Phase-stepping algorithms for synchronous demodulation of nonlinear phase-shifted fringes

Manuel Servin; Moises Padilla; Ivan Choque; Sotero Ordones

doi:10.1364/OE.27.005824

1. Introduction

Linear reference phase-shifting algorithms (PSAs) have been used in optical metrology to demodulate linear-carrier temporal fringes since the pioneering work by Bruning et al. [1–3]. To generate linear phase-shifting fringes one normally uses well-calibrated optical phase-shifters. However in practice the phase shifters may drift from its nominal phase step producing wideband, spread-spectrum fringes [4,5]. In these cases the PSA must also be wideband to deal with nonuniform/nonlinear phase-shifted fringes [6–13]. Hibino et al. indicated that an spurious piston appears in the estimated phase when a linear-reference PSA is used to demodulate non-uniform phase-shifted fringes [6–13]. This spurious-piston is a numeric artifact of the linear-reference PSA, which may be wrongly interpreted as physical optical thickness [6–13]. Real optical thickness measuring is fundamental when testing optical material slabs in semiconductor and display equipment [6–13]. Many systematic errors in linear-carrier fringes (such as phase-shift miscalibration, fringe harmonics, experiment vibrations) have been attenuated using linear-reference PSAs [2,3]. For precision thickness measurement, and nonlinear phase-shifted fringes, linear-reference PSAs with no spurious-piston have been proposed [6–13]. Recently, Kim and Hibino have pointed-out that this numerical/spurious piston has received little attention because it does not give a waving profiling error (such as detuning or harmonic distortion) when an optical surface is profiled [10–13]. However, when the central interest is to measure absolute optical thickness of transparent slabs by wavelength tuning (for example), this numerical/spurious piston translates into errors in thickness [6–13]. This error is given by the product of the demodulated phase and the synthetic wavelength, which is much greater than the wavelength used [13]. Linear-reference PSAs for demodulating wideband, nonlinear-carrier fringes have been developed using the Taylor series expansion of the arc-tangent of the phase-error [6–13]. The linear-reference PSA's coefficients are then calculated to set the first terms of this Taylor expansion to zero [6–13].

In this work we propose a temporal PSA for nonlinear-carrier fringes using a synchronous, nonlinear/nonuniform phase-shifted reference. This is similar to the theory behind chirp-carrier radars where the radio-frequency (RF) phase varies quadratically with time [14,15] (see Fig. 1). When the RF chirp-pulse bounces back from the target, the incoming signal is correlated with a synchronous, local chirp waveform. In the case of wideband chirp radar, one is interested in timing the amplitude of the correlation peak between the returning chirp signal and the chirped local oscillator. Timing this correlation peak gives the round-trip distance to target [14,15].

Fig. 1 Panel (a) shows the standard case of using a linear-carrier reference r₁(t) for phase demodulation of a chirp (quadratic phase) data I(t). Panel (b) shows the herein proposed strategy, which consist on synchronously following the nonlinear phase variations of the fringe data I(t) by a chirped reference r₂(t).

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Mathematical models for nonlinear-phase information carriers have been applied to a wide number of fields, such as: radio-frequency pulse compression, anti-jamming wideband communications, spread-spectrum secure communications, wideband pulse-compression chirp radar, attosecond laser pulse compression, and low-power home/mobile gigabyte communications [14–17]. Thus, we believe it is convenient to list some possible synonyms for nonlinear carrier digital interferometry, such as: nonuniform phase-shifting, nonlinear phase shifting, wideband nonlinear carrier, spread-spectrum fringes, and spread-spectrum synchronous reference. As mentioned, the special case of quadratic phase-variation is called a chirp. Synonymous may include: chirp-reference, chirp-carrier, chirp-pulse, chirp-waveform, chirp-wavelet pulse, chirp-fringes [14–17].

2. Linear and nonlinear phase-shifted fringe patterns

Let us first show the usual mathematical models for continuous-time linear and nonlinear (nonuniform) phase-shifted fringes. The model for linear-carrier fringe patterns is,

J (t) = a + b \cos (φ + ω_{0} t); t \in [0, T] ​ .

