Carré 4-Step Phase-Shifting Algorithm

• The Carré algorithm is a self-tuning phase demodulation method that recovers phase from four interferograms without preset phase-step values.
• It decomposes into a tunable linear 4-step phase-shifting filter and a nonlinear phase-step estimator, enabling detailed spectral and noise analysis.
• Its optimal performance at a 90° phase step makes it crucial in high-precision optical metrology where phase-shift uncertainties and harmonic distortions are common.

Searching arXiv for the specified papers and any related Carré phase-shifting references. arxiv_search({"query":"Carré 4-step phase shifting algorithm arXiv (Servin et al., 2012) phase-shifting interferometry", "max_results": 5, "sort_by": "relevance"}) arxiv_search({"query":"(Servin et al., 2012)", "max_results": 5, "sort_by": "relevance"}) arxiv_search({"query":"(Lippa et al., 2015, Xu et al., 2016, Zhou et al., 2018) phase shifting interferometry", "max_results": 10, "sort_by": "relevance"}) The Carré 4-step phase-shifting algorithm is a four-interferogram phase-shifting algorithm in optical metrology that estimates the modulating phase without prior knowledge of the phase step, and is therefore treated as a self-tuning or self-calibrating phase-shifting algorithm. In the arXiv literature considered here, its most precise contemporary characterization is the decomposition of the nominally nonlinear Carré procedure into a tunable linear 4-step phase-shifting algorithm and a nonlinear phase-step estimator, which makes it possible to analyze spectrum, signal-to-noise ratio, and harmonic rejection with linear-systems tools (Servin et al., 2012). The same literature also shows that the Carré algorithm is often conflated with the standard four-step $\pi/2$ algorithm, even though those methods differ in whether the phase steps are assumed known and equally spaced (Xu et al., 2016).

1. Conceptual definition and measurement model

In its canonical formulation, the Carré algorithm operates on four interferometric samples acquired at temporal steps

$$t=-\frac{3}{2},\;-\frac{1}{2},\;\frac{1}{2},\;\frac{3}{2},$$

with interferograms modeled as

$$\begin{aligned} I(x, y, -\tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{3}{2} \omega_0],\ I(x, y, -\tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{3}{2} \omega_0]. \end{aligned}$$

Here $a(x,y)$ is the background term, $b(x,y)$ is the modulation, $\varphi(x,y)$ is the target phase, and $\omega_0$ is the carrier frequency or phase step (Servin et al., 2012).

The defining distinction is that $\omega_0$ is not assumed known a priori. This separates the Carré algorithm from the standard four-step phase-shifting formula used when the phase shifts are precisely controlled at $0,\pi/2,\pi,3\pi/2$. In the ESSPPI literature, the latter is written as

$$\phi = \arctan\left(\frac{I_4 - I_2}{I_1 - I_3}\right),$$

and is explicitly described as not being the Carré algorithm because it presumes known equal phase steps supplied by a PZT actuator (Xu et al., 2016).

This difference is not merely terminological. It determines whether the algorithm solves only for $$t=-\frac{3}{2},\;-\frac{1}{2},\;\frac{1}{2},\;\frac{3}{2},$$0 under a calibrated stepping law, or jointly accommodates the unknown phase step through an embedded estimation stage. A plausible implication is that the Carré algorithm is best understood as a self-calibrating demodulator rather than simply as a four-sample arctangent identity.

2. Self-tuning structure and phase recovery

The 2012 linear analysis shows that the nonlinear Carré algorithm can be decomposed into two building blocks: a tunable linear 4-step phase-shifting algorithm and a nonlinear phase-step estimator (Servin et al., 2012). This decomposition is the key result that resolves the apparent paradox of analyzing a nonlinear phase-shifting algorithm with linear-systems theory.

With samples $$t=-\frac{3}{2},\;-\frac{1}{2},\;\frac{1}{2},\;\frac{3}{2},$$1 corresponding to the four temporal positions, the filtered phase extraction can be written as

$$t=-\frac{3}{2},\;-\frac{1}{2},\;\frac{1}{2},\;\frac{3}{2},$$2

In this representation, the estimated or tuned $$t=-\frac{3}{2},\;-\frac{1}{2},\;\frac{1}{2},\;\frac{3}{2},$$3 enters the demodulation formula, but the spectral, noise, and harmonic properties are governed entirely by the linear block rather than by the nonlinear carrier estimator (Servin et al., 2012).

This decomposition has two important consequences. First, the Carré algorithm can be analyzed as a filter family parameterized by $$t=-\frac{3}{2},\;-\frac{1}{2},\;\frac{1}{2},\;\frac{3}{2},$$4. Second, the self-tuning property is isolated in the estimator of the phase step, rather than being diffused across the whole demodulation expression. This suggests a modular interpretation of Carré-type processing: one mechanism estimates the step, and another performs the phase-sensitive filtering.

