Abstract

We define the width, divergence, and curvature radius for non-Gaussian and nonspherical light beams. A complex beam parameter is also defined as a function of the three previous ones. We then prove that the ABCD law remains valid for transforming the new complex beam parameter when a non-Gaussian and nonspherical, orthogonal, or cylindrical symmetric laser beam passes through a real ABCD optical system. The product of the minimum width multiplied by the divergence of the beam is invariant under ABCD transformations. Some examples are given.

© 1992 Optical Society of America

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  1. H. Kogelnik, “Imaging of optical mode-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  2. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  3. J. P. Taché, “Derivation of the ABCD law for Laguerre–Gaussian beams,” Appl. Opt. 26, 2698–2700 (1987).
    [CrossRef] [PubMed]
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    [CrossRef]
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  8. J. P. Campbell, L. G. DeShazer, “Near fields of truncated-Gaussian apertures,” J. Opt. Soc. Am. 59, 1427–1429 (1969).
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  9. P. Belland, J. P. Crenn, “Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture,” Appl. Opt. 21, 522–527 (1982).
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  12. F. Gori, M. Santarsiero, A. Sona, “The change of the width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
    [CrossRef]
  13. J. Alda, S. Wang, E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
    [CrossRef]
  14. The most usual definition of the Fourier transform is with the minus in the exponential (see Ref. 18 below). In addition, in paraxial wave optics the most usual definition of the plane wave traveling toward positive x is with the minus in the exponential (see Ref. 17 below). The consequence of these choices is the minus in this expression.
  15. P. A. Bélanger, P. Mathieu, “On an extremum property of Gaussian beams and Gaussian pulses,” Opt. Commun. 67, 396–398 (1988).
    [CrossRef]
  16. A. E. Siegman, E. A. Sziklas, “Mode calculations in unstable resonators with flowing saturable gain. 1: Hermite–Gaussian expansion,” Appl. Opt. 13, 2775–2791 (1974).
    [CrossRef] [PubMed]
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  18. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), pp. 323–330.
  19. S. M. Selby, ed., Standard Mathematical Tables (CRC, Boca Raton, Fla., 1972), p. 449.

1991

F. Gori, M. Santarsiero, A. Sona, “The change of the width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

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

1988

P. A. Bélanger, P. Mathieu, “On an extremum property of Gaussian beams and Gaussian pulses,” Opt. Commun. 67, 396–398 (1988).
[CrossRef]

1987

1983

1982

1980

1978

S. G. Roper, “Beam divergence of a highly multimode CO2 laser,” J. Phys. E 11, 1102–1103 (1978).
[CrossRef]

1977

1976

1974

1969

1966

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1965

H. Kogelnik, “Imaging of optical mode-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Alda, J.

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

Andrews, L. C.

Bélanger, P. A.

P. A. Bélanger, P. Mathieu, “On an extremum property of Gaussian beams and Gaussian pulses,” Opt. Commun. 67, 396–398 (1988).
[CrossRef]

Belland, P.

Bernabeu, E.

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

Burlakoff, M.

Butts, R. R.

Campbell, J. P.

Carter, W. H.

Crenn, J. P.

DeShazer, L. G.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), pp. 323–330.

Gori, F.

F. Gori, M. Santarsiero, A. Sona, “The change of the width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

Hogge, C. B.

Hunt, J. T.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

H. Kogelnik, “Imaging of optical mode-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Mathieu, P.

P. A. Bélanger, P. Mathieu, “On an extremum property of Gaussian beams and Gaussian pulses,” Opt. Commun. 67, 396–398 (1988).
[CrossRef]

Nelson, R. G.

Phillips, R. L.

Renard, P. A.

Roper, S. G.

S. G. Roper, “Beam divergence of a highly multimode CO2 laser,” J. Phys. E 11, 1102–1103 (1978).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, A. Sona, “The change of the width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

Siegman, A. E.

Simmons, W. W.

Sona, A.

F. Gori, M. Santarsiero, A. Sona, “The change of the width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

Sziklas, E. A.

Taché, J. P.

Wang, S.

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

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, “Imaging of optical mode-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

J. Opt. Soc. Am.

J. Phys. E

S. G. Roper, “Beam divergence of a highly multimode CO2 laser,” J. Phys. E 11, 1102–1103 (1978).
[CrossRef]

Opt. Commun.