Where a is the background and b the contrast of the fringes. The linear carrier-frequency

ω_{0}

is given in radians per second, and

φ \in [- π, π)

is the measured phase. On the other hand, nonlinear-carrier fringes are formalized by,

I (t) = a + b \cos [φ + ω_{0} t + Δ (t)]; t \in [0, T] .

We are assuming that the phase nonlinearity

Δ (t)

is continuous and smooth, and can be determined experimentally; three practical methods to do this are given in [4,5]. Typically

Δ (t)

is approximated up to the quadratic term in a Taylor series [4–12]. Here we require that the derivative of

[ω_{0} t + Δ (t)]

be bandlimited to

(0, π)

as,

[ω_{0} + \frac{d Δ (t)}{d t}] \in (0, π); t \in [0, T] .

In Fig. 2 we show linear (in blue) and nonlinear (in red) phase-shifted fringes.

Fig. 2 Panel (a) shows in blue linear phase-shifting, and in red nonlinear phase-shifting. Panel (b) shows linear carrier fringes. Panel (c) shows nonlinear carrier fringes.

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As we prove next, the nonlinear phase-shifting $Δ (t)$ generate a spurious-numerical piston when a linear-reference PSA is used as phase-demodulator.

2.1 Fourier spectrum for linear and nonlinear phase-shifted fringes

From Eq. (1), linear carrier fringes are single-frequency signals having a spectrum given by,

F {a + \frac{b}{2} e^{i [φ + ω_{0} t]} + \frac{b}{2} e^{- i [φ + ω_{0} t]}} = a δ (ω) + \frac{b}{2} e^{i φ} δ (ω - ω_{0}) + \frac{b}{2} e^{- i φ} δ (ω + ω_{0}) .

Where

F [\cdot]

is the Fourier transform operator, and

δ (ω)

is the Dirac delta function. In contrast, nonlinear phase-shifted fringes (Eq. (2)) are wideband, and its spectrum is given by,

F {a + \frac{b}{2} e^{i φ} e^{i [ω_{0} t + Δ (t)]} + \frac{b}{2} e^{- i φ} e^{- i [ω_{0} t + Δ (t)]}} = a δ (ω) + \frac{b}{2} e^{i φ} C (ω) + \frac{b}{2} e^{- i φ} C^{*} (- ω),

Where,

C^{*} (- ω) = F {e^{- i [ω_{0} t + Δ (t)]}}; C (ω) = F {e^{i [ω_{0} t + Δ (t)]}} .

Figure 3 schematically shows the spectrum of linear and wideband nonlinear-carrier fringes.

Fig. 3 Panel (a) shows the Fourier spectra of linear-carrier fringes. In panel (b), the red rectangle schematically/abstractly represents wideband spectral lobes.

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Note that $C^{*} (- ω)$ and $C (ω)$ are wideband lobes which replace the narrowband Dirac deltas.

3. Linear and nonlinear reference digital PSAs

Now we show the discrete-time mathematical form of PSAs using linear and nonlinear-reference for demodulating nonlinear-carrier fringes. In digital interferometry, we usually sample the continuous-time interferograms J(t), I(t), (t ∈[0,T]), at uniform spaced times nT_s for n∈{0, 1, …, N–1}, being T_s = T/N the constant sampling period. Therefore, let’s introduce the following definitions and notation: θ₀ = ω₀T_s, Δ_n = Δ(nT_s), and I_n = I(nT_s).

The demodulated analytic signal using synchronous linear-reference PSA is [1–3],

A e^{i φ} = \sum_{n = 0}^{N - 1} \overset{L i n e a r R e f e r e n c e}{\overset{︷}{[c_{n} e^{- i n θ_{0}}]}} \overset{L i n e a r C a r r i e r F r i n g e s}{\overset{︷}{[a + b \cos (φ + n θ_{0})]}}; θ_{0} = ω_{0} T / N, (c_{n} \in ℂ) ​ .

