  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.
