1. Introduction

One of the most used techniques to demodulate fringe patterns in optical metrology is Phase Shifting Interferometry (PSI), either in the spatial or temporal domain. In the temporal approach, PSI techniques require a mobile piston element in the interferometer to introduce a uniform phase step between successive samples, in order to obtain three or more phase-shifted interferograms to process. Typically, Phase-Shift Algorithms (PSAs) are constructed on the assumption that fringe patterns present a sinusoidal profile. In practice, non-sinusoidal fringes are due to nonlinear photodetector response, gain saturation, multiple beams interferences, etc. These systematic errors lead to the generation of harmonics of the fundamental signal (distorted fringes) which degrades the quality of the recovered phase map [1].

Many-steps PSAs allow good phase estimations even with non-sinusoidal fringe patterns. However, stability of the experimental set-up usually limits the number of samples that may be obtained. Several authors have described spatial techniques for simultaneous acquisition of phase-shifted interferograms [2–6]. This allows data acquisition faster than one-at-a-time temporal PSI. In the technique proposed by Miller et al. [6], a periodic phase-mask array of micro-polarizers is physically attached to a CCD sensor to obtain a unique phase-shift over each pixel. The physical principles of this approach are review in Ref [6].

A useful mathematical model for a Pixelated Phase-Mask (PPM) interferogram is

 

Fig. 1 Periodic distribution for the basic building cell of the phase-mask $pm(x,y)$.

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Fourier phase demodulation of pixelated phase-mask interferograms was shown independently by Servin & Estrada [7] and Kimbrough & Millerd [8], and it is reviewed below.

2. Fourier demodulation of pixelated phase-mask interferograms

In spatial-carrier interferometry, the frequency of the carrier must be higher than the highest frequency component of the signal of interest. If this condition is satisfied the components of the interferogram can be isolated by linear filtering [9]. For PPM interferograms this condition must be satisfied in two dimensions, which may be formally stated as

The first step to demodulate a pixelated phase-mask interferogram [7,8] is to multiply Eq. (1) by the complex pixelated carrier $\exp [-ipm(x,y)]$,

Applying a low-pass filter $h(x,y)$ to Eq. (3), preferably in the Fourier domain [7,8], the high frequency terms are rejected (henceforth, the “hat” denotes estimation):

Solving Eq. (4) for the estimated phase $\widehat{\varphi }(x,y)$, the spatial PSA results as

3 Fourier demodulation with distorted-fringe pixelated interferograms

Experimental data frequently have systematic errors (nonlinear photodetector response, gain saturation, multiple beams interferences, etc.) which adds harmonics to the signal. A distorted-fringes interferogram may be represented by the following Fourier series,

Multiplying the distorted interferogram in Eq. (6) by the complex carrier, one obtains

Since $pm(x,y)$ is periodic, its symmetry allows developing $\exp [\pm i(n\mp 1)pm(x,y)]$ for each 2x2 fundamental cell to obtain the spectral shift for every harmonic as:

Substituting Eq. (8) into Eq. (7), and grouping terms with the same spatial carrier, results

Note that the terms in the 1^st row do not undergo any frequency shift; they are all located at the spectral origin along with $\exp [i\varphi (x,y)]$. The terms in the 2^nd row are multiplied by $\exp [i\pi (x+y)]$ and displaced to the spectral corners $(\pm \pi ,\mp \pi )$. The terms in the 3rd row are shifted by the complex carrier $\exp [ipm(x,y)]$ to the cardinal points $(0,\pm \pi )$, and $(\pm \pi ,0)$.

Applying a low-pass filter $h(x,y)$ in the Fourier domain, the terms in the 2^nd and 3^rd rows in Eq. (9) are supressed:

In Fig. 2 and Fig. 3 we illustrate several stages of this demodulation process. (The spectral plots are in pseudo-color logarithmic intensity scale with the spectral origin at the center).

 

Fig. 2 (Color online) (a) Non-sinusoidal fringe pattern having many harmonics overlapped at the spectral origin. (b) The same measuring phase but now modulated by the pixelated phase-mask over the CCD. As shown in Eqs. (6-9), some harmonics are spectrally displaced.

