Abstract

Based on the Legendre polynomials expressions and its properties, this article proposes a new approach to reconstruct the distorted wavefront under test of a laser beam over square area from the phase difference data obtained by a RSI system. And the result of simulation and experimental results verifies the reliability of the method proposed in this paper. The formula of the error propagation coefficients is deduced when the phase difference data of overlapping area contain noise randomly. The matrix T which can be used to evaluate the impact of high-orders Legendre polynomial terms on the outcomes of the low-order terms due to mode aliasing is proposed, and the magnitude of impact can be estimated by calculating the F norm of the T. In addition, the relationship between ratio shear, sampling points, terms of polynomials and noise propagation coefficients, and the relationship between ratio shear, sampling points and norms of the T matrix are both analyzed, respectively. Those research results can provide an optimization design way for radial shearing interferometry system with the theoretical reference and instruction.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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2012 (1)

2011 (1)

2010 (2)

2008 (2)

2007 (4)

2005 (1)

Y. Y. Yang, Y. B. Lu, Y. J. Chen, Y. M. Zhuo, X. M. Zhang, B. Chen, and X. W. Qing, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[Crossref]

2004 (1)

M. L. Spaeth, K. R. Manes, C. C. Widmayer, W. H. Williams, P. K. Whitman, M. A. Henesian, I. F. Stowers, and J. Honig, “National Ignition Facility wavefront requirements and optical architecture,” Opt. Eng. 43(12), 2854–2865 (2004).
[Crossref]

2002 (1)

D. H. Li, H. X. Chen, and Z. P. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1–6 (2002).
[Crossref]

1999 (4)

D. Fan and X. He, “Inertial confinement fusion energy and laser driver,” Exploration Nature 18(67), 31–35 (1999).

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. Williams, and B. M. Van Wonterghem, “Wavefront and divergence of the Beamlet prototype laser,” Proc. SPIE 3492, 1019–1030 (1999).
[Crossref]

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

V. Laude, S. Olivier, C. Dirson, and J. P. Huignard, “Hartmann wave-front scanner,” Opt. Lett. 24(24), 1796–1798 (1999).
[Crossref] [PubMed]

1995 (1)

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English., “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

1982 (1)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

1980 (1)

1961 (1)

P. Hariharan, D. Sen, and M. Sc, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961).

Aikens, D. M.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English., “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

Bao, B.

Chen, B.

Y. Y. Yang, Y. B. Lu, Y. J. Chen, Y. M. Zhuo, X. M. Zhang, B. Chen, and X. W. Qing, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[Crossref]

Chen, H. X.

D. H. Li, H. X. Chen, and Z. P. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1–6 (2002).
[Crossref]

Chen, Y. J.

Y. Y. Yang, Y. B. Lu, Y. J. Chen, Y. M. Zhuo, X. M. Zhang, B. Chen, and X. W. Qing, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[Crossref]

Chen, Z. P.

D. H. Li, H. X. Chen, and Z. P. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1–6 (2002).
[Crossref]

Dai, F.

Dai, G. M.

Dirson, C.

English, R. E.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English., “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

Fan, D.

D. Fan and X. He, “Inertial confinement fusion energy and laser driver,” Exploration Nature 18(67), 31–35 (1999).

Feng, G. Y.

Feng, P.

Gu, N.

Hariharan, P.

P. Hariharan, D. Sen, and M. Sc, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961).

He, X.

D. Fan and X. He, “Inertial confinement fusion energy and laser driver,” Exploration Nature 18(67), 31–35 (1999).

Henesian, M. A.

M. L. Spaeth, K. R. Manes, C. C. Widmayer, W. H. Williams, P. K. Whitman, M. A. Henesian, I. F. Stowers, and J. Honig, “National Ignition Facility wavefront requirements and optical architecture,” Opt. Eng. 43(12), 2854–2865 (2004).
[Crossref]

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. Williams, and B. M. Van Wonterghem, “Wavefront and divergence of the Beamlet prototype laser,” Proc. SPIE 3492, 1019–1030 (1999).
[Crossref]

Honig, J.

