Abstract

A new (to our knowledge) kind of light beam called the controllable elliptical dark-hollow beam (CEDHB), is introduced to describe dark-hollow beams with axially rotational asymmetry by using the tensor method. The propagation formulas of CEDHBs through paraxial aligned and misaligned nonsymmetrical optical systems are derived through vector integration. With the derived formulas, the propagation properties of CEDHBs in free-space propagation and through a misaligned thin lens are studied graphically. The CEDHBs provide a convenient model to describe and treat dark-hollow beams with axially rotational asymmetry and can be used conveniently to analyze atoms manipulated with a dark-hollow beam.

© 2006 Optical Society of America

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  1. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
    [CrossRef]
  2. Yu. B. Ovchinnikov, I. Manek, and R. Grimm, "Surface trap for Cs atoms based on evanescent-wave cooling," Phys. Rev. Lett. 79, 2225-2228 (1997).
    [CrossRef]
  3. Y. Song, D. Milam, and W. T. Hill, "Long, narrow all-light atom guide," Opt. Lett. 24, 1805-1807 (1999).
    [CrossRef]
  4. J. Soding, R. Grimm, and Yu. B. Ovchinnikov, "Gravitational laser trap for atoms with evanescent-wave cooling," Opt. Commun. 119, 652-662 (1995).
    [CrossRef]
  5. X. Xu, Y. Wang, and W. Jhe, "Theory of atom guidance in a hollow laser beam: dressed-atom approach," J. Opt. Soc. Am. B 17, 1039-1050 (2002).
    [CrossRef]
  6. J. Yin, Y. Zhu, W. Wang, Y. Wang, and W. Jhe, "Optical potential for atom guidance in a dark hollow laser beam," J. Opt. Soc. Am. B 15, 25-33 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  9. H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
    [CrossRef] [PubMed]
  10. C. Paterson and R. Smith, "Higher-order Bessel waves produced by axicon-type computer-generated holograms," Opt. Commun. 124, 121-130 (1996).
    [CrossRef]
  11. S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, "Coherent atomic waveguides from hollow optical fibers: quantized atomic motion," Phys. Rev. A 50, 2680-2690 (1994).
    [CrossRef] [PubMed]
  12. J. Arlt and K. Dholakia, "Generation of high-order Bessel beams by use of an axicon," Opt. Commun. 177, 297-301 (2000).
    [CrossRef]
  13. I. Manek, Y. B. Ovchinnikov, and R. Grimm, "Generation of a hollow laser beam for atom trapping using an axion," Opt. Commun. 147, 67-70 (1998).
    [CrossRef]
  14. M. de Angelis, L. Cacciapuoti, G. Pierattini, and G. M. Tino, "Axially symmetric hollow beams using refractive conical lenses," Opt. Lasers Eng. 39, 283-291 (2003)
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970).
    [CrossRef]
  22. 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).
  23. S. Wang and L. Ronchi, "Principles and design of optical arrays," in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

2005

2004

2003

M. de Angelis, L. Cacciapuoti, G. Pierattini, and G. M. Tino, "Axially symmetric hollow beams using refractive conical lenses," Opt. Lasers Eng. 39, 283-291 (2003)
[CrossRef]

Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beams and their propagation properties," Opt. Lett. 28, 1084-1086 (2003).
[CrossRef] [PubMed]

2002

2000

J. Arlt and K. Dholakia, "Generation of high-order Bessel beams by use of an axicon," Opt. Commun. 177, 297-301 (2000).
[CrossRef]

1999

1998

J. Yin, Y. Zhu, W. Wang, Y. Wang, and W. Jhe, "Optical potential for atom guidance in a dark hollow laser beam," J. Opt. Soc. Am. B 15, 25-33 (1998).
[CrossRef]

I. Manek, Y. B. Ovchinnikov, and R. Grimm, "Generation of a hollow laser beam for atom trapping using an axion," Opt. Commun. 147, 67-70 (1998).
[CrossRef]

1997

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, "Surface trap for Cs atoms based on evanescent-wave cooling," Phys. Rev. Lett. 79, 2225-2228 (1997).
[CrossRef]

1996

C. Paterson and R. Smith, "Higher-order Bessel waves produced by axicon-type computer-generated holograms," Opt. Commun. 124, 121-130 (1996).
[CrossRef]

