Abstract

The generalized analytical expression for the propagation of flat-topped multi-Gaussian beams through a misaligned apertured ABCD optical system is derived. Using this analytical expression, the propagation characteristics of flat-topped multi-Gaussian beams through a spatial filter are investigated. The analytical formula of the electric field distribution in the focal plane is also derived for revealing the effects of the misalignment parameters clearly. It is found that different misalignment parameters have different influences on the electric field distributions of the beam focus spot and the output beam characteristics. The intensity distribution of the beam is mainly determined by the misalignment matrix element E, and the phase distribution is affected by the misalignment matrix elements G and E.

© 2009 Optical Society of America

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2009 (7)

2008 (5)

2007 (4)

2006 (2)

Y. J. Cai and S. L. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am. A 23, 2623-2628 (2006).
[CrossRef]

I. Moreno, C. Ferreira, and M. M. Sánchez-López, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A, Pure Appl. Opt. 8, 427-435 (2006).
[CrossRef]

2005 (2)

P. Wu, B. Lü, and T. Chen, “Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function,” Chin. Phys. 14, 1130-1135 (2005).
[CrossRef]

Z. Mei and D. Zhao, “Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams,” Appl. Opt. 44, 1381-1386 (2005).
[CrossRef] [PubMed]

2004 (2)

D. Ge, Y. J. Cai, and Q. Lin, “Partially coherent flat-topped beam and its propagation,” Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Y. J. Cai and Q. Lin, “Partially coherent flat-topped multi-Gaussian-Schell-model beam and its propagation,” Opt. Commun. 239, 33-41 (2004).
[CrossRef]

2003 (1)

X. L. Jim and B. D. Lü, “Propagation of a flattened Gaussian beam through muti-apertured optical ABCD systems,” Optik (Stuttgart) 114, 394-400 (2003).
[CrossRef]

2002 (2)

B. D. Lü and B. Z. Li, “Study of the correspondence between flat-topped multi-Gaussian beams and super-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 4, 509-513 (2002).
[CrossRef]

J. E. Harvey and A. Krywonos, “Axial irradiance distribution throughout the whole space behind an annular aperture,” Appl. Opt. 41, 3790-3795 (2002).
[CrossRef] [PubMed]

2001 (2)

A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A 18, 1897-1904 (2001).
[CrossRef]

B. Lü and S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2619-2718 (2001).

2000 (3)

1998 (2)

1996 (1)

1995 (1)

1994 (2)

1988 (2)

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

S. D. Silvestri, P. Laporta, V. Magni, O. Svelto, and B. Majocchi, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201-203 (1988).
[CrossRef] [PubMed]

1978 (1)

1973 (1)

A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85-87 (1973).
[CrossRef]

1970 (1)

Agresti, J.

Barmashova, T. V.

Boley, C. D.

Borghi, R.

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Cai, Y. J.

Caird, J. A.

Campillo, A. J.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85-87 (1973).
[CrossRef]

Cao, Q.

Celliers, P. M.

Chen, J. N.

Chen, T.

P. Wu, B. Lü, and T. Chen, “Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function,” Chin. Phys. 14, 1130-1135 (2005).
[CrossRef]

Chi, S.

Collins, S. A.

D'Ambrosio, E.

Dan, Y. Q.

DeSalvo, R.

Estabrook, K. G.

Eyyuboglu, H. T.

Ferreira, C.

I. Moreno, C. Ferreira, and M. M. Sánchez-López, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A, Pure Appl. Opt. 8, 427-435 (2006).
[CrossRef]

Forest, D.

Gan, X. S.

Gao, Y. Q.

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48, 1591-1597 (2009).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18, 215-220 (2009).
[CrossRef]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, and Z. Q. Lin, “Propagation of flat-topped multi-Gaussian beams through a double-lens system with apertures,” Opt. Express 17, 12753-12766 (2009).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, Z. Y. Peng, and Z. Q. Lin, “Study of mathematical model for auto-alignment in four-pass amplifier,” Chin. Phys. B 57, 6992-6997 (2008).

Ge, D.

Glaze, J. A.

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335-341 (1994).
[CrossRef]

Gu, M.

Harvey, J. E.

He, S. L.

Hunt, J. T.

Jim, X. L.

X. L. Jim and B. D. Lü, “Propagation of a flattened Gaussian beam through muti-apertured optical ABCD systems,” Optik (Stuttgart) 114, 394-400 (2003).
[CrossRef]

Khazanov, E. A.

Kirsanov, A. V.

Krywonos, A.

Lagrange, B.

Laporta, P.

Li, B. Z.

