Generation of linearly polarized orbital angular momentum modes in a side-hole ring fiber with tunable topology numbers

Ya Han; Yan-Ge Liu; Wei Huang; Zhi Wang; Jun-Qi Guo; Ming-Ming Luo

doi:10.1364/OE.24.017272

1. Introduction

A light beam carrying orbital angular momentum (OAM) is one with a phase ϕ(r, φ) = exp(ilφ) in the transverse plane, where φ is azimuth angle and l is an integer called topological charge number. Such beams have a doughnut intensity profile and helical phase front with a 2lπ azimuthal phase change. The OAM modes with different charge number l and + or – charge is spatially orthogonal to each other. Since Allen first published a detailed study of OAM in 1992, the researches on OAM beams have been in the spotlight. It has been found that the OAM beams have a variety of applications in atom manipulation [1–3], nano-scale microscopy, quantum information processing, and large-capacity optics communication [4–6]. Recently, various OAM fibers, such as step index [7–9] and grade index ring core fibers [10], have been designed and applied to transmit OAM. While the OAM generation methods are still based on the discrete free space optic system, such as spiral phase plates, cylindrical lenses [11], diffractive optical elements [12,13], Laguerre–Gaussian beams [14]. Compared to the traditional generation methods, the fiber based generation methods, such as multimode interference [15], spin-obit interaction [16], acoustic-optic interaction [17], effective index matching coupler [18–20], stress-induced phase difference [21–23], are compact, cheap and easy to operate. While the aforementioned fiber based generation methods are hard to realize tunable, broadband OAM modes generation. Recently, using an OAM mode sorter, H. Huang et al. realized the MUX/DEMUX of the LP-OAM _{± 1} modes over a graded-index, few-mode optical fiber (FMF) [11]. The pivotal OAM mode sorter is composed of two custom refractive optical elements, which isn’t benefit for the compact OAM system. Utilizing stress-induced phase difference between two orthogonal LP₁₁ modes in two mode fiber (TMF), S. H. Li et al. demonstrated the selective conversion from the LP₁₁ mode to the LP-OAM _{± 1} mode [22]. The selective mode conversion rely on the manual adjustment of the stress on the TMF, which hinders the online remote monitoring. Additionally, H. Huang and S. H. Li just generated the low order OAM modes, and didn’t discuss the bandwidth of the generated OAM modes. L. X. Wang et al. proposed a polarization-maintaining FMF that features an elliptical ring shaped core, whose eigenmodes degenerated into LP modes with a higher values of ellipticity [24]. While the reason of the conversion from hybrid mode such as TE₀₁, TM₀₁, HE₂₁^even and HE₂₁^odd mode, to the corresponding LP₁₁^even, ^y, LP₁₁^odd, ^y, LP₁₁^even, ^x and LP₁₁^odd, ^x mode isn’t explained.

In this paper, we proposed a tunable side-hole ring fiber (SHRF) to generate 10 linearly polarized (LP) OAM states with broad operation bandwidth and high purity. The mode analysis are based on the full-vector finite-element method (FV-FEM). The SHRF is composed of an air core and two side holes infiltrated with the refractive index (RI) tunable functional materials. The two infiltrated side holes break the circular symmetry, thus introduce birefringence between the even and odd LP_l,1 mode in the ring core. With a certain n_a (RI value of the infiltrated functional material), the phase difference of the even and odd LP_l,1 core modes accumulated to ± π/2 after a fixed propagation distance which is odd times of the SHRF quarter fiber beat length. Finally, tunable LP OAM states can be generated at the output of the SHRF. In section 2, we theoretically demonstrated that the eigenmodes of the SHRF are the LP modes rather than the hybrid mode in perfect annular fibers. In section 3, three important factors of the generated OAM modes: purity, bandwidth and tunable n_a range are analyzed in detail. Additionally, the experiment feasibility is discussed. In section 4, the conclusions are given.

