Half-slit phase-step diffraction factorization ($I_{\pi }/I_{0,\mathrm{peak}}={\mathrm{sinc}}^{2}s{\sin }^{2}s$)

Two adjacent half-slits of width h with opposite field transmission have centres separated by h. The aperture transform is $h\mathrm{sinc}(qh/2)[e^{-iqh/2}-e^{iqh/2}]$. Thus the phase-reversed pair factor multiplies the finite-half-slit envelope, giving the displayed normalized intensity with $s=qh/2$ and $\mathrm{sinc}s=\sin s/s$. The on-axis field cancels. Positive side maxima satisfy $\tan s=2s$, and total integrated transmission is unchanged by a lossless phase plate even though the angular distribution changes.