M. L. Spaeth, K. R. Manes, C. C. Widmayer, W. H. Williams, P. K. Whitman, M. A. Henesian, I. F. Stowers, and J. Honig, “National Ignition Facility wavefront requirements and optical architecture,” Opt. Eng. 43(12), 2854–2865 (2004).
[Crossref]

Huang, L.

Huignard, J. P.

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

Jeong, T. M.

Ko, D. K.

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

Laude, V.

Lawson, J. K.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English., “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

Lee, J.

Li, D.

Li, D. H.

D. H. Li, X. P. Qi, Q. H. Wang, X. Y. Liu, G. Y. Feng, and S. H. Zhou, “Accurate retrieval algorithm of amplitude from radial-shearing interferogram,” Opt. Lett. 35(18), 3054–3056 (2010).
[Crossref] [PubMed]

D. H. Li, H. X. Chen, and Z. P. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1–6 (2002).
[Crossref]

Li, F.

Liu, D.

Liu, X. Y.

Lu, Y. B.

Y. Y. Yang, Y. B. Lu, Y. J. Chen, Y. M. Zhuo, X. M. Zhang, B. Chen, and X. W. Qing, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[Crossref]

Luo, Q.

Mahajan, V. N.

Manes, K. R.

M. L. Spaeth, K. R. Manes, C. C. Widmayer, W. H. Williams, P. K. Whitman, M. A. Henesian, I. F. Stowers, and J. Honig, “National Ignition Facility wavefront requirements and optical architecture,” Opt. Eng. 43(12), 2854–2865 (2004).
[Crossref]

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English., “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

Marchetti, E.

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

Olivier, S.

Qi, X. P.

Qing, X. W.

Y. Y. Yang, Y. B. Lu, Y. J. Chen, Y. M. Zhuo, X. M. Zhang, B. Chen, and X. W. Qing, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[Crossref]

Ragazzoni, R.

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

Rao, C.

Rigaut, F.

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

Salmon, J. T.

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. Williams, and B. M. Van Wonterghem, “Wavefront and divergence of the Beamlet prototype laser,” Proc. SPIE 3492, 1019–1030 (1999).
[Crossref]

Sasaki, O.

Sc, M.

P. Hariharan, D. Sen, and M. Sc, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961).

Sen, D.

P. Hariharan, D. Sen, and M. Sc, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961).

Seppala, L. G.

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. Williams, and B. M. Van Wonterghem, “Wavefront and divergence of the Beamlet prototype laser,” Proc. SPIE 3492, 1019–1030 (1999).
[Crossref]

Southwell, W. H.

Spaeth, M. L.

M. L. Spaeth, K. R. Manes, C. C. Widmayer, W. H. Williams, P. K. Whitman, M. A. Henesian, I. F. Stowers, and J. Honig, “National Ignition Facility wavefront requirements and optical architecture,” Opt. Eng. 43(12), 2854–2865 (2004).
[Crossref]

Stowers, I. F.

M. L. Spaeth, K. R. Manes, C. C. Widmayer, W. H. Williams, P. K. Whitman, M. A. Henesian, I. F. Stowers, and J. Honig, “National Ignition Facility wavefront requirements and optical architecture,” Opt. Eng. 43(12), 2854–2865 (2004).
[Crossref]

Takeda, M.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

Tang, F.

Trenholme, J. B.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English., “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

Van Wonterghem, B. M.

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. Williams, and B. M. Van Wonterghem, “Wavefront and divergence of the Beamlet prototype laser,” Proc. SPIE 3492, 1019–1030 (1999).
[Crossref]

Wang, L.

Wang, Q.

Wang, Q. H.

Wang, X.

Wegner, P. J.

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. Williams, and B. M. Van Wonterghem, “Wavefront and divergence of the Beamlet prototype laser,” Proc. SPIE 3492, 1019–1030 (1999).
[Crossref]

Weiland, T. L.