The searched phase is given modulo 2π as

\arg (A e^{i φ})

. These are the standard PSAs in use since 1974 [1–3]. Typically, PSAs are restricted to a single temporal period (Nθ₀< 2π). However in this work the fringes may have several temporal periods (Nθ₀> 2π).

Linear-reference PSAs has been proposed to demodulate nonlinear fringes as [6–13],

A_{1} e^{i (φ + P i s t o n)} = \sum_{n = 0}^{N - 1} \overset{L i n e a r R e f e r e n c e}{\overset{︷}{[d_{n} e^{- i n θ_{0}}]}} \overset{N o n l i n e a r C a r r i e r F r i n g e s}{\overset{︷}{[a + b \cos (φ + n θ_{0} + Δ_{n})]}}; (d_{n} \in ℂ) ​ .

As we show next, using a linear-reference PSA, we generally obtain a spurious piston,

P i s t o n \neq 0

. Hibino et al. proposed linear-reference PSAs to eliminate this spurious piston by imposing quadrature conditions on the PSA's coefficients [6–12]. Here we are proposing an alternative solution, a more natural way (we believe), to eliminate this spurious-piston.

Here we specifically propose the use of a nonlinear-reference PSA, given by

A_{2} e^{i φ} = \sum_{n = 0}^{N - 1} \overset{N o n l i n e a r R e f e r e n c e}{\overset{︷}{[w_{n} e^{- i (n θ_{0} + Δ_{n})}]}} \overset{N o n l i n e a r C a r r i e r F r i n g e s}{\overset{︷}{[a + b \cos (φ + n θ_{0} + Δ_{n})]}} ​; (w_{n} \in ℝ) .

Note that the nonlinear-reference

\exp [i (n θ_{0} + Δ_{n})]

synchronously match the nonlinear-carrier

\cos (φ + n θ_{0} + Δ_{n})

of the fringes; this fact makes the spurious piston disappear (Piston = 0). The weighting coefficients

w_{n}

are chosen to approximate a Hilbert quadrature filter [3].

3.1 Spurious-piston using uniform phase-shifted reference PSAs

Using linear-reference PSAs to demodulate nonlinear-carrier fringes (Eq. (8)) one obtains,

A_{1} e^{i [φ + P i s t o n]} = \sum_{n = 0}^{N - 1} d_{n} e^{- i n θ_{0}} I_{n} = \sum_{n = 0}^{N - 1} d_{n} e^{- i n θ_{0}} {a + \frac{b}{2} e^{i φ} e^{i (n θ_{0} + Δ_{n})} + \frac{b}{2} e^{- i φ} e^{- i (n θ_{0} + Δ_{n})}}

Performing the indicated multiplications one obtains,

A_{1} e^{i [φ + P i s t o n]} = a [\sum_{n = 0}^{N - 1} ​ d_{n} e^{- i n θ_{0}}] + \frac{b}{2} e^{i φ} [\sum_{n = 0}^{N - 1} d_{n} e^{i Δ_{n}}] + \frac{b}{2} e^{- i φ} [\sum_{n = 0}^{N - 1} d_{n} e^{- i (2 n θ_{0} + Δ_{n})}]

The coefficients

​ d_{n}

are chosen such that the first and third square-brackets are set to zero as,

[\sum_{n = 0}^{N - 1} ​ d_{n} e^{- i n θ_{0}}] = 0, and [\sum_{n = 0}^{N - 1} d_{n} e^{- i (2 n θ_{0} + Δ_{n})}] = 0 .

Obtaining the desired analytic signal as,

A_{1} e^{i (φ + P i s t o n)} = \frac{b}{2} e^{i φ} [\sum_{n = 0}^{N - 1} d_{n} e^{i Δ_{n}}]; P i s t o n = \arg {\sum_{n = 0}^{N - 1} d_{n} e^{i Δ_{n}}} .

