Abstract

A single-shot common-path phase-stepping radial shearing interferometer (RSI) is proposed for wavefront measurements. In the proposed RSI, three quarter-wave plates are used as phase shifters to produce four spatially separated phase-stepping fringe patterns that are recorded simultaneously by a single CCD camera. The proposed RSI can measure the wavefront under test in real-time, and it is also insensitive to environmental vibration due to its common-path structure. Experimentally the proposed RSI is applied to detect the distorted wavefronts generated by a liquid crystal spatial light modulator. The measured aberrations are in good agreement with that obtained with (by) a Hartmann-Shack wavefront sensor, indicating that the proposed RSI is a useful tool for wavefront measurements.

© 2011 OSA

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    [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).
  3. M. Wang, B. Zhang, and N. Shouping, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
    [CrossRef]
  4. T. Kohno, D. Matsumoto, and T. Yazawa, “Radial Shearing Interferometer For In-process Measurement of Diamond Turning,” Proc. SPIE 3173, 280–285 (1997).
    [CrossRef]
  5. B. Zhang, L. Ma, M. Wang, and A. He, “Aspheric lens testing by means of compact radial shearing interferometer with two zone plates,” Laser Technol. 31, 37–46 (2007) (in Chinese).
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    [CrossRef]
  7. W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part II – experiment results,” Optik (Stuttg.) 113(1), 46–50 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  13. Y. Yang, Y. Lu, and Y. Chen, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  37. X. Li and C. Wang, “H. XIAN and W. JIANG, “Real-time modal reconstruction algorithm for adaptive optics systems,” Laser and Part. Beams 11, 53–56 (2002) (in Chinese).

2010

F. Bai and C. Rao, “Experimental validation of closed-loop adaptive optics based on a self-referencing interferometer wavefront sensor and a liquid-crystal spatial light modulator,” Opt. Commun. 283(14), 2782–2786 (2010).
[CrossRef]

2009

2007

2006

2005

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, and Y. Chen, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[CrossRef]

2004

2003

C. Dunsby, Y. Gu, and P. French, “Single-shot phase-stepped wide-field coherencegated imaging,” Opt. Express 11(2), 105–115 (2003).
[CrossRef] [PubMed]

W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) 114(5), 199–206 (2003).
[CrossRef]

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

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), 46–50 (2002).
[CrossRef]

M. Wang, B. Zhang, and N. Shouping, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
[CrossRef]

X. Li and C. Wang, “H. XIAN and W. JIANG, “Real-time modal reconstruction algorithm for adaptive optics systems,” Laser and Part. Beams 11, 53–56 (2002) (in Chinese).

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

2000

1999

1997

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

1990

1989

1986

1985

1974

1973

1972

1970

1964

1961

P. Hariharan and D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38(11), 428–432 (1961).
[CrossRef]

Bai, F.

F. Bai and C. Rao, “Experimental validation of closed-loop adaptive optics based on a self-referencing interferometer wavefront sensor and a liquid-crystal spatial light modulator,” Opt. Commun. 283(14), 2782–2786 (2010).
[CrossRef]

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.

Bryngdahl, O.

Cai, L. Z.

Chang, C. C.

Chen, H.

Chen, S. J.

Chen, Y.

Y. Yang, Y. Lu, and Y. Chen, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion 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]

Cho, K. C.

Chung, C. Y.

Collier, J. L.

Danson, C. N.

Delisle, C.

Dong, G. Y.

Dunsby, C.

Edwards, C. B.

Ezawa, T.

French, P.

Garncarz, B.

W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) 114(5), 199–206 (2003).
[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), 46–50 (2002).
[CrossRef]

Gu, Y.

Hariharan, P.

P. Hariharan and D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38(11), 428–432 (1961).
[CrossRef]

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.

Hawkes, S. J.

He, A.

