Abstract

Explicit solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation are applied to wave fields with invariant features, such as oscillating laser beams in a parabolic waveguide and spiral light beams in varying media. A similar effect of superfocusing of particle beams in a thin monocrystal film, harmonic oscillations of cold trapped atoms, and motion in magnetic field are also mentioned.

© 2013 Optical Society of America

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  1. R. Cordero-Soto, R. M. Lopez, E. Suazo, and S. K. Suslov, Lett. Math. Phys. 84, 159 (2008).
    [CrossRef]
  2. N. Lanfear, R. M. López, and S. K. Suslov, J. Russ. Laser Res. 32, 352 (2011).
    [CrossRef]
  3. E. Suazo and S. K. Suslov, J. Russ. Laser Res. 33, 63 (2012).
    [CrossRef]
  4. S. K. Suslov, Proc. Am. Math. Soc. 140, 3067 (2012).
    [CrossRef]
  5. V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, 1965).
  6. S. N. Vlasov and V. I. Talanov, Radiophys. Quantum Electron. 38, 1 (1995).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999).
  8. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Theory of Waves (Nauka, 1979) [in Russian].
  9. V. V. Dodonov and V. I. Man’ko, in Invariants and the Evolution of Nonstationary Quantum Systems (Nova Science, 1989), p. 103.
  10. A. Mahalov and S. K. Suslov, “Solution of paraxial wave equation for inhomogeneous media in linear and quadratic approximation,” Proc. Am. Math. Soc. (to be published).
  11. A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer, 1991).
  12. R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, Phys. Scr. 87, 038112 (2013).
    [CrossRef]
  13. R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, J. Differ. Equ. Appl. 19, 543 (2013).
    [CrossRef]
  14. D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003).
    [CrossRef]
  15. M. E. Marhic, Lett. Nuovo Cimento Soc. Ital. Fis. 22, 376 (1978).
    [CrossRef]
  16. G. P. Agrawal, A. K. Ghatak, and C. L. Mehtav, Opt. Commun. 12, 333 (1974).
    [CrossRef]
  17. Y. Gu and G. Gbur, Opt. Lett. 35, 3456 (2010).
    [CrossRef]
  18. E. G. Abramochkin and E. Razueva, Opt. Lett. 36, 3732 (2011).
    [CrossRef]
  19. A. Mahalov and S. K. Suslov, Phys. Lett. A 377, 33 (2012).
    [CrossRef]
  20. V. Fock, Zs. für Phys. 47, 446 (1928).
    [CrossRef]
  21. V. A. Fock, Selected Works: Quantum Mechanics and Quantum Field Theory (Chapman & Hall/CRC, 2004), p. 29.
  22. M. Meiler, R. Cordero-Soto, and S. K. Suslov, J. Math. Phys. 49, 072102 (2008).
    [CrossRef]
  23. R. Cordero-Soto and S. K. Suslov, Theor. Math. Phys. 162, 286 (2010).
    [CrossRef]
  24. R. Piestun, Y. Y. Schechner, and J. Shamir, J. Opt. Soc. Am. 17, 294 (2000).
    [CrossRef]
  25. E. G. Abramochkin and V. G. Volostnikov, Phys. Usp. 47, 1177 (2004).
    [CrossRef]
  26. M. R. Hatzvi and Y. Y. Schechner, Opt. Lett. 37, 3207 (2012).
    [CrossRef]
  27. Y. N. Demkov, Phys. Atomic Nuclei 72, 779 (2009).
    [CrossRef]
  28. S. I. Kryuchkov, S. K. Suslov, and J. M. Vega-Guzmán, J. Phys. B 46, 104007 (2013).
    [CrossRef]
  29. E. A. Kuznetsov and S. K. Turitsyn, Phys. Lett. A 112, 273 (1985).
    [CrossRef]
  30. A. Y. Okulov, J. Phys. B 41, 101001 (2008).
    [CrossRef]
  31. O. Korotkova, N. Farwell, and A. Mahalov, Waves Random Media 19, 692 (2009).
    [CrossRef]
  32. X. Pang, G. Gbur, and T. D. Visser, Opt. Lett. 36, 2492 (2011).
    [CrossRef]
  33. Y. Gu, J. Opt. Soc. Am. A 30, 708 (2013).
    [CrossRef]

