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For monochromatic solutions of D'Alembert's wave equation and Maxwell'sequations, we obtain sharp bounds on the sup norm as a function of the farfield energy. The extremizer in the scalar case is radial. In the case ofMaxwell's equation, the electric field maximizing the value at the originfollows longitude lines on the sphere at infinity. In dimension $d=3$ thehighest electric field for Maxwell's equation is smaller by a factor 2/3 thanthe 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 thewavelength. The extremizing fields are identical to those that attain themaximum field intensity at the origin.

Optimal Focusing for Monochromatic Scalar and Electromagnetic Waves