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Fig. 3 (Color online) (a) Multiplying the pixelated interferogram by the complex carrier leave the complex harmonics {-3^rd, 5^th, −7^th, 9^th,…} at the spectral origin distorting $\exp [i\varphi (x,y)]$. (b) The phase estimation obtained by filtering-out outside the circle shown in (a), (Eq. (10)).

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As can be seen in Fig. 3b and Eq. (10), the complex harmonics not-rejected by the low-pass filtering $h(x,y)$ are {-3^rd, +5^th, −7^th, +9^th, −11^th,...}. Please note that, these harmonics coincides with those not-rejected in a standard 4-steps temporal least-squares PSA [10]. The next section shows our second contribution, namely how to remove some of these complex harmonics using few-steps temporal PSI techniques (a quadrature temporal filter).

4. Spatio-temporal demodulation of non-sinusoidal pixelated interferograms

Whenever possible, one may introduce a linear phase-step ${\omega }_0$ (radians/step) between successive samples of a distorted-fringes pixelated interferogram (Eq. (6)):

Applying the spatial demodulation in Section 2, to these phase-shifted distorted-fringes interferograms (Eq. (11)), the following 3 single-image phase estimations are obtained:

Which are the phase-shifted versions of Eq. (10). Then, one forms the temporal signal:

The signal in Eq. (13) still contains the remaining complex harmonics {-3^rd, 5^th, −7^th, 9^th,…}, but we can use temporal quadrature filters (PSAs) to reject them [1,10,11]. Specifically, we propose the following 2-steps PSA to reject the −3^rd and 5^th complex harmonics:

Similarly, the following 3-steps PSA rejects the −3^rd, 5^th, −7^th, and 9^th complex harmonics:

 

Fig. 4 (Color online) Normalized spectral response of the 2-step (solid curve) and 3-step (dotted curve) temporal quadrature filters (Eqs. (14,15)). The complex harmonics marked with an “x” were suppressed through single-image spatial demodulation, while the vertical arrows represent those complex harmonics that survived this single-image demodulation process (Eq. (10)).

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Using the 2-step PSA (Eqs. (13,14)) the estimated phase ${\widehat{\varphi }}_2(x,y)$ is given by:

Similarly, the 3-step PSA (Eqs. (13,15) with ${\omega }_0=\pi /3$) give the estimation ${\widehat{\varphi }}_3(x,y)$:

Finally, the most general expression for our spatio-temporal phase-demodulation algorithms is given by

Figure 5 shows the application of our spatio-temporal algorithms (Eqs. (16,17)) to the non-sinusoidal pixelated fringe pattern presented in Fig. 2(b).

 

Fig. 5 (Color online) Estimated phase and spectra obtained by spatio-temporal demodulation using (a) 2-step and (b) 3-step phase-shifted temporal samples (Eqs. (16,17)).

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For comparative purposes, Fig. 6 shows slices of the estimated phase maps using single-image demodulation, and spatio-temporal demodulation with 2, and 3 steps PSAs.

 

Fig. 6 (Color online) Slices of the estimated phase using: (a) single-image pixelated interferogram demodulation (Eq. (10)), (b) 2-step PSA phase demodulation (Eq. (16)), and (c) 3-step PSA phase demodulation (Eq. (17)).

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These simulations were done using a resolution of 512x512 pixels, but higher resolution provides greater spectral separation allowing better phase-demodulation. Today, pixelated phase-mask interferograms have resolutions almost sixteen times higher (4 mega-pixels).

5 Conclusions

We have shown that single-image Fourier demodulation of 2x2 pixelated interferogram leaves the complex harmonics {-3^rd, +5^th, −7^th, +9^th, −11^th,...} overlapping with the fundamental signal $\exp [i\varphi (x,y)]$, distorting it. Considering this, we proposed few-steps temporal PSI techniques to reject some of these remaining complex harmonics. Specifically, we used a 2-step temporal PSA designed to remove (at least) the −3^rd, and +5^th complex harmonics, and a 3-step one to remove (at least) the −3^rd, +5^th, −7^th, and +9^th complex harmonics. Note that temporal demodulation of N-step, 2x2 pixelated interferograms (N=1,2,3,…) rejects the same harmonics as a temporal 4N-step least-squares PSA. In other words, N-steps temporal interferometry of 2x2 pixelated fringe patterns is 4 times faster, preserving the harmonics rejection robustness of a 4N-step least-squares PSA.