Abstract

The mode expansion method (MEM) models the propagation of an apertured beam by expressing the diffracted field as a finite series of Laguerre–Gaussian or Hermite–Gaussian modes. An optimal expansion parameter set (beam waist of the modes and its location) reduces the number of modes, which is difficult to derive, especially for high-order incident beams. We propose a user-friendly version of the MEM in which the expansion parameter set and the suitable number of modes are simply deduced from the approximation of the apertured incident beam.

© 2010 Optical Society of America

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References

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  1. K. Tanaka, M. Shibukawa, and O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. 20, 749-755 (1972).
    [CrossRef]
  2. N. S. Petrovic and A. D. Rakic, “Modeling diffraction in free space optical interconnects by the mode expansion method,” Appl. Opt. 42, 5308-5318 (2003).
    [CrossRef] [PubMed]
  3. J. J. Snyder, “Modeling laser beam diffraction and propagation by the mode-expansion method,” Appl. Opt. 46, 5056-5061 (2007).
    [CrossRef] [PubMed]
  4. G. Stephan and M. Truemper, “Inhomogeneity effects in a gas laser,” Phys. Rev. A 28, 2344-2362 (1983).
    [CrossRef]
  5. M. Traïche and A. Kellou, “Calculation of the resonant eigenfield in the basis of Laguerre-Gaussian modes for rotationally symmetric plano-concave stable resonators,” Opt. Commun. 208, 391-399 (2002).
    [CrossRef]
  6. H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Quasi-Optics: Proceedings of the Symposium on Quasi-Optics, J.Fox, ed., Vol. 14 of the Microwave Research Institute Symposia Series (Polytechnic, 1964), pp. 333-347.
  7. N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagation factor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51, 2279-2286 (2004).
  8. N. S. Petrovic, “Modelling diffraction in optical interconnects,” Ph.D. thesis (University of Queensland, School of Information Technology and Electrical Engineering, 2004).
  9. A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express 13, 4952-4962 (2005).
    [CrossRef] [PubMed]
  10. R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325-386 (2001).
    [CrossRef]
  11. G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
    [CrossRef]
  12. R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncation series,” Opt. Commun. 125, 197-203 (1996).
    [CrossRef]
  13. R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack. A Subroutine Package for Automatic Integration (Springer Verlag, 1983).
  14. Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured abcd systems,” Opt. Commun. 281, 1384-1395 (2008).
    [CrossRef]
  15. E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
    [CrossRef]
  16. Y. Hida, X. S. Li, and D. H. Bailey, “Algorithms for quad-double precision floating point arithmetic,” in 15th IEEE Symposium on Computer Arithmetic (ARITH-15 '01), N.Burgess and L.Ciminiera, eds. (IEEE Computer Society, 2001), pp. 0155-162.
    [CrossRef]

2008 (3)

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured abcd systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
[CrossRef]

2007 (1)

2005 (1)

2004 (1)

N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagation factor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51, 2279-2286 (2004).

2003 (1)

2002 (1)

M. Traïche and A. Kellou, “Calculation of the resonant eigenfield in the basis of Laguerre-Gaussian modes for rotationally symmetric plano-concave stable resonators,” Opt. Commun. 208, 391-399 (2002).
[CrossRef]

2001 (1)

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325-386 (2001).
[CrossRef]

1996 (1)

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncation series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

1983 (1)

G. Stephan and M. Truemper, “Inhomogeneity effects in a gas laser,” Phys. Rev. A 28, 2344-2362 (1983).
[CrossRef]

1972 (1)

K. Tanaka, M. Shibukawa, and O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. 20, 749-755 (1972).
[CrossRef]

Ait-Ameur, K.

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured abcd systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

Aït-Ameur, K.

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
[CrossRef]

N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagation factor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51, 2279-2286 (2004).

Bailey, D. H.

Y. Hida, X. S. Li, and D. H. Bailey, “Algorithms for quad-double precision floating point arithmetic,” in 15th IEEE Symposium on Computer Arithmetic (ARITH-15 '01), N.Burgess and L.Ciminiera, eds. (IEEE Computer Society, 2001), pp. 0155-162.
[CrossRef]

Borghi, R.

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncation series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

Cagniot, E.

