Abstract

A new mathematical model called “controllable dark-hollow beams” is introduced to describe hollow beams. The central dark size of this beam can be controlled easily by the beam order N and parameter ϵ. An analytical formula is derived for the propagation of a controllable dark-hollow beam through a paraxial optical system, and some numerical calculations are carried out. Some important propagation characteristics of this beam, such as the beam propagation factor and the kurtosis parameter, are studied in detail, and their variation rules versus the beam order N and parameter ϵ are presented and plotted.

© 2005 Optical Society of America

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  1. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, 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, 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, W. T. Hill, “Long narrow all-light atom guide,” Opt. Lett. 24, 1805–1807 (1999).
    [CrossRef]
  4. J. Soding, R. Grimm, Yu. B. Ovchinnikov, “Gravitational laser trap for atoms with evanescent-wave cooling,” Opt. Commun. 119, 652–662 (1995).
    [CrossRef]
  5. X. Xu, Y. Wang, 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, W. Jhe, “Optical potential for atom guidance in a dark hollow laser beam,” J. Opt. Soc. Am. B 15, 25–33 (1998).
    [CrossRef]
  7. Y. Cai, X. Lu, Q. Lin, “Hollow Gaussian Beams and their propagation properties,” Opt. Lett. 28, 1084–1086 (2003).
    [CrossRef] [PubMed]
  8. X. Wang, M. G. Littman, “Laser cavity for generation of variable-radius rings of light,” Opt. Lett. 18, 767–768 (1993).
    [CrossRef] [PubMed]
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    [CrossRef]
  10. H. S. Lee, B. W. Atewart, K. Choi, H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922–4927 (1994).
    [CrossRef] [PubMed]
  11. C. Paterson, R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
    [CrossRef]
  12. S. Marksteiner, C. M. Savage, P. Zoller, S. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50, 2680–2690 (1994).
    [CrossRef] [PubMed]
  13. J. Arlt, K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
    [CrossRef]
  14. I. Manek, Y. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axion,” Opt. Commun. 147, 67–70 (1998).
    [CrossRef]
  15. M. de Angelis, L. Cacciapuoti, G. Pierattini, 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]
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    [CrossRef]
  21. A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.
  22. R. D. Bock, Multivariate Statistical Method in Behavioral Research (McGraw-Hill, 1975).
  23. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
    [CrossRef]
  24. R. Martinez-Herrero, G. Piquero, P. M. Mejias, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
    [CrossRef]

2003 (2)

Y. Cai, X. Lu, Q. Lin, “Hollow Gaussian Beams and their propagation properties,” Opt. Lett. 28, 1084–1086 (2003).
[CrossRef] [PubMed]

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

2002 (1)

2000 (1)

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

1999 (1)

1998 (2)

J. Yin, Y. Zhu, W. Wang, Y. Wang, 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, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axion,” Opt. Commun. 147, 67–70 (1998).
[CrossRef]

1997 (2)

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

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

1996 (1)

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

1995 (2)

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

R. Martinez-Herrero, G. Piquero, P. M. Mejias, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

1994 (3)

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

S. Marksteiner, C. M. Savage, P. Zoller, S. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50, 2680–2690 (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 (1)

1992 (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

1991 (1)

1987 (1)

V. I. Balykin, 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 (1)

Arlt, J.

J. Arlt, 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, H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922–4927 (1994).
[CrossRef] [PubMed]

Balykin, V. I.

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

Bock, R. D.

R. D. Bock, Multivariate Statistical Method in Behavioral Research (McGraw-Hill, 1975).

Cacciapuoti, L.

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

Dholakia, K.

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

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Fenichel, H.

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

Grimm, R.

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

J. Soding, R. Grimm, 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, 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, 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, H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922–4927 (1994).
[CrossRef] [PubMed]

Letokhov, V. S.

V. I. Balykin, 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.

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Manek, I.

I. Manek, Y. B. Ovchinnikov, 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, 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, S. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50, 2680–2690 (1994).
[CrossRef] [PubMed]

Martinez-Herrero, R.

R. Martinez-Herrero, G. Piquero, P. M. Mejias, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

Mejias, P. M.

R. Martinez-Herrero, G. Piquero, P. M. Mejias, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

Milam, D.

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Ovchinnikov, Y. B.

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

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

Paterson, C.

C. Paterson, 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, G. M. Tino, “Axially symmetric hollow beams using refractive conical lenses,” Opt. Lasers Eng. 39, 283–291 (2003)
[CrossRef]

Piquero, G.

R. Martinez-Herrero, G. Piquero, P. M. Mejias, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

Rolston, S.

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

Sasada, H.

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

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.

Smith, R.

C. Paterson, 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, 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, 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, H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Wang, W.

Wang, X.

Wang, Y.

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

Wiggins, T. A.

Xu, X.

Yin, J.

Zhu, Y.

Zoller, P.

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

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

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

Opt. Commun. (6)

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

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

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

J. Arlt, 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, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axion,” Opt. Commun. 147, 67–70 (1998).
[CrossRef]

R. Martinez-Herrero, G. Piquero, P. M. Mejias, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

Opt. Lasers Eng. (1)

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

Opt. Lett. (3)

Opt. Quantum Electron. (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

Phys. Rev. A (2)

S. Marksteiner, C. M. Savage, P. Zoller, 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, H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922–4927 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

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

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

Other (4)

A. Erdelyi, W. Magnus, F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.

R. D. Bock, Multivariate Statistical Method in Behavioral Research (McGraw-Hill, 1975).

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Figures (4)

Fig. 1
Fig. 1

Normalized intensity distributions of the CDHBs as a function of parameter r, (a) for different order N; (b) for different parameter ϵ.

