Abstract

The axicon-based-Bessel–Gauss resonator (ABGR) has been proposed for the production of Bessel–Gauss beams. To analyze eigenfields of the ABGR with a plane or spherical output coupler, we present and demonstrate the transfer-matrix method. Since the method is slow to converge to eigenmodes of the ABGR by use of the Fox and Li iterative algorithm, in this paper the Huygens–Fresnel diffraction integral equations associated with ray matrices are converted into finite-sum matrix equations, and mode-fields and corresponding losses are described as eigenvectors and eigenvalues of a transfer matrix according to the self-reproducing principle of the laser field. By solving the transfer matrix for eigenvectors and eigenvalues, we obtain field distributions and losses of the dominant eigenmodes. Moreover, eigenfields across arbitrary interfaces between the axicon and the output coupler, and the propagation of output beams, are simulated by using the fast-Fourier transform (FFT). The calculation results reveal that because of the ABGR’s poor transverse mode discrimination the ABGR should be improved to produce good-quality Bessel–Gauss beams.

© 2006 Optical Society of America

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  1. J. Durnin, J. J. Micely, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  2. J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  3. F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  4. J. K. Jabczynski, "A 'diffraction-free' resonator," Opt. Commun. 77, 292-294 (1990).
    [CrossRef]
  5. J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
    [CrossRef]
  6. J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Bessel-Gauss resonators," in Laser Resonators IV, A.V.Kudryashov and A.H.Paxton, eds., Proc. SPIE 4270, 52-63 (2001).
  7. A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, "Axicon-based Bessel resonator: analytical description and experiment," J. Opt. Soc. Am. A 18, 1986-1992 (2001).
    [CrossRef]
  8. J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, "Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis," J. Opt. Soc. Am. A 20, 2113-2122 (2003).
    [CrossRef]
  9. P. Baues, "The connection of geometric optics with the propagation of Gaussian beams and the theory of optical resonators," Opto-electronics (London) 1, 103-118 (1969).
    [CrossRef]
  10. P. Baues, "Huygens's principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators," Opto-electronics (London) 1, 37-44 (1969).
    [CrossRef]
  11. S. A. Collins, Jr., "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970).
    [CrossRef]
  12. B. Lü, Propagation and Control of High-power Lasers (National Defense Industry Press, Beijing, 1999), p. 23.
  13. L. Dongxiong, L. Junchang, and L. Xingyi, "Numerical simulation of laser field across the diffraction-limited optics system," Laser Technol. 26, 284-286 (2002).
  14. A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-458 (1961).
  15. A. G. Fox and T. Li, "Resonant modes in an optical maser," Proc. IRE 48, 1904-1905 (1960).
  16. W. Zaifu, W. Runwen, and W. Zhijiang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).
  17. D. Ling, Y. Fu, D. Xu, and Y. Guan, "Finite-sum matrix analysis of eigen-mode fields of the Gaussian-reflectivity plano-concave resonator," in High-Power Lasers and Applications II, DianyuanFan, KeithA.Truesdell, and KojiYasui, eds., Proc. SPIE 4914, 371-381 (2002).

2003 (1)

2002 (1)

L. Dongxiong, L. Junchang, and L. Xingyi, "Numerical simulation of laser field across the diffraction-limited optics system," Laser Technol. 26, 284-286 (2002).

2001 (2)

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, "Axicon-based Bessel resonator: analytical description and experiment," J. Opt. Soc. Am. A 18, 1986-1992 (2001).
[CrossRef]

1995 (1)

W. Zaifu, W. Runwen, and W. Zhijiang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

1990 (1)

J. K. Jabczynski, "A 'diffraction-free' resonator," Opt. Commun. 77, 292-294 (1990).
[CrossRef]

1987 (3)

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

J. Durnin, J. J. Micely, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

1970 (1)

1969 (2)

P. Baues, "The connection of geometric optics with the propagation of Gaussian beams and the theory of optical resonators," Opto-electronics (London) 1, 103-118 (1969).
[CrossRef]

P. Baues, "Huygens's principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators," Opto-electronics (London) 1, 37-44 (1969).
[CrossRef]

1961 (1)

A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-458 (1961).

1960 (1)

A. G. Fox and T. Li, "Resonant modes in an optical maser," Proc. IRE 48, 1904-1905 (1960).

Baues, P.

P. Baues, "The connection of geometric optics with the propagation of Gaussian beams and the theory of optical resonators," Opto-electronics (London) 1, 103-118 (1969).
[CrossRef]

P. Baues, "Huygens's principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators," Opto-electronics (London) 1, 37-44 (1969).
[CrossRef]

Chávez-Cerda, S.

