Abstract

Theories of moiré deflectometry are presented based on scalar diffraction theory. It is shown that the moiré effect is not a pure geometric phenomenon but actually the result of multishearing interference. By performing zeroth-order or first-order filtering, the field in the plane of observation is seen to be the result of double- or triple-shearing interference, respectively. With first-order filtering, the intensity distribution is proved to be a strict cosinusoidal intensity distribution, and the diffraction effect, which depends on the distance between two gratings, affects just the phase shift of the moiré fringes. Compared with previous research, a more precise relation between the unwrapped phase and the deflection angles is obtained. The results will be very useful for image processing of moiré patterns with Fourier transform profilometry and phase-shift methods.

© 2009 Optical Society of America

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References

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  1. J. D. Posner and D. Dunn-Rankin, “Temperature field measurements of small, nonpremixed flames with use of an Abel inversion of holographic interferograms,” Appl. Opt. 42, 952-959 (2003).
    [CrossRef] [PubMed]
  2. X. Wan, S. Yu, G. Cai, Y. Gao, and J. Yi, “Three-dimensional plasma field reconstruction with multiobjective optimization emission spectral tomography,” J. Opt. Soc. Am. A 21, 1161-1171 (2004).
    [CrossRef]
  3. H. Thayyullathil, R. M. Vasu, and R. Kanhirodan, “Quantitative flow visualization in supersonic jets through tomographic inversion of wavefronts estimated through shadow casting,” Appl. Opt. 45, 5010-5019 (2006).
    [CrossRef] [PubMed]
  4. D. A. Feikema, “Quantitative rainbow schlieren deflectometry as a temperature diagnostic for nonsooting spherical flames,” Appl. Opt. 45, 4826-4832 (2006).
    [CrossRef] [PubMed]
  5. O. Kafri, “Noncoherent methods for mapping phase objects,” Opt. Lett. 5, 555-557 (1980).
    [CrossRef] [PubMed]
  6. O. Kafri and I. Glatt, “Moiré deflectometry: a ray deflection approach to optical testing,” Opt. Eng. (Bellingham) 24, 944-960 (1985).
  7. 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] [PubMed]
  8. 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] [PubMed]
  9. A. K. Agrawal, N. K. Butuk, S. R. Gollahalli, and D. Griffin, “Three-dimensional rainbow schlieren tomography of a temperature field in gas flows,” Appl. Opt. 37, 479-485 (1998).
    [CrossRef]
  10. X. Xiao, I. K. Puri, and A. K. Agrawal, “Temperature measurements in steady axisymmetric partially premixed flames by use of rainbow schlieren deflectometry,” Appl. Opt. 41, 1922-1928 (2002).
    [CrossRef] [PubMed]
  11. E. Goldhahn and J. Seume, “The background oriented schlieren technique: sensitivity, accuracy, resolution and application to a three-dimensional density field,” Exp. Fluids 43, 241-249 (2007).
    [CrossRef]
  12. M. Servin, R. Rodriguez-Vera, M. Carpio, and A. Morales, “Automatic fringe detection algorithm used for moiré deflectometry,” Appl. Opt. 29, 3266-3270 (1990).
    [CrossRef] [PubMed]
  13. H. Canabal, J. A. Quiroga, and E. Bernabeu, “Automatic processing in moiré deflectometry by local fringe direction calculation,” Appl. Opt. 37, 5894-5901 (1998).
    [CrossRef]
  14. J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. (Bellingham) 38, 974-982 (1999).
    [CrossRef]
  15. M. Wang, “Fourier transform moiré tomography for high-sensitivity mapping asymmetric 3-D temperature field,” Opt. Laser Technol. 34, 679-685 (2002).
    [CrossRef]
  16. J. Zhong and M. Wang, “Fourier transform moiré deflectometry for the automatic measurement of phase objects,” Proc. SPIE 2899, 311-318 (1996).
    [CrossRef]
  17. S. Ranjbar, H. R. Khalesifard, and S. Rasouli, “Nondestructive measurement of refractive index profile of optical fiber preforms using moiré technique and phase shift method,” Proc. SPIE 6025, 602520 (2006).
    [CrossRef]
  18. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977-3982 (1983).
    [CrossRef] [PubMed]
  19. M. D. Pritt, “Weighted least squares phase unwrapping by means of multigrid techniques,” Proc. SPIE 2584, 278-288 (1995).
  20. J. A. Quiroga, A. González-Cano, and E. Bernabeu, “Phase-unwrapping algorithm based on adaptive criterion,” Appl. Opt. 34, 2560-2563 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
  22. E. Bar-Ziv, “Effect of diffraction on the moiré image for temperature mapping in flames,” Appl. Opt. 23, 4040-4044 (1984).
    [CrossRef] [PubMed]
  23. E. Bar-Ziv, “Effect of diffraction on the moiré image I. Theory,” J. Opt. Soc. Am. A 2, 371-379 (1985).
    [CrossRef]
  24. E. Bar-Ziv, S. Sgulim, and D. Manor, “Effect of Diffraction on the moiré image. II. Experiment,” J. Opt. Soc. Am. A 2, 380-385 (1985).
    [CrossRef]
  25. A. Dahan, G. Ben-Dor, and E. Bar-Ziv, “Fourier transform deflection mapping,” Opt. Eng. (Bellingham) 32, 1094-1100 (1993).
    [CrossRef]
  26. D. W. Sweeny and C. M. Vest, “Reconstruction of three-dimensional refractive index field from multi-direction interferometric data,” Appl. Opt. 12, 2649-2664 (1973).
    [CrossRef]
  27. B. J. Pelliccia-Kraft and D. W. Watt, “Three-dimensional imaging of a turbulent jet using shearing interferometry and optical tomography,” Exp. Fluids 29, 573-581 (2000).
    [CrossRef]
  28. W. Merzkirch and Y. Egami, “Density-based techniques,” in Springer Handbook of Experimental Fluid Mechanics, C.Tropea, A.L.Yarin, and J. F. Foss, eds. (Springer, 2007), pp. 473-486
  29. G. W. Faris and R. L. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202-5212 (1988).
    [CrossRef] [PubMed]
  30. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  31. D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11, 2613-2624 (1972).
    [CrossRef] [PubMed]