Fig. 2
Fig. 2

Normalized three-dimensional intensity distributions of a CDHB in free space propagation at two propagation distances: (a) z = 1 m ; (b) z = 10 m .

Fig. 3
Fig. 3

(a) The M 2 factor of the CDHBs versus ϵ for different beam order N; (b) the M 2 factor of the CDHBs versus N for different parameter ϵ.

Fig. 4
Fig. 4

(a) The kurtosis parameter of the CDHBs at the plane z = 0 versus ϵ for different beam order N; (b) the kurtosis parameter of the CDHBs at the plane z = 0 versus N for different parameter ϵ.

Equations (18)

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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 )
( N n )
E ( r , θ , z ) = i λ B exp ( i k z ) 0 2 π 0 E 0 ( r 0 , θ 0 , 0 ) exp { i k 2 B [ A r 0 2 2 r 0 r cos ( θ θ 0 ) + D r 2 ] } r 0 d r 0 d θ 0 ,
J 0 ( x ) = 1 2 π 0 2 π exp ( i x cos θ ) d θ ,
0 exp ( p t ) J 0 ( 2 α 1 2 t 1 2 ) d t = p 1 exp ( α p ) ,
E ( r , z ) = i k 2 B exp ( i k z ) exp ( i k D r 2 2 B ) n = 1 N a n { ( n w 0 2 + i k A 2 B ) 1 exp [ ( k r 2 B ) 2 n w 0 2 + i k A 2 B ] ( n v 0 2 + i k A 2 B ) 1 exp [ ( k r 2 B ) 2 n v 0 2 + i k A 2 B ] } .
M 2 = { [ 0 r 2 E ( r , 0 ) 2 r d r ] [ 0 E ( r , 0 ) 2 r d r ] } 1 2 0 E ( r , 0 ) 2 r d r .
0 e s x x n d x = n ! s n + 1 , n = 0 , 1 , 2 , , s > 0 ,
M 2 = { n 1 = 1 N n 2 = 1 N a n 1 a n 2 [ 1 + ϵ 4 ( n 1 + n 2 ) 2 ϵ 4 ( ϵ 2 n 1 + n 2 ) 2 ϵ 4 ( n 1 + ϵ 2 n 2 ) 2 ] } 1 2 × { n 1 = 1 N n 2 = 1 N a n 1 a n 2 n 1 n 2 [ 2 ( n 1 + n 2 ) 2 ϵ 2 ( ϵ 2 n 1 + n 2 ) 2 ϵ 2 ( n 1 + ϵ 2 n 2 ) 2 ] } 1 2 × { 1 2 n 1 = 1 N n 2 = 1 N a n 1 a n 2 [ 1 + ϵ 2 n 1 + n 2 ϵ 2 ϵ 2 n 1 + n 2 ϵ 2 n 1 + ϵ 2 n 2 ] } 1 .
K = [ 0 r 4 E ( r , z ) 2 r d r ] [ 0 E ( r , z ) 2 r d r ] [ 0 r 2 E ( r , z ) 2 r d r ] 2 .
K = 2 { n 1 = 1 N n 2 = 1 N a n 1 a n 2 [ ( b n 1 b n 2 ) 2 ( b n 1 + b n 2 ) 3 ( b n 1 c n 2 ) 2 ( b n 1 + c n 2 ) 3 ( c n 1 b n 2 ) 2 ( c n 1 + b n 2 ) 3 + ( c n 1 c n 2 ) 2 ( c n 1 + c n 2 ) 3 ] } × { n 1 = 1 N n 2 = 1 N a n 1 a n 2 [ 1 b n 1 + b n 2 1 b n 1 + c n 2 1 c n 1 + b n 2 + 1 c n 1 + c n 2 ] } × { n 1 = 1 N n 2 = 1 N a n 1 a n 2 [ b n 1 b n 2 ( b n 1 + b n 2 ) 2 b n 1 c n 2 ( b n 1 + c n 2 ) 2 c n 1 b n 2 ( c n 1 + b n 2 ) 2 + c n 1 c n 2 ( c n 1 + c n 2 ) 2 ] } 2 ,
b n 1 = n 1 w 0 2 + i k A 2 B ,
b n 2 = n 2 w 0 2 i k A 2 B ,
c n 1 = n 1 ϵ 2 w 0 2 + i k A 2 B ,
c n 2 = n 2 ϵ 2 w 0 2 i k A 2 B .
k = 2 { n 1 = 1 N n 2 = 1 N a n 1 a n 2 [ 1 + ϵ 6 ( n 1 + n 2 ) 3 ϵ 6 ( ϵ 2 n 1 + n 2 ) 3 ϵ 6 ( n 1 + ϵ 2 n 2 ) 3 ] } × { n 1 = 1 N n 2 = 1 N a n 1 a n 2 [ 1 + ϵ 2 n 1 + n 2 ϵ 2 ϵ 2 n 1 + n 2 ϵ 2 n 1 + ϵ 2 n 2 ] } × { n 1 = 1 N n 2 = 1 N a n 1 a n 2 [ 1 + ϵ 4 ( n 1 + n 2 ) 2 ϵ 4 ( ϵ 2 n 1 + n 2 ) 2 ϵ 4 ( n 1 + ϵ 2 n 2 ) 2 ] } 2 .

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