Abstract

An analytical propagation expression for a generalized astigmatic elliptical Gaussian beam (EGB) modulated by an elliptical annular aperture and passing through an axially nonsymmetrical optical system is obtained by the use of vector integration. The derived analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual method of using the diffraction integral directly, and the efficiency of the numerical calculation is significantly improved. Some numerical simulations are illustrated for the propagation properties of EGBs passing through a free space with an elliptical annular aperture, an elliptical screen, or an elliptical aperture. Further extensions are also pointed out.

© 2007 Optical Society of America

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  1. S. A. Collins, "Lens-systems diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970).
    [CrossRef]
  2. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).
  3. S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).
  4. X. Du and D. Zhao, "Propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems," J. Opt. Soc. Am. A 23, 625-631 (2006).
    [CrossRef]
  5. F. I. Baida and D. Van Labeke, "Light transmission by subwavelength annular aperture arrays in metallic films," Opt. Commun. 209, 17-22 (2002).
    [CrossRef]
  6. F. I. Baida and D. Van Labeke, "Three-dimensional structures for enhanced transmission through a metallic film: annular aperture arrays," Phys. Rev. B 67, 155314 (2003).
    [CrossRef]
  7. C. Liu and S. H. Park, "Numerical analysis of an annular-aperture solid immersion lens," Opt. Lett. 29, 1742-1744 (2004).
    [CrossRef] [PubMed]
  8. M. J. Lockyear, A. P. Hibbins, J. R. Sambles, and C. R. Lawrence, "Microwave transmission through a single subwavelength annular aperture in a metal plate," Phys. Rev. Lett. 94, 193902 (2005).
    [CrossRef] [PubMed]
  9. H. Caglayan, I. Bulu, and E. Ozbay, "Extraordinary grating-coupled microwave transmission through a subwavelength annular aperture," Opt. Express 13, 1666-1671 (2005).
    [CrossRef] [PubMed]
  10. H. Caglayan, I. Bulu, and E. Ozbay, "Beaming of electromagnetic waves emitted through a subwavelength annular aperture," J. Opt. Soc. Am. B 23, 419-422 (2006).
    [CrossRef]
  11. J. Gu and D. Zhao, "Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture," J. Mod. Opt. 52, 1065-1072 (2005).
    [CrossRef]
  12. Z. Mei, D. Zhao, and J. Gu, "Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system," Opt. Commun. 240, 337-343 (2004).
    [CrossRef]
  13. P. Liu, B. Lü, and K. Duan, "Propagation of vectorial nonparaxial Gaussian beams through an annular aperture," Opt. Laser Technol. 38, 133-137 (2006).
    [CrossRef]
  14. J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
    [CrossRef]
  15. J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
    [CrossRef]
  16. H. Mao and D. Zhao, "Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system," J. Opt. Soc. Am. A 22, 647-653 (2005).
    [CrossRef]

2006 (3)

2005 (4)

H. Caglayan, I. Bulu, and E. Ozbay, "Extraordinary grating-coupled microwave transmission through a subwavelength annular aperture," Opt. Express 13, 1666-1671 (2005).
[CrossRef] [PubMed]

H. Mao and D. Zhao, "Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system," J. Opt. Soc. Am. A 22, 647-653 (2005).
[CrossRef]

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, and C. R. Lawrence, "Microwave transmission through a single subwavelength annular aperture in a metal plate," Phys. Rev. Lett. 94, 193902 (2005).
[CrossRef] [PubMed]

J. Gu and D. Zhao, "Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture," J. Mod. Opt. 52, 1065-1072 (2005).
[CrossRef]

2004 (2)

Z. Mei, D. Zhao, and J. Gu, "Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system," Opt. Commun. 240, 337-343 (2004).
[CrossRef]

C. Liu and S. H. Park, "Numerical analysis of an annular-aperture solid immersion lens," Opt. Lett. 29, 1742-1744 (2004).
[CrossRef] [PubMed]

2003 (1)

F. I. Baida and D. Van Labeke, "Three-dimensional structures for enhanced transmission through a metallic film: annular aperture arrays," Phys. Rev. B 67, 155314 (2003).
[CrossRef]

2002 (1)

F. I. Baida and D. Van Labeke, "Light transmission by subwavelength annular aperture arrays in metallic films," Opt. Commun. 209, 17-22 (2002).
[CrossRef]

1991 (1)

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

1990 (1)

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

1988 (1)

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

1970 (1)

Alda, J.