B. Zhang, L. Ma, M. Wang, and A. He, “Aspheric lens testing by means of compact radial shearing interferometer with two zone plates,” Laser Technol. 31, 37–46 (2007) (in Chinese).

Hernandez-Gomez, C.

Honda, T.

Jeong, T. M.

Joenathan, C.

Joenathan, C. J.

Kasprzak, H.

W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) 114(5), 199–206 (2003).
[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), 46–50 (2002).
[CrossRef]

Ko, D. K.

Kohno, T.

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

Kothiyal, M. P.

Kowalik, W.

W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) 114(5), 199–206 (2003).
[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), 46–50 (2002).
[CrossRef]

Lee, J.

Li, D.

Li, X.

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]

X. Li and C. Wang, “H. XIAN and W. JIANG, “Real-time modal reconstruction algorithm for adaptive optics systems,” Laser and Part. Beams 11, 53–56 (2002) (in Chinese).

Lin, C. H.

Liu, D.

Liu, J. P.

Lu, Y.

Y. Yang, Y. Lu, and Y. Chen, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[CrossRef]

Ma, L.

B. Zhang, L. Ma, M. Wang, and A. He, “Aspheric lens testing by means of compact radial shearing interferometer with two zone plates,” Laser Technol. 31, 37–46 (2007) (in Chinese).

Matsumoto, D.

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

Meng, X. F.

Mihaylova, E.

Miyamoto, Y.

Mohanty, R. K.

Murty, M.

Murty, M. V.

Naik, D. N.

Notaras, J.

Ohyama, N.

Paterson, C.

Pepler, D. A.

Poon, T. C.

Rao, C.

F. Bai and C. Rao, “Experimental validation of closed-loop adaptive optics based on a self-referencing interferometer wavefront sensor and a liquid-crystal spatial light modulator,” Opt. Commun. 283(14), 2782–2786 (2010).
[CrossRef]

Ross, I. N.

Ru, Q.-S.

Sen, D.

P. Hariharan and D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38(11), 428–432 (1961).
[CrossRef]

Shen, X. X.

Shirai, T.

Shouping, N.

M. Wang, B. Zhang, and N. Shouping, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
[CrossRef]

Shukla, R. P.

Silva, D. E.

Sirohi, R. S.

Smartt, R. N.

Strand, J.

Takeda, M.

Taxt, T.

Toal, V.

Torroba, R.

Tsujiuchi, J.

Wang, C.

X. Li and C. Wang, “H. XIAN and W. JIANG, “Real-time modal reconstruction algorithm for adaptive optics systems,” Laser and Part. Beams 11, 53–56 (2002) (in Chinese).

Wang, L.

Wang, M.

B. Zhang, L. Ma, M. Wang, and A. He, “Aspheric lens testing by means of compact radial shearing interferometer with two zone plates,” Laser Technol. 31, 37–46 (2007) (in Chinese).

M. Wang, B. Zhang, and N. Shouping, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
[CrossRef]

Wang, P.

Wang, Y. R.

Whelan, M.

Winstone, T. B.

Wizinowich, P. L.

Xu, X. F.

Yang, H.

Yang, X. L.

Yang, Y.

D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. 46(34), 8305–8314 (2007).
[CrossRef] [PubMed]

Y. Yang, Y. Lu, and Y. Chen, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[CrossRef]

Yazawa, T.

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

Yen, W. C.

Zhang, B.

B. Zhang, L. Ma, M. Wang, and A. He, “Aspheric lens testing by means of compact radial shearing interferometer with two zone plates,” Laser Technol. 31, 37–46 (2007) (in Chinese).

M. Wang, B. Zhang, and N. Shouping, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
[CrossRef]

Zhuo, Y.

Appl. Opt.

