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 the last decade, light beams endowed with orbital angular momentum (OAM) have received increasing attention. Such beams have been found to be useful in several applications, such as: optical lithography, medical imaging and surgery, microscopy, optical trapping and optical tweezers [15]. In complex notation, the electric field of a beam carrying a well defined OAM is written as E(ρ, φ) = E0(ρ)exp(ilφ), where ρ and φ are the radial and azimuthal polar coordinates in the xy plane, respectively, and l is an integer. As predicted by Allen et al. [6], in fields like this, besides the intrinsic spin angular momentum ±ħ, related to the light polarization, each photon carries a quantized OAM along the z-axis given by , related to the topology of the wavefront. Moreover, these beams present a topological phase singularity (“optical vortex”) at the beam axis. In this context, since OAM is inserted in a multidimensional space [7], more information can be encoded in a single light pulse. This kind of quantum information could cause a future revolution in communication systems [8]. With these motivations, efforts have been devoted to the study and generation of light beams other than the well-known Laguerre-Gaussian (LG) modes, which also present these special properties. The paraxial Helmholtz equation admits solutions with separable variables in 17 coordinate systems [9]. Many of these solutions have been studied both theoretically and experimentally with the purpose of scientific and technological applications. The Bessel-Gauss (BG), Hypergeometric (HyG), Hypergeometric-Gaussian (HyGG), fractional elegant Laguerre-Gauss (fr-eLG), Ince-Gaussian (IG), Laplace-Gauss and Mathieu beams are the most recent examples [1012].

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) [1113]. 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),
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]) 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,
with k = 2πn 0/λ as the wavenumber far from the optical axis. At this point, when the paraxial approximation is assumed [15, 16], 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]. 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),
and writing ψ(ρ, φ) = R(ρ)Φ(φ), the wave equation becomes
1R1ρρ(ρRρ)+1Φρ22Φφ2+k2n1n0ρ+(k2β2)=0.
Using separation of variables we obtain
d2dφ2Φ(φ)=l2Φ(φ),
which gives
Φ(φ)=1(2π)1/2eilφ,l=0,±1,±2,±3,
and
d2dρ2R(ρ)+1ρddρR(ρ)+[(k2β2)+k2n1n0ρl2ρ2]R(ρ)=0.
Notice that Eq. (7) is the radial Schrödinger equation for the 2D hydrogen atom [18]. At first, we analyze the case in which k 2 > β 2. Let us define
m[2(k2β2)n02k4n12]1/2,
sk2n12n0,
b=i/m,
x=i2msρ,
then, we write Eq. (7) as
d2dx2R(x)+1xddxR(x)+[(14+bx)l2x2]R(x)=0.
Its regular solution is
Rml(ρ)=Cml(2msρ)|l|exp(imsρ)1F1(i/m+|l|+1/2;2|l|+1,i2msρ).
The constant Cml can be evaluated from the δ-function normalization condition
0Rml(ρ)Rml(ρ)ρdρ=δ(mm),
to give [19]
Cml=s[2m1+e2π/m]1/2p=0|l|1[(p+1/2)2+1/m2]1/2,
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,
ρ=Nn0k2n1x,
and
R(x)=x|l|ex/2G(x),
Eq.(7) becomes
xd2Gdx2+[(2|l|+1)x]dGdx(N+|l|+1/2)G=0.
The solution of Eq. (11) is the confluent hypergeometric function [20].
G(x)=1F1(N+|l|+1/2;2|l|+1,x).
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,
For a given n, |l| can assume the values
|l|=0,1,2,,n1.
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.
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),
where
αn=1(n1/2)k2n1n0.
Then, the total complex field is given by
Enl(ρ,φ,z)=1(2π)1/2Rnl(ρ)exp[i(βnzlφ)].
Notice that, unlike the homogeneous medium solution (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 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)].
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.

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3. Transmission Properties

The advantages of optical fiber communication systems for long distances are an undeniable fact. However, two factors normally limit the number of optical pulses that can be transmitted per unit time: the modal dispersion and the group velocity dispersion. Now we analyze them for the case of a hyperbolic-index fiber.

1. Modal Dispersion. The dependence of β on the mode index n causes the different modes to have different phase velocities vn = ω/βn, as well as different group velocities (vg)n = /n. If we consider that the region of high index is very thin so that

1(n1/2)kn12n01,
we can approximate Eq. (23) as
βn=k+k38[1(n1/2)n1n0]2.
Thus, since k = ωn 0/c, we obtain the expression for the group velocity
(vg)n=c/n01+38k2[1(n1/2)n1n0]2.
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 (vg)n. Considering that modes from n = 1 to n = nmax are excited, the output pulse obtained at z = L will broaden to
ΔτL[1(vg)11(vg)nmax].
Now we use Eq. (30) to obtain
Δτ3k2n128cn0L[491(nmax1/2)2].
The maximum number of pulses per second transmitted without a significant overlap of neighboring pulses is fmax ∼ 1/Δτ. For example, for a 1-km-long hyperbolic-index fiber with n 0 = 1.5 and n 1 = 1.1μm, if one sends pulses at λ = 1μm, exciting a large number of modes, no matter if tens or hundreds of modes, from Eq. (32), we have Δτ ≈ 4 × 10−5 s, and fmax ∼ 2.5 × 104 pulses per second for the maximum pulse rate. This is significantly less than the capacity of a quadratic-index fiber (fmax ∼ 107 pulses per second), which supports Hermite-Gaussian modes (HG) [17].

2. Group velocity dispersion. The modal pulse spreading can be removed, for example, if only a single mode is excited. In this case, despite the possibility of high data transmission, pulse spreading still remains due to the group velocity dispersion. If we consider a pulse with a spectral width Δω, it will spread in a distance L by [17]

Δτ2Lvg2|dvgdω|Δω.
Since vg depends implicitly on ω, due to the dependence of the index of refraction of the material n 0 on ω, which gives dvg/ = ∂vg/∂ω + ∂vg/∂n 0(dn 0/), we have from Eqs. (30) and (33) that
Δτ2Lc[3n12k4c(n1/2)2+dn0dω]Δω,
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/. Therefore, for single-mode excitation, both quadratic and hyperbolic-index media present a group velocity dispersion of the same order [17, 21]. However, only the last can support OAM transmission. Observe that since the modal dispersion in Eq. (32) does not depend on 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

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]  

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)

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)

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.

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.

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]

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.

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]

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.

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]

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)

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]

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]

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).

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

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

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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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