Abstract

In order to measure the phase projection in moiré tomography, in this paper we present a new spatial phase-shifting shearing interferometry that consists only of a crossed grating and a linear grating. With it, six phase-shifted interferograms can be acquired simultaneously. The intensity distributions of these six interferograms are derived based on the scalar diffraction theory. Analytical results indicate that phase shifts are symmetrically and consistently distributed in the vertical and horizontal directions. Moreover, phase-shift values depend primarily on the parameters of the optical structure. And a six-step phase-shifting algorithm with arbitrary phase-shift values in two perpendicular directions is proposed to extract the phase information. The phase retrieval results of the spherical wave have verified the validity of the proposed method. Finally, an experiment with a plane incident wave is performed to measure the first-order derivative of the phase projection of a propane flame.

© 2013 Optical Society of America

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  1. O. Kafri, “Noncoherent method for mapping phase objects,” Opt. Lett. 5, 555–557 (1980).
    [CrossRef]
  2. E. Keren, E. Bar-Ziv, I. Glatt, and O. Kafri, “Measurements of temperature distribution of flames by moiré deflectometry,” Appl. Opt. 20, 4263–4266 (1981).
    [CrossRef]
  3. Y. Song, B. Zhang, and A. He, “Algebraic iterative algorithm for deflection tomography and its application to density flow fields in a hypersonic wind tunnel,” Appl. Opt. 45, 8092–8101 (2006).
    [CrossRef]
  4. Y. Chen, Y. Song, A. He, and Z. Li, “Applicability of moiré deflection tomography for diagnosing arc plasmas,” Appl. Opt. 48, 489–496 (2009).
    [CrossRef]
  5. Y. Chen, S. Yang, Z. Li, and A. He, “A model for arc plasma’s optical diagnosis by the measurement of the refractive index,” Opt. Commun. 284, 2648–2652 (2011).
    [CrossRef]
  6. N. Sun, Y. Song, J. Wang, Z. Li, and A. He, “Volume moiré tomography based on double cross gratings for real three-dimension flow field diagnosis,” Appl. Opt. 51, 8081–8089 (2012).
    [CrossRef]
  7. B. Zhang, Y. Song, and A. He, “New reconstruction algorithm for moiré tomography in flow field measurements,” Opt. Eng. 45, 117002 (2006).
    [CrossRef]
  8. J. Stricker, E. Keren, and O. Kafri, “Axisymmetric density field measurements by moiré deflectometry,” AIAA J. 21, 1767–1769 (1983).
    [CrossRef]
  9. V. Vlad, D. Popa, and I. Apostol, “Computer moiré deflectometry using the Talbot effect,” Opt. Eng. 30, 300–306 (1991).
    [CrossRef]
  10. M. Servin, R. Rodriguez-Vera, M. Carpio, and A. Morale, “Automatic fringe detection algorithm used for moiré deflectometry,” Appl. Opt. 29, 3266–3270 (1990).
    [CrossRef]
  11. J. Zhong and M. Wang, “Fourier transform moiré deflectometry for the automatic measurement of phase objects,” Proc. SPIE 2899, 311–318 (1996).
    [CrossRef]
  12. J. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
    [CrossRef]
  13. C. Koliopoulos, “Phase shifting techniques applied to unique applications,” Proc. SPIE 2861, 86–93 (1996).
    [CrossRef]
  14. C. Koliopoulos, “Simultaneous phase shift interferometer,” Proc. SPIE 1531, 119–127 (1991).
    [CrossRef]
  15. O. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9, 59–61 (1984).
    [CrossRef]
  16. H. Bruning, D. Herriott, J. Gallagher, D. Rosenfeld, A. White, and D. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef]
  17. H. Schreiber and J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36, 5321–5324 (1997).
    [CrossRef]
  18. R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 234361 (1984).
    [CrossRef]
  19. L. Deck, “Vibration-resistant phase-shifting interometry,” Appl. Opt. 35, 6655–6662 (1996).
    [CrossRef]
  20. S. Wolfling and E. Lanzmann, “Wavefront reconstruction by spatial-phase-shift imaging interometry,” Appl. Opt. 45, 2586–2596 (2006).
    [CrossRef]
  21. B. Ngoi, K. Venkatakrishnan, and N. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. 40, 3211–3214 (2001).
    [CrossRef]
  22. M. Kujawinska, L. Salbut, and K. Patorski, “Three-channel phase stepped system for moire interferometry,” Appl. Opt. 30, 1633–1637 (1991).
    [CrossRef]
  23. N. I. Toto-Arellano, A. Martínez-García, G. Rodríguez-Zurita, J. Rayas-Álvarez, and A. Montes-Perez, “Slope measurement of a phase object using a polarizing phase-shifting high-frequency Ronchi grating interferometer,” Appl. Opt. 49, 6402–6408 (2010).
    [CrossRef]
  24. Y. Song, Y. Chen, A. He, and Z. Zhao, “Spatial phase-shifting characteristic of double grating interferometer,” Opt. Express 17, 20415–20429 (2009).
    [CrossRef]
  25. Y. Song, Y. Chen, J. Wang, N. Sun, and A. He, “Four-step spatial phase-shifting shearing interferometry from moire configuration by triple gratings,” Opt. Lett. 37, 1922–1924 (2012).
    [CrossRef]
  26. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  27. M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
    [CrossRef]

