Abstract

Wavefront reconstruction in radial shearing interferometry with general aperture shapes is challenging because the problem may be ill-conditioned. Here we propose a Gram-Schmidt orthogonalization method to cope with off-axis wavefront reconstruction with any aperture type. The proposed method constructs a set of orthogonal basis functions and computes the corresponding expansion coefficients, which are converted into another set of expansion coefficients to reproduce the original wavefront. The method can effectively alleviate the ill-conditioning of the problem, and is numerically stable compared with the classic least-squares method, especially for non-circular apertures and in the presence of noise. Computer simulation and experimental results are presented to demonstrate the performance of the algorithm.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
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2015 (3)

2013 (1)

2012 (1)

2011 (4)

2009 (1)

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
[Crossref]

2008 (2)

2007 (1)

2004 (1)

2002 (2)

D. Li, H. Chen, and Z. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1893–1898 (2002).
[Crossref]

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

2000 (2)

T. Shirai, T. H. Barnes, and T. G. Haskell, “Adaptive wave-front correction by means of all-optical feedback interferometry,” Opt. Lett. 25(11), 773–775 (2000).
[Crossref] [PubMed]

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
[Crossref]

1994 (1)

1990 (1)

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

1986 (1)

1982 (1)

Bao, B.

Barnes, T. H.

Bovik, A. C.

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

Carpio-Valadez, M.

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

Chen, H.

D. Li, H. Chen, and Z. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1893–1898 (2002).
[Crossref]

Chen, L.

Chen, Z.

D. Li, H. Chen, and Z. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1893–1898 (2002).
[Crossref]

Chow, W. W.

Dai, F.

de la Fuente, R.

Ellerbroek, B.

Feng, P.

Gamiz, V. L.

Gu, C.

Gu, N.

Haskell, T. G.

Huang, L.

Ina, H.

Jeong, T. M.

Kantun-Montiel, R.

R. Kantun-Montiel and C. Meneses-Fabian, “Carrier fringes and a non-conventional rotational shear in a triangular cyclic-path interferometer,” J. Opt. 17(4), 045602 (2015).
[Crossref]

Kewei, E.

Ko, D.-K.

Kobayashi, S.

Kohler, D. R.

Kohno, T.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
[Crossref]

Lee, J.

Li, D.

Li, F.

Li, M.

Ling, T.

Liu, D.

T. Ling, D. Liu, Y. Yang, L. Sun, C. Tian, and Y. Shen, “Off-axis cyclic radial shearing interferometer for measurement of centrally blocked transient wavefront,” Opt. Lett. 38(14), 2493–2495 (2013).
[Crossref] [PubMed]

C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” in 5th International Symposium on Advanced Optical Manufacturing and Testing Technologies, (SPIE, 2010), 765612.
[Crossref]

Liu, S.

C. Tian and S. Liu, “Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization,” Opt. Express (To be published).

López Lago, E.

Luo, Q.

Luo, Y.

C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” in 5th International Symposium on Advanced Optical Manufacturing and Testing Technologies, (SPIE, 2010), 765612.
[Crossref]

Malacara-Hernandez, D.

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

Matsumoto, D.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
[Crossref]

Meneses-Fabian, C.

R. Kantun-Montiel and C. Meneses-Fabian, “Carrier fringes and a non-conventional rotational shear in a triangular cyclic-path interferometer,” J. Opt. 17(4), 045602 (2015).
[Crossref]

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
[Crossref]

Rao, C.

Rodriguez-Zurita, G.

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
[Crossref]

Sanchez-Mondragon, J. J.

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

Sasaki, O.

Shen, Y.

Shirai, T.

Sun, L.

Swantner, W.

Takeda, M.

Tang, F.

Tian, C.

Toto-Arellano, N. I.

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
[Crossref]

Uda, Y.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
[Crossref]

Upton, R.

Vázquez-Castillo, J. F.

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
[Crossref]

Wan, J.

Wang, Q.

Wang, X.

Wang, Z.

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

Wei, T.

Wen, F.

Xiong, Z.

Yang, Y.

Yang, Z.

Yazawa, T.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
[Crossref]

Zhang, C.

