Abstract

The normalization of energy divergent Weber waves and finite energy Weber–Gauss beams is reported. The well-known Bessel and Mathieu waves are used to derive the integral relations between circular, elliptic, and parabolic waves and to present the Bessel and Mathieu wave decomposition of the Weber waves. The efficiency to approximate a Weber–Gauss beam as a finite superposition of Bessel–Gauss beams is also given.

© 2010 Optical Society of America

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References

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  1. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge Univ. Press, 1927).
  2. J. A. Stratton, Electromagnetic Theory, International Series in Pure and Applied Physics (Read Books, 1941).
  3. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1, International Series in Pure and Applied Physics (McGraw-Hill, 1953).
  4. W. Miller, Symmetry and Separation of Variables, Encyclopedia of Mathematics and Its Applications (Cambridge Univ. Press, 1984).
  5. J. Durnin, “Exact solutions for nondiffracting beams. I. the scalar theory,” J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  6. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  7. A. A. Makarov, J. A. Smorodinsky, K. Valiev, and P. Winternitz, “A systematic search for nonrelativistic systems with dynamical symmetries,” Il Nuovo Cimento LII A, 1061-1084 (1967).
  8. C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35-80 (1976).
  9. R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
    [CrossRef]
  10. K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A, Pure Appl. Opt. 8, 867-877 (2006).
    [CrossRef]
  11. B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry,” Phys. Rev. A 78, 033813 (2008).
    [CrossRef]
  12. B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants of structured photons with parabolic-cylindrical symmetry,” Phys. Rev. A 79, 055806 (2009).
    [CrossRef]
  13. D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Elsevier, 2009).
    [PubMed]
  14. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289-298 (2005).
    [CrossRef]
  15. C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 456, 068001 (2006).
    [CrossRef]
  16. M. B. Alvarez-Elizondo, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, “Generation of Mathieu-Gauss modes with an axicon-based laser resonator,” Opt. Express 16, 18770-18775 (2008).
    [CrossRef]
  17. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44-46 (2004).
    [CrossRef] [PubMed]
  18. J. C. Gutiérrez-Vega and M. A. Bandres, “Normalization of the Mathieu-Gauss optical beams,” J. Opt. Soc. Am. A 24, 215-220 (2007).
    [CrossRef]
  19. A. P. Prudnikov, J. A. Brychkov, and O. I. Marichev, Integrals and Series, Vols. 1-3 (Mockba, 1981).
  20. A.Erdélyi, ed., Higher Trascendental Functions, Vol. 1 (McGraw-Hill, 1985).
  21. A. A. Inayat-Hussain, “Mathieu integral transforms,” J. Math. Phys. 32, 669-675 (1991).
    [CrossRef]
  22. N. N. Lebedev, Special Functions and Their Applications (Prentice-Hall, 1965).
  23. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Courier Dover, 1970).

2009 (1)

B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants of structured photons with parabolic-cylindrical symmetry,” Phys. Rev. A 79, 055806 (2009).
[CrossRef]

2008 (2)

B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry,” Phys. Rev. A 78, 033813 (2008).
[CrossRef]

M. B. Alvarez-Elizondo, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, “Generation of Mathieu-Gauss modes with an axicon-based laser resonator,” Opt. Express 16, 18770-18775 (2008).
[CrossRef]

2007 (1)

2006 (2)

C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 456, 068001 (2006).
[CrossRef]

K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A, Pure Appl. Opt. 8, 867-877 (2006).
[CrossRef]

2005 (2)

J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289-298 (2005).
[CrossRef]

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
[CrossRef]

2004 (1)

1991 (1)

A. A. Inayat-Hussain, “Mathieu integral transforms,” J. Math. Phys. 32, 669-675 (1991).
[CrossRef]

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. the scalar theory,” J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

1976 (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35-80 (1976).

1967 (1)

A. A. Makarov, J. A. Smorodinsky, K. Valiev, and P. Winternitz, “A systematic search for nonrelativistic systems with dynamical symmetries,” Il Nuovo Cimento LII A, 1061-1084 (1967).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Courier Dover, 1970).

