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

Standard phase-stepping algorithms (PSAs) estimate the measuring phase of linear carrier temporal-fringes with respect to a linear-reference. Linear-carrier fringes are normally obtained using feedback, closed-loop, optical phase-shifting devices. On the other hand, open-loop, phase-shifting devices, usually give fringe patterns with nonlinear phase-shifts. The Fourier spectrum of linear-carrier fringes is composed by Dirac deltas only. In contrast, nonlinear phase-shifted fringes are wideband, spread-spectrum signals. It is well known that using linear-phase reference PSA to demodulate nonlinear phase-shifted fringes, one obtains an spurious-piston. The problem with this spurious-piston, is that it may wrongly be taken as a real optical thickness. Here we mathematically find the origin of this spurious-piston and design nonlinear phase-stepping PSAs to cope with open-loop, nonlinear phase-shifted interferometric fringes. We give a general theory to tailor nonlinear phase-stepping PSAs to demodulate nonlinear phase-shifted wideband fringes.

In this work we are proposing a different approach for phase demodulating temporal nonlinear-carrier fringes using wideband, synchronous, nonlinear-reference PSA. This is similar to the theory behind chirp-carrier radars [10,11]. In chirp-radars the wideband radio-frequency (RF) pulse varies quadratically with time. When the RF chirp-pulse bounce back from the radar target, the incoming RF-signal is correlated with a synchronous, local chirpwaveform. In the case of wideband chirp-radar, one is interested in timing the amplitude of the correlation peak between the incoming RF chirp-signal and the chirped local-oscillator. Timing this correlation peak give us the round-trip target distance [10,11]. As Fig. 1 shows, here we are using the same concept of synchronously following the nonlinear phase-shifted carrier fringes using the same nonlinearity phase-shifting as reference.

Linear and nonlinear phase-shifted interferometric fringes
Let us first show the usual mathematical models for linear and nonlinear phase-shifted interferometric fringes. The model for linear-carrier interferometric fringes is, is the measuring phase. On the other hand, nonlinear-carrier fringes are formalized by, We are assuming that the nonlinearity ( ) t  is smooth, and can be determined experimentally [3][4][5][6][7][8][9]. Previous papers assume that ( ) t  can be approximated by few Taylor series terms [3,9]. Here we relax this condition, by requiring only that the derivative of In Fig. 2 we show linear (in blue) and nonlinear (in red) phase-shifted fringes. As we prove in the next sections, the nonlinear phase-shifting ( ) t  generate a spuriousnumerical piston when a linear-reference PSA is used as phase-demodulator [4][5][6][7][8][9]. Fig.2 shows an example of linear and nonlinear carrier fringes,

Fourier spectrum for linear and nonlinear phase-shifted fringes
From Eq. (1), linear fringes are single-frequency at 0  , having a spectrum given by [2], Where   F  is the Fourier transform operator (see Fig. 3(a)). In contrast, highly nonlinear phase-shifted fringes (Eq. (2)) are wideband, and its spectrum may be modeled as, Where,  Summarizing, linear-phase carrier fringes have a three delta spectrum (Fig 3(a)); while nonlinear-phase carrier fringes have two spread-spectrum components ( Fig. 3(b)).

Linear and nonlinear reference PSAs
Let us now show the mathematical form of phase-shifting algorithms (PSAs) using linear and nonlinear-reference for demodulating nonlinear-carrier fringes

Standard linear-reference PSA for demodulating linear-carrier fringes
The general form for standard linear-carrier, linear-reference PSA is [2], These are the standard linear-carrier, linear-reference PSAs in use since 1974 [1,2].

Nonlinear-reference PSA for demodulating nonlinear-carrier fringes
We specifically propose the use of a nonlinear-reference PSA which has the following form, Note that the nonlinear-reference

Spurious-piston using linear phase-shifted reference PSAs
Using a linear-reference PSAs to demodulate nonlinear-carrier fringes (Eq. (8)) one obtains, Performing the indicated multiplications one obtains, The coefficients n d are chosen such that the first and third square-brackets are set to zero as, Obtaining the desired analytic signal as, As we see, in general, the spurious-piston is non-zero ( 0 Piston  ), and it may give erroneous absolute optical thickness measurements [4][5][6][7][8][9].

