Abstract

We study the existence of a novel complete family of exact and orthogonal solutions of the paraxial wave equation. The complex amplitude of these beams is proportional to the confluent hypergeometric function, which we name hypergeometric modes of type-II (HyG-II). It is formally demonstrated that hyperbolic-index medium can generate and support the propagation of such a class of beams. Since these modes are eigenfunctions of the photon orbital angular momentum, we conclude that an optical fiber with hyperbolic-index profile could take advantage over other graded-index fibers by the capacity of data transmission.

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References

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  1. J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
    [CrossRef]
  2. G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano. Lett. 7, 415–420 (2007).
    [CrossRef] [PubMed]
  3. J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform—theory and experiments,” Opt. Lett. 25, 99–101 (2000).
    [CrossRef]
  4. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer-generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
    [CrossRef]
  5. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901–073904 (2007).
    [CrossRef] [PubMed]
  6. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
  7. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
    [CrossRef]
  8. A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, “Entanglement of the angular momentum states of photons,” Nature 412, 313–315 (2001).
    [CrossRef] [PubMed]
  9. W. Miller, Symmetry and Separation of Variables (Addison-Wesley, 1977).
  10. M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. 33, 177–179 (2008).
    [CrossRef] [PubMed]
  11. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. 32, 742–744 (2007).
    [CrossRef] [PubMed]
  12. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric–Gaussian modes,” Opt. Lett. 32, 3053–3055 (2007).
    [CrossRef] [PubMed]
  13. E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Improved focusing with hypergeometric-Gaussian type-II optical modes,” Opt. Express 16, 21069–21075 (2007).
    [CrossRef]
  14. C. Yeh, Handbook of Fiber Optics: Theory and Applications (Academic Press, 1990).
  15. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A , 11, 1365–1370 (1975).
    [CrossRef]
  16. S. J. van Enk and G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94147–158 (1992).
    [CrossRef]
  17. A. Yariv, Optical Electronics (Saunders College Publishing, 1991).
  18. X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, and W. Y. Ching, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory,” Phys. Rev. A 43, 1186–1196 (1991).
    [CrossRef] [PubMed]
  19. L. S. Davityan, G. S. Pogosyan, A. N. Sisakyan, and V. M. Ter-Antonyan, “Transformations between parabolic bases of the two-dimensional hydrogen atom in the continuous spectrum,” Theor. Math. Phys. 74, 240–246 (1988).
  20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).
  21. L. G. Cohen and H. M. Presby, “Shuttle pulse measurements of pulse spreading in a low loss graded-index fiber,” Appl. Opt. 14, 1361–1363 (1975)
    [CrossRef] [PubMed]

2008 (1)

2007 (6)

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Improved focusing with hypergeometric-Gaussian type-II optical modes,” Opt. Express 16, 21069–21075 (2007).
[CrossRef]

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. 32, 742–744 (2007).
[CrossRef] [PubMed]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric–Gaussian modes,” Opt. Lett. 32, 3053–3055 (2007).
[CrossRef] [PubMed]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901–073904 (2007).
[CrossRef] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano. Lett. 7, 415–420 (2007).
[CrossRef] [PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, “Entanglement of the angular momentum states of photons,” Nature 412, 313–315 (2001).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
[CrossRef]

1995 (1)

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer-generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

1992 (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

S. J. van Enk and G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94147–158 (1992).
[CrossRef]

1991 (1)

X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, and W. Y. Ching, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory,” Phys. Rev. A 43, 1186–1196 (1991).
[CrossRef] [PubMed]

1988 (1)

L. S. Davityan, G. S. Pogosyan, A. N. Sisakyan, and V. M. Ter-Antonyan, “Transformations between parabolic bases of the two-dimensional hydrogen atom in the continuous spectrum,” Theor. Math. Phys. 74, 240–246 (1988).

