Abstract

A new method of phase determination from a single undersampled interferogram is described. Two low-fringe-density synthetic interferograms corresponding to the phase differences along orthogonal directions are obtained from neighboring pixels of the aliased measured data. The only assumption is that the illumination background, the modulation intensity, and the searched phase are smooth and continuous functions. The synthetic interferograms are demodulated by use of either standard frequency or spatial-domain procedures to obtain the phase differences. The phase is then recovered by integration of the phase differences with a least-squares method. The proposed method is demonstrated to be noise tolerant.

© 2003 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  32. M. Pirga, M. Kujawinska, “Two-directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
    [CrossRef]
  33. G. Páez, M. Strojnik, “Convergent, recursive phase reconstruction from noisy, modulated intensity patterns by use of synthetic interferograms,” Opt. Lett. 23, 406–408 (1998).
    [CrossRef]
  34. M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
    [CrossRef] [PubMed]

2001 (1)

G. Garcia-Torales, G. Páez, M. Strojnik, “Simulations and experimental results with a vectorial shearing interferometer,” Opt. Eng. 40, 767–773 (2001).
[CrossRef]

2000 (1)

1999 (3)

1998 (1)

1997 (2)

1996 (5)

D. C. Ghiglia, L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999–2013 (1996).
[CrossRef]

J. E. Greivenkamp, A. E. Lowman, R. J. Palum, “Sub-Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35, 2962–2969 (1996).
[CrossRef]

M. Servin, D. Malacara, F. J. Cuevas, “Path-independent phase unwrapping of subsampled phase maps,” Appl. Opt. 35, 1643–1649 (1996).
[CrossRef] [PubMed]

M. Servin, D. Malacara, “Sub-Nyquist interferometry using a computer-stored reference,” J. Mod. Opt. 43, 1723–1729 (1996).
[CrossRef]

M. Servin, D. Malacara, J. L. Marroquin, F. J. Cuevas, “New technique for ray aberration detection in Hartmanngrams based on regularized bandpass filters,” Opt. Eng. 35, 1677–1683 (1996).
[CrossRef]

1995 (2)

M. Pirga, M. Kujawinska, “Two-directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
[CrossRef]

1994 (3)

1988 (1)

1987 (1)

1986 (1)

1984 (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

1982 (2)

1979 (1)

1978 (1)

1977 (1)

1974 (1)

1972 (1)

1970 (1)

Bell, B. W.

Brangaccio, D. J.

Bruning, J. H.

Cuevas, F. J.

Fried, D. L.

Gallagher, J. E.

Garcia-Torales, G.

G. Garcia-Torales, G. Páez, M. Strojnik, “Simulations and experimental results with a vectorial shearing interferometer,” Opt. Eng. 40, 767–773 (2001).
[CrossRef]

G. Páez, M. Strojnik, G. Garcia-Torales, “Vectorial shearing interferometer,” Appl. Opt. 39, 5172–5178 (2000).
[CrossRef]

Ghiglia, D. C.

Greivenkamp, J. E.

J. E. Greivenkamp, A. E. Lowman, R. J. Palum, “Sub-Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35, 2962–2969 (1996).
[CrossRef]

J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5257 (1987).
[CrossRef] [PubMed]

Herriott, D. R.

Hudgin, R. H.

Hunt, B. R.

Ina, H.

Itoh, K.

Katsuyuki, O.

Kobayashi, S.

Koliopoulos, C. L.

Kreis, T.

Kujawinska, M.

M. Pirga, M. Kujawinska, “Two-directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

M. Kujawinska, J. Schmidt, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. SPIE1508, 61–67 (1991).
[CrossRef]

Lowman, A. E.

J. E. Greivenkamp, A. E. Lowman, R. J. Palum, “Sub-Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35, 2962–2969 (1996).
[CrossRef]

Lyuboshenko, I.

Maitre, H.

Malacara, D.

Malacara, Z.

Marroquin, J. L.

Muñoz, J.

J. Muñoz, M. Strojnik, G. Páez, “Detection and interpretation of high-frequency spatial interferograms,” in Infrared Spaceborne Remote Sensing IX, M. Strojnik, B. F. Andersen, eds., Proc. SPIE4486, 523–532 (2002).
[CrossRef]

J. Muñoz, G. Páez, M. Strojnik, “Phase unwrapping of subsampled phase-shifted interferograms,” in Interferometry XI: Techniques and Analysis, K. Creath, J. Schmit, eds., Proc. SPIE4777, 288–299 (2002).
[CrossRef]

Murty, M. V. R. K.

