## Abstract

Based on the phased-shifted interference between supermodes, a novel method that can directly convert LP_{01} mode to orbital angular momentum (OAM) mode in a dual-ring microstructure optical fiber is proposed. In this fiber, the resonance between even and odd HE_{11} modes in inner ring and higher order mode in outer ring will form two pairs of supermodes, and the intensities and phases of the complete superposition mode fields for the involved supermodes created by the resonance at different wavelengths and propagating lengths are investigated and exhibited in this paper. We demonstrate that OAM mode can be generated from π/2-phase-shifted linear combinations of supermodes, and the phase difference of the even and odd higher order eigenmodes can accumulate to π/2 during the coupling process, which is defined as “phase-shifted” conversion. We build a complete theoretical model and systematically analyze the phase-shifted coupling mechanism, and the design principle and optimization method of this fiber are also illustrated in detail. The proposed microstructure fiber is compact, and the OAM mode conversion method is simple and flexible, which could provide a new approach to generate OAM states.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Orbital angular momentum beams, which are characterized by a doughnut intensity profile and helical phase front, have drawn immense research interests in the recent past [1,2]. Due to the unique intrinsic spatial orthogonal property, OAM modes have found promising applications in large-capacity fiber optical communications [3–5]. In various fiber-based OAM applications, the generation of multiple OAM states is always highly desirable. Different approaches to generate OAM states in optical fiber have been proposed and demonstrated, such as helical or twisted special fibers and gratings [6–12], polarization controller or pressure assistant phase control methods [13–17], and fused biconical fiber tapers [18–21]. The OAM states are generated through the twisted structures or phase-control devices in the aforementioned approaches. In twisted structures, the helical perturbation of refractive index brings forth the coupling between normal fiber mode and OAM mode. However, the helical index-modulation is not easy to replicate due to the fabrication process. OAM modes can also be generated with the assistant of phase control devices: polarization controllers (PCs) or pressure slabs [13–21]. The stress and the PCs change the phase velocities of the two orthogonal components of OAM modes. By adjusting the pressure or the PCs appropriately, the two orthogonal modes can achieve a π/2 phase difference. Acousto-optics gratings which constitute a low loss high purity technique for OAM generation have also been reported [10]. However, due to the phase-control devices and acousto-optic modulation equipment, the experiment setups in the related approaches are relatively complex. Novel compact all-fiber OAM couplers which can convert circularly polarized HE_{11} mode ($HE_{11}^{even} \pm iHE_{11}^{odd}$) to different OAM modes are also proposed [22,23], while the generation is seriously affected by the polarization state of incident light. Other special polarization-maintaining fibers which can convert LP_{m1} modes to $OA{M_{ {\pm} m}}$ modes are reported [24,25], and the π/2 phase difference between even and odd LP_{m1} modes comes from the large effective refractive index difference between two orthogonally decomposed LP_{m1} modes. Thus, this kind of polarization-maintaining fiber can only generate the same order linearly polarized $OA{M_{ {\pm} m}}$ modes, and the incident light is confined to LP_{m1} modes. Hence, compact and simple all-fiber OAM devices which offer the potential for direct generation of OAM modes in optical fiber from a regular fiber-mode input are still highly desirable and deeply significant.

In this paper, we propose a novel method to directly convert LP_{01} mode to OAM mode in a dual-ring microstructure optical fiber based on the phased-shifted interference between supermodes. The proposed microstructure optical fiber is composed of two high refractive index rings and two axisymmetric rods between the inner and outer rings. The LP_{01} mode in inner ring can be coupled to higher order modes in outer ring under certain conditions, and the resonance between even and odd HE_{11} modes and higher order mode will form two pairs of supermodes. We demonstrate that OAM mode can be generated from π/2-phase-shifted linear combinations of supermodes, and the phase difference of the even and odd higher order eigenmodes can accumulate to π/2 during the coupling process, which is defined as “phase-shifted” conversion. The effective refractive indices for different resonant eigenmodes and the corresponding supermodes are investigated, and the intensities and phases of the complete superposition mode fields for the involved supermodes created by the resonance between HE_{11} and higher order modes at different wavelengths and propagating lengths are investigated and exhibited in this paper. We build a complete theoretical model and systematically analyze the phase-shifted coupling mechanism, and the design principle and optimization method of this fiber are also illustrated in detail. The proposed microstructure fiber is compact, and the OAM mode conversion method is simple and flexible, which could provide a new approach to generate OAM states.

