Synthesis of multi-wavelength temporal phase-shifting algorithms optimized for high signal-to-noise ratio and high detuning robustness using the frequency transfer function

Synthesis of single-wavelength temporal phase-shifting algorithms (PSA) for interferometry is well-known and firmly based on the frequency transfer function (FTF) paradigm. Here we extend the single-wavelength FTF-theory to dual and multi-wavelength PSA-synthesis when several simultaneous laser-colors are present. The FTF-based synthesis for dual-wavelength PSA (DW-PSA) is optimized for high signal-to-noise ratio and minimum number of temporal phase-shifted interferograms. The DW-PSA synthesis herein presented may be used for interferometric contouring of discontinuous industrial objects. Also DW-PSA may be useful for DW shop-testing of deep free-form aspheres. As shown here, using the FTF-based synthesis one may easily find explicit DW-PSA formulae optimized for high signal-to-noise and high detuning robustness. To this date, no general synthesis and analysis for temporal DW-PSAs has been given; only had-hoc DW-PSAs formulas have been reported. Consequently, no explicit formulae for their spectra, their signal-to-noise, their detuning and harmonic robustness has been given. Here for the first time a fully general procedure for designing DW-PSAs (or triple-wavelengths PSAs) with desire spectrum, signal-to-noise ratio and detuning robustness is given. We finally generalize DW-PSA to higher number of wavelength temporal PSAs.


Introduction
As far as we know, the first researcher to use dual-wavelength (DW) interferometry was Wyant  ). Double-wavelength (DW) interferometry was improved by Polhemus [3] and Cheng [4,5] using digital temporal phase-shifting.
On the other hand, Onodera et al. [6] used spatial-carrier, double-wavelength digitalholography (DW-DH) and Fourier interferometry for phase-demodulation. This in turn was followed by a large number of multi-wavelength digital-holographic (DH) Fourier phasedemodulation methods in such diverse applications as interferometric contouring [7], phaseimaging [8], chromatic aberration compensation in microscopy [9]; single hologram DW microscopy [10]; comb multi-wavelength laser for extended range optical metrology [11], and a two-steps digital-holography for image quality improvement [12].
More recently temporal dual-wavelength phase-shifting algorithms (DW-PSAs) have been reworked by Abdelsalam et al. [14]. Even though Abdelsalam et al. give working PSA formulas they do not estimate their spectra, their signal-to noise ratio, or their detuning and harmonics robustness. Kumar [15] and Baranda [16] also provided valid temporal PSA formulas but also failed to characterize their PSAs in terms of signal-to-noise, detuning and harmonic rejection. Another different approach was followed by Kulkarni and Rastogi [16] in which they have demodulated the two interesting phases by fitting a low-order polynomial to each phase. Their approach [17] worked well for the example provided but we think their method could easily cross-talk between fitted polynomials for complicated modulating phases [17]. Yet another approach by Zhang et al. was published [18][19]. Zhang used a simultaneous two-steps [18], and principal component interferometry [19] to solve the dual-wavelength phase-shifting measurement. Zhang et al. used 32 randomly phase-shifted interferograms [19]. Even though Zhang [19] could demodulate the two phases, they used 32 phase-shifted temporal interferograms. All these works on temporal DW-PSA [2][3][4][5][14][15][16][17][18][19] have given just specific DW-PSAs without explicit formulae for their spectra, signal-to-noise, detuning and harmonic robustness.
In contrast to previous ad-hoc temporal DW-PSA formulas, here we give a general theory for synthesizing DW-PSAs formalizing their spectrum, their signal-to-noise, and their detuning-harmonic robustness. At the risk of being repetitive, we emphasize that we are not just giving particular DW-PSAs formulas as previously done [2][3][4][5][14][15][16][17][18][19]. Here we are giving a general FTF-theory for synthesizing DW-PSA giving with explicit formulae for the most important characteristics of any PSA: spectra, signal-to-noise, detuning and harmonic robustness.

Spatial-carrier phase-demodulation for Dual-wavelength (DW) interferometry
Dual-wavelength digital-holography (DW-DH) is well understood and widely used [6][7][8][9][10]. As shown in Fig. 1, in DW-DH the two lasers beams are tilted to introduce spatial-carrier fringes [7]. In Fig. 1 both lasers beams are tilted in the x direction, but in general, for a better use of the Fourier space, one may tilt them independently along the x and y directions [11][12][13][14]. Fig.1 Schematics for DW-DH with a single tilted reference mirror [6]. The orange-light corresponds to the spatial superposition of the red and green lasers.
The DW-DH obtained at the CCD camera in Fig.1 may be modeled by, are the spatial-carriers of the DW-DH. The reference mirror angle along the x axis is  . The searched phases are ; being 1 ( , ) W x y and 2 ( , ) W x y the measuring wavefronts. Figure 2 shows a Fourier spectrum of Eq. (1). The two hexagons in Fig. 2 are the quadrature filters that passband the desired analytic signals. After filtering, the inverse Fourier transform find the demodulated phases [1]. The advantage of DW-DH is that only one digital-hologram is needed to obtain 1 2 { , }   ; however its drawback is that just a fraction of the Fourier space ( , ) [ , ] [ , ] u v         is used (Fig  2). This limitation makes DW-DH not suitable for measuring discontinuous industrial objects [7]. In contrast, in DW-PSAs the full Fourier spectrum ( , ) [ , ] [ , ] u v         may be used.