B. D. Lü and B. Z. Li, “Study of the correspondence between flat-topped multi-Gaussian beams and super-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 4, 509-513 (2002).
[CrossRef]

Lin, Q.

Y. J. Cai and Q. Lin, “Partially coherent flat-topped multi-Gaussian-Schell-model beam and its propagation,” Opt. Commun. 239, 33-41 (2004).
[CrossRef]

D. Ge, Y. J. Cai, and Q. Lin, “Partially coherent flat-topped beam and its propagation,” Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Lin, Z. Q.

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18, 215-220 (2009).
[CrossRef]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, and Z. Q. Lin, “Propagation of flat-topped multi-Gaussian beams through a double-lens system with apertures,” Opt. Express 17, 12753-12766 (2009).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48, 1591-1597 (2009).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, Z. Y. Peng, and Z. Q. Lin, “Study of mathematical model for auto-alignment in four-pass amplifier,” Chin. Phys. B 57, 6992-6997 (2008).

Liu, D. J.

Liu, D. Z.

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48, 1591-1597 (2009).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, and Z. Q. Lin, “Propagation of flat-topped multi-Gaussian beams through a double-lens system with apertures,” Opt. Express 17, 12753-12766 (2009).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18, 215-220 (2009).
[CrossRef]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, Z. Y. Peng, and Z. Q. Lin, “Study of mathematical model for auto-alignment in four-pass amplifier,” Chin. Phys. B 57, 6992-6997 (2008).

Liu, X. F.

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48, 1591-1597 (2009).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18, 215-220 (2009).
[CrossRef]

Lu, X. H.

Lü, B.

P. Wu, B. Lü, and T. Chen, “Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function,” Chin. Phys. 14, 1130-1135 (2005).
[CrossRef]

B. Lü and S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2619-2718 (2001).

B. Lü and S. Luo, “General propagation equation of flatted Gaussian beams,” J. Opt. Soc. Am. A 17, 2001-2004 (2000).
[CrossRef]

Lü, B. D.

X. L. Jim and B. D. Lü, “Propagation of a flattened Gaussian beam through muti-apertured optical ABCD systems,” Optik (Stuttgart) 114, 394-400 (2003).
[CrossRef]

B. D. Lü and B. Z. Li, “Study of the correspondence between flat-topped multi-Gaussian beams and super-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 4, 509-513 (2002).
[CrossRef]

Luo, S.

B. Lü and S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2619-2718 (2001).

B. Lü and S. Luo, “General propagation equation of flatted Gaussian beams,” J. Opt. Soc. Am. A 17, 2001-2004 (2000).
[CrossRef]

MacGowan, B. J.

Mackowsky, J. M.

Magni, V.

Majocchi, B.

Manes, K. R.

Martyanov, M. A.

Mei, Z.

Mendlovic, D.

Michel, C.

Milam, D.

Miller, J.

Montorio, J. L.

Moreno, I.

I. Moreno, C. Ferreira, and M. M. Sánchez-López, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A, Pure Appl. Opt. 8, 427-435 (2006).
[CrossRef]

Morgado, N.

Murray, J. E.

Nix, M.

Ozaktas, H. M.

Pan, P. P.

Pan, W. Q.

W. Q. Pan and Y. J. Zhu, “The impulse response function of apertured and misaligned ABCD optical system in phase-space,” Opt. Commun. 282, 752-756 (2009).
[CrossRef]

Pearson, J. E.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85-87 (1973).
[CrossRef]

Pellat-Finet, P.

Peng, Z. Y.

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, Z. Y. Peng, and Z. Q. Lin, “Study of mathematical model for auto-alignment in four-pass amplifier,” Chin. Phys. B 57, 6992-6997 (2008).

Pinard, L.

Potemkin, A. K.

Remilleux, A.

Renard, P. A.

Ronchi, L.

S. Wang and L. Ronchi, “Principles and design of optical array,” in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier, 1988), pp. 284-323.

Sánchez-López, M. M.

I. Moreno, C. Ferreira, and M. M. Sánchez-López, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A, Pure Appl. Opt. 8, 427-435 (2006).
[CrossRef]

Santarsiero, M.

Shapiro, S. L.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85-87 (1973).
[CrossRef]

Shaykin, A. A.

Silva, L. B. D.

Silvestri, S. D.

Simmons, W. W.

Simoni, B.

Svelto, O.

Tarallo, M. G.

Terrell, N. J.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85-87 (1973).
[CrossRef]

Tian, Z. B.

Tovar, A. A.

Wallace, R. J.

Wang, F.

Wang, S.

S. Wang and L. Ronchi, “Principles and design of optical array,” in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier, 1988), pp. 284-323.