1975 (2)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A , 11, 1365–1370 (1975).
[CrossRef]

L. G. Cohen and H. M. Presby, “Shuttle pulse measurements of pulse spreading in a low loss graded-index fiber,” Appl. Opt. 14, 1361–1363 (1975)
[CrossRef] [PubMed]

Abramowitz, M.

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

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Bandres, M. A.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Campos, J.

Chan, F. T.

X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, and W. Y. Ching, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory,” Phys. Rev. A 43, 1186–1196 (1991).
[CrossRef] [PubMed]

Ching, W. Y.

X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, and W. Y. Ching, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory,” Phys. Rev. A 43, 1186–1196 (1991).
[CrossRef] [PubMed]

Chiu, D. T.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901–073904 (2007).
[CrossRef] [PubMed]

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano. Lett. 7, 415–420 (2007).
[CrossRef] [PubMed]

Cohen, L. G.

Cottrell, D. M.

Davis, J. A.

Davityan, L. S.

L. S. Davityan, G. S. Pogosyan, A. N. Sisakyan, and V. M. Ter-Antonyan, “Transformations between parabolic bases of the two-dimensional hydrogen atom in the continuous spectrum,” Theor. Math. Phys. 74, 240–246 (1988).

Edgar, J. S.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano. Lett. 7, 415–420 (2007).
[CrossRef] [PubMed]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901–073904 (2007).
[CrossRef] [PubMed]

Fagerholm, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
[CrossRef]

Fong, C.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano. Lett. 7, 415–420 (2007).
[CrossRef] [PubMed]

Friberg, A. T.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
[CrossRef]

Guo, S. H.

X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, and W. Y. Ching, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory,” Phys. Rev. A 43, 1186–1196 (1991).
[CrossRef] [PubMed]

Gutiérrez-Vega, J. C.

He, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer-generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Heckenberg, N. R.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer-generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Jeffries, G. D. M.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano. Lett. 7, 415–420 (2007).
[CrossRef] [PubMed]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901–073904 (2007).
[CrossRef] [PubMed]

Karimi, E.

Khonina, S. N.

Kotlyar, V. V.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A , 11, 1365–1370 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A , 11, 1365–1370 (1975).
[CrossRef]

Mair, A.

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, “Entanglement of the angular momentum states of photons,” Nature 412, 313–315 (2001).
[CrossRef] [PubMed]

Marrucci, L.

McGloin, D.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901–073904 (2007).
[CrossRef] [PubMed]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A , 11, 1365–1370 (1975).
[CrossRef]

McNamara, D. E.

Miller, W.

W. Miller, Symmetry and Separation of Variables (Addison-Wesley, 1977).

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Nienhuis, G.

S. J. van Enk and G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94147–158 (1992).
[CrossRef]

Piccirillo, B.

Pogosyan, G. S.

L. S. Davityan, G. S. Pogosyan, A. N. Sisakyan, and V. M. Ter-Antonyan, “Transformations between parabolic bases of the two-dimensional hydrogen atom in the continuous spectrum,” Theor. Math. Phys. 74, 240–246 (1988).

Presby, H. M.

Rubinsztein-Dunlop, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer-generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Salo, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
[CrossRef]

Salomaa, M. M.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
[CrossRef]

Santamato, E.

Shelby, J. P.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano. Lett. 7, 415–420 (2007).
[CrossRef] [PubMed]

Sisakyan, A. N.

L. S. Davityan, G. S. Pogosyan, A. N. Sisakyan, and V. M. Ter-Antonyan, “Transformations between parabolic bases of the two-dimensional hydrogen atom in the continuous spectrum,” Theor. Math. Phys. 74, 240–246 (1988).

Skidanov, R. V.

Soifer, V. A.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Stegun, I. A.

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

Ter-Antonyan, V. M.

L. S. Davityan, G. S. Pogosyan, A. N. Sisakyan, and V. M. Ter-Antonyan, “Transformations between parabolic bases of the two-dimensional hydrogen atom in the continuous spectrum,” Theor. Math. Phys. 74, 240–246 (1988).

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

van Enk, S. J.

