 Optical path delay in crossed-beam interference

For the right-angle geometry of temporal coherence from crossed beams, a physical arm-path delay $\Delta \ell$ gives $I(y)=I_0+\mathrm{Re}\Gamma (\sqrt{2}y/c+\Delta \ell /c)$, with the sign set by the delayed arm. A quarter-cycle optical phase delay requires $\Delta \ell =\lambda /4$; it shifts the fringe pattern by a quarter of its period, $\lambda /(4\sqrt{2})$. A path delay equal to that latter length is a different operation. Replacing the shifted correlation by a phase multiplier additionally requires the delay to be small compared with the envelope optical coherence time.