Theory of Electromagnetic Wave Scattering and Dispersion in Exponential Materials

The comprehension and control of wave propagation and energy transfer within exponential materials remain an open question to multiple research disciplines. Exponential materials, ubiquitous in nature due to gravitational effects, are found in various contexts including the earth's atmosphere, ground soil, and marine sediments. In this work, we propose a universal theory for predicting the electromagnetic scattering spectra of unidirectional exponential materials characterized by inhomogeneous exponential permittivity and/or permeability. The core concept of our approach revolves around the derivation of modified transfer and scattering matrices through the analytical generalization of plane-wave eigenmodes from uniform to exponential materials. This model's universality and precision demonstrate a superior performance when juxtaposed with two widely recognized theories — the small reflection theory and the transfer matrix method. Moreover, our theory enables the analytical exploration of dispersive group/phase velocities instigated by exponential properties. Theoretical predictions are validated through numerical examples, which are used for predicting scattering spectra. To underscore the efficacy of the proposed theory, we first compute the functionality of field modulation (either amplification or attenuation), and secondly, we analytically establish the correlation between the broadband impedance matching effects of exponential materials and spatial inhomogeneity. Based on these findings, we propose an inverse design scheme for crafting planar layered metamaterials with exceptional broadband anti-reflection performance in the microwave band. This holds potential for resolving impedance matching challenges in diverse systems. The insights garnered from this research serve to deepen our understanding of inhomogeneous gradient materials and to improve their efficient modeling and design.