2. Fiber structure and theoretical model

The proposed tunable LP OAM generator is based on a segment of functional material infiltrated SHRF. As is shown in Fig. 1(a), the SHRF is composed of an air core and two side holes. The air core acts as a repulsive barrier, forcing the mode field to encounter the large index step between ring and cladding. The air core can increase the RI contrast between the ring and cladding, and eliminate the mode with a higher radial order (i.e. m ≥ 2) which are more difficult to multiplex and demultiplex in data transmission. The two side holes aligned along the y axis induce a birefringence RI distribution, thus the eigenmodes are linearly polarized along the optical axes x and y axis respectively. The two side holes are natural micro fluid channels which can be filled with functional materials whose refractive index (RI) can be modulated by external temperature, magnetic, electric intensity, such as RI matching liquid [20, 25], magnetic fluid [26, 27], liquid crystal [28, 29] and nonlinear liquid [30]. As an example, we choose the RI matching fluid which is produced by Cargille Laboratories Inc. as the infiltrated functional materials [25]. The RI of the liquid is 1.444 for 1550 nm at 25°C and has a thermal-optic coefficient of −0.0004 refractive index unit per centigrade (RIU/°C). Thus the birefringence of the fiber can be regulated and controlled by external temperature. The clad material is pure silica, and the RI difference between the ring and the background silica is 0.01. It should be noted that the thermo-optic coefficient of the pure silica is 8.6 × 10⁻⁶/K, thus the temperature induced RI variation for silica is minor and we have ignored this minor effect in the following simulation. The size parameters of the proposed fiber are listed in Table.1:

Fig. 1 Cross section (a) and composition (b) of the SHRF.

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Table 1. Parameters of the Side Hole Fiber

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To analyze the eigenmodes of the SHRF, the SHRF is divided into two parts in Fig. 1(b). Part ① is composed of two side holes, and part ② is an air core ring fiber. Without part ①, the eigenmodes of the air core ring fiber are the hybrid modes, such as the TE₀₁, TM₀₁, HE_l_+1,1, EH_l_-1,1 (l = 1,2,3) modes. For the SHRF, part ① introduces spatially dependent perturbation, and changes the symmetry of the fiber from circular symmetry to axis symmetry, which means the SHRF is an anisotropic fiber. Thus the eigenmodes of the SHRF is varied with the position of the two side holes and the RI value of the infiltrated liquid.

For an unperturbed, perfectly round optical fiber, the propagation constants of normal modes is determined from the solution of the vector wave equation [31–33]:

\begin{array}{l} (\nabla_{t}^{2} + n^{2} k^{2} - β^{2}) e_{t} = - \nabla_{t} (e_{t} \cdot \nabla_{t} \ln n^{2}) \\ n^{2} (x, y) = n_{c o}^{2} (1 - 2 Δ f (x, y)) Δ = \frac{n_{c o}^{2} - n_{c l a d}^{2}}{2 n_{c o}^{2}} \\ f (x, y) = {\begin{cases} 1 (x^{2} + y^{2}) / ρ^{2} \geq 1 \\ 0 (x^{2} + y^{2}) / ρ^{2} \leq 1 \end{cases} \end{array}

where β and field e_t denotes the exact propagation constant and the transverse field, n_co and n_clad is the RI of the fiber core and clad respectively, ρ is the radius of fiber core, and k = 2π/λ. The vector operators

\nabla_{t}

and

\nabla_{t}^{2}

are defined as

\nabla_{t} = \hat{x} \partial / \partial x + \hat{y} \partial / \partial y

and

\nabla_{t}^{2} = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}

respectively.

Assuming that the effective core RI of the SHRF along the x and y axis are n_x and n_y respectively, thus the RI of the fiber in Eq. (1) is changed to

\begin{array}{l} n^{2} (x, y) = [\begin{array}{l} n_{x}^{2} 0 \\ 0 n_{y}^{2} \end{array}] (1 - 2 Δ f (x, y)) = {\tilde{n}}^{2} (x, y) \hat{I} + δ n^{2} {\hat{σ}}_{z} \\ {\tilde{n}}^{2} (x, y) = n_{c o}^{2} (1 - 2 Δ f (x, y)) \\ n_{c o}^{2} = (n_{x}^{2} + n_{y}^{2}) / 2 δ n^{2} = (n_{x}^{2} - n_{y}^{2}) / 2 \\ \hat{I} = [\begin{array}{l} 1 0 \\ 0 1 \end{array}] {\hat{σ}}_{z} = [\begin{array}{l} 1 0 \\ 0 - 1 \end{array}] \end{array}

