Abstract
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 (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 ad 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.
© 2016 Optical Society of America
1. Introduction
Throughout this paper we assume that the frequency transfer function (FTF) paradigm is known [1]. As far as we know, the first researcher to use dual-wavelength (DW) interferometry was Wyant in 1971 [2]. Wyant used two fixed laser-wavelengths and to test an optical surface with an equivalent wavelength of [2]. Thus typically is much larger than either or (). Dual-wavelength (DW) interferometry was improved by Polhemus [3], and Cheng and Wyant [4,5] using digital temporal phase-shifting.
On the other hand, Onodera et al. [6] used spatial-carrier double-wavelength digital-holography (DW-DH) and Fourier interferometry for phase-demodulation. This in turn was followed by many multi-wavelength digital-holographic (DH) Fourier phase-demodulation methods in such diverse applications as interferometric contouring [7], phase-imaging [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]. DW-DH is already well understood.
Switching back to temporal DW phase-shifting algorithms (DW-PSAs), Abdelsalam et al. [14] have recently reworked this technique. Even though Abdelsalam et al. [14] give working PSA formulas they do not estimate their spectra, their signal-to-noise ratio, or their detuning and harmonics robustness. Kumar et al. [15] and Baranda et al. [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 [17] 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 [2–5,14–19] 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 [2–5, 14–19], here we give a general theory for synthesizing DW-PSAs mathematically formalizing their spectrum, their signal-to-noise, and their detuning-harmonic robustness; these are the most important characteristics of any PSA.
2. Spatial-carrier phase-demodulation for Dual-wavelength (DW) interferometry
Dual-wavelength digital-holography (DW-DH) is well understood and widely used [6–10]. 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 [11–14].
The DW-DH obtained at the CCD camera in Fig. 1 may be modeled by,
Here and are the spatial-carriers of the DW-DH. The reference mirror-angle with respect to the axis is . The searched phases are and ; being and the measuring wavefronts. Figure 2 shows a schematic of the Fourier spectrum of Eq. (1).The two hexagons in Fig. 2 are the spatial 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 ; however its drawback is that just a fraction of the Fourier space 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 may be used.
3. Temporal dual-wavelength (DW) phase-shifting interferometry
From now on only temporal interferometry is discussed. The temporal phase-shifting fringes for double-wavelength interferometry may be modeled as,
Here , and , are the measuring phases. The parameter is the PZT-step. The fringes background is and their contrasts are and . Figure 3 shows one possible set-up for a DW temporal phase-shifting interferometer.With 2-wavelengths measurements one can synthesize an equivalent wavelength [2–19],
With large one may measure deeper surface discontinuities or topographies than using either or [2–19]. For a given PZT-step , 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 phase-shifted interferograms (5-equations) to obtain a solution for ; these are,For clarity, most coordinates were omitted.4. Fourier-spectrum for temporal DW-PSAs
The Fourier transform of the temporal interferogram (with) in Eq. (5) is:
All were omitted. As mentioned, and are the two temporal-carrier frequencies in radians/interferogram; Fig. 4 shows this spectrum.Figure 5 shows two ideal frequency transfer functions (FTF), and , that could passband the desired analytic signals and . Note how each filter is able to passband the desired signals from the same N temporal interferograms.
