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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 20 — Sep. 26, 2011
  • pp: 19508–19513
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Harmonics rejection in pixelated interferograms using spatio-temporal demodulation

J. M. Padilla, M. Servin, and J. C. Estrada  »View Author Affiliations


Optics Express, Vol. 19, Issue 20, pp. 19508-19513 (2011)
http://dx.doi.org/10.1364/OE.19.019508


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Abstract

Pixelated phase-mask interferograms have become an industry standard in spatial phase-shifting interferometry. These pixelated interferograms allow full wavefront encoding using a single interferogram. This allows the study of fast dynamic events in hostile mechanical environments. Recently an error-free demodulation method for ideal pixelated interferograms was proposed. However, non-ideal conditions in interferometry may arise due to non-linear response of the CCD camera, multiple light paths in the interferometer, etc. These conditions generate non-sinusoidal fringes containing harmonics which degrade the phase estimation. Here we show that two-dimensional Fourier demodulation of pixelated interferograms rejects most harmonics except the complex ones at {-3rd, +5th, −7th, +9th, −11th,…}. We propose temporal phase-shifting to remove these remaining harmonics. In particular, a 2-step phase-shifting algorithm is used to eliminate the −3rd and +5th complex harmonics, while a 3-step one is used to remove the −3rd, +5th, −7th and +9th complex harmonics.

© 2011 OSA

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

1. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

].

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

2. R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

6

6. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004). [CrossRef]

]. This allows data acquisition faster than one-at-a-time temporal PSI. In the technique proposed by Miller et al. [6

6. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004). [CrossRef]

], 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

6. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004). [CrossRef]

].

A useful mathematical model for a Pixelated Phase-Mask (PPM) interferogram is
I(x,y)=a(x,y)+b(x,y)cos[ϕ(x,y)+pm(x,y)],
(1)
where a(x,y), b(x,y), ϕ(x,y) are respectively the background, local contrast and the searched phase function. The pixelated carrier pm(x,y) is a periodic 2-D phase-array. In the following analysis we work with the so-called “2x2 circular orientation”, illustrated in Fig. 1
Fig. 1 Periodic distribution for the basic building cell of the phase-mask pm(x,y).
, however the “stacked” 2x2 spatial configuration [6

6. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004). [CrossRef]

] may also be analyzed in a similar way.

Fourier phase demodulation of pixelated phase-mask interferograms was shown independently by Servin & Estrada [7

7. M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express 18(17), 18492–18497 (2010). [CrossRef] [PubMed]

] and Kimbrough & Millerd [8

8. B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 1–12 (2010).

], 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

9. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982). [CrossRef]

]. For PPM interferograms this condition must be satisfied in two dimensions, which may be formally stated as
|pm(x,y)|>max|ϕ(x,y)|,
(2)
where ∇ is the gradient operator. This is a correction of the one presented in [7

7. M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express 18(17), 18492–18497 (2010). [CrossRef] [PubMed]

].

The first step to demodulate a pixelated phase-mask interferogram [7

7. M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express 18(17), 18492–18497 (2010). [CrossRef] [PubMed]

,8

8. B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 1–12 (2010).

] is to multiply Eq. (1) by the complex pixelated carrier exp[ipm(x,y)],

I(x,y)exp[ipm(x,y)]=[a+bcos(ϕ+pm)]exp(ipm),=aexp(ipm)+(b/2){exp(iϕ)+exp[i(ϕ+2pm)]}.
(3)

Applying a low-pass filter h(x,y) to Eq. (3), preferably in the Fourier domain [7

7. M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express 18(17), 18492–18497 (2010). [CrossRef] [PubMed]

,8

8. B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 1–12 (2010).

], the high frequency terms are rejected (henceforth, the “hat” denotes estimation):

(1/2)b(x,y)exp[iϕ^(x,y)]={exp[ipm(x,y)]I(x,y)}*h(x,y).
(4)

Solving Eq. (4) for the estimated phase ϕ^(x,y), the spatial PSA results as
tan[ϕ^(x,y)]=Im{[exp(ipm(x,y))I(x,y)]*h(x,y)}Re{[exp[ipm(x,y)]I(x,y)]*h(x,y)},
(5)
where the operators Im[] and Re[] take the imaginary and real parts of their argument.

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,
I(x,y)=a(x,y)+n=1bn(x,y)cos{n[ϕ(x,y)+pm(x,y)]+θn},
(6)
where bn(x,y) are the fringe contrast for the n-th harmonics, and θn are irrelevant angular displacements which arise from the Fourier series expansion of the fringes.