1995

J. Soding, R. Grimm, and Yu. B. Ovchinnikov, "Gravitational laser trap for atoms with evanescent-wave cooling," Opt. Commun. 119, 652-662 (1995).
[CrossRef]

1994

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, "Coherent atomic waveguides from hollow optical fibers: quantized atomic motion," Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

J. J. Chang, "Time-resolved beam-quality characterization of copper-vapor lasers with unstable resonators," Appl. Opt. 33, 2255-2265 (1994).
[CrossRef] [PubMed]

1993

1991

R. M. Herman and T. A. Wiggins, "Production and uses of diffractionless beams," J. Opt. Soc. Am. A 8, 932-942 (1991).
[CrossRef]

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

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).

1987

V. I. Balykin and V. S. Letokhov, "The possibility of deep laser focusing of an atomic beam into the A-region," Opt. Commun. 64, 151-156 (1987).
[CrossRef]

1970

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).

Arlt, J.

J. Arlt and K. Dholakia, "Generation of high-order Bessel beams by use of an axicon," Opt. Commun. 177, 297-301 (2000).
[CrossRef]

Atewart, B. W.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Balykin, V. I.

V. I. Balykin and V. S. Letokhov, "The possibility of deep laser focusing of an atomic beam into the A-region," Opt. Commun. 64, 151-156 (1987).
[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).

Cacciapuoti, L.

M. de Angelis, L. Cacciapuoti, G. Pierattini, and G. M. Tino, "Axially symmetric hollow beams using refractive conical lenses," Opt. Lasers Eng. 39, 283-291 (2003)
[CrossRef]

Cai, Y.

Chang, J. J.

Choi, K.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Collins, S. A.

de Angelis, M.

M. de Angelis, L. Cacciapuoti, G. Pierattini, and G. M. Tino, "Axially symmetric hollow beams using refractive conical lenses," Opt. Lasers Eng. 39, 283-291 (2003)
[CrossRef]

Dholakia, K.

J. Arlt and K. Dholakia, "Generation of high-order Bessel beams by use of an axicon," Opt. Commun. 177, 297-301 (2000).
[CrossRef]

Fenichel, H.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Grimm, R.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, "Generation of a hollow laser beam for atom trapping using an axion," Opt. Commun. 147, 67-70 (1998).
[CrossRef]

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, "Surface trap for Cs atoms based on evanescent-wave cooling," Phys. Rev. Lett. 79, 2225-2228 (1997).
[CrossRef]

J. Soding, R. Grimm, and Yu. B. Ovchinnikov, "Gravitational laser trap for atoms with evanescent-wave cooling," Opt. Commun. 119, 652-662 (1995).
[CrossRef]

Herman, R. M.

Hill, W. T.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Jhe, W.

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Lee, H. S.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Letokhov, V. S.

V. I. Balykin and V. S. Letokhov, "The possibility of deep laser focusing of an atomic beam into the A-region," Opt. Commun. 64, 151-156 (1987).
[CrossRef]

Lin, Q.

Littman, M. G.

Lu, X.

Manek, I.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, "Generation of a hollow laser beam for atom trapping using an axion," Opt. Commun. 147, 67-70 (1998).
[CrossRef]

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, "Surface trap for Cs atoms based on evanescent-wave cooling," Phys. Rev. Lett. 79, 2225-2228 (1997).
[CrossRef]

Marksteiner, S.

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, "Coherent atomic waveguides from hollow optical fibers: quantized atomic motion," Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

Mei, Z.

Milam, D.

Ovchinnikov, Y. B.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, "Generation of a hollow laser beam for atom trapping using an axion," Opt. Commun. 147, 67-70 (1998).
[CrossRef]

Ovchinnikov, Yu. B.

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, "Surface trap for Cs atoms based on evanescent-wave cooling," Phys. Rev. Lett. 79, 2225-2228 (1997).
[CrossRef]

J. Soding, R. Grimm, and Yu. B. Ovchinnikov, "Gravitational laser trap for atoms with evanescent-wave cooling," Opt. Commun. 119, 652-662 (1995).
[CrossRef]

Paterson, C.

C. Paterson and R. Smith, "Higher-order Bessel waves produced by axicon-type computer-generated holograms," Opt. Commun. 124, 121-130 (1996).
[CrossRef]

Pierattini, G.

