Interferometry

Measurement using wave interference: relative path or phase changes are inferred from the resulting signal. The interacting fields must have sufficient optical coherence in the measured modes for the intended interference signal.

Phase unwrapping reconstructs a wave phase field from measurements that determine only its value modulo $2\pi$. If a detector or complex-argument calculation returns ${\theta }_j=\mathrm{Arg}(e^{i{\varphi }_j})$, then all values ${\varphi }_j+2\pi n_j$ with integers $n_j$ give the same wrapped observation. Recovering a continuous wave phase therefore requires information beyond each isolated value. A primary 1982 analysis discusses this problem in wave phase curves obtained from image Fourier transforms before averaging them.^[ref]

Define $W(a)=\mathrm{Arg}(e^{ia})$ with principal interval $(-\pi ,\pi ]$. If adjacent true differences satisfy $|{\varphi }_{j+1}-{\varphi }_j|<\pi$, thenStarting from one reference wave phase, adding these wrapped differences reconstructs the sampled wave phase up to a global multiple of $2\pi$. The proof is direct: wrapping the two endpoint phases changes their difference only by an integer multiple of $2\pi$, and the strict bound selects its unique representative.

For an original example, true phases $0,0.8\pi ,1.6\pi ,2.4\pi$ wrap to $0,0.8\pi ,-0.4\pi ,0.4\pi$. Their wrapped increments are all $0.8\pi$, so reconstruction succeeds. But true increments $1.5\pi$ produce exactly the same wrapped samples as increments $-0.5\pi$. No algorithm can distinguish those two possibilities from the samples alone without a further assumption. Increasing sampling density can restore the bound only when the field's variation and noise allow it.

In two dimensions, different integration paths introduce a consistency test. A nonzero integer multiple of $2\pi$ in the sum of wrapped increments around a closed sampling loop is a phase-unwrapping residue. Noise, undersampling or true wave phase singularities can obstruct a single consistent smooth reconstruction. Choosing an integration path is therefore part of the inference.

For interferometry, low interference visibility can make the wave phase poorly determined even before unwrapping. Unwrapping does not recover information lost where the measured complex amplitude vanishes, nor does it distinguish an actual discontinuity from a smooth but insufficiently sampled change. Report the wave phase reference, principal interval, sampling and any continuity assumptions alongside the reconstructed field.

Phase-shifting interferometry measures an optical phase difference by recording light intensities at controlled relative wave phase shifts. For two sufficiently coherent fields at one detector point, a common model iswhere $C$ is a common background, $A$ a modulation amplitude and ${\delta }_j$ a known applied step. Several measurements separate the unknown wave phase from light intensity background and contrast; a single bright or dark frame generally cannot do so.

A concrete primary implementation was reported in 1974. A piezoelectrically varied reference arm and a 1024-element photodiode array supplied sampled light intensities to a minicomputer, which recovered wave phase by synchronous Fourier-series detection. The paper describes optical-surface and lens testing, repeated sampling and compensation of instrumental wavefront errors.^[ref] This is an experimental implementation of controlled wave phase modulation; it is not a claim that every such system uses the same number of steps.

Writing the model asturns wave phase recovery into a linear fit followed by $\varphi =atan2(-D,B)$. The step choices must make the three columns $1,\cos {\delta }_j,\sin {\delta }_j$ independent. Repeating only the same wave phase cannot identify all three unknowns. With more steps, least squares can fit the coefficients while exposing model residuals.

A reflecting reference mirror moved axially by $\Delta z$ changes its round-trip optical path by $2\Delta z$ in the simple normal-incidence geometry, giving $\delta =4\pi \Delta z/\lambda$. A quarter-cycle wave phase step thus corresponds to $\Delta z=\lambda /8$; at 632.8 nm this is 79.1 nm. These are geometric model values, not a calibration result for the historical apparatus.

Phase-step calibration error, light intensity drift, motion between frames and detector nonlinearity can all break the common-intensity model. The retrieved wave phase remains modulo $2\pi$, so phase unwrapping requires additional spatial or temporal information. The useful observable is the relative wave phase under controlled acquisition conditions, not an absolute optical wave phase inferred from brightness alone.

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.

Intensity interferometry infers source coherence from correlations between measured intensities, rather than directly combining electric-field amplitudes. For stationary chaotic Gaussian light, the normalized correlations satisfy $g^{(2)}(\tau )=1+|g^{(1)}(\tau ){|}^2$, under the relevant statistical assumptions.

Separated detectors can therefore measure the magnitude of spatial coherence, which is related to the source's angular structure. Optical carrier phase need not be stabilized as in amplitude interferometry, but fast timing and sufficient photon statistics are crucial. The squared magnitude does not by itself retain every phase needed for unrestricted image reconstruction.

Measurement using interfering light fields. A phase difference changes the detected intensity, allowing measurements of displacement, optical path length or a spectrum. The round-trip phase in a Michelson interferometer gives one explicit path-to-phase calibration; Fourier transform spectroscopy recovers spectral information from an interferogram.

A nearly uniform wavefront phase over one collector is called its piston. Differential atmospheric piston changes the pairwise optical phase without requiring a change in the source. Delay tracking and closure phase address different consequences of this phase error; intensity correlations are unchanged by an ideal pure phase multiplication at either detector.

fringe tracking estimates changing optical delay or phase and adjusts delay compensation to keep an astronomical interferometer near a useful interference condition. The delay must first lie inside the finite-bandwidth coherence envelope; rapid tracking then controls atmospheric piston. This is optical-path stabilization, with sensitivity and reference-source limitations.