Theory and fabrication of colloidal-crystal Bragg filters
A theory, based on material parameters, is developed for colloidal-crystal Bragg filters. The coupling constant, for coupled-wave equations, is derived to produce filter spectra. The spectra are modified by convolving with a Gaussian function to represent crystal defects. Scattering losses are calculated to attenuate the out-of-band transmission. The theory is tested by comparing theoretical spectra with experimental spectra from colloidal-crystal Bragg filters fabricated with polystyrene and poly(methyl methacrylate) colloid spheres. A novel cell, for containing the colloidal crystal, is presented in addition to crystal growing techniques. Coupled-wave theory spectra are compared with experimental filter spectra for filters with variable colloid sphere diameters, filter thicknesses, liquid refractive indices, and Bragg wavelengths. Spectral comparisons are based on the notch bandwidth, optical density, and out-of-band scattering losses. The bandwidths agree to within one nanometer provided the criteria for the Rayleigh-Gans scattering approximation are satisfied. The optical densities correlate with spectrophotometer-measured optical densities. And, scattering losses correspond to within 10 percent for all material parameters. Thus, the coupled-wave theory is consistent with the data and provides an excellent tool for evaluating colloidal-crystal Bragg filter performance.
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