## 2. Principle and structure

The OAM generation in the proposed fiber is based on the phased-shifted interference between supermodes. The proposed fiber can be core-aligned and spliced with the single mode fiber. In single mode fiber, linearly polarized LP_{01} mode consists of two degenerate orthogonal modes: $HE_{11}^{even}$ and $HE_{11}^{odd}$ mode. When LP_{01} mode propagates from single mode fiber to the proposed fiber, the two HE_{11} modes can be coupled to higher order HE or EH modes under certain conditions. Here we take HE mode as an example to systematically analyze and explain the phase-shifted coupling mechanism in the proposed fiber. As each of the HE mode has two-fold orthogonal degeneracy, $HE_{11}^{even}$ (or $HE_{11}^{odd}$) mode can only be coupled to $HE_{mn}^{even}$ (or $HE_{mn}^{odd}$) mode, and the coupling mechanism between the two pair of orthogonal modes are similar.

When the incident light propagates from the single mode fiber to the proposed fiber core, according to the coupled-mode theory [26], the electric fields of the $HE_{11}^{even}$ mode and higher order $HE_{mn}^{even}$ mode can be expressed as:

Thus, the complete field profile across the proposed fiber can be obtained as a combination of the two mode fields:

*z*. Equation (3) show that the total field ${E_{even}}(r)$ is a linear combination of two independent normal mode fields ${E_A}(x,y)$ and ${E_B}(x,y)$ with different propagation constants ${\beta _A}$ and ${\beta _B}$, which are also known as the supermodes in this fiber [23]. If we consider the inner ring and the outer ring as two different optical waveguides, the HE

_{11}modes and the HE

_{mn}modes are approximate normal modes for the inner and outer rings respectively, and the variation of the mode field amplitudes can be expressed as the coupling between HE

_{11}and HE

_{mn}modes. But for the entire dual-ring fiber, the supermodes are exact the normal mode solutions of this fiber, and the variation of the mode field amplitudes in the inner ring and outer ring can also be expressed as the interference of this pair of supermodes.

Based on coupled-mode theory, the maximum energy transfer between $HE_{11}^{even}$ and $HE_{mn}^{even}$ mode occurs at the coupling length:

And the complete energy transfer will happen at a particular optical frequency (defined as the coupling wavelength ${\lambda _c}$) when the phase matching condition is satisfied: ${\delta _{even}} = 0$. Thus, after propagating a distance of $L = (2n + 1) \cdot {l_0}$, the total mode field in the proposed fiber at the coupling wavelength ${\lambda _c}$ can be expressed as:Similarly, the variation of the electric fields for $HE_{11}^{odd}$ and $HE_{mn}^{odd}$ can also be expressed as the interference of supermodes, as shown in Fig. 1:

Therefore, when $L{P_{01}} = HE_{11}^{even} + HE_{11}^{odd}$ mode propagates from single mode fiber to the proposed fiber, the complete field profile can be expressed as:

If the phase matching conditions for even and odd modes are satisfied at the same wavelength ${\lambda _c}$, and the propagation constants for each mode at ${\lambda _c}$ satisfy:

*q*can be any positive integers. Here we define Eq. (13) as “phase-shifted coupling condition”. After propagating a distance of

*L*, the complete mode field in the proposed fiber at the coupling wavelength ${\lambda _c}$ can be calculated:

As analyzed above, the phase difference of the even and odd $H{E_{mn}}$ modes can accumulate to π/2 during the coupling process, and we define this phenomenon as “phase-shifted” conversion. The conclusion also indicate that OAM mode can be generated from π/2-phase-shifted linear combinations of supermodes, which could provide a new approach to generate OAM states.

