Data transmission by hypergeometric modes through a hyperbolic-index medium

Bertúlio de Lima Bernardo; Fernando Moraes

doi:10.1364/OE.19.011264

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 [1–5]. 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 lħ, 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 [10–12].

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) [11–13]. 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

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

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

\nabla^{2} E + k^{2} (1 + \frac{n_{1}}{n_{0} ρ}) 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)e^ikz, which are slowly dependent on z. The complex amplitude takes the form ψ = exp{−i[P(z) + kρ 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 (- i β z),

and writing ψ(ρ, φ) = R(ρ)Φ(φ), the wave equation becomes

\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.

Using separation of variables we obtain

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

which gives

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

and

\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.

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 \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,

then, we write Eq. (7) as

\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.

Its regular solution is

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) .

The constant C_ml can be evaluated from the δ-function normalization condition

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

to give [19]

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},

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

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,

and

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

Eq.(7) becomes

x \frac{d^{2} G}{d x^{2}} + [(2 | l | + 1) - x] \frac{dG}{dx} - (- N + | l | + 1 / 2) G = 0.

The solution of Eq. (11) is the confluent hypergeometric function [20].

G (x) =_{1} F_{1} (- 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 = \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \dots

. Let us define an integer n as

n = N + 1 / 2 = 1, 2, 3, \dots

For a given n, |l| can assume the values

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

The propagation constant β_n of the n mode is obtained from Eqs. (16) and (21)

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

The normalized radial solution is given by

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),

where

α_{n} = \frac{1}{(n - 1 / 2)} \frac{k^{2} n_{1}}{n_{0}} .

Then, the total complex field is given by

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

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 E_nl explicitly:

\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}

At a given z plane, the intensity distribution of the HyG-II beams are given by I_nl (ρ, φ, z) = 2πρ|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.

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 v_n = ω/β_n, as well as different group velocities (v_g)_n = dω/dβ_n. If we consider that the region of high index is very thin so that

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

we can approximate Eq. (23) as

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

Thus, since k = ωn ₀/c, we obtain the expression for the group velocity

{(v_{g})}_{n} = \frac{c / n_{0}}{1 + \frac{3}{8} k^{2} {[\frac{1}{(n - 1 / 2)} \frac{n_{1}}{n_{0}}]}^{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 (v_g)_n. Considering that modes from n = 1 to n = n_max are excited, the output pulse obtained at z = L will broaden to

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

Now we use Eq. (30) to obtain

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

The maximum number of pulses per second transmitted without a significant overlap of neighboring pulses is f_max ∼ 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 f_max ∼ 2.5 × 10⁴ pulses per second for the maximum pulse rate. This is significantly less than the capacity of a quadratic-index fiber (f_max ∼ 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]

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

Since v_g depends implicitly on 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

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,

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]. 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.

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Data transmission by hypergeometric modes through a hyperbolic-index medium

Abstract

1. Introduction

2. Beam Equations

3. Transmission Properties

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (1)

Equations (34)

Optics Express