广义惠更斯原理与脉冲束小波
Generalized Huygens principle with pulsed-beam wavelets
摘要：
惠更斯原理由于次级小波的球形性质而具有关于反向传播的众所周知的问题。我们通过解析地继续积分曲面来解决这个问题。如果曲面是半径为$R$的球体，则通过将$R$复化为$R+ia$来完成。由此产生的复杂的球体被证明是一个真正的盘束半径$a$相切的领域。惠更斯的“次级源点”因此被圆盘所取代，而他的球面小波则被向外传播的聚焦良好的脉冲光束所取代。这解决了反向传播问题。扩展惠更斯原理是脉冲光束的完备性关系，给出了一般辐射场作为这种光束的叠加的表示。此外，它自然产生一个非常有效的方法来计算辐射场，因为所有的脉冲光束错过了一个给定的观察者可以忽略不计。增加$a$会使脉冲光束的焦点锐化，这反过来又会提高表示的压缩。
英文摘要：
Huygens' principle has a well-known problem with back-propagation due to the spherical nature of the secondary wavelets. We solve this by analytically continuing the surface of integration. If the surface is a sphere of radius $R$, this is done by complexifying $R$ to $R+ia$. The resulting complex sphere is shown to be a real bundle of disks with radius $a$ tangent to the sphere. Huygens' "secondary source points" are thus replaced by disks, and his spherical wavelets by well-focused pulsed beams propagating outward. This solves the back-propagation problem. The extended Huygens principle is a completeness relation for pulsed beams, giving a representation of a general radiation field as a superposition of such beams. Furthermore, it naturally yields a very efficient way to compute radiation fields because all pulsed beams missing a given observer can be ignored. Increasing $a$ sharpens the focus of the pulsed beams, which in turn raises the compression of the representation.
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