Alvarez-Elizondo, M. B.

Andrews, D. L.

D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Elsevier, 2009).
[PubMed]

Bandres, M. A.

Boyer, C. P.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35-80 (1976).

Brychkov, J. A.

A. P. Prudnikov, J. A. Brychkov, and O. I. Marichev, Integrals and Series, Vols. 1-3 (Mockba, 1981).

Chávez-Cerda, S.

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. the scalar theory,” J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1, International Series in Pure and Applied Physics (McGraw-Hill, 1953).

Gutiérrez-Vega, J. C.

Hacyan, S.

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
[CrossRef]

Inayat-Hussain, A. A.

A. A. Inayat-Hussain, “Mathieu integral transforms,” J. Math. Phys. 32, 669-675 (1991).
[CrossRef]

Jáuregui, R.

B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants of structured photons with parabolic-cylindrical symmetry,” Phys. Rev. A 79, 055806 (2009).
[CrossRef]

B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry,” Phys. Rev. A 78, 033813 (2008).
[CrossRef]

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
[CrossRef]

Kalnins, E. G.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35-80 (1976).

Lebedev, N. N.

N. N. Lebedev, Special Functions and Their Applications (Prentice-Hall, 1965).

Ley-Koo, E.

K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A, Pure Appl. Opt. 8, 867-877 (2006).
[CrossRef]

López-Mariscal, C.

C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 456, 068001 (2006).
[CrossRef]

Makarov, A. A.

A. A. Makarov, J. A. Smorodinsky, K. Valiev, and P. Winternitz, “A systematic search for nonrelativistic systems with dynamical symmetries,” Il Nuovo Cimento LII A, 1061-1084 (1967).

Marichev, O. I.

A. P. Prudnikov, J. A. Brychkov, and O. I. Marichev, Integrals and Series, Vols. 1-3 (Mockba, 1981).

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Miller, W.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35-80 (1976).

W. Miller, Symmetry and Separation of Variables, Encyclopedia of Mathematics and Its Applications (Cambridge Univ. Press, 1984).

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1, International Series in Pure and Applied Physics (McGraw-Hill, 1953).

Prudnikov, A. P.

A. P. Prudnikov, J. A. Brychkov, and O. I. Marichev, Integrals and Series, Vols. 1-3 (Mockba, 1981).

Rodríguez-Lara, B. M.

B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants of structured photons with parabolic-cylindrical symmetry,” Phys. Rev. A 79, 055806 (2009).
[CrossRef]

B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry,” Phys. Rev. A 78, 033813 (2008).
[CrossRef]

Rodríguez-Masegosa, R.

Smorodinsky, J. A.

A. A. Makarov, J. A. Smorodinsky, K. Valiev, and P. Winternitz, “A systematic search for nonrelativistic systems with dynamical symmetries,” Il Nuovo Cimento LII A, 1061-1084 (1967).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Courier Dover, 1970).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory, International Series in Pure and Applied Physics (Read Books, 1941).

Valiev, K.

A. A. Makarov, J. A. Smorodinsky, K. Valiev, and P. Winternitz, “A systematic search for nonrelativistic systems with dynamical symmetries,” Il Nuovo Cimento LII A, 1061-1084 (1967).

Volke-Sepulveda, K.

K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A, Pure Appl. Opt. 8, 867-877 (2006).
[CrossRef]

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge Univ. Press, 1927).

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge Univ. Press, 1927).

Winternitz, P.

A. A. Makarov, J. A. Smorodinsky, K. Valiev, and P. Winternitz, “A systematic search for nonrelativistic systems with dynamical symmetries,” Il Nuovo Cimento LII A, 1061-1084 (1967).

Il Nuovo Cimento (1)

A. A. Makarov, J. A. Smorodinsky, K. Valiev, and P. Winternitz, “A systematic search for nonrelativistic systems with dynamical symmetries,” Il Nuovo Cimento LII A, 1061-1084 (1967).