3. Linear-systems formulation

The same analysis derives the impulse response of the linear block as

$$t=-\frac{3}{2},\;-\frac{1}{2},\;\frac{1}{2},\;\frac{3}{2},$$5

where $$t=-\frac{3}{2},\;-\frac{1}{2},\;\frac{1}{2},\;\frac{3}{2},$$6 is the Dirac delta and $$t=-\frac{3}{2},\;-\frac{1}{2},\;\frac{1}{2},\;\frac{3}{2},$$7 (Servin et al., 2012).

Its Fourier-domain frequency transfer function is given explicitly by

$$t=-\frac{3}{2},\;-\frac{1}{2},\;\frac{1}{2},\;\frac{3}{2},$$8

With $$t=-\frac{3}{2},\;-\frac{1}{2},\;\frac{1}{2},\;\frac{3}{2},$$9, this can also be written as the Surrel characteristic polynomial

$$\begin{aligned} I(x, y, -\tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{3}{2} \omega_0],\ I(x, y, -\tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{3}{2} \omega_0]. \end{aligned}$$0

The factorization exposes three filter zeros, located at

$$\begin{aligned} I(x, y, -\tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{3}{2} \omega_0],\ I(x, y, -\tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{3}{2} \omega_0]. \end{aligned}$$1

The zero at $$\begin{aligned} I(x, y, -\tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{3}{2} \omega_0],\ I(x, y, -\tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{3}{2} \omega_0]. \end{aligned}$$2 is tunable, whereas the zeros at $$\begin{aligned} I(x, y, -\tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{3}{2} \omega_0],\ I(x, y, -\tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{3}{2} \omega_0]. \end{aligned}$$3 and $$\begin{aligned} I(x, y, -\tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{3}{2} \omega_0],\ I(x, y, -\tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{3}{2} \omega_0]. \end{aligned}$$4 are fixed (Servin et al., 2012). In interferometric terms, the filter therefore suppresses DC, the Nyquist component, and a movable carrier-symmetric component. This supplies an explicit spectral interpretation that earlier discussions of the Carré algorithm lacked.

A central significance of this result is methodological. Once the Carré algorithm is expressed through $$\begin{aligned} I(x, y, -\tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{3}{2} \omega_0],\ I(x, y, -\tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{3}{2} \omega_0]. \end{aligned}$$5, one can discuss tuning, selectivity, harmonic rejection, and white-noise behavior using the same formalism used for linear phase-shifting algorithms, while preserving the self-tuning character of the original method.

4. Signal-to-noise ratio and harmonic rejection

The signal-to-noise ratio gain for the Carré filter under a white-noise assumption is

$$\begin{aligned} I(x, y, -\tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{3}{2} \omega_0],\ I(x, y, -\tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{3}{2} \omega_0]. \end{aligned}$$6

The reported maximum value is $$\begin{aligned} I(x, y, -\tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{3}{2} \omega_0],\ I(x, y, -\tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{3}{2} \omega_0]. \end{aligned}$$7, and this maximum occurs when the algorithm is tuned at

$$\begin{aligned} I(x, y, -\tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{3}{2} \omega_0],\ I(x, y, -\tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{3}{2} \omega_0]. \end{aligned}$$8

The same analysis states that prior literature had incorrectly placed the optimum at $$\begin{aligned} I(x, y, -\tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{3}{2} \omega_0],\ I(x, y, -\tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) - \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{1}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{1}{2} \omega_0],\ I(x, y, \tfrac{3}{2}) &= a(x, y) + b(x, y) \cos[\varphi(x, y) + \tfrac{3}{2} \omega_0]. \end{aligned}$$9 or $a(x,y)$0; the explicit linear treatment corrects that claim and identifies $a(x,y)$1 as the optimal tuning point (Servin et al., 2012).

Harmonic rejection follows directly from the transfer-function zeros. When the Carré algorithm is tuned to $a(x,y)$2, it completely rejects even harmonics and the component at $a(x,y)$3. When tuned away from that value, harmonic rejection degrades (Servin et al., 2012). The practical interpretation is straightforward: the self-tuning property does not imply uniform performance over all possible phase steps. The algorithm remains operational away from $a(x,y)$4, but its noise gain and harmonic immunity are not invariant with tuning.

This spectral result clarifies a recurring misconception in interferometric practice. The Carré algorithm is often valued because it does not require prior calibration of the phase step; however, that autonomy does not eliminate the existence of an optimal operating region. Self-calibration and optimal SNR are distinct properties.

5. Relation to standard four-step algorithms and terminological ambiguity

The arXiv record shows a persistent ambiguity in the use of “four-step” nomenclature. One strand of the literature uses four equally spaced phase shifts and the standard arctangent reconstruction formula. Another reserves “Carré” for the self-calibrating case in which the phase step is not required to be known in advance.

