Abstract

In a radial shearing interferometer, a portion of the test beam is magnified and used as the reference for the tested wavefront. However, the reference portion is always off center (lateral shear), which complicates the wavefront reconstruction. A modal method for solving this problem is presented here. This method uses orthogonal Zernike polynomials and its matrix formalism to calculate the Zernike coefficient of the wavefront under test. This approach has easier implementation, is better filtering, and has a more adaptive practical situation. The corresponding mathematical formula is deduced, and a computer simulation is also made to verify operation of the algorithm. The result of simulation analysis shows that the proposed method is correct and accurate.

© 2011 Optical Society of America

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References

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2011 (1)

2008 (2)

2007 (2)

B. Zhang, L. Ma, M. Wang, and A. He, Laser Technol. 31, 37 (2007) (in Chinese).

D. Liu, Y. Ying, L. Wang, and Y. Zhuo, Appl. Opt. 46, 8305(2007).
[CrossRef] [PubMed]

2005 (1)

2002 (1)

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, Optik 113, 39 (2002).
[CrossRef]

2000 (1)

1999 (1)

R. Ragazzoni, E. Marchetti, and F. Rigaut, Astron. Astrophys. 342, L53 (1999).

1986 (2)

1961 (1)

P. Hariharan and D. Sen, J. Sci. Instrum. 38, 428 (1961).
[CrossRef]

Bao, B.

Barnes, T. H.

Chen, H.

Freischlad, K.

Fuente, R.

Gamiz, V.

Garncarz, B. E.

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, Optik 113, 39 (2002).
[CrossRef]

Gu, N.

Hariharan, P.

P. Hariharan and D. Sen, J. Sci. Instrum. 38, 428 (1961).
[CrossRef]

Haskell, T. G.

He, A.

B. Zhang, L. Ma, M. Wang, and A. He, Laser Technol. 31, 37 (2007) (in Chinese).

Huang, L.

Kasprzak, H. T.

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, Optik 113, 39 (2002).
[CrossRef]

Kohler, D.

Koliopoulos, C.

Kowalik, W. W.

W. W. Kowalik, B. E. Garncarz, and H. T. Kasprzak, Optik 113, 39 (2002).
[CrossRef]

Lago, E.

Li, D.

Li, F.

Li, X.

Liu, D.

Ma, L.

B. Zhang, L. Ma, M. Wang, and A. He, Laser Technol. 31, 37 (2007) (in Chinese).

Marchetti, E.

R. Ragazzoni, E. Marchetti, and F. Rigaut, Astron. Astrophys. 342, L53 (1999).

Ragazzoni, R.

R. Ragazzoni, E. Marchetti, and F. Rigaut, Astron. Astrophys. 342, L53 (1999).

Rao, C.

Rigaut, F.

R. Ragazzoni, E. Marchetti, and F. Rigaut, Astron. Astrophys. 342, L53 (1999).

Sen, D.

P. Hariharan and D. Sen, J. Sci. Instrum. 38, 428 (1961).
[CrossRef]

Shirai, T.

Wang, L.

Wang, M.

B. Zhang, L. Ma, M. Wang, and A. He, Laser Technol. 31, 37 (2007) (in Chinese).

Wang, P.

Wang, Q.

Wen, F.

Yang, H.

Yang, Z.

Ying, Y.

Zhang, B.

B. Zhang, L. Ma, M. Wang, and A. He, Laser Technol. 31, 37 (2007) (in Chinese).

Zhao, Y.

Zhuo, Y.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the radial shearing interferogram without and with lateral shear. (a) Shearogram without lateral shear. (b) Shearogram with lateral shear.

Fig. 2
Fig. 2

Simulated wavefront under test and the corresponding wavefront difference. (a) Random wavefront under test φ 0 ( x , y ) . (b) Weighting coefficient vector A 0 for each order of Zernike polynomials. (c) Wavefront difference Δ φ ( x , y ) calculated from φ 0 ( x , y ) ; the RMS and PTV of Δ φ ( x , y ) are 2.091 λ and 18.057 λ , respectively. (d) Coefficient vector C of Zernike polynomials for Δ φ ( x , y ) .

Fig. 3
Fig. 3

Wavefront reconstruction and the corresponding residual error. (a) Two-dimensional plot of coefficient matrix B. (b) Coefficient vector A calculated from Eq. (13). (c) Reconstructed wavefront under test by substituting the vector A into Eq. (4). (d) Residual error between the origin wavefront and the reconstructed wavefront.

Equations (13)

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Δ φ ( x , y ) = φ 1 ( x , y ) φ 2 ( x , y ) .
{ φ 1 ( x , y ) = φ 0 ( x , y ) , φ 2 ( x , y ) = φ 0 ( x / s 2 , y / s 2 ) , ( x , y ) circle ( d ) .
{ φ 1 ( x , y ) = φ 0 ( x , y ) , φ 2 ( x , y ) = φ 0 ( x / s 2 x 0 , y / s 2 y 0 ) .
φ 0 ( x , y ) = k = 1 N a k · Z k ( x , y ) ,
φ 2 ( x , y ) = k = 1 N a k · P k ( x , y ) ,
P k ( x , y ) = j = 1 k b j k · Z j ( x , y ) .
Δ φ ( x , y ) = k = 1 N a k Z k ( x , y ) k = 1 N a k [ j = 1 k b j k Z j ( x , y ) ] .
Δ φ = [ a 1 , a 2 , , a k , a N ] [ 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 ] [ Z 1 Z 2 Z k Z N ] [ a 1 , a 2 , , a k , a N ] [ b 1 1 0 0 0 b 1 2 b 2 2 0 0 0 0 b 1 k b 2 k b k k 0 0 b 1 N b 2 N b k N b N N ] [ Z 1 Z 2 Z k Z N ] ,
Δ φ = ABZ ,
B = [ 1 b 1 1 0 0 0 b 1 2 1 b 2 2 0 0 b 1 k b 2 k 1 b k k 0 b 1 N b 2 N b k N 1 b N N ] .
Δ φ = CZ ,
AB = C ,
A = CB + ,

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