2007 (1)

E. Goldhahn and J. Seume, “The background oriented schlieren technique: sensitivity, accuracy, resolution and application to a three-dimensional density field,” Exp. Fluids 43, 241-249 (2007).
[CrossRef]

2006 (4)

2004 (1)

2003 (1)

2002 (2)

M. Wang, “Fourier transform moiré tomography for high-sensitivity mapping asymmetric 3-D temperature field,” Opt. Laser Technol. 34, 679-685 (2002).
[CrossRef]

X. Xiao, I. K. Puri, and A. K. Agrawal, “Temperature measurements in steady axisymmetric partially premixed flames by use of rainbow schlieren deflectometry,” Appl. Opt. 41, 1922-1928 (2002).
[CrossRef] [PubMed]

2000 (1)

B. J. Pelliccia-Kraft and D. W. Watt, “Three-dimensional imaging of a turbulent jet using shearing interferometry and optical tomography,” Exp. Fluids 29, 573-581 (2000).
[CrossRef]

1999 (1)

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

1998 (2)

1996 (1)

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

1995 (2)

M. D. Pritt, “Weighted least squares phase unwrapping by means of multigrid techniques,” Proc. SPIE 2584, 278-288 (1995).

J. A. Quiroga, A. González-Cano, and E. Bernabeu, “Phase-unwrapping algorithm based on adaptive criterion,” Appl. Opt. 34, 2560-2563 (1995).
[CrossRef] [PubMed]

1993 (1)

A. Dahan, G. Ben-Dor, and E. Bar-Ziv, “Fourier transform deflection mapping,” Opt. Eng. (Bellingham) 32, 1094-1100 (1993).
[CrossRef]

1990 (1)

1988 (1)

1985 (4)

1984 (1)

1983 (1)

1981 (1)

1980 (1)

1973 (1)

1972 (1)

Agrawal, A. K.