Meanwhile the vector wave equation for SHRF is

\begin{array}{l} ({\hat{H}}_{0} + δ n^{2} {\hat{σ}}_{z} + \hat{U}) | e_{t} 〉 = β^{2} | e_{t} 〉 \\ {\hat{H}}_{0} = (\nabla^{2} + k^{2} {\tilde{n}}^{2} (x, y)) \hat{I} \\ \hat{U} = (\begin{array}{l} \nabla_{x} \frac{\partial {lnn}^{2} (x, y)}{\partial x} \nabla_{x} \frac{\partial {lnn}^{2} (x, y)}{\partial y} \\ \nabla_{y} \frac{\partial {lnn}^{2} (x, y)}{\partial x} \nabla_{y} \frac{\partial {lnn}^{2} (x, y)}{\partial y} \end{array}) \end{array}

where

| e_{t} 〉 = (\begin{array}{l} e_{x} \\ e_{y} \end{array})

represents the linear polarization basis. It is somewhat convenient to pass Eq. (3) to the basis of circular polarizations defined as:

| Ψ 〉 = \frac{1}{\sqrt{2}} (\begin{array}{l} e_{x} - i e_{y} \\ e_{x} + i e_{y} \end{array}) = \frac{1}{\sqrt{2}} (\begin{array}{l} 1 - i \\ 1 i \end{array}) (\begin{array}{l} e_{x} \\ e_{y} \end{array}) = C | e_{t} 〉

Thus the unitary transformation matrix S = C^T and the resulting equation is

\begin{array}{l} ({\hat{H}}_{0} + δ n^{2} {\hat{σ}}_{x} + \hat{V}) | Ψ 〉 = β^{2} | Ψ 〉 \\ \hat{V} = S^{+} \hat{U} S {\hat{σ}}_{x} = [\begin{array}{l} 0 1 \\ 1 0 \end{array}] \end{array}

We can seek for the solution of Eq. (5) using the perturbation theory: ${\hat{H}}_{0}$ is supposed to be the zero-order operator and ${\hat{H}}_{p e r} = δ n^{2} {\hat{σ}}_{x} + \hat{V}$ is treated as a perturbation. Setting the eigenvectors ${| Ψ_{l, 1}^{0} 〉}_{i}$ satisfies the scalar wave equation [32, 33]

\begin{array}{l} (\nabla_{t}^{2} + n^{2} k^{2} - {\tilde{β}}_{}^{2}) Ψ_{}^{0} = 0 \\ {\hat{H}}_{0} {| Ψ_{}^{0} 〉}_{i} = {\tilde{β}}_{} {| Ψ_{}^{0} 〉}_{i} \end{array}

where

\tilde{β}

and

Ψ_{}^{0}

denotes the propagation constant and the transverse field for the scalar wave equation. In the weak guidance limit it is convenient to choose the eigenfunctions

| Ψ 〉

in the form of OVs

| σ, m 〉

which in the basis of circular polarizations

| Ψ 〉 = \frac{1}{\sqrt{2}} (\begin{array}{l} e_{x} - i e_{y} \\ e_{x} + i e_{y} \end{array})

can be represented as

\begin{array}{l} | Ψ_{1} 〉 = | 1, l 〉 = (\begin{array}{l} 1 \\ 0 \end{array}) e^{i l φ} F_{l} (R), | Ψ_{2} 〉 = | 1, - l 〉 = (\begin{array}{l} 1 \\ 0 \end{array}) e^{- i l φ} F_{l} (R) \\ | Ψ_{3} 〉 = | - 1, - l 〉 = (\begin{array}{l} 0 \\ 1 \end{array}) e^{- i l φ} F_{l} (R), | Ψ_{4} 〉 = | - 1, l 〉 = (\begin{array}{l} 0 \\ 1 \end{array}) e^{i l φ} F_{l} (R) \end{array}

where R = r/ρ, F_l(R) is the radial function of the step index fiber

F_{l} (R) = {\begin{cases} \frac{J_{l} (U R)}{J_{l} (U)} R \leq 1 \\ \frac{K_{l} (W R)}{K_{l} (W)} R \geq 1 \end{cases}