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

Baida, F. I.

F. I. Baida and D. Van Labeke, "Three-dimensional structures for enhanced transmission through a metallic film: annular aperture arrays," Phys. Rev. B 67, 155314 (2003).
[CrossRef]

F. I. Baida and D. Van Labeke, "Light transmission by subwavelength annular aperture arrays in metallic films," Opt. Commun. 209, 17-22 (2002).
[CrossRef]

Bernabeu, E.

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

Breazeale, M. A.

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

Bulu, I.

Caglayan, H.

Collins, S. A.

Du, X.

Duan, K.

P. Liu, B. Lü, and K. Duan, "Propagation of vectorial nonparaxial Gaussian beams through an annular aperture," Opt. Laser Technol. 38, 133-137 (2006).
[CrossRef]

Gu, J.

J. Gu and D. Zhao, "Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture," J. Mod. Opt. 52, 1065-1072 (2005).
[CrossRef]

Z. Mei, D. Zhao, and J. Gu, "Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system," Opt. Commun. 240, 337-343 (2004).
[CrossRef]

Hibbins, A. P.

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, and C. R. Lawrence, "Microwave transmission through a single subwavelength annular aperture in a metal plate," Phys. Rev. Lett. 94, 193902 (2005).
[CrossRef] [PubMed]

Lawrence, C. R.

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, and C. R. Lawrence, "Microwave transmission through a single subwavelength annular aperture in a metal plate," Phys. Rev. Lett. 94, 193902 (2005).
[CrossRef] [PubMed]

Lin, Q.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

Liu, C.

Liu, P.

P. Liu, B. Lü, and K. Duan, "Propagation of vectorial nonparaxial Gaussian beams through an annular aperture," Opt. Laser Technol. 38, 133-137 (2006).
[CrossRef]

Lockyear, M. J.

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, and C. R. Lawrence, "Microwave transmission through a single subwavelength annular aperture in a metal plate," Phys. Rev. Lett. 94, 193902 (2005).
[CrossRef] [PubMed]

Lü, B.

P. Liu, B. Lü, and K. Duan, "Propagation of vectorial nonparaxial Gaussian beams through an annular aperture," Opt. Laser Technol. 38, 133-137 (2006).
[CrossRef]

Mao, H.

Mei, Z.

Z. Mei, D. Zhao, and J. Gu, "Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system," Opt. Commun. 240, 337-343 (2004).
[CrossRef]

Ozbay, E.

Park, S. H.

Sambles, J. R.

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, and C. R. Lawrence, "Microwave transmission through a single subwavelength annular aperture in a metal plate," Phys. Rev. Lett. 94, 193902 (2005).
[CrossRef] [PubMed]

Van Labeke, D.

F. I. Baida and D. Van Labeke, "Three-dimensional structures for enhanced transmission through a metallic film: annular aperture arrays," Phys. Rev. B 67, 155314 (2003).
[CrossRef]

F. I. Baida and D. Van Labeke, "Light transmission by subwavelength annular aperture arrays in metallic films," Opt. Commun. 209, 17-22 (2002).
[CrossRef]

Wang, S.

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).

Wen, J. J.

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

Zhao, D.

