Abstract

Signal processing methods based on maximum-likelihood theory, discrete chirp Fourier transform, and spectral estimation methods have enabled accurate measurement of phase in phase-shifting interferometry in the presence of nonlinear response of the piezoelectric transducer to the applied voltage. We present the statistical study of these generalized nonlinear phase step estimation methods to identify the best method by deriving the Cramér–Rao bound. We also address important aspects of these methods for implementation in practical applications and compare the performance of the best-identified method with other bench marking algorithms in the presence of harmonics and noise.

© 2007 Optical Society of America

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References

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  1. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, "Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts," J. Opt. Soc. Am. A 4, 918-930 (1997).
    [CrossRef]
  2. P. de Groot, "Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window," Appl. Opt. 34, 4723-4730 (1995).
    [CrossRef]
  3. J. Schmit and K. Creath, "Extended averaging technique for derivation of error compensating algorithms in phase shifting interferometry," Appl. Opt. 34, 3610-3619 (1995).
    [CrossRef] [PubMed]
  4. R. Langoju, A. Patil, and P. Rastogi, "Phase-shifting interferometry in the presence of nonlinear phase steps, harmonics, and noise," Opt. Lett. 31, 1058-1060 (2006).
    [CrossRef] [PubMed]
  5. R. Langoju, A. Patil, and P. Rastogi, "Accurate nonlinear phase step estimation in phase shifting interferometry," Opt. Commun. 266, 638-647 (2006).
    [CrossRef]
  6. R. Langoju, A. Patil, and P. Rastogi, "Chirp estimation in phase-shifting interferometry," Opt. Lett. 31, 1982-1984 (2006).
    [CrossRef] [PubMed]
  7. C. Rathjen, "Statistical properties of the phase shifting algorithms," J. Opt. Soc. Am. A 12, 1997-2008 (1995).
    [CrossRef]
  8. S. M. Kay, "Sinusoidal parameter estimation," in Modern Spectral Estimation: Theory and Application (Prentice Hall, 1988), pp. 407-415.
  9. B. Raphael and I. F. C. Smith, "A direct stochastic algorithm for global search," Appl. Math. Comput. 146, 729-758 (2003).
    [CrossRef]
  10. V. Cizek, "Discrete Hilbert transform," IEEE Trans. Audio Electroacoust. 18, 340-343 (1970).
    [CrossRef]
  11. X.-G. Xia, "Discrete chirp-Fourier Transform and its application to chirp rate estimation," IEEE Trans. Signal Process. 13, 3122-3133 (2000).
  12. D. W. Marquardt, "An algorithm for least squares estimation of nonlinear parameters," J. Soc. Ind. Appl. Math. 11, 431-444 (1963).
    [CrossRef]
  13. A. Patil, P. Rastogi, and B. Raphael, "Phase-shifting interferometry by a covariance-based method," Appl. Opt. 44, 5778-5785 (2005).
    [CrossRef] [PubMed]
  14. A. Patil and P. Rastogi, "Subspace-based method for phase retrieval in interferometry," Opt. Express 13, 1240-1248 (2005).
    [CrossRef] [PubMed]
  15. P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, 1997).
  16. D. C. Rife and R. R. Boorstyn, "Single-tone parameter estimation from discrete-time observations," IEEE Trans. Inf. Theory IT-20, 591-598 (1974).
    [CrossRef]

2006 (3)

2005 (2)

2003 (1)

B. Raphael and I. F. C. Smith, "A direct stochastic algorithm for global search," Appl. Math. Comput. 146, 729-758 (2003).
[CrossRef]

2000 (1)

X.-G. Xia, "Discrete chirp-Fourier Transform and its application to chirp rate estimation," IEEE Trans. Signal Process. 13, 3122-3133 (2000).