Zhao, Y.

Zheng, D.

Zhu, W.

Zhuo, Y.

Appl. Opt. (5)

IEEE Signal Process. Lett. (1)

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

J. Opt. (1)

R. Kantun-Montiel and C. Meneses-Fabian, “Carrier fringes and a non-conventional rotational shear in a triangular cyclic-path interferometer,” J. Opt. 17(4), 045602 (2015).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Eng. (3)

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
[Crossref]

D. Li, H. Chen, and Z. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1893–1898 (2002).
[Crossref]

Opt. Express (3)

Opt. Lett. (7)

Other (5)

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007).

M. Wang, B. Zhang, S. Nie, and J. Fu, “Radial shearing interferometer for aspheric surface testing,” in Photonics Asia 2002 (SPIE, 2002), 673–676.

C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” in 5th International Symposium on Advanced Optical Manufacturing and Testing Technologies, (SPIE, 2010), 765612.
[Crossref]

C. Tian and S. Liu, “Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization,” Opt. Express (To be published).

C. D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, 2000).

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Figures (6)

Fig. 1
Fig. 1 Schematic diagram of one type of off-axis RSI using two identical Galilean telescopes and the coordinate system.
Fig. 2
Fig. 2 Flow chart of the least-squares method (a) and the Gram-Schmidt orthogonalization method (b). Zj and ∆W are known, and W is the unknown to be determined.
Fig. 3
Fig. 3 Generation of a differential wavefront with an elliptic aperture. (a) Original wavefront W, contracted wavefront Wc, expanded wavefront We and their overlapping area (red curve), (b) three-dimensional (3D) and two-dimensional (2D) differential wavefront ∆W with a small additive noise. Colorbar unit: rad.
Fig. 4
Fig. 4 Reconstruction results by the least-squares method (1st row) and the proposed method (2nd row). (a) - (c) Computed coefficients aj, reconstructed wavefront and residual error by least squares; (d) - (f) computed coefficients aj and bj, reconstructed wavefront and residual error by Gram-Schmidt orthogonalization. Colorbar unit: rad.
Fig. 5
Fig. 5 Wavefront reconstruction from a single-shot off-axis RSI interferogram. (a) and (b) An experimental interferogram and its enlarged view, (c) demodulated phase, (d) computed coefficients aj and bj, (e) and (f) reconstructed wavefronts using the proposed method and the iterative method in [14], respectively, and (g) difference map. Colorbar unit: rad.
Fig. 6
Fig. 6 Wavefront reconstruction from a single-shot off-axis RSI interferogram. (a) and (b) An experimental interferogram and the ROI, (c) demodulated phase, (d) - (f) reconstructed wavefronts using the least-squares method, the proposed method and the iterative method in [14], respectively. Colorbar unit: rad.

Equations (42)