J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, "Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis," J. Opt. Soc. Am. A 20, 2113-2122 (2003).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Bessel-Gauss resonators," in Laser Resonators IV, A.V.Kudryashov and A.H.Paxton, eds., Proc. SPIE 4270, 52-63 (2001).

Collins, S. A.

Dongxiong, L.

L. Dongxiong, L. Junchang, and L. Xingyi, "Numerical simulation of laser field across the diffraction-limited optics system," Laser Technol. 26, 284-286 (2002).

Durnin, J.

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

J. Durnin, J. J. Micely, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Micely, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Fox, A. G.

A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-458 (1961).

A. G. Fox and T. Li, "Resonant modes in an optical maser," Proc. IRE 48, 1904-1905 (1960).

Fu, Y.

D. Ling, Y. Fu, D. Xu, and Y. Guan, "Finite-sum matrix analysis of eigen-mode fields of the Gaussian-reflectivity plano-concave resonator," in High-Power Lasers and Applications II, DianyuanFan, KeithA.Truesdell, and KojiYasui, eds., Proc. SPIE 4914, 371-381 (2002).

Gori, F.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Guan, Y.

D. Ling, Y. Fu, D. Xu, and Y. Guan, "Finite-sum matrix analysis of eigen-mode fields of the Gaussian-reflectivity plano-concave resonator," in High-Power Lasers and Applications II, DianyuanFan, KeithA.Truesdell, and KojiYasui, eds., Proc. SPIE 4914, 371-381 (2002).

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, "Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis," J. Opt. Soc. Am. A 20, 2113-2122 (2003).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Bessel-Gauss resonators," in Laser Resonators IV, A.V.Kudryashov and A.H.Paxton, eds., Proc. SPIE 4270, 52-63 (2001).

Jabczynski, J. K.

J. K. Jabczynski, "A 'diffraction-free' resonator," Opt. Commun. 77, 292-294 (1990).
[CrossRef]

Junchang, L.

L. Dongxiong, L. Junchang, and L. Xingyi, "Numerical simulation of laser field across the diffraction-limited optics system," Laser Technol. 26, 284-286 (2002).

Katranji, E. G.

Khilo, A. N.

Li, T.

A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-458 (1961).

A. G. Fox and T. Li, "Resonant modes in an optical maser," Proc. IRE 48, 1904-1905 (1960).

Ling, D.

D. Ling, Y. Fu, D. Xu, and Y. Guan, "Finite-sum matrix analysis of eigen-mode fields of the Gaussian-reflectivity plano-concave resonator," in High-Power Lasers and Applications II, DianyuanFan, KeithA.Truesdell, and KojiYasui, eds., Proc. SPIE 4914, 371-381 (2002).

Lü, B.

B. Lü, Propagation and Control of High-power Lasers (National Defense Industry Press, Beijing, 1999), p. 23.

Micely, J. J.

J. Durnin, J. J. Micely, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Muys, P.

J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Bessel-Gauss resonators," in Laser Resonators IV, A.V.Kudryashov and A.H.Paxton, eds., Proc. SPIE 4270, 52-63 (2001).

New, G. H. C.

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Bessel-Gauss resonators," in Laser Resonators IV, A.V.Kudryashov and A.H.Paxton, eds., Proc. SPIE 4270, 52-63 (2001).

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Rodríguez-Masegosa, R.

Rogel-Salazar, J.

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Bessel-Gauss resonators," in Laser Resonators IV, A.V.Kudryashov and A.H.Paxton, eds., Proc. SPIE 4270, 52-63 (2001).

Runwen, W.

W. Zaifu, W. Runwen, and W. Zhijiang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

Ryzhevich, A. A.

Xingyi, L.

L. Dongxiong, L. Junchang, and L. Xingyi, "Numerical simulation of laser field across the diffraction-limited optics system," Laser Technol. 26, 284-286 (2002).

Xu, D.

D. Ling, Y. Fu, D. Xu, and Y. Guan, "Finite-sum matrix analysis of eigen-mode fields of the Gaussian-reflectivity plano-concave resonator," in High-Power Lasers and Applications II, DianyuanFan, KeithA.Truesdell, and KojiYasui, eds., Proc. SPIE 4914, 371-381 (2002).

Zaifu, W.

W. Zaifu, W. Runwen, and W. Zhijiang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

Zhijiang, W.

W. Zaifu, W. Runwen, and W. Zhijiang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

Acta Opt. Sin. (1)

W. Zaifu, W. Runwen, and W. Zhijiang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-458 (1961).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Laser Technol. (1)

L. Dongxiong, L. Junchang, and L. Xingyi, "Numerical simulation of laser field across the diffraction-limited optics system," Laser Technol. 26, 284-286 (2002).

