Optimal Focusing for Monochromatic Scalar and Electromagnetic Waves

TL;DR

This paper derives sharp bounds on the maximum amplitude of monochromatic scalar and electromagnetic waves based on far field energy, identifying extremizers and their geometric properties in various dimensions.

Contribution

It provides new sharp bounds and characterizes extremizers for monochromatic wave solutions, revealing differences between scalar and electromagnetic cases and their geometric configurations.

Findings

Radial extremizer for scalar waves.

Electric field extremizer follows longitude lines at infinity.

Maximum electric field in Maxwell's equations is smaller by 2/3 than scalar waves.

Abstract

For monochromatic solutions of D'Alembert's wave equation and Maxwell's equations, we obtain sharp bounds on the sup norm as a function of the far field energy. The extremizer in the scalar case is radial. In the case of Maxwell's equation, the electric field maximizing the value at the origin follows longitude lines on the sphere at infinity. In dimension $d=3$ the highest electric field for Maxwell's equation is smaller by a factor 2/3 than the highest corresponding scalar waves. The highest electric field densities on the balls $B_R(0)$ occur as $R\to 0$. The density dips to half max at $R$ approximately equal to one third the wavelength. The extremizing fields are identical to those that attain the maximum field intensity at the origin.