1997 (1)

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, "Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts," J. Opt. Soc. Am. A 4, 918-930 (1997).
[CrossRef]

1995 (3)

1974 (1)

D. C. Rife and R. R. Boorstyn, "Single-tone parameter estimation from discrete-time observations," IEEE Trans. Inf. Theory IT-20, 591-598 (1974).
[CrossRef]

1970 (1)

V. Cizek, "Discrete Hilbert transform," IEEE Trans. Audio Electroacoust. 18, 340-343 (1970).
[CrossRef]

1963 (1)

D. W. Marquardt, "An algorithm for least squares estimation of nonlinear parameters," J. Soc. Ind. Appl. Math. 11, 431-444 (1963).
[CrossRef]

Appl. Math. Comput. (1)

B. Raphael and I. F. C. Smith, "A direct stochastic algorithm for global search," Appl. Math. Comput. 146, 729-758 (2003).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Audio Electroacoust. (1)

V. Cizek, "Discrete Hilbert transform," IEEE Trans. Audio Electroacoust. 18, 340-343 (1970).
[CrossRef]

IEEE Trans. Inf. Theory (1)

D. C. Rife and R. R. Boorstyn, "Single-tone parameter estimation from discrete-time observations," IEEE Trans. Inf. Theory IT-20, 591-598 (1974).
[CrossRef]

IEEE Trans. Signal Process. (1)

X.-G. Xia, "Discrete chirp-Fourier Transform and its application to chirp rate estimation," IEEE Trans. Signal Process. 13, 3122-3133 (2000).

J. Opt. Soc. Am. A (2)

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, "Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts," J. Opt. Soc. Am. A 4, 918-930 (1997).
[CrossRef]

C. Rathjen, "Statistical properties of the phase shifting algorithms," J. Opt. Soc. Am. A 12, 1997-2008 (1995).
[CrossRef]

J. Soc. Ind. Appl. Math. (1)

D. W. Marquardt, "An algorithm for least squares estimation of nonlinear parameters," J. Soc. Ind. Appl. Math. 11, 431-444 (1963).
[CrossRef]

Opt. Commun. (1)

R. Langoju, A. Patil, and P. Rastogi, "Accurate nonlinear phase step estimation in phase shifting interferometry," Opt. Commun. 266, 638-647 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Other (2)

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

S. M. Kay, "Sinusoidal parameter estimation," in Modern Spectral Estimation: Theory and Application (Prentice Hall, 1988), pp. 407-415.

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Figures (5)

Fig. 1
Fig. 1

Comparison of MSE of phase φ with CRB for the nonsymmetric case with φ = 2 π / 3 and (a) κ = 1 , N = 10 , ϵ 2 = 0.2 ; (b) κ = 1 , N = 10 , ϵ 2 = 0.4 ; (c) κ = 2 , N = 15 , ϵ 2 = 0.2 ; and (d) κ = 2 , N = 15 , ϵ 2 = 0.4 .

Fig. 2
Fig. 2

Comparison of MSE of phase φ with CRB for the symmetric case with φ = 2 π / 3 and (a) κ = 1 , N = 10 , ϵ 2 = 0.2 ; (b) κ = 1 , N = 10 , ϵ 2 = 0.4 ; (c) κ = 2 , N = 15 , ϵ 2 = 0.2 ; and (d) κ = 2 , N = 15 , ϵ 2 = 0.4 .

Fig. 3
Fig. 3

(a) PZT response to applied voltage 0 150   V shown in Eq. (17) with coefficients A = 7.8 and B = 0.0154 . (b) Comparison of the MSE of the retrieved phase φ for various sampling cases.

Fig. 4
Fig. 4

Comparison of MSE of φ obtained using HRM with Hibino method for φ = 2 π / 3 , κ = 1 , and (a) ϵ 1 = 0.1 , ϵ 2 = 0.05 ; (b) ϵ 1 = 0.1 , ϵ 2 = 0.1 ; (c) ϵ 1 = 0.1 , ϵ 2 = 0.2 ; and (d) ϵ 1 = 0.1 , ϵ 2 = 0.4 .

