Abstract

A compact multiple-wave radial shearing interferometer (MWRSI) with strong adaptability, which is based on a Fresnel zone plate (FZP), is proposed. The nominally plane beam under test passes through the FZP and diffracts into multiple orders, including nominally plane beam, converging spherical beams, and diverging spherical beams. After propagating a distance, different apertures of diffraction orders cause multiple beams to meet and interfere in the imaging plane and form radial shear. The single interference pattern contains multiple wavefront differences in the overlapping region of beams. Then, a method that is employed in the wavefront difference retrieval from the MWRSI’s interferogram has been given. This MWRSI needs only an FZP and a CCD; therefore, it can be made very compact. It is suitable to measure the continuous and transient wavefront. The numerical simulations and experiments validate the proposed method.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. M. Wang, B. Zhang, S. Nie, and J. Fu, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
    [Crossref]
  2. D. Liu, Y. Yang, and Y. Shen, “System optimization of radial shearing interferometer for aspheric testing,” Proc. SPIE 6834, 68340U_1–8 (2007).
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    [Crossref] [PubMed]
  4. T. Shirai, “Liquid-crystal adaptive optics based on feedback interferometry for high-resolution retinal imaging,” Appl. Opt. 41(19), 4013–4023 (2002).
    [Crossref] [PubMed]
  5. D. Cheung, T. Barnes, and T. Haskell, “Feedback interferometry with membrane mirror for adaptive optics,” Opt. Commun. 218(1–3), 33–41 (2003).
    [Crossref]
  6. Y. Yang, Y. Lu, Y. Chen, Y. Zhuo, X. Zhang, B. Chen, and X. Qing, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distor-tion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
    [Crossref]
  7. W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part I – theoretical consideration,” Optik (Stuttg.) 113(1), 39–45 (2002).
    [Crossref]
  8. W. Kowalik, B Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part II – experiment results,” Optik (Stuttg.) 113(1), 39–45 (2002).
    [Crossref]
  9. W. Kowalik, B Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) 113(1), 39–45 (2002).
    [Crossref]
  10. M. V. R. K. Murty, “A Compact Radial Shearing Interferometer Based on the Law of Refraction,” Appl. Opt. 3(7), 853–858 (1964).
    [Crossref]
  11. D. Li, P. Wang, X. Li, H. Yang, and H. Chen, “Algorithm for near-field reconstruction based on radial-shearing interferometry,” Opt. Lett. 30(5), 492–494 (2005).
    [Crossref] [PubMed]
  12. N. Gu, L. Huang, Z. Yang, and C. Rao, “A single-shot common-path phase-stepping radial shearing interferometer for wavefront measurements,” Opt. Express 19(5), 4703–4713 (2011).
    [Crossref] [PubMed]
  13. E. Kewei, C. Zhang, M. Li, Z. Xiong, and D. Li, “Wavefront reconstruction algorithm based on Legendre polynomials for radial shearing interferometry over a square area and error analysis,” Opt. Express 23(16), 20267–20279 (2015).
    [Crossref] [PubMed]
  14. N. Gu, B. Yao, L. Huang, and C. Rao, “Compact single-shot radial shearing interferometer with random phase shift,” Opt. Lett. 42(18), 3622–3625 (2017).
    [Crossref] [PubMed]
  15. 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]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  21. T. M. Jeong, D. K. Ko, and J. Lee, “Method of reconstructing wavefront aberrations by use of Zernike polynomials in radial shearing interferometers,” Opt. Lett. 32(3), 232–234 (2007).
    [Crossref] [PubMed]
  22. N. Gu, L. Huang, Z. Yang, Q. Luo, and C. Rao, “Modal wavefront reconstruction for radial shearing interferometer with lateral shear,” Opt. Lett. 36(18), 3693–3695 (2011).
    [Crossref] [PubMed]
  23. D. Li, F. Wen, Q. Wang, Y. Zhao, F. Li, and B. Bao, “Improved formula of wavefront reconstruction from a radial shearing interferogram,” Opt. Lett. 33(3), 210–212 (2008).
    [Crossref] [PubMed]
  24. R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

2017 (1)

2015 (2)

2013 (1)

2011 (2)

2008 (1)

2007 (1)

2005 (2)

D. Li, P. Wang, X. Li, H. Yang, and H. Chen, “Algorithm for near-field reconstruction based on radial-shearing interferometry,” Opt. Lett. 30(5), 492–494 (2005).
[Crossref] [PubMed]

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

2003 (1)

D. Cheung, T. Barnes, and T. Haskell, “Feedback interferometry with membrane mirror for adaptive optics,” Opt. Commun. 218(1–3), 33–41 (2003).
[Crossref]

2002 (5)