The exact eigenvalues β ² follows [32, 33]:

\det | {({\hat{H}}_{p e r})}_{i j} - δ_{i j} β^{2} | = 0

As $〈 Ψ_{i} | {\hat{H}}_{p e r} | Ψ_{j} 〉 = 〈 Ψ_{i} | {\hat{H}}_{p e r}^{+} | Ψ_{j} 〉$ in basis Eq. (7), the Hermitian operator $δ \hat{H} = \frac{1}{2} ({\hat{H}}_{p e r} + {\hat{H}}_{p e r}^{+})$ can instead the operator of ${\hat{H}}_{p e r}$ . Thus the perturbation matrix is constructed as follows [32, 33]:

δ {\hat{H}}_{i j} = 〈 Ψ_{i} | δ \hat{H} | Ψ_{j} 〉

Using Eq. (7) – Eq. (10), we obtain the following perturbation matrix determining the structure of the exact modes with the azimuthal number l [33]:

\begin{array}{l} δ \hat{H} = (\begin{array}{l} A_{l} 0 0 E \\ 0 B_{l} E B_{l} \\ 0 E A_{l} 0 \\ E B_{l} 0 B_{l} \end{array}) (l = 1) δ \hat{H} = (\begin{array}{l} A_{l} 0 0 E \\ 0 - A_{l} E 0 \\ 0 E A_{l} 0 \\ E 0 0 - A_{l} \end{array}) (l > 1) \\ A_{l} = - \frac{2 π Δ}{r_{0}^{2}} {(F_{l}^{2} - F_{l} F_{l}^{'})}_{R = 1} B_{l} = \frac{2 π Δ}{r_{0}^{2}} {(F_{l}^{2} + F_{l} F_{l}^{'})}_{R = 1} \\ E = n_{c o} k^{2} Δ n = n_{c o} k^{2} (n_{x} - n_{y}) \end{array}

Thus the normal modes Ψ_i in the linear polarization basis of the anisotropic fibers are [33]

\begin{array}{l} (\begin{array}{l} Ψ_{1} \\ Ψ_{2} \\ Ψ_{3} \\ Ψ_{4} \end{array}) = (\begin{array}{l} c o s θ_{1} \sin l ϕ \sin θ_{1} c o s l ϕ \\ - \sin θ_{1} \sin l ϕ c o s θ_{1} c o s l ϕ \\ c o s θ_{3} c o s l ϕ - \sin θ_{3} \sin l ϕ \\ \sin θ_{3} c o s l ϕ c o s θ_{3} \sin l ϕ \end{array}) (\begin{array}{l} e_{x} \\ e_{y} \end{array}) \\ \tan (2 θ_{1}) = \frac{A_{l}}{2 E}, \tan (2 θ_{3}) = \frac{A_{l} + 2 B_{l}}{2 E}, 2 θ_{i} \in [0, π] \end{array}

From Eq. (12), the fields of these modes can be obtained as a superposition of the hybrid modes with specified weighting factors in the unperturbed fiber. For a given structure $\tan (2 θ_{i}) \propto (Δ / ρ^{2}) / (n_{c o} Δ n k^{2}) \propto (Δ {(λ / ρ)}^{2}) / Δ n$ , where ∆n = n_x– n_y. The criterion ∆ (λ/ρ)² represents the polarization correction, ∆n represents the anisotropy. The trade-off between this two parameters influence the final state of the normal modes Ψ_i. Here we consider radial number as m = 1. Thus, it is reasonable to consider two limiting cases:

