Data transmission by hypergeometric modes through a hyperbolic-index medium

Bertúlio de Lima Bernardo; Fernando Moraes

doi:10.1364/OE.19.011264

Optics Express
Vol. 19,
Issue 12,
pp. 11264-11270
(2011)
•https://doi.org/10.1364/OE.19.011264

Data transmission by hypergeometric modes through a hyperbolic-index medium

Bertúlio de Lima Bernardo and Fernando Moraes

Open Access

Get PDF
Email
Share
Get Citation
Copy Citation Text
Bertúlio de Lima Bernardo and Fernando Moraes, "Data transmission by hypergeometric modes through a hyperbolic-index medium," Opt. Express 19, 11264-11270 (2011)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

Related Topics
Table of Contents Category
- Fiber Optics and Optical Communications
Optics & Photonics Topics
?

The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fiber optics and optical communications (060.0060)
- Wave propagation (070.7345)
- Laser beam shaping (140.3300)

About this Article
History
- Original Manuscript: March 2, 2011
- Revised Manuscript: May 13, 2011
- Manuscript Accepted: May 19, 2011
- Published: May 25, 2011

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.

Full Article | PDF Article

More Like This

Improved focusing with Hypergeometric-Gaussian type-II optical modes

Ebrahim Karimi, Bruno Piccirillo, Lorenzo Marrucci, and Enrico Santamato
Opt. Express 16(25) 21069-21075 (2008)

Propagation properties of hypergeometric-Gaussian type-II beams through the quadratic-index medium

Lirong Bian and Bin Tang
Appl. Opt. 57(17) 4735-4742 (2018)

Generating hypergeometric laser beams with a diffractive optical element

V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, S. N. Khonina, and J. Turunen
Appl. Opt. 47(32) 6124-6133 (2008)

Previous Article Next Article

References

View by:
Article Order
Year
Author
Publication

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]
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]
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]
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]
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]
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]
G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]
A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, “Entanglement of the angular momentum states of photons,” Nature 412, 313–315 (2001).
[Crossref] [PubMed]
W. Miller, Symmetry and Separation of Variables (Addison-Wesley, 1977).
M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. 33, 177–179 (2008).
[Crossref] [PubMed]
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]
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]
C. Yeh, Handbook of Fiber Optics: Theory and Applications (Academic Press, 1990).
M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A, 11, 1365–1370 (1975).
[Crossref]
S. J. van Enk and G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94147–158 (1992).
[Crossref]
A. Yariv, Optical Electronics (Saunders College Publishing, 1991).
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. 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).
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).
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)

M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. 33, 177–179 (2008).
[Crossref] [PubMed]

2007 (6)

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]

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]

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]

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

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)

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]

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)

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]

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

Abramowitz, M.

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

Allen, L.

Bandres, M. A.

M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. 33, 177–179 (2008).
[Crossref] [PubMed]

Beijersbergen, M. W.

Campos, J.

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]

Chan, F. T.

Ching, W. Y.

Chiu, D. T.

Davityan, L. S.

Edgar, J. S.

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.

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.

Gutiérrez-Vega, J. C.

M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. 33, 177–179 (2008).
[Crossref] [PubMed]

He, H.

Heckenberg, N. R.

Jeffries, G. D. M.

Karimi, E.

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

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]

Khonina, S. N.

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

Kotlyar, V. V.

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

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.

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]

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

McGloin, D.

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.

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]

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.

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

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]

Pogosyan, G. S.

Presby, H. M.

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]

Rubinsztein-Dunlop, H.

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.

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]

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

Shelby, J. P.

Sisakyan, A. N.

Skidanov, R. V.

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

Soifer, V. A.

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

Spreeuw, R. J. C.

Stegun, I. A.

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

Ter-Antonyan, V. M.

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.

Wong, K. W.

Yang, X. L.

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.

Zito, G.

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

Appl. Opt. (1)

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]

J. Mod. Opt. (1)

Nano. Lett. (1)

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)

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]

Opt. Lett. (4)

M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. 33, 177–179 (2008).
[Crossref] [PubMed]

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]

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]

Phys. Rev. A (3)

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]

Theor. Math. Phys. (1)

Other (4)

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

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

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

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

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

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.

Download Full Size | PPT Slide | PDF

Equations (34)

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

𝒩^{2} (r) = n_{0}^{2} (1 + \frac{n_{1}}{n_{0} ρ}),

\nabla^{2} E + k^{2} (1 + \frac{n_{1}}{n_{0} ρ}) E = 0,

E (ρ, φ, z) = ψ (ρ, φ) exp (- i β z),

\frac{1}{R} \frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ \frac{\partial R}{\partial ρ}) + \frac{1}{Φ ρ^{2}} \frac{\partial^{2} Φ}{\partial φ^{2}} + \frac{k^{2} n_{1}}{n_{0} ρ} + (k^{2} - β^{2}) = 0.