J. Math. Phys. (1)

A. A. Inayat-Hussain, “Mathieu integral transforms,” J. Math. Phys. 32, 669-675 (1991).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A, Pure Appl. Opt. 8, 867-877 (2006).
[CrossRef]

J. Opt. Soc. Am. A (3)

Nagoya Math. J. (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35-80 (1976).

Opt. Eng. (1)

C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 456, 068001 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (3)

B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry,” Phys. Rev. A 78, 033813 (2008).
[CrossRef]

B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants of structured photons with parabolic-cylindrical symmetry,” Phys. Rev. A 79, 055806 (2009).
[CrossRef]

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Other (9)

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge Univ. Press, 1927).

J. A. Stratton, Electromagnetic Theory, International Series in Pure and Applied Physics (Read Books, 1941).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1, International Series in Pure and Applied Physics (McGraw-Hill, 1953).

W. Miller, Symmetry and Separation of Variables, Encyclopedia of Mathematics and Its Applications (Cambridge Univ. Press, 1984).

D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Elsevier, 2009).
[PubMed]

A. P. Prudnikov, J. A. Brychkov, and O. I. Marichev, Integrals and Series, Vols. 1-3 (Mockba, 1981).

A.Erdélyi, ed., Higher Trascendental Functions, Vol. 1 (McGraw-Hill, 1985).

N. N. Lebedev, Special Functions and Their Applications (Prentice-Hall, 1965).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Courier Dover, 1970).

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Figures (3)

Fig. 1
Fig. 1

Ideal Weber waves with wave number k = 2 π λ and z component γ = 0.31756 rad leading to k z = 0.95 k . (a) Ψ e , k , γ , a = 2 ( W ) . (b) Ψ o , k , γ , a = 2 ( W ) . (c) Ψ k , γ , a = 2 ( W ) . (d) Ψ e , k , γ , a = 2 ( W ) . (e) Ψ o , k , γ , a = 2 ( W ) . (f) Ψ k , γ , a = 2 ( W ) .

Fig. 2
Fig. 2

Behavior of the truncated normalization coefficient j = 0 n S j for a given wave number k = 2 π λ and Euler angle γ = 0.31756 rad leading to k z = 0.95 k for (a) even Weber fields Ψ e , k , γ , a ( W ) and (b) Ψ o , k , γ , a = 2 ( W ) , with eigenvalues a = 11.2821 (circle), a = 2.2827 (square), a = 0 (diamond), a = 10 (triangle), a = 23.7902 (inverted triangle).

Fig. 3
Fig. 3

Behavior of the absolute value of the normalized Bessel decomposition coefficients of (a) even and (b) odd Weber waves with a given wave number k = 2 π λ , Euler angle γ = 0.31756 rad leading to k z = 0.95 k , and eigenvalues a = 11.2821 (circle), a = 2.2827 (square), a = 0 (diamond), a = 10 (triangle), a = 23.7902 (inverted triangle).

Equations (54)