5. Synthesis of DW-PSAs using the FTF and 5-step temporal interferograms
As we know from the FTF-based PSA theory, the rectangular filters in Fig. 5 require a large number N of temporal interferograms [1]. However we can synthesize 5-step bandpass quadrature filters by allocating just 4 spectral-zeroes at frequencies for the FTF , and 4-zeroes at for the FTF as,
From Eqs. (7)-(8) one sees that passband the signal , while bandpass . Their impulse responses and are,Here and are the 5 complex-valued coefficients that depend on the frequencies . Having the searched DW-PSAs are,Where are the 5 interferograms. The explicit 5-step DW-PSA to estimate is,With . Conversely the 5-step DW-PSA to estimate is:Being . This is the basics for synthesizing DW-PSAs grounded on the FTF paradigm [1]. Previous papers on DW-PSAs [2–5,14–19] stop much shorter than this. They just show particular pairs of DW-PSAs [2–5,14–19] that work for just particular carriers, i.e. . In this section, we offered DW-PSAs (Eqs. (11)-(12)) which work well (find and) for infinitely-many frequency-pairs . Even if the theory of this paper would stop right here, this paper contains a substantial improvement against current ad hoc state of the art in DW-PSA [2–5,14–19].6. Signal-to-noise power-ratio (SNR) for the FTFs and
Here we review the signal-to-noise power-ratio formulas for PSA quadrature filters [1]. The signal-to-noise power-ratios (SNR) for the FTFs and are given by [1]:
These SNR-formulas give the power of the signals and divided by their total noise-power and .7. Non-optimized DW FTF-based design for and
Let us assume that we use a typical temporal frequency of radians per sample for the algorithm . Having made this choice for , the frequency is set to
Giving a PZT-step of . The DW-FTFs for the two frequencies are:Figure 6 shows the magnitude plot of these two quadrature filters .The signal-to-noise [1] for the signals and are:
For comparison, a 5-step least-squares PSA has a signal-to-noise power-ratio of 5 [1]. Thus and were a bad choice. Even though we can estimate without cross-talking, from Eqs. (11)-(12), they are going to have poor SNR. Previous efforts in DW-PSAs [2–5,14–19] only provided numeric-specific formulas to obtain . However, they were absolutely silent about their Fourier spectra, their cross-talk, their signal-to-noise, their harmonics and detuning robustness. All this useful and practical formulae are given here for the first time in terms of the FTFs for designing DW-PSAs. Moreover, in contrast to previous art in DW-PSAs, Eq. (11) and Eq. (12) give infinitely many DW-PSA formulas for continuous pairs of temporal frequencies .8. Synthesis of DW-PSAs optimized for signal-to-noise ratio
To find a better selection for and , we construct a joint product signal-to-noise ratio as,
has many local maxima, but fortunately it is one-dimensional. Then plot , look for a good maximum and take the PZT-step d. This PZT-step d is used to find , and the two specific DW-PSA (Eqs. (11)-(12)) which solves the DW interferometric problem.9. Example of SNR-optimized synthesis for and
The graph for the signal-to-noise power-ratio product with , and is shown next (Fig. 7).
The first good local maximum is (in blue), being or . Note that most of this graph is less than 20; i.e. . This means that taking a PZT-step within at random, the probability of landing in a very low signal-to-noise point is very high. The FTF graphs for are shown in Fig. 8.
Here we have shown that there is a high probability of having a low SNR for the demodulated phases and without optimizing for (Eq. (17)).
10. Example for DW-PSA phase-demodulation for and
Figure 9 shows five computer-simulated interferograms to test the DW-PSAs found in previous section. The PZT-step is , giving a good signal-to-noise ratio. As mentioned, for large PZT-steps, the angular frequencies are wrapped and given by,
Using these angular frequencies in Eq. (11), the specific formula to estimate is,Also, from Eq. (12), the specific 5-step DW-PSA to estimate the signal is,Figure 10 shows the demodulated signals and .Figure 10(a) shows the noiseless demodulated phases, while Fig. 10(b) shows the demodulated phases degraded with a phase noise uniformly distributed within. Note that absolutely no cross-talking between the demodulated phases and appears.
11. Detuning-robust and SNR-optimized DW-PSA synthesis
Let us assume that our PZT is poorly calibrated. Thus instead of having well-tuned frequencies at we have detuned frequencies at, being the amount of detuning. As Fig. 11 shows, the estimated (erroneous) phase is now given by,
The estimated phase thus have cross-talking from the signals ; conversely will have distorting cross-talking from .To have good detuning robustness we need double-zeroes at the rejected frequencies. Therefore, we transform the FTFs in Eq. (8) (5-steps) to detuning-robust FTFs (8-steps) as,
Proceeding as before, we need to plot and look for a local signal-to-noise maximum. This is shown in Fig. 12 for and .We choose the second maximum (in blue) where , with nm. Each 8-step DW-PSA filter in Eq. (22) has a signal-to-noise ratio of about . Figure 13 shows the two 8-step detuning-robust FTFs. The spectral second-order zeroes are flatter, so they are frequency detuning tolerant.
