Effective Medium Theory

In order to describe and predict the optical properties of a medium consisting different phases, one should consider structural geometry and aggregate topology of the medium which will make the model derivation a challenging issue. To solve this problem, these systems can be took into account as an optically quasi-homogeneous effective-medium materials. The main idea of such an assumption is to replace the inhomogeneous composite by a homogeneous material with specific dielectric function and consequently calculation of optical properties such as reflectance and transmittance as a linear response of this structure.

effective_medium

Precisely, to introduce a model for determining the dielectric function of a medium first we should start by examining the case of homogeneous media. In this case, one can describe the dielectric function by classical Clausius-Mossotti equation

eq1

Where α is the linear microscopic polarizability, ℵ is density of microscopic constituents and ε is dielectric coefficient of medium. Furthermore, for a two-phase media, Clausius-Mossotti equation can be extended to a first order effective medium approximation where the polarizabilities of component a is a and polarizabilities of component b is b. In this case we can
rewrite the Clausius-Mossotti equation as:

eq2

Finally we can write:

eq3

Where f is the volume fraction of the component i within the matrix.

References:

[1] Huimin Su, Yingshun Li, Xiao-Yuan Li, and Kam Sing Wong. Optical and electrical properties of au nanoparticles in two-dimensional networks: an effective cluster model. Optics Express, 17:22223–22234, 2009.
[2] V. Lucarini, J.J. Saarinen, K.-E. Peiponen, and E.M. Vartiainen. KramersKronig Relations in Optical Materials Research. Springer, Berlin, 2005.
[3] Michael Quinten. Optical properties of nanoparticle systems : Mie and beyond. Wiley-VCH, 1st Edition, 2011.

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