M. Murty, “A Compact Radial Shearing Interferometer Based on the Law of Refraction,” Appl. Opt. 3(7), 853–858 (1964).
[CrossRef]

M. V. Murty and R. P. Shukla, “Radial shearing interferometers using a laser source,” Appl. Opt. 12(11), 2765–2767 (1973).
[CrossRef] [PubMed]

R. N. Smartt, “Zone plate interferometer,” Appl. Opt. 13(5), 1093–1099 (1974).
[CrossRef] [PubMed]

M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24(24), 4439–4447 (1985).
[CrossRef] [PubMed]

R. K. Mohanty, C. J. Joenathan, and R. S. Sirohi, “High sensitivity tilt measurement by speckle shear interferometry,” Appl. Opt. 25(10), 1661–1664 (1986).
[CrossRef] [PubMed]

Q.-S. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, “Constant radial shearing interferometry with circular gratings,” Appl. Opt. 28(16), 3350–3353 (1989).
[CrossRef] [PubMed]

P. L. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29(22), 3271–3279 (1990).
[CrossRef] [PubMed]

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

C. Hernandez-Gomez, J. L. Collier, S. J. Hawkes, C. N. Danson, C. B. Edwards, D. A. Pepler, I. N. Ross, and T. B. Winstone, “Wave-front control of a large-aperture laser system by use of a static phase corrector,” Appl. Opt. 39(12), 1954–1961 (2000).
[CrossRef]

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

D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11(11), 2613–2624 (1972).
[CrossRef] [PubMed]

C. Y. Chung, K. C. Cho, C. C. Chang, C. H. Lin, W. C. Yen, and S. J. Chen, “Adaptive-optics system with liquid-crystal phase-shift interferometer,” Appl. Opt. 45(15), 3409–3414 (2006).
[CrossRef] [PubMed]

D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. 46(34), 8305–8314 (2007).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Sci. Instrum.

P. Hariharan and D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38(11), 428–432 (1961).
[CrossRef]

Laser and Part. Beams

X. Li and C. Wang, “H. XIAN and W. JIANG, “Real-time modal reconstruction algorithm for adaptive optics systems,” Laser and Part. Beams 11, 53–56 (2002) (in Chinese).

Laser Technol.

B. Zhang, L. Ma, M. Wang, and A. He, “Aspheric lens testing by means of compact radial shearing interferometer with two zone plates,” Laser Technol. 31, 37–46 (2007) (in Chinese).

Opt. Commun.

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

F. Bai and C. Rao, “Experimental validation of closed-loop adaptive optics based on a self-referencing interferometer wavefront sensor and a liquid-crystal spatial light modulator,” Opt. Commun. 283(14), 2782–2786 (2010).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Stuttg.)

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), 46–50 (2002).
[CrossRef]

W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) 114(5), 199–206 (2003).
[CrossRef]

Proc. SPIE

M. Wang, B. Zhang, and N. Shouping, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002).
[CrossRef]

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

Y. Yang, Y. Lu, and Y. Chen, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005).
[CrossRef]

Other

D. Yu, and H. Tan, Engineering Optics, China machine press, Beijing, chapter 15th, pp: 310, 422, 440–444(2006)(in Chinese).

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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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the experimental setup with the proposed single shot common-path phase-stepping RSI. L1-L8, lenses; PBS1 and PBS2, polarizing beam splitters; BS1-BS4, beam splitters; M1-M6, reflectors; QW1-QW3, quarter-wave plates; A1 and A2, diaphragms; P1 and P2, polarizers; HS, Hartmann-Shack wavefront sensor.

Fig. 2
Fig. 2

the change of fringe visibility K with the angle θ of P2 for s = 1.2.

Fig. 3
Fig. 3

(a).the change of the best angle θ with different radial shear s changing from 1.1 to 5.0. (b). the change of fringe visibility K with the angle θ of P2 for some different radial shear values.

Fig. 4
Fig. 4

phase shift of 0° through 270° in 90°step in the phase stepped SRI.

Fig. 5
Fig. 5

The wavefront difference calculated from the four interferograms (shown in Fig. 4).(a). The wrapped wavefront difference; (b). the unwrapped wavefront difference.