2012

2011

Y. Chen, S. Yang, Z. Li, and A. He, “A model for arc plasma’s optical diagnosis by the measurement of the refractive index,” Opt. Commun. 284, 2648–2652 (2011).
[CrossRef]

2010

2009

2006

2001

1999

J. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

1997

1996

L. Deck, “Vibration-resistant phase-shifting interometry,” Appl. Opt. 35, 6655–6662 (1996).
[CrossRef]

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
[CrossRef]

C. Koliopoulos, “Phase shifting techniques applied to unique applications,” Proc. SPIE 2861, 86–93 (1996).
[CrossRef]

J. Zhong and M. Wang, “Fourier transform moiré deflectometry for the automatic measurement of phase objects,” Proc. SPIE 2899, 311–318 (1996).
[CrossRef]

1991

V. Vlad, D. Popa, and I. Apostol, “Computer moiré deflectometry using the Talbot effect,” Opt. Eng. 30, 300–306 (1991).
[CrossRef]

C. Koliopoulos, “Simultaneous phase shift interferometer,” Proc. SPIE 1531, 119–127 (1991).
[CrossRef]

M. Kujawinska, L. Salbut, and K. Patorski, “Three-channel phase stepped system for moire interferometry,” Appl. Opt. 30, 1633–1637 (1991).
[CrossRef]

1990

1984

O. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9, 59–61 (1984).
[CrossRef]

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 234361 (1984).
[CrossRef]

1983

J. Stricker, E. Keren, and O. Kafri, “Axisymmetric density field measurements by moiré deflectometry,” AIAA J. 21, 1767–1769 (1983).
[CrossRef]

1981

1980

1974

Apostol, I.

V. Vlad, D. Popa, and I. Apostol, “Computer moiré deflectometry using the Talbot effect,” Opt. Eng. 30, 300–306 (1991).
[CrossRef]

Bar-Ziv, E.

Bernabeu, E.

J. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Brangaccio, D.

Bruning, H.

Carpio, M.

Chen, Y.

Crespo, D.

J. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

Deck, L.

Gallagher, J.

Glatt, I.

He, A.

Herriott, D.

Kafri, O.

Keren, E.

J. Stricker, E. Keren, and O. Kafri, “Axisymmetric density field measurements by moiré deflectometry,” AIAA J. 21, 1767–1769 (1983).
[CrossRef]

E. Keren, E. Bar-Ziv, I. Glatt, and O. Kafri, “Measurements of temperature distribution of flames by moiré deflectometry,” Appl. Opt. 20, 4263–4266 (1981).
[CrossRef]

Koliopoulos, C.

C. Koliopoulos, “Phase shifting techniques applied to unique applications,” Proc. SPIE 2861, 86–93 (1996).
[CrossRef]

C. Koliopoulos, “Simultaneous phase shift interferometer,” Proc. SPIE 1531, 119–127 (1991).
[CrossRef]

Kujawinska, M.

