Abstract

Phase-shifting profilometry requires projection of sinusoidal fringes on a 3D object. We analyze the visibility and frequency content of fringes created by a sinusoidal phase grating at coherent illumination. We derive an expression for the intensity of fringes in the Fresnel zone and measure their visibility and frequency content for a grating that has been interferometrically recorded on a holographic plate. Both evaluation of the systematic errors due to the presence of higher harmonics by simulation of a profilometric measurement and measurement of 3D coordinates of test objects confirm the good performance of the sinusoidal phase grating as a projective element. In addition, we prove theoretically that in comparison with a sinusoidal amplitude grating this grating produces better quality of fringes in the near-infrared region. Sinusoidal phase gratings are fabricated easily, and their implementation in fringe projection profilometry facilitates construction of portable, small size, and low-cost devices.

© 2009 Optical Society of America

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    [CrossRef]
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  17. C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang, “Shape measurement by use of liquid-crystal display fringe projection with two-step phase shifting,” Appl. Opt. 42, 2329-2335(2003).
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    [CrossRef]
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2009 (2)

T. Anna, S. K. Dubey, C. Shakher, A. Roy, and D. S. Mehta, “Sinusoidal fringe projection system based on compact and non-mechanical scanning low coherence Michelson interferometer for three-dimensional shape measurement,” Opt. Commun. 282, 1237-1242 (2009).
[CrossRef]

E. Stoykova, J. Harizanova, and V. Sainov, “Pattern projection with a sinusoidal phase grating,” EURASIP J. Adv. Signal Process. 2009, 351626 (2009).

2007 (2)

S. K. Dubey, D. S. Mehta, A. Roy, and C. Shakher, “Wavelength scanning Talbot effect: Wavelength-scanning Talbot effect and its application for arbitrary threedimensional step-height measurement,” Opt. Commun. 279, 13-19(2007).
[CrossRef]

S. Teng, X. Chen, T. Zhou, and C. Cheng, “Quasi-Talbot effect of a grating in the deep Fresnel diffraction region,” J. Opt. Soc. Am. A 24, 1656-1665 (2007).
[CrossRef]

2006 (2)

2005 (5)

2004 (2)

2003 (4)

S. Yoneyama, Y. Morimoto, M. Fujigaki, and Y. Ikeda, “Three-dimensional surface profile measurement of a moving object by a spatial-offset phase stepping method,” Opt. Eng. 42, 137-142 (2003).

P. S. Huang, C. Zhang, and F.-P. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42, 163-168 (2003).

C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang, “Shape measurement by use of liquid-crystal display fringe projection with two-step phase shifting,” Appl. Opt. 42, 2329-2335(2003).
[CrossRef]

S. Teng, L. Liu, J. Zu, Z. Luan, and D. Liu, “Uniform theory of the Talbot effect with partially coherent light illumination,” J. Opt. Soc. Am. A 20, 1747-1754 (2003).
[CrossRef]

2001 (3)

2000 (1)

F. Chen, G. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10-22 (2000).

1999 (2)

B. Dorrio and J. Fernandez, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

Y. Hao, Y. Zhao, and D. Li, “Multifrequency grating projection profilometry based on the nonlinear excess fraction method,” Appl. Opt. 38, 4106-4111 (1999).
[CrossRef]

1998 (1)

1997 (1)

1996 (2)

1994 (1)

1992 (3)

P. Latimer and R. F. Crouse, “Talbot effect reinterpreted,” Appl. Opt. 31, 80-89 (1992).
[CrossRef]

K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740-1748 (1992).
[CrossRef]

X. Su, W.-S. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561-573 (1992).
[CrossRef]

1991 (1)

1990 (2)

J. Schwider, “Advanced evaluation techniques in interferometry,” Prog. Opt. 28, 271-359 (1990).
[CrossRef]

A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337-4340(1990).
[CrossRef]

1989 (1)

1988 (2)

1985 (1)

1984 (2)

1982 (1)

Anna, T.