As we see, in general, the spurious-piston is non-zero (

P i s t o n \neq 0

), and it may give erroneous absolute optical thickness measurements. This Piston does not depend on the object phase

φ

, nor on the phase steps

n θ_{0}

. Also we have assumed no linear-detuning between the fringe carrier and the PSA's reference. We next consider the PSA's reference synchronously following the nonlinear phase variations of the fringes.

3.2 No spurious-piston using nonuniform phase-shifted reference PSA

Now using a synchronous (matched-phase) nonlinear reference PSA (Eq. (9)) one gets,

A_{2} e^{i φ} = \sum_{n = 0}^{N - 1} w_{n} e^{- i (n θ_{0} + Δ_{n})} I_{n} = \sum_{n = 0}^{N - 1} w_{n} e^{- i (n θ_{0} + Δ_{n})} [a + \frac{b}{2} e^{i φ} e^{i (n θ_{0} + Δ_{n})} + \frac{b}{2} e^{- i φ} e^{- i (n θ_{0} + Δ_{n})}]

Performing the multiplications one obtains,

A_{2} e^{i φ} = a [\sum_{n = 0}^{N - 1} ​ w_{n} e^{- i (n θ_{0} + Δ_{n})}] + \frac{b}{2} e^{i φ} [\sum_{n = 0}^{N - 1} w_{n}] + \frac{b}{2} e^{- i φ} [\sum_{n = 0}^{N - 1} w_{n} e^{- i (2 n θ_{0} + Δ_{n})}]

As before, the real-valued coefficients w_n are chosen to satisfy the quadrature conditions,

[\sum_{n = 0}^{N - 1} ​ w_{n} e^{- i (n θ_{0} + Δ_{n})}] = 0; and [\sum_{n = 0}^{N - 1} ​ w_{n} e^{- i (2 n θ_{0} + Δ_{n})}] = 0 .

Finally, the searched phase-demodulated analytic signal is given by

A_{2} e^{i φ} = \frac{b}{2} e^{i φ} \sum_{n = 0}^{N - 1} w_{n}; (w_{n} \in ℝ) .

The spurious piston has naturally disappeared (Piston = 0) thanks to the use of a synchronous nonlinear-phase reference

\exp [- i (n θ_{0} + Δ_{n})]

in the PSA which match the phase nonlinearity variation of the fringes.

4. Spectral design for nonuniform phase-shifted reference PSAs

Previously we presented an algebraic approach for calculating the coefficients $w_{n} \in ℝ$ for nonlinear phase-shifted reference PSAs. However, we believe that designing the spectral shape of the frequency transfer function (FTF) of the PSA is more intuitive. The impulse response of the nonlinear-phase reference PSA (Eq. (9)) is [3],

h (t) = \sum_{n = 0}^{N - 1} w_{n} e^{i [ω_{0} t + Δ (t)]} δ (t - n T_{s}); (w_{n} \in ℝ) .

Then its FTF is

H (ω) = F [h (t)]

(for more details about the FTF see chapter 2 in [3]),

H (ω) = \sum_{n = 0}^{N - 1} w_{n} e^{i [ω_{0} n T_{s} + Δ (n T_{s})]} e^{- i n ω} = \sum_{n = 0}^{N - 1} w_{n} e^{i (n θ_{0} + Δ_{n})} e^{- i n ω} .

The coefficients

(w_{n} \in ℝ)

of

H_{2} (ω)

are chosen to obtain a wideband quadrature filter as,

\begin{matrix} H (ω) \approx 0 f o r ω \in [- π, 0] \\ H (ω) \neq 0 f o r ω \in (0, π) \end{matrix} .

Figure 4(a) shows that a square window (

w_{n} = 1.0

) produces a non-zero

H (ω)

for

ω \in [- π, 0]

resulting in an erroneous estimated phase. One solution to this is to use a Gaussian window

w_{n} = e^{- g {[n - 0.5 (N - 1)]}^{2}}, (g < 1.0)

resulting in the FTF shown in Fig. 4(b). Of course other weightings windows

(w_{n} \neq 1)

may be used [18,19].