F. Gori, M. Santarsiero, A. Sona, “The change of the width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

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

P. A. Bélanger, P. Mathieu, “On an extremum property of Gaussian beams and Gaussian pulses,” Opt. Commun. 67, 396–398 (1988).
[CrossRef]

Proc. IEEE

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other

A. E. Siegman, Lasers, (Oxford U. Press, Mill Valley, Calif., 1986), pp. 777–782.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), pp. 323–330.

S. M. Selby, ed., Standard Mathematical Tables (CRC, Boca Raton, Fla., 1972), p. 449.

The most usual definition of the Fourier transform is with the minus in the exponential (see Ref. 18 below). In addition, in paraxial wave optics the most usual definition of the plane wave traveling toward positive x is with the minus in the exponential (see Ref. 17 below). The consequence of these choices is the minus in this expression.

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Figures (4)

Fig. 1
Fig. 1

Optical paraxial system illuminated by an arbitrary beam Ψ1(x1). The ABCD matrix elements are real, Ψ2(x2) is the output beam, and θ01), ω0(ϕ), ω0(Ψ), and ω02) are the divergences and minimum widths of the input and output beams, respectively.

Fig. 2
Fig. 2

Divergence θ0(ϕ) of the beam Ψ(x) and divergence θ0 p ) of the collimated beam Ψp(x). An on-axis and nontilted beam is depicted for clarity.

Fig. 3
Fig. 3

(a) Intensity profiles of some Ψ s and Ψ s ( a ). (b) Some far-field intensity patterns of the unaberrated beams ϕ s and (c) of the aberrated beams ϕ s ( a ). These patterns are shown in logarithmic scale for best viewing of the sidelobes. On the top of (c) the comparison between aberrated and unaberrated far-field patterns is shown in lineal scale. (d) Conserved products for the Ψ s beams (dashed curve) and for the Ψ s ( a ) beams (solid curve) as a function of s. The magnitude of the aberration is L = 0.75λ.

Fig. 4
Fig. 4

Evolution of the (a) width and (b) radius of curvature of the diffracted Gaussian beam: ω0 is the width of the input Gaussian beam and z R is its Rayleigh range. The open circles are numerical data, based on the Fresnel integral, with a 99.4% criterion; the solid curve is the ABCD law prediction; the dashed curve is the evolution of the nondiffracted Gaussian beam.

Equations (82)

