单色标量波和电磁波的最佳聚焦
Optimal Focusing for Monochromatic Scalar and Electromagnetic Waves
摘要：
对于D 'Alembert波动方程和Maxwell方程的单色解，我们得到了作为远场能量函数的超模的精确界。标量情况下的极值化器是径向的。在麦克斯韦方程的情况下，在原点处最大化值的电场沿着无穷远处球体上的经线。在维度$d=3$中，麦克斯韦方程的最高电场比最高对应标量波小2/3倍。球$B_R(0)$上的最高电场密度以$R\to 0$出现。在$R$处，密度下降到半最大值，约等于波长的三分之一。极值场与在原点处达到最大场强的场是相同的。
英文摘要：
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.
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