2013 (4)

R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, Phys. Scr. 87, 038112 (2013).
[CrossRef]

R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, J. Differ. Equ. Appl. 19, 543 (2013).
[CrossRef]

S. I. Kryuchkov, S. K. Suslov, and J. M. Vega-Guzmán, J. Phys. B 46, 104007 (2013).
[CrossRef]

Y. Gu, J. Opt. Soc. Am. A 30, 708 (2013).
[CrossRef]

2012 (4)

M. R. Hatzvi and Y. Y. Schechner, Opt. Lett. 37, 3207 (2012).
[CrossRef]

A. Mahalov and S. K. Suslov, Phys. Lett. A 377, 33 (2012).
[CrossRef]

E. Suazo and S. K. Suslov, J. Russ. Laser Res. 33, 63 (2012).
[CrossRef]

S. K. Suslov, Proc. Am. Math. Soc. 140, 3067 (2012).
[CrossRef]

2011 (3)

2010 (2)

Y. Gu and G. Gbur, Opt. Lett. 35, 3456 (2010).
[CrossRef]

R. Cordero-Soto and S. K. Suslov, Theor. Math. Phys. 162, 286 (2010).
[CrossRef]

2009 (2)

Y. N. Demkov, Phys. Atomic Nuclei 72, 779 (2009).
[CrossRef]

O. Korotkova, N. Farwell, and A. Mahalov, Waves Random Media 19, 692 (2009).
[CrossRef]

2008 (3)

M. Meiler, R. Cordero-Soto, and S. K. Suslov, J. Math. Phys. 49, 072102 (2008).
[CrossRef]

A. Y. Okulov, J. Phys. B 41, 101001 (2008).
[CrossRef]

R. Cordero-Soto, R. M. Lopez, E. Suazo, and S. K. Suslov, Lett. Math. Phys. 84, 159 (2008).
[CrossRef]

2004 (1)

E. G. Abramochkin and V. G. Volostnikov, Phys. Usp. 47, 1177 (2004).
[CrossRef]

2003 (1)

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003).
[CrossRef]

2000 (1)

1995 (1)

S. N. Vlasov and V. I. Talanov, Radiophys. Quantum Electron. 38, 1 (1995).
[CrossRef]

1985 (1)

E. A. Kuznetsov and S. K. Turitsyn, Phys. Lett. A 112, 273 (1985).
[CrossRef]

1978 (1)

M. E. Marhic, Lett. Nuovo Cimento Soc. Ital. Fis. 22, 376 (1978).
[CrossRef]

1974 (1)

G. P. Agrawal, A. K. Ghatak, and C. L. Mehtav, Opt. Commun. 12, 333 (1974).
[CrossRef]

1928 (1)

V. Fock, Zs. für Phys. 47, 446 (1928).
[CrossRef]

Abramochkin, E. G.

E. G. Abramochkin and E. Razueva, Opt. Lett. 36, 3732 (2011).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, Phys. Usp. 47, 1177 (2004).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, A. K. Ghatak, and C. L. Mehtav, Opt. Commun. 12, 333 (1974).
[CrossRef]

Blatt, R.

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999).

Cordero-Soto, R.

R. Cordero-Soto and S. K. Suslov, Theor. Math. Phys. 162, 286 (2010).
[CrossRef]

R. Cordero-Soto, R. M. Lopez, E. Suazo, and S. K. Suslov, Lett. Math. Phys. 84, 159 (2008).
[CrossRef]

M. Meiler, R. Cordero-Soto, and S. K. Suslov, J. Math. Phys. 49, 072102 (2008).
[CrossRef]

Demkov, Y. N.

