Abstract

The paper proposes a super-resolution Fourier transform method for phase estimation in phase shifting interferometry. Incorporation of a super-resolution technique before the application of Fourier transform significantly enhances the resolution capability of the proposed method. The other salient features of the method lie in its ability to handle multiple harmonics, PZT miscalibration, and arbitrary phase steps in the optical configuration. The method does not need addition of any carrier fringes to separate the spectral contents in the intensity fringes. The proposed concept thus overcomes the limitations of other methods based on Fourier transform techniques. The robustness of the proposed method is studied in the presence of noise.

© 2005 Optical Society of America

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References

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  1. T. Kreis, “Holographic interferometry Principles and Methods,” Akademie Verlag, 1996, pp. 101–170.
  2. J. E. Greivenkamp and J. H. Bruning, Phase shifting interferometry Optical Shop Testing ed D. Malacara (New York: Wiley) 501–598 (1992).
  3. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [Crossref] [PubMed]
  4. Y. Zhu and T. Gemma, “Method for designing error-compensating phase-calculation algorithms for phase-shifting interferometry,” Appl. Opt. 40, 4540–4546 (2001).
    [Crossref]
  5. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [Crossref] [PubMed]
  6. J. Schwider, O. Falkenstorfer, H. Schreiber, and A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [Crossref]
  7. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [Crossref] [PubMed]
  8. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [Crossref] [PubMed]
  9. Y. -Y. Cheng and J. C. Wyant, “Phase-shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [Crossref] [PubMed]
  10. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [Crossref]
  11. K. G. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Exp. 9, 236–253 (2001).
    [Crossref]
  12. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [Crossref]
  13. C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [Crossref] [PubMed]
  14. J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
  15. G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8, 822–827 (1991).
    [Crossref]
  16. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [Crossref]
  17. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40, 2886–2894 (2001).
    [Crossref]
  18. C. S. Guo, Z. Y. Rong, J. L. He, H. T. Wang, L. Z. Cai, and Y. R. Wang, “Determination of global phase shifts between interferograms by use of an energy-minimum algorithm,” Appl. Opt. 42, 6514–6519 (2003).
    [Crossref] [PubMed]
  19. L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24, 905–907 (1999).
    [Crossref]
  20. P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, New Jersey, 1997).
  21. T. Söderström and P. Stoica, “Accuracy of high-order Yule-Walker methods for frequency estimation of complex sine waves,” IEEE Proceedings-F 140, 71–80 (1993).
  22. J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing 36, 1846–1853 (1988).
    [Crossref]
  23. P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt. 32, 3669–3675 (1993).
    [Crossref] [PubMed]
  24. R. Roy and T. Kailath, “ESPRIT-Estimation of signal parameters via rotational invariance techniques,” IEEE Transactions on Acoustics, Speech, and Signal Processing 37, 984–995 (1989).
    [Crossref]

2003 (1)

2001 (3)

1999 (1)

1996 (1)

1995 (1)

1993 (4)

T. Söderström and P. Stoica, “Accuracy of high-order Yule-Walker methods for frequency estimation of complex sine waves,” IEEE Proceedings-F 140, 71–80 (1993).

J. Schwider, O. Falkenstorfer, H. Schreiber, and A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[Crossref] [PubMed]

P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt. 32, 3669–3675 (1993).
[Crossref] [PubMed]

1992 (1)

1991 (1)

1989 (1)

R. Roy and T. Kailath, “ESPRIT-Estimation of signal parameters via rotational invariance techniques,” IEEE Transactions on Acoustics, Speech, and Signal Processing 37, 984–995 (1989).
[Crossref]

1988 (1)

J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing 36, 1846–1853 (1988).
[Crossref]

1987 (1)

1985 (1)

1984 (1)

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

1983 (1)

1982 (2)

Barnes, T. H.

Bokor, J.

Bruning, J. H.

J. E. Greivenkamp and J. H. Bruning, Phase shifting interferometry Optical Shop Testing ed D. Malacara (New York: Wiley) 501–598 (1992).

Burow, R.

Cai, L. Z.

Cheng, Y. -Y.

Eiju, T.

Elssner, K. E.

Falkenstorfer, O.

J. Schwider, O. Falkenstorfer, H. Schreiber, and A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Farrant, D. I.

Fuchs, J. J.

J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing 36, 1846–1853 (1988).
[Crossref]

Gemma, T.

Goldberg, K. A.

Greivenkamp, J. E.

J. E. Greivenkamp and J. H. Bruning, Phase shifting interferometry Optical Shop Testing ed D. Malacara (New York: Wiley) 501–598 (1992).

Grievenkamp, J. E.

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Grzanna, J.

Guo, C. S.

Hariharan, P.

He, J. L.

Hibino, K.

Ina, H.

Kailath, T.

R. Roy and T. Kailath, “ESPRIT-Estimation of signal parameters via rotational invariance techniques,” IEEE Transactions on Acoustics, Speech, and Signal Processing 37, 984–995 (1989).
[Crossref]

Kobayashi, S.

