Abstract

In this manuscript, some interesting properties for generalized or nonuniform phase-shifting algorithms are shown in the Fourier frequency space. A procedure to find algorithms with equal amplitudes for their sampling function transforms is described. We also consider in this procedure the finding of algorithms that are orthogonal for all possible values in the frequency space. This last kind of algorithms should closely satisfy the first order detuning insensitive condition. The procedure consists of the minimization of functionals associated with the desired insensitivity conditions.

© 2011 Optical Society of America

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2010

2009

2008

2005

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).
[CrossRef]

2004

K. Madsen, H. B. Nielsen, and O. Tingleff, Methods for Non-Linear Least Squares Problems, 2nd ed. (IMM, Technical University of Denmark, 2004).

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29, 183–185 (2004).
[CrossRef] [PubMed]

2001

E. H. Lieb and M. Loss, Analysis, 2nd ed. (American Mathematical Society, 2001).

2000

1999

J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 1999).
[CrossRef]

1998

1997

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 14, 918–930 (1997).
[CrossRef]

M. Servín, D. Malacara, J. L. Marroquín, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278(1997).
[CrossRef]

1995

1991

1990

1988

1987

1983

1982

Burow, R.

Cai, L. Z.

Chai, L.

Cheng, X. C.

Cornejo-Rodriguez, A.

Creath, K.

Cuevas, F. J.

M. Servín, D. Malacara, J. L. Marroquín, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278(1997).
[CrossRef]

Dong, G. Y.

Dorrío, B. V.

Elssner, K. E.

Eyui, T.

Farrant, D. I.

Freischlad, K.

Gao, P.

Geist, E.

Grzanna, J.

Harder, I.

Hariharan, P.

Hibino, K.

Honda, T.

Kinoshita, S.

Koliopoulos, C. L.

Lai, G.

Larkin, K. G.

Lieb, E. H.

E. H. Lieb and M. Loss, Analysis, 2nd ed. (American Mathematical Society, 2001).

Lindlein, N.

Liu, Q.

Loss, M.

E. H. Lieb and M. Loss, Analysis, 2nd ed. (American Mathematical Society, 2001).

Madsen, K.

K. Madsen, H. B. Nielsen, and O. Tingleff, Methods for Non-Linear Least Squares Problems, 2nd ed. (IMM, Technical University of Denmark, 2004).

Malacara, D.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).
[CrossRef]

M. Servín, D. Malacara, J. L. Marroquín, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278(1997).
[CrossRef]

Malacara, Z.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).
[CrossRef]

Malacara-Doblado, D.

Mantel, K.

Marroquín, J. L.

M. Servín, D. Malacara, J. L. Marroquín, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278(1997).
[CrossRef]

Meng, X. F.

Merkel, K.

Morgan, C. J.

Nielsen, H. B.

K. Madsen, H. B. Nielsen, and O. Tingleff, Methods for Non-Linear Least Squares Problems, 2nd ed. (IMM, Technical University of Denmark, 2004).

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 1999).
[CrossRef]

Ohyama, N.

Oreb, B. F.

Schmit, J.

Schwider, J.

Servín, M.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).
[CrossRef]

M. Servín, D. Malacara, J. L. Marroquín, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278(1997).
[CrossRef]

Shen, X. X.

Spolaczyk, R.

Sun, W. J.

Surrel, Y.

Téllez-Quiñones, A.

Tingleff, O.

K. Madsen, H. B. Nielsen, and O. Tingleff, Methods for Non-Linear Least Squares Problems, 2nd ed. (IMM, Technical University of Denmark, 2004).

Tsujiuchi, J.

Vogel, C. R.

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2000).

Wang, Y. R.

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 1999).
[CrossRef]

Xu, J.

Xu, Q.

Xu, X. F.

Yang, X. L.

Yao, B.

Yatagai, T.

Zhang, H.

Appl. Opt.

J. Mod. Opt.

M. Servín, D. Malacara, J. L. Marroquín, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278(1997).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Other

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).
[CrossRef]

E. H. Lieb and M. Loss, Analysis, 2nd ed. (American Mathematical Society, 2001).

J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 1999).
[CrossRef]

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2000).

K. Madsen, H. B. Nielsen, and O. Tingleff, Methods for Non-Linear Least Squares Problems, 2nd ed. (IMM, Technical University of Denmark, 2004).

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Figures (8)

Fig. 1
Fig. 1

Frequency dependent sampling functions for an unequal-step algorithm with five steps [continuous and dashed lines correspond to the numerator and denominator, respectively, in Eq. (6)]. (a) Amplitudes for normalized frequencies less than 10 (the numerator and denominator coincide), (b) phases for normalized frequencies in the neighborhood of the reference one ( f r = 1 ).

Fig. 2
Fig. 2

Frequency dependent sampling functions for an unequal-step algorithm with six steps [continuous and dashed lines correspond to the numerator and denominator, respectively, in Eq. (6)]. (a) Amplitudes for normalized frequencies less than 10 (the numerator and denominator coincide), (b) phases for normalized frequencies in the neighborhood of the reference one ( f r = 1 ).

