Weyl plane-wave representation ($\frac{e^{ikR}}{R}=\frac{ik}{2\pi }\int \frac{e^{ik[p(x-x^{\prime })+q(y-y^{\prime })+m|z-z^{\prime }|]}}{m}dpdq$)

Here $m=\sqrt{1-p^2-q^2}$ has nonnegative imaginary part. This representation of the outgoing Helmholtz equation kernel separates propagating plane waves from evanescent waves. The absolute vertical separation is required for fields on either side of a source; choosing its sign fixes the upward or downward angular spectrum.