Bar-Ziv, E.

Ben-Dor, G.

A. Dahan, G. Ben-Dor, and E. Bar-Ziv, “Fourier transform deflection mapping,” Opt. Eng. (Bellingham) 32, 1094-1100 (1993).
[CrossRef]

Bernabeu, E.

Born, M.

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

Butuk, N. K.

Byer, R. L.

Cai, G.

Canabal, H.

Carpio, M.

Crespo, D.

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

Dahan, A.

A. Dahan, G. Ben-Dor, and E. Bar-Ziv, “Fourier transform deflection mapping,” Opt. Eng. (Bellingham) 32, 1094-1100 (1993).
[CrossRef]

Dunn-Rankin, D.

Egami, Y.

W. Merzkirch and Y. Egami, “Density-based techniques,” in Springer Handbook of Experimental Fluid Mechanics, C.Tropea, A.L.Yarin, and J. F. Foss, eds. (Springer, 2007), pp. 473-486

Faris, G. W.

Feikema, D. A.

Foss, J. F.

W. Merzkirch and Y. Egami, “Density-based techniques,” in Springer Handbook of Experimental Fluid Mechanics, C.Tropea, A.L.Yarin, and J. F. Foss, eds. (Springer, 2007), pp. 473-486

Gao, Y.

Glatt, I.

O. Kafri and I. Glatt, “Moiré deflectometry: a ray deflection approach to optical testing,” Opt. Eng. (Bellingham) 24, 944-960 (1985).

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] [PubMed]

Goldhahn, E.

E. Goldhahn and J. Seume, “The background oriented schlieren technique: sensitivity, accuracy, resolution and application to a three-dimensional density field,” Exp. Fluids 43, 241-249 (2007).
[CrossRef]

Gollahalli, S. R.

González-Cano, A.

Griffin, D.

He, A.

Kafri, O.

Kanhirodan, R.

Keren, E.

Khalesifard, H. R.

S. Ranjbar, H. R. Khalesifard, and S. Rasouli, “Nondestructive measurement of refractive index profile of optical fiber preforms using moiré technique and phase shift method,” Proc. SPIE 6025, 602520 (2006).
[CrossRef]

Manor, D.

Merzkirch, W.

W. Merzkirch and Y. Egami, “Density-based techniques,” in Springer Handbook of Experimental Fluid Mechanics, C.Tropea, A.L.Yarin, and J. F. Foss, eds. (Springer, 2007), pp. 473-486

Morales, A.

Mutoh, K.

Pelliccia-Kraft, B. J.

B. J. Pelliccia-Kraft and D. W. Watt, “Three-dimensional imaging of a turbulent jet using shearing interferometry and optical tomography,” Exp. Fluids 29, 573-581 (2000).
[CrossRef]

Posner, J. D.

Pritt, M. D.

M. D. Pritt, “Weighted least squares phase unwrapping by means of multigrid techniques,” Proc. SPIE 2584, 278-288 (1995).

Puri, I. K.

Quiroga, J. A.

Ranjbar, S.

S. Ranjbar, H. R. Khalesifard, and S. Rasouli, “Nondestructive measurement of refractive index profile of optical fiber preforms using moiré technique and phase shift method,” Proc. SPIE 6025, 602520 (2006).
[CrossRef]

Rasouli, S.

S. Ranjbar, H. R. Khalesifard, and S. Rasouli, “Nondestructive measurement of refractive index profile of optical fiber preforms using moiré technique and phase shift method,” Proc. SPIE 6025, 602520 (2006).
[CrossRef]

Rodriguez-Vera, R.

Servin, M.

Seume, J.

E. Goldhahn and J. Seume, “The background oriented schlieren technique: sensitivity, accuracy, resolution and application to a three-dimensional density field,” Exp. Fluids 43, 241-249 (2007).
[CrossRef]

Sgulim, S.

Silva, D. E.

Song, Y.

Sweeny, D. W.

Takeda, M.

Thayyullathil, H.

Vasu, R. M.

Vest, C. M.

Wan, X.