(1) Weak anisotropy, $Δ n ≪ Δ {(ρ / λ)}^{2} \to \tan (2 θ_{i}) \approx \infty \to θ_{i} = π / 4$ , the normal modes Ψ_i of the fibers with weak anisotropy are that of the unperturbed fibers: $(\begin{array}{l} Ψ_{1} \\ Ψ_{2} \\ Ψ_{3} \\ Ψ_{4} \end{array}) = {\begin{cases} (\begin{array}{l} H E_{2, 1}^{o d d} \\ T E_{0, 1}^{} \\ H E_{2, 1}^{e v e n} \\ T M_{0, 1}^{} \end{array}) = (\begin{array}{l} \sin ϕ \cos ϕ \\ - \sin ϕ \cos ϕ \\ \cos ϕ - \sin ϕ \\ \cos ϕ \sin ϕ \end{array}) (\begin{array}{l} e_{x} \\ e_{y} \end{array}) l = 1 \\ (\begin{array}{l} H E_{l + 1, 1}^{o d d} \\ E H_{l - 1, 1}^{o d d} \\ H E_{l + 1, 1}^{e v e n} \\ E H_{l - 1, 1}^{e v e n} \end{array}) = (\begin{array}{l} \sin l ϕ \cos l ϕ \\ - \sin l ϕ \cos l ϕ \\ \cos l ϕ - \sin l ϕ \\ \cos l ϕ \sin l ϕ \end{array}) (\begin{array}{l} e_{x} \\ e_{y} \end{array}) l \geq 2 \end{cases}$
(2) Strong anisotropy, $Δ {(ρ / λ)}^{2} ≪ Δ n ≪ 1 \to \tan (2 θ_{i}) \approx 0 \to θ_{i} = 0$ ,the normal modes Ψ_i of the fibers with strong anisotropy are the linearly polarized (LP) modes: $(\begin{array}{l} Ψ_{1} \\ Ψ_{2} \\ Ψ_{3} \\ Ψ_{4} \end{array}) = (\begin{array}{l} L P_{l, 1}^{o d d, x} \\ L P_{l, 1}^{e v e n, y} \\ L P_{l, 1}^{e v e n, x} \\ L P_{l, 1}^{o d d, y} \end{array}) = (\begin{array}{l} \sin l ϕ 0 \\ 0 \cos l ϕ \\ \cos l ϕ 0 \\ 0 \sin l ϕ \end{array}) (\begin{array}{l} e_{x} \\ e_{y} \end{array}) l \geq 1$

With the parameters in Table 1, the value of ∆n in the SHRF is around 10⁻⁵ which represents the strong anisotropy. Thus the eigenmodes of the SHRF is the LP modes with different effective RI value of the LP_l_,1^{even, (}^{x, y}⁾ and LP_l_,1^{odd, (}^{x, y}⁾. At a certain length L = nL_B/4 = nλ/(4B), B = n_eff (LP_l_,1^{even, (}^{x, y}⁾)-n_eff (LP_l_,1^{odd, (}^{x, y}⁾), the phase difference of this two modes is ± π/2. In this condition, the normal modes of the SHRF, i.e., the LP mode, could constitute the linearly polarized OAM:

Ψ_{l}^{(x, y)} = O A M_{l}^{(x, y)} = L P_{l, 1}^{e v e n, (x, y)} \pm i L P_{l, 1}^{o d d, (x, y)} = (\begin{array}{l} e_{x} \\ e_{y} \end{array}) e^{\pm i l ϕ} F_{l, 1}