Wen, J. J.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Willems, P.

Wonterghem, B. M. V.

Wu, P.

P. Wu, B. Lü, and T. Chen, “Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function,” Chin. Phys. 14, 1130-1135 (2005).
[CrossRef]

Yam, S. S.-H.

Zhang, B.

Zhao, C. L.

Zhao, D.

Zhao, D. M.

Zheng, C. W.

C. W. Zheng, “Off-axis flat-topped multi-Gaussian beam and its propagation through a paraxial optical system,” Optik (Stuttgart) 118, 552-556 (2007).
[CrossRef]

Zhou, Z. X.

Zhu, B. Q.

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48, 1591-1597 (2009).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, and Z. Q. Lin, “Propagation of flat-topped multi-Gaussian beams through a double-lens system with apertures,” Opt. Express 17, 12753-12766 (2009).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18, 215-220 (2009).
[CrossRef]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, Z. Y. Peng, and Z. Q. Lin, “Study of mathematical model for auto-alignment in four-pass amplifier,” Chin. Phys. B 57, 6992-6997 (2008).

Zhu, Y. B.

Zhu, Y. J.

W. Q. Pan and Y. J. Zhu, “The impulse response function of apertured and misaligned ABCD optical system in phase-space,” Opt. Commun. 282, 752-756 (2009).
[CrossRef]

Appl. Opt. (9)

P. M. Celliers, K. G. Estabrook, R. J. Wallace, J. E. Murray, L. B. D. Silva, B. J. MacGowan, B. M. V. Wonterghem, and K. R. Manes, “Spatial filter pinhole for high-energy pulsed lasers,” Appl. Opt. 37, 2371-2378 (1998).
[CrossRef]

J. E. Murray, D. Milam, C. D. Boley, K. G. Estabrook, and J. A. Caird, “Spatial filter pinhole development for the national ignition facility,” Appl. Opt. 39, 1405-1420 (2000).
[CrossRef]

J. E. Harvey and A. Krywonos, “Axial irradiance distribution throughout the whole space behind an annular aperture,” Appl. Opt. 41, 3790-3795 (2002).
[CrossRef] [PubMed]

D. Ge, Y. J. Cai, and Q. Lin, “Partially coherent flat-topped beam and its propagation,” Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Z. Mei and D. Zhao, “Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams,” Appl. Opt. 44, 1381-1386 (2005).
[CrossRef] [PubMed]

J. T. Hunt, J. A. Glaze, W. W. Simmons, and P. A. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. 17, 2053-2057 (1978).
[CrossRef] [PubMed]

A. K. Potemkin, T. V. Barmashova, A. V. Kirsanov, M. A. Martyanov, E. A. Khazanov, and A. A. Shaykin, “Spatial filters for high-peak-power multistage laser amplifiers,” Appl. Opt. 46, 4423-4430 (2007).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Misaligned optical system with aperture.

Fig. 2
Fig. 2

Spatial filter: (a) Aligned spatial filter; (b) one of the misaligned lenses in (a).

Fig. 3
Fig. 3

Electric field intensity distributions in the focal plane for different misaligned parameters: (a), (b), (c) Three-dimensional figures with the misaligned parameters ϵ 1 x = ϵ 1 y = 0 , ϵ 1 x = ϵ 1 y = 0.1 mm , and ϵ 1 x = ϵ 1 y = 0.2 mm , respectively. (d) profiles of (a), (b), (c).

Fig. 4
Fig. 4

Output of the spatial filter for different misalignment parameters of the left lens L 1 , (a), (b), (c), (d) Three-dimensional intensity distributions when ϵ 1 x = ϵ 1 y = 0 , ϵ 1 x = ϵ 1 y = 0.1 mm , ϵ 1 x = ϵ 1 y = 0.2 mm , and ϵ 1 x = ϵ 1 y = 0.27 mm , respectively, (e) profiles of (a), (b), (c), (d).

Fig. 5
Fig. 5

Profiles of the output beam for different misalignment parameters ϵ 2 x and ϵ 2 y when z 2 = f 2 = 2000 mm and the diameter of the pinhole is 5 times the diffraction limit.

