## Data transmission by hypergeometric modes through a hyperbolic-index medium |

Optics Express, Vol. 19, Issue 12, pp. 11264-11270 (2011)

http://dx.doi.org/10.1364/OE.19.011264

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### 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.

© 2011 OSA

## 1. Introduction

11. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. **32**, 742–744 (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]

*ρ*. We name them “hypergeometric modes of type-II” (HyG-II). For the generation of these modes, we demonstrate that light propagating in a hyperbolic-index media decay on them. Nevertheless, due to the OAM carried by the modes, we argue that optical fibers with this profile can transfer additional information.

## 2. Beam Equations

*ρ*

_{2}=

*x*

^{2}+

*y*

^{2}is the radial transverse coordinate. To obtain this index profile, approximately, the medium must have a very thin opaque band along the z-axis, since an infinite refractive index implies absence of light propagation. Surrounding this region, one can have silica glass with high doping concentration (e.g., Germanium dioxide [14]) in order to raise the refractive index of the glass to its maximum and, as the radius increases, the doping concentration must diminish with cylindrical symmetry, in such a way that the hyperbolic profile of the index is verified. Therefore, the wave equation for the electric field takes the following form with

*k*= 2

*πn*

_{0}/

*λ*as the wavenumber far from the optical axis. At this point, when the paraxial approximation is assumed [15

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. **94**147–158 (1992). [CrossRef]

*E*(

*ρ*,

*φ*,

*z*) =

*ψ*(

*ρ*,

*φ*,

*z*)

*e*, which are slowly dependent on

^{ikz}*z*. The complex amplitude takes the form

*ψ*=

*exp*{−

*i*[

*P*(

*z*) +

*kρ*

^{2}/2

*q*(

*z*)]}, where the phase shift factor

*P*(

*z*) and the complex beam parameter

*q*(

*z*) are functions to be determined. However, as we are more interested in the properties of light in what concerns data transmission, we limit our attention to steady-state solutions, that is,

*q*(

*z*) = const. and

*P*(

*z*) = 0, without loss of generality [17]. In this manner, by taking some component of Eq. (2), assuming a solution in the paraxial form and writing

*ψ*(

*ρ*,

*φ*) =

*R*(

*ρ*)Φ(

*φ*), the wave equation becomes Using separation of variables we obtain which gives and Notice that Eq. (7) is the radial Schrödinger equation for the 2D hydrogen atom [18

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]

*k*

^{2}>

*β*

^{2}. Let us define then, we write Eq. (7) as Its regular solution is The constant

*C*can be evaluated from the

_{ml}*δ*-function normalization condition to give [19] for

*l*= 0, the product can be replaced by unity. The solutions given by Eq. (13) are not confined states. Thus, in this case, the index profile causes diffraction of the beam.

*k*

^{2}<

*β*

^{2}. Defining the transformations and Eq.(7) becomes The solution of Eq. (11) is the confluent hypergeometric function [20]. This solution is well behaved at

*x*= 0 and exponentially divergent at

*x*→ ∞. The quadratic integrability of the radial function, necessary for the beam to have finite energy, is guaranteed only with zero or negative values of –

*N*+|

*l*| + 1/2. Thus we have

*n*as For a given

*n*, |

*l*| can assume the values The propagation constant

*β*of the

_{n}*n*mode is obtained from Eqs. (16) and (21) The normalized radial solution is given by

*n*

_{1}= 0) [17], the mode spot size of the beam is independent of

*z*. This fact is due to the focusing action of the index variation, which opposes the natural tendency of a beam to spread. For the sake of clarification, we show the first few electric field modes

*E*explicitly:

_{nl}*z*plane, the intensity distribution of the HyG-II beams are given by

*I*(

_{nl}*ρ*,

*φ*,

*z*) = 2

*πρ*|

*E*(

_{nl}*ρ*,

*φ*,

*z*)|

^{2}, so that they preserve their structure upon propagation. The intensity profiles for the first four modes are depicted in Fig. 1.