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

I(x,y)exp[ipm]=aexp(ipm)+n=1(bn/2){exp[i(n1)pm]exp[i(nϕ+θn)]+                   exp[i(n+1)pm]exp[i(nϕ+θn)]}.
(7)

Since pm(x,y) is periodic, its symmetry allows developing exp[±i(n1)pm(x,y)] for each 2x2 fundamental cell to obtain the spectral shift for every harmonic as:
k=1,exp[±i(1)(0π/23π/2π)]=exp[±ipm(x,y)],k=2,exp[±i(2)(0π/23π/2π)]=exp[±i(0ππ2π)]=exp[±iπ(x+y)],k=3,exp[±i(3)(0π/23π/2π)]=exp[±i(03π/2π/2π)]=exp[±ipm(x,y)T],k=4,exp[±i(4)(0π/23π/2π)]=exp[±i(02π6π4π)]=[1111],
(8)
where k labels the harmonic carriers in Eq. (7). From Eq. (8), it is clear that for higher than the 4th harmonics the carrier is chosen from the above 4 possibilities using k=(n1)mod4.

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

I(x,y)exp(ipm)=(1/2){b1exp(iϕ)+b3exp(i3ϕ)+b5exp(i5ϕ)+b7exp(i7ϕ)+}+(1/2){b1exp(iϕ)+b3exp(i3ϕ)+b5exp(5iϕ)+}exp[iπ(x+y)]+{a+b2cos(2ϕ)+b4cos(4ϕ)+b6cos(6ϕ)+b8cos(8ϕ)+}exp(ipm).
(9)

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

Applying a low-pass filter h(x,y) in the Fourier domain, the terms in the 2nd and 3rd rows in Eq. (9) are supressed:
Aexp[iϕ^1(x,y)]=[exp(ipm)I(x,y)]h(x,y),=b1exp[iϕ(x,y)]+n=1b2n+1exp[i(1)n(2n+1)ϕ(x,y)],
(10)
where A is a proportionality constant and ϕ^1(x,y) is our single-image phase estimation. To our knowledge, no prior harmonic analysis has been made on (distorted-fringe) pixelated interferogram demodulation. Therefore, Eq. (10) represents our first contribution in this paper.

In Fig. 2
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.
and Fig. 3
Fig. 3 (Color online) (a) Multiplying the pixelated interferogram by the complex carrier leave the complex harmonics {-3rd, 5th, −7th, 9th,…} at the spectral origin distorting exp[iϕ(x,y)]. (b) The phase estimation obtained by filtering-out outside the circle shown in (a), (Eq. (10)).
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).

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

Whenever possible, one may introduce a linear phase-step ω0 (radians/step) between successive samples of a distorted-fringes pixelated interferogram (Eq. (6)):

I(x,y,t)=a(x,y)+n=1bn(x,y)cos{n[ϕ(x,y)+pm(x,y)+ω0t]+θn},t=0,1,2,...
(11)

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:

exp[iϕ^1]         =b1exp(iϕ)+b3exp(i3ϕ)+b5exp(i5ϕ)+b7exp(i7ϕ)+exp[i(ϕ^1+ω0)]     =b1exp[i(ϕ+ω0)]+b3exp[i3(ϕ+ω0)]+b5exp[i5(ϕ+ω0)]+exp[i(ϕ^1+2ω0)]=b1exp[i(ϕ+2ω0)]+b3exp[i3(ϕ+2ω0)]+b5exp[i5(ϕ+2ω0)]+
(12)

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

s(t)=exp[iϕ^1]δ(t)+exp[i(ϕ^1+ω0)]δ(t1)+exp[i(ϕ^1+2ω0)]δ(t2).
(13)

The signal in Eq. (13) still contains the remaining complex harmonics {-3rd, 5th, −7th, 9th,…}, but we can use temporal quadrature filters (PSAs) to reject them [1

1. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

,10

10. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef] [PubMed]

,11

11. A. Gonzalez, M. Servin, J. C. Estrada, and J. A. Quiroga, “Design of phase-shifting algorithms by fine-tuning spectral shaping,” Opt. Express 19(11), 10692–10697 (2011). [CrossRef] [PubMed]

]. Specifically, we propose the following 2-steps PSA to reject the −3rd and 5th complex harmonics:

         H2(ω)=1exp[i(ω+3ω0)],h2(t)=F1[H2(ω)]=δ(t)exp(i3ω0)δ(t+1);ω0=π/4.
(14)

Similarly, the following 3-steps PSA rejects the −3rd, 5th, −7th, and 9th complex harmonics:
H3(ω)={1exp[i(ω+3ω0)]}{1exp[i(ω5ω0)]},h3(t)=F1[H3(ω)]=δ(t)+[1exp(iω0)]δ(t+1)+exp(i2ω0)δ(t+2);ω0=π/3,
(15)
where F1[] is the inverse Fourier transform operator. The normalized spectral plots of these two quadrature filters (Eqs. (14,15)) are shown in Fig. 4
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)).
.

