OPTIMAL FOCUSING FOR MONOCHROMATIC SCALAR AND ELECTROMAGNETIC WAVES

DOI10.1142/S0129055X11004448zbMATH Open1229.35289arXiv1008.3571OpenAlexW1974336260MaRDI QIDQ3098867

Jeffrey Rauch

Publication date: 18 November 2011

Published in: Reviews in Mathematical Physics (Search for Journal in Brave)

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 $Ro0$. 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.

Full work available at URL: https://arxiv.org/abs/1008.3571

zbMATH Keywords

Maxwell equationsenergy densityfocusingextreme light initiative

Mathematics Subject Classification ID

PDEs in connection with optics and electromagnetic theory (35Q60) Estimates of eigenvalues in context of PDEs (35P15) Higher-order hyperbolic equations (35L25) First-order hyperbolic systems (35L40) Maxwell equations (35Q61)

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