The phase-shifted coupling mechanism and the variation of the electric fields between $H{E_{\textrm{11}}}$ and $E{H_{mn}}$ mode are similar, and LP_{01} mode can also be converted into $OAM_{ {\pm} m + 1,n}^ \mp$ mode through $E{H_{mn}}$ mode when its phase-shifted coupling condition is satisfied.

Based on the phase-shifted coupling mechanism analyzed above, we specially design and optimize a microstructure optical fiber, whose phase matching conditions for even and odd modes are satisfied at the same wavelength and the propagation constants for each coupled-mode meet the phase-shifted coupling condition, as shown in Fig. 2.

The proposed microstructure optical fiber is composed of two high refractive index rings and two axisymmetric rods between the inner and outer rings. The background material of this fiber is made of pure silica, which is more compatible to be integrated with conventional optical fibers. As we know, the fundamental mode and higher order modes are orthogonal to each other in circularly symmetric fibers. The two axisymmetric high refractive index rods are specially designed to break the circular symmetry of the dual-ring fiber and act as spatially dependent perturbation for this fiber so that the LP_{01} mode in the inner ring can be coupled to the higher order modes in the outer ring.

In the design process, the physical parameters of the inner and outer rings are adjusted firstly to ensure that the effective refractive indices (ERI) of the inner LP_{01} mode and the corresponding outer higher order mode are close to each other. The coupling coefficients and conversion properties between fundamental mode and higher order mode are closely related to the physical parameters of the two axisymmetric rods. The physical parameters *n*_{3} and *R*_{hole} of the axisymmetric rods are related to the symmetry of the proposed fiber. The cross section of this fiber is close to circular symmetry when *R*_{hole} is small and *n*_{3} approaches the background refractive index. When *n*_{3} gets higher and the physical size *R*_{hole} is bigger, the cross section of this fiber is close to axial symmetry, and the corresponding coupling coefficients become larger, so LP_{01} mode in inner ring can be coupled to higher order mode more easily at a relatively wide wavelength range. However, LP_{01} mode could be converted to EH and HE modes simultaneously at the same wavelength when the cross section tends to axial symmetry. Thus, the physical parameters *n*_{3} and *R*_{hole} should be adjusted carefully to ensure the coupling wavelengths of EH and HE modes are not in the same region.

Next, we have optimized the position *y* of the axisymmetric rods. As we illustrated before, $HE_{11}^{even}$ (or $HE_{11}^{odd}$) mode can only be coupled to $HE_{mn}^{even}$ (or $HE_{mn}^{odd}$) mode, and we should ensure that the phase matching conditions for even and odd modes are satisfied at the same wavelength. Figure 3 shows the optimization method to unify the coupling wavelengths for even and odd modes, and we also take HE modes as an example here. In the proposed fiber, the ERI of fundamental mode always decreases more slowly than higher order modes as wavelength increases, and the coupling wavelengths for even and odd modes can be calculated respectively through a commercial finite element code (COMSOL). Figures 3(a) and (b) indicates that ERI difference between HE_{11} modes is too large if the coupling wavelength for even mode is shorter than odd mode, while the ERI difference in HE_{mn} modes should be reduced when the coupling wavelength for even mode is longer than odd mode.

The ERI difference between even and odd modes depends on the position *y* of the two axisymmetric rods. The ERI difference between $HE_{11}^{even}$ and $HE_{11}^{odd}$ mode gets larger if the axisymmetric rods are close to inner ring, while the ERI difference between $HE_{mn}^{even}$ and $HE_{mn}^{odd}$ mode becomes larger if the axisymmetric rods are close to outer ring. Therefore, we can adjust the position *y* to ensure that the phase matching conditions for even and odd modes are satisfied at the same wavelength.

Finally, we calculate the propagation constants for the involved four supermodes (${\beta _A}$, ${\beta _B}$, ${\beta _C}$ and ${\beta _D}$) and try to find an appropriate constant *L* to satisfy the phase-shifted coupling condition.

Based on the design principle and optimization method illustrated above, we have designed a series of fiber structure parameters for the generation of different OAM modes, as shown in Table 1 (*n*_{0} represents the refractive index of pure silica).