T. Anna, S. K. Dubey, C. Shakher, A. Roy, and D. S. Mehta, “Sinusoidal fringe projection system based on compact and non-mechanical scanning low coherence Michelson interferometer for three-dimensional shape measurement,” Opt. Commun. 282, 1237-1242 (2009).
[CrossRef]

Bao, C.

Brohinsky, W. R.

Brown, G.

F. Chen, G. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10-22 (2000).

Cartwright, S. L.

Chavel, P.

Chen, F.

F. Chen, G. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10-22 (2000).

Chen, M.

Chen, X.

Cheng, C.

Chiang, F.-P.

P. S. Huang, C. Zhang, and F.-P. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42, 163-168 (2003).

Chicharo, J.

Cohen-Sabban, Y.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349-393 (1988).
[CrossRef]

Crouse, R. F.

Dorrio, B.

B. Dorrio and J. Fernandez, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

Dorsch, R. G.

Dubey, S. K.

T. Anna, S. K. Dubey, C. Shakher, A. Roy, and D. S. Mehta, “Sinusoidal fringe projection system based on compact and non-mechanical scanning low coherence Michelson interferometer for three-dimensional shape measurement,” Opt. Commun. 282, 1237-1242 (2009).
[CrossRef]

S. K. Dubey, D. S. Mehta, A. Roy, and C. Shakher, “Wavelength scanning Talbot effect: Wavelength-scanning Talbot effect and its application for arbitrary threedimensional step-height measurement,” Opt. Commun. 279, 13-19(2007).
[CrossRef]

D. S. Mehta, S. K. Dubey, M. M. Hossain, and C. Shakher, “Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry,” Appl. Opt. 44, 7515-7521 (2005).
[CrossRef]

Farrant, D.

Fernandez, J.

B. Dorrio and J. Fernandez, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

Fernandez-Pousa, C. R.

Flores-Arias, M. T.

Fujigaki, M.

S. Yoneyama, Y. Morimoto, M. Fujigaki, and Y. Ikeda, “Three-dimensional surface profile measurement of a moving object by a spatial-offset phase stepping method,” Opt. Eng. 42, 137-142 (2003).

Gasvik, K. J.

K. J. Gasvik, Optical Metrology (Wiley, 2002).

Ghiglia, D.

D. Ghiglia and M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Gomez-Reino, C.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Guo, H.

Hao, Y.

Harding, K. G.

Harizanova, J.

E. Stoykova, J. Harizanova, and V. Sainov, “Pattern projection with a sinusoidal phase grating,” EURASIP J. Adv. Signal Process. 2009, 351626 (2009).

J. Harizanova and A. Kolev, “Comparative study of fringe generation in two-spacing phase-shifting profilometry,” Proc. SPIE 6252, 625223 (2005).

E. Stoykova, J. Harizanova, and V. Sainov, “Pattern projection profilometry for 3D coordinates measurement of dynamic scenes,” in Three Dimensional Television, H. M. Ozaktas and L. Onural, eds. (Springer, 2008), pp. 85-164.

He, H.

He, X. Y.

C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang, “Shape measurement by use of liquid-crystal display fringe projection with two-step phase shifting,” Appl. Opt. 42, 2329-2335(2003).
[CrossRef]

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, and H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase-shifting,” Opt. Commun. 189, 21-29 (2001).
[CrossRef]

Hibino, K.

Hossain, M. M.

Hu, Y.

Huang, P. S.

P. S. Huang, C. Zhang, and F.-P. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42, 163-168 (2003).

Ikeda, Y.

S. Yoneyama, Y. Morimoto, M. Fujigaki, and Y. Ikeda, “Three-dimensional surface profile measurement of a moving object by a spatial-offset phase stepping method,” Opt. Eng. 42, 137-142 (2003).

Ishii, Y.

Y. Ishii, “Laser-diode interferometry,” Prog. Opt. 46, 243-309(2004).

Jahns, J.