Y. N. Demkov, Phys. Atomic Nuclei 72, 779 (2009).
[CrossRef]

Dodonov, V. V.

V. V. Dodonov and V. I. Man’ko, in Invariants and the Evolution of Nonstationary Quantum Systems (Nova Science, 1989), p. 103.

Farwell, N.

O. Korotkova, N. Farwell, and A. Mahalov, Waves Random Media 19, 692 (2009).
[CrossRef]

Fock, V.

V. Fock, Zs. für Phys. 47, 446 (1928).
[CrossRef]

Fock, V. A.

V. A. Fock, Selected Works: Quantum Mechanics and Quantum Field Theory (Chapman & Hall/CRC, 2004), p. 29.

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, 1965).

Gbur, G.

Ghatak, A. K.

G. P. Agrawal, A. K. Ghatak, and C. L. Mehtav, Opt. Commun. 12, 333 (1974).
[CrossRef]

Gu, Y.

Hatzvi, M. R.

Korotkova, O.

O. Korotkova, N. Farwell, and A. Mahalov, Waves Random Media 19, 692 (2009).
[CrossRef]

Kryuchkov, S. I.

S. I. Kryuchkov, S. K. Suslov, and J. M. Vega-Guzmán, J. Phys. B 46, 104007 (2013).
[CrossRef]

Kuznetsov, E. A.

E. A. Kuznetsov and S. K. Turitsyn, Phys. Lett. A 112, 273 (1985).
[CrossRef]

Lanfear, N.

N. Lanfear, R. M. López, and S. K. Suslov, J. Russ. Laser Res. 32, 352 (2011).
[CrossRef]

Leibfried, D.

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003).
[CrossRef]

Lopez, R. M.

R. Cordero-Soto, R. M. Lopez, E. Suazo, and S. K. Suslov, Lett. Math. Phys. 84, 159 (2008).
[CrossRef]

López, R. M.

R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, Phys. Scr. 87, 038112 (2013).
[CrossRef]

R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, J. Differ. Equ. Appl. 19, 543 (2013).
[CrossRef]

N. Lanfear, R. M. López, and S. K. Suslov, J. Russ. Laser Res. 32, 352 (2011).
[CrossRef]

Mahalov, A.

A. Mahalov and S. K. Suslov, Phys. Lett. A 377, 33 (2012).
[CrossRef]

O. Korotkova, N. Farwell, and A. Mahalov, Waves Random Media 19, 692 (2009).
[CrossRef]

A. Mahalov and S. K. Suslov, “Solution of paraxial wave equation for inhomogeneous media in linear and quadratic approximation,” Proc. Am. Math. Soc. (to be published).

Man’ko, V. I.

V. V. Dodonov and V. I. Man’ko, in Invariants and the Evolution of Nonstationary Quantum Systems (Nova Science, 1989), p. 103.

Marhic, M. E.

M. E. Marhic, Lett. Nuovo Cimento Soc. Ital. Fis. 22, 376 (1978).
[CrossRef]

Mehtav, C. L.

G. P. Agrawal, A. K. Ghatak, and C. L. Mehtav, Opt. Commun. 12, 333 (1974).
[CrossRef]

Meiler, M.

M. Meiler, R. Cordero-Soto, and S. K. Suslov, J. Math. Phys. 49, 072102 (2008).
[CrossRef]

Monroe, C.

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003).
[CrossRef]

Nikiforov, A. F.

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer, 1991).

Okulov, A. Y.

A. Y. Okulov, J. Phys. B 41, 101001 (2008).
[CrossRef]

Pang, X.

Piestun, R.

Razueva, E.

Rudenko, O. V.

M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Theory of Waves (Nauka, 1979) [in Russian].

Schechner, Y. Y.

Shamir, J.