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. Williams, and B. M. Van Wonterghem, “Wavefront and divergence of the Beamlet prototype laser,” Proc. SPIE 3492, 1019–1030 (1999).
[Crossref]

Wen, F.

Whitman, P. K.

M. L. Spaeth, K. R. Manes, C. C. Widmayer, W. H. Williams, P. K. Whitman, M. A. Henesian, I. F. Stowers, and J. Honig, “National Ignition Facility wavefront requirements and optical architecture,” Opt. Eng. 43(12), 2854–2865 (2004).
[Crossref]

Widmayer, C. C.

M. L. Spaeth, K. R. Manes, C. C. Widmayer, W. H. Williams, P. K. Whitman, M. A. Henesian, I. F. Stowers, and J. Honig, “National Ignition Facility wavefront requirements and optical architecture,” Opt. Eng. 43(12), 2854–2865 (2004).
[Crossref]

Williams, W. H.

M. L. Spaeth, K. R. Manes, C. C. Widmayer, W. H. Williams, P. K. Whitman, M. A. Henesian, I. F. Stowers, and J. Honig, “National Ignition Facility wavefront requirements and optical architecture,” Opt. Eng. 43(12), 2854–2865 (2004).
[Crossref]

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. Williams, and B. M. Van Wonterghem, “Wavefront and divergence of the Beamlet prototype laser,” Proc. SPIE 3492, 1019–1030 (1999).
[Crossref]

Wolfe, C. R.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English., “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

Yang, Y.

Yang, Y. Y.

Y. Y. Yang, Y. B. Lu, Y. J. Chen, Y. M. Zhuo, X. M. Zhang, B. Chen, and X. W. Qing, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[Crossref]

Yang, Z.

Zhang, X. M.

Y. Y. Yang, Y. B. Lu, Y. J. Chen, Y. M. Zhuo, X. M. Zhang, B. Chen, and X. W. Qing, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[Crossref]

Zhao, Y.

Zhou, S. H.

Zhuo, Y.

Zhuo, Y. M.

Y. Y. Yang, Y. B. Lu, Y. J. Chen, Y. M. Zhuo, X. M. Zhang, B. Chen, and X. W. Qing, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[Crossref]

Appl. Opt. (3)

Astron. Astrophys. (1)

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

Exploration Nature (1)

D. Fan and X. He, “Inertial confinement fusion energy and laser driver,” Exploration Nature 18(67), 31–35 (1999).

J. Opt. Soc. Am. (1)

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

V. N. Mahajan and G. M. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24(9), 2994–3016 (2007).
[Crossref] [PubMed]

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

J. Sci. Instrum. (1)

P. Hariharan, D. Sen, and M. Sc, “Radial shearing interferometer,” J. Sci. Instrum. 38, 428–432 (1961).

Opt. Eng. (2)

D. H. Li, H. X. Chen, and Z. P. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1–6 (2002).
[Crossref]

M. L. Spaeth, K. R. Manes, C. C. Widmayer, W. H. Williams, P. K. Whitman, M. A. Henesian, I. F. Stowers, and J. Honig, “National Ignition Facility wavefront requirements and optical architecture,” Opt. Eng. 43(12), 2854–2865 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (6)

Proc. SPIE (3)

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. Williams, and B. M. Van Wonterghem, “Wavefront and divergence of the Beamlet prototype laser,” Proc. SPIE 3492, 1019–1030 (1999).
[Crossref]

Y. Y. Yang, Y. B. Lu, Y. J. Chen, Y. M. Zhuo, X. M. Zhang, B. Chen, and X. W. Qing, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[Crossref]