Fig. 4 FTF spectra of two nonlinear-reference PSAs. Panel (a) shows (in green) the FTF of a square-window, nonlinear-reference PSA. Panel (b) shows the FTF of a nonlinear-reference PSA with Gaussian window [18,19]. The red rectangles schematically represent the wideband spectrum of nonlinear-carrier fringes. The FTF in panel (b) is a smooth approximation of a Hilbert quadrature filter (for more details see chapter 5, page 229 in [3]).

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Figure 5 shows the (normalized frequency) harmonic response of this FTF [3].

Fig. 5 Fundamental (in green) and harmonic (in red) FTF response for the nonlinear-reference PSA centered at π/2. The surviving distorting harmonics are {...,-7,-3,5,9,...} [3].

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5. Signal-to-noise ratio (SNR) for linear and nonlinear reference PSA

Here we find the SNR for the phase-demodulated nonlinear-carrier fringes corrupted by additive white Gaussian-noise (AWGN) $η (t)$ [3]. The noisy fringes may be represented as,

I (t) = \sum_{n = 0}^{N - 1} {a + b \cos [φ + ω_{0} t + Δ (t)] + η (t)} δ (t - n T_{s}) .

The spectral density of

η (t)

is flat, and it is given by [3],

S (ω) = F [R_{η η} (τ)] = F [E {η (t) η (t + τ)}] = \frac{η_{0}}{2}; ω \in [- π, π] .

Being

R_{η η} (τ) = E {η (t) η (t + τ)}

the ensemble autocorrelation function of

η (t)

[3]. The flat noise power-spectrum of

η (t)

is modified to

(η_{0} / 2) | H (ω) |^{2}

at the output of the PSA [3].

For nonlinear-carrier fringes, and linear-reference PSA, the SNR is given by,

{SNR}_{1} = \frac{Signal Energy}{Noise Energy} = \frac{{(\frac{b}{2})}^{2} {| \sum_{n = 0}^{N - 1} d_{n} e^{i Δ_{n}} |}^{2}}{(\frac{η_{0}}{2}) \sum_{n = 0}^{N - 1} {| d_{n} |}^{2}} .

On the other hand for nonlinear-carrier and reference, the SNR is given,

{SNR}_{2} = \frac{Signal Energy}{Noise Energy} = \frac{{(\frac{b}{2})}^{2} {| \sum_{n = 0}^{N - 1} w_{n} |}^{2}}{(\frac{η_{0}}{2}) \sum_{n = 0}^{N - 1} {| w_{n} |}^{2}} .

Where

(η_{0} / 2) \sum {| d_{n} |}^{2}

and

(η_{0} / 2) \sum {| w_{n} |}^{2}

are the total filtered-noise energy. For a fair comparison let us assume that

\sum {| d_{n} |}^{2} = \sum {| w_{n} |}^{2}

. The energy of

A_{2} e^{i φ}

using the nonlinear-reference PSA is generally higher than the energy of

A_{1} e^{i [φ + P i s t o n]}

using a linear-reference PSA; this is because,

{| \sum_{n = 0}^{N - 1} w_{n} |}^{2} \geq {| \sum_{n = 0}^{N - 1} d_{n} e^{i Δ_{n}} |}^{2},

obtaining,

{SNR}_{2} \geq {SNR}_{1} .

As conclusion, when demodulating nonlinear phase-shifted fringes, a phase-matched nonlinear reference PSA produces a higher SNR compared to a linear reference PSA.

6. Computer simulation with 13-step nonlinear-reference PSA

6.1 Nonlinear reference PSA with Gaussian window

Here we give a computer simulation example of a 13-step nonlinear-reference PSA applied to nonlinear-carrier fringes. The most usual phase-shifting nonlinearity is quadratic, $Δ_{n} = ε_{2} n^{2} T_{s}^{2}$ [4–13]. Then we start by considering nonlinear-carrier fringes as,

I_{n} = [1 + \cos (φ + n θ_{0} + ε_{2} n^{2} T_{s}^{2})]; n \in {0, 1, ..., 12}, (θ_{0} = 0.35 π, ε_{2} = 0.055 \frac{r a d}{\sec^{2}}) .

