 Half-slit phase-reversal diffraction

Reversing the phase on one half of an otherwise uniform slit makes its aperture transmission odd. Its Fourier amplitude is $4{\sin }^2(qb/4)/(iq)$, where $q=k\sin \theta$, so the axis is dark despite the removable denominator singularity. The largest symmetric lobes satisfy $\tan s=2s$, with first positive root approximately $1.16556$. Doubling an ideal half-wave layer restores equal phases and the ordinary bright-centred single-slit pattern. Total transmitted power is conserved if the layer has no absorption or reflection.