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1 q = 1 R - i λ π ω 2 ,
ω ( Ψ ) = 2 { - Ψ Ψ * [ x - x ( Ψ ) ] 2 d x I ( Ψ ) } 1 / 2 = 2 [ - Ψ Ψ * x 2 d x I ( Ψ ) - x 2 ( Ψ ) ] 1 / 2 ,
I ( Ψ ) = - Ψ Ψ * d x
x ( Ψ ) = 1 I ( Ψ ) - Ψ Ψ * x d x
ϕ ( ξ ) = - Ψ ( x ) exp ( - i 2 π x ξ ) d x
θ 0 ( ϕ ) = 2 λ { - ϕ ϕ * [ ξ - ξ ( ϕ ) ] 2 d ξ I ( ϕ ) } 1 / 2 = 2 λ [ - ϕ ϕ * ξ 2 d ξ I ( ϕ ) - ξ 2 ( ϕ ) ] 1 / 2 ,
ξ ( ϕ ) = 1 I ( ϕ ) - ϕ ϕ * ξ d ξ
- k 2 R ( x - x 0 ) 2 + t ,
[ arg ( Ψ ) ] - k R ( x - x 0 ) .
- Ψ Ψ * { [ arg ( Ψ ) ] + k R ( x - x 0 ) } 2 d x .
1 2 i - ( Ψ Ψ * - Ψ Ψ * ) ( x - x 0 ) d x + k R - Ψ Ψ * ( x - x 0 ) 2 d x = 0 ,
1 2 i - ( Ψ Ψ * - Ψ Ψ * ) d x + k R - Ψ Ψ * ( x - x 0 ) d x = 0 ,
Ψ Ψ * [ arg ( Ψ ) ] = Ψ Ψ * d d x [ tan - 1 Im ( Ψ ) Re ( Ψ ) ] = 1 2 i ( Ψ Ψ * - Ψ Ψ * ) .
1 R ( Ψ ) = i λ π I ( Ψ ) ω 2 ( Ψ ) - ( Ψ Ψ * - Ψ Ψ * ) × [ x - x ( Ψ ) ] d x .
2 π i - ϕ ϕ * ξ d ξ = - Ψ Ψ * d x = - - Ψ Ψ * d x ,
1 R ( Ψ ) = i λ π I ( Ψ ) ω 2 ( Ψ ) - ( Ψ Ψ * - Ψ Ψ * ) x d x + 4 λ x ( Ψ ) ξ ( ϕ ) ω 2 ( Ψ ) .
x 0 ( Ψ ) = x ( Ψ ) + λ ξ ( ϕ ) R ( Ψ ) = x ( Ψ ) - α R ( Ψ ) .
Ψ ( x ) = n = 0 N - 1 c n Ψ n ( x ; ω ) ,
Ψ n ( x ; ω ) = ( 2 π ) 1 / 4 1 ( 2 n n ! ω ) 1 / 2 H n ( 2 x ω ) exp ( - x 2 ω 2 ) ,
x ( Ψ ) = ω I ( Ψ ) Re [ n = 0 N - 1 c n c n + 1 * n + 1 ] ,
α = λ π ω I ( ϕ ) Im [ n = 0 N - 1 c n c n + 1 * n + 1 ] ,
I ( Ψ ) = n = 0 N - 1 c n 2 .
ω 2 ( Ψ ) = ω 2 I ( Ψ ) ( n = 0 N - 1 ( 2 n + 1 ) c n 2 + 2 Re { n = 0 N - 1 [ ( n + 1 ) ( n + 2 ) ] 1 / 2 c n c n + 2 * } ) - 4 x 2 ( Ψ ) ,
θ 0 2 ( ϕ ) = λ 2 π 2 ω 2 I ( ϕ ) ( n = 0 N - 1 ( 2 n + 1 ) c n 2 - 2 Re { n = 0 N - 1 [ ( n + 1 ) ( n + 2 ) ] 1 / 2 c n c n + 2 * } ) - 4 α 2 ,
1 R ( Ψ ) = λ π ω 2 ( Ψ ) I ( Ψ ) Im { n = 0 N - 1 [ ( n + 1 ) ( n + 2 ) ] 1 / 2 × ( c n c n + 2 * - c n + 1 c n - 1 * ) } - 4 x ( Ψ ) α ω 2 ( Ψ ) .
ω 2 ( Ψ ) = ω 2 I ( Ψ ) n = 0 N - 1 ( 2 n + 1 ) c n 2 ω 2 M ,
θ 0 2 ( ϕ ) = λ 2 π 2 ω 2 I ( ϕ ) n = 0 N - 1 ( 2 n + 1 ) c n 2 λ 2 π 2 ω 2 M .