X. Du and D. Zhao, "Propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems," J. Opt. Soc. Am. A 23, 625-631 (2006).
[CrossRef]

J. Gu and D. Zhao, "Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture," J. Mod. Opt. 52, 1065-1072 (2005).
[CrossRef]

H. Mao and D. Zhao, "Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system," J. Opt. Soc. Am. A 22, 647-653 (2005).
[CrossRef]

Z. Mei, D. Zhao, and J. Gu, "Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system," Opt. Commun. 240, 337-343 (2004).
[CrossRef]

S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).

J. Acoust. Soc. Am. (1)

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

J. Mod. Opt. (1)

J. Gu and D. Zhao, "Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture," J. Mod. Opt. 52, 1065-1072 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

Z. Mei, D. Zhao, and J. Gu, "Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system," Opt. Commun. 240, 337-343 (2004).
[CrossRef]

F. I. Baida and D. Van Labeke, "Light transmission by subwavelength annular aperture arrays in metallic films," Opt. Commun. 209, 17-22 (2002).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

P. Liu, B. Lü, and K. Duan, "Propagation of vectorial nonparaxial Gaussian beams through an annular aperture," Opt. Laser Technol. 38, 133-137 (2006).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

Phys. Rev. B (1)

F. I. Baida and D. Van Labeke, "Three-dimensional structures for enhanced transmission through a metallic film: annular aperture arrays," Phys. Rev. B 67, 155314 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, and C. R. Lawrence, "Microwave transmission through a single subwavelength annular aperture in a metal plate," Phys. Rev. Lett. 94, 193902 (2005).
[CrossRef] [PubMed]

Other (1)

S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).

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Figures (7)

Fig. 1
Fig. 1

(a) Elliptical annular aperture function with a x = 1 mm , a y = 2 mm , b x = 0.5 mm , b y = 1 mm , and θ = 0 . (b) Elliptical annular aperture function with a x = 1 mm , a y = 2 mm , b x = 0.5 mm , b y = 1 mm , and θ = π 6 . (c) Elliptical screen function with ( a x , a y ) , b x = 0.5 mm , b y = 1 mm , and θ = 0 . (d) Elliptical aperture function with a x = 1 mm , a y = 2 mm , ( b x , b y ) 0 , and θ = 0 .

Fig. 2
Fig. 2

Schematic of an EGB modulated by an annular aperture and passing through a free space.

Fig. 3
Fig. 3

Amplitude sections with y = 0 of an EGB modulated by an elliptical annular aperture with a x = 1 mm , a y = 2 mm , b x = 0.5 mm , b y = 1 mm , θ = 0 , and passing through a free space. Solid curves denote results with the diffraction integral Eq. (6); dotted curves denote results with the approximate analytical Eq. (11).

Fig. 4
Fig. 4

Amplitude distributions of an EGB modulated by an elliptical annular aperture with a x = 1 mm , a y = 2 mm , b x = 0.5 mm , b y = 1 mm , θ = 0 , and passing through a free space.

Fig. 5
Fig. 5

As Fig. 4, but the azimuthal angle θ = π 6 .

Fig. 6
Fig. 6

Amplitude distributions of an EGB modulated by an elliptical screen with ( a x , a y ) , b x = 0.5 mm , b y = 1 mm , θ = 0 , and passing through a free space.

Fig. 7
Fig. 7

Amplitude distributions of an EGB modulated by an elliptical aperture with a x = 1 mm , a y = 2 mm , ( b x , b y ) 0 , θ = 0 , and passing through a free space.

Equations (26)