Joyeux, D.

Kang, X.

Kolev, A.

J. Harizanova and A. Kolev, “Comparative study of fringe generation in two-spacing phase-shifting profilometry,” Proc. SPIE 6252, 625223 (2005).

Kozak, S.

Lai, G.

Langoju, R.

Larkin, K.

Larkin, K. G.

Latimer, P.

Li, D.

Li, E.

Liu, D.

Liu, L.

Lohmann, A. W.

Luan, Z.

Mehta, D. S.

T. Anna, S. K. Dubey, C. Shakher, A. Roy, and D. S. Mehta, “Sinusoidal fringe projection system based on compact and non-mechanical scanning low coherence Michelson interferometer for three-dimensional shape measurement,” Opt. Commun. 282, 1237-1242 (2009).
[CrossRef]

S. K. Dubey, D. S. Mehta, A. Roy, and C. Shakher, “Wavelength scanning Talbot effect: Wavelength-scanning Talbot effect and its application for arbitrary threedimensional step-height measurement,” Opt. Commun. 279, 13-19(2007).
[CrossRef]

D. S. Mehta, S. K. Dubey, M. M. Hossain, and C. Shakher, “Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry,” Appl. Opt. 44, 7515-7521 (2005).
[CrossRef]

Morimoto, Y.

S. Yoneyama, Y. Morimoto, M. Fujigaki, and Y. Ikeda, “Three-dimensional surface profile measurement of a moving object by a spatial-offset phase stepping method,” Opt. Eng. 42, 137-142 (2003).

Oreb, B.

Oreb, B. F.

Patil, A.

R. Langoju, A. Patil, and P. Rastogi, “Phase-shifting interferometry in the presence of nonlinear phase steps, harmonics and noise,” Opt. Lett. 31, 1058-1060 (2006).
[CrossRef]

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43, 475-490(2005).

Patorski, K.

Peng, X.

Perez, M. V.

Pritt, M.

D. Ghiglia and M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Qiu, W.

Quan, C.

Rastogi, P.

R. Langoju, A. Patil, and P. Rastogi, “Phase-shifting interferometry in the presence of nonlinear phase steps, harmonics and noise,” Opt. Lett. 31, 1058-1060 (2006).
[CrossRef]

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43, 475-490(2005).

Roy, A.

T. Anna, S. K. Dubey, C. Shakher, A. Roy, and D. S. Mehta, “Sinusoidal fringe projection system based on compact and non-mechanical scanning low coherence Michelson interferometer for three-dimensional shape measurement,” Opt. Commun. 282, 1237-1242 (2009).
[CrossRef]

S. K. Dubey, D. S. Mehta, A. Roy, and C. Shakher, “Wavelength scanning Talbot effect: Wavelength-scanning Talbot effect and its application for arbitrary threedimensional step-height measurement,” Opt. Commun. 279, 13-19(2007).
[CrossRef]

Sainov, V.

E. Stoykova, J. Harizanova, and V. Sainov, “Pattern projection with a sinusoidal phase grating,” EURASIP J. Adv. Signal Process. 2009, 351626 (2009).

E. Stoykova, J. Harizanova, and V. Sainov, “Pattern projection profilometry for 3D coordinates measurement of dynamic scenes,” in Three Dimensional Television, H. M. Ozaktas and L. Onural, eds. (Springer, 2008), pp. 85-164.

Schwider, J.

J. Schwider, “Advanced evaluation techniques in interferometry,” Prog. Opt. 28, 271-359 (1990).
[CrossRef]

Shakher, C.

