Abstract

We propose a novel synchronous phase-demodulation of pixelated interferograms using squared 3x3 phase-shifted unit-cells. This 3x3 unit-cell is tiled over the CCD image sensor to create a two-dimensional (2D) pixelated carrier. Our synchronous phase-demodulation uses this 2D carrier to demodulate the pixelated interferogram as in the standard 2x2 unit-cell case. The main motivation behind the use of a 3x3 pixelated carrier (instead of the usual 2x2) is its higher harmonic robustness, allowing one to demodulate intensity-distorted fringe patterns. The harmonic rejection robustness of our spatial 3x3 configuration equals the robustness of the temporal least-squares 9-step phase-shifting algorithm (PSA). In other words, extending from the usual 2x2 phase-shifting unit-cell to 3x3 unit-cells, one extends the harmonic rejection of the demodulation algorithm. Finally we also prove that our proposed 9-step, 3x3 pixelated carrier uses the 2D available spectral space more efficiently than using these 9-steps in a linear spatial-carrier configuration.

© 2012 OSA

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References

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  1. H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, NJ, 2007), Chap. 14, doi: 10.1002/9780470135976.ch14
    [CrossRef]
  2. 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]
  3. Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
    [CrossRef]
  4. 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]
  5. B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. 45(19), 4554–4562 (2006).
    [CrossRef] [PubMed]
  6. G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. 33(23), 2788–2790 (2008).
    [CrossRef] [PubMed]
  7. B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 77900K, 77900K-12 (2010).
    [CrossRef]
  8. M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express 18(17), 18492–18497 (2010).
    [CrossRef] [PubMed]
  9. J. M. Padilla, M. Servin, and J. C. Estrada, “Harmonics rejection in pixelated interferograms using spatio-temporal demodulation,” Opt. Express 19(20), 19508–19513 (2011).
    [CrossRef] [PubMed]
  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. 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]
  12. J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express 19(6), 5126–5133 (2011).
    [CrossRef] [PubMed]

2011 (2)

2010 (2)

B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 77900K, 77900K-12 (2010).
[CrossRef]

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

2009 (1)

2008 (1)

2006 (1)

2004 (2)

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[CrossRef]

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]

1996 (1)

1982 (1)

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]

Awatsuji, Y.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[CrossRef]

Brock, N.

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]

Estrada, J. C.

Hayes, J.

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]

Ina, H.

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]

Kimbrough, B.

B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 77900K, 77900K-12 (2010).
[CrossRef]

Kimbrough, B. T.

Kobayashi, S.

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]

Kubota, T.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[CrossRef]

Meneses-Fabian, C.

Millerd, J.

B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 77900K, 77900K-12 (2010).
[CrossRef]

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]

North-Morris, M.

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]

Novak, M.

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]

Padilla, J. M.

Quiroga, J. A.

Rodriguez-Zurita, G.

Sasada, M.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[CrossRef]

Servin, M.

Surrel, Y.

Takeda, M.

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]

Toto-Arellano, N. I.

Vázquez-Castillo, J. F.

Wyant, J. C.

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]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[CrossRef]

J. Opt. Soc. Am. A (1)

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]

Opt. Express (4)

Opt. Lett. (1)

Proc. SPIE (2)

B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 77900K, 77900K-12 (2010).
[CrossRef]

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]

Other (1)

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, NJ, 2007), Chap. 14, doi: 10.1002/9780470135976.ch14
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Periodic distribution for our 9-steps (3x3 unit-cell) pixelated carrier phase-shifts with spiral configuration.

Fig. 2
Fig. 2

The panels on this figure show our 3x3 synchronous demodulation herein proposed applied to harmonic-free interferograms. Detailed panel description is presented in the text.

Fig. 3
Fig. 3

Magnitude of the FTF for the temporal 9-step least-squares PSA.

Fig. 4
Fig. 4

The panels on this figure show the synchronous demodulating process herein proposed applied to intensity-distorted interferograms. Detailed panel description is presented in the text.

Fig. 5
Fig. 5

Spectral distribution of distorted, non-sinusoidal interferograms using: the 2x2, the 3x3, and the linear 9-step pixelated modulating spatial carriers.

Fig. 6
Fig. 6

Panel (a) shows the central slice of the non-sinusoidal fringe pattern. The central slices of the estimated phase using: (b) 2x2 pixelated carrier, (c) 9-step linear carrier, and (d) 3x3 pixelated carrier interferograms.

Equations (11)

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I(x,y)=a(x,y)+b(x,y)cos[φ(x,y)+pm(x,y)],
exp[ipm(x,y)]=[ n m δ(x3m,y3n) ]exp{ i2π 9 [ 1 2 3 8 9 4 7 6 5 ] },
| pm(x,y) |>max| φ(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)] }.
(1/2)b(x,y)exp[i φ ^ (x,y)]=LP[ exp(ipm)I(x,y) ].
tan[ φ ^ (x,y)]= Im{ LP[ exp(ipm)I(x,y) ] } Re{ LP[ exp(ipm)I(x,y) ] } ,
max| φ(x,y) |<2π/6.
I(x,y)=a(x,y)+ n=1 b n (x,y)cos{ n[φ(x,y)+pm(x,y)] } ,
exp(ipm)I(x,y)=aexp(ipm)+ n=1 ( b n /2){ exp[i(n1)pm]exp[i(nφ)] +exp[i(n+1)pm]exp[i(nφ)] }.
exp[+i(n1)pm(x,y)]=1,(x,y)ifn=1,10,19,28,... exp[i(n+1)pm(x,y)]=1,(x,y)ifn=8,17,26,35,...
Aexp[i φ ^ (x,y)]={ b 1 exp(iφ)+ b 8 exp(i8φ)+ b 10 exp(i10φ)+ b 17 exp(i17φ)+...},

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