Noll, R. J.

Páez, G.

G. Garcia-Torales, G. Páez, M. Strojnik, “Simulations and experimental results with a vectorial shearing interferometer,” Opt. Eng. 40, 767–773 (2001).
[CrossRef]

G. Páez, M. Strojnik, G. Garcia-Torales, “Vectorial shearing interferometer,” Appl. Opt. 39, 5172–5178 (2000).
[CrossRef]

G. Páez, M. Strojnik, “Convergent, recursive phase reconstruction from noisy, modulated intensity patterns by use of synthetic interferograms,” Opt. Lett. 23, 406–408 (1998).
[CrossRef]

J. Muñoz, G. Páez, M. Strojnik, “Phase unwrapping of subsampled phase-shifted interferograms,” in Interferometry XI: Techniques and Analysis, K. Creath, J. Schmit, eds., Proc. SPIE4777, 288–299 (2002).
[CrossRef]

J. Muñoz, M. Strojnik, G. Páez, “Detection and interpretation of high-frequency spatial interferograms,” in Infrared Spaceborne Remote Sensing IX, M. Strojnik, B. F. Andersen, eds., Proc. SPIE4486, 523–532 (2002).
[CrossRef]

Palum, R. J.

J. E. Greivenkamp, A. E. Lowman, R. J. Palum, “Sub-Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35, 2962–2969 (1996).
[CrossRef]

Pirga, M.

M. Pirga, M. Kujawinska, “Two-directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

Rivera, M.

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 195.

Rodríguez, R.

Romero, L. A.

Rosenfeld, D. P.

Schmidt, J.

M. Kujawinska, J. Schmidt, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. SPIE1508, 61–67 (1991).
[CrossRef]

Schwider, J.

Servin, M.

Strojnik, M.

G. Garcia-Torales, G. Páez, M. Strojnik, “Simulations and experimental results with a vectorial shearing interferometer,” Opt. Eng. 40, 767–773 (2001).
[CrossRef]

G. Páez, M. Strojnik, G. Garcia-Torales, “Vectorial shearing interferometer,” Appl. Opt. 39, 5172–5178 (2000).
[CrossRef]

G. Páez, M. Strojnik, “Convergent, recursive phase reconstruction from noisy, modulated intensity patterns by use of synthetic interferograms,” Opt. Lett. 23, 406–408 (1998).
[CrossRef]

J. Muñoz, G. Páez, M. Strojnik, “Phase unwrapping of subsampled phase-shifted interferograms,” in Interferometry XI: Techniques and Analysis, K. Creath, J. Schmit, eds., Proc. SPIE4777, 288–299 (2002).
[CrossRef]

J. Muñoz, M. Strojnik, G. Páez, “Detection and interpretation of high-frequency spatial interferograms,” in Infrared Spaceborne Remote Sensing IX, M. Strojnik, B. F. Andersen, eds., Proc. SPIE4486, 523–532 (2002).
[CrossRef]

Takeda, M.

Toyohiko, Y.

Vlad, V. I.

White, A. D.

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Appl. Opt. (12)

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, R. Rodríguez, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
[CrossRef]

M. V. R. K. Murty, “A compact lateral shearing interferometer based on the Michelson interferometer,” Appl. Opt. 9, 1146–1152 (1970).
[CrossRef] [PubMed]

G. Páez, M. Strojnik, G. Garcia-Torales, “Vectorial shearing interferometer,” Appl. Opt. 39, 5172–5178 (2000).
[CrossRef]

K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
[CrossRef] [PubMed]

D. Malacara, “Hartmann test of aspherical mirrors,” Appl. Opt. 11, 99–101 (1972).
[PubMed]

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, “Direct ray aberration estimation in Hartmanngrams by use of a regularized phase-tracking system,” Appl. Opt. 38, 2862–2869 (1999).
[CrossRef]

O. Katsuyuki, Y. Toyohiko, “Phase measuring Ronchi test,” Appl. Opt. 27, 523–528 (1988).
[CrossRef]

J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5257 (1987).
[CrossRef] [PubMed]

M. Servin, D. Malacara, F. J. Cuevas, “Path-independent phase unwrapping of subsampled phase maps,” Appl. Opt. 35, 1643–1649 (1996).
[CrossRef] [PubMed]

