Theory of light diffusion in disordered media with linear absorption or gain

A. Lubatsch, J. Kroha, and K. Busch

Phys. Rev. B 71, 184201 – Published 10 May 2005

Abstract

We present a detailed, microscopic transport theory for light in strongly scattering disordered systems whose constituent materials exhibit linear absorption or gain. Starting from Maxwell’s equations, we derive general expressions for transport quantities such as energy transport velocity, transport mean free path, diffusion coefficient, and absorption/gain length. The approach is based on a fully vectorial treatment of the generalized kinetic equation and utilizes an exact Ward identity (WI). While for loss- and gainless media the WI reflects local energy conservation, the effects of absorption or coherent gain are implemented exactly by additional terms in the WI. As a result of resonant (Mie) scattering from the individual scatterers, all transport quantities acquire strong, frequency-dependent renormalizations, which are, in addition, characteristically modified by absorption or gain. We illustrate the influence of various experimentally accessible parameters on these quanitities for dilute systems. The transport theory presented here may set the stage for a theory of random lasing in three-dimensional disordered media.

Received 25 November 2004

DOI:https://doi.org/10.1103/PhysRevB.71.184201

Authors & Affiliations

A. Lubatsch and J. Kroha

Physikalisches Institut, Universität Bonn, 53115 Bonn, Germany

K. Busch

Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany, and Department of Physics and College of Optics & Photonics: CREOL & FPCE, University of Central Florida, Orlando, Florida 32816, USA

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Vol. 71, Iss. 18 — 1 May 2005

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Images

Figure 1
Diagrammatic representation of the Bethe-Salpeter equation, Eq. (9), indicating the tensorial structure as well as the full index notation.Reuse & Permissions
Figure 2
Frequency dependence of the energy transport velocity $v_{E}$ units of the vacuum speed of light for a dilute ( $n = 6 %$ by volume) collection of spherical scatterers in air $(ϵ_{h} = 1)$ with different dielectric constants $ϵ_{s}$ (as indicated). The frequency unit is $ω_{0} = 2 π c ∕ r_{0}$ where $c$ is the vacuum speed of light and $r_{0}$ the radius of the scatterers. The pronounced dips correspond to the Mie resonances of the individual sphere and represent the “dwell time” effect discussed in the text.Reuse & Permissions
Figure 3
Frequency dependence of the phase velocity $c_{P}$ units of the vacuum speed of light for a dilute ( $n = 6 %$ by volume) collection of spherical scatterers in air $(ϵ_{h} = 1)$ with different dielectric constants $ϵ_{s}$ (as indicated). The frequency unit is $ω_{0} = 2 π c ∕ r_{0}$ where $c$ is the vacuum speed of light and $r_{0}$ the radius of the scatterers. Near Mie resonances, this velocity significantly exceeds the vacuum speed of light. The corresponding physical energy transport velocity is displayed in Fig. 2.Reuse & Permissions
Figure 4
Frequency dependence of the energy transport velocity $v_{E}$ units of the vacuum speed of light for a dilute ( $n = 6 %$ by volume) collection of spherical scatterers in air $(ϵ_{h} = 1)$ where the scatterer dielectric constant $ϵ_{s}$ exhibits different amounts of linear gain (as indicated). The frequency unit is $ω_{0} = 2 π c ∕ r_{0}$ where $c$ is the vacuum speed of light and $r_{0}$ the radius of the scatterers. For increasing gain, the dips in the energy transport velocity that are associated with the individual scatterers’ Mie resonances exhibit considerable narrowing. This may be interpreted as a sign for the incipient onset of random lasing action in this system.Reuse & Permissions
Figure 5
Frequency dependence of the energy transport velocity $v_{E}$ units of the vacuum speed of light for a dilute ( $n = 6 %$ by volume) collection of spherical scatterers in air $(ϵ_{h} = 1)$ where the scatterer dielectric constant $ϵ_{s}$ exhibits different amounts of linear absorption (as indicated). The frequency unit is $ω_{0} = 2 π c ∕ r_{0}$ where $c$ is the vacuum speed of light and $r_{0}$ the radius of the scatterers. For increasing absorption, the dips in the energy transport velocity that are associated with the individual scatterers’ Mie resonances become washed out and disappear altogether for sufficiently large absorption, as the wave propagation becomes overdamped and interference effects become impossible.Reuse & Permissions

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