Suazo, E.

E. Suazo and S. K. Suslov, J. Russ. Laser Res. 33, 63 (2012).
[CrossRef]

R. Cordero-Soto, R. M. Lopez, E. Suazo, and S. K. Suslov, Lett. Math. Phys. 84, 159 (2008).
[CrossRef]

Sukhorukov, A. P.

M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Theory of Waves (Nauka, 1979) [in Russian].

Suslov, S. K.

R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, Phys. Scr. 87, 038112 (2013).
[CrossRef]

S. I. Kryuchkov, S. K. Suslov, and J. M. Vega-Guzmán, J. Phys. B 46, 104007 (2013).
[CrossRef]

R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, J. Differ. Equ. Appl. 19, 543 (2013).
[CrossRef]

E. Suazo and S. K. Suslov, J. Russ. Laser Res. 33, 63 (2012).
[CrossRef]

A. Mahalov and S. K. Suslov, Phys. Lett. A 377, 33 (2012).
[CrossRef]

S. K. Suslov, Proc. Am. Math. Soc. 140, 3067 (2012).
[CrossRef]

N. Lanfear, R. M. López, and S. K. Suslov, J. Russ. Laser Res. 32, 352 (2011).
[CrossRef]

R. Cordero-Soto and S. K. Suslov, Theor. Math. Phys. 162, 286 (2010).
[CrossRef]

R. Cordero-Soto, R. M. Lopez, E. Suazo, and S. K. Suslov, Lett. Math. Phys. 84, 159 (2008).
[CrossRef]

M. Meiler, R. Cordero-Soto, and S. K. Suslov, J. Math. Phys. 49, 072102 (2008).
[CrossRef]

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer, 1991).

A. Mahalov and S. K. Suslov, “Solution of paraxial wave equation for inhomogeneous media in linear and quadratic approximation,” Proc. Am. Math. Soc. (to be published).

Talanov, V. I.

S. N. Vlasov and V. I. Talanov, Radiophys. Quantum Electron. 38, 1 (1995).
[CrossRef]

Turitsyn, S. K.

E. A. Kuznetsov and S. K. Turitsyn, Phys. Lett. A 112, 273 (1985).
[CrossRef]

Uvarov, V. B.

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer, 1991).

Vega-Guzmán, J. M.

R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, J. Differ. Equ. Appl. 19, 543 (2013).
[CrossRef]

S. I. Kryuchkov, S. K. Suslov, and J. M. Vega-Guzmán, J. Phys. B 46, 104007 (2013).
[CrossRef]

R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, Phys. Scr. 87, 038112 (2013).
[CrossRef]

Vinogradova, M. B.

M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Theory of Waves (Nauka, 1979) [in Russian].

Visser, T. D.

Vlasov, S. N.

S. N. Vlasov and V. I. Talanov, Radiophys. Quantum Electron. 38, 1 (1995).
[CrossRef]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, Phys. Usp. 47, 1177 (2004).
[CrossRef]

Wineland, D.

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999).

J. Differ. Equ. Appl. (1)

R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, J. Differ. Equ. Appl. 19, 543 (2013).
[CrossRef]

J. Math. Phys. (1)

M. Meiler, R. Cordero-Soto, and S. K. Suslov, J. Math. Phys. 49, 072102 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. B (2)

S. I. Kryuchkov, S. K. Suslov, and J. M. Vega-Guzmán, J. Phys. B 46, 104007 (2013).
[CrossRef]

A. Y. Okulov, J. Phys. B 41, 101001 (2008).
[CrossRef]

J. Russ. Laser Res. (2)

N. Lanfear, R. M. López, and S. K. Suslov, J. Russ. Laser Res. 32, 352 (2011).
[CrossRef]

E. Suazo and S. K. Suslov, J. Russ. Laser Res. 33, 63 (2012).
[CrossRef]

Lett. Math. Phys. (1)

