---
title: Carré 4-Step Phase-Shifting Algorithm
url: https://www.emergentmind.com/topics/carre-4-step-phase-shifting-algorithm
type: topic
---

# Carré 4-Step Phase-Shifting Algorithm

Searching arXiv for the specified papers and any related Carré phase-shifting references.
arxiv_search({"query":"Carré 4-step phase shifting algorithm arXiv 1203.1947 phase-shifting interferometry", "max_results": 5, "sort_by": "relevance"})
arxiv_search({"query":"1203.1947", "max_results": 5, "sort_by": "relevance"})
arxiv_search({"query":"1501.04738 1602.06975 1807.01399 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 [1203.1947]. 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 [1602.06975].

## 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 [1203.1947].

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 [1602.06975].

This difference is not merely terminological. It determines whether the algorithm solves only for $\varphi$ 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 [1203.1947]. This decomposition is the key result that resolves the apparent paradox of analyzing a nonlinear phase-shifting algorithm with linear-systems theory.

With samples $s_1,s_2,s_3,s_4$ corresponding to the four temporal positions, the filtered phase extraction can be written as
$$
\tan[\hat{\varphi}(x, y)] =
\frac{ (s_1 + s_2 - s_3 - s_4) \cdot \sin\left(\frac{\omega_0}{2}\right) }
{ (-s_1 + s_2 + s_3 - s_4)\cdot \cos\left(\frac{\omega_0}{2}\right) }.
$$
In this representation, the estimated or tuned $\omega_0$ 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 [1203.1947].

This decomposition has two important consequences. First, the Carré algorithm can be analyzed as a filter family parameterized by $\omega_0$. 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
$$
\begin{aligned}
h(t) = & \left[ -\delta(t + 3/2) + \delta(t + 1/2) + \delta(t - 1/2) - \delta(t - 3/2) \right] \cos\left(\frac{\omega_0}{2}\right) \\
& + i \left[ \delta(t + 3/2) + \delta(t + 1/2) - \delta(t - 1/2) - \delta(t - 3/2) \right] \sin\left(\frac{\omega_0}{2}\right),
\end{aligned}
$$
where $\delta(\cdot)$ is the Dirac delta and $i=\sqrt{-1}$ [1203.1947].

Its Fourier-domain frequency transfer function is given explicitly by
$$
H_2(\omega) = \left[1 - e^{-i\omega}\right]
              \left[1 - e^{-i(\omega + \omega_0)}\right]
              \left[1 - e^{-i(\omega + \pi)}\right].
$$
With $x=e^{-i\omega}$, this can also be written as the Surrel characteristic polynomial
$$
P(x) = (1-x)(1-e^{-i\omega_0}x)(1+x).
$$

The factorization exposes three filter zeros, located at
$$
\omega = 0,\quad \omega = -\omega_0,\quad \omega = \pi.
$$
The zero at $-\omega_0$ is tunable, whereas the zeros at $0$ and $\pi$ are fixed [1203.1947]. 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 $H_2(\omega)$, 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
$$
G(\omega_0) =
\frac{ |H_2(\omega_0)|^2 }
{ \frac{1}{2\pi} \int_{-\pi}^{\pi} |H_2(\omega)|^2\, d\omega }.
$$
The reported maximum value is $4$, and this maximum occurs when the algorithm is tuned at
$$
\omega_0 = 0.5\pi \; (90^\circ).
$$
The same analysis states that prior literature had incorrectly placed the optimum at $65^\circ$ or $110^\circ$; the explicit linear treatment corrects that claim and identifies $90^\circ$ as the optimal tuning point [1203.1947].

Harmonic rejection follows directly from the transfer-function zeros. When the Carré algorithm is tuned to $\omega_0=0.5\pi$, it completely rejects even harmonics and the component at $-\omega_0$. When tuned away from that value, harmonic rejection degrades [1203.1947]. 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 $90^\circ$, 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 $0,\pi/2,\pi,3\pi/2$ | $\phi = \arctan\left(\frac{I_4-I_2}{I_1-I_3}\right)$ [1602.06975] |
| Carré 4-step PSA | Unknown; self-tuning | Decomposes into a tunable linear PSA plus nonlinear phase-step estimator [1203.1947] |

The ESSPPI paper is explicit: the four-step method implemented with a PZT mirror and phase steps of $0,\pi/2,\pi,3\pi/2$ 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” [1602.06975]. By contrast, the dynamic 3D shape-measurement paper describes four $\pi/2$-shifted moiré patterns and states that this arrangement matches the “classic four-step phase-shifting algorithm (Carré algorithm)” [1807.01399].

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 $\pi/2$ formula is the standard four-step PSA [1602.06975].

## 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 $10$ microarcseconds [1501.04738]. The underlying four-step ABCD concept uses nominal phase steps
$$
\alpha_i=\left\{0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\right\},
$$
with intensity model
$$
I(x, y, \alpha_i) = I_1 + I_2 + 2\sqrt{I_1 I_2}\sin(\Phi(x, y) + \alpha_i),
$$
and phase estimate
$$
\Phi = \arctan\left( \frac{I_A - I_C}{I_B - I_D} \right)
$$
when the steps are exactly at $\pi/2$ intervals [1501.04738].

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 $2$ nm. To overcome this, the paper introduces a phase-step-insensitive calibration based on ellipse fitting, using
$$
I_i(\beta)=a_i+b_i\sin(\beta+\alpha_i),
$$
the conic model
$$
EI_i^2 + F I_i I_j + G I_j^2 + H I_i + K I_j + L = 0,
$$
and the extracted phase difference
$$
\Delta_{ij} = \arccos\left( \frac{-F}{\sqrt{4EG}} \right).
$$
With a $\chi^2$-minimization routine and inclusion of time-dependent power fluctuations,
$$
I_i(t)=P(t)\left[a_i+b_i\sin(\beta(t)+\alpha_i)\right],
$$
the reported phase-step calibration repeatability is $0.2^\circ$ and the systematic error is also approximately $1$ nm, corresponding to $\lambda/2000$ at the metrology wavelength [1501.04738].

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 $1$ nm contributes to a total metrology phase extraction error of $1.2$ nm, corresponding to an astrometric error of $10.3\,\mu\mathrm{as}$, which lies within the science requirement [1501.04738]. 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 [1203.1947]. That perspective resolves older uncertainty about its noise and harmonic behavior, establishes that its optimal tuning is at $90^\circ$, 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 [1602.06975], [1807.01399]. Yet the GRAVITY metrology study demonstrates that real phase-shifting devices exhibit nonlinearities, transmission modulation, drift, and vibration, so the assumption of ideal $\pi/2$ steps may be insufficient in high-accuracy regimes [1501.04738]. 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 [1203.1947].

Source: https://www.emergentmind.com/topics/carre-4-step-phase-shifting-algorithm