Partially Coherent Spectral Transmittance of Dielectric Thin Films with Rough Surfaces

Abstract

The effects of scattering and partial coherence on the transmittance of thin films with rough surfaces are investigated. For the rough side, the rms roughness is between 0.3 and 0.7 µm, whereas the autocorrelation length ranges from 1 to 36 µm. Scalar scattering theory (SST) is used to account for the scattering losses in the specular direction due to surface roughness, and then the calculated transmittance is spectrally averaged over the coherence spectral width. The spectral averaging method takes into consideration the effect of partial coherence. A Fouriertransform infrared spectrometer measures the near-normal transmittance in the midinfrared region from 2- to 20-µ mw avelengths. Comparison of the calculated transmittance with that of the measured transmittance shows that the combination of SST with spectral averaging can correctly predict the measured fringe contrast and fringe flipping, whereas SST alone does not result in quantitative agreement with the experiments. A coherence function is introduced to characterize the degree of coherence as a function of wave number for samples with or without ar ough surface. Furthermore, it is shown that SST is not applicable to surfaces whose autocorrelation length is greater than the wavelength of the incident radiation. Nomenclature d = thickness, m i = √ −1 k = imaginary part of the complex refractive index n = real part of the complex refractive index R = reflectance r = Fresnel reflection coefficient S = scattering factor T = transmittance ¯ T = spectrally averaged transmittance t = Fresnel transmission coefficient β = phase shift, rad �ν = free spectral range, m −1 δν = coherence spectral width, m −1 θ = polar angle, rad λ =w avelength of the incident radiation in vacuum, m ν =w ave number, m −1 σ = rms roughness, m τ = internal transmissivity � = fringe contrast φ = coherence function Subscripts coh = coherent p = p polarization r = reflection s = s polarization t = transmission