The structure parameters to generate every OAM mode in each group are not unique, and Table 1 only shows some typical examples of the design. The radius of the inner ring in Table 1 is close to the fiber core of single mode fiber, which makes it more compatible to be integrated with conventional optical fibers. The materials *n*_{1}, *n*_{2}, and *n*_{3} can also be replaced by functional material whose refractive index can be modulated.

## 3. Results and properties

The mode analysis and the propagation properties in the proposed fiber are investigated by full-vector finite-element method. We first built the proposed fiber structure model according to Table 1 and accurately calculated the dispersion curves of each involved supermode. Figure 4 shows the dispersion curves and modal energy distributions of the four supermodes (${\beta _A}$, ${\beta _B}$, ${\beta _C}$ and ${\beta _D}$) created by the resonances between HE_{11} and HE_{51} modes. The phase matching conditions for even and odd modes are both satisfied at 1550.3 nm. To verify the possibility to generate OAM mode through the proposed phase-shifted conversion method, the intensities and phases of the superpositions of different supermodes are investigated.

As we illustrated before, the energy exchange between $HE_{11}^{even}$ and higher order even mode can be expressed as the interference of supermode A and B, so after propagating a distance of $L = \pi /({\beta _A} - {\beta _B})$, the superposition mode field of A and B can be expressed as:

If we can find a constant *L* which satisfies the condition: $L = \frac{\pi }{{{\beta _A} - {\beta _B}}}\textrm{ = }\frac{\pi }{{{\beta _C} - {\beta _D}}}\textrm{ = }\frac{{\pi /2}}{{{\beta _B} - {\beta _D}}}$, and after propagating *L*, the superposition mode field of the whole four supermodes at 1550.3 nm can be expressed as:

We also simulated the intensities and phases of superposition mode field of ${E_D}(x,y) + i{E_B}(x,y)$ and ${E_C}(x,y) + i{E_A}(x,y)$, as shown in Figs. 5(a) and (c). The results also indicate that OAM mode could be generated from π/2-phase-shifted linear combinations of supermodes, which could provide a new approach to generate OAM states.

Next, we calculated the propagation constants for the involved four supermodes created by the resonance between HE_{11} and HE_{51} modes at 1550.3 nm and try to find an appropriate length *L.* Table 2 shows the matching coefficients under different transmission lengths according to Eq. (13). Meanwhile, the complete superposition mode fields at 1550.3 nm are simulated by Eq. (12) with the corresponding different propagation distances, as shown in Fig. 6. We calculated the mode purity for the generated OAM mode in outer ring based on Fourier series expansion method [27]. The purity of OAM_{41} mode generated in outer ring reaches the highest value 99.02% at the propagation length of 273 mm. However, the optical intensity in outer ring is relatively low at 273 mm. Considering both the purity and energy conversion efficiency, we think 268 mm is an appropriate propagation length to meet the phase-shifted coupling condition. This result indicate that the proposed fiber can convert LP_{01} mode to OAM_{41} mode based on the theoretical model.

Moreover, we investigate the influence of wavelength deviation on the performance of OAM mode conversion. The coupling wavelength between HE_{11} and HE_{51} mode is 1550.3 nm, where the complete energy transfer can happen. When the wavelength deviates from 1550.3 nm, the OAM mode conversion will be affected. The energy transfers between the two pairs of even and odd modes are incomplete, and the transfer ratios could be different.

On the other hand, the propagation constants of the four supermodes vary with wavelength, so the corresponding matching coefficients and the appropriate propagation length will also change. Figures 7(a)-(c) show the intensity profiles, phase distributions, purities and azimuthal phase variations of the generated OAM mode in outer ring at different wavelengths under the same propagation distance of 268 mm. The purity of OAM mode decreases fast and the generated mode in outer ring tends to become HE_{51} mode when the wavelength is away from 1550.3 nm. We also calculated the intensity profiles and phase distributions of the superposition mode fields of $E(r) = {E_D}(x,y) - {E_C}(x,y) + i{E_B}(x,y) - i{E_A}(x,y)$ at the corresponding different wavelengths, as shown in Figs. 7(d) and (e). The generated mode in outer ring of $E(r)$ still presents a doughnut profile and the $\exp ({\pm} 4i\phi )$ phase dependence in a certain wavelength range. This result indicates that the matching coefficients and the propagation length can no longer satisfy the OAM mode coupling condition, and we can still obtain OAM mode if the propagation length can be adjusted accordingly in a certain wavelength range.