T. Anna, S. K. Dubey, C. Shakher, A. Roy, and D. S. Mehta, “Sinusoidal fringe projection system based on compact and non-mechanical scanning low coherence Michelson interferometer for three-dimensional shape measurement,” Opt. Commun. 282, 1237-1242 (2009).
[CrossRef]

S. K. Dubey, D. S. Mehta, A. Roy, and C. Shakher, “Wavelength scanning Talbot effect: Wavelength-scanning Talbot effect and its application for arbitrary threedimensional step-height measurement,” Opt. Commun. 279, 13-19(2007).
[CrossRef]

D. S. Mehta, S. K. Dubey, M. M. Hossain, and C. Shakher, “Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry,” Appl. Opt. 44, 7515-7521 (2005).
[CrossRef]

Shang, H. M.

C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang, “Shape measurement by use of liquid-crystal display fringe projection with two-step phase shifting,” Appl. Opt. 42, 2329-2335(2003).
[CrossRef]

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, and H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase-shifting,” Opt. Commun. 189, 21-29 (2001).
[CrossRef]

Song, M.

F. Chen, G. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10-22 (2000).

Stetson, K. A.

Stoykova, E.

E. Stoykova, J. Harizanova, and V. Sainov, “Pattern projection with a sinusoidal phase grating,” EURASIP J. Adv. Signal Process. 2009, 351626 (2009).

E. Stoykova, J. Harizanova, and V. Sainov, “Pattern projection profilometry for 3D coordinates measurement of dynamic scenes,” in Three Dimensional Television, H. M. Ozaktas and L. Onural, eds. (Springer, 2008), pp. 85-164.

Strand, T. C.

Su, X.

T. Xian and X. Su, “Area modulation grating for sinusoidal structure illumination on phase-measuring profilometry,” Appl. Opt. 40, 1201-1206 (2001).
[CrossRef]

X. Su, W.-S. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Surrel, Y.

Tay, C.

Tay, C. J.

C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang, “Shape measurement by use of liquid-crystal display fringe projection with two-step phase shifting,” Appl. Opt. 42, 2329-2335(2003).
[CrossRef]

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, and H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase-shifting,” Opt. Commun. 189, 21-29 (2001).
[CrossRef]

Teng, S.

Terrillon, J.-C.

Testorf, M.

Thakur, M.

Thomas, J. A.

Tian, J.

von Bally, G.

X. Su, W.-S. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Vukicevic, D.

X. Su, W.-S. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Wang, C. F.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, and H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase-shifting,” Opt. Commun. 189, 21-29 (2001).
[CrossRef]

Wei, L.

Xi, J.

Xian, T.

Yang, Z.

Yatagai, T.

Yoneyama, S.

S. Yoneyama, Y. Morimoto, M. Fujigaki, and Y. Ikeda, “Three-dimensional surface profile measurement of a moving object by a spatial-offset phase stepping method,” Opt. Eng. 42, 137-142 (2003).

Zhang, C.

P. S. Huang, C. Zhang, and F.-P. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42, 163-168 (2003).

Zhang, D.

Zhang, P.

Zhao, Y.

Zhou, T.

Zhou, W.-S.

X. Su, W.-S. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Zu, J.

Appl. Opt. (18)

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51-60(1996).
[CrossRef]

D. S. Mehta, S. K. Dubey, M. M. Hossain, and C. Shakher, “Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry,” Appl. Opt. 44, 7515-7521 (2005).
[CrossRef]

C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang, “Shape measurement by use of liquid-crystal display fringe projection with two-step phase shifting,” Appl. Opt. 42, 2329-2335(2003).
[CrossRef]

Y. Hao, Y. Zhao, and D. Li, “Multifrequency grating projection profilometry based on the nonlinear excess fraction method,” Appl. Opt. 38, 4106-4111 (1999).
[CrossRef]

C. Tay, M. Thakur, and C. Quan, “Grating projection system for surface contour measurement,” Appl. Opt. 44, 1393-1400(2005).
[CrossRef]

T. Xian and X. Su, “Area modulation grating for sinusoidal structure illumination on phase-measuring profilometry,” Appl. Opt. 40, 1201-1206 (2001).
[CrossRef]

H. Guo, H. He, and M. Chen, “Gamma correction for digital fringe projection profilometry,” Appl. Opt. 43, 2906-2914(2004).
[CrossRef]

