0 Time - Reversed Diffraction

In the usual formulation of the Kirchhoff diffraction integral, a scalar field with harmonic time dependence at frequency ω is deduced at the interior of a charge-free volume from knowledge of the field (or its normal derivative) on the bounding surface. In particular, the field is propagated forwards in time from the boundary to the desired observation point. Construct a time-reversed version of the Kirchhoff integral in which the knowledge of the field on the boundary is propagated backwards in time into the interior of the volume. Consider the example of an optical focus at the origin for a system with the z axis as the optic axis. In the far field beyond the focus a Gaussian beam has cone angle θ0 ≡ √ 2σθ, and the x component of the electric field in a spherical coordinate system is given approximately by Ex(r, θ, φ, t) = E(r)e e /θ 0 , (1)