S. J. van Enk and G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94147–158 (1992).
[CrossRef]

Vaziri, A.

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, “Entanglement of the angular momentum states of photons,” Nature 412, 313–315 (2001).
[CrossRef] [PubMed]

Welhs, G.

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, “Entanglement of the angular momentum states of photons,” Nature 412, 313–315 (2001).
[CrossRef] [PubMed]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Wong, K. W.

X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, and W. Y. Ching, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory,” Phys. Rev. A 43, 1186–1196 (1991).
[CrossRef] [PubMed]

Yang, X. L.

X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, and W. Y. Ching, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory,” Phys. Rev. A 43, 1186–1196 (1991).
[CrossRef] [PubMed]

Yariv, A.

A. Yariv, Optical Electronics (Saunders College Publishing, 1991).

Yeh, C.

C. Yeh, Handbook of Fiber Optics: Theory and Applications (Academic Press, 1990).

Zeilinger, A.

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, “Entanglement of the angular momentum states of photons,” Nature 412, 313–315 (2001).
[CrossRef] [PubMed]

Zhao, Y.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano. Lett. 7, 415–420 (2007).
[CrossRef] [PubMed]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901–073904 (2007).
[CrossRef] [PubMed]

Zito, G.

Appl. Opt. (1)

J. Mod. Opt. (1)

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer-generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Nano. Lett. (1)

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu, “Using polarization-shaped optical vortex traps for single-cell nanosurgery,” Nano. Lett. 7, 415–420 (2007).
[CrossRef] [PubMed]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, “Entanglement of the angular momentum states of photons,” Nature 412, 313–315 (2001).
[CrossRef] [PubMed]

Opt. Commun. (1)

S. J. van Enk and G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94147–158 (1992).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (3)

X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, and W. Y. Ching, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory,” Phys. Rev. A 43, 1186–1196 (1991).
[CrossRef] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A , 11, 1365–1370 (1975).
[CrossRef]

Phys. Rev. Lett. (2)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
[CrossRef]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901–073904 (2007).
[CrossRef] [PubMed]

Theor. Math. Phys. (1)

L. S. Davityan, G. S. Pogosyan, A. N. Sisakyan, and V. M. Ter-Antonyan, “Transformations between parabolic bases of the two-dimensional hydrogen atom in the continuous spectrum,” Theor. Math. Phys. 74, 240–246 (1988).

Other (4)

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

A. Yariv, Optical Electronics (Saunders College Publishing, 1991).

W. Miller, Symmetry and Separation of Variables (Addison-Wesley, 1977).

C. Yeh, Handbook of Fiber Optics: Theory and Applications (Academic Press, 1990).

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Figures (1)

Fig. 1
Fig. 1

Transverse field distributions of some HyG-II modes at the same scale. They are characterized by either a single or concentric brilliant rings with a singularity at the center. Because the fast radial intensity decay, higher order profiles are not significantly different from these.

Equations (34)