Ψ 2 ( x 2 ) = i / B λ - d x 1 Ψ 1 ( x 1 ) × exp [ - i π B λ ( A x 1 2 - 2 x 1 x 2 + D x 2 2 ) ] .
[ x ( Ψ 2 ) - λ ξ ( ϕ 2 ) ] = [ A B C D ] [ x ( Ψ 1 ) - λ ξ ( ϕ 1 ) ] .
x ( Ψ 2 ) = 1 B λ I ( Ψ 1 ) - - d x 1 d x 1 Ψ 1 ( x 1 ) Ψ 1 * ( x 1 ) × exp [ - i π B λ A ( x 1 2 - x 1 2 ) ] × - d x 2 x 2 exp [ 2 i π B λ ( x 1 - x 1 ) x 2 ] .
x ( Ψ 2 ) = A I ( Ψ 1 ) - d x x Ψ 1 ( x ) Ψ 1 * ( x ) - B λ I ( Ψ 1 ) 2 π i - d x Ψ 1 ( x ) Ψ 1 * ( x ) .
x ( Ψ 2 ) = A x ( Ψ 1 ) - B λ ξ ( ϕ 1 ) = A x ( Ψ 1 ) + B α 1 ,
α 2 = - λ ξ ( Ψ 2 ) = C x ( Ψ 1 ) - D λ ξ ( ϕ 1 ) = C x ( Ψ 1 ) + D α 1 ,
ω 2 ( Ψ 2 ) = 4 B λ I ( Ψ 1 ) - - d x 1 d x 1 Ψ 1 ( x 1 ) Ψ 1 * ( x 1 ) × exp [ - i π B λ A ( x 1 2 - x 1 2 ) ] × - d x 2 x 2 2 exp [ 2 i π B λ ( x 1 - x 1 ) x 2 ] .
ω 2 ( Ψ 2 ) = 4 A 2 I ( Ψ 1 ) - d x x 2 Ψ 1 ( x ) Ψ 1 * ( x ) - B 2 λ 2 I ( Ψ 1 ) π 2 × - d x Ψ 1 ( x ) Ψ 1 * ( x ) + 2 i λ A B π I ( Ψ 1 ) × - d x [ 2 Ψ 1 * ( x ) x Ψ 1 ( x ) + Ψ 1 * ( x ) Ψ 1 ( x ) ] .
- 4 π 2 - ϕ * ϕ ξ 2 d ξ = - Ψ * Ψ d x = - Ψ * Ψ d x = - - Ψ * Ψ d x ,
2 Ψ 1 * x Ψ 1 + Ψ 1 * Ψ 1 = Ψ 1 Ψ 1 * x - Ψ 1 Ψ 1 * x + ( Ψ 1 Ψ 1 * x ) .
ω 2 ( Ψ 2 ) = A 2 ω 2 ( Ψ 1 ) + B 2 θ 0 2 ( ϕ 1 ) + 2 A B ω 2 ( Ψ 1 ) R ( Ψ 1 ) ,
ω ( Ψ 2 ) = ω ( Ψ 1 ) { [ A + B R ( Ψ ) ] 2 + B 2 [ θ 0 2 ( ϕ 1 ) ω 2 ( Ψ 1 ) - 1 R 2 ( Ψ 1 ) ] } 1 / 2 ,
θ 0 ( ϕ 1 ) = θ 0 = λ π ω 0 = ω 1 R 1 [ 1 + ( λ R 1 π ω 1 2 ) 2 ] 1 / 2 ,
[ θ 0 2 ( ϕ 1 ) ω 2 ( Ψ 1 ) - 1 R 2 ( Ψ 1 ) ] 1 / 2 = λ π ω 1 2 .
z 0 ( Ψ ) = - ω 2 ( Ψ 1 ) θ 0 2 ( ϕ 1 ) R ( Ψ 1 ) ,
ω 0 2 ( Ψ ) = ω 2 ( Ψ 1 ) [ 1 - ω 2 ( Ψ 1 ) θ 0 2 ( ϕ 1 ) R 2 ( Ψ 1 ) ] .
θ 0 2 ( ϕ 2 ) = 4 λ 2 I ( ϕ 2 ) - ϕ 2 ϕ 2 * ξ 2 d ξ = λ 2 π 2 I ( Ψ 1 ) - Ψ 2 ( x 2 ) Ψ 2 * ( x 2 ) d x 2 .
θ 0 2 ( ϕ 2 ) = 4 I ( Ψ 1 ) 1 B 3 λ × - - - d x 1 d x 1 d x 2 Ψ 1 ( x 1 ) Ψ 1 * ( x 1 ) × ( x 1 x 1 - x 1 x 2 D - x 1 x 2 D + D x 2 2 ) × exp { - i π B λ [ A ( x 1 2 - x 1 2 ) - 2 ( x 1 - x 1 ) x 2 ] } .
θ 0 2 ( ϕ 2 ) = 4 I ( Ψ 1 ) 1 B 3 λ - - d x 1 d x 1 Ψ 1 ( x 1 ) Ψ 1 * ( x 1 ) × exp [ - i π B λ A ( x 1 2 - x 1 2 ) ] × [ x 1 x 1 δ ( x 1 - x 1 B λ ) - D x 1 2 π i δ ( 1 ) ( x 1 - x 1 B λ ) - D x 1 2 π i δ ( 1 ) ( x 1 - x 1 B λ ) - D 4 π 2 δ ( 2 ) ( x 1 - x 1 B λ ) ] .
