Comment: 5 pages, 4 figure

Study of transmission and reflection from a disordered lasing medium

A numerical study of the statistics of transmission ($t$) and reflection ($r$) of quasi-particles from a one-dimensional disordered lasing or amplifying medium is presented. The amplification is introduced via a uniform imaginary part in the site energies in the disordered segment of the single-band tight binding model. It is shown that $t$ is a non-self-averaging quantity. The cross-over length scale above which the amplification suppresses the transmittance is studied as a function of amplification strength. A new cross-over length scale is introduced in the regime of strong disorder and weak amplification. The stationary distribution of the backscattered reflection coefficient is shown to differ qualitatively from the earlier analytical results obtained within the random phase approximation.Comment: 5 pages RevTex (twocolumn format), 5 EPS figures, considerably modifie

Berry phase and adiabaticity of a spin diffusing in a non-uniform magnetic field

An electron spin moving adiabatically in a strong, spatially non-uniform magnetic field accumulates a geometric phase or Berry phase, which might be observable as a conductance oscillation in a mesoscopic ring. Two contradicting theories exist for how strong the magnetic field should be to ensure adiabaticity if the motion is diffusive. To resolve this controversy, we study the effect of a non-uniform magnetic field on the spin polarization and on the weak-localization effect. The diffusion equation for the Cooperon is solved exactly. Adiabaticity requires that the spin-precession time is short compared to the elastic scattering time - it is not sufficient that it is short compared to the diffusion time around the ring. This strong condition severely complicates the experimental observation.Comment: 16 pages REVTEX, including 3 figure

Dynamic effect of phase conjugation on wave localization

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 frequency omega_0 of the phase-conjugating mirror. The decay in time of the average power spectrum of the reflected pulse is delayed for frequencies near omega_0. In the presence of localization the resonance width is tau_s^{-1}exp(-L/l), with L the length of the waveguide, l the mean free path, and tau_s the scattering time. Inside this frequency range the decay of the average power spectrum is delayed up to times t simeq tau_s exp(L/l).Comment: 10 pages including 2 figure

Statistical Properties of the Reflectance and Transmittance of an Amplifying Random Media

Statistical properties of the transmittance ($T$) and reflectance ($R$) of an amplifying layer with one-dimensional disorder are investigated analytically. Whereas the transmittance at typical realizations decreases exponentially with the layer thickness $L$ just as it does in absorbing media, the average $\left\langle T\right\rangle$ and $\left\langle R\right\rangle$\ are shown to be infinite even for finite $L$ due to the contribution of low-probable resonant realizations corresponding to the non-Gaussian tail of the distribution of $\ln T$. This tail differs drastically from that in the case of absorption. The physical meaning of typical and resonant realizations is discussed.Comment: 5 pages (RevTeX