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured abcd systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
[CrossRef]

Davidson, N.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express 13, 4952-4962 (2005).
[CrossRef] [PubMed]

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325-386 (2001).
[CrossRef]

de Doncker-Kapenga, E.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack. A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

Derrar-Kaddour, Z.

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured abcd systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
[CrossRef]

Friesem, A. A.

A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express 13, 4952-4962 (2005).
[CrossRef] [PubMed]

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325-386 (2001).
[CrossRef]

Fromager, M.

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
[CrossRef]

Fukumitsu, O.

K. Tanaka, M. Shibukawa, and O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. 20, 749-755 (1972).
[CrossRef]

Gori, F.

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncation series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

Hasman, E.

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325-386 (2001).
[CrossRef]

Hida, Y.

Y. Hida, X. S. Li, and D. H. Bailey, “Algorithms for quad-double precision floating point arithmetic,” in 15th IEEE Symposium on Computer Arithmetic (ARITH-15 '01), N.Burgess and L.Ciminiera, eds. (IEEE Computer Society, 2001), pp. 0155-162.
[CrossRef]

Ishaaya, A. A.

Jackel, S.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Kahaner, D. K.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack. A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

Kellou, A.

M. Traïche and A. Kellou, “Calculation of the resonant eigenfield in the basis of Laguerre-Gaussian modes for rotationally symmetric plano-concave stable resonators,” Opt. Commun. 208, 391-399 (2002).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Quasi-Optics: Proceedings of the Symposium on Quasi-Optics, J.Fox, ed., Vol. 14 of the Microwave Research Institute Symposia Series (Polytechnic, 1964), pp. 333-347.

Li, X. S.

Y. Hida, X. S. Li, and D. H. Bailey, “Algorithms for quad-double precision floating point arithmetic,” in 15th IEEE Symposium on Computer Arithmetic (ARITH-15 '01), N.Burgess and L.Ciminiera, eds. (IEEE Computer Society, 2001), pp. 0155-162.
[CrossRef]

Lumer, Y.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Machavariani, G.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Martel, G.

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured abcd systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagation factor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51, 2279-2286 (2004).

Meir, A.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Moshe, I.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Oron, R.

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325-386 (2001).
[CrossRef]

Passilly, N.

N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagation factor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51, 2279-2286 (2004).

Petrovic, N. S.

N. S. Petrovic and A. D. Rakic, “Modeling diffraction in free space optical interconnects by the mode expansion method,” Appl. Opt. 42, 5308-5318 (2003).
[CrossRef] [PubMed]

N. S. Petrovic, “Modelling diffraction in optical interconnects,” Ph.D. thesis (University of Queensland, School of Information Technology and Electrical Engineering, 2004).

Piessens, R.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack. A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

Rakic, A. D.

Santarsiero, M.

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncation series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

Shibukawa, M.

K. Tanaka, M. Shibukawa, and O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. 20, 749-755 (1972).
[CrossRef]

Snyder, J. J.

Stephan, G.

G. Stephan and M. Truemper, “Inhomogeneity effects in a gas laser,” Phys. Rev. A 28, 2344-2362 (1983).
[CrossRef]

Taleb, A.

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured abcd systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

Tanaka, K.

K. Tanaka, M. Shibukawa, and O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. 20, 749-755 (1972).
[CrossRef]

Traïche, M.

M. Traïche and A. Kellou, “Calculation of the resonant eigenfield in the basis of Laguerre-Gaussian modes for rotationally symmetric plano-concave stable resonators,” Opt. Commun. 208, 391-399 (2002).
[CrossRef]

Truemper, M.

G. Stephan and M. Truemper, “Inhomogeneity effects in a gas laser,” Phys. Rev. A 28, 2344-2362 (1983).
[CrossRef]

Überhuber, C. W.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack. A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

Appl. Opt. (2)

IEEE Trans. Microwave Theory Tech. (1)

K. Tanaka, M. Shibukawa, and O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. 20, 749-755 (1972).
[CrossRef]

J. Mod. Opt. (1)

N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagation factor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51, 2279-2286 (2004).