3. Properties of the generated LP-OAM modes

There are four LP mode groups in the proposed SHRF, i.e. LP₀₁, LP₁₁, LP₂₁ and LP₃₁ groups. As the OAM of the LP₀₁ groups is 0, we just present the LP₁₁, LP₂₁ and LP₃₁ groups in Fig. 2. The input wavelength λ and the RI of the filled liquid n_a are 1.55 μm and 1.43 respectively. The mode intensity of the LP modes in SHRF are obtained with the FV-FEM. The phase distribution of each LP mode and the generated OAM modes are calculated by the scripting language. As the intensity of the generated OAM modes are all doughnut profile, it’s replaced by the interferogram with a reference Gaussian beam. From the interferogram, the topology number and handedness of the generated mode is clearly presented. As is depicted in Fig. 2, if the phase difference between the LP_l_,1 ^even and LP_l_,1 ^odd mode is 0 and π, the corresponding mode field is LP_l_,1 ^even + LP_l_,1 ^odd and LP_l_,1 ^even - LP_l_,1 ^odd, which presents a rotation of the LP_l_,1 ^even or LP_l_,1 ^odd mode. Once the phase difference between this two orthogonal mode reached ± π/2, the phase and the interferogram of the synthesis modes LP_l_,1 ^even ± iLP_l_,1 ^odd, i.e., the OAM _± _l modes, present the exp( ± ilϕ) phase dependence. It should be noted that the modal profile variation in the normal FMF is different from that in the proposed fiber. In the normal FMF, the LP modes are linear combination of the degenerated vector modes [34]. Thus the modal profile of the combined LP modes varies along the propagation axis. While the LP modes in the proposed fiber is eigenmodes with stable modal profiles as is depicted in section 2. For the LP_l_,1 mode group, the variation of phase difference between LP_l_,1^even and LP_l_,1^odd mode results in the mode conversion from LP mode to the OAM mode. Thus after a fixed propagation distance of L = nL_B/4, the phase difference of this two modes is ± π/2. The LP-OAM _± _l states are generated and can be stable propagated in the OAM ring core fiber.

Fig. 2 The intensity and phase of the LP_l_,1^even, LP_l_,1^odd, LP_l_,1^even + LP_l_,1^odd (0), OAM_l (π/2), LP_l_,1^even-LP_l_,1^odd (π), OAM_-l (-π/2), for l = 1 (a), l = 2 (b) and l = 3(c). The intensity of the OAM _± _l is replaced by that of the interference patterns with a reference Gaussian beam.

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In order to choose a proper SHRF length which can generate different order OAM mode, we first calculated the quarter fiber beat length L_B/₄ of every LP_l_,1 mode for different value of n_a. The L_B/₄ is expressed as:

L_{B / 4} = \frac{L_{B}}{4} = \frac{λ}{4 B} = \frac{λ}{4 (n_{e f f} (L P_{l, 1}^{e v e n, (x, y)}) {-n}_{e f f} (L P_{l, 1}^{o d d, (x, y)}))}

As the phase difference of the LP_l_,1 ^even and LP_l_,1 ^odd mode is π/2 after propogation distance L_B/₄ in the SHRF, the designed fiber length L should be odd times of L_B/₄. The relationship between the L_B/₄ and n_a can be regard as the relationship between the fiber length L and n_a (Fig. 3). In Fig. 3(a), the slope of each curve shows a sharp rise when n_a is greater than 1.40, thus we enlarge Fig. 3(a) to present the details of the curves in Fig. 3(b). The designed fiber length L should not only generate OAM_l mode but also the OAM_-l modes. Thus we choose L = 0.015 m to generate enough OAM modes in Fig. 3(b). If the L_B/₄ is 0.005 m (L/3) for a group of n_a (the orange horizontal dotted line), the accumulated phase difference after propagating a fiber length of L (0.015m) is –π/2, thus the output beam is the OAM_-l modes. If the L_B/₄ is 0.015 m (L) for another group of n_a (the purple dotted horizontal line), the accumulated phase difference after propagating a fiber length of L is π/2, thus the output beam is the OAM_l modes. The corresponding values of n_a for different OAM modes are listed in Table 2. Finally, the proposed fiber with a length of 0.015 m can generate 10 LP-OAM modes (OAM _{± 1}^x, OAM _{± 2}^x, OAM _{± 2}^y, OAM _{± 3}^x, OAM _{± 3}^y) at the output port.

Fig. 3 The relationship between the fiber length L and the RI of the tunable liquid n_a for x and y polarized OAM_l mode with n_a ranging from 1.35 to 1.444 (a) and 1.40 to 1.444 (b). The fixed fiber length L is selected as 0.015m, thus the OAM_-l modes is generated when L = 3L_B/4 (L_B/4 = 0.005 m), and the OAM_l modes is generated when L = L_B/4 (L_B/4 = 0.015 m). The intersection points between the purple dotted horizontal line and the 6 curves are the corresponding value of n_a for the OAM_l modes. The intersection points between the orange dotted horizontal line and the 6 curves are the corresponding value of n_a for the OAM_-l modes.