The nonlinear quadratic-phase (chirp) and the interferometric waveform are shown in Fig. 6.

Fig. 6 Panel (a) shows the linear (in blue) and a strong chirp-phase (in red). Panel (b) shows the chirp-carrier fringes sampled at a constant sampling rate. Note that the time interval [0,T] has more than three temporal fringes and has been sampled 13 times .

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The specific form of the 13-steps nonlinear, chirp-reference PSA is then given by,

A_{2} e^{i φ} = \sum_{n = 0}^{12} w_{n} e^{- i (θ_{0} n + ε_{2} n^{2} T_{s}^{2})} I_{n}; w_{n} = e^{- 0.1 {[n - 6]}^{2}} .

Here we are assuming no linear detuning (ε₁ = 0 in [6]). The PSA's nonlinear reference

\exp {- i [θ_{0} n + ε_{2} n^{2} T_{s}^{2}]}

is phase-matched or synchronous with the chirp fringes. The FTF of this nonlinear-reference PSA is,

H (ω) = F [h (t)] = F [\sum_{n = 0}^{12} w_{n} e^{i (ω_{0} t + ε_{2} t^{2})} δ (t - n T_{s})] = \sum_{n = 0}^{12} w_{n} e^{i [θ_{0} n + ε_{2} n^{2} T_{s}^{2}]} e^{- i n ω} .

The temporal impulse response and spectral FTF graph are shown in Fig. 7.

Fig. 7 Panel (a) shows the real component of the complex-valued chirp impulse response. Panel (b) shows the corresponding FTF which smoothly approximate a Hilbert quadrature filter [3].

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Figure 8 shows superimposed, the spectra of the fringe-data and chirp-reference PSA.

Fig. 8 Schematic of nonlinear-carrier fringe spectrum (in red), and the FTF of the Gaussian-window PSA (in green) having negligible response at the negative frequencies.

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We evaluate our nonlinear-reference PSA's demodulation error by the following formula,

φ_{E r r o r} = φ - \arg [\sum_{n = 0}^{12} w_{n} e^{- i (n θ_{0} + ε_{2} n^{2} T_{s}^{2})} I_{n}]; φ \in [0, 2 π] .

This equation compares the angle given by the PSA against its “true” computer-simulated value. Figure 9 shows the continuous plot of

φ_{E r r o r}

.

Fig. 9 Phase error given by Eq. (30). Note the vertical scale is within [-0.03,0.03] radians. This phase error has about the same magnitude as the one obtained in [12] using a linear reference PSA.

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Note that the peak phase demodulation error $φ_{E r r o r}$ is about 0.004 radians.

6.2 Nonlinear reference PSA with square window

Next we apply a square-window $(w_{n} = 1.0)$ nonlinear-reference PSA for demodulating the same fringes previously presented. Then our square-window PSA is,

A_{2} e^{i φ_{S q u a r e}} = \sum_{n = 0}^{12} e^{- i [n θ_{0} + ε_{2} n^{2} T_{s}^{2}]} {1 + \cos [φ + n θ_{0} + ε_{2} n^{2} T_{s}^{2}]} .

The FTF spectral plot associated to this PSA is shown in Fig. 10.

Fig. 10 Spectral response (FTF) for the square-window, nonlinear-phase reference PSA. This square window (w_n = 1.0) has large response at the left side of the fringe spectrum. This FTF is a bad approximation of a one-sided Hilbert quadrature filter [3].

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As Fig. 10 shows, the DC background at $(ω = 0)$ is not fully filtered-out, and also large energy from the unwanted conjugated signal leaks into the desired analytic signal $A_{2} e^{i φ_{S q u a r e}}$ .