Opt. Commun. (4)

M. Traïche and A. Kellou, “Calculation of the resonant eigenfield in the basis of Laguerre-Gaussian modes for rotationally symmetric plano-concave stable resonators,” Opt. Commun. 208, 391-399 (2002).
[CrossRef]

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncation series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured abcd systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
[CrossRef]

Opt. Express (1)

Opt. Mater. (1)

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Phys. Rev. A (1)

G. Stephan and M. Truemper, “Inhomogeneity effects in a gas laser,” Phys. Rev. A 28, 2344-2362 (1983).
[CrossRef]

Prog. Opt. (1)

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325-386 (2001).
[CrossRef]

Other (4)

N. S. Petrovic, “Modelling diffraction in optical interconnects,” Ph.D. thesis (University of Queensland, School of Information Technology and Electrical Engineering, 2004).

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Quasi-Optics: Proceedings of the Symposium on Quasi-Optics, J.Fox, ed., Vol. 14 of the Microwave Research Institute Symposia Series (Polytechnic, 1964), pp. 333-347.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack. A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

Y. Hida, X. S. Li, and D. H. Bailey, “Algorithms for quad-double precision floating point arithmetic,” in 15th IEEE Symposium on Computer Arithmetic (ARITH-15 '01), N.Burgess and L.Ciminiera, eds. (IEEE Computer Society, 2001), pp. 0155-162.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Gaussian beam and its approximation using 31 expansion modes at the aperture plane. The beam truncation ratio and the accuracy required for the approximation are, respectively, κ = 10 % and ϵ = 1 % .

Fig. 2
Fig. 2

Radial intensity distributions computed by QAG and myMEM at N f = 7 for an incident Gaussian beam.

Fig. 3
Fig. 3

TEM 10 beam and its approximation using 37 expansion modes at the aperture plane. The beam truncation ratio and the accuracy required for the approximation are, respectively κ = 120 % , and ϵ = 1 % .

Fig. 4
Fig. 4

Radial intensity distributions computed by QAG and myMEM at N f = 9 for an incident TEM 10 beam.

Fig. 5
Fig. 5

TEM 30 beam and its approximation using 35 expansion modes at the aperture plane. The beam truncation ratio and the accuracy required for the approximation are, respectively, κ = 140 % and ϵ = 1 % .

Fig. 6
Fig. 6

Radial intensity distributions computed by QAG and myMEM at the focal plane of a lens for an incident TEM 30 beam.

Tables (2)

Tables Icon

Table 1 Radial Intensity Distributions Corresponding to an Apertured Gaussian Beam Expanded onto 31 Modes a

Tables Icon

Table 2 Radial Intensity Distributions Corresponding to an Apertured TEM 10 Beam Expanded onto 37 Modes a

Equations (42)