Fig. 3
Fig. 3

Frequency dependent sampling functions for an equal-step algorithm with five steps [continuous and dashed lines correspond to the numerator and denominator, respectively, in Eq. (6)]. (a) Amplitudes for normalized frequencies less than 10, (b) phases for normalized frequencies in the neighborhood of the reference one ( f r = 1 ).

Fig. 4
Fig. 4

Frequency dependent sampling functions for an equal-step algorithm with seven steps [continuous and dashed lines correspond to the numerator and denominator, respectively, in Eq. (6)]. (a) Amplitudes for normalized frequencies less than 10, (b) phases for normalized frequencies in the neighborhood of the reference one ( f r = 1 ).

Fig. 5
Fig. 5

Frequency dependent sampling functions for an unequal-step algorithm with five steps [continuous and dashed lines correspond to the numerator and denominator, respectively, in Eq. (6)]. (a) Amplitudes for normalized frequencies less than 10, (b) phases for normalized frequencies in the neighborhood of the reference one ( f r = 1 ).

Fig. 6
Fig. 6

Frequency dependent sampling functions for an unequal-step algorithm with four steps [continuous and dashed lines correspond to the numerator and denominator, respectively, in Eq. (6)]. (a) Amplitudes for normalized frequencies less than 10, (b) phases for normalized frequencies in the neighborhood of the reference one ( f r = 1 ).

Fig. 7
Fig. 7

Frequency dependent sampling functions for an unequal-step algorithm with six steps [continuous and dashed lines correspond to the numerator and denominator, respectively, in Eq. (6)]. (a) Amplitudes for normalized frequencies less than 10, (b) phases for normalized frequencies in the neighborhood of the reference one ( f r = 1 ).

Fig. 8
Fig. 8

Peak-to-valley phase error as a function of the normalized frequency. (a)  Schwider–Hariharan algorithm and those given by Eqs. (14, 15), (b) algorithms given by Eqs. (16, 17, 18).

Equations (22)

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tan ϕ = B k Δ s k k + 1 / B k Δ sin ( α ) k k + 1 A k Δ s k k + 1 / A k Δ cos ( α ) k k + 1 ,
B ˜ k = B k B j Δ sin ( α ) j j + 1 = B k D C , A ˜ k = A k A j Δ cos ( α ) j j + 1 = A k N C ,
tan ϕ = B ˜ k Δ s k k + 1 A ˜ k Δ s k k + 1 .
tan ϕ = s ( t ) g N ( t ) d t s ( t ) g D ( t ) d t ,
tan ϕ = S ( f ) G ¯ N ( f ) d f S ( f ) G ¯ D ( f ) d f
G ¯ N ( f ) = B ˜ k [ Δ cos ( f α / f r ) k k + 1 + i Δ sin ( f α / f r ) k k + 1 ] , G ¯ D ( f ) = A ˜ k [ Δ cos ( f α / f r ) k k + 1 + i Δ sin ( f α / f r ) k k + 1 ] .
G ¯ N ( 0 ) = G ¯ D ( 0 ) = 0 ,
G ¯ N ( f r ) = i , G ¯ D ( f r ) = 1 ,
Am ( G ¯ N ( f r ) ) = Am ( G ¯ D ( f r ) ) = 1 ,
F ( x ) = p = 1 n k = 1 n 1 [ ( B ˜ k α p ) 2 + ( A ˜ k α p ) 2 ] ,
F ( x ) = k = 1 n 1 ( B ˜ k A ˜ k ) 2 ,
x * , 5 = [ 0.657 , 1.524 , 3.927 , 6.329 , 8.511 ] .
x * , 6 = [ 0.159 , 0.785 , 1.73 , 2.982 , 3.927 , 4.871 ]
F ( x , y ) = ( A ˜ k Δ ( α cos α ) k k + 1 B ˜ k Δ ( α sin α ) k k + 1 ) 2 + ( A ˜ k Δ ( α cos α ) k k + 1 + B ˜ k Δ ( α sin α ) k k + 1 ) 2 .
y = [ 1 , 2 , 1 , 1 , 2 , 1 ] .
γ N ( f ) = arctan [ B k Δ sin ( f α / f r ) k k + 1 B k Δ cos ( f α / f r ) k k + 1 ] , γ D ( f ) = arctan [ A k Δ sin ( f α / f r ) k k + 1 A k Δ cos ( f α / f r ) k k + 1 ] ,
y * , 5 = [ 0.4 , 1.3 , 1.3 , 0.4 , 0.4 , 1.3 , 1.3 , 0.4 ] .
y * , 7 = [ 0.6 , 1.3 , 0.9 , 0.9 , 1.3 , 0.6 , 0.4 , 1.4 , 1.9 , 1.9 , 1.4 , 0.4 ] .
y * , 5 = [ 1 , 1 , 1 , 1 , 0 , 1 , 1 , 0 ] .
y = [ 1 , 2 , 1 , 1 , 0 , 1 ] ,
y = [ 0.9 , 0.4 , 1 , 1.1 , 0.6 , 1 , 0 , 1 , 1 , 1 ] ,
Δ ϕ ( f ) = 1 2 [ 1 ρ ( f ) ] sin ( 2 ϕ ) ,

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