# Sequences of Exact Analytical Solutions for Plane-Waves in Graded Media

## TL;DR

This paper introduces a novel method using Darboux Transformations to generate exact analytical solutions for electromagnetic waves in graded media, enabling precise modeling of optical devices with simple, closed-form expressions.

## Contribution

It adapts the Property-and-Field Darboux Transformations from heat diffusion to Maxwell equations, providing a systematic way to create solvable profiles for complex optical structures.

## Key findings

- Exact analytical expressions for EM fields in graded media.
- Flexible construction of complex refractive index profiles.
- Application to design of optical devices like filters and gratings.

## Abstract

We present a new method for building sequences of solvable profiles of the electromagnetic (EM) admittance in lossless isotropic materials with 1D graded permittivity and permeability (in particular profiles of the optical refractive-index). These solvable profiles lead to analytical closed-form expressions of the EM fields, for both TE and TM modes. The Property-and-Field Darboux Transformations method, initially developed for heat diffusion modelling, is here transposed to the Maxwell equations in the optical-depth space. Several examples are provided, all stemming from a constant seed-potential, which makes them based on elementary functions only. Solvable profiles of increasingly complex shape can be obtained by iterating the process or by assembling highly flexible canonical profiles. Their implementation for modelling optical devices like matching layers, rugate filters, Bragg gratings, chirped mirrors or 1D photonic crystals, offers an exact and cost-effective alternative to the classical approaches

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Source: https://tomesphere.com/paper/1704.08929