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( 2 u 2 + k 2 u 2 2 k a ) U ( u ) = 0 ,
( 2 v 2 + k 2 v 2 + 2 k a ) V ( v ) = 0 .
Ψ e , k , γ , a ( W ) ( u , v , z ) = | Γ [ 1 4 + ı a 2 ] | 2 π 2 sin γ U e , k , γ , a ( u ) V e , k , γ , a ( v ) e ı ( k z z ω t ) ,
Ψ o , k , γ , a ( W ) ( u , v , z ) = 2 | Γ [ 3 4 + ı a 2 ] | 2 π sin γ U o , k , γ , a ( u ) V o , k , γ , a ( v ) e ı ( k z z ω t ) .
d 3 x Ψ p ̃ , k , γ ̃ , a ̃ ( W ) * Ψ p , k , γ , a ( W ) = δ p ̃ , p δ ( γ ̃ γ ) δ ( a ̃ a ) , p , p ̃ = e , o .
U e , k , γ , a ( u ) = e ı k u 2 2 F 1 1 ( 1 4 ı a 2 , 1 2 , ı k u 2 ) ,
V e , k , γ , a ( v ) = e ı k v 2 2 F 1 1 ( 1 4 + ı a 2 , 1 2 , ı k v 2 ) ,
U o , k , γ , a ( u ) = 2 k u e ı k u 2 2 F 1 1 ( 3 4 ı a 2 , 3 2 , ı k u 2 ) ,
V o , k , γ , a ( v ) = 2 k v e ı k v 2 2 F 1 1 ( 3 4 + ı a 2 , 3 2 , ı k v 2 ) .
P z Ψ p , k , γ , a = k z Ψ p , k , γ , a ,
{ J z , P y } Ψ p , k , γ , a = 2 a k Ψ p , k , γ , a ,
Ψ p , k , γ , a ( x , y , z ) = π π d ϕ A p , k , γ , a ( W ) ( ϕ ) e ı k ( x cos ϕ + y sin ϕ ) e ı k z z ,
A e , k , γ , a ( W ) ( ϕ ) = e ı a ln | tan ϕ 2 | 2 π sin γ | sin ϕ | ,
A o , k , γ , a ( W ) ( ϕ ) = ı sgn ϕ A e , k , γ , a ( ϕ ) .
Ψ e o , k , γ , a ( W ) ( r , φ , z ) = 1 2 [ Ψ k , γ , a ( W ) ( r , φ , z ) ± Ψ k , γ , a ( W ) ( r , φ , z ) ] ,
A k , γ , a ( W ) ( ϕ ) = A e , k , γ , a ( ϕ ) Θ ( ϕ ) ,
A k , γ , a ( W ) ( ϕ ) = ı A k , γ , a ( ϕ ) ,
Ψ p , k , γ , a ( W ) ( r , φ , z ) = n = 0 ψ p , k , γ , a ( B ) ( n ) Ψ p , k , γ , n ( B ) ( r , φ , z ) ,
ψ e , k , γ , a ( B ) ( n ) = 1 2 π e π ( 2 a + ı ) 4 [ C ( n , a ) + C ( n , a ) ] ,
ψ o , k , γ , a ( B ) ( n ) = ı 2 π e π ( 2 a + ı ) 4 [ C ( n , a ) C ( n , a ) ] ,
ψ k , γ , ± a ( B ) ( n ) = 1 2 π e π ( 2 a + ı ) 4 C ( n , a ) ,
C ( n , a ) = Γ ( n + 1 2 ) [ ( 1 ) n f ( n , a ) + ı sgn a f ( n , a ) ] ,
f ( n , a ) = 2 ı a Γ ( 1 2 ı a ) Γ ( 1 + n ı a ) × F 1 2 ( 1 2 ı a , 1 2 ı a , 1 + n ı a , 1 2 ) .
Ψ e o , k , γ , n ( B ) ( r , φ , z ) = 1 2 [ Ψ k , γ , n ( B ) ( r , φ , z ) ± Ψ k , γ , n ( B ) ( r , φ , z ) ] ,
n = 0 , 1 , 2 , 3
Ψ k , γ , n ( B ) ( r , φ , z ) = ( 2 π ) ( 1 2 ) ı n J n ( k r ) e ı ( n φ + k z z ) ,
n = 0 , ± 1 , ± 2 , ,
A k , γ , n ( B ) ( ϕ ) = e ı n ϕ 2 π sin γ .
