  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.