Opt. Commun. (3)

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

J. K. Jabczynski, "A 'diffraction-free' resonator," Opt. Commun. 77, 292-294 (1990).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

Opto-electronics (London) (2)

P. Baues, "The connection of geometric optics with the propagation of Gaussian beams and the theory of optical resonators," Opto-electronics (London) 1, 103-118 (1969).
[CrossRef]

P. Baues, "Huygens's principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators," Opto-electronics (London) 1, 37-44 (1969).
[CrossRef]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Micely, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Proc. IRE (1)

A. G. Fox and T. Li, "Resonant modes in an optical maser," Proc. IRE 48, 1904-1905 (1960).

Other (3)

D. Ling, Y. Fu, D. Xu, and Y. Guan, "Finite-sum matrix analysis of eigen-mode fields of the Gaussian-reflectivity plano-concave resonator," in High-Power Lasers and Applications II, DianyuanFan, KeithA.Truesdell, and KojiYasui, eds., Proc. SPIE 4914, 371-381 (2002).

B. Lü, Propagation and Control of High-power Lasers (National Defense Industry Press, Beijing, 1999), p. 23.

J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Bessel-Gauss resonators," in Laser Resonators IV, A.V.Kudryashov and A.H.Paxton, eds., Proc. SPIE 4270, 52-63 (2001).

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Figures (10)

Fig. 1
Fig. 1

Scheme of the axicon-based Bessel–Gauss resonator.

Fig. 2
Fig. 2

Equivalent resonator with two optics subsystems. Interfaces 1 and 3 are placed just before the output coupler; interface 2 is placed just before the axicon.

Fig. 3
Fig. 3

Field distributions across the axicon plane of the ABGR with a plane output mirror: (a) E 1 ( r 1 , φ 1 ) 00 , (b) E 1 ( r 1 , φ 1 ) 10 , (c) E 1 ( r 1 , φ 1 ) 20 .

Fig. 4
Fig. 4

Field distributions across the output plane of the ABGR with a plane output mirror: (a) E 2 ( r 2 , φ 2 ) 00 , (b) E 2 ( r 2 , φ 2 ) 10 , (c) E 2 ( r 2 , φ 2 ) 20 .

Fig. 5
Fig. 5

Field distributions across the axicon plane of the ABGR with a concave spherical output mirror: (a) E 1 ( r 1 , φ 1 ) 00 , (b) E 1 ( r 1 , φ 1 ) 10 , (c) E 1 ( r 1 , φ 1 ) 20 .

Fig. 6
Fig. 6

Field distributions across the output plane of the ABGR with a concave spherical output mirror: (a) E 2 ( r 2 , φ 2 ) 00 , (b) E 2 ( r 2 , φ 2 ) 10 , (c) E 2 ( r 2 , φ 2 ) 20 .

Fig. 7
Fig. 7

(Color online) Amplitude profiles across the axicon plane of the ABGR with a concave spherical mirror calculated by (a) the transfer-matrix method, (b) the method proposed by Gutiérrez-Vega et al.[6]

Fig. 8
Fig. 8

Radial field amplitudes inside the ABGR with a concave spherical mirror: (a) E ( x , z ) 00 , (b) E ( x , z ) 10 , (c) E ( x , z ) 20 .

Fig. 9
Fig. 9

Radial field amplitudes of output beams produced by the ABGR with a concave spherical mirror: (a) E ( x , z ) 00 , (b) E ( x , z ) 10 , (c) E ( x , z ) 20 .

Fig. 10
Fig. 10

Three-dimensional distributions of E ( x , y ) 00 : (a) the intracavity distribution at the center of the ABGR, (b) output beams at a distance L 2 from the output coupler, (c) output beams at a distance L from the output coupler.

Tables (1)

Tables Icon

Table 1 Eigenvalues ( γ ) and Losses ( δ = 1 γ 2 ) of Dominant Modes

Equations (27)