K. A. Stetson and W. R. Brohinsky, “Electrooptic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631-3637 (1985).
[CrossRef]

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, “Three-dimensional profilometry based on shift estimation of projected fringe patterns,” Appl. Opt. 45, 678-687 (2006).
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Figures (14)

Fig. 1
Fig. 1

Intensity distribution (a) and spatial frequency spectrum (b) of the light transmitted by a SPG in the Fresnel zone as a function of the distance z from the grating at plane-wave illumination. For convenience, the part of the spectrum at zero frequency is omitted. The grating parameters are spacing L = 0.025 cm , modulation parameter m = 0.2 ; the wavelength is λ = 660 nm .

Fig. 2
Fig. 2

Fringe contrast in the Fresnel zone as a function of the distance from a SPG at divergent illumination: L = 0.025 cm , m = 0.2 .

Fig. 3
Fig. 3

Diffraction pattern (top) and spectral content (bottom) of the fringes along the z axis at divergent illumination for d = 12 cm , λ = 830 nm , L = 0.025 cm , m = 0.2 . The zero-order term is omitted for clarity.

Fig. 4
Fig. 4

Distance z m f h at which the first harmonic is missing as a function of the location of the illuminating point source; the meaning of the parameter n is clarified in Eq. (16).

Fig. 5
Fig. 5

Fringe contrast in the Fresnel zone as a function of the distance from a SAG at divergent illumination; L = 0.025 cm .

Fig. 6
Fig. 6

The ratio between the energy concentrated in the second and first harmonics in the fringes created by a SAG with A 0 = 4 A 1 and a SPG with m = 0.2 as a function of the distance from the grating; for both gratings L = 0.025 cm .

Fig. 7
Fig. 7

Grey scale 8   bit maps of (top) wrapped phase distributions φ ^ ( x , y ) and φ ^ R ( x , y ) corresponding to the object (left) and the reference plane (right); (bottom) reconstructed dome surface h ^ ( x , y ) (left) and distribution of the reconstruction error h ^ ( x , y ) h ( x , y ) (right); L = 0.025 cm , m = 0.2 , z = 1 m , λ = 830 nm , d = 12 cm , θ = 36 ° .

Fig. 8
Fig. 8

Deviation of the reconstructed surface from the real surface averaged in a row as a function of the row number; L = 0.025 cm , m = 0.2 , θ = 36 ° .

Fig. 9
Fig. 9

Mean value of the maximum spread of the deviation between the reconstructed surface and the real surface as a function of the angle θ; L = 0.025 cm , m = 0.2 .

Fig. 10
Fig. 10

(top) Optical arrangement for determination of frequency content of patterns projected by a sinusoidal phase grating: DL, diode laser; L 1 , L 2 , lenses; SPG, sinusoidal phase grating; GGS, ground glass screen. (bottom) Cross-axes optical arrangement for profilometric measurement; RP, reference plane.

Fig. 11
Fig. 11

Visibility of the fringes created by a SPG with L = 0.025 cm (experiment).

Fig. 12
Fig. 12

Experimentally determined frequency content of fringes created by a SPG with L = 0.025 cm : (a), (b) λ = 660 nm ; (c), (d) λ = 830 nm ; ( a), ( c) d * = 9 cm , ( b), ( d) d * = 10.5 cm ; the zero-order term is omitted for clarity.

Fig. 13
Fig. 13

Reconstructed 3D surface of a dome (the five-step algorithm has been used for phase retrieval).

Fig. 14
Fig. 14

Comparison of the reconstructed profile through the dome apex with the real one measured with a stylus.