Using the 2-step PSA (Eqs. (13,14)) the estimated phase ϕ^2(x,y) is given by:

H2(ω0)exp(iϕ^2)={s(t)h2(t)}t=2=exp(iϕ^1)ei3ω0exp[i(ϕ^1+ω0)];ω0=π/4.
(16)

Similarly, the 3-step PSA (Eqs. (13,15) with ω0=π/3) give the estimation ϕ^3(x,y):

H3(ω0)exp(iϕ^3)={s(t)h3(t)}t=3=exp(iϕ^1)+(1eiω0)exp[i(ϕ^1+ω0)]+ei2ω0exp[i(ϕ^1+2ω0)].
(17)

Finally, the most general expression for our spatio-temporal phase-demodulation algorithms is given by
Aexp[iϕ^N(x,y)]={exp[ipm(x,y)][I(x,y,t)*hN(t)]t=N}*h(x,y),
(18)
where A is a proportionality constant. The convolutions order in Eq. (18) may be interchanged, but for numerical efficiency we suggest to first apply the temporal convolution.

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

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

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 {-3rd, +5th, −7th, +9th, −11th,...} overlapping with the fundamental signal exp[iϕ(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 −3rd, and +5th complex harmonics, and a 3-step one to remove (at least) the −3rd, +5th, −7th, and +9th 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.

Acknowledgements

The authors wish to thank the useful reviewers’ comments and the financial support of the Mexican National Council for Science and Technology (CONACyT).

References and Links

1.

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

2.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

3.

O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984). [CrossRef] [PubMed]

4.

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1992). [CrossRef]

5.

B. K. A. Ngoi, K. Venkatakrishnan, and N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. 40(19), 3211–3214 (2001). [CrossRef] [PubMed]

6.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004). [CrossRef]

7.

M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express 18(17), 18492–18497 (2010). [CrossRef] [PubMed]

8.

B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 1–12 (2010).

9.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982). [CrossRef]

10.

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef] [PubMed]

11.

A. Gonzalez, M. Servin, J. C. Estrada, and J. A. Quiroga, “Design of phase-shifting algorithms by fine-tuning spectral shaping,” Opt. Express 19(11), 10692–10697 (2011). [CrossRef] [PubMed]

OCIS Codes
(120.2650) Instrumentation, measurement, and metrology : Fringe analysis
(120.3180) Instrumentation, measurement, and metrology : Interferometry

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: July 13, 2011
Revised Manuscript: August 12, 2011
Manuscript Accepted: August 15, 2011
Published: September 22, 2011

Citation
J. M. Padilla, M. Servin, and J. C. Estrada, "Harmonics rejection in pixelated interferograms using spatio-temporal demodulation," Opt. Express 19, 19508-19513 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19508


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References

  1. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express17(24), 21867–21881 (2009). [CrossRef] [PubMed]
  2. R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng.23, 361–364 (1984).
  3. O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett.9(2), 59–61 (1984). [CrossRef] [PubMed]
  4. C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE1531, 119–127 (1992). [CrossRef]
  5. B. K. A. Ngoi, K. Venkatakrishnan, and N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt.40(19), 3211–3214 (2001). [CrossRef] [PubMed]
  6. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE5531, 304–314 (2004). [CrossRef]
  7. M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express18(17), 18492–18497 (2010). [CrossRef] [PubMed]
  8. B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE7790, 1–12 (2010).
  9. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A72(1), 156–160 (1982). [CrossRef]
  10. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt.35(1), 51–60 (1996). [CrossRef] [PubMed]
  11. A. Gonzalez, M. Servin, J. C. Estrada, and J. A. Quiroga, “Design of phase-shifting algorithms by fine-tuning spectral shaping,” Opt. Express19(11), 10692–10697 (2011). [CrossRef] [PubMed]

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