Kwon, O.

Lanzmann, E.

Li, Z.

Martínez-García, A.

Montes-Perez, A.

Moore, R.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 234361 (1984).
[CrossRef]

Morale, A.

Ngoi, B.

Patorski, K.

Popa, D.

V. Vlad, D. Popa, and I. Apostol, “Computer moiré deflectometry using the Talbot effect,” Opt. Eng. 30, 300–306 (1991).
[CrossRef]

Pritt, M. D.

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
[CrossRef]

Quiroga, J.

J. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

Rayas-Álvarez, J.

Rodriguez-Vera, R.

Rodríguez-Zurita, G.

Rosenfeld, D.

Salbut, L.

Schreiber, H.

Schwider, J.

Servin, M.

Sivakumar, N.

Smythe, R.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 234361 (1984).
[CrossRef]

Song, Y.

Stricker, J.

J. Stricker, E. Keren, and O. Kafri, “Axisymmetric density field measurements by moiré deflectometry,” AIAA J. 21, 1767–1769 (1983).
[CrossRef]

Sun, N.

Toto-Arellano, N. I.

Venkatakrishnan, K.

Vlad, V.

V. Vlad, D. Popa, and I. Apostol, “Computer moiré deflectometry using the Talbot effect,” Opt. Eng. 30, 300–306 (1991).
[CrossRef]

Wang, J.

Wang, M.

J. Zhong and M. Wang, “Fourier transform moiré deflectometry for the automatic measurement of phase objects,” Proc. SPIE 2899, 311–318 (1996).
[CrossRef]

White, A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Wolfling, S.

Yang, S.

Y. Chen, S. Yang, Z. Li, and A. He, “A model for arc plasma’s optical diagnosis by the measurement of the refractive index,” Opt. Commun. 284, 2648–2652 (2011).
[CrossRef]

Zhang, B.

Zhao, Z.

Zhong, J.

J. Zhong and M. Wang, “Fourier transform moiré deflectometry for the automatic measurement of phase objects,” Proc. SPIE 2899, 311–318 (1996).
[CrossRef]

AIAA J.

J. Stricker, E. Keren, and O. Kafri, “Axisymmetric density field measurements by moiré deflectometry,” AIAA J. 21, 1767–1769 (1983).
[CrossRef]

Appl. Opt.

H. Bruning, D. Herriott, J. Gallagher, D. Rosenfeld, A. White, and D. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef]

E. Keren, E. Bar-Ziv, I. Glatt, and O. Kafri, “Measurements of temperature distribution of flames by moiré deflectometry,” Appl. Opt. 20, 4263–4266 (1981).
[CrossRef]

M. Kujawinska, L. Salbut, and K. Patorski, “Three-channel phase stepped system for moire interferometry,” Appl. Opt. 30, 1633–1637 (1991).
[CrossRef]

H. Schreiber and J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36, 5321–5324 (1997).
[CrossRef]

L. Deck, “Vibration-resistant phase-shifting interometry,” Appl. Opt. 35, 6655–6662 (1996).
[CrossRef]

M. Servin, R. Rodriguez-Vera, M. Carpio, and A. Morale, “Automatic fringe detection algorithm used for moiré deflectometry,” Appl. Opt. 29, 3266–3270 (1990).
[CrossRef]

B. Ngoi, K. Venkatakrishnan, and N. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. 40, 3211–3214 (2001).
[CrossRef]

S. Wolfling and E. Lanzmann, “Wavefront reconstruction by spatial-phase-shift imaging interometry,” Appl. Opt. 45, 2586–2596 (2006).
[CrossRef]

Y. Song, B. Zhang, and A. He, “Algebraic iterative algorithm for deflection tomography and its application to density flow fields in a hypersonic wind tunnel,” Appl. Opt. 45, 8092–8101 (2006).
[CrossRef]

Y. Chen, Y. Song, A. He, and Z. Li, “Applicability of moiré deflection tomography for diagnosing arc plasmas,” Appl. Opt. 48, 489–496 (2009).
[CrossRef]

