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arXiv:0801.4409

The equivalent medium theory within the linear response region

Li‐Gang Wang

Published Jan 29, 2008• 0 citations
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Abstract

In this paper, we present the equivalent medium theory by using the linear response theory. It is found that, under the condition of the linear response, a series of different media with different refractive indices $n_{i}(ω)$ and lengths $d_{i}$ can be equivalent to an effective medium with the volume-averaged refractive index $\frac{1}{D}\sum_{i=1}^{N}n_{i}(ω)d_{i}$ and the total length $D=\sum_{i=1i}^{N}d_{i}$,where $N$ is the number of different media. Based on this equivalent theory, it is a simple but very useful method to design the effective medium with any desirable dispersion properties. As an example, we present a proposal to obtain the enhancement or reduction of the refractive index without absorption and the large dispersion without obvious absorption by assembling different linear dispersive gain and absorptive media.

DOI
10.48550/arxiv.0801.4409

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In this paper, we present the equivalent medium theory by using the linear response theory. It is found that, under the condition of the linear response, a series of different media with different refractive indices ni(ω)n_{i}(ω)ni​(ω) and lengths did_{i}di​ can be equivalent to an effective medium with the volume-averaged refractive index 1D∑i=1Nni(ω)di\frac{1}{D}\sum_{i=1}^{N}n_{i}(ω)d_{i}D1​∑i=1N​ni​(ω)di​ and the total length D=∑i=1iNdiD=\sum_{i=1i}^{N}d_{i}D=∑i=1iN​di​,where NNN is the number of different media. Based on this equivalent theory, it is a simple but very useful method to design the effective medium with any desirable dispersion properties. As an example, we present a proposal to obtain the enhancement or reduction of the refractive index without absorption and the large dispersion without obvious absorption by assembling different linear dispersive gain and absorptive media.
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A method for controlling the refractive index that is based on the backward rescattering of the scattered and amplified electromagnetic waves is proposed and analysed. The resulting field is a sum of the initial (probe) wave and the rescattered wave. The effective wave number of the total wave can be varied by varying the gain. This variation in the wave number can be treated as a variation in the refractive index, although the physical properties of the medium do not change. The effective refractive index for the total wave propagating in such a device was called the virtual refractive index.
It is well known that, for static fields, the local microscopic electric field EL, that couples to the atomic dipole moment p, is related to the macroscopic field E and volume polarization P, in a dense dielectric medium, by the Lorentz-Lorenz (LL) relation.1 For dense media, this relation implies a nonlinear relationship between the microscopic molecular polarizability α and the macroscopic susceptibility χ, for a material volume density N. The LL relation also holds for arbitrary time-dependent fields in a nonlinear medium.2 What this implies is that all of quantum optics that uses an external electric field rather than the local field is internally inconsistent and will yield only approximate results for an optically dense medium. We consider the effect of the local field interaction on a simple three-level A system predicted to exhibit lasing without inversion and a high index of refraction at zero absorption.3 A threshold transition allows us to choose a value of this parameter that enhances inversionless gain and absorptionless dispersion by at least two orders of magnitude over that foundin usual lasing without inversion.4
By utilizing the interference between an absorptive resonance and an amplifying resonance, one can achieve an enhanced refractive index without an increase in absorption to the beam. We analyze noise added to the beam due to spontaneous emission while propagating through such an index enhanced medium. We find that, for a medium with a refractive index of nnn and of length LLL, $\ensuremath{\approx}(n\ensuremath{-}1)(2\ensuremath{\pi}∕{\ensuremath{\lambda}}_{0})L$ noise photons are added to the beam.
Linear media are predicted to exist whose relative permiability is an operator in the space of quantum states of light. Such media are characterized by a photon statistics--dependent refractive index. This indicates a new type of optical dispersion -- the photon statistics dispersion. Interaction of quantum light with such media modifies the photon number distribution and, in particular, the degree of coherence of light. An excitonic composite -- a collection of noninteracting quantum dots -- is considered as a realization of the medium with the photon statistics dispersion. Expressions are derived for generalized plane waves in an excitonic composite and input--output relations for a planar layer of the material. Transformation rules for different photon initial states are analyzed. Utilization of the photon statistics dispersion in potential quantum--optical devices is discussed.