Abstract

In this work, we have developed a different algorithm than the classical one on phase-shifting interferometry. These algorithms typically use constant or homogeneous phase displacements and they can be quite accurate and insensitive to detuning, taking appropriate weight factors in the formula to recover the wrapped phase. However, these algorithms have not been considered with variable or inhomogeneous displacements. We have generalized these formulas and obtained some expressions for an implementation with variable displacements and ways to get partially insensitive algorithms with respect to these arbitrary error shifts.

© 2010 Optical Society of America

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References

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2000

1998

M. Constantini, “A novel phase unwrapping method based on network,” IEEE Trans. Geosci. Remote Sens. 36, 813–821(1998).
[CrossRef]

Y. Surrel, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts: comment,” J. Opt. Soc. Am. A 15, 1227–1233 (1998).
[CrossRef]

1996

1987

Constantini, M.

M. Constantini, “A novel phase unwrapping method based on network,” IEEE Trans. Geosci. Remote Sens. 36, 813–821(1998).
[CrossRef]

Dorrío, B. V.

Eiju, T.

Hariharan, P.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2001).

Higham, D. J.

D. J. Higham and N. J. Higham, MATLAB Guide (SIAM, 2005).
[CrossRef]

Higham, N. J.

D. J. Higham and N. J. Higham, MATLAB Guide (SIAM, 2005).
[CrossRef]

Luong, B.

B. Luong, “Constantini phase unwrapping,” http://www.mathworks.com/mathlabcentral/fileexchange /25154-constantini-phase-unwrapping.

Malacara, D.

D. Malacara and Z. Malacara, Handbook of Optical Design (CRC, 2004).

D. Malacara, Optical Shop Testing (Wiley-Interscience, 2007).
[CrossRef]

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

Malacara, Z.

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

D. Malacara and Z. Malacara, Handbook of Optical Design (CRC, 2004).

Malacara-Doblado, D.

Nocedal, J.

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

Oreb, B. F.

Servín, M.

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

Surrel, Y.

Voguel, C. R.

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

Wright, S. J.

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

Appl. Opt.

IEEE Trans. Geosci. Remote Sens.

M. Constantini, “A novel phase unwrapping method based on network,” IEEE Trans. Geosci. Remote Sens. 36, 813–821(1998).
[CrossRef]

J. Opt. Soc. Am. A

Other

B. Luong, “Constantini phase unwrapping,” http://www.mathworks.com/mathlabcentral/fileexchange /25154-constantini-phase-unwrapping.

D. J. Higham and N. J. Higham, MATLAB Guide (SIAM, 2005).
[CrossRef]

D. Malacara, Optical Shop Testing (Wiley-Interscience, 2007).
[CrossRef]

D. Malacara and Z. Malacara, Handbook of Optical Design (CRC, 2004).

E. Hecht, Optics (Addison-Wesley, 2001).

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

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

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

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Figures (7)

Fig. 1
Fig. 1

(a) Synthetic phase, (b) the offset distribution and (c) the amplitude.

Fig. 2
Fig. 2

Intensity patterns s k of the five step Schwider–Hariharan algorithm.

Fig. 3
Fig. 3

(a) Wrapped and (b) unwrapped phase for the Schwider–Hariharan algorithm, (c) the synthetic phase, and (d) their corresponding absolute errors.

Fig. 4
Fig. 4

Intensity patterns s k of the four-step inhomogeneous algorithm.

Fig. 5
Fig. 5

(a) Wrapped and (b) unwrapped phase for the four-step inhomogeneous algorithm, (c) the synthetic phase, and (d) their corresponding absolute errors.

Fig. 6
Fig. 6

Intensity patterns s k of the five-step inhomogeneous algorithm.

Fig. 7
Fig. 7

(a) Wrapped and (b) unwrapped phase for the five-step inhomogeneous algorithm, (c) the synthetic phase, and (d) their corresponding absolute errors.

