Optimization of periodic composite structures for sub-wavelength focusing

TL;DR

This paper formulates an optimization approach to design periodic dielectric composite structures that achieve sub-wavelength focusing, surpassing traditional photonic crystal limitations, with solutions validated through numerical examples.

Contribution

It introduces an optimization framework for designing dielectric composites with enhanced focusing capabilities, demonstrating the potential for smaller image spots than existing photonic crystals.

Findings

Existence of optimal solutions with low absorption

Numerical examples show improved focusing performance

Design method can produce sub-wavelength image spots

Abstract

Recently, there has been plenty of work in designing and fabricating materials with an effective negative refractive index. Veselago realized that a slab of material with a refractive index of -1 would act as a lens. Pendry suggested that the Veselago lens would act as a superlens, providing a perfect image of an object in contrast to conventional lenses which are only able to focus a point source to an image having a diameter of the order of the wavelength of the incident field. Recent work has shown that similar focusing effects can be obtained with certain slabs of ``conventional'' periodic composite materials: photonic crystals. The present work seeks to answer the question of what periodic dielectric composite medium (described by dielectric coefficient with positive real part) gives an optimal image of a point source. An optimization problem is formulated and it is shown that a solution exists provided the medium has small absorption. Solutions are characterized by an adjoint-state gradient condition, and several numerical examples illustrate both the plausibility of this design approach, and the possibility of obtaining smaller image spot sizes than with typical photonic crystals.