However, when the wavelength is far away from 1550.3 nm, the coupling between HE_{11} and HE_{51} modes becomes weak, and the energy transfer ratios decrease fast and could be different for odd and even modes. As a result, the purities of the generated OAM modes in outer ring are very low even if the phase-shifted coupling condition is satisfied, and the energy transfer ratios become the leading factor to affect the generation of OAM mode.

Next, we built several theoretical models to generate different OAM modes according to the physical parameters in Table 1. We calculated each group of supermodes and coupling wavelengths. Considering both the purity and energy conversion efficiency, the corresponding appropriate propagation lengths and matching coefficients of each model are proposed, as shown in Table 3.

The complete superposition mode fields at the coupling wavelengths are simulated by Eq. (12) with different propagation distances, and the OAM mode purities and azimuthal phase variations for the generated modes in outer ring are also calculated, as shown in Figs. 8–12. $OAM_{ {\pm} \textrm{61}}^ \pm$, $OAM_{ {\pm} \textrm{81}}^ \pm$, $OAM_{ {\pm} \textrm{41}}^ \mp$, $OAM_{ {\pm} \textrm{61}}^ \mp$ and $OAM_{ {\pm} \textrm{81}}^ \mp$ modes can be generated respectively in the proposed fiber after propagating a certain distance at the corresponding coupling wavelength. The physical parameters in each group can also be further optimized to achieve better performance based on the proposed phase-shifted coupling mechanism and design principle. This result indicates that the proposed method could convert LP_{01} mode to different OAM modes through the interference of supermodes, which provides a new approach to generate OAM states.

The proposed fiber device is controllable and flexible, for the physical parameters to generate different OAM modes are not unique. Based on the proposed phase-shifted coupling mechanism and design principle, the structure and parameters of this fiber can be adjusted accordingly to realize the generation of different OAM modes, and the coupling wavelength and conversion length can also be modified. Moreover, the materials of the high refractive index rings and axisymmetric rods can also be replaced by optical functional materials whose refractive indices can be modulated by physical parameters. The proposed OAM mode conversion method and fiber structure have advantages of simplicity, compactness and flexibility, which have potential applications in fiber-based OAM generation and conversion systems.

## 4. Conclusion

In conclusion, we proposed a simple and flexible microstructure optical fiber to convert LP_{01} mode directly to orbital angular momentum mode. The proposed microstructure optical fiber is composed of two high refractive index rings and two axisymmetric rods between the inner and outer rings. The LP_{01} mode in inner ring can be coupled to higher order modes in outer ring under certain conditions, and the phase difference of the even and odd higher order eigenmodes can accumulate to π/2 during the coupling process, which is defined as “phase-shifted” conversion. We build a theoretical model and systematically analyze the phase-shifted coupling mechanism and illustrate the design principle and optimization method. The effective refractive indices for different resonant eigenmodes and the corresponding supermodes are investigated, and the intensities and phases of the complete superposition mode fields for the involved four supermodes created by the resonance between HE_{11} and higher order modes at different wavelengths and propagating lengths are investigated and exhibited in this paper. We also demonstrated that OAM mode can be generated from π/2-phase-shifted linear combinations of supermodes, which could provide a new approach to generate OAM states. The proposed microstructure fiber is compact, and the OAM mode conversion method is simple, which are expected to be applied in fiber-based OAM systems.

## Funding

National Natural Science Foundation of China (11704283, 11804250, U1509207); Natural Science Foundation of Tianjin City (19JCQNJC01500, 18JCQNJC71300); National Key Research and Development Program of China (2018YFB1305200).

## Disclosures

The authors declare no conflicts of interest.

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