Wang, M.

M. Wang, “Fourier transform moiré tomography for high-sensitivity mapping asymmetric 3-D temperature field,” Opt. Laser Technol. 34, 679-685 (2002).
[CrossRef]

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

Watt, D. W.

B. J. Pelliccia-Kraft and D. W. Watt, “Three-dimensional imaging of a turbulent jet using shearing interferometry and optical tomography,” Exp. Fluids 29, 573-581 (2000).
[CrossRef]

Wolf, E.

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

Xiao, X.

Yi, J.

Yu, S.

Zhang, B.

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]

Appl. Opt. (15)

J. D. Posner and D. Dunn-Rankin, “Temperature field measurements of small, nonpremixed flames with use of an Abel inversion of holographic interferograms,” Appl. Opt. 42, 952-959 (2003).
[CrossRef] [PubMed]

H. Thayyullathil, R. M. Vasu, and R. Kanhirodan, “Quantitative flow visualization in supersonic jets through tomographic inversion of wavefronts estimated through shadow casting,” Appl. Opt. 45, 5010-5019 (2006).
[CrossRef] [PubMed]

D. A. Feikema, “Quantitative rainbow schlieren deflectometry as a temperature diagnostic for nonsooting spherical flames,” Appl. Opt. 45, 4826-4832 (2006).
[CrossRef] [PubMed]

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] [PubMed]

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] [PubMed]

A. K. Agrawal, N. K. Butuk, S. R. Gollahalli, and D. Griffin, “Three-dimensional rainbow schlieren tomography of a temperature field in gas flows,” Appl. Opt. 37, 479-485 (1998).
[CrossRef]

X. Xiao, I. K. Puri, and A. K. Agrawal, “Temperature measurements in steady axisymmetric partially premixed flames by use of rainbow schlieren deflectometry,” Appl. Opt. 41, 1922-1928 (2002).
[CrossRef] [PubMed]

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

H. Canabal, J. A. Quiroga, and E. Bernabeu, “Automatic processing in moiré deflectometry by local fringe direction calculation,” Appl. Opt. 37, 5894-5901 (1998).
[CrossRef]

M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977-3982 (1983).
[CrossRef] [PubMed]

J. A. Quiroga, A. González-Cano, and E. Bernabeu, “Phase-unwrapping algorithm based on adaptive criterion,” Appl. Opt. 34, 2560-2563 (1995).
[CrossRef] [PubMed]

E. Bar-Ziv, “Effect of diffraction on the moiré image for temperature mapping in flames,” Appl. Opt. 23, 4040-4044 (1984).
[CrossRef] [PubMed]

D. W. Sweeny and C. M. Vest, “Reconstruction of three-dimensional refractive index field from multi-direction interferometric data,” Appl. Opt. 12, 2649-2664 (1973).
[CrossRef]

G. W. Faris and R. L. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202-5212 (1988).
[CrossRef] [PubMed]

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

Exp. Fluids (2)

B. J. Pelliccia-Kraft and D. W. Watt, “Three-dimensional imaging of a turbulent jet using shearing interferometry and optical tomography,” Exp. Fluids 29, 573-581 (2000).
[CrossRef]

E. Goldhahn and J. Seume, “The background oriented schlieren technique: sensitivity, accuracy, resolution and application to a three-dimensional density field,” Exp. Fluids 43, 241-249 (2007).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Eng. (Bellingham) (3)

A. Dahan, G. Ben-Dor, and E. Bar-Ziv, “Fourier transform deflection mapping,” Opt. Eng. (Bellingham) 32, 1094-1100 (1993).
[CrossRef]

O. Kafri and I. Glatt, “Moiré deflectometry: a ray deflection approach to optical testing,” Opt. Eng. (Bellingham) 24, 944-960 (1985).