θ 0 2 ( ϕ 2 ) = 1 B 2 ω 2 ( Ψ 1 ) + D 2 B 2 ω 2 ( Ψ 2 ) - 4 I ( Ψ 1 ) 1 B 3 λ D × { - - d x 1 d x 1 Ψ 1 ( x 1 ) Ψ 1 * ( x 1 ) × exp [ - i π B λ A ( x 1 2 - x 1 2 ) ] × 1 2 π i δ ( 1 ) ( x 1 - x 1 B λ ) + C . C . } ,
θ 0 2 ( ϕ 2 ) = 1 B 2 ω 2 ( Ψ 1 ) + D 2 B 2 ω 2 ( Ψ 2 ) - 2 A D B 2 ω 2 ( Ψ 1 ) - 2 D B ω 2 ( Ψ 1 ) R ( Ψ 1 ) .
θ 0 ( ϕ 2 ) = ω ( Ψ 1 ) { [ C + D R ( Ψ 1 ) ] 2 + D 2 [ θ 0 2 ( ϕ 1 ) ω 2 ( Ψ 1 ) - 1 R 2 ( Ψ 1 ) ] } 1 / 2 ,
1 R ( Ψ 2 ) = ω 2 ( Ψ 1 ) ω 2 ( Ψ 2 ) { [ A + B R ( Ψ 1 ) ] [ C + D R ( Ψ 1 ) ] + B D [ θ 0 2 ( ϕ 1 ) ω 2 ( Ψ 1 ) - 1 R 2 ( Ψ 1 ) ] } ,
z 0 ( Ψ ) - f f = - 1 1 + [ ω ( Ψ 1 ) / θ 0 ( ϕ 1 ) f ] 2 ,
[ θ 0 2 ( ϕ ) ω 2 ( Ψ ) - 1 R 2 ( Ψ ) ] 1 / 2 ,
Ψ p = Ψ exp [ i k 2 R ( Ψ ) x 2 ] .
θ 0 ( ϕ p ) = [ θ 0 2 ( ϕ ) - ω 2 ( Ψ ) R 2 ( Ψ ) ] 1 / 2 .
ω ( Ψ 1 ) [ θ 0 2 ( ϕ 1 ) - ω 2 ( Ψ 1 ) R 2 ( Ψ 1 ) ] 1 / 2 = ω ( Ψ 2 ) [ θ 0 2 ( ϕ 2 ) - ω 2 ( Ψ 2 ) R 2 ( Ψ 2 ) ] 1 / 2 .
1 q ( Ψ ) = 1 R ( Ψ ) - i [ θ 0 2 ( ϕ ) ω 2 ( Ψ ) - 1 R 2 ( Ψ ) ] 1 / 2 = 1 R ( Ψ ) - i θ 0 ( ϕ p ) ω ( Ψ ) .
C + D / q ( Ψ 1 ) A + B / q ( Ψ 1 ) ,
q ( Ψ 2 ) = A q ( Ψ 1 ) + B C q ( Ψ 1 ) + D .
I ( Ψ ) = 0 2 π d θ 0 Ψ ( r ) Ψ * ( r ) r d r = 2 π 0 Ψ ( r ) Ψ * ( r ) r d r .
ω ( Ψ ) = 2 [ 1 I ( Ψ ) 0 2 π cos 2 θ d θ 0 Ψ ( r ) Ψ * ( r ) r 3 d r ] 1 / 2 = 2 [ π I ( Ψ ) 0 Ψ ( r ) Ψ * ( r ) r 3 d r ] 1 / 2 ,
0 2 π 0 { [ arg Ψ ( r ) ] + k R r } 2 r d r d θ ,
1 R ( Ψ ) = i λ I ( Ψ ) ω 2 ( Ψ ) 0 ( Ψ Ψ * - Ψ Ψ * ) r 2 d r ,
θ 0 ( ϕ ) = 2 λ [ π I ( ϕ ) 0 ϕ ( ρ ) ϕ * ( ρ ) ρ 3 d ρ ] 1 / 2 ,
Ψ 2 ( r 2 ) = i 2 π λ B 0 Ψ 1 ( r 1 ) exp [ - i π λ B ( A r 1 2 + D r 2 2 ) ] × r 1 J 0 ( 2 π r 1 r 2 λ B ) d r 1 ,
Ψ s ( a ) ( r ) = Ψ s ( r ) exp ( - i k L Ψ s ( r ) 2 ) ,
ω 2 ( Ψ s ) = 2 2 / s π Γ ( 2 s + 1 2 ) ω 2 ,
θ 0 2 ( ϕ s ) = s 2 2 2 / s 8 Γ ( 2 / s ) λ 2 π 2 ω 2 , θ 0 2 ( ϕ s ( a ) ) = θ 0 2 ( ϕ s ) [ 1 + ( 2 k L 3 ) 2 ] ,