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ΔW(x,y)= W c (x,y) W e (x,y)=W(x,y)W(βx+ x 0 ,βy+ y 0 ),
W(x,y)= j=1 N a j Z j (x,y),
Z j ={ (n+1) R n m (ρ),m=0, 2(n+1) R n m (ρ)cosmθ,m0andevenj, 2(n+1) R n m (ρ)sinmθ,m0andoddj,
R n m (ρ)= s=0 (nm)/2 (1) s (ns)! s![(n+m)/2s]![(nm)/2s]! ρ n2s ,
ΔW(x,y)= j=1 N a j Z j (x,y) j=1 N a j Z j (βx+ x 0 ,βy+ y 0 )= j=1 N a j U j (x,y),
U j (x,y)= Z j (x,y) Z j (βx+ x 0 ,βy+ y 0 ),
Ua=ΔW,
U=[ U 1 ( x 1 , y 1 ) U 2 ( x 1 , y 1 ) U N ( x 1 , y 1 ) U 1 ( x 2 , y 2 ) U 2 ( x 2 , y 2 ) U N ( x 2 , y 2 ) U 1 ( x M , y M ) U 2 ( x M , y M ) U N ( x M , y M ) ],
a=[ a 1 a 2 a N ],ΔW=[ ΔW( x 1 , y 1 ) ΔW( x 2 , y 2 ) ΔW( x M , y M ) ],
U T Ua= U T ΔW.
a= ( U T U) 1 U T ΔW.
a+δa= ( U T U) 1 U T (ΔW+ϵ).
δa= ( U T U) 1 U T ϵ=(S Λ 1 S T ) U T ϵ,
Δ a j i=1 M U ji ε i / λ j ,
i=1 M V p ( x i , y i ) V q ( x i , y i )={ 0,ifpq, 1,ifp=q,
ΔW( x i , y i )= k=1 N b k V k ( x i , y i ),
U j ( x i , y i )= k=1 j α kj V k ( x i , y i ).
V j ( x i , y i )= 1 α jj [ U j ( x i , y i ) k=1 j1 α kj V k ( x i , y i ) ],
α kj ={ i=1 M [ U j ( x i , y i ) V k ( x i , y i ) ],k<j, { i=1 M [ U j ( x i , y i ) ] 2 k=1 j1 α kj 2 } 1/2 ,k=j.
ΔW( x i , y i )= j=1 N a j U j ( x i , y i )= j=1 N a j [ k=1 j α kj V k ( x i , y i ) ]= k=1 N j=k N a j α kj V k ( x i , y i ).
b k = j=k N a j α kj .
[ α 11 α 12 α 13 α 1N 0 α 22 α 23 α 2N 0 0 α 33 α 3N 0 0 0 0 α NN ][ a 1 a 2 a 3 a N ]=[ b 1 b 2 b 3 b N ],
αa=b.
a= α 1 b,
b k = i=1 M [ ΔW( x i , y i ) V k ( x i , y i ) ],
ΔW= b 1 V 1 + b 2 V 2 + b 3 V 3 .
U 1 = α 11 V 1 , α 11 = ( i=1 M U 1 2 ) 1/2 , V 1 = U 1 / α 11 .
U 2 = α 12 V 1 + α 22 V 2 , α 12 = i=1 M ( U 2 V 1 ), α 22 = ( i=1 M U 2 2 α 12 2 ) 1/2 , V 2 =( U 2 α 12 V 1 )/ α 22 .
U 3 = α 13 V 1 + α 23 V 2 + α 33 V 3 , α 13 = i=1 M ( U 3 V 1 ), α 23 = i=1 M ( U 3 V 2 ), α 33 = [ i=1 M U 3 2 ( α 13 2 + α 23 2 ) ] 1/2 , V 3 =( U 3 α 13 V 1 α 23 V 2 )/ α 33 .
ΔW= a 1 U 1 + a 2 U 2 + a 3 U 3 =( α 11 a 1 + α 12 a 2 + α 13 a 3 ) V 1 +( α 22 a 2 + α 23 a 3 ) V 2 + α 33 a 3 V 3 .
[ α 11 α 12 α 13 0 α 22 α 23 0 0 α 33 ][ a 1 a 2 a 3 ]=[ b 1 b 2 b 3 ].
a 3 = b 3 / α 33 , a 2 =( b 2 a 23 b 3 )/ α 22 , a 1 =( b 1 α 12 a 2 α 13 a 3 )/ α 11 ,
b 3 = i=1 M (ΔW V 3 ), b 2 = i=1 M (ΔW V 2 ), b 1 = i=1 M (ΔW V 1 ),
Vb=ΔW,
V=[ V 1 ( x 1 , y 1 ) V 2 ( x 1 , y 1 ) V N ( x 1 , y 1 ) V 1 ( x 2 , y 2 ) V 2 ( x 2 , y 2 ) V N ( x 2 , y 2 ) V 1 ( x M , y M ) V 2 ( x M , y M ) V N ( x M , y M ) ],b=[ b 1 b 2 b N ].
V(b+δb)=(ΔW+ϵ).
Vδb=ϵ,
δb= V T ϵ,
a+δa= α 1 (b+δb).
δa= α 1 δb= α 1 V T ϵ.
δW=Zδa=Z α 1 V T ϵ.
Q= 4 σ AB μ A μ B ( σ A 2 + σ B 2 )( μ A 2 + μ B 2 ) ,

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