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

Opt. Laser Technol. (1)

M. Wang, “Fourier transform moiré tomography for high-sensitivity mapping asymmetric 3-D temperature field,” Opt. Laser Technol. 34, 679-685 (2002).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (3)

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

S. Ranjbar, H. R. Khalesifard, and S. Rasouli, “Nondestructive measurement of refractive index profile of optical fiber preforms using moiré technique and phase shift method,” Proc. SPIE 6025, 602520 (2006).
[CrossRef]

M. D. Pritt, “Weighted least squares phase unwrapping by means of multigrid techniques,” Proc. SPIE 2584, 278-288 (1995).

Other (2)

W. Merzkirch and Y. Egami, “Density-based techniques,” in Springer Handbook of Experimental Fluid Mechanics, C.Tropea, A.L.Yarin, and J. F. Foss, eds. (Springer, 2007), pp. 473-486

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

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Figures (7)

Fig. 1
Fig. 1

Coordinate setup.

Fig. 2
Fig. 2

Optical schematic diagram of moiré deflectometry.

Fig. 3
Fig. 3

Spectrum distribution of u 2 + ( x , y ) : (a) α 0 , (b) α 0 .

Fig. 4
Fig. 4

Propane flame as object with zeroth-order filtering: (a) Talbot distance, (b) sub-Talbot distance.

Fig. 5
Fig. 5

Simulated intensity distributions with zeroth-order filtering.

Fig. 6
Fig. 6

Propane flame as object with first-order filtering.

Fig. 7
Fig. 7

Simulated intensity distribution.

Equations (33)