M. de Angelis, L. Cacciapuoti, G. Pierattini, and G. M. Tino, "Axially symmetric hollow beams using refractive conical lenses," Opt. Lasers Eng. 39, 283-291 (2003)
[CrossRef]

Rolston, S.

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, "Coherent atomic waveguides from hollow optical fibers: quantized atomic motion," Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

Ronchi, L.

S. Wang and L. Ronchi, "Principles and design of optical arrays," in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Savage, C. M.

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, "Coherent atomic waveguides from hollow optical fibers: quantized atomic motion," Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Smith, R.

C. Paterson and R. Smith, "Higher-order Bessel waves produced by axicon-type computer-generated holograms," Opt. Commun. 124, 121-130 (1996).
[CrossRef]

Soding, J.

J. Soding, R. Grimm, and Yu. B. Ovchinnikov, "Gravitational laser trap for atoms with evanescent-wave cooling," Opt. Commun. 119, 652-662 (1995).
[CrossRef]

Song, Y.

Tino, G. M.

M. de Angelis, L. Cacciapuoti, G. Pierattini, and G. M. Tino, "Axially symmetric hollow beams using refractive conical lenses," Opt. Lasers Eng. 39, 283-291 (2003)
[CrossRef]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[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 L. Ronchi, "Principles and design of optical arrays," in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

Wang, W.

Wang, X.

Wang, Y.

Wiggins, T. A.

Xu, X.

Yin, J.

Zhao, D.

Zhu, Y.

Zoller, P.

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, "Coherent atomic waveguides from hollow optical fibers: quantized atomic motion," Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

J. Soding, R. Grimm, and Yu. B. Ovchinnikov, "Gravitational laser trap for atoms with evanescent-wave cooling," Opt. Commun. 119, 652-662 (1995).
[CrossRef]

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]

V. I. Balykin and V. S. Letokhov, "The possibility of deep laser focusing of an atomic beam into the A-region," Opt. Commun. 64, 151-156 (1987).
[CrossRef]

C. Paterson and R. Smith, "Higher-order Bessel waves produced by axicon-type computer-generated holograms," Opt. Commun. 124, 121-130 (1996).
[CrossRef]

J. Arlt and K. Dholakia, "Generation of high-order Bessel beams by use of an axicon," Opt. Commun. 177, 297-301 (2000).
[CrossRef]

I. Manek, Y. B. Ovchinnikov, and R. Grimm, "Generation of a hollow laser beam for atom trapping using an axion," Opt. Commun. 147, 67-70 (1998).
[CrossRef]

Opt. Lasers Eng.

M. de Angelis, L. Cacciapuoti, G. Pierattini, and G. M. Tino, "Axially symmetric hollow beams using refractive conical lenses," Opt. Lasers Eng. 39, 283-291 (2003)
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

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. A

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

S. Marksteiner, C. M. Savage, P. Zoller, and S. Rolston, "Coherent atomic waveguides from hollow optical fibers: quantized atomic motion," Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, "Surface trap for Cs atoms based on evanescent-wave cooling," Phys. Rev. Lett. 79, 2225-2228 (1997).
[CrossRef]

Other

S. Wang and L. Ronchi, "Principles and design of optical arrays," in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

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Figures (5)

Fig. 1
Fig. 1

Three-dimensional normalized intensity distributions and corresponding contour graphs of CEDHBs for various N and ϵ at z = 0 . (a) N = 5 , ϵ = 0.4 ; (b) N = 30 , ϵ = 0.4 ; (c) N = 5 , ϵ = 0.8 ; (d) N = 30 , ϵ = 0.8 .

Fig. 2
Fig. 2

Misaligned optical system.

Fig. 3
Fig. 3

Three-dimensional normalized intensity distributions and corresponding contour graphs of CEDHBs in free space at various propagation distances. (a) z = 0 , (b) z = 0.2 z x , (c) z = 3 z x , (d) z = 10 z x .

Fig. 4
Fig. 4

Optical system with a misaligned lens.

Fig. 5
Fig. 5

Three-dimensional normalized intensity distributions and corresponding contour graphs of a CEDHB passing through a misaligned thin lens at different propagation distances. (a) z = 60 mm , (b) z = 80 mm .