N. I. Toto-Arellano, A. Martínez-García, G. Rodríguez-Zurita, J. Rayas-Álvarez, and A. Montes-Perez, “Slope measurement of a phase object using a polarizing phase-shifting high-frequency Ronchi grating interferometer,” Appl. Opt. 49, 6402–6408 (2010).
[CrossRef]

N. Sun, Y. Song, J. Wang, Z. Li, and A. He, “Volume moiré tomography based on double cross gratings for real three-dimension flow field diagnosis,” Appl. Opt. 51, 8081–8089 (2012).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
[CrossRef]

Opt. Commun.

Y. Chen, S. Yang, Z. Li, and A. He, “A model for arc plasma’s optical diagnosis by the measurement of the refractive index,” Opt. Commun. 284, 2648–2652 (2011).
[CrossRef]

Opt. Eng.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 234361 (1984).
[CrossRef]

V. Vlad, D. Popa, and I. Apostol, “Computer moiré deflectometry using the Talbot effect,” Opt. Eng. 30, 300–306 (1991).
[CrossRef]

J. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

B. Zhang, Y. Song, and A. He, “New reconstruction algorithm for moiré tomography in flow field measurements,” Opt. Eng. 45, 117002 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

C. Koliopoulos, “Phase shifting techniques applied to unique applications,” Proc. SPIE 2861, 86–93 (1996).
[CrossRef]

C. Koliopoulos, “Simultaneous phase shift interferometer,” Proc. SPIE 1531, 119–127 (1991).
[CrossRef]

J. Zhong and M. Wang, “Fourier transform moiré deflectometry for the automatic measurement of phase objects,” Proc. SPIE 2899, 311–318 (1996).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

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Figures (10)

Fig. 1.
Fig. 1.

Optical configuration of a crossed and a linear grating interferometer.

Fig. 2.
Fig. 2.

Structure of CG1.

Fig. 3.
Fig. 3.

Shift b between G1 and G2.

Fig. 4.
Fig. 4.

Frequency spectrum distribution of the interferograms.

Fig. 5.
Fig. 5.

Interferograms with sub-Talbot distance.

Fig. 6.
Fig. 6.

Variations of the coefficient ratio | k P 1 y / k P 1 x | with different parameters: (a)  N , (b)  d , and (c)  β .

Fig. 7.
Fig. 7.

Interferograms and phase distribution: (a) reference moiré patterns and (b) spherical wave moiré patterns.

Fig. 8.
Fig. 8.

Comparison between experimental and simulation results: (a) experiment, (b) simulation, and (c) comparison in a line.

Fig. 9.
Fig. 9.

Interferograms and phase distribution with propane flame.

Fig. 10.
Fig. 10.

First-order partial derivative of propane flame phase projection.

Tables (1)

Tables Icon

Table 1. Coefficients before the Partial Derivatives in and k P 1 , k P 2 and k P 3

Equations (34)