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E 2 ( r 2 ) = i λ exp ( i k L 0 ) [ det ( B ) ] 1 2 E 1 ( r 1 ) exp [ i k 2 ( r 1 T B 1 Ar 1 2 r 1 T B 1 r 2 + r 2 T DB 1 r 2 ) ] d r 1 ,
( r 2 r 2 ) = [ A B C D ] ( r 1 r 1 ) .
E 1 ( r 1 ) = exp [ i k 2 r 1 T Q 1 1 r 1 ] ,
Q 1 1 = [ q x x 1 q x y 1 q x y 1 q y y 1 ] = i λ π [ w 0 x 2 w 0 x y 2 w 0 x y 2 w 0 y 2 ] .
A p ( r 1 ) = { 1 , inside the elliptical annular aperture 0 , outside the elliptical annular aperture ,
E 2 ( r 2 ) = i λ exp ( i k L 0 ) [ det ( B ) ] 1 2 A p ( r 1 ) E 1 ( r 1 ) × exp [ i k 2 ( r 1 T B 1 Ar 1 2 r 1 T B 1 r 2 + r 2 T DB 1 r 2 ) ] d r 1 .
A p ( r 1 ) = A p a ( r 1 ) A p b ( r 1 ) ,
A p ( r 1 ) = n = 1 N A n exp ( r 1 T R T P n a Rr 1 ) n = 1 N A n exp ( r 1 T R T P n b Rr 1 ) ,
P n a = [ B n a x 2 0 0 B n a y 2 ] , P n b = [ B n b x 2 0 0 B n b y 2 ] ,
R = [ cos θ sin θ sin θ cos θ ] ,
E 2 ( r 2 ) = exp ( i k L 0 ) { n = 1 N A n [ det ( A + BQ 1 n a 1 ) ] 1 2 exp ( i k 2 r 2 T Q 2 n a 1 r 2 ) ) n = 1 N A n [ det ( A + BQ 1 n b 1 ) ] 1 2 exp ( i k 2 r 2 T Q 2 n b 1 r 2 ) } ,
Q 2 n a 1 = ( C + DQ 1 n a 1 ) ( A + BQ 1 n a 1 ) 1 ,
Q 2 n b 1 = ( C + DQ 1 n b 1 ) ( A + BQ 1 n b 1 ) 1 ,
Q 1 n a 1 = Q 1 1 + ( 2 i k ) R T P n a R , Q 1 n b 1 = Q 1 1 + ( 2 i k ) R T P n b R .
E 2 ( r 2 ) = exp ( i k L 0 ) { [ det ( A + BQ 1 1 ) ] 1 2 exp ( i k 2 r 2 T Q 2 1 r 2 ) ) n = 1 N A n [ det ( A + BQ 1 n b 1 ) ] 1 2 exp ( i k 2 r 2 T Q 2 n b 1 r 2 ) } ,
Q 2 1 = ( C + DQ 1 1 ) ( A + BQ 1 1 ) 1 .
E 2 ( r 2 ) = exp ( i k L 0 ) n = 1 N A n [ det ( A + BQ 1 n a 1 ) ] 1 2 exp [ i k 2 r 2 T Q 2 n a 1 r 2 ] .
A = [ 1 0 0 1 ] , B = [ z 0 0 z ] , C = [ 0 0 0 0 ] , D = [ 1 0 0 1 ] .
E 2 ( r 2 ) = i λ exp ( i k L 0 ) [ det ( B ) ] 1 2 exp [ i k 2 r 1 T Q 1 1 r 1 ] ( n = 1 N A n exp ( r 1 T R T P n a Rr 1 ) n = 1 N A n exp ( r 1 T R T P n b Rr 1 ) ) exp ( i k 2 ( r 1 T B 1 Ar 1 2 r 1 T B 1 r 2 + r 2 T DB 1 r 2 ) ) d r 1 = n = 1 N A n i λ exp ( i k L 0 ) [ det ( B ) ] 1 2 exp ( ik 2 r 1 T [ Q 1 1 + 2 i k R T P n a R ] r 1 ) exp ( i k 2 ( r 1 T B 1 Ar 1 2 r 1 T B 1 r 2 + r 2 T DB 1 r 2 ) ) d r 1 n = 1 N A n i λ exp ( i k L 0 ) [ det ( B ) ] 1 2 exp ( i k 2 r 1 T [ Q 1 1 + 2 i k R T P n b R ] r 1 ) exp ( i k 2 ( r 1 T B 1 Ar 1 2 r 1 T B 1 r 2 + r 2 T DB 1 r 2 ) ) d r 1 .
