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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 12 — Jun. 6, 2011
  • pp: 11264–11270
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Data transmission by hypergeometric modes through a hyperbolic-index medium

Bertúlio de Lima Bernardo and Fernando Moraes  »View Author Affiliations


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

In this Letter we introduce a new family of laser beams forming an orthogonal basis that are solutions to the paraxial wave equation, as well as eigenstates of the photon OAM. The transverse mode profiles are proportional to the confluent hypergeometric function. Modes with similar profiles has been studied previously (HyG, HyGG, HyGG-II) [11

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

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]

]. However, unlike those, we obtained solutions both orthogonal and carrying finite power, whose amplitude drops exponentially towards the periphery when increasing radial variable 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

Consider a cylindrically symmetric medium of infinite extent, whose index of refraction 𝒩 can be described by
𝒩2(r)=n02(1+n1n0ρ),
(1)
where 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

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

]) 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
2E+k2(1+n1n0ρ)E=0,
(2)
with k = 2πn 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

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]

], one generally looks for solutions of the type E(ρ, φ, z) = ψ(ρ, φ, z)eikz, which are slowly dependent on z. The complex amplitude takes the form ψ = exp{−i[P(z) + 2/2q(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

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

]. In this manner, by taking some component of Eq. (2), assuming a solution in the paraxial form
E(ρ,φ,z)=ψ(ρ,φ)exp(iβz),
(3)
and writing ψ(ρ, φ) = R(ρ)Φ(φ), the wave equation becomes
1R1ρρ(ρRρ)+1Φρ22Φφ2+k2n1n0ρ+(k2β2)=0.
(4)
Using separation of variables we obtain
d2dφ2Φ(φ)=l2Φ(φ),
(5)
which gives
Φ(φ)=1(2π)1/2eilφ,l=0,±1,±2,±3,
(6)
and
d2dρ2R(ρ)+1ρddρR(ρ)+[(k2β2)+k2n1n0ρl2ρ2]R(ρ)=0.
(7)
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]

]. At first, we analyze the case in which k 2 > β 2. Let us define
m[2(k2β2)n02k4n12]1/2,
(8)
sk2n12n0,
(9)
b=i/m,
(10)
x=i2msρ,
(11)
then, we write Eq. (7) as
d2dx2R(x)+1xddxR(x)+[(14+bx)l2x2]R(x)=0.
(12)
Its regular solution is
Rml(ρ)=Cml(2msρ)|l|exp(imsρ)1F1(i/m+|l|+1/2;2|l|+1,i2msρ).
(13)
The constant Cml can be evaluated from the δ-function normalization condition
0Rml(ρ)Rml(ρ)ρdρ=δ(mm),
(14)
to give [19

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).

]
Cml=s[2m1+e2π/m]1/2p=0|l|1[(p+1/2)2+1/m2]1/2,
(15)
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.

Now let us analyze the case in which k 2 < β 2. Defining the transformations
k2β2=k4n124n02N2,
(16)
ρ=Nn0k2n1x,
(17)
and
R(x)=x|l|ex/2G(x),
(18)
Eq.(7) becomes
xd2Gdx2+[(2|l|+1)x]dGdx(N+|l|+1/2)G=0.
(19)
The solution of Eq. (11) is the confluent hypergeometric function [20

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

].
G(x)=1F1(N+|l|+1/2;2|l|+1,x).
(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=12,32,52,. Let us define an integer n as
n=N+1/2=1,2,3,
(21)
For a given n, |l| can assume the values
|l|=0,1,2,,n1.
(22)
The propagation constant βn of the n mode is obtained from Eqs. (16) and (21)
βn=k(1+[1(n1/2)kn12n0]2)1/2.
(23)
The normalized radial solution is given by
Rnl(ρ)=αn(2|l|)![(n+|l|1)!(2n1)(n|l|1)!]1/2(αnρ)|l|exp(αnρ/2)1F1(n+|l|+1;2|l|+1,αnρ),
(24)
where
αn=1(n1/2)k2n1n0.
(25)
Then, the total complex field is given by
Enl(ρ,φ,z)=1(2π)1/2Rnl(ρ)exp[i(βnzlφ)].
(26)
Notice that, unlike the homogeneous medium solution (n 1 = 0) [17

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

], 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 Enl explicitly:
E10=[α1/(2π)1/2]eα1ρ/2exp[i(β1z)],E20=[α2/(6π)1/2](1α2ρ)eα2ρ/2exp[i(β2z)],E21=(α22/(12π)1/2)ρeα2ρ/2exp[i(β2zφ)],E30=[α3/2(10π)1/2](24α3ρ+α32ρ2)eα3ρ/2exp[i(β3z)],E31=(α32/(60π)1/2)ρ(3α3ρ)eα3ρ/2exp[i(β3zφ)],E32=[α33/(5!2π)1/2]ρ2eα3ρ/2exp[i(β3z2φ)].
(27)
At a given z plane, the intensity distribution of the HyG-II beams are given by Inl (ρ, φ, z) = 2πρ|Enl (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.

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.

3. Transmission Properties

4. Conclusion

In conclusion, we studied a novel family of optical vortices, called HyG-II modes, which are solutions of the two-dimensional Helmholtz equation in a hyperbolic-index medium. This family constitutes an orthogonal set of modes, whose intensity profile is characterized by either a single or concentric brilliant rings with an exponential radial decay of intensity. We found that, in general, the hyperbolic-index fibers cause larger broadening to multimode light pulses when compared to quadratic-index fibers. For the single-mode excitation, on the other hand, both types of fibers behave almost equally. However, it is noteworthy that, due to the presence of OAM states, specific multimode excitations in hyperbolic-index fibers could carry more information than in quadratic-index fibers.

Acknowledgments

This work was supported by CNPq and CAPES (NANOBIOTEC BRASIL) and INCT-FC (Instituto Nacional de Ciência e Tecnologia de Fluidos Complexos). We thank J. B. Formiga for giving us instructions to make the figure.

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. 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 Jr., 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]

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

  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]

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