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

𝒩 2 ( r ) = n 0 2 ( 1 + n 1 n 0 ρ ) ,
2 E + k 2 ( 1 + n 1 n 0 ρ ) E = 0 ,
E ( ρ , φ , z ) = ψ ( ρ , φ ) exp ( i β z ) ,
1 R 1 ρ ρ ( ρ R ρ ) + 1 Φ ρ 2 2 Φ φ 2 + k 2 n 1 n 0 ρ + ( k 2 β 2 ) = 0.
d 2 d φ 2 Φ ( φ ) = l 2 Φ ( φ ) ,
Φ ( φ ) = 1 ( 2 π ) 1 / 2 e il φ , l = 0 , ± 1 , ± 2 , ± 3 ,
d 2 d ρ 2 R ( ρ ) + 1 ρ d d ρ R ( ρ ) + [ ( k 2 β 2 ) + k 2 n 1 n 0 ρ l 2 ρ 2 ] R ( ρ ) = 0.
m [ 2 ( k 2 β 2 ) n 0 2 k 4 n 1 2 ] 1 / 2 ,
s k 2 n 1 2 n 0 ,
b = i / m ,
x = i 2 ms ρ ,
d 2 d x 2 R ( x ) + 1 x d dx R ( x ) + [ ( 1 4 + b x ) l 2 x 2 ] R ( x ) = 0.
R ml ( ρ ) = C ml ( 2 ms ρ ) | l | exp ( ims ρ ) 1 F 1 ( i / m + | l | + 1 / 2 ; 2 | l | + 1 , i 2 ms ρ ) .
0 R ml ( ρ ) R m l ( ρ ) ρ d ρ = δ ( m m ) ,
C ml = s [ 2 m 1 + e 2 π / m ] 1 / 2 p = 0 | l | 1 [ ( p + 1 / 2 ) 2 + 1 / m 2 ] 1 / 2 ,
k 2 β 2 = k 4 n 1 2 4 n 0 2 N 2 ,
ρ = N n 0 k 2 n 1 x ,
R ( x ) = x | l | e x / 2 G ( x ) ,
x d 2 G d x 2 + [ ( 2 | l | + 1 ) x ] dG dx ( N + | l | + 1 / 2 ) G = 0.
G ( x ) = 1 F 1 ( N + | l | + 1 / 2 ; 2 | l | + 1 , x ) .
n = N + 1 / 2 = 1 , 2 , 3 ,
| l | = 0 , 1 , 2 , , n 1 .
β n = k ( 1 + [ 1 ( n 1 / 2 ) k n 1 2 n 0 ] 2 ) 1 / 2 .
R nl ( ρ ) = α n ( 2 | l | ) ! [ ( n + | l | 1 ) ! ( 2 n 1 ) ( n | l | 1 ) ! ] 1 / 2 ( α n ρ ) | l | exp ( α n ρ / 2 ) 1 F 1 ( n + | l | + 1 ; 2 | l | + 1 , α n ρ ) ,
α n = 1 ( n 1 / 2 ) k 2 n 1 n 0 .
E nl ( ρ , φ , z ) = 1 ( 2 π ) 1 / 2 R nl ( ρ ) exp [ i ( β n z l φ ) ] .
E 10 = [ α 1 / ( 2 π ) 1 / 2 ] e α 1 ρ / 2 exp [ i ( β 1 z ) ] , E 20 = [ α 2 / ( 6 π ) 1 / 2 ] ( 1 α 2 ρ ) e α 2 ρ / 2 exp [ i ( β 2 z ) ] , E 21 = ( α 2 2 / ( 12 π ) 1 / 2 ) ρ e α 2 ρ / 2 exp [ i ( β 2 z φ ) ] , E 30 = [ α 3 / 2 ( 10 π ) 1 / 2 ] ( 2 4 α 3 ρ + α 3 2 ρ 2 ) e α 3 ρ / 2 exp [ i ( β 3 z ) ] , E 31 = ( α 3 2 / ( 60 π ) 1 / 2 ) ρ ( 3 α 3 ρ ) e α 3 ρ / 2 exp [ i ( β 3 z φ ) ] , E 32 = [ α 3 3 / ( 5 ! 2 π ) 1 / 2 ] ρ 2 e α 3 ρ / 2 exp [ i ( β 3 z 2 φ ) ] .
1 ( n 1 / 2 ) k n 1 2 n 0 1 ,
β n = k + k 3 8 [ 1 ( n 1 / 2 ) n 1 n 0 ] 2 .
( v g ) n = c / n 0 1 + 3 8 k 2 [ 1 ( n 1 / 2 ) n 1 n 0 ] 2 .
Δ τ L [ 1 ( v g ) 1 1 ( v g ) n max ] .
Δ τ 3 k 2 n 1 2 8 c n 0 L [ 4 9 1 ( n max 1 / 2 ) 2 ] .
Δ τ 2 L v g 2 | d v g d ω | Δ ω .
Δ τ 2 L c [ 3 n 1 2 k 4 c ( n 1 / 2 ) 2 + d n 0 d ω ] Δ ω ,

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