Effective Medium Expression for the Optical Properties of Periodic Multilayer Films

Abstract

We derive an expression for the normal incidence transmittance of light through a multilayer film consisting of an arbitrary number of sequential layers of two materials with different optical constants. For wavelengths large compared to the thicknesses of the two materials, d_a and d_b, the optical properties of the multilayer film are determined by a complex effective medium dielectric constant ε_e=ε_ad_a/(d_a+d_b)+ε_bd_b/(d_a+d_b) where ε_a and ε_b are the complex dielectric constants of the two materials. We verify the correctness of the effective medium expression by comparing the optical spectra of multilayer films predicted by this formula with numerical calculations obtained by solving the boundary value problem for all layers. After establishing the validity of the effective medium expression, we then use it to analyze optical transmittance data of multilayer films consisting of hydrogenated amorphous silicon (a-Si:H) and amorphous silicon nitride (a-SiNx:H). The optical gap E_o of the a-Si:H layers is found to increase with decreasing a-Si:H layer thickness whereas the slope B of the Tauc plot remains constant.