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English., “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Principle diagram of cyclic radial shearing interferometry with square aperture.
Fig. 2
Fig. 2 Schematic of the RS with lateral shearing and the coordinate system in square aperture.
Fig. 3
Fig. 3 Original data of radial shearing interference with s = 0.75, x 0 = 0.1and y 0 = 0.3. (a) Contracted wavefront phase. (b) Expanded wavefront phase. (c) The phase difference Δ W .
Fig. 4
Fig. 4 Reconstruction of radial shearing interference. (a) Coefficients transform matrix B. (b) Reconstructed wavefront. (c) Difference between original wavefront and reconstructed wavefront. (d) Relative error of Legendre coefficients between original wavefront and reconstructed wavefront.
Fig. 5
Fig. 5 Experiment results. (a) CRSI interferogram. (b) Phase difference. (c) Phase difference after remove piston and tilt. (d) Reconstructed phase by the iterative algorithm proposed in [6]. (e) Reconstructed phase using Legendre polynomials. (f) The difference between the two reconstructed wavefronts by using two different methods.
Fig. 6
Fig. 6 The relationship between the noise coefficients and sampling points using Legendre polynomials over square area. Curves A, B, C, D corresponding to the shear ratio s = 0.75,0.6, 0.45,0.3, respectively.
Fig. 7
Fig. 7 The relationship between noise coefficients and terms of Legendre polynomials, shear ratio s = 0.75, 0.6, 0.45, 0.3,sampling points = 50*50.
Fig. 8
Fig. 8 The relationship between F matrix norm of T and sampling points. Curves A,B,C,D correspond to the shear ratio s = 0.75,0.6,0.45,0.3,respectively.

Tables (1)

Tables Icon

Table 1 The first 15 orthonormal Legendre polynomials for a system with square pupil.

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

Δ W ( x / s , y / s ) = W ( x / s , y / s ) W ( x s x 0 , y s y 0 ) .
Δ W ( x , y ) = W 1 ( x , y ) W 2 ( x , y ) .
1 1 P 1 ( x ) P k ( x ) d x = 2 2 l + 1 δ k l .
L j ( x , y ) = L l ( x ) L k ( y ) .
1 4 1 1 1 1 L j ( x , y ) L j ( x , y ) d x d y = δ j j .
W 1 ( x , y ) = k = 1 N a k L k ( x , y ) .
W 2 ( x , y ) = k = 1 N a k P k ( x , y ) .
P k ( x , y ) = j = 1 k b j L j ( x , y ) .
Δ W ( x , y ) = k = 1 N a k L k ( x , y ) k = 1 N a k j = 1 k b j L j ( x , y ) .
Δ W = L B A .
B T = [ 1 b 1 1 0 0 0 b 1 1 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 ] .
B = I - L + P .
C = B A .
A = B + C .
( Δ W + σ ) = L ( C + ξ ) .
ξ = L + σ .
A + ε = B + ( C + ξ ) .
ε = B + ξ = B + L + σ .
δ = L ε = L B + L + σ .
E w 1 = σ N ( i k E i k 2 ) = σ E F N .
e p = E F N .
Δ W = L C .
Δ W = [ L f L r ] [ C f C r ] .
C ^ f = L f + Δ W = L f + ( L f C f + L r C r ) = C f + L f + L r C r = C f + D C r .
D = L f + L r .
C = [ C f C r ] = [ B f 1 B f 2 B r 1 B r 2 ] [ a f a r ] = [ B f 1 B f 2 0 B r 2 ] [ a f a r ] .
C f = B f 1 a f + B f 2 a r . C r = B r 2 a r
a ^ f = B f 1 + C ^ f .
a ^ f = B f 1 + C ^ f = B f 1 + ( C f + L f + L r C r ) = B f 1 + ( B f 1 a f + B f 2 a r + L f + L r C r ) = a f + B f 1 + B f 2 a r + B f 1 + L f + L r B r 2 a r . = a f + B f 1 + B f 2 a r + B f 1 + D B r 2 a r
T = B f 1 + B f 2 + B f 1 + D B r 2 .
T = B f 1 + B f 2 .
a ^ f a f = T a r .
( Δ a 1 Δ a 2 Δ a f ) = ( T 11 T 12 T 1 r T 21 T 22 T 2 r T f 1 T f 2 T f r ) ( a f + 1 a f + 2 a f + r ) .
T n o r m = T F = ( i , j = 1 f , r ( T i j ) 2 ) 1 2 .

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