Kreis, T.

T. Kreis, “Holographic interferometry Principles and Methods,” Akademie Verlag, 1996, pp. 101–170.

Lai, G.

Larkin, K. G.

Merkel, K.

Morgan, C. J.

Moses, R.

P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, New Jersey, 1997).

Oreb, B. F.

Rastogi, P. K.

Rong, Z. Y.

Roy, R.

R. Roy and T. Kailath, “ESPRIT-Estimation of signal parameters via rotational invariance techniques,” IEEE Transactions on Acoustics, Speech, and Signal Processing 37, 984–995 (1989).
[Crossref]

Schreiber, H.

J. Schwider, O. Falkenstorfer, H. Schreiber, and A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Schwider, J.

J. Schwider, O. Falkenstorfer, H. Schreiber, and A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[Crossref] [PubMed]

Söderström, T.

T. Söderström and P. Stoica, “Accuracy of high-order Yule-Walker methods for frequency estimation of complex sine waves,” IEEE Proceedings-F 140, 71–80 (1993).

Spolaczyk, R.

Stoica, P.

T. Söderström and P. Stoica, “Accuracy of high-order Yule-Walker methods for frequency estimation of complex sine waves,” IEEE Proceedings-F 140, 71–80 (1993).

P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, New Jersey, 1997).

Surrel, Y.

Takeda, M.

Tan, S. M.

Wang, H. T.

Wang, Y. R.

Watkins, L. R.

Wyant, J. C.

Yatagai, T.

Zhu, Y.

Zoller, A.

J. Schwider, O. Falkenstorfer, H. Schreiber, and A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Appl. Opt. (9)

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[Crossref] [PubMed]

Y. Zhu and T. Gemma, “Method for designing error-compensating phase-calculation algorithms for phase-shifting interferometry,” Appl. Opt. 40, 4540–4546 (2001).
[Crossref]

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[Crossref] [PubMed]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[Crossref] [PubMed]

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

Y. -Y. Cheng and J. C. Wyant, “Phase-shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[Crossref] [PubMed]

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40, 2886–2894 (2001).
[Crossref]

C. S. Guo, Z. Y. Rong, J. L. He, H. T. Wang, L. Z. Cai, and Y. R. Wang, “Determination of global phase shifts between interferograms by use of an energy-minimum algorithm,” Appl. Opt. 42, 6514–6519 (2003).
[Crossref] [PubMed]

P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt. 32, 3669–3675 (1993).
[Crossref] [PubMed]

IEEE Proceedings-F (1)

T. Söderström and P. Stoica, “Accuracy of high-order Yule-Walker methods for frequency estimation of complex sine waves,” IEEE Proceedings-F 140, 71–80 (1993).

IEEE Transactions on Acoustics, Speech, and Signal Processing (2)

J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing 36, 1846–1853 (1988).
[Crossref]

R. Roy and T. Kailath, “ESPRIT-Estimation of signal parameters via rotational invariance techniques,” IEEE Transactions on Acoustics, Speech, and Signal Processing 37, 984–995 (1989).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Eng. (2)

J. Schwider, O. Falkenstorfer, H. Schreiber, and A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Opt. Exp. (1)

K. G. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Exp. 9, 236–253 (2001).
[Crossref]

Opt. Lett. (2)

Other (3)

P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, New Jersey, 1997).

T. Kreis, “Holographic interferometry Principles and Methods,” Akademie Verlag, 1996, pp. 101–170.

J. E. Greivenkamp and J. H. Bruning, Phase shifting interferometry Optical Shop Testing ed D. Malacara (New York: Wiley) 501–598 (1992).

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Figures (6)

Fig. 1.
Fig. 1.

Plots of power spectrum obtained for κ = 1 using a) Fourier transform b) Zero padded Fourier transform and c) super-resolution Fourier transform for determining the phase step values α.

Fig. 2.
Fig. 2.

Plots of power spectrum obtained for κ = 2 using a) Fourier transform b) Zero padded Fourier transform and c) super-resolution Fourier transform for determining the phase step values α.

Fig. 3.
Fig. 3.

Plots of phase step α (in degrees) versus SNR obtained using Eq. (15) at an arbitrary pixel location on a data frame for (a) - (d) κ = 1; N = 6,8,10, and 12, respectively, and (e) - (h) κ= 2; N = 10,12,14, and 16, respectively.

Fig. 4.
Fig. 4.

Plots of power spectrum for N = 12 data frames and κ= 1 for various orders of nonlinearity in the PZT response to the applied voltage.

Fig. 5
Fig. 5

(a) Fringe pattern and (b) retrieved phase φ.

Fig. 6.
Fig. 6.

Typical phase error in computation of phase distributions φ (in radians), for the phase steps obtained from Fig 3(d) for SNR=30dB.

Tables (1)

Tables Icon

Table 1. Phase step estimation for various values of NOF and N = 8 and 12. During simulation the phase step α is taken as 33°, κ=1, and SNR = 40 dB.