Equations (21)

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τ ( x , y ) = exp [ j m sin ( 2 π x L ) ] = q = J q ( m ) exp ( j 2 π q x L ) ,
U ( x , z ) = exp ( j k z ) j λ z exp ( j k 2 z x 2 ) U ( x 0 , 0 ) exp ( j k 2 z x 0 2 ) exp ( j k z x x 0 ) d x 0 ,
U ( x 0 , 0 ) = A 0 τ ( x 0 ) exp ( j k 2 d x 0 2 ) ,
U ( x , z ) = A 0 exp ( j k z ) j z λ η exp ( j k 2 z x 2 ) q = J q ( m ) { ( 1 + j ) cos [ 2 ( π σ q ) 2 k η ] + ( 1 j ) sin [ 2 ( π σ q ) 2 k η ] } ,
I ( x , z ) = 2 A 0 2 d z λ ( z + d ) [ S 1 2 ( x , z ) + S 2 2 ( x , z ) ] = 2 A 0 2 d z λ ( z + d ) I S ( x , z ) ,
S 1 = q = J q ( m ) cos [ 2 ( π σ q ) 2 k η ] , S 2 = q = J q ( m ) sin [ 2 ( π σ q ) 2 k η ] .
I S = q = J q 2 + q = p = p q J q J p cos [ ( q 2 p 2 ) α 2 ( q p ) β ] ,
α = π λ z d L 2 ( z + d ) , β = π x d L ( z + d ) .
I ( x , z ) = 2 A 0 2 d z λ ( z + d ) { I 0 + I V ( z ) sin 2 π x d L ( z + d ) + Θ ( x , z ) } ,
I 0 = q = J q 2 I V ( z ) = 2 { q = 0 J 2 q J 2 q + 1 sin ( 2 q + 1 ) α + q = 1 J 2 q J 2 q 1 sin ( 2 q 1 ) α } Θ ( x , z ) = 2 q = 0 p = 1 [ J 2 q J 2 q + 2 p + 1 sin ( 4 p β ) × sin { α ( 2 p + 1 ) ( 2 p + 4 q + 1 ) } + cos ( 4 p β ) × ( J 2 q J 2 q + 2 p cos { 4 α ( q + 2 p ) } + J 2 q + 1 J 2 q + 2 p + 1 cos { 4 α p ( q + 2 p + 1 ) } ) ] .
z n = 2 n L 2 λ , n = 1 , 2 ; z n = ( 2 n + 1 ) L 2 λ , n = 0 , 1 , 2 .
z n = ( 2 n + 1 / 2 ) L 2 λ , n = 1 , 2 ,
1 z + 1 d = λ n L 2
V ( z ) = I ( 0 , z ) I [ L / 2 M ( z ) , z ] I ( 0 , z ) + I [ L / 2 M ( z ) , z ] .
τ A ( x ) = A 0 + 2 A 1 cos ( 2 π x / L ) ,
I A ( x , z ) = A 0 2 + 2 A 1 2 + 4 A 0 A 1 cos [ π λ z d L 2 ( z + d ) ] cos [ 2 π x L ( 1 + z / d ) ] + 2 A 1 2 cos [ 4 π x L ( 1 + z / d ) ] = A 0 2 + 2 A 1 2 + 4 A 0 A 1 cos α cos 2 β + 2 A 1 2 cos 4 β .
z m f h = n L 2 2 λ d n L 2 , n = 2 k + 1 , k = 0 , 1 , 2 ,
I ( x , y ) = I 0 ( x , y ) + I V ( x , y ) f [ φ ( x , y ) + ϕ ] ,
h ( x , y ) = L ( l p d + 1 ) { sin θ + [ l c ( l p + d ) ] cos θ l c ( l p + d ) } 1 [ 1 + x sin θ l p + d ] 2 Δ φ ( x , y ) ,
φ ^ ( x , y ) = arctan I 4 ( x , y ) I 2 ( x , y ) I 1 ( x , y ) I 3 ( x , y ) ,
φ ^ ( x , y ) = arctan 2 [ I 4 ( x , y ) I 2 ( x , y ) ] I 1 ( x , y ) 2 I 3 ( x , y ) + I 5 ( x , y ) ,

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