Q 1 n a 1 = Q 1 1 + ( 2 i k ) R T P n a R , Q 1 n b 1 = Q 1 1 + ( 2 i k ) R T P n b R ,
E 2 ( r 2 ) = n = 1 N A n i λ exp ( i k L 0 ) [ det ( B ) ] 1 2 exp { i k 2 [ r 1 T ( Q 1 n a 1 + B 1 A ) r 1 2 r 1 T B 1 r 2 + r 2 T DB 1 r 2 ] } d r 1 n = 1 N A n i λ exp ( i k L 0 ) [ det ( B ) ] 1 2 exp { i k 2 [ r 1 T ( Q 1 n b 1 + B 1 A ) r 1 2 r 1 T B 1 r 2 + r 2 T DB 1 r 2 ] } d r 1 = n = 1 N A n i λ exp ( i k L 0 ) [ det ( B ) ] 1 2 exp [ i k 2 ( Q 1 n a 1 + B 1 A ) 1 2 r 1 ( Q 1 n a 1 + B 1 A ) 1 2 B 1 r 2 2 ] d r 1 × exp [ i k 2 r 2 T B 1 T ( Q 1 n a 1 + B 1 A ) 1 B 1 r 2 ] exp ( i k 2 r 2 T DB 1 r 2 ) n = 1 N A n i λ exp ( i k L 0 ) [ det ( B ) ] 1 2 exp [ i k 2 ( Q 1 n b 1 + B 1 A ) 1 2 r 1 ( Q 1 n b 1 + B 1 A ) 1 2 B 1 r 2 2 ] d r 1 exp [ i k 2 r 2 T B 1 T ( Q 1 n b 1 + B 1 A ) 1 B 1 r 2 ] exp ( i k 2 r 2 T DB 1 r 2 ) .
exp ( a x 2 ) d x = π a ,
E 2 ( r 2 ) = n = 1 N A n exp ( i k L 0 ) [ det ( B ) ] 1 2 [ det ( Q 1 n a 1 + B 1 A ) ] 1 2 exp { i k 2 r 2 T [ DB 1 B 1 T ( Q 1 n a 1 + B 1 A ) 1 B 1 ] r 2 } n = 1 N A n exp ( i k L 0 ) [ det ( B ) ] 1 2 [ det ( Q 1 n b 1 + B 1 A ) ] 1 2 exp { i k 2 r 2 T [ DB 1 B 1 T ( Q 1 n b 1 + B 1 A ) 1 B 1 ] r 2 } = n = 1 N A n exp ( i k L 0 ) [ det ( A + BQ 1 n a 1 ) ] 1 2 exp [ i k 2 r 2 T ( C + DQ 1 n a 1 ) ( A + BQ 1 n a 1 ) 1 r 2 ] n = 1 N A n exp ( i k L 0 ) [ det ( A + BQ 1 n b 1 ) ] 1 2 exp [ i k 2 r 2 T ( C + DQ 1 n b 1 ) ( A + BQ 1 n b 1 ) 1 r 2 ] = exp ( i k L 0 ) { n = 1 N A n [ det ( A + BQ 1 n a 1 ) ] 1 2 exp ( i k 2 r 2 T Q 2 n a 1 r 2 ) ) n = 1 N A n [ det ( A + BQ 1 n b 1 ) ] 1 2 exp ( i k 2 r 2 T Q 2 n b 1 r 2 ) } ,
Q 2 n a 1 = ( C + DQ 1 n a 1 ) ( A + BQ 1 n a 1 ) 1 ,
Q 2 n b 1 = ( C + DQ 1 n b 1 ) ( A + BQ 1 n b 1 ) 1 .
( B 1 A ) T = B 1 A , ( B 1 ) T = C DB 1 A , ( DB 1 ) T = DB 1 .

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