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φ ( y , θ ) = + n ( x , y ) d x .
φ d ( y , θ ) = + 1 n c ( x , y ) n c ( x , y ) y d x 1 n 0 + n ( x , y ) y d x .
+ φ d ( y , θ ) e j 2 π y Y d y = 1 n 0 + [ + n ( x , y ) y d x ] e j 2 π y Y d y .
n 0 j 2 π Y Φ d ( Y , θ ) = Φ ( Y , θ ) .
φ ( y , θ ) = n 0 2 F 1 [ Φ d ( Y , θ ) 1 j π Y ] = n 0 2 [ φ d ( y , θ ) sgn ( y ) ] .
φ ( y , θ ) = + n ( x , y ) d x = n 0 y φ d ( τ , θ ) d τ .
φ d ( y , θ ) = 1 n 0 φ ( y , θ ) y .
u 1 ( x , y ) exp [ i k φ ( x , y ) ] ,
g 1 ( x , y ) = ( m ) a m exp [ i 2 π m d ( x cos α 2 y sin α 2 ) ] .
u 1 + ( x , y ) = u 1 ( x , y ) ( m ) a m exp [ i 2 π m d ( x cos α 2 y sin α 2 ) ] = F 1 [ ( m ) a m U 1 ( u m d cos α 2 , v + m d sin α 2 ) ] ,
U 1 + ( u , v ) = ( m ) a m U 1 ( u m d cos α 2 , v + m d sin α 2 ) .
U 2 ( u , v ) = exp [ i k Δ 1 λ 2 ( u 2 + v 2 ) ] ( m ) a m U 1 ( u m d cos α 2 , v + m d sin α 2 ) .
g 2 ( x , y ) = ( n ) a n exp [ i 2 π n d ( x cos α 2 + y sin α 2 ) ] .
U 2 + ( u , v ) = ( m ) ( n ) a m a n U 1 ( u m + n d cos α 2 , v + m n d sin α 2 ) × exp { i k Δ [ 1 λ 2 2 ( u 2 + v 2 ) ] } exp ( i Δ λ n 2 π d 2 ) exp [ i 2 π Δ λ n d ( u cos α 2 + v sin α 2 ) ] .
u 2 + ( x , y ) = ( m ) ( n ) a m a n exp { i 2 π d [ ( m + n ) x cos α 2 ( m n ) y sin α 2 ] } exp ( i π λ Δ m 2 d 2 ) × ( u 1 ( x λ Δ m d cos α 2 , y + λ Δ m d sin α 2 ) { exp ( i k Δ ) i λ Δ exp [ i π λ Δ ( x 2 + y 2 ) ] } )
u 2 + ( x , y ) = exp ( i k Δ ) ( m ) ( n ) a m a n exp { i 2 π d [ ( m + n ) x cos α 2 ( m n ) y sin α 2 ] } × exp ( i π λ Δ m 2 d 2 ) u 1 ( x λ Δ m d cos α 2 , y + λ Δ m d sin α 2 ) .
Δ = K d 2 λ ( K is an integer ) ,
φ ( x λ Δ m d cos α 2 , y + λ Δ m d sin α 2 ) = φ ( x , y ) + φ ( x , y ) x ( λ Δ m d cos α 2 ) + φ ( x , y ) y ( λ Δ m d sin α 2 ) + 1 2 2 φ ( x , y ) x 2 ( λ Δ m d cos α 2 ) 2 + 1 2 2 φ ( x , y ) y 2 ( λ Δ m d sin α 2 ) 2 + 2 φ ( x , y ) x y ( λ Δ m d cos α 2 ) ( λ Δ m d sin α 2 ) +
( λ Δ m d ) = K d m .
u 2 + ( x , y ) = exp ( i k Δ ) ( m ) ( n ) a m a n exp { i 2 π d [ ( m + n ) x cos α 2 ( m n ) y sin α 2 ] } × exp ( i π λ Δ m 2 d 2 ) exp [ i k φ ( x , y ) ] exp [ i k φ ( x , y ) x λ Δ m d cos α 2 ] .
u ( x , y ) = exp ( i k Δ ) ( m ) a m a m exp [ i 2 π d ( 2 m y sin α 2 ) ] × exp ( i π λ Δ m 2 d 2 ) exp [ i k φ ( x , y ) ] exp [ i k φ ( x , y ) x λ Δ m d cos α 2 ] .
I ( x , y ) = u ( x , y ) u * ( x , y ) = a 0 4 + 4 a 0 2 a 1 2 cos ( π Δ λ d 2 ) cos [ φ ( x , y ) x 2 π Δ d cos α 2 + 4 π d y sin α 2 ] + 4 a 1 4 cos 2 [ φ ( x , y ) x 2 π Δ d cos α 2 + 4 π d y sin α 2 ] .
I ( x , y ) = { a 0 2 ± 2 a 1 2 cos [ φ ( x , y ) x 2 π K d λ cos α 2 + 4 π d y sin α 2 ] } 2 .
I ( x , y ) = a 0 4 + 4 a 1 4 cos 2 [ φ ( x , y ) x 2 π Δ d cos α 2 + 4 π d y sin α 2 ] .
y = { Q d 2 sin ( α 2 ) φ ( x , y ) x K d 2 2 λ ctg α 2 , K is even Q d 2 sin ( α 2 ) + d 4 sin ( α 2 ) φ ( x , y ) x K d 2 2 λ ctg α 2 , K is odd } ,
p = d 2 sin ( α 2 ) .
p m = φ d ( x , y ) K d 2 2 λ sin ( α 2 ) n 0 cos α 2 = φ d ( x , y ) Δ p d n 0 cos α 2 .
p m = φ d ( x , y ) Δ p d .
u ( x , y ) = a 0 a 1 exp ( i 2 π d x cos α 2 ) exp { i k [ Δ + φ ( x , y ) ] } × { exp ( i 2 π d y sin α 2 ) exp ( i Δ π λ d 2 ) exp [ i φ ( x , y ) x 2 π Δ d cos α 2 ] + exp ( i 2 π d y sin α 2 ) } .
I ( x , y ) = u ( x , y ) u * ( x , y ) = 2 a 0 2 a 1 2 { 1 + cos [ 4 π d y sin α 2 + Δ π λ d 2 + φ ( x , y ) x 2 π Δ d cos α 2 ] } .
φ d ( x , y ) = d 2 π Δ ϕ ( x , y ) [ 1 n 0 cos ( α 2 ) ] = α p 2 π Δ ϕ ( x , y ) [ 1 n 0 cos ( α 2 ) ] .
φ d ( x , y ) = α p 2 π Δ ϕ ( x , y ) .
y = Q d 2 sin ( α 2 ) K d 4 sin ( α 2 ) φ ( x , y ) x K d 2 2 λ ctg α 2 .

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