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

Δ 2 = β F d λ , β 1 ,
u 1 ( x , y ) exp [ i k φ ( x , y ) ] ,
g 1 = p = + m = + a p a m exp [ i 2 π p d ( sin α 2 x cos α 2 y ) + i 2 π m d ( cos α 2 x sin α 2 y ) ] ,
u 1 + ( x , y ) = u 1 ( x , y ) · p = + m = + a p a m exp [ i 2 π p d ( sin α 2 x cos α 2 y ) + i 2 π m d ( cos α 2 x sin α 2 y ) ] .
U 1 + ( u , v ) = F [ u 1 ( x , y ) · g 1 ( x , y ) ] = p = + m = + a p a m U 1 ( u + p d sin α 2 m d cos α 2 , v + p d cos α 2 + m d sin α 2 ) .
U 2 ( u , v ) = exp [ i k Δ 1 ( 1 λ 2 ( u 2 + v 2 ) ) 1 / 2 ] U 1 + ( u , v ) = exp [ i k Δ 1 ( 1 λ 2 ( u 2 + v 2 ) ) 1 / 2 ] × p = + m = + a p a m U 1 ( u + p d sin α 2 m d cos α 2 , v + p d cos α 2 + m d sin α 2 ) .
g 2 ( x , y ) = n = n = + a n exp [ i 2 π n d ( cos α 2 x + sin α 2 y b ) ] ,
U 3 ( u , v ) = ( p ) ( m ) ( n ) a p a m a n exp ( i 2 π n b d ) × U 1 ( u n d cos α 2 m d cos α 2 + p d sin α 2 , v n d sin α 2 + m d sin α 2 + p d cos α 2 ) × exp ( i π Δ 1 λ n 2 d 2 ) × exp { i k ( Δ 1 + Δ 2 ) [ 1 1 2 λ 2 ( u 2 + v 2 ) ] } × exp [ i 2 π λ Δ 1 n d u cos α 2 + i 2 π λ Δ 1 n d v sin α 2 ] .
u 3 ( x , y ) = U 3 ( u , v ) exp ( i 2 π ( u x + v y ) ) d u d v = ( p ) ( m ) ( n ) a p a m a n exp ( i 2 π n b d ) × exp ( i π λ Δ p 2 d 2 ) exp { i π λ Δ 1 ( m 2 d 2 ) } × exp { i π λ Δ 2 ( n 2 + m 2 2 n p sin α + 2 m n cos α d 2 ) } × exp { i 2 π [ ( n + m d cos α 2 p d sin α 2 ) x + ( n m d sin α 2 p d cos α 2 ) y ] } × u 1 ( x + Δ λ p d sin α 2 Δ λ m d cos α 2 Δ 2 λ n d cos α 2 , y + Δ λ p d cos α 2 + Δ λ m d sin α 2 Δ 2 λ n d sin α 2 ) .
u 3 ( x , y ) = ( p ) ( m ) ( n ) a p a m a n exp ( i k C ) exp ( i 2 π n b d ) × exp ( i π λ Δ p 2 d 2 ) exp { i π λ Δ 1 ( m 2 d 2 ) } × exp { i π λ Δ 2 ( n 2 + m 2 2 n p sin α + 2 m n cos α d 2 ) } × exp { i 2 π [ ( n + m d cos α 2 p d sin α 2 ) x + ( n m d sin α 2 p d cos α 2 ) y ] } .
I 1 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y + η ξ 2 π b d ] } , I 2 ( x , y ) = 2 a 1 2 a 0 4 { 1 + cos [ 4 π d sin α 2 y ξ 2 π b d ] } , I 3 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y η ξ 2 π b d ] } , I 4 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y + η + ξ 2 π b d ] } , I 5 ( x , y ) = 2 a 1 2 a 0 4 { 1 + cos [ 4 π d sin α 2 y + ξ 2 π b d ] } , I 6 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y η + ξ 2 π b d ] } ,
