Temporal coherence from crossed beams ($I(y)=I_0+\mathrm{Re}\Gamma (\sqrt{2}y/c)$)

Split one stationary scalar optical field into equal copies travelling at right angles with common transverse linear polarization. For wavevectors $k(\hat{\mathbf{x}}\pm \hat{\mathbf{y}})/\sqrt{2}$, the relative geometric delay is $\sqrt{2}y/c$ in vacuum, while the delay on the $x$ bisector is zero. The electric-field intensity convention gives the displayed pattern and $I(x)=2I_0$. The mean Poynting vector points along $+x$, with magnitude $I(y)/\sqrt{2}$ in that convention. A transverse detector can record the $y$ profile if it resolves the period $\lambda /\sqrt{2}$; a longitudinal $x$ profile is not a transverse camera image.

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.