= Half-slit phase-reversal diffraction
{title2=$I/I_0=\sin^4s/s^2,\quad s=kb\sin\theta/4$}
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>.
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