Method	Phase-step assumption	Representative formula or property
Standard four-step PSA	Known, typically $a(x,y)$5	$a(x,y)$6 (Xu et al., 2016)
Carré 4-step PSA	Unknown; self-tuning	Decomposes into a tunable linear PSA plus nonlinear phase-step estimator (Servin et al., 2012)

The ESSPPI paper is explicit: the four-step method implemented with a PZT mirror and phase steps of $a(x,y)$7 is “the classic four-step phase-shifting algorithm,” and “not the Carré algorithm, which is four-step but self-calibrating and does not require equal phase steps” (Xu et al., 2016). By contrast, the dynamic 3D shape-measurement paper describes four $a(x,y)$8-shifted moiré patterns and states that this arrangement matches the “classic four-step phase-shifting algorithm (Carré algorithm)” (Zhou et al., 2018).

Taken together, these sources show that the label “Carré” is sometimes used loosely for any four-image tangent-form phase retrieval, even when the steps are known and equally spaced. For strict usage, the evidence here favors the ESSPPI distinction: Carré denotes the self-calibrating four-step method, whereas the equal-step $a(x,y)$9 formula is the standard four-step PSA (Xu et al., 2016).

6. Instrumental relevance and calibration practice

The practical importance of self-calibration becomes particularly visible in precision interferometric instrumentation. In the GRAVITY metrology system, phase-shifting interferometry is used to monitor internal differential optical path differences at the nanometer level for narrow-angle astrometry with a target precision of $b(x,y)$0 microarcseconds (Lippa et al., 2015). The underlying four-step ABCD concept uses nominal phase steps

$b(x,y)$1

with intensity model

$b(x,y)$2

and phase estimate

$b(x,y)$3

when the steps are exactly at $b(x,y)$4 intervals (Lippa et al., 2015).

That work then emphasizes why such ideal stepping cannot simply be presumed in real hardware. The applied phase shifts deviate from nominal values because of non-linearities in the voltage-to-phase relationship, transmission modulation with voltage, laser-power fluctuations, and environmental drifts and vibrations. A naive linear calibration based on full-wave voltage ramps yields a typical phase calibration error of approximately $b(x,y)$5 nm. To overcome this, the paper introduces a phase-step-insensitive calibration based on ellipse fitting, using

$b(x,y)$6

the conic model

$b(x,y)$7

and the extracted phase difference

$b(x,y)$8

With a $b(x,y)$9-minimization routine and inclusion of time-dependent power fluctuations,

$\varphi(x,y)$0

the reported phase-step calibration repeatability is $\varphi(x,y)$1 and the systematic error is also approximately $\varphi(x,y)$2 nm, corresponding to $\varphi(x,y)$3 at the metrology wavelength (Lippa et al., 2015).

Although this GRAVITY method is not identified as the Carré algorithm, it addresses the same fundamental problem: phase retrieval under imperfect or insufficiently known phase steps. The comparison is instructive. Carré embeds the step estimation inside a four-frame demodulation algorithm; the GRAVITY approach performs an external calibration through a phase-step-insensitive ellipse fit and then uses the measured step values in reduction. A plausible implication is that modern high-precision interferometry treats phase-step uncertainty as a first-order systems issue, whether it is handled algorithmically, as in Carré, or procedurally, as in explicit calibration.

The astrometric consequence in GRAVITY is stated quantitatively. A phase-shifter calibration error of $\varphi(x,y)$4 nm contributes to a total metrology phase extraction error of $\varphi(x,y)$5 nm, corresponding to an astrometric error of $\varphi(x,y)$6, which lies within the science requirement (Lippa et al., 2015). This illustrates why the Carré problem—recovering phase when the step is not perfectly known—remains consequential far beyond textbook interferogram demodulation.

7. Significance in optical metrology

Within phase-shifting interferometry, the Carré 4-step algorithm occupies a specific niche: it is a four-frame, self-tuning phase demodulator whose nonlinear appearance conceals a tunable linear filter with explicit spectral structure (Servin et al., 2012). That perspective resolves older uncertainty about its noise and harmonic behavior, establishes that its optimal tuning is at $\varphi(x,y)$7, and shows that its harmonic rejection is strongest at the same operating point.

The surrounding literature also indicates why the algorithm continues to matter. Standard equal-step four-frame methods remain common in systems with tight actuator control, including PZT-driven interferometers and projected-fringe profilometry (Xu et al., 2016, Zhou et al., 2018). Yet the GRAVITY metrology study demonstrates that real phase-shifting devices exhibit nonlinearities, transmission modulation, drift, and vibration, so the assumption of ideal $\varphi(x,y)$8 steps may be insufficient in high-accuracy regimes (Lippa et al., 2015). The Carré algorithm addresses precisely that vulnerability by avoiding prior commitment to a fixed step.

For that reason, the Carré 4-step phase-shifting algorithm is best regarded not simply as one member of the four-frame PSA family, but as a reference point for self-calibrating phase demodulation. Its enduring relevance lies in the combination of minimal frame count, explicit tunability, and a now well-defined linear-systems description of its spectrum, SNR, and harmonic response (Servin et al., 2012).