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

T 1 = [ A 1 B 1 C 1 D 1 ] = [ 1 2 L R L 2 R 1 ] .
T 2 = [ A 2 B 2 C 2 D 2 ] = [ 1 2 θ 0 L r 1 L 2 θ 0 r 1 1 ] ,
θ 0 = arcsin ( n sin α ) α ( n 1 ) α .
E 2 ( r 2 , φ 2 ) = i k exp ( i k L ) 2 π B S 1 E 1 ( r 1 , φ 1 ) exp { i k 2 B [ A r 1 2 + D r 2 2 2 r 1 r 2 cos ( φ 1 φ 2 ) ] } r 1 d r 1 d φ 1 ,
E 1 ( r 1 , φ 1 ) = i k exp ( i k L ) 2 π B 1 S 2 E 2 ( r 2 , φ 2 ) exp { i k 2 B 1 [ A 1 r 2 2 + D 1 r 1 2 2 r 1 r 2 cos ( φ 1 φ 2 ) ] } r 2 d r 2 d φ 2 .
E 2 ( r 2 , φ 2 ) = i k exp ( i k L ) 2 π B 2 S 1 E 1 ( r 1 , φ 1 ) exp { i k 2 B 2 [ A 2 r 1 2 + D 2 r 2 2 2 r 1 r 2 cos ( φ 1 φ 2 ) ] } r 1 d r 1 d φ 1 .
E 1 ( r 1 , φ 1 ) = E 1 ( r 1 ) exp ( i n φ 1 ) ,
E 2 ( r 2 , φ 2 ) = E 2 ( r 2 ) exp ( i n φ 2 ) ,
E 2 ( r 2 , φ 2 ) = E 2 ( r 2 ) exp ( i n φ 2 ) ,
E 1 ( r 1 ) = ( 1 ) n + 1 k exp ( i k L ) B 1 0 a E 2 ( r 2 ) J n ( k r 1 r 2 B 1 ) exp [ i k 2 B 1 ( A 1 r 2 2 + D 1 r 1 2 ) ] r 2 d r 2 ,
E 2 ( r 2 ) = ( i ) n + 1 k exp ( i k L ) B 2 0 a E 1 ( r 1 ) J n ( k r 1 r 2 B 2 ) exp [ i k 2 B 2 ( A 2 r 1 2 + D 2 r 2 2 ) ] r 1 d r 1 ,
E 1 ( r 1 ) m = n = 1 M X m n E 2 ( r 2 ) n ,
E 2 ( r 2 ) m = n = 1 M Y m n E 1 ( r 1 ) n ,
X m n = ( 1 ) n n + 1 k n a 2 exp ( i k L ) B 1 M 2 J n n ( k m n a 2 B 1 M 2 ) exp [ i a 2 π B 1 λ M 2 ( A 1 n 2 + D 1 m 2 ) ] ,
Y m n = ( 1 ) n n + 1 k n a 2 exp ( i k L ) B 2 M 2 J n n ( k m n a 2 B 2 M 2 ) exp [ i a 2 π B 2 λ M 2 ( A 2 n 2 + D 2 m 2 ) ] ,
A 2 ( r 1 ) = 1 2 θ 0 L M ( n a ) .
Y m n = ( i ) n n + 1 k n a 2 exp ( i k L ) B 2 M 2 exp ( i k θ 0 n a M ) J n n ( k m n a 2 B 2 M 2 ) exp [ i a 2 π B 2 λ M 2 ( A 2 n 2 + D 2 m 2 ) ] ,
T 2 = [ A 2 B 2 C 2 D 2 ] = [ 1 L 0 1 ] .
E 2 = γ E 2 ,
γ E 2 = Z E 2 = ( Y X ) E 2 ,
X m n = ( i ) n n + 1 k n a 2 exp ( i k L ) B 1 M 2 exp ( i k θ 0 n a M ) J n n ( k m n a 2 B 1 M 2 ) exp [ i a 2 π B 1 λ M 2 ( A 1 n 2 + D 1 m 2 ) ] ,
Y m n = ( i ) n n + 1 k n a 2 exp ( i k L ) B 2 M 2 J n n ( k m n a 2 B 1 M 2 ) exp [ i a 2 π B 2 λ M 2 ( A 2 n 2 + D 2 m 2 ) ] ,
T 1 = [ A 1 B 1 C 1 D 1 ] = [ 1 L 0 1 ] ,
T 2 = [ A 2 B 2 C 2 D 2 ] = [ 1 L 2 R 1 ] .
E 2 ( x 2 , y 2 ) = i k exp ( i k L ) 2 π B S 1 E 1 ( x 1 , y 1 ) exp { i k 2 B [ A ( x 1 2 + y 1 2 ) 2 ( x 1 x 2 + y 1 y 2 ) + D ( x 2 2 + y 2 2 ) ] } d x 1 d y 1 .
E 2 ( x 2 , y 2 ) = i k exp ( i k L ) 2 π B 1 exp [ i k D 1 2 B 1 ( x 2 2 + y 2 2 ) ] × { E 1 ( x 1 , y 1 ) exp [ i k A 1 2 B 1 ( x 1 2 + y 1 2 ) ] } exp [ i k B 1 ( x 1 x 2 + y 1 y 2 ) ] d x 1 d y 1 .
E 2 ( x 2 , y 2 ) = i λ B 1 exp ( i k L ) exp [ i k D 1 2 B 1 ( x 2 2 + y 2 2 ) ] F { E 1 ( λ ξ 1 B 1 , λ η 1 B 1 ) exp [ i π λ A 1 B 1 ( ξ 1 2 + η 1 2 ) ] } .

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