## Abstract

We propose a helically twisted pig-nose-shaped core microstructured optical fiber (HPC-MOF) for orbital angular momentum (OAM) mode generation. It comprises seven air-hole rings hexagonally arranged with two air holes and one air-hole ring replaced, forming two cores in a line 3 µm from the fiber center and one ring-shaped core. The fiber is helically twisted along the rotation axis. In this fiber, supermodes in inner dual-core can be coupled to high-order modes in outer ring-core, yielding OAM ring-shaped modes at different certain wavelengths, and various OAM modes at different twist rates were investigated in this paper. We demonstrate the distinct coupling differences of symmetric and antisymmetric supermodes in inner dual-core when the supermode coupled to ring-core mode. A modal matching rule is presented to characterize the coupling differences, which is suitable for describing supermode coupling characteristics in HPC-MOFs. Compared to conventional methods, these properties indicate that the fiber can generate higher-order OAM modes and more easily integrate into all-fiber optical communication systems, with potential in OAM generators, light-controlling devices, and integrated optics applications.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

With the rapid development of optical communications, conventional optical fibers can no longer meet the high channel capacity and transmission speed demands. Space division multiplexing (SDM) has great potential in future optical communications. Optical light beams carrying orbital angular momentum may play an important role in SDM owing to their theoretically infinite topological charge. Presently, free-space coupling methods such as cylindrical-lens mode converters [1], spatial light modulators [2], integrated silicon devices [3], and micrometer-scale metamaterials [4] are typically used to generate OAM modes. However, the coupling loss from free-space OAM generators into optical fibers is a significant issue during application. In recent years, various all-fiber OAM generators have been developed and reported using single-mode fibers [5], few-mode fibers [6–12], photonic crystal fibers (PCFs) [13,14], and other types of special fibers [15–21]. Among these methods, helically twisted microstructured fibers are considered to be a convenient and promising approach. Wong et al. reported the excitation of OAM modes in helically twisted photonic crystal fibers, providing a new perspective on generating OAM modes [13]. Xi et al. demonstrated robust transmission of OAM modes of order +1 and −1 in twisted PCFs with a three-bladed Y-shaped core [14]. Fu et al. experimentally verified the generation of +5 and +6 order OAM using a helically twisted PCF [22]. Compared to other in-fiber OAM generators, twisted microstructured optical fibers have the advantage of generating rich OAM modes owing to their structural parameters. Higher-order OAM modes will support higher data transmission capacity in an all-fiber optical communication system. However, to the best of our knowledge, no higher-order OAM mode generators based on helically twisted microstructured fibers have been reported.

In this paper, a novel high-order OAM generator based on a helically twisted pig-nose-shaped core microstructured optical fiber (HPC-MOF) is proposed and investigated numerically. The proposed fiber supports four supermodes in inner dual-core, two pairs of circular, namely, left-circular (LC) and right-circular (RC), polarization modes, each contains one symmetric (even) and one antisymmetric (odd) supermodes. The supermodes can be denoted as LC-even, LC-odd, RC-even and RC-odd. Based on the coupled-mode theory in a helically twisted microstructured fiber, the supermodes in inner dual-core can be coupled to the higher-order modes in outer ring-core that carry OAM within a certain wavelength range, yielding a series of resonant peaks in the loss spectra. We demonstrate the significant coupling differences between the supermodes, and the loss spectra for different supermodes at various twist rates are investigated. The phase distributions and mode profiles as well as the coupling lengths of OAM modes corresponding to the resonant peaks in outer ring-core at twist rate *α *= 3141.59 rad/m are also investigated and exhibited in this paper. Based on the analysis of the above results, we present a modal matching rule for describing the supermode couplings in helically twisted microstructured optical fibers, which could be considered important for understanding supermode couplings in the helical fibers. Remarkably, these properties indicate that the generation of OAM mode in the proposed fiber can be more flexible and tunable, providing a much more convenient approach for OAMs generation.

## 2. Structure and simulation method

The cross-section of the proposed fiber and the sketch of fiber cores are shown in Fig. 1(a) and (b), respectively. It consists of hexagonally arranged air holes with diameter *d* = 1.2 µm and lattice pitch Λ_{L} = 3 µm. There are seven air hole rings. Two air holes and the fourth hexagonally arranged air hole ring are missing to form a symmetric dual core and a ring-shaped core, giving it the appearance of a pig nose. The fiber is twisted along the propagation axis with a twist rate well defined by *α *= 2π/Λ_{H}, where Λ_{H} is the helix pitch and the helical fibers can be fabricated by preform spinning during the fiber drawing process. The refractive index of silica glass was determined by the Sellmeier equation [23].