T. Shirai, “Liquid-crystal adaptive optics based on feedback interferometry for high-resolution retinal imaging,” Appl. Opt. 41(19), 4013–4023 (2002).
[Crossref] [PubMed]

M. Wang, B. Zhang, S. Nie, and J. Fu, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
[Crossref]

W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part I – theoretical consideration,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

W. Kowalik, B Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part II – experiment results,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

W. Kowalik, B Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

2000 (1)

1999 (2)

J. Strand and T. Taxt, “Performance evaluation of two-dimensional phase unwrapping algorithms,” Appl. Opt. 38(20), 4333–4344 (1999).
[Crossref] [PubMed]

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

1997 (1)

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial Shearing Interferometer For In-process Measurement of Diamond Turning,” Proc. SPIE 3173, 280–285 (1997).
[Crossref]

1974 (1)

1964 (1)

Bao, B.

Barnes, T.

D. Cheung, T. Barnes, and T. Haskell, “Feedback interferometry with membrane mirror for adaptive optics,” Opt. Commun. 218(1–3), 33–41 (2003).
[Crossref]

Barnes, T. H.

Chen, B.

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

Chen, H.

Chen, Y.

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

Cheung, D.

D. Cheung, T. Barnes, and T. Haskell, “Feedback interferometry with membrane mirror for adaptive optics,” Opt. Commun. 218(1–3), 33–41 (2003).
[Crossref]

Fu, J.

M. Wang, B. Zhang, S. Nie, and J. Fu, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
[Crossref]

Garncarz, B

W. Kowalik, B Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part II – experiment results,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

W. Kowalik, B Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

Garncarz, B.

W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part I – theoretical consideration,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

Gu, N.

Haskell, T.

D. Cheung, T. Barnes, and T. Haskell, “Feedback interferometry with membrane mirror for adaptive optics,” Opt. Commun. 218(1–3), 33–41 (2003).
[Crossref]

Haskell, T. G.

Huang, L.

Jeong, T. M.

Kasprzak, H.

W. Kowalik, B Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part II – experiment results,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part I – theoretical consideration,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

W. Kowalik, B Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

Kewei, E.

Ko, D. K.

Kohno, T.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial Shearing Interferometer For In-process Measurement of Diamond Turning,” Proc. SPIE 3173, 280–285 (1997).
[Crossref]

Kowalik, W.

W. Kowalik, B Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

W. Kowalik, B Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part II – experiment results,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part I – theoretical consideration,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

Lee, J.

Li, D.

Li, F.

Li, M.

Li, X.

Ling, T.

Liu, D.

Lu, Y.

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

Luo, Q.

Marchetti, E.

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

Matsumoto, D.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial Shearing Interferometer For In-process Measurement of Diamond Turning,” Proc. SPIE 3173, 280–285 (1997).
[Crossref]

Murty, M. V. R. K.

Nie, S.

M. Wang, B. Zhang, S. Nie, and J. Fu, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
[Crossref]

Qing, X.

Y. Yang, Y. Lu, Y. Chen, Y. Zhuo, X. Zhang, B. Chen, and X. Qing, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distor-tion 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).

Shen, Y.

Shirai, T.

Smartt, R. N.

Strand, J.

Sun, L.

Taxt, T.

Tian, C.

Uda, Y.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial Shearing Interferometer For In-process Measurement of Diamond Turning,” Proc. SPIE 3173, 280–285 (1997).
[Crossref]

Wang, M.

M. Wang, B. Zhang, S. Nie, and J. Fu, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
[Crossref]

Wang, P.

Wang, Q.

Wen, F.

Xie, C.

Xiong, Z.

Yang, H.

Yang, Y.

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]

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

Yang, Z.

Yao, B.

Yazawa, T.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial Shearing Interferometer For In-process Measurement of Diamond Turning,” Proc. SPIE 3173, 280–285 (1997).
[Crossref]

Zhang, B.

M. Wang, B. Zhang, S. Nie, and J. Fu, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
[Crossref]

Zhang, C.

Zhang, X.

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

Zhang, Y.

Zhao, Y.

Zhuo, Y.