Fig. 5
Fig. 5

Comparison of MSE of φ obtained using HRM with Hibino method for φ = 2 π / 3 , κ = 2 , and (a) ϵ 1 = 0.1 , ϵ 2 = 0.05 ; (b) ϵ 1 = 0.1 , ϵ 2 = 0.1 ; (c) ϵ 1 = 0.1 , ϵ 2 = 0.2 ; and (d) ϵ 1 = 0.1 , ϵ 2 = 0.4 .

Tables (2)

Tables Icon

Table 1 Categorization of the Conventional Bench Marking Nonlinear Phase-Shifting Algorithms and Our Proposed Methods

Tables Icon

Table 2 Comparison of HRM and Hibino Method. Phase Error (in rad) in Computation of Phase φ

Equations (25)

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I ¯ n = I d c { 1 + k = 1 κ γ k cos [ k ( φ + α n ) ] } + η n ,
α n = n α ( 1 + ϵ 1 ) + ϵ 2 ( n α ) 2 / π ,
α n = n α ϵ + n 2 ϖ ,
I ¯ = I + η = S ( ξ ) C + η ,
p ( I ¯ ; ξ ) = 1 π N σ N exp { 1 2 σ 2 [ I ¯ S ( ξ ) C ] H [ I ¯ S ( ξ ) C ] } ,
L ( I ¯ , ξ ) = constant 1 2 σ 2 [ I ¯ S ( ξ ) C ] H [ I ¯ S ( ξ ) C ] .
D C = D C [ I ¯ H I ¯ I ¯ H S C C H S H I ¯ + C H S H S C ] , = 2 S H [ I ¯ S C ] .
C = { [ S ( ξ ) ] H S ( ξ ) } 1 S ( ξ ) H I ¯ .
D ( ξ ) = { I ¯ S [ S H S ] 1 S H I ¯ } H { I ¯ S [ S H S ] 1 S H I ¯ } , = { I ¯ H I H S [ S H S ] 1 S H } { I ¯ S [ S H S ] 1 S H I ¯ } , = I ¯ H I ¯ I ¯ H S [ S H S ] 1 S H I ¯ .
ξ = max I ¯ H S ( ξ ) { [ S ( ξ ) ] H S ( ξ ) } 1 S ( ξ ) H I ¯ .
I d ( n ) = A exp [ j ( φ + n α ϵ + n 2 ϖ ) ] + ζ n .
I DCFT ( k , l ) = 1 N n = 0 N 1 I d ( n ) exp [ j ( n α ^ k + n 2 ϖ ^ l ) ] ,
0 k , l N 1 ,
| I DCFT ( k 0 , l 0 ) | = A N .
I DCFT ( k , l ) = 1 N n = 0 N 1 I d ( n ) exp [ j ( n α ^ k + n 2 ϖ ^ l ) ] ,
0 k , l N 1 1 ,
I d ( n ) = A exp ( φ + n α ϵ + n 2 ϖ ) + ζ n .
I d ( n ) I d * ( n τ ) = A 2 exp [ j ( 2 n τ ϖ + τ α ϵ τ 2 ϖ ) ] ,
E { Ψ ^ Ψ ^ T } J 1 ,
J = E { [ Ψ log p ( I ¯ ) ] [ Ψ log p ( I ¯ ) ] T } .
E { Ψ ^ r 2 } J r , r 1 , for   r = 1 , 2 , 3 , 4 ,
J r , s = 1 σ 2 n = 0 N 1 I n ψ r I n ψ s ,
J = [ J I d c , I d c J I d c , γ J I d c , α ϵ J I d c , ϖ J I d c , φ J γ , I d c J γ , γ J γ , α ϵ J γ , ϖ J γ , φ J α ϵ , I d c J α ϵ , γ J α ϵ , α ϵ J α ϵ , ϖ J α ϵ , φ J ϖ , I d c J ϖ , γ J ϖ , α ϵ J ϖ , ϖ J ϖ , φ J φ , I d c J φ , γ J φ , α ϵ J φ , ϖ J φ , φ ] ,
α = A V + B V 2 ,
α n = A ( n V step ) + B ( n V step ) 2 , = ( A V step ) n + ( B V step 2 ) n 2 , = α ϵ n + ϖ n 2 ,

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