Prediction of metallo-dielectric transmission filter performance based on underlying dispersion relations

Abstract

The dispersion relation for electromagnetic/optical wave propagation based on the Helmholtz equation for an infinite one-dimensional metallo-dielectric structure is derived using the Bloch theorem and heuristically modified to include material dispersion. We investigate the connection between the dispersion relation of an infinite metallo-dielectric structure with the transmittance characteristics of finite metallo-dielectric structures. The dispersion relation is used to determine the center wavelength and bandwidth as a function of the material properties and the thicknesses of the metal and dielectric layers. These estimates are found to be in excellent agreement with the values obtained from numerically calculated transmittance spectra using the transfer matrix method for finite metallo-dielectric structures with the same building units. The dispersion relation calculations and simulations for the transmittance are done for an ideal case where the real part for the refractive index of the metal and imaginary part for the refractive index of the dielectric are zero, and also with actual values of the real and imaginary parts of the refractive index for the metal and dielectric obtained from literature, instead of using the canonical Drude model for the metal. It is shown that the real part of the dispersion relation for the actual case is almost identical to the ideal case in the visible and NIR range, implying that essential information on the center wavelength and bandwidth can be obtained from the ideal dispersion relation. It is also found that an ideal metal and dielectric give near-unity transmittance in the passband. It is predominantly the presence of a finite real part of the refractive index of the metal that introduces attenuation. The effective refractive index of the structure can also be determined. Oscillations present in the transmittance spectrum can be explained as a Fabry–Perot effect. Approximate simple estimates of the center wavelength and bandwidth can be useful in initiating intelligent designs of finite metallo-dielectric filter structures for fabrication and characterization.