12. Harmonic rejection for DW-PSAs
The main source of fringe-distorting harmonics is the non-linear response of the CCD-camera used to digitize the interferograms [1]. Therefore instead of having perfect-sinusoidal fringe-profile we may have saturated-distorted fringes containing high harmonic power [1]. Figure 14 shows the harmonic response for the FTFs in Eq. (8). The red-sticks are the fringe harmonics at, and the green ones are the fringe harmonics at , .
The power of the desired analytic signals and with respect to the sum of their distorting harmonic power is given by,
We assumed that the harmonics amplitude decreases as , so their power decreases as . With this assumption the PSA-filters have about 10-times more power than the total power-sum of their harmonics .Figure 15 shows five saturated phase-shifted interferograms. These five temporal interferograms are phase demodulated using DW-PSAs, Eqs. (11)-(12).
Figure 16 shows the distorted demodulated-phases of the saturated fringes in Fig. 15.
13. Multi-wavelength FTF-based phase-shifting algorithms synthesis
Here DW-PSA is generalized to 3-walengths. A simplified schematic of an interferometer simultaneously illuminated with 3-wavelengths is shown in Fig. 17.
The continuous-time phase-shifted interferogram is,
Now Eq. (24) have 7 unknowns ; being the searched phases. Thus we need at least 7 phase-shifted interferograms (7-equations) to find . Figure 18 shows the spectrum (for ) of this 3-wavelengths temporal-interferograms.Therefore we need to construct 3-FTFs having at least 6 first-order zeroes (7-steps) as,
The FTF rejects the analytic signals at ; the FTF rejects the Dirac deltas at ; and the FTF rejects the deltas at . Therefore isolates ; isolates , and finally obtains .The joint-product signal-to-noise ratio (SNR) optimizing criterion now reads,
We then find a high local maximum for , obtaining a fixed PZT-step , and three angular-frequencies as,The three impulse responses are then given by,Here , , are the complex coefficients of the PSAs, which now depend on the three temporal-carrier frequencies .We now digitally capture 7 phase-shifted interferograms given by:
With these 7 interferograms we obtain the three searched quadrature analytic signals as,where . By mathematical induction, one may see that a 4-wavelength phase-shifting algorithm would need at least 9 phase-shifted interferograms, requiring FTFs having 8 first–order zeroes, et cetera.14. Conclusions
The problem that was solved here may be stated as follows: Having a laser interferometer simultaneously illuminated with fixed wavelengths and a single PZT phase-shifter, find K phase-shifting algorithms (PSAs) which phase-demodulate for each laser-color, with high signal-to-noise and no cross-taking among these phases.
This was solved as follows (for K = 2 sections 3-12, and K = 3 in section 13),
- c) Having an optimum PZT-step , we then calculated the tuning frequencies , , which substituted back into gave us the specific DW-PSAs that demodulate and (Eqs. (11)-(12)).
- e) We used the SNR-optimized FTF-designs to phase-demodulate 5 phase-shifted interferograms (Figs. 9–10) with high signal-to-noise and no phase cross-talking.
- f) For poor PZT-calibration we modified the FTFs by raising the first-order zeroes to second-order ones, i.e. , , etc.; making robust to detuning at the rejected frequencies (Fig. 13).
- g) With the SNR-optimized FTFs we quantified the harmonic-rejection capacity for each using Eq. (23).
- h) Finally in section 13, we extended the DW FTF-based theory to 3-wavelengths ; further K-wavelengths generalization of this FTF-based multi-wavelength PSA theory is just a matter of mathematical induction.
As far as we know, previous art on DW-PSAs [2–5,14–19] only provided ad hoc multi-wavelength PSA designs. Thus, this is the first time that a general theory for synthesizing and analyzing multi-wavelength temporal phase-shifting algorithms is presented, and from which one may derive quantifying formulas for: (a) the PSAs spectra for each wavelength, (b) the PSAs signal-to-noise robustness for each wavelength, (c) the PSAs detuning sensitivity, and (d) the PSAs harmonics rejection for each wavelength. Finally, we presented two computer simulated examples of 5 DW phase-shifted interferograms with and in order to illustrate the behavior of our synthesized FTF-based DW-PSAs.
Acknowledgments
The authors acknowledge the financial support of the Mexican National Council for Science and Technology (CONACYT), grant 157044. Also the authors acknowledge Cornell University for supporting the e-print repository arXiv.org and the Optical Society of America for permitting OSA’s contributors to post their manuscript at arXiv.
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