Equations (23)

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[ A 1 x B 1 x α T 1 x ϵ 1 x β T 1 x ϵ 1 x C 1 x D 1 x γ T 1 x ϵ 1 x η T 1 x ϵ 1 x 0 0 1 0 0 0 0 1 ] = [ a b x b b x 0 0 c b x d b x 0 0 0 0 1 0 0 0 0 1 ] [ a m x b m x α m x ϵ m x β m x ϵ m x c m x d m x γ m x ϵ m x η m x ϵ m x 0 0 1 0 0 0 0 1 ] [ a f x b f x 0 0 c f x d f x 0 0 0 0 1 0 0 0 0 1 ] ,
α m x = 1 a m x , β m x = l b m x ,
γ m x = c m x , η m x = ± 1 d m x ,
E ( x 2 , y 2 ) = i k 2 π B 1 x B 1 y exp ( i k L 1 ) E ( x 1 , y 1 ) exp [ i k 2 B 1 x ( A 1 x x 1 2 2 x 1 x 2 + D 1 x x 2 2 + E 1 x x 2 + G 1 x x 2 ) ] exp [ i k 2 B 1 y ( A 1 y y 1 2 2 y 1 y 2 + D 1 y y 2 2 + E 1 y y 1 + G 1 y y 2 ) ] d x 1 d y 1 ,
E = 2 ( α T ϵ + β T ϵ ) ,
G = 2 ( B γ T D α T ) ϵ + 2 ( B η T D β T ) ϵ .
E ( x 1 , y 1 ) = n 1 = N 1 N 1 m 1 = M 1 M 1 exp [ ( x 1 n 1 ω 1 ) 2 + ( y 1 m 1 ω 1 ) 2 ω 1 2 ] n 1 = N 1 N 1 m 1 = M 1 M 1 exp [ ( n 1 2 + m 1 2 ) ] ,
W 1 = ω 1 ( N 1 + { 1 ln [ n 1 = N 1 N 1 exp ( n 1 2 ) ] } 1 2 ) .
P ( x 2 , y 2 ) = n 2 = N 2 N 2 m 2 = M 2 M 2 exp [ ( x 2 n 2 ω 2 x p ) 2 + ( y 2 m 2 ω 2 y p ) 2 ω 2 2 ] n 2 = N 2 N 2 m 2 = M 2 M 2 exp [ ( n 2 2 + m 2 2 ) ] ,
E ( x 3 , y 3 ) = i k 2 π B 2 x B 2 y i k 2 π B 1 x B 1 y exp ( i k L 2 ) exp ( i k L 1 ) P ( x 2 , y 2 ) E ( x 1 , y 1 ) × exp [ i k 2 B 1 x ( A 1 x x 1 2 2 x 1 x 2 + D 1 x x 2 2 + E 1 x x 1 + G 1 x x 2 ) ] exp [ i k 2 B 1 y ( A 1 y y 1 2 2 y 1 y 2 + D 1 y y 2 2 + E 1 y y 1 + G 1 y y 2 ) ] d x 1 d y 1 × exp [ i k 2 B 2 x ( A 2 x x 2 2 2 x 2 x 3 + D 2 x x 3 2 + E 2 x x 2 + G 2 x x 3 ) ] exp [ i k 2 B 2 y ( A 2 y y 2 2 2 y 2 y 3 + D 2 y y 3 2 + E 2 y y 2 + G 2 y y 3 ) ] d x 2 d y 2 .
exp ( p 2 x 2 ± q x ) d x = exp ( q 2 4 p 2 ) π p , ( Re p 2 > 0 ) ,
E ( x 3 , y 3 ) = i k 2 π B 2 x B 2 y i k 2 π B 1 x B 1 y exp ( i k L 1 ) n 1 = N 1 N 1 m 1 = M 1 M 1 exp [ ( n 1 2 + m 1 2 ) ] exp ( i k L 2 ) n 2 = N 2 N 2 m 2 = M 2 M 2 exp [ ( n 2 2 + m 2 2 ) ] π Q 1 x Q 1 y × π Q 2 x Q 2 y n 2 = N 2 N 2 m 2 = M 2 M 2 n 1 = N 1 N 1 m 1 = M 1 M 1 exp { [ ( n 2 ω 2 + x p ) 2 ω 2 2 n 1 2 + 1 4 Q 1 x ( 2 n 1 ω 1 + i k E 1 x 2 B 1 x ) 2 + i k 2 B 2 x ( D 2 x x 3 2 + G 2 x x 3 ) ] + [ ( m 2 ω 2 + y p ) 2 ω 2 2 m 1 2 + 1 4 Q 1 y ( 2 m 1 ω 1 + i k E 1 y 2 B 1 y ) 2 + i k 2 B 2 y ( D 2 y y 3 2 + G 2 y y 3 ) ] + 1 4 Q 2 x [ 2 ( n 2 ω 2 + x p ) ω 2 2 + i k G 1 x 2 B 1 x i k 2 Q 1 x B 1 x ( 2 n 1 ω 1 + i k E 1 x 2 B 1 x ) i k 2 B 2 x ( 2 x 3 E 2 x ) ] 2 + 1 4 Q 2 y [ 2 ( m 2 ω 2 + y p ) ω 2 2 + i k G 1 y 2 B 1 y i k 2 Q 1 y B 1 y ( 2 m 1 ω 1 + i k E 1 y 2 B 1 y ) i k 2 B 2 y ( 2 y 3 E 2 y ) ] 2 } ,