## 3. Transmission Properties

*β*on the mode index

*n*causes the different modes to have different phase velocities

*v*=

_{n}*ω*/

*β*, as well as different group velocities (

_{n}*v*)

_{g}*=*

_{n}*dω*/

*dβ*. If we consider that the region of high index is very thin so that we can approximate Eq. (23) as Thus, since

_{n}*k*=

*ωn*

_{0}/

*c*, we obtain the expression for the group velocity For the transmission of information onto trains of optical pulses through an hyperbolic-index fiber, it is important to obtain information about the modal dispersion of such a medium. Then, if the pulses fed into the input end of the fiber excite several modes, we have that each mode will propagate with a group velocity (

*v*)

_{g}*. Considering that modes from*

_{n}*n*= 1 to

*n*

**=**

*n*are excited, the output pulse obtained at

_{max}*z*=

*L*will broaden to Now we use Eq. (30) to obtain The maximum number of pulses per second transmitted without a significant overlap of neighboring pulses is

*f*∼ 1/Δ

_{max}*τ*. For example, for a 1-km-long hyperbolic-index fiber with

*n*

_{0}= 1.5 and

*n*

_{1}= 1.1

*μm*, if one sends pulses at

*λ*= 1

*μm*, exciting a large number of modes, no matter if tens or hundreds of modes, from Eq. (32), we have Δ

*τ*≈ 4 × 10

^{−5}

*s*, and

*f*∼ 2.5 × 10

_{max}^{4}pulses per second for the maximum pulse rate. This is significantly less than the capacity of a quadratic-index fiber (

*f*∼ 10

_{max}^{7}pulses per second), which supports Hermite-Gaussian modes (HG) [17].

*ω*, it will spread in a distance

*L*by [17] Since

*v*depends implicitly on

_{g}*ω*, due to the dependence of the index of refraction of the material

*n*

_{0}on

*ω*, which gives

*dv*/

_{g}*dω*=

*∂v*/

_{g}*∂ω*+

*∂v*/

_{g}*∂n*

_{0}(

*dn*

_{0}/

*dω*), we have from Eqs. (30) and (33) that where in the second term we assumed

*k*

^{2}[(

*n*

_{1}/

*n*

_{0})/(

*n*+ 1/2)]

^{2}≪ 1. In typical fibers, the pulse spreading is dominated by the material dispersion term

*dn*

_{0}/

*dω*. Therefore, for single-mode excitation, both quadratic and hyperbolic-index media present a group velocity dispersion of the same order [17, 21

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]

*l*, for each mode number

*n*excited, there exists a (

*n*− 1)-dimensional subspace of modes contained on

*l*, which is available to carry additional information without modal dispersion.

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. |

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. |

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. |

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. |

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. |

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 |

7. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. |

8. | A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, “Entanglement of the angular momentum states of photons,” Nature |

9. | W. Miller Jr., |

10. | M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. |

11. | V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. |

12. | E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric–Gaussian modes,” Opt. Lett. |

13. | E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Improved focusing with hypergeometric-Gaussian type-II optical modes,” Opt. Express |

14. | C. Yeh, |

15. | M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A , |

16. | S. J. van Enk and G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. |

17. | A. Yariv, |

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 |

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. |

20. | M. Abramowitz and I. A. Stegun, |

21. | L. G. Cohen and H. M. Presby, “Shuttle pulse measurements of pulse spreading in a low loss graded-index fiber,” Appl. Opt. |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(140.3300) Lasers and laser optics : Laser beam shaping

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: March 2, 2011

Revised Manuscript: May 13, 2011

Manuscript Accepted: May 19, 2011

Published: May 25, 2011

**Citation**

Bertúlio de Lima Bernardo and Fernando Moraes, "Data transmission by hypergeometric modes through a hyperbolic-index medium," Opt. Express **19**, 11264-11270 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11264

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### References

- 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]

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