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Table 2. Values of n_a and Purity for Each OAM_l Mode with L = 0.015m

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Purity is an important parameter for OAM mode. It’s a stability metric in a fiber with respect to the relative power in unwanted modes. An impure OAM states has a narrow angular distribution and generates many OAM sidebands, whereas a pure OAM state corresponds to a uniform angular distribution with distinct, well defined effective RI, group delays, and field distribution patterns. The amplitudes of the OAM states C_l is expressed as [35]:

C_{l} = \frac{1}{2 π} \int_{0}^{2 π} f (ϕ) \exp (- i l ϕ) d ϕ

where f(ϕ) is the phase function of an OAM mode, and can be expressed as:

p h a s e (ϕ) |^{r = a} = f (ϕ) = \sum_{l = - \infty}^{\infty} C_{l} \exp (i l ϕ) \sum_{l = - \infty}^{\infty} {| C_{l} |}^{2} = 1

Thus the purity of the l^th OAM mode is ${| C_{l} |}^{2}$ . Using Eq. (18), we calculate the purity of generated OAM mode with a fiber length of 0.015 m as shown in Table 2. The purity of each OAM mode is almost higher than 95%, which means the proposed fiber can generate 10 LP-OAM modes with high purity.

Bandwidth and tunable n_a range are another two important parameter for a tunable OAM generator. As the effective bandwidth and n_a range are meaningful only if the fiber length of the SHRF is fixed, we parametric sweep the input wavelength from 1.25 to 1.65 μm and the infiltrated n_a from 1.404 to 1.444 with all combinations. Then the quarter fiber beat length L_B/₄ as variations of input wavelength and n_a is shown in Fig. 4. The white and black curves represent the contour lines of the L_B/₄. Referring to Fig. 3, the L_B/₄ of the black contour lines are set as 3 times longer than the white contour lines, then both OAM _± _l with a corresponding fiber length of the black contour lines can be generated. For the OAM _±1 modes, the L_B/₄ of the y-polarized modes (b1) are much shorter than that of the x-polarized mode (a1), thus the y-polarized OAM _{± 1} modes can’t be generated with a same fiber length as the x-polarized OAM _{± 1} modes. The range of L_B/4 for the y polarized OAM₁ group is from 0.0013m to 0.0034m. As 0.0034m/0.0013m<3, the proposed fiber can only generate OAM₁^y mode with a fixed fiber length. The corresponding range of L_B/4 for the x polarized OAM₁ group is from 0.0031m to 0.0604m. As 0.0604m/0.0031m>3, the proposed fiber can generate OAM₁^x and OAM_-1^x mode with a fixed fiber length. The three yellow contour curves in Fig. 4 (b1) represents three value of L_B/₄: 0.0033m, 0.002m and 0.0016m. A meaningful fiber length should not only support a maximum number of OAM modes but also obtain a broad bandwidth. Thus for the OAM₁^y modes, the fixed fiber length of the SHRF can be 0.002 m, the corresponding bandwidth and n_a range are 244 nm and 0.0319 respectively. For the OAM_±1^x modes, the fixed fiber length of the SHRF can be 0.03 m, the corresponding bandwidth and n_a range are 380 nm and 0.028 respectively. For the OAM _{± 2} modes, the fixed fiber length of the SHRF can be 0.009 m, the corresponding bandwidth and n_a range are 100 nm and 0.028 respectively. For the OAM _{± 3} modes, the fixed fiber length of the SHRF can be 0.012 m, the corresponding bandwidth and n_a range are 80 nm and 0.028 respectively.

Fig. 4 The quarter fiber beat length L_B/₄ as variations of input wavelength and n_a for x-polarized (a1) and y-polarized (b1) OAM _{± 1} modes, x-polarized (a2) and y-polarized (b2) OAM _{± 2} modes, x-polarized (a3) and y-polarized (b3) OAM _{± 3} modes. The white and black curves of (a1) represents the L_B/₄ are 0.01 and 0.03 m respectively. The yellow curves of (b1) represents the L_B/₄ are 0.0033m, 0.002m and 0.0016m respectively. The white and black curves of (a2) and (b2) represent the L_B/₄ are 0.003 and 0.009 m respectively. The white and black curves of (a3) and (b3) represents the L_B/₄ are 0.004 and 0.012 m respectively.