Q 1 x = 1 ω 1 2 i k A 1 x 2 B 1 x , Q 1 y = 1 ω 1 2 i k A 1 y 2 B 1 y ,
Q 2 x = 1 ω 2 2 i k D 1 x 2 B 1 x 1 4 Q 1 x ( i k B 1 x ) 2 i k A 2 x 2 B 2 x , Q 2 y = 1 ω 2 2 i k D 1 y 2 B 1 y 1 4 Q 1 y ( i k B 1 y ) 2 i k A 2 y 2 B 2 y .
E ( x 3 , y 3 ) = i k 2 π B 2 x B 2 y i k 2 π B 1 x B 1 y exp ( i k L 1 ) n 1 = N 1 N 1 m 1 = M 1 M 1 exp [ ( n 1 2 + m 1 2 ) ] exp ( i k L 2 ) n 2 = N 2 N 2 m 2 = M 2 M 2 exp [ ( n 2 2 + m 2 2 ) ] × π Q 1 x Q 1 y π Q 2 x Q 2 y n 2 = N 2 N 2 m 2 = M 2 M 2 n 1 = N 1 N 1 m 1 = M 1 M 1 exp { [ ( n 2 ω 2 + x p ) 2 ω 2 2 n 1 2 + n 1 2 Q 1 x ω 1 2 + i k D 2 x x 3 2 2 B 2 x ] + [ ( m 2 ω 2 + y p ) 2 ω 2 2 m 1 2 + m 1 2 Q 1 y ω 1 2 + i k D 2 y y 3 2 2 B 2 y ] + 1 4 Q 2 x [ 2 ( n 2 ω 2 + x p ) ω 2 2 i k n 1 Q 1 x B 1 x ω 1 i k x 3 B 2 x ] 2 + 1 4 Q 2 y [ 2 ( m 2 ω 2 + y p ) ω 2 2 i k m 1 Q 1 y B 1 y ω 1 i k y 3 B 2 y ] 2 } .
[ A 1 x B 1 x α T 1 x ϵ 1 x β T 1 x ϵ 1 x C 1 x D 1 x γ T 1 x ϵ 1 x η T 1 x ϵ 1 x 0 0 1 0 0 0 0 1 ] = [ 1 f 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] × [ 1 0 0 0 1 f 1 1 ϵ 1 x f 1 0 0 0 1 0 0 0 0 1 ] [ 1 z 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] = [ 0 f 1 ϵ 1 x 0 1 f 1 1 z 1 f 1 ϵ 1 x f 1 0 0 0 1 0 0 0 0 1 ] ,
α T 1 x = 1 , β T 1 x = 0 , γ T 1 x = 1 f 1 , η T 1 x = 0 .
A 1 x = 0 , B 1 x = f 1 , C 1 x = 1 f 1 ,
D 1 x = 1 z 1 f 1 , E 1 x = 2 ϵ 1 x , G 1 x = 2 z 1 ϵ 1 x f 1 .
α T 2 x = z 2 f 2 , β T 2 x = 0 , γ T 2 x = 1 f 2 , η T 2 x = 0 .
A 2 x = 1 z 2 f 2 , B 2 x = f 2 , C 2 x = 1 f 2 ,
D 2 x = 0 , E 2 x = 2 ϵ 2 x z 2 f 2 , G 2 x = 2 ϵ 2 x .
E ( x 2 , y 2 ) = i k 2 π B 1 x B 1 y exp ( i k L 1 ) n 1 = N 1 N 1 m 1 = M 1 M 1 exp [ ( n 1 2 + m 1 2 ) ] exp [ i k 2 B 1 x ( D 1 x x 2 2 + G 1 x x 2 ) + i k 2 B 1 y ( D 1 y y 2 2 + G 1 y y 2 ) ] × π Q 1 x Q 1 y n 1 = N 1 N 1 m 1 = M 1 M 1 exp [ n 1 2 m 1 2 + 1 4 Q 1 x ( 2 n 1 ω 1 i k x 2 B 1 x + i k E 1 x 2 B 1 x ) 2 + 1 4 Q 1 y ( 2 m 1 ω 1 i k y 2 B 1 y + i k E 1 y 2 B 1 y ) 2 ] .

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