Fig. 6
Fig. 6

The wavefront under test measured by the proposed RSI. (a). the 3D plot of the wavefront under test reconstructed from the wavefront difference shown in Fig. 5(b); (b). the coefficients of the first 45order Zernike polynomials.

Fig. 7
Fig. 7

The experiment results measured by HS WFS. (a). the wavefront under test reconstructed from the spot array graph; (b). the coefficients of the first 45order Zernike polynomials for the wavefront under test measured by HS WFS.

Fig. 8
Fig. 8

The residual errors between the wavefronts under test measured by the proposed RSI and the HS WFS. (a). 3D plot of the residual errors; (b). the coefficients of the first 45order Zernike polynomials for the residual errors.

Equations (15)

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E 0 ( x , y ) = A 0 ( x , y ) exp ( i φ 0 ( x , y ) ) .
{ E 1 ( x , y ) = E 0 ( x / s , y / s ) = A 0 ( x / s , y / s ) exp ( i φ 0 ( x / s , y / s ) ) = A 1 ( x , y ) exp ( i φ 1 ( x , y ) ) E 2 ( x , y ) = E 0 ( x s , y s ) = A 0 ( x s , y s ) exp ( i φ 0 ( x s , y s ) ) = A 2 ( x , y ) exp ( i φ 2 ( x , y ) ) ,
{ A 1 ( x , y ) = A 0 ( x / s , y / s ) , φ 1 ( x , y ) = φ 0 ( x / s , y / s ) A 2 ( x , y ) = A 0 ( x s , y s ) , φ 2 ( x , y ) = φ 0 ( x s , y s ) .
P 0 = [ 1 0 0 0 ] , P 90 = [ 0 0 0 1 ] , Q 0 = [ 1 0 0 i ] , Q 45 = 1 2 [ 1 i i 1 ] .
a 1 = P 0 Q 45 Q 0 , b 1 = P 0 Q 45 , a 2 = P 90 Q 45 Q 0 , b 2 = P 90 Q 45 .
a 1 = 1 2 [ 1 1 0 0 ] , b 1 = 1 2 [ 1 i 0 0 ] , a 2 = 1 2 [ 0 0 i i ] , b 2 = 1 2 [ 0 0 i 1 ] .
Δ φ ( x , y ) = φ 2 ( x , y ) φ 1 ( x , y ) = φ 0 ( x s , y s ) φ 0 ( x / s , y / s ) .
I a 1 = A 1 2 / 2 + A 2 2 / 2 + A 1 A 2 cos ( Δ φ ) I b 1 = A 1 2 / 2 + A 2 2 / 2 + A 1 A 2 cos ( Δ φ + π / 2 ) I a 2 = A 1 2 / 2 + A 2 2 / 2 + A 1 A 2 cos ( Δ φ + π ) I b 2 = A 1 2 / 2 + A 2 2 / 2 + A 1 A 2 cos ( Δ φ + 3 π / 2 ) ,
Δ φ ( x , y ) = arctan [ I b 2 ( x , y ) I b 1 ( x , y ) I a 1 ( x , y ) I a 2 ( x , y ) ] .
{ I t = I 0 cos 2 ( θ ) I r = I 0 cos 2 ( π 2 θ ) = I 0 sin 2 ( θ ) .
{ I t ' = I 0 cos 2 ( θ ) s 2 η I r ' = I 0 sin 2 ( θ ) / s 2 η
I = I t ' + I r ' + 2 I t ' I r ' cos ( Δ φ ) ,
K = I m a x I m i n I m a x + I m i n = 2 I t ' I r ' I t ' + I r ' .
K = 2 cos ( θ ) sin ( θ ) cos 2 ( θ ) s 2 + sin 2 ( θ ) / s 2 = 2 s 2 tan ( θ ) + tan ( θ ) s 2 .
s 2 tan ( θ ) = tan ( θ ) s 2 , i . e . , θ = arctan ( s 2 ) .

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