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

Appl. Opt. (4)

Astron. Astrophys. (1)

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

Opt. Commun. (1)

D. Cheung, T. Barnes, and T. Haskell, “Feedback interferometry with membrane mirror for adaptive optics,” Opt. Commun. 218(1–3), 33–41 (2003).
[Crossref]

Opt. Express (2)

Opt. Lett. (8)

N. Gu, B. Yao, L. Huang, and C. Rao, “Compact single-shot radial shearing interferometer with random phase shift,” Opt. Lett. 42(18), 3622–3625 (2017).
[Crossref] [PubMed]

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]

T. M. Jeong, D. K. Ko, and J. Lee, “Method of reconstructing wavefront aberrations by use of Zernike polynomials in radial shearing interferometers,” Opt. Lett. 32(3), 232–234 (2007).
[Crossref] [PubMed]

N. Gu, L. Huang, Z. Yang, Q. Luo, and C. Rao, “Modal wavefront reconstruction for radial shearing interferometer with lateral shear,” Opt. Lett. 36(18), 3693–3695 (2011).
[Crossref] [PubMed]

D. Li, F. Wen, Q. Wang, Y. Zhao, F. Li, and B. Bao, “Improved formula of wavefront reconstruction from a radial shearing interferogram,” Opt. Lett. 33(3), 210–212 (2008).
[Crossref] [PubMed]

D. Li, P. Wang, X. Li, H. Yang, and H. Chen, “Algorithm for near-field reconstruction based on radial-shearing interferometry,” Opt. Lett. 30(5), 492–494 (2005).
[Crossref] [PubMed]

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]

Y. Zhang and C. Xie, “Differential-interference-contrast digital in-line holography microscopy based on a single-optical-element,” Opt. Lett. 40(21), 5015–5018 (2015).
[Crossref] [PubMed]

Optik (Stuttg.) (3)

W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part I – theoretical consideration,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

W. Kowalik, B Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part II – experiment results,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

W. Kowalik, B Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) 113(1), 39–45 (2002).
[Crossref]

Proc. SPIE (3)

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

M. Wang, B. Zhang, S. Nie, and J. Fu, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
[Crossref]

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial Shearing Interferometer For In-process Measurement of Diamond Turning,” Proc. SPIE 3173, 280–285 (1997).
[Crossref]

Other (2)

D. Liu, Y. Yang, and Y. Shen, “System optimization of radial shearing interferometer for aspheric testing,” Proc. SPIE 6834, 68340U_1–8 (2007).

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, Second Edition, (2005).

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Figures (9)

Fig. 1
Fig. 1 Principle diagram of multiple-wave radial shearing interferometry based on a Fresnel zone plate (Only 0, ± 1, and ± 3 orders are drawn in the diagram.).
Fig. 2
Fig. 2 Optical layout of the RSI for wavefront testing. (a) Schematic diagram of the Mach-Zehnder RSI, (b) schematic diagram of the simplified path of compressed wavefront in the Mach-Zehnder RSI, (c) schematic diagram of the mth diffraction order in MWRSI (m > 0 and z < f/m).
Fig. 3
Fig. 3 The relationship between s and z.
Fig. 4
Fig. 4 One-dimensional normalized frequency distribution diagram of I / ' ( r , θ , z ) .
Fig. 5
Fig. 5 Wavefront reconstruction results by MWRSI method (unit: λ). (a1) - (a4) Random wavefronts under test. (b1) - (b4) Corresponding interferograms generated by angular spectrum theory. (c1) - (c4) Unwrapped wavefront differences after removing tilt. (d1) - (d4) Reconstructed results by MWRSI. (e1) - (e4) Residual errors between the origin wavefronts and the reconstructed wavefronts.
Fig. 6
Fig. 6 Wavefronts coefficient vectors. (a) Incident wavefront coefficient vector for each order of Zernike polynomials. (b) Coefficient vector of Zernike polynomials for wavefront difference. (c) Coefficient vector of Zernike polynomials for reconstruction wavefront. (d) Coefficient vector of Zernike polynomials for residual error.
Fig. 7
Fig. 7 Experiment setup for wavefront retrieval based on an FZP.
Fig. 8
Fig. 8 Interferogram generated by MWRSI. (a) The interferogram of the collimated beam, (b) the interferogram of the collimated beam passing through the optical flat.
Fig. 9
Fig. 9 Comparative experiment result with a ZYGO interferometer (unit: λ). (a) Wavefront difference extracted from interferogram generated by MWRSI, (b) reconstructed wavefront by MWRSI, (c) the testing result of the same optical flat with a ZYGO interferometer.

Equations (25)