Equations (37)

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s k = a + b cos ( ϕ α k ) ,
Δ cos α k k + 1 = cos α k + 1 cos α k 0 , Δ sin α k k + 1 = sin α k + 1 sin α k 0 ,
k = 1 n 1 A k = k = 1 n 1 B k = 0 ,
k = 1 n 1 A k Δ sin α k k + 1 = k = 1 n 1 B k Δ cos α k k + 1 = 0 ,
Δ s k k + 1 = s k + 1 s k = b [ Δ cos α k k + 1 cos ϕ + Δ sin α k k + 1 sin ϕ ] ,
ϕ = arctan [ C N D ] ,
N = k = 1 n 1 B k Δ s k k + 1 , D = k = 1 n 1 A k Δ s k k + 1 , C = k = 1 n 1 A k Δ cos α k k + 1 k = 1 n 1 B k Δ sin α k k + 1 .
α k = α 0 k + δ α k ,
( N / D ) α k = ( tan ϕ C ) α k = tan ϕ C 2 C α k = 0 ,
C α k | x 0 = 0 ,
( N k α k C D k α k ) | x 0 = 0 ,
N k = ε k 1 0 A k 1 Δ cos α k 1 k + ε k n A k Δ cos α k k + 1 , D k = ε k 1 0 B k 1 Δ sin α k 1 k + ε k n B k Δ sin α k k + 1 , N k α k = ε k 1 0 A k 1 τ 1 + ε k n A k τ 2 , D k α k = ε k 1 0 B k 1 σ 1 ε k n B k σ 2 ,
τ 1 = cos α k Δ cos α k 1 k Δ sin α k 1 k + sin α k , τ 2 = cos α k Δ cos α k k + 1 Δ sin α k k + 1 + sin α k , σ 1 = sin α k Δ sin α k 1 k Δ cos α k 1 k + cos α k , σ 2 = sin α k Δ sin α k k + 1 Δ cos α k k + 1 + cos α k ,
ε q p = 1 δ p q ,
k = 1 n ( N k α k C D k α k ) 2 | x 0 0 .
C ^ = k = 1 n ( N k / α k ) ( D k / α k ) k = 1 n ( D k / α k ) 2 ,
C = N C D C ,
N C = j = 1 n 1 A j Δ cos α j j + 1 , D C = j = 1 n 1 B j Δ sin α j j + 1 ,
C α k = N ¯ k D ¯ ,
N ¯ k = D C N k α k N C D k α k , D ¯ = D C 2 .
2 C α p α k = D ¯ ( N ¯ k / α p ) N ¯ k ( D ¯ / α p ) D ¯ 2 ,
D ¯ N ¯ k α p = N ¯ k D ¯ α p ,
D C 2 2 N k α p α k D C N p α p D k α k D C N C 2 D k α p α k = D C D p α p N k α k 2 N C D p α p D k α k ,
ϕ α k = [ 1 1 + [ C ( N / D ) ] 2 ] ( N / D ) C α k = ξ ( N / D ) C α k ,
x = ( α 1 , , α n ) = x 0 + k = 1 n e k δ α k = x 0 + δ x ,
2 ϕ α p α k = ( N / D ) [ ξ α p C α k + ξ 2 C α p α k ] ,
2 ϕ α p α k | x 0 = ( N / D ) ξ 2 C α p α k | x 0 = 2 C α p α k sin ( 2 ϕ ) 2 C | x 0 .
ϕ ( x 0 + δ x ) = ϕ ( x 0 ) + 1 2 p = 1 n k = 1 n 2 ϕ α p α k | x 0 ( δ α p ) ( δ α k ) ,
δ ϕ = p = 1 n k = 1 n 2 C α p α k sin ( 2 ϕ ) 4 C | x 0 ( δ α p ) ( δ α k ) ,
ϕ ( x , y ) = 2 x y + 4 [ 2 ( x 2 + y 2 ) 1 ] + 2 [ 3 ( x 2 + y 2 ) y 2 y ] ,
a ( x , y ) = 0 . 0001 + 0 . 0001 [ sin ( 2 π x / L x ) + cos ( π y / L y ) ] ,
b ( x , y ) = 0 . 0002 + 0 . 0001 exp [ 0 . 001 ( x 2 + y 2 ) ] .
0 , π / 2 , π , 3 π / 2 , 2 π ,
π / 2 , 3 π / 4 , 7 π / 8 , 15 π / 16 ,
- 0 . 2212 , 2 . 549 , 3 . 9212 , 5 . 3202 , 8 . 0875 ,
x ( j ) = L x + ( j 1 ) Δ x , j = 1 , , N , y ( i ) = L y + ( i - 1 ) Δ y , i = 1 , , M .
MSE = 1 M × N i = 1 M j = 1 N | ϕ ( x ( j ) , y ( i ) ) ϕ ^ ( x ( j ) , y ( i ) ) | 2 ,

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