The modal characteristics of the HPC-MOF were obtained using a full vector finite-element method (FVFEM) combined with transformation optics formalism. It is suggested that the calculations of HPC-MOFs applied in a helicoidal coordinate system, rather than Cartesian space, would be more beneficial [24–26], while the proposed fiber is invariant with respect to one of the helicoidal coordinates (*ξ*_{1}, *ξ*_{2}, *ξ*_{3}). The transformation relationship between the Cartesian and helicoidal coordinate systems is given by

The materials, which are isotropic and homogeneous in Cartesian coordinates, can be transformed into equivalent materials using the following relation:

*T*is the metric tensor, and its inverse matrix is given by

As the above equation shows, the material permittivity (ɛ) and permeability (µ) tensors are independent of the axial coordinate *z* = ξ_{3} through all the coordinate changes. Moreover, a twisted perfectly matched layer (PML) surrounding the fiber is built to absorb the scattering light wave and estimate the confinement loss of propagating modes [26]. It is worth mentioning that such an approach makes the calculations more convenient.

## 3. Results and discussion

First, mode profiles and transverse electric fields of the four dual-core supermodes in the HPC-MOF with a twist rate α = 3141.59 rad/m are illustrated in Fig. 2. We have also calculated the effective refractive indices and loss spectra of the supermodes, as shown in Fig. 3(a) and Fig. 4. The modal effective refractive index in HPC-MOFs can be approximated by the following equation [23]:

where*n*is the modal effective refractive index of the untwisted fiber,

_{eff}*λ*is the wavelength and

*J*represents the angular momentum of the mode. As shown in Fig. 2, the four supermodes exhibit distinct differences. They are left and right circular polarization in pairs, respectively. According to the symmetry of modal electric fields, the supermodes can be divided into symmetric (even) and antisymmetric (odd) supermodes. The even supermodes can be seen as the lowest order modes and the odd supermodes can be seen as higher-order modes, which can be denoted as zero and first-order supermodes, respectively [27]. The phase-matching condition for resonant coupling between the dual-core and ring-core modes is given by [18,28]: where

*n*and

_{dc}*n*represent the effective refractive indices of the dual-core and ring-core modes, respectively. According to the coupled-mode theory in helically twisted photonic crystal fibers (PCFs) [29], core modes can be coupled to cladding modes, yielding a series of resonant peaks under certain conditions. There are two kinds of resonant couplings between core and cladding modes, one is the same handedness coupling, the other is the opposite handedness coupling [28]. For the same handedness coupling, it means that the core mode is coupled to the cladding mode with same circular polarization rotation whereas the opposite handedness coupling means the core mode is coupled to the cladding mode with opposite circular polarization rotation. Analogously, in HPC-MOFs, supermodes in inner dual-core can be coupled to high-order modes in outer ring-core at certain wavelengths, and both the same and opposite handedness couplings occurred in the fiber. To explain the coupling characteristics in details, we choose the LC-even supermodes at twist rate

_{rc}*α*= 3141.59 rad/m as an example illustrated in Fig. 3. It can be seen that two resonant peaks appear to the LC-even supermode at the wavelength range of 1000–2000nm. The same handedness coupling between LC-even supermode and

*HE*ring-core mode yields the first resonant peak at 1084 nm while the opposite handedness coupling between LC-even supermode and

_{11 1}^{+}*EH*ring-core mode yields the second resonant peak at 1348 nm. Note that + sign of the mode represents the same rotation direction as the LC polarization and the fiber twist direction while – sign represents the opposite. Phase distributions show that the

_{71}^{-}*HE*and

_{11 1}^{+}*EH*can carry OAM order of

_{71}^{-}*l*=+10 and +8, respectively, demonstrating a great performance of HPC-MOF for OAM mode generation.

Effective refractive indices of the coupling modes and loss spectra of four dual-core supermodes are presented in Fig. 4. Effective refractive indices of the four supermodes separate gradually with the increasing wavelength and the splitting of the refractive index curves exhibit a high circular birefringence induced by the helically twisted structure [30]. The resonant couplings between the dual-core supermodes and the ring-core modes manifest themselves as sharp maxima in the loss characteristics shown in Fig. 4(b).

There are ten resonant peaks in total that occur in