Dual-wavelength (DW) temporal-carrier phase-shifting interferometry
The temporal phase-shifting double-wave interferogram may be modeled as, .
Where ( , ) t   , and 1 1 1 are the measuring phases. The parameter d is the PZT-step. The fringes background is ( , ) a x y and the lasers power must be about the same to obtain high fringe contrast: 1 Figure 3 shows one possible set-up for a DW temporal phase-shifting interferometer. The motivation of using 2-wavelengths 1  and 2  (in spatial or temporal interferometry) is that interferometric measurements can be made with an equivalent wavelength eq  [2-19], With large eq  one may measure deeper surface discontinuities or topographies than using either 1  or 2  alone [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. For a given PZT-step d , the two angular-frequencies (in radians per interferogram) are given by, Using this equation one may rewrite Eq. (2) as, Here we have 5 unknowns: namely, Therefore we need at least 5 phaseshifted interferograms to obtain a solution for 1 ( , ) x y  and 2 ( , ) x y  . These are given by: For clarity, most ( , ) x y coordinates were omitted.

Fourier-spectrum for temporal DW-PSAs
The Fourier transform of the temporal interferogram (with ( , ) t    ) in Eq. (5) is: All ( , ) x y were omitted. As mentioned, 1 are the two temporal-carrier frequencies in radians/interferogram; Fig. 4 shows this spectrum.
. Note how each filter is able to passband the desired signals from the same temporal interferograms.

Optimized joint signal-to-noise ratio synthesis for DW-PSAs
To find a better selection for 1 , we construct a joint signalto-noise ratio as,   The function d is complicated and has many local maxima but, fortunately, it is onedimensional. Thus we plot   / S N G d and look for a good maximum, and take the PZT-step d. This PZT-step d is used to find the two specific DW-PSA (Eqs. (11)- (12)) which solves the dual-wavelength interferometric problem.
9. Example of optimized DW-PSA synthesis for 1 λ = 632.8nm and 2 λ = 532nm The graph for the joint signal-to-noise ratio is shown next. at random, the probability of landing in a very low signal-to-noise point is very high.
Therefore in this section we have shown that even though the correct phases 1 ( , ) x y  and 2 ( , ) x y  can be found using Eq. (11) and Eq. (12), without plotting   / S N G d these DW-PSAs designs will have a low signal-to-noise power-ratio with high probability. 9. Example for DW-PSA phase-demodulation for 1 λ = 632.8nm and 2 λ = 532.0nm Figure 9 shows five computer-simulated interferograms to test the DW-PSAs found in previous section. The PZT-step is 751nm d  , giving a good signal-to-noise ratio. As mentioned, for large PZT-steps, the angular frequencies 1 2 ( , )   are wrapped and given by, Using these angular frequencies in Eq. (11), the specific formula to estimate 1 ( , )  The noisy fringes were low-pass filtered by a 3x3 averaging window. Figure 10 shows the demodulated signals 1 ( , ) x y  and 2 ( , ) x y  . Fig. 10. The demodulated phases 1(x,y) and 2(x,y) corresponding to the noiseless (panel (a)) and noisy (panel (b)) 5-steps interferograms in Fig. 9. Please note that there is no crosstacking between the two demodulated phases 1(x,y) and 2(x,y). Figure 10(a) shows the noiseless demodulated phases, while Fig. 10(b) shows the demodulated phases degraded with a phase noise uniformly distributed within[0, ]  . Note that absolutely no cross-talking between the demodulated phases 1  and 2  appears.

Detuning-robust DW-PSA synthesis for 1 λ = 632.8nm and 2 λ = 458nm
Let us assume that our PZT is poorly calibrated. Thus instead of having well-tuned frequencies at 1 2 { , }   we have detuned frequencies at 1 being  the amount of detuning. As Fig. 11 shows, the estimated phase 2 ( , ) x y  is now be given by, The estimated phase 2 ( , ) x y  then have cross-talking from  Figure 13 shows the two 8-step DW-PSA detuning-robust FTFs. The spectral second-order zeroes are flatter, so they are frequency detuning  tolerant. Fig. 13. Spectra of detuning-robust DW-PSA tuned at 1 =2.5rad  and 2 =1.05rad  . The second-order zeroes tolerate a fair amount of frequency detuning .  The power of the desired analytic signals  Figure 15 shows five saturated phase-shifted interferograms. These 5 temporal interferograms are then phase demodulated using DW-PSAs, Eqs (11)-(12). Fig. 15. Five DW phase-shifted temporal interferograms with amplitude saturation. Figure 16 shows the demodulated phases 1  and 2  of the interferograms in Fig. 15.