# Maximal Free-Space Concentration of Electromagnetic Waves

## TL;DR

This paper establishes fundamental limits on the maximum intensity concentration of electromagnetic waves in free space, revealing potential for significant enhancements and guiding the design of metasurfaces for various optical applications.

## Contribution

It derives universal upper bounds for free-space electromagnetic concentration, applicable to any spot size or device, and demonstrates near-limit metasurface designs using inverse methods.

## Key findings

- Bounds suggest orders-of-magnitude intensity enhancements are possible.
- Inverse-designed metasurfaces operate near theoretical concentration limits.
- Bounds are applicable across various metrics without symmetry or scattering assumptions.

## Abstract

We derive upper bounds to free-space concentration of electromagnetic waves, mapping out the limits to maximum intensity for any spot size and optical beam-shaping device. For sub-diffraction-limited optical beams, our bounds suggest the possibility for orders-of-magnitude intensity enhancements compared to existing demonstrations, and we use inverse design to discover metasurfaces operating near these new limits. We also demonstrate that our bounds may surprisingly describe maximum concentration defined by a wide variety of metrics. Our bounds require no assumptions about symmetry, scalar waves, or weak scattering, instead relying primarily on the transformation of a quadratic program via orthogonal-projection methods. The bounds and inverse-designed structures presented here can be useful for applications from imaging to 3D printing.