R. Cordero-Soto, R. M. Lopez, E. Suazo, and S. K. Suslov, Lett. Math. Phys. 84, 159 (2008).
[CrossRef]

Lett. Nuovo Cimento Soc. Ital. Fis. (1)

M. E. Marhic, Lett. Nuovo Cimento Soc. Ital. Fis. 22, 376 (1978).
[CrossRef]

Opt. Commun. (1)

G. P. Agrawal, A. K. Ghatak, and C. L. Mehtav, Opt. Commun. 12, 333 (1974).
[CrossRef]

Opt. Lett. (4)

Phys. Atomic Nuclei (1)

Y. N. Demkov, Phys. Atomic Nuclei 72, 779 (2009).
[CrossRef]

Phys. Lett. A (2)

E. A. Kuznetsov and S. K. Turitsyn, Phys. Lett. A 112, 273 (1985).
[CrossRef]

A. Mahalov and S. K. Suslov, Phys. Lett. A 377, 33 (2012).
[CrossRef]

Phys. Scr. (1)

R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, Phys. Scr. 87, 038112 (2013).
[CrossRef]

Phys. Usp. (1)

E. G. Abramochkin and V. G. Volostnikov, Phys. Usp. 47, 1177 (2004).
[CrossRef]

Proc. Am. Math. Soc. (1)

S. K. Suslov, Proc. Am. Math. Soc. 140, 3067 (2012).
[CrossRef]

Radiophys. Quantum Electron. (1)

S. N. Vlasov and V. I. Talanov, Radiophys. Quantum Electron. 38, 1 (1995).
[CrossRef]

Rev. Mod. Phys. (1)

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003).
[CrossRef]

Theor. Math. Phys. (1)

R. Cordero-Soto and S. K. Suslov, Theor. Math. Phys. 162, 286 (2010).
[CrossRef]

Waves Random Media (1)

O. Korotkova, N. Farwell, and A. Mahalov, Waves Random Media 19, 692 (2009).
[CrossRef]

Zs. für Phys. (1)

V. Fock, Zs. für Phys. 47, 446 (1928).
[CrossRef]

Other (7)

V. A. Fock, Selected Works: Quantum Mechanics and Quantum Field Theory (Chapman & Hall/CRC, 2004), p. 29.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999).

M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Theory of Waves (Nauka, 1979) [in Russian].

V. V. Dodonov and V. I. Man’ko, in Invariants and the Evolution of Nonstationary Quantum Systems (Nova Science, 1989), p. 103.

A. Mahalov and S. K. Suslov, “Solution of paraxial wave equation for inhomogeneous media in linear and quadratic approximation,” Proc. Am. Math. Soc. (to be published).

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer, 1991).

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, 1965).

Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1.

Breathing Gaussian mode.

Fig. 2.
Fig. 2.

Bending and breathing Gaussian mode.

Fig. 3.
Fig. 3.

Breathing Gaussian mode: surface where the intensity |A|2 changes by the factor e.

Fig. 4.
Fig. 4.

Breathing and rotating Gaussian mode: surface where the intensity |A|2 changes by the factor e.

Equations (26)