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As to the experimental feasibility, the proposed SHRF is easy to be fabricated with the mature microstructure optical fiber drawing technology: the background material is pure silica, the core-clad RI contrast is 0.01. The two side holes of the SHRF are natural microfluid channels which can be selectively filled with functional materials [25].The input mode of the FMF should be the LP_l,₁ modes which can be generated by the long period fiber gratings (LPFG) [36] or the couplers [37]. Then the LP_l,₁ mode in FMF is rotated by an angle of π/(4l) to equally excite the LP_l,₁^even and LP_l,₁^odd modes in the SHRF [22]. As the even and odd mode of LP_l,₁ are not circularly symmetric, the two modes experience different phase velocities in the infiltrated SHRF (see Fig. 5). When the input wavelength and the length of the SHRF are 1.55 um and 0.015 m respectively, the OAM₊₁^x and OAM_-1^x modes can be generated with a n_a value of 1.4369 and 1.4203 respectively; the OAM₊₂^x, OAM_-2^x, OAM₊₂^y, OAM_-2^y modes can be generated with a n_a value of 1.4417, 1.4332, 1.4420, 1.4358 respectively; the OAM₊₃^x, OAM_-3^x, OAM₊₃^y, OAM_-3^y modes can be generated with a n_a value of 1.4407, 1.4242, 1.4413, 1.4301 respectively. Finally, the resultant field distribution is a single ring annular beam with an azimuthal phase structure of exp( ± ilϕ) and can be transmitted in the OAM fiber.

Fig. 5 Tunable generation of LP-OAM modes based on the SHRF.

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4. Conclusion

In summary, a functional material infiltrated SHRF for generating 10 LP OAM states with tunable topology numbers, ultra-broad operation bandwidth and high purity is proposed by simulation. The SHRF is composed of an air ring core with two big air holes aligned along the y axis, which result in the LP mode basis. After a fixed propagation distance of 0.03 m, 0.009 m and 0.012 m, the phase difference between the odd and even LP₁₁^x, LP₂₁^{x, y}, LP₃₁^{x, y} modes in the SHRF accumulate to ± π/2 respectively with n_a ranging from 1.412 to 1.44. Correspondingly, the output states are OAM _{± 1}^x, OAM _{± 2} ^{x, y}, OAM _{± 3} ^{x, y} with a bandwidth of 380 nm, 100 nm and 80 nm respectively. As to the preparation process of the SHRF, the background material is pure silica, the core-clad RI contrast is 0.01 and the microstructure are three air holes, which are easy to be fabricated with the mature silica fiber drawing technology. The two side holes are natural microfluid channels which can be selectively filled with functional materials. The presented SHRF with favorable performance may find wide potential use in all fiber based OAM multiplexing transmission systems and other OAM communication applications.

Funding

National Natural Science Foundation of China (NSFC) (61322510); Tianjin Natural Science Foundation (14JCZDJC31300, 16JCZDJC31000).

Acknowledgments

Sincerely thanks Shuhui Li and Chen Li in Huazhong University of Science and technology, Taixing Liu in Nankai University and professor C. N. Alexeyev in V. I. Vernadsky Crimean Federal University for their kindly discussion with authors.

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Generation of linearly polarized orbital angular momentum modes in a side-hole ring fiber with tunable topology numbers

Abstract

1. Introduction

2. Fiber structure and theoretical model

3. Properties of the generated LP-OAM modes

4. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (5)

Tables (2)

Equations (18)

Optics Express

	L(m)	OAM₁		OAM₂		OAM₃
	L(m)	x	y	x	y	x	y
n_a	0.005 × 3 (−)	1.4203	–	1.4332	1.4358	1.4242	1.4301
n_a	0.015( + )	1.4369	–	1.4417	1.4420	1.4407	1.4413
purity	0.005 × 3 (−)	99.62%	–	98.42%	94.86%	96.58%	99.48%
purity	0.015( + )	99.82%	–	99.52%	97.11%	99.06%	95.32%