\frac{d^{2}}{d φ^{2}} Φ (φ) = - l^{2} Φ (φ),

Φ (φ) = \frac{1}{{(2 π)}^{1 / 2}} e^{il φ}, l = 0, \pm 1, \pm 2, \pm 3, \dots

\frac{d^{2}}{d ρ^{2}} R (ρ) + \frac{1}{ρ} \frac{d}{d ρ} R (ρ) + [(k^{2} - β^{2}) + \frac{k^{2} n_{1}}{n_{0} ρ} - \frac{l^{2}}{ρ^{2}}] R (ρ) = 0.

m \equiv {[\frac{2 (k^{2} - β^{2}) n_{0}^{2}}{k^{4} n_{1}^{2}}]}^{1 / 2},

s \equiv \frac{k^{2} n_{1}}{2 n_{0}},

b = - i / m,

x = i 2 ms ρ,

\frac{d^{2}}{d x^{2}} R (x) + \frac{1}{x} \frac{d}{dx} R (x) + [(- \frac{1}{4} + \frac{b}{x}) - \frac{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) .

\int_{0}^{\infty} R_{ml} (ρ) R_{m' l} (ρ) ρ d ρ = δ (m - m'),

C_{ml} = s {[\frac{2 m}{1 + e^{- 2 π / m}}]}^{1 / 2} \prod_{p = 0}^{| l | - 1} {[{(p + 1 / 2)}^{2} + 1 / m^{2}]}^{1 / 2},

k^{2} - β^{2} = - \frac{k^{4} n_{1}^{2}}{4 n_{0}^{2} N^{2}},

ρ = \frac{N n_{0}}{k^{2} n_{1}} x,

R (x) = x^{| l |} e^{- x / 2} G (x),

x \frac{d^{2} G}{d x^{2}} + [(2 | l | + 1) - x] \frac{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, \dots

| l | = 0, 1, 2, \dots, n - 1 .

β_{n} = k {(1 + {[\frac{1}{(n - 1 / 2)} \frac{k n_{1}}{2 n_{0}}]}^{2})}^{1 / 2} .

R_{nl} (ρ) = \frac{α_{n}}{(2 | l |)!} {[\frac{(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} = \frac{1}{(n - 1 / 2)} \frac{k^{2} n_{1}}{n_{0}} .

E_{nl} (ρ, φ, z) = \frac{1}{{(2 π)}^{1 / 2}} R_{nl} (ρ) exp [- i (β_{n} z - l φ)] .

\begin{array}{l} 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)] . \end{array}

\frac{1}{(n - 1 / 2)} \frac{k n_{1}}{2 n_{0}} ≪ 1,

β_{n} = k + \frac{k^{3}}{8} {[\frac{1}{(n - 1 / 2)} \frac{n_{1}}{n_{0}}]}^{2} .

{(v_{g})}_{n} = \frac{c / n_{0}}{1 + \frac{3}{8} k^{2} {[\frac{1}{(n - 1 / 2)} \frac{n_{1}}{n_{0}}]}^{2}} .

Δ τ ≅ L [\frac{1}{{(v_{g})}_{1}} - \frac{1}{{(v_{g})}_{n_{max}}}] .

Δ τ ≅ \frac{3 k^{2} n_{1}^{2}}{8 c n_{0}} L [\frac{4}{9} - \frac{1}{{(n_{max} - 1 / 2)}^{2}}] .

Δ τ^{'} \approx \frac{2 L}{v_{g}^{2}} | \frac{d v_{g}}{d ω} | Δ ω .

Δ τ^{'} \approx \frac{2 L}{c} [\frac{3 n_{1}^{2} k}{4 c {(n - 1 / 2)}^{2}} + \frac{d n_{0}}{d ω}] Δ ω,

Abstract

References

2008 (1)

2007 (6)

2001 (1)

2000 (1)

1999 (1)

1995 (1)

1992 (2)

1991 (1)

1988 (1)

1975 (2)

Abramowitz, M.

Allen, L.

Bandres, M. A.

Beijersbergen, M. W.

Campos, J.

Chan, F. T.

Ching, W. Y.

Chiu, D. T.

Cohen, L. G.

Cottrell, D. M.

Davis, J. A.

Davityan, L. S.

Edgar, J. S.

Fagerholm, J.

Fong, C.

Friberg, A. T.

Guo, S. H.

Gutiérrez-Vega, J. C.

He, H.

Heckenberg, N. R.

Jeffries, G. D. M.

Karimi, E.

Khonina, S. N.

Kotlyar, V. V.

Lax, M.

Louisell, W. H.

Mair, A.

Marrucci, L.

McGloin, D.

McKnight, W. B.

McNamara, D. E.

Miller, W.

Molina-Terriza, G.

Nienhuis, G.

Piccirillo, B.

Pogosyan, G. S.

Presby, H. M.

Rubinsztein-Dunlop, H.

Salo, J.

Salomaa, M. M.

Santamato, E.

Shelby, J. P.

Sisakyan, A. N.

Skidanov, R. V.

Soifer, V. A.

Spreeuw, R. J. C.

Stegun, I. A.

Ter-Antonyan, V. M.

Torner, L.

Torres, J. P.

van Enk, S. J.

Vaziri, A.

Welhs, G.

Woerdman, J. P.

Wong, K. W.

Yang, X. L.

Yariv, A.

Yeh, C.

Zeilinger, A.

Zhao, Y.

Zito, G.

Appl. Opt. (1)

J. Mod. Opt. (1)

Nano. Lett. (1)

Nat. Phys. (1)

Nature (1)

Opt. Commun. (1)

Opt. Express (1)