In previous work, we have introduced an analytical approach that utilizes the dispersion relation for an infinite periodic multilayer structure to predict the performance of finite multilayer structures. We have validated the accuracy of our predictions by demonstrating numerical agreement with other established simulation methods, such as the transfer matrix method, and through experimental confirmation. In this work, we employ dispersion relations to first illustrate that metallo-dielectric structures, as opposed to multilayer dielectric-dielectric structures, can efficiently yield a sharp-edge transmittance spectrum profile, with control over both sides of the bandpass cutoff edges. Our approach also enables the calculation of effective permittivity without relying on traditional homogenization techniques. Next, utilizing the concept of effective permittivity, we illustrate that increasing the thickness of specific dielectric layers within MD structures leads to narrower passbands without significant loss in transmission, demonstrating the potential of this approach for engineering the transmittance spectrum of bandpass filters in the visible and near-IR regions. The capability to achieve a sharp-edge filter with a limited number of layers further underscores the cost-effectiveness of such bandpass filters.
We show applications of our analytical approach to predict the performance of multilayer metallo-dielectric bandpass filters, which also enables estimation of their effective permittivity without relying on homogenization techniques. The approach is based on the one-dimensional dispersion relation for an infinite metallo-dielectric structure that accounts for the complex nature of the permittivities for the metal and dielectric constituents. The dispersion relation clearly reveals the band structure (often comprising multiple passbands), directly provides transmittance characteristics such as center wavelengths and bandwidths and enables the calculation of effective propagation constant and effective attenuation. In this work, we evaluate the dispersion relations for metallo-dielectric structures with complex refractive index data for the metal, viz., Ag, acquired from different sources to show the differences in the center wavelength and the cutoff wavelengths. We verify the accuracy of our method numerically by comparing the transmittance spectrum of finite metallo-dielectric structures using the transfer matrix method. We also plot the dispersion relation using Al as the metal and show the differences in the dispersion relations of the infinite structure and the transmittances of the finite structures relative to Ag. Extension to determination of dispersion relations for other polarizations, viz., transverse magnetic, is discussed, along with corresponding transmittance spectra for oblique incidence.
We extend the transfer matrix method to study the propagation of beams and arbitrary profiled fields through anisotropic metamaterial slabs, and to demonstrate the negative refractive index property resulting in linear self-focusing of beams in hyperbolic metamaterials. Specifically, the transfer matrix method, commonly used to analyze bi-directional plane wave propagation, is developed to analyze beam propagation. By expressing a Gaussian beam as an angular spectrum of plane waves, an anisotropic transfer matrix, which is also obtained using the eigenvalues mentioned above, can be applied to calculate the beam spectrum at an arbitrary distance of propagation through a hyperbolic metamaterial. With given incident and emergent media, say, air, linear self-focusing within the metamaterial slab and subsequent reimaging in the emergent medium are numerically investigated for one transverse dimensional TM polarized Gaussian beam. Simulation results are compared with results from the unidirectional transfer function approach. The anisotropic transfer matrix method can be used to study beam transmission and reflection at the interfaces, and can be applied to analyze optical propagation through anisotropic metamaterial on uniaxial electro-optic substrates. The technique can be extended to arbitrary initial optical field profiles in one transverse dimension to assess the imaging quality of metamaterial slabs.
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