I 7 ( x , y ) = a 1 2 { 4 a 0 2 a 1 2 cos ( ξ π λ 2 Δ 2 d 2 + 2 π λ Δ 2 cos α d 2 ) cos ( 2 π b d + η + 4 π sin ( α / 2 ) d y ) + 4 a 1 4 cos 2 ( 2 π b d + η + 4 π sin ( α / 2 ) d y ) + a 0 4 } I 8 ( x , y ) = a 0 2 { 4 a 0 2 a 1 2 cos ( ξ π λ 2 Δ 2 d 2 + 2 π λ Δ 2 cos α d 2 ) cos ( 2 π b d + 4 π sin ( α / 2 ) d y ) + 4 a 1 4 cos 2 ( 2 π b d + 4 π sin ( α / 2 ) d y ) + a 0 4 } , I 9 ( x , y ) = a 1 2 { 4 a 0 2 a 1 2 cos ( ξ π λ 2 Δ 2 d 2 + 2 π λ Δ 2 cos α d 2 ) cos ( 2 π b d η + 4 π sin ( α / 2 ) d y ) + 4 a 1 4 cos 2 ( 2 π b d η + 4 π sin ( α / 2 ) d y ) + a 0 4 } ,
η = 2 π β F d sin α .
sin α = N d F ,
2 π λ d 2 Δ 2 sin α = 2 π β N .
[ I 1 I 7 I 4 I 2 I 8 I 5 I 3 I 9 I 6 ] = [ ξ + η ξ + η ξ ξ ξ η ξ η ] .
cos ( π λ ( Δ 1 + 2 Δ 2 ) d 2 + 2 π λ Δ 2 cos α d 2 ) cos ( π λ Δ 1 d 2 ) .
I 1 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y + 2 π β N K π 2 π b d ] } , I 2 ( x , y ) = 2 a 1 2 a 0 4 { 1 + cos [ 4 π d sin α 2 y K π 2 π b d ] } , I 3 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y 2 π β N K π 2 π b d ] } , I 4 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y + 2 π β N + K π 2 π b d ] } , I 5 ( x , y ) = 2 a 1 2 a 0 4 { 1 + cos [ 4 π d sin α 2 y + K π 2 π b d ] } , I 6 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y 2 π β N + K π 2 π b d ] } ,
I 7 ( x , y ) = a 1 2 { a 0 2 ± 2 a 1 2 cos ( 4 π d sin α 2 y + 2 π β N 2 π b d ) } 2 , I 8 ( x , y ) = a 1 2 { a 0 2 ± 2 a 1 2 cos ( 4 π d sin α 2 y 2 π b d ) } 2 , I 9 ( x , y ) = a 1 2 { a 0 2 ± 2 a 1 2 cos ( 4 π d sin α 2 y 2 π β N 2 π b d ) } 2 .
I 1 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y + 2 π β N K π π 2 2 π b d ] } , I 2 ( x , y ) = 2 a 1 2 a 0 4 { 1 + cos [ 4 π d sin α 2 y K π π 2 2 π b d ] } , I 3 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y 2 π β N K π π 2 2 π b d ] } , I 4 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y + 2 π β N + K π + π 2 2 π b d ] } , I 5 ( x , y ) = 2 a 1 2 a 0 4 { 1 + cos [ 4 π d sin α 2 y + K π + π 2 2 π b d ] } , I 6 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y 2 π β N + K π + π 2 2 π b d ] } ,
I 7 ( x , y ) = a 1 2 { a 0 4 + 2 a 1 4 + 2 a 1 4 cos [ 2 ( 4 π d sin α 2 y + 2 π β N 2 π b d ) ] } , I 8 ( x , y ) = a 0 2 { a 0 4 + 2 a 1 4 + 2 a 1 4 cos [ 2 ( 4 π sin ( α / 2 ) d y 2 π b d ) ] } , I 9 ( x , y ) = a 1 2 { a 0 4 + 2 a 1 4 + 2 a 1 4 cos [ 2 ( 4 π d sin α 2 y 2 π β N 2 π b d ) ] } .
u 3 ( x , y ) = ( p ) ( m ) ( n ) a p a m a n exp ( i 2 π n b d ) × exp ( i π λ Δ p 2 d 2 ) exp { i π λ Δ 1 ( m 2 d 2 ) } × exp { i π λ Δ 2 ( n 2 + m 2 2 n p sin α + 2 m n cos α d 2 ) } × exp { i 2 π [ ( n + m d cos α 2 p d sin α 2 ) x + ( n m d sin α 2 p d cos α 2 ) y ] } × exp [ i k φ ( x + Δ λ p d sin α 2 Δ λ m d cos α 2 Δ 2 λ n d cos α 2 , y + Δ λ p d cos α 2 + Δ λ m d sin α 2 Δ 2 λ n d sin α 2 ) ] .