1 R ( Ψ s ) = 0 ,             1 R ( Ψ s ( a ) ) = - 2 π 2 4 / s Γ [ ( 2 / s ) + ( 1 / 2 ) ] ω 2 L .
K 2 ( Ψ s ) = λ 2 π 2 s 2 4 Γ ( 4 / s ) Γ 2 ( 2 / s ) ,
K 2 ( Ψ s ( a ) ) = K 2 ( Ψ s ) { 1 + ( 2 k L 3 ) 2 [ 1 - ( 3 2 2 / s s ) 2 Γ 2 ( 2 / s ) Γ ( 4 / s ) ] } .
ω 2 ( Ψ ) = ω 2 ( Ψ 0 ) [ 1 + z 2 θ 0 2 ( ϕ 0 ) ω 2 ( Ψ 0 ) ] ,
R ( Ψ ) = z [ 1 + ω 2 ( Ψ 0 ) z 2 θ 0 2 ( ϕ 0 ) ] .
ω 2 ( Ψ 2 ) = 8 π 2 λ 2 B 2 I ( Ψ 1 ) 0 0 d r 1 d r 1 r 1 r 1 Ψ 1 ( r 1 ) Ψ 1 * ( r 1 ) × exp [ - i π λ B A ( r 1 2 - r 1 2 ) ] × 2 π 0 r 2 2 J 0 ( 2 π r 1 r 2 ) λ B ) J 0 ( 2 π r 1 r 2 λ B ) r 2 d r 2 .
ω 2 ( Ψ 2 ) = - 1 π λ 2 B 2 I ( Ψ 1 ) 0 0 d r 1 d r 1 r 1 r 1 Ψ 1 ( r 1 ) Ψ 1 * ( r 1 ) × exp [ - i π λ B A ( r 1 2 - r 1 2 ) ] × λ B r 1 [ δ ( 2 ) ( r 1 - r 1 λ B ) + λ B r 1 δ ( 1 ) ( r 1 - r 1 λ B ) ] .
ω 2 ( Ψ 2 ) = 1 π I ( Ψ 1 ) { - λ 2 B 2 0 Ψ 1 ( r ) Ψ 1 * ( r ) d r - λ 2 B 2 0 Ψ 1 ( r ) Ψ 1 * ( r ) r d r - 4 π i λ A B 0 [ Ψ 1 ( r ) Ψ 1 * ( r ) r + Ψ 1 ( r ) Ψ 1 * ( r ) ] + 4 π 2 A 2 0 Ψ 1 ( r ) Ψ 1 * ( r ) r 3 d r } .
( r 2 Ψ Ψ * ) = 2 r Ψ Ψ * + r 2 Ψ Ψ * + r 2 Ψ Ψ *
- 4 π 2 0 ϕ ϕ * ρ 3 d ρ = 0 Ψ Ψ * d r + 0 Ψ Ψ * r d r ,
Ψ 2 ( r 2 ) = 4 π 2 λ 2 B 2 0 Ψ 1 ( r 1 ) r 1 [ r 2 D J 0 ( 2 π r 1 r 2 λ B ) - i r 1 J 1 ( 2 π r 1 r 2 λ B ) ] exp [ - i π λ B ( A r 1 2 + D r 2 2 ) ] d r 1 .
0 Ψ 2 ( r 2 ) Ψ 2 * ( r 2 ) r 2 2 d r 2 = - 8 π 3 i λ 3 B 3 0 0 d r 1 d r 1 r 1 r 1 × Ψ 1 ( r 1 ) Ψ 1 * ( r 1 ) exp [ - i π λ B A ( r 1 2 - r 1 2 ) ] × 0 d r 2 r 2 2 [ D r 2 J 0 ( 2 π r 1 r 2 λ B ) J 0 ( 2 π r 1 r 2 λ B ) - i r 1 J 1 ( 2 π r 1 r 2 λ B ) J 0 ( 2 π r 1 r 2 λ B ) ] .
2 π 0 d r 2 r 2 2 J 1 ( 2 π r 1 r 2 λ B ) J 0 ( 2 π r 1 r 2 λ B ) = r 1 4 π 2 λ B [ δ ( 1 ) ( r 1 - r 1 λ B ) + λ B r 1 δ ( r 1 - r 1 λ B ) ] .
0 Ψ 2 ( r 2 ) Ψ 2 * ( r 2 ) r 2 2 d r 2 = - i D 2 λ B I ( Ψ 1 ) ω 2 ( Ψ 2 ) + 0 d r r 2 Ψ 1 ( r ) Ψ 1 * ( r ) + 2 π i A λ B 0 d r r 3 Ψ 1 ( r ) Ψ 1 * ( r ) .
1 R ( Ψ 2 ) = D B - ω 2 ( Ψ 1 ) ω 2 ( Ψ 2 ) R ( Ψ 1 ) - A B ω 2 ( Ψ 1 ) ω 2 ( Ψ 2 ) .

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