Figure 11 shows in the blue trace the phase-error for the square-window $(w_{n} = 1.0)$ nonlinear-reference PSA. We summarize this section by remarking the fact that synchronously following the nonlinear-phase variations of the fringes is not enough; one must also apply a non-flat weighting window $(w_{n} \neq 1.0)$ to the nonlinear-reference PSA [13,14,16,19].

Fig. 11 The blue trace shows the phase-error for the 13-step square-window nonlinear-reference PSA. For comparison, the red-trace is the phase error corresponding to the Gaussian window PSA seen previously in Fig. 9. Note that the vertical scale is now [-0.1,0.1] radians.

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7. Discussion of the proposed nonlinear-reference PSA theory

Before our summary, we make a short exposition of our contribution. For the reader's convenience we rewrite the equations for our PSA theory. The fringes are modeled as,

I (t) = a + b \cos [φ + ω_{0} t + Δ (t)]; t \in [0, T] .

Where the only restriction about the phase-shifted variation

[ω_{0} t + Δ (t)]

is,

[ω_{0} + \frac{d Δ (t)}{d t}] \in (0, π); t \in [0, T] .

Using T_s = T/N, θ₀ = ω₀T_s, Δ_n = Δ(nT_s), and I_n = I(nT_s), our nonlinear-reference PSA reads as,

A_{2} e^{i φ} = \sum_{n = 0}^{N - 1} w_{n} e^{- i [n θ_{0} + Δ_{n}]} I_{n} ​; (w_{n} \in ℝ) .

The weighting coefficients

w_{n}

shape the PSA’s FTF as [18,19],

H (ω) = F [h (t)] = F {\sum_{n = 0}^{N - 1} w_{n} e^{i [ω_{0} n + Δ_{n}]} δ (t - n T_{s})} = \sum_{n = 0}^{N - 1} w_{n} e^{i [ω_{0} n + Δ_{n}]} e^{- i n ω} .

This FTF must filter-out the left hand-side spectrum of the fringes as,

\begin{matrix} H (ω) \approx 0 f o r ω \in [- π, 0], \\ H (ω) \neq 0 f o r ω \in (0, π) . \end{matrix}

These equations show our PSA design strategy for demodulation of nonuniform temporal phase-shifted fringes.

8. Summary

We have studied the phase-demodulation of nonlinear phase-stepped fringe patterns. We mathematically proved the origin of the spurious piston due to the use of (standard) linear reference PSAs for phase demodulation of nonlinear carrier fringes. This numerical/spurious piston is normally irrelevant for most applications in phase metrology; however, this is paramount for absolute phase measurements, such as optical thickness. We demonstrated that this piston naturally disappears when the PSA synchronously follows (matches) the nonlinear phase steps of the fringes.

Here we have given a frequency transfer function (FTF) approach for designing this nonuniform phase-shifting algorithms (PSAs). We then find the real-valued PSA coefficients w_n that shapes the FTF spectrum of the PSA to smoothly approximate a Hilbert quadrature filter. As such, this spectral FTF shaping must render almost zero the left hand side (including zero) of the fringes' spectrum.

We think that using a synchronous nonlinear-phase reference is a more natural solution to deal with non-linear phase-stepped fringe patterns.

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Phase-stepping algorithms for synchronous demodulation of nonlinear phase-shifted fringes

Abstract

1. Introduction

2. Linear and nonlinear phase-shifted fringe patterns

2.1 Fourier spectrum for linear and nonlinear phase-shifted fringes

3. Linear and nonlinear reference digital PSAs

3.1 Spurious-piston using uniform phase-shifted reference PSAs

3.2 No spurious-piston using nonuniform phase-shifted reference PSA

4. Spectral design for nonuniform phase-shifted reference PSAs

5. Signal-to-noise ratio (SNR) for linear and nonlinear reference PSA

6. Computer simulation with 13-step nonlinear-reference PSA

6.1 Nonlinear reference PSA with Gaussian window

6.2 Nonlinear reference PSA with square window

7. Discussion of the proposed nonlinear-reference PSA theory

8. Summary

References

Cited By

Figures (11)

Equations (36)

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