 Four-step phase retrieval

Four-step phase retrieval estimates relative wave phase from four wave interference frames separated by quarter-cycle wave phase steps. Use the explicit signal-arm conventionwith a common background $C$ and modulation amplitude $A>0$. Opposite-frame differences giveso $\hat{\varphi }=atan2(Y,X)$. The order of the subtractions fixes the sign. Changing which arm receives the wave phase step requires a correspondingly changed convention.

The same data estimate $C=(I_0+I_{\pi /2}+I_{\pi }+I_{3\pi /2})/4$ and $A=\sqrt{X^2+Y^2}/2$. A simple model-consistency residual is $(I_0+I_{\pi })-(I_{\pi /2}+I_{3\pi /2})$, which is zero for the exact stated model. It does not detect every possible systematic error.

For an original example, let $C=3$, $A=2$ and $\varphi =\pi /3$. The four light intensities are $4$, $3-\sqrt{3}$, $2$ and $3+\sqrt{3}$. Then $X=2$, $Y=2\sqrt{3}$, and the recovered wave phase is $\pi /3$. Common background cancels, whereas changing background separately between frames need not cancel.

If independent additive errors in the four light intensities have common variance ${\sigma }_I^2$, first-order propagation givesThis follows from $d\varphi =(XdY-YdX)/(X^2+Y^2)$ and difference variances $2{\sigma }_I^2$. It applies away from vanishing modulation and assumes the specified independent equal-noise model. wave phase uncertainty grows when interference visibility is weak.

The method is a discrete instance of phase-shifting interferometry, whose early digital apparatus used controlled reference-path modulation and synchronous wave phase extraction.^[ref] The result is still a principal wave phase; phase unwrapping cannot be replaced by choosing a different arctangent branch at each pixel without a consistency rule.