M. Servin, D. Malacara, Z. Malacara, V. I. Vlad, “Sub-Nyquist null aspheric testing using a computer-stored compensator,” Appl. Opt. 33, 4103–4108 (1994).
[CrossRef] [PubMed]

M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

M. Servin, D. Malacara, “Sub-Nyquist interferometry using a computer-stored reference,” J. Mod. Opt. 43, 1723–1729 (1996).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (4)

Opt. Eng. (5)

M. Servin, D. Malacara, J. L. Marroquin, F. J. Cuevas, “New technique for ray aberration detection in Hartmanngrams based on regularized bandpass filters,” Opt. Eng. 35, 1677–1683 (1996).
[CrossRef]

G. Garcia-Torales, G. Páez, M. Strojnik, “Simulations and experimental results with a vectorial shearing interferometer,” Opt. Eng. 40, 767–773 (2001).
[CrossRef]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

M. Pirga, M. Kujawinska, “Two-directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

J. E. Greivenkamp, A. E. Lowman, R. J. Palum, “Sub-Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35, 2962–2969 (1996).
[CrossRef]

Opt. Lett. (3)

Other (4)

J. Muñoz, M. Strojnik, G. Páez, “Detection and interpretation of high-frequency spatial interferograms,” in Infrared Spaceborne Remote Sensing IX, M. Strojnik, B. F. Andersen, eds., Proc. SPIE4486, 523–532 (2002).
[CrossRef]

M. Kujawinska, J. Schmidt, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. SPIE1508, 61–67 (1991).
[CrossRef]

J. Muñoz, G. Páez, M. Strojnik, “Phase unwrapping of subsampled phase-shifted interferograms,” in Interferometry XI: Techniques and Analysis, K. Creath, J. Schmit, eds., Proc. SPIE4777, 288–299 (2002).
[CrossRef]

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 195.

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Figures (15)

Fig. 1
Fig. 1

High-fringe-density simulated interferograms. (a) With defocus aberration according to Eq. (6) and (b) without defocus. The low-contrast moiré patterns at the edges indicate undersampling and pixel intensity averaging.

Fig. 2
Fig. 2

Synthetic interferograms, (a) and (c), calculated from the approximated formulas for the x and y directions, respectively. Ideal results calculated by use of the cosines of the exact phase differences: (b) with respect to the x direction and (d) with respect to the y direction.

Fig. 3
Fig. 3

Wrapped phase differences, (a) and (c), obtained from the synthetic interferograms in the x and y directions. Ideal results, shown for comparison purposes: (b) with respect to the x direction and (d) with respect to the y direction.

Fig. 4
Fig. 4

Phase reconstructed with the developed algorithm. The maximum value is 698.462 rad, and the minimum is -7.562 rad.

Fig. 5
Fig. 5

Phase error ∊ = ϕ i,j - ϕi,jr. The peak-valley error is 1.365% of the phase, and its mean-square value is 1.628 rad rms.

Fig. 6
Fig. 6

Noisy interferogram corresponding to the cosine of the phase. The noise added to the phase ranges from λ/8 to -λ/8 rad. Some regions of the interferogram appear as a speckle pattern owing to the high fringe density and the noise.

Fig. 7
Fig. 7

Synthetic interferogram with respect to the x direction calculated from the noisy fringe pattern shown in Fig. 6. It can be observed that the noise level increases; however, the phase-difference information is still present.

Fig. 8
Fig. 8

Synthetic interferogram with respect to the y direction calculated from the noisy fringe pattern shown in Fig. 6. It can be observed that the noise level increases; however, the phase-difference information can be still recognized.

Fig. 9
Fig. 9

Recovered phase from a noisy undersampled interferogram. It looks identical to that shown in Fig. 4 reconstructed from an interferogram with no noise.

Fig. 10
Fig. 10

Phase-reconstruction error by use of a noisy undersampled interferogram. The peak-valley error is 2.711% of the phase, and the mean-square error is 2.416 rad rms. These values show a close estimation of the recovered wave front.

Fig. 11
Fig. 11

Synthetic interferograms that correspond to the undersampled interferogram shown in Fig. 1(b): with respect to (a) the x direction and (b) the y direction. The absence of the carrier function makes the Fourier method developed by Takeda inapplicable.

