# Second-Harmonic Generation in Nano-Structured Metamaterials

## TL;DR

This paper presents a theoretical and numerical investigation into second-harmonic generation in nano-structured metamaterials, demonstrating how geometric design enables efficient nonlinear optical responses in centrosymmetric material systems.

## Contribution

It introduces a novel formalism for calculating nonlinear polarization in complex metamaterials, allowing for tunable SH responses through geometric modifications.

## Key findings

- Efficient SH generation achieved by geometric design of inclusions.
- The formalism is adaptable to various materials and geometries.
- Tunable optical spectra via geometrical configuration changes.

## Abstract

We conduct a theoretical and numerical study on the second-harmonic (SH) optical response of a nano-structured metamaterial composed of a periodic array of inclusions. Both the inclusions and their surrounding matrix are made of centrosymmetrical materials, for which SH is strongly suppressed, but by appropriately choosing the shape of the inclusions, we may produce a geometrically non-centrosymmetric system which does allow efficient SH generation. Variations in the geometrical configuration allows tuning the linear and quadratic spectra of the optical response of the system. We develop an efficient scheme for calculating the nonlinear polarization, extending a formalism for the calculation of the macroscopic dielectric function using Haydock's recursion method. We apply the formalism developed here to an array of holes within an Ag matrix, but it can be readily applied to any metamaterial made of arbitrary materials and for inclusions of any geometry within the long-wavelength regime.