Equations (35)

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I ( t ) = I dc + k = 1 κ a k exp ( jkφ ) k u k t + k = 1 κ a k exp ( jkφ ) k * ( u k * ) t
+ η ( t ) ; for t = 0,1 , , N 1
r ( p ) = E [ I ( t ) I * ( t p ) ] = k = 0 2 κ A k 2 exp ( j ω k p ) + σ 2 δ p , 0 .
R I = E [ I ( t ) I * ( t ) ] = [ r ( 0 ) r * ( 1 ) . r * ( m 1 ) r ( 1 ) r ( 0 ) . r * ( m 2 ) . . . . r ( m 1 ) . . r ( 0 ) ]
R I = SPS H R s + σ 2 I d R ε
P = [ A 0 2 0 . 0 0 A 1 2 . . . . . . 0 . . A 2 κ 2 ]
R I = [ υ ν ] [ λ 0 0 0 λ m 1 ] [ υ ν ] H
R I υ = SPS H υ + σ 2 υ
= υ [ λ 0 . . . 0 . λ 1 . . . . . . . . . . . . . 0 . . . λ 2 κ ]
= υγ + σ 2 υ
γ = [ λ 0 σ 2 . . . 0 . λ 1 σ 2 . . . . . . . . . . . . . 0 . . . λ 2 κ σ 2 ]
υ = S ( PS H υ γ 1 ) SC
S H ν = [ s 0 H . . s 2 κ H ] ν = 0
= s k H ν i = 0 i = ( 0,1,2 , …. , m 2 κ 2 ) and k = ( 0,1,2 , …. , 2 κ )
s k , ν i = n = 0 m 1 s k H ( n ) ν i ( n )
s k , ν i = 0 i = ( 0,1,2 , , m 2 κ 2 ) , k = ( 0,1 , ….. , 2 κ )
s k , ν i = n = 0 m 1 exp ( jn ω k ) ν i ( n )
= s , ν i | ω = ω k
= 0
P i [ exp ( j ω f ) ] = 1 s f , ν i 2 ; ω f = 0,2 π / NOF , ……….. , 2 π ( NOF 1 ) / NOF
P [ exp ( j ω f ) ] = 1 i = 0 m 2 κ 2 1 λ 2 κ + 1 + i s f , ν i 2 ; ω f = 0,2 π / NOF , ……….. , 2 π ( NOF 1 ) / NOF
I ̂ ( k ) = t = 0 N 1 I ( t ) exp ( j 2 πkt N ) ; k = 0,1,2 , …. , N 1
I ̂ ZP ( k ) = t = 0 ZP 1 I ( t ) exp ( j 2 πkt N ) ; k = 0,1,2 , . . N 1 , , ZP 1
[ 1 1 . . 1 exp ( jκα ) exp ( jκα ) . . 1 exp ( 2 jκα ) exp ( 2 jκα ) . . 1 . . . . . . . . . . exp [ ( N 1 ) jκα ] . . . 1 ] [ κ κ * κ 1 . . I dc ] = [ I 0 I 1 . . . I N 1 ]
I ( t ) = I dc + k = 1 κ a k exp ( jkφ ) exp ( jαkt ) + k = 1 κ a k exp ( jkφ ) exp ( jαkt ) + η ( t ) ;
for t = 0,1,2 , , m , N 1
r ( p ) = E [ I ( t ) I * ( t p ) ]
I ( t ) = I dc + a 1 exp ( ) exp ( jαt ) + a 1 exp ( ) exp ( jαt ) + η ( t )
I * ( t p ) = I dc + a 1 exp ( ) exp [ ( t p ) ] + a 1 exp ( ) exp [ ( t p ) ] + η * ( t p )
r ( p ) = E [ I ( t ) I * ( t p ) ] = E { I dc 2 + I dc a 1 exp ( ) exp ( jαt ) + I dc a 1 exp ( ) exp ( jαt ) + exp ( jαp ) [ a 1 2 + I dc a 1 exp ( ) exp ( jαt ) + a 1 2 exp ( 2 ) exp ( 2 jαt ) ] + exp ( jαp ) [ a 1 2 + I dc a 1 exp ( ) exp ( jαt ) + a 1 2 exp ( 2 ) exp ( 2 jαt ) ] + η ( t ) η * ( t p ) }
r ( p ) = E [ I dc 2 + c 1 + exp ( jαp ) ( a 1 2 + c 2 ) + exp ( jαp ) ( a 1 2 + c 3 ) + η ( t ) η * ( t p ) ]
r ( p ) = A 0 2 + A 1 2 exp ( jαp ) + A 2 2 exp ( jαp ) + σ 2 δ p , 0
E [ η ( g ) η * ( h ) ] = σ 2 δ g , h
0 2 π exp ( ) = 0
r ( p ) = E [ I ( t ) I * ( t p ) ] = n = 0 2 κ A n 2 exp ( j ω n p ) + σ 2 δ p , 0

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