Fig. 12
Fig. 12

(a) Experimentally obtained interferogram from a commercial Fizeau interferometer. (b) Undersampled interferogram resulting from our taking every fourth pixel from the original fringe pattern; additionally, these data were multiplied by a circular pupil function. Note the low-contrast moiré patterns at the undersampled interferogram edges and at the regions surrounding the low-density central fringes.

Fig. 13
Fig. 13

Synthetic interferograms, (a) and (b), obtained from the undersampled data corresponding to the cosines of the phase differences with respect to the x and y directions. Recovered phase differences, (c) and (d), with respect to the x and y coordinates.

Fig. 14
Fig. 14

Recovered phase obtained by integration of the recovered phase differences shown in Figs. 13(c) and 13(d). The maximum and minimum values are 63.946 and -27.722 rad.

Fig. 15
Fig. 15

(a) Undersampled interferogram. (b) Interferogram resulting by our taking the cosine of the recovered phase. The recovered interferogram shows good agreement with the sampled data.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

Ix, y=Ibx, y+Imx, ycosϕx, y.
Δϕxx, yϕx+1, y-ϕx, yϕx, y-ϕx-1, y, Δϕyx, yϕx, y+1-ϕx, yϕx, y-ϕx, y-1, Δϕxx, y+Δϕyx, yϕx+1, y+1-ϕx, yϕx, y-ϕx-1, y-1, Δϕxx, y-Δϕyx, yϕx+1, y-1-ϕx, yϕx, y-ϕx-1, y+1.
Ii,j+1Ibi,j+Imi,jcosϕi,j+Δϕi,jxIbi,j+Imi,jcosϕi,jcosΔϕi,jx-sinϕi,jsinΔϕi,jx, Ii,j-1Ibi,j+Imi,jcosϕi,j-Δϕi,jxIbi,j+Imi,jcosϕi,jcosΔϕi,jx+sinϕi,jsinΔϕi,jx.
A1i,j=Ii-2,j-2-Ii+2,j+2+Ii+1,j-2-Ii-2,j+24Imi,j sinϕi,jsin2Δϕi,jxcos2Δϕi,jy, A2i,j=Ii-2,j-1-Ii+2,j+1+Ii+2,j-1-Ii-2,j+14Imi,j sinϕi,jsinΔϕi,jxcos2Δϕi,jy, A3i,j=Ii-1,j-2-Ii+2,j+1+Ii+1,j-2-Ii-1,j+24Imi,j sinϕi,jsin2Δϕi,jxcosΔϕi,jy, A4i,j=Ii-1,j-1-Ii+1,j+1+Ii+1,j-1-Ii-1,j+14Imi,j sinϕi,jsinΔϕi,jxcosΔϕi,jy, A5i,j=Ii,j-2-Ii,j+22Imi,j sinϕi,jsin2Δϕi,jx, A6i,j=Ii,j-1-Ii,j+12Imi,j sinϕi,jsinΔϕi,jx, B1i,j=Ii-2,j-2-Ii+2,j+2-Ii+2,j-2+Ii-2,j+24Imi,j sinϕi,jcos2Δϕi,jxsin2Δϕi,jy, B2i,j=Ii-2,j-1-Ii+2,j+1-Ii+2,j-1+Ii-2,j+14Imi,j sinϕi,jcosΔϕi,jxsin2Δϕi,jy, B3i,j=Ii-1,j-2-Ii+2,j+1-Ii+1,j-2+Ii-1,j+24Imi,j sinϕi,jcos2Δϕi,jxsinΔϕi,jy, B4i,j=Ii-1,j-1-Ii+1,j+1-Ii+1,j-1+Ii-1,j+14Imi,j sinϕi,jcosΔϕi,jxsinΔϕi,jy, B5i,j=Ii-2,j-Ii+2,j2Imi,j sinϕi,jsin2Δϕi,jy, B6i,j=Ii-1,j-Ii+1,j2Imi,j sinϕi,jsinΔϕi,jy.
cosΔϕi,jxA1i,j2A2i,jA3i,j2A4i,jA5i,j2A6i,jB4i,j2B6i,jB2i,j2B5i,j, cosΔϕi,jyB1i,j2B2i,jB3i,j2B4i,jB5i,j2B6i,jA4i,j2A6i,jA2i,j2A5i,j.
Ii,j=Ibi,j+Imi,j cosϕi,j,
ϕi,j=2π16xi2+yj22+20xi2+yj2-14xiyj+4yj-2xi+1.
φi,j=φi,jr-arctansinφi,jr-φi,jwcosφi,jr-φi,jw.

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