Equations (29)

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

E ( r , 0 ) = n = 1 N a n [ exp ( n r 2 w 0 2 ) exp ( n r 2 v 0 2 ) ] , N = 1 , 2 , 3 , ,
a n = ( 1 ) n 1 N ( N n )
E ( r , 0 ) = n = 1 N a n [ exp ( i k 2 r T Q 1 n 1 r ) exp ( i k 2 r T Q 1 n ϵ 1 r ) ] , N = 1 , 2 , 3 ,
Q 1 1 = [ q 1 x x 1 q 1 x y 1 q 1 x y 1 q 1 y y 1 ] ,
w 0 x = 1.0 mm , w 0 y = 1.5 mm ,
w 0 x y = 2.0 mm , λ = 632.8 nm ,
Q 1 1 = [ 0.2014 i 0.0503 i 0.0503 i 0.0895 i ] m 1 .
θ = 1 2 arctan ( 2 w 0 x y w 0 x w 0 y ) .
E ( r 2 ) = i n 1 λ [ det ( B ) ] 1 2 exp ( i k l 0 ) E 0 ( r 1 ) exp ( i k l 1 ) d r 1 ,
l 1 = 1 2 ( r 1 r 2 ) T [ n 1 B 1 A n 1 B 1 n 2 ( C D B 1 A ) n 2 D B 1 ] ( r 1 r 2 ) ,
( r 2 r 2 ) = [ A B C D ] ( r 1 r 1 ) .
E ( r 2 ) = n = 1 N a n { [ det ( A + B Q 1 n 1 ) ] 1 2 exp [ i k 2 r 2 T Q 2 n 1 r 2 ] [ det ( A + B Q 1 n ϵ 1 ) ] 1 2 exp [ i k 2 r 2 T Q 2 n ϵ 1 r 2 ] } exp ( i k l 0 ) ,
Q 2 n 1 = ( C + D Q 1 n 1 ) ( A + B Q 1 n 1 ) ,
Q 2 n ϵ 1 = ( C + D Q 1 n ϵ 1 ) ( A + B Q 1 n ϵ 1 ) .
E ( r 2 ) = i λ [ det ( B ) ] 1 2 exp ( i k l 0 ) E 0 ( 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 ) ] exp [ i k 2 ( r 1 T B 1 e f + r 2 T B 1 g h ) ] d r 1 ,
A = [ a 0 0 a ] , B = [ b 0 0 b ] , C = [ c 0 0 c ] , D = [ d 0 0 d ] ,
e = 2 ( α T ϵ x + β T ϵ x ) ,
f = 2 ( α T ϵ y + β T ϵ y ) ,
g = 2 ( b γ T d α T ) ϵ x + 2 ( b δ T d β T ) ϵ x ,
h = 2 ( b γ T d α T ) ϵ y + 2 ( b δ T d β T ) ϵ y ,
α T = 1 a , β T = 1 b , γ T = c , δ T = ± 1 d ,
E ( r 2 ) = n = 1 N a n exp ( i k l 0 ) { 1 [ det ( A + B Q 1 n 1 ) ] 1 2 exp [ i k 2 ( r 2 T Q 2 n 1 r 2 + r 2 T B 1 g h + r 2 T B 1 T ( A + B Q 1 n 1 ) 1 e f ) ] × exp [ i k 8 e f T B 1 T ( A + B Q 1 n 1 ) 1 e f ] 1 [ det ( A + B Q 1 n ϵ 1 ) ] 1 2 × exp [ i k 2 ( r 2 T Q 2 n ϵ 1 r 2 + r 2 T B 1 g h + r 2 T B 1 T ( A + B Q 1 n ϵ 1 ) 1 e f ) ] × exp [ i k 8 e f T B 1 T ( A + B Q 1 n ϵ 1 ) 1 e f ] } ,
A = [ 1 0 0 1 ] , B = [ z 0 0 z ] , C = [ 0 0 0 0 ] , D = [ 1 0 0 1 ] .
A = [ 1 ( z z 1 ) f 0 0 1 ( z z 1 ) f ] ,
B = [ z z 1 ( z z 1 ) f 0 0 z z 1 ( z z 1 ) f ] ,
C = [ 1 f 0 0 1 f ] , D = [ 1 z 1 f 0 0 1 z 1 f ] ,
α T = z z 1 f , β T = z 1 ( z z 1 ) f ,
γ T = 1 f , δ T = z 1 f ,
e = 2 ( z z 1 ) ϵ x f , f = 0 , g = 2 z 1 ϵ x f , h = 0 .

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