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Annual Meeting Optical Society of America

Effective medium theory of nonlinear optical properties of composites of semiconductor microstallites

Chi C. Sung, Y. Q. Li, R. Ingura

Published Jan 1, 1987• 0 citations
Publisher

Abstract

Nonlinear optical properties of microstructures immersed in a linear dielectric are investigated. The microstructures are assumed to be composed of simple saturable two-level atoms. The local field corrections are included in a self-consistent manner leading to multiple solutions for the absorptive part of the dielectric constant of the microstructure as noted earlier by Chemla and Miller.1 For a given structure we use the effective medium theory and average over the random positions of the grains to obtain the effective medium dielectric constant in the quasistatic limit (λ ≪ grain size) as a function of the volume fraction of the grains. The results exhibit sharp hysteresis effects. However, in the study of the propagation in the medium these effects are modified considerably. While defined switches are observed in the mean field at certain points in the medium where the absorption jumps from the upper to lower branch, the optical intensity undergoes only a change of slope.

DOI
10.1364/oam.1987.tuj10

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Nonlinear optical properties of microstructures immersed in a linear dielectric are investigated. The microstructures are assumed to be composed of simple saturable two-level atoms. The local field corrections are included in a self-consistent manner leading to multiple solutions for the absorptive part of the dielectric constant of the microstructure as noted earlier by Chemla and Miller.1 For a given structure we use the effective medium theory and average over the random positions of the grains to obtain the effective medium dielectric constant in the quasistatic limit (λ ≪ grain size) as a function of the volume fraction of the grains. The results exhibit sharp hysteresis effects. However, in the study of the propagation in the medium these effects are modified considerably. While defined switches are observed in the mean field at certain points in the medium where the absorption jumps from the upper to lower branch, the optical intensity undergoes only a change of slope.
The propagation of electromagnetic waves in a nonlinear medium composed of semiconductor microcrystallites embedded in a host medium has been investigated. Explicit results are given for the case of cadmium sulfide grains in glass when a saturable two-level approximation is used for the nonlinear dielectric function. The bistable behavior, noticed earlier by Chemla and Miller [ Opt. Lett.11, 522 ( 1986)], for single grains is manifested in the propagation studies conducted here by the occurrence of a discontinuity in the index of refraction in the medium. Various approximations are used to obtain the optical bistability properties. Possible applications of this feature are pointed out.
We examine the nonlinear dielectric response of small spherical grains ϵ m embedded in a linear dielectric ϵ d . The models for the spherical grains are a Kerr medium1,2 and a two-level system.3 The expression for the dielectric function ϵ m is dependent of the grain size; in addition it also depends on the local electric field E L . This field is related to the external applied field E0 by the equation E L = (3ϵ d /ϵ m + 2ϵ d )E0. The steady-state equation for the averaged external field is ∇2E0 + (ω2/c2)ϵ*E0 = 0, where ω is the frequency of the field, ϵ* is the dielectric function from the effective medium approximation, and it is obtained from the self-consistency relation4 〈ϵ(r) − ϵ*/ϵ(r) + 2ϵ*〉 = 0; the angular brackets denote an average taken over the ensemble of dielectric spheres distributed in a small volume. Using the slowly varying-envelope approximation we solve these equations for a distribution of grain radii and the affect this has on the effective optical nonlinearities. In particular, we show its effect on the optical bistability for these composite systems.
An effective-medium theory for nonlinear spherical particles embedded in a linear host medium is developed. The formulation is based on a T-matrix approach. Introduction of a field-dependent T matrix enables one to obtain an exact expression for effective dielectric constant of a medium with spherical grains. In the special cases of low-field values, the results obtained previously for the third- and fifth-order nonlinear susceptibilities of the medium are recovered. The resonance behavior of the effective dielectric constant and intrinsic optical bistability are also discussed.
Abstract Optically bistable behaviour is demonstrated for systems consisting of optically nonlinear microcrystallites embedded in an optically linear metal or dielectric. The most simple version of the local‐field correction serves as feedback mechanism thus avoiding external mirrors. The dielectric function for macroscopic samples is calculated with Garnett's effective medium theory. Numerical examples for the reflectivity and the transmissivity are given for an embedded dielectric with exciton‐like nonlinearity.
The effect of level degeneracy on saturated absorption in a resonant medium is considered using a simple but realistic three-level system to model $J=1\ensuremath{\rightarrow}J=0$ atomic transition. Assuming homogeneous broadening and exact resonance, an analytic nonperturbative expression for the medium susceptibility is obtained and used to study propagation of an arbitrarily polarized optical field. Optical saturation affects considerably Zeeman-coherence-induced circular dichroism. The analysis may find application in high-resolution spectroscopy and other branches of nonlinear optics.
I recent years, significant attention has been given to nonlinear optical phenomena, such as nonlinear optical absorption and dispersion, derived from a driven quantum transition of semiconductor nanostructures, such as semiconductor quantum wells, quantum disks and mainly quantum dots. An important issue that is widely studied is the effect of probe electromagnetic field intensity on the total (nonlinear) absorption coefficient and on the total (nonlinear) index of refraction. Some of the phenomena that are commonly found for large electromagnetic field intensities are the creation of a strong dip in the absorption spectrum near or at resonance (the so-called bleaching effect), negative absorption near or at resonance that leads to optical gain and the change of the slope of the total index of refraction near or at resonance from negative to positive that changes the behavior of the system from fast to slow light. Unfortunately, these phenomena do not exist and are artifacts of a common abuse of the theoretical methodology for the study of the total absorption coefficient and the total index of refraction. The main error is that the total absorption coefficient and the total index of refraction was initially derived from density matrix equations using perturbation theory and includes first-order (linear) as well as third-order (nonlinear) terms. This method gives correct results for small electromagnetic field intensities and it leads to saturation of absorption and dispersion in this regime. However, for larger intensities it gives the misleading results mentioned above due to either failure of the order of perturbation theory considered in the derivation of the total absorption coefficient and the total index of refraction or even failure of perturbation theory at all in the treatment of the problem. In this work, we revisit the problem of the total absorption coefficient and the total index of refraction in a symmetric semiconductor quantum dot structure (we choose the example of a core-shell quantum dot) under a strong probe field excitation. We use the two-level model, solve the relevant density matrix equations under the rotating wave approximation and under steady state conditions, and obtain the correct form of the nonlinear optical susceptibility that is then used for the derivation of the formulae of the total absorption coefficient and the total index of refraction under the interaction with a strong probe field. Actually, the commonly used perturbation theory formulae can be obtained by our formulae using a series expansion and keeping terms up to second order in electric field amplitude. We also derive the electronic structure of the quantum dot system by solving numerically the time-independent Schrodinger equation, under the effective mass approximation, using the potential morphing method and obtain the energy difference and the electric dipole matrix element for the relevant transitions of the quantum dot system. Then, for the specific quantum dot system we compare the results of the total absorption coefficient and the total index of refraction for the two methodologies (the perturbation result and our result) for different electromagnetic field intensities.