Ψ e , k , γ , a ( M ) ( ξ , η , z ) = s e , n , q C e m ( ξ , q ) c e m ( η , q ) e ı ( k z z ω t ) ,
Ψ o , k , γ , a ( M ) ( ξ , η , z ) = s o , n , q S e m ( ξ , q ) s e m ( η , q ) e ı ( k z z ω t ) ,
s e , 2 n , q = ce 2 n ( 0 , q ) ce 2 n ( π 2 , q ) A 0 ( 2 n ) ( q ) ,
s e , 2 n + 1 , q = ce 2 n + 1 ( 0 , q ) ce 2 n + 1 ( π 2 , q ) q 1 2 A 1 ( 2 n + 1 ) ( q ) ,
s o , 2 n + 2 , q = se 2 n + 2 ( 0 , q ) se 2 n + 2 ( π 2 , q ) q B 2 ( 2 n + 2 ) ( q ) ,
s o , 2 n + 1 , q = se 2 n + 1 ( 0 , q ) se 2 n + 1 ( π 2 , q ) q 1 2 B 1 ( 2 n + 1 ) ( q ) .
A e , k , γ , m ( M ) ( ϕ ) = ce m ( φ , q ) 2 π sin γ ,
A o , k , γ , m ( M ) ( ϕ ) = se m ( φ , q ) 2 π sin γ .
ce m ( φ , q ) = n = 0 m A n ̃ ( m ) ( q ) cos n ̃ φ ,
ce m ( φ , q ) = n = 0 m B n ̃ ( m ) ( q ) sin n ̃ φ , n ̃ = 2 n + m mod 2 ,
ψ e , k , γ , a ( M ) ( m ) = n = 0 m A n ̃ ( m ) ( q ) ψ e , k , γ , a ( B ) ( n ̃ ) ,
ψ o , k , γ , a ( M ) ( m ) = n = 0 m B n ̃ ( m ) ( q ) ψ o , k , γ , a ( B ) ( n ̃ ) , n ̃ = 2 n + m mod 2 ,
Ψ p , k , γ , a ( W ) ( ξ , η , z ) = m = 0 ψ p , k , γ , a ( M ) ( m ) Ψ p , k , γ , m ( M ) ( ξ , η , z ) .
Φ p , k , γ , a ( r ) = 1 μ ( z ) e ı ( k k 2 2 k μ ( z ) ) z e r 2 ω 0 2 μ ( z ) Ψ p , k , γ , a ( u ̃ , v ̃ ) .
d 2 x | Φ p , k , γ , a ( r , φ ) | 2 = π ω ̃ 2 e k 2 ω ̃ 2 n = 0 ( 1 + δ n , 0 ) | ψ p , k , γ , a ( B ) ( n ) | 2 I n ( k 2 ω ̃ 2 ) , p = e , o ,
0 d r r e 2 r 2 ω 0 2 J n ( k r ) J n ( k r ) = ω ̃ 2 e k 2 ω ̃ 2 I n ( k 2 ω ̃ 2 ) , n 0 .
lim n S n S n 1 < 1
S n = π ω ̃ 2 e k 2 ω ̃ 2 ( 1 + δ n , 0 ) | ψ p , k , γ , a ( B ) ( n ) | 2 I n ( k 2 ω ̃ 2 ) , n 0 .
lim n S n S n 2 I n ( k 2 ω ̃ 2 ) I n 2 ( k 2 ω ̃ 2 ) ( e k 2 ω ̃ 2 8 n ) 2 , log e = 1 ,
| ψ e , k , γ , 0 ( 0 ) | 2 = 4 Γ 2 ( 5 / 4 ) π Γ 2 ( 3 / 4 ) ,
| ψ e , k , γ , 0 ( n ) | 2 = ( 1 n mod 2 ) Γ 2 ( 2 n + 1 4 ) 4 π Γ 2 ( 2 n + 3 4 ) , n 1 ,
| ψ o , k , γ , 0 ( 1 ) | 2 = 4 Γ 2 ( 3 / 4 ) π Γ 2 ( 1 / 4 ) ,
| ψ o , k , γ , 0 ( n ) | 2 = ( n mod 2 ) Γ 2 ( 2 n + 1 4 ) 4 π Γ 2 ( 2 n + 3 4 ) , n 2 ,
I n ( z ) 1 2 ı e ı n π 2 + n + n log ( z e ı π 2 2 ) ( n + 1 2 ) log n , n ,
Γ ( z ) 2 π e z z z 1 2 ( 1 + 1 12 z ) , z .
lim n | ψ p , k , γ , a ( B ) ( n ) ψ p , k , γ , a ( B ) ( n 1 ) | 1 .

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