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

iψt=a(t)ψxx+b(t)x2ψic(t)xψxid(t)ψf(t)xψ+ig(t)ψx,
ψ(x,t)=G(x,y,t)ψ(y,0)dy,
G(x,y,t)=[2πμ0(t)]1/2×exp[i(α0(t)x2+β0(t)xy+γ0(t)y2+δ0(t)x+ε0(t)y+κ0(t))],
iAs=a(Axx+Ayy)+b(x2+y2)Aic(xAx+yAy)2idA(xf1+yf2)A+i(g1Ax+g2Ay),
iχτ+χξξ+χηη=c0(ξ2+η2)χ,
A=μ1ei(α(x2+y2)+δ1x+δ2y+κ1+κ2)χ(ξ,η,τ),
Anm(r,s)=ei(κ1+κ2)+2i(n+m+1)γ2n+mn!m!πβ×ei(α(x2+y2)+δ1x+δ2y)(βx+ε1)2/2(βy+ε2)2/2×Hn(βx+ε1)Hm(βy+ε2),
2iAs+Axxx2A=0,
An(x,s)=ei(αx2+δx+κ)+i(2n+1)γβ2nn!πe(βx+ε)2/2Hn(βx+ε),
α(s)=α0cos2s+sin2s(β04+4α021)/4β04sin2s+(2α0sins+coss)2,
β(s)=β0β04sin2s+(2α0sins+coss)2,
γ(s)=12arctanβ02tans1+2α0tans,
δ(s)=δ0(2α0sins+coss)+ε0β03sinsβ04sin2s+(2α0sins+coss)2,
ε(s)=ε0(2α0sins+coss)β0δ0sinsβ04sin2s+(2α0sins+coss)2,
κ(s)=sin2sε0β02(α0ε0β0δ0)α0δ02β04sin2s+(2α0sins+coss)2+14sin2sε02β02δ02β04sin2s+(2α0sins+coss)2.
2iBs+Bxx=0,
B(x,s)=1(s2+1)1/4exp(isx22(s2+1))A(xs2+1,arctans),
Bn(x,s)=[((2α0s+1)2+β04s2)]1/4×β02nn!πexp(ix2((4α02+β04)s+2α0)2((2α0s+1)2+β04s2))×exp(ix(2α0s+1)δ0+sβ03ε0(2α0s+1)2+β04s2)×exp(is(2α0s+1)(β02ε02δ02)2sβ03δ0ε02((2α0s+1)2+β04s2))×exp(i(n+12)arctan(β02s2α0s+1))×exp((β0(xδ0s)+ε0(2α0s+1))22((2α0s+1)2+β04s2))×Hn(β0(xδ0s)+ε0(2α0s+1)(2α0s+1)2+β04s2).
(XY)=(cosωτsinωτsinωτcosωτ)(ξη),
iΨT+ΨXX+ΨYY=(X2+Y2)Ψ+iω(XΨYYΨX).
Ψ(R,Θ,T)=n!π(n+|m|)!eiET×eimΘR|m|eR2/2Ln|m|(R2),E=4n+2(|m|+1)mω,
2iAs+Axx+Ayy=(x2+y2)A,
Anm(x,y,s)=βn!π(n+m)!×ei(α(x2+y2)+δ1x+δ2y+κ1+κ2)ei(2n+m+1)γ×(β(x±iy)+ε1±iε2)me(βx+ε1)2/2(βy+ε2)2/2×Lnm((βx+ε1)2+(βy+ε2)2),m0,
2iBs+Bxx+Byy=0,
B(x,y,s)=1(s2+1)1/2exp(is(x2+y2)2(s2+1))×A(xs2+1,ys2+1,arctans).
Bnm(x,y,s)=eis(δ0(1)2+δ0(2)2)/(2(1+2αs))(2α0s+1)2+β04s2×exp(i(1+m+2n)arctan(sβ021+2α0s))×exp(iα0(x2+y2)+xδ0(1)+yδ0(2)2α0s+1)×exp[(β0(xδ0(1)s)+ε0(1)(2α0s+1))22(2α0s+1+iβ02s)(1+2α0s)]×exp[(β0(yδ0(2)s)+ε0(2)(2α0s+1))22(2α0s+1+iβ02s)(1+2α0s)]×[β0(x+iy)(δ0(1)+iδ0(2))s(2α0s+1)2+β04s2+(ε0(1)+iε0(2))(2α0s+1)(2α0s+1)2+β04s2]m×Lnm[(β0(xδ0(1)s)+ε0(1)(2α0s+1))2(2α0s+1)2+β04s2+(β0(yδ0(2)s)+ε0(2)(2α0s+1))2(2α0s+1)2+β04s2].

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