u 3 ( x , y ) = ( p ) ( m ) ( n ) a p a m a n exp ( i 2 π n b d ) × exp ( i π λ Δ p 2 d 2 ) exp { i π λ Δ 1 ( m 2 d 2 ) } × exp { i π λ Δ 2 ( n 2 + m 2 2 n p sin α + 2 m n cos α d 2 ) } × exp { i 2 π [ ( n + m d cos α 2 p d sin α 2 ) x + ( n m d sin α 2 p d cos α 2 ) y ] } × exp { i k [ φ ( x , y ) + φ ( x , y ) x ( Δ λ p d sin α 2 Δ λ m d cos α 2 Δ 2 λ n d cos α 2 ) + φ ( x , y ) y ( Δ λ p d cos α 2 + Δ λ m d sin α 2 Δ 2 λ n d sin α 2 ) + D ] } ,
I 1 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y + k P 1 ξ + η 2 π b d ] } , I 2 ( x , y ) = 2 a 1 2 a 0 4 { 1 + cos [ 4 π d sin α 2 y + k P 1 + η 2 π b d ] } , I 3 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y + k P 1 + ξ + η 2 π b d ] } , I 4 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y + k P 1 ξ η 2 π b d ] } , I 5 ( x , y ) = 2 a 1 2 a 0 4 { 1 + cos [ 4 π d sin α 2 y + k P 1 η 2 π b d ] } , I 6 ( x , y ) = 2 a 1 4 a 0 2 { 1 + cos [ 4 π d sin α 2 y + k P 1 + ξ η 2 π b d ] } ,
k P 1 = 2 π d ( φ ( x , y ) x Δ 1 cos α 2 φ ( x , y ) y ( Δ 1 + 2 Δ 2 ) sin α 2 ) , ξ = k P 2 2 π β N , η = k P 3 π λ Δ 1 d 2 ,
k P 2 = π λ ( 2 φ ( x , y ) x [ 2 Δ Δ 1 d 2 sin α 2 cos α 2 ] + 2 φ ( x , y ) y [ 2 Δ ( Δ 1 + 2 Δ 2 ) d 2 cos α 2 sin α 2 ] 2 2 φ ( x , y ) x y ( Δ 2 d 2 cos α Δ Δ 2 d 2 ) ) , k P 3 = π λ ( 2 φ ( x , y ) x [ Δ 1 2 + 2 Δ 1 Δ 2 d 2 cos 2 α 2 ] + 2 φ ( x , y ) y [ Δ 1 2 + 2 Δ 1 Δ 2 d 2 sin 2 α 2 ] 2 2 φ ( x , y ) x y ( Δ 2 + Δ 2 2 d 2 cos α 2 sin α 2 ) ) .
k P 1 y = 2 π d ( Δ 1 + 2 Δ 2 ) sin α 2 , k P 1 x = 2 π d Δ 1 cos α 2 .
| k P 1 y k P 1 x | = ( 1 + 2 Δ 2 Δ 1 ) tan ( α 2 ) .
tan ( α 2 ) sin ( α 2 ) = N d 2 F .
| k P 1 y k P 1 x | = N d 2 F + β N d 2 λ Δ 1 .
I 1 ( x , y ) = a ( x , y ) + b ( x , y ) cos ( ϕ + θ + δ ) , I 2 ( x , y ) = μ [ a ( x , y ) + b ( x , y ) cos ( ϕ + θ ) ] , I 3 ( x , y ) = a ( x , y ) + b ( x , y ) cos ( ϕ + θ δ ) , I 4 ( x , y ) = a ( x , y ) + b ( x , y ) cos ( ϕ θ + δ ) , I 5 ( x , y ) = μ [ a ( x , y ) + b ( x , y ) cos ( ϕ θ ) ] , I 6 ( x , y ) = a ( x , y ) + b ( x , y ) cos ( ϕ θ δ ) ,
I 1 = a + b cos ( ϕ + θ + δ ) , I 2 = a + b cos ( ϕ + θ ) , I 3 = a + b cos ( ϕ + θ δ ) , I 4 = a + b cos ( ϕ θ + δ ) , I 5 = a + b cos ( ϕ θ ) , I 6 = a + b cos ( ϕ θ δ ) .
ϕ = arctan ( I 1 + I 3 I 4 I 6 I 1 I 3 I 4 + I 6 tan [ arccos ( I 1 + I 3 I 4 I 6 2 ( I 2 I 5 ) ) ] ) .
φ ( x , y ) x = ( z 2 + x 2 + y 2 ) x = x ( z 2 + x 2 + y 2 ) .

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