No spurious-piston using nonlinear phase-shifted reference PSA
Now using a synchronous (matched-phase) nonlinear reference PSA (Eq. (9)) one gets, Performing the multiplications one obtains,  

Spectral design for nonlinearly phase-shifted reference PSAs
In previous section we gave an algebraic approach for calculating the coefficients ( ) n w   for nonlinear phase-shifted reference PSAs. Here we develop a more intuitive spectral design. The impulse response of the nonlinear reference PSA (Eq. (9)) is.
The coefficients ( )  [12,13]. In red we show the wideband spectrum of the nonlinearcarrier fringes. The FTF in panel (b) is a smooth approximation of a Hilbert quadrature filter.
As Fig. 4 Fig. 4(b). Of course other weightings windows ( 1) n w  may be used [12,13]. In Fig. 5 we show the (normalized frequency) harmonic response of the apodized, nonlinear reference PSA.

Signal-to-noise ratio (SNR) for linear and nonlinear reference PSA
Here we find the SNR [2] of the phase-demodulated nonlinear-carrier fringes corrupted by additive white Gaussian-noise (AWGN) . The noisy fringes are, Where the noise spectral density ( ) S  is flat, and it is given by, As conclusion, the SNR is generally higher for a PSA with nonlinear-reference.

Example of a 13-steps Gaussian-window nonlinear-reference PSA
Here we are given a computer simulation example of a 13-step nonlinear-reference PSA applied to nonlinear-carrier fringes. The most usual phase-shifted nonlinearity is quadratic, [3][4][5][6][7][8][9]. We start by considering nonlinear-carrier fringes as, 12 2 The non-linear phase and the interferometric chirp-waveform is shown in Fig. 6. The specific form of the 13-steps nonlinear, chirp-reference PSA is given by, And its temporal and spectral graphs are shown in Fig. 7. Next Fig. 8 shows, superimposed, the fringe-data and the chirp-reference PSA spectra.  Figure 9 shows that the peak phase demodulation error Error  is about 0.04 radians.

Example of a 13-steps square-window nonlinear-reference PSA
Here we analyze a square-window nonlinear-reference PSA for the same fringes used in previous section The spectral graph of the FTF associated to this PSA is shown in Fig. 10. Fig. 10. Spectral response (FTF) for the square-window, nonlinear-reference PSA. This square window cannot be used because it has large response in the origin and the left side of the fringe spectrum. This FTF is a bad approximation of a one-sided Hilbert quadrature filter.
As Fig. 11 shows, the DC background of the fringes is not fully filtered-out, and also large energy from the unwanted conjugate-signal leaks into the desired analytic signal. Fig. 11. The blue trace shows the phase-error for the 13-step, square-window, nonlinearreference PSA. For comparison, the red-trace is the phase error corresponding to the Gaussian window seen in previous section. Note that the vertical scale is now [-0.1,0.1] radians. Figure 11 shows in the blue trace the phase-error of the square-window nonlinear-reference PSA. We summarize this section by remarking the fact that synchronously following the nonlinear-carrier variations of the fringes is not enough. One must also apply an apodizing weighting window to the nonlinear-reference PSA [12,13].

Summary
Here we have given a frequency transfer function (FTF) approach for designing nonlinear phase-shifting algorithms (PSAs) applied to demodulate nonlinear phase-shifted fringes. The estimated nonlinear phase-step variations of the fringes [4][5][6][7][8][9], constitute also our nonlinear phase-step reference for the PSA. We then find the real-valued PSA coefficients ( ) n w that shapes the FTF spectrum of the PSA. The spectral shape of the nonlinear reference PSA smoothly approximate a Hilbert quadrature filter. As such, the spectral FTF shaping must render almost zero the left side (including zero) of the fringes spectrum.