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t F Z P ( r ) = n = c n exp ( j n π r 2 d ) ,
U i ( r , θ ) = A i ( r , θ ) exp [ j k W ( r , θ ) ] ,
U t ( r , θ ) = U i ( r , θ ) t F Z P ( r ) .
U t ( r , θ ) = n = c n A i ( r , θ ) exp [ j k W ( r , θ ) ] exp ( j n π r 2 d ) .
U 1 ( r , θ ) = A 1 ( r , θ ) exp [ j k W ( r , θ ) ] exp ( j k r 2 2 f a ) ,
U 2 ( r , θ ) = A 2 ( r , θ ) exp [ j k W ( α r , θ ) ] exp ( j k r 2 2 f b ) , = A 2 ( r , θ ) exp [ j k W ( α r , θ ) ] exp ( j k α r 2 2 f a )
U t m ( r , θ ) = c m A i ( r , θ ) exp [ j k W ( r , θ ) ] exp ( j m π r 2 d ) .
U z m ( r , θ , z ) = c m A m ( r , θ , z ) exp [ j k W ( α m r , θ ) ] exp ( j α m m π r 2 d ) ,
U z ( r , θ , z ) = n = c n A n ( r , θ , z ) exp [ j k W ( α n r , θ ) ] exp ( j α n n π r 2 d ) ,
α n = { 1 , n = 0 ( f / n ) / ( f / n z ) , n 0 .
I ( r , θ , z ) = U z ( r , θ , z ) U z * ( r , θ , z ) = n = , m = n c n A n ( r , θ , z ) c m A m ( r , θ , z ) + n = , m < n 2 c n A n ( r , θ , z ) c m A m ( r , θ , z ) . cos [ k W ( α n r , θ ) k W ( α m r , θ ) α n n π r 2 d + α m m π r 2 d ] ,
I ( r , θ , z ) = a ( r , θ , z ) + b 1 , 1 ( r , θ , z ) cos [ k Δ W 1 , 1 ( r , θ , z ) α 1 π r 2 d α 1 π r 2 d ] , + n = , m < n , [ n , m ] [ 1 , 1 ] b n , m ( r , θ , z ) cos [ k Δ W n , m ( r , θ , z ) α n n π r 2 d + α m m π r 2 d ]
{ a ( r , θ , z ) = n = , m = n c n A n ( r , θ , z ) c m A m ( r , θ , z ) , b n , m ( r , θ , z ) = 2 c n A n ( r , θ , z ) c m A m ( r , θ , z ) , Δ W n , m ( r , θ , z ) = W ( α n r , θ ) W ( α m r , θ ) .
I l ( r , θ , z ) = 1 + cos ( φ l α 1 π r 2 d α 1 π r 2 d ) ,
I / ' ( r , θ , z ) = a ( r , θ , z ) + 1 2 b 1 , 1 ( r , θ , z ) cos [ k Δ W 1 , 1 ( r , θ , z ) φ l ] + a ( r , θ , z ) cos [ φ l ( α 1 π d + α 1 π d ) r 2 ] + b 1 , 1 ( r , θ , z ) cos [ k Δ W 1 , 1 ( r , θ , z ) ( α 1 π d + α 1 π d ) r 2 ] + n = , m < n , [ n , m ] [ 1 , 1 ] b n , m ( r , θ , z ) cos [ k Δ W n , m ( r , θ , z ) ( α n n π d α m m π d ) r 2 ] . + 1 2 b 1 , 1 ( r , θ , z ) cos [ k Δ W 1 , 1 ( r , θ , z ) ( 2 α 1 π d + 2 α 1 π d ) r 2 + φ l ] + n = , m < n , [ n , m ] [ 1 , 1 ] 1 2 b n , m ( r , θ , z ) cos [ k Δ W n , m ( r , θ , z ) ( α n n π d α m m π d + α 1 π d + α 1 π d ) r 2 + φ l ] + n = , m < n , [ n , m ] [ 1 , 1 ] 1 2 b n , m ( r , θ , z ) cos [ k Δ W n , m ( r , θ , z ) ( α n n π d α m m π d α 1 π d α 1 π d ) r 2 φ l ]
I / ' ' ( r , θ , z ) = H ( r , θ ) I / ' ( r , θ , z ) ,
I / ' ' ( r , θ , z ) = a ( r , θ , z ) + 1 2 b 1 , 1 ( r , θ , z ) cos [ k Δ W 1 , 1 ( r , θ , z ) φ l ] .
Δ W 1 , 1 ( r , θ , z ) = 1 k arc tan ( I 2 ' ' I 4 ' ' I 1 ' ' I 3 ' ' ) .
{ W 1 ( r , θ ) = μ = 1 N a μ Z μ ( r , θ ) W 2 ( r , θ ) = μ = 1 N a μ P μ ( r , θ ) ,
P μ ( r , θ ) = ν = 1 μ b ν μ Z ν ( r , θ ) .
Δ W ( r , θ ) = W 1 ( r , θ ) W 2 ( r , θ ) = μ = 1 N a μ Z μ ( r , θ ) μ = 1 N a μ [ ν = 1 μ b ν μ Z ν ( r , θ ) ] .
Δ W ( r , θ ) = A B Z ,
B = [ 1 b 1 1 0 0 0 b 1 2 1 b 2 2 0 0 0 0 b 1 μ b 2 μ 1 b μ μ 0 0 b 1 N b 2 N b μ N 1 b N N ] .
Δ W ( r , θ ) = C Z ,
A = C B + ,

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