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Ψ p ( r , z ) = 2 π 1 w ( z ) exp [ r 2 w ( z ) 2 ] L p [ 2 r 2 w ( z ) 2 ] exp [ j k r 2 2 R ( z ) ] exp [ j ( 2 p + 1 ) γ ( z ) ] exp [ j k ( z z 0 ) ] ,
w ( z ) = w 0 [ 1 + ( z z 0 z R ) 2 ] 1 2 ,
R ( z ) = ( z z 0 ) [ 1 + ( z R z z 0 ) 2 ] ,
γ ( z ) = arctan ( z z 0 z R ) ,
U ( r , z ) = n = 0 + C n ( z ) Ψ ¯ n ( r , z ) .
Ψ ¯ n ( r , z ) = 2 π 1 w ¯ ( z ) exp [ r 2 w ¯ ( z ) 2 ] L n [ 2 r 2 w ¯ ( z ) 2 ] exp [ j k r 2 2 R ¯ ( z ) ] exp [ j ( 2 n + 1 ) Δ γ ¯ ( z ) ] exp [ j k ( z z ¯ 0 ) ] exp [ j φ ] ,
φ = k ( z ¯ 0 z 0 ) ( 2 p + 1 ) γ ( 0 ) ,
Δ γ ¯ ( z ) = γ ¯ ( z ) γ ¯ ( 0 ) .
C n = 0 2 π 0 + U ( r , z ) Ψ ¯ n * ( r , z ) r d r d θ = 2 π 0 + U ( r , z ) Ψ ¯ n * ( r , z ) r d r ,
U ( r , z = 0 ) = τ ( r ) Ψ p ( r , z = 0 ) ,
C p n = 2 w ( 0 ) w ¯ ( 0 ) 0 + τ ( X ) L p ( α X ) L n ( β X ) exp ( Q X ) d X ,
α = 2 w ( 0 ) 2 , β = 2 w ¯ ( 0 ) 2 ,
Q = 1 w ( 0 ) 2 + 1 w ¯ ( 0 ) 2 + j k 2 [ 1 R ( 0 ) 1 R ¯ ( 0 ) ] .
Q = 1 w ( 0 ) 2 + 1 w ¯ ( 0 ) 2 + j k 2 [ 1 R ( 0 ) 1 f 1 R ¯ ( 0 ) ] .
U ( r , z ) n = 0 N 1 C p n Ψ ¯ n ( r , z ) ,
circ ( r a ) = { 1 0 r a 0 r > a } ,
C p n = 2 w ( 0 ) w ¯ ( 0 ) 0 Γ L p ( α X ) L n ( β X ) exp ( Q X ) d X ,
1 R ¯ ( 0 ) = 1 R ( 0 ) 1 f
1 R ¯ ( 0 ) = 1 R ( 0 )
U ( r ) = circ ( r a ) Ψ p ( r ) ,
U ( r ) U N ( r ) = n = 0 N 1 C p n Ψ ¯ n ( r ) ,
ϵ N = U U N 2 U 2 ,
U 2 = 0 2 π 0 + | U ( r ) | 2 r d r d θ = 2 π 0 + | U ( r ) | 2 r d r = 2 π 0 1 | Ψ p ( r ) | 2 r d r .
ϵ N = 1 1 U 2 n = 0 N 1 C p n 2 .
C p n = 2 w ( 0 ) w ¯ ( 0 ) I ( p , n ) .
U 2 = α 0 1 L p ( α X ) L p ( α X ) exp ( α X ) d X ,
U 2 = α I ( p , p ) ,
err = i | I 1 ( i ) I 2 ( i ) | i I 1 ( i ) × 100.
N f = [ 1 R ( 0 ) + 1 z m ] a 2 λ .
I Γ ( p , q ) = 0 Γ L p ( μ x ) L q ( ν x ) exp ( R x ) d x ,
I Γ ( p , q ) = Γ 0 1 L p ( α y ) L q ( β y ) exp ( Q y ) d y = Γ I ( p , q ) ,
p 2 , p L p ( x ) = ( 2 p 1 x ) L p 1 ( x ) ( p 1 ) L p 2 ( x ) ,
I ( 0 , 0 ) = 1 Q [ 1 exp ( Q ) ] ,
I ( 0 , 1 ) = ( 1 β Q ) I ( 0 , 0 ) + β Q exp ( Q ) ,
q 2 , I ( 0 , q ) = ( 2 q 1 q β Q ) I ( 0 , q 1 ) + q 1 q ( β Q 1 ) I ( 0 , q 2 ) + β q Q exp ( Q ) L q 1 ( β ) ,
I ( 1 , 0 ) = ( 1 α Q ) I ( 0 , 0 ) + α Q exp ( Q ) ,
q 1 , I ( 1 , q ) = [ 1 α ( q + 1 ) Q ] I ( 0 , q ) + α q Q I ( 0 , q 1 ) + α Q L q ( β ) exp ( Q ) ,
p 2 , I ( p , 0 ) = ( 2 p 1 p α Q ) I ( p 1 , 0 ) + p 1 p ( α Q 1 ) I ( p 2 , 0 ) + α p Q exp ( Q ) L p 1 ( α ) ,
p 2 , q 1 , I ( p , q ) = 1 p [ T 1 ( p , q ) + T 2 ( p , q ) + T 3 ( p , q ) + α Q exp ( Q ) L p 1 ( α ) L q ( β ) ] ,
T 1 ( p , q ) = [ 2 p 1 α ( p + q ) Q ] I ( p 1 , q ) ,
T 2 ( p , q ) = ( p 1 ) ( α Q 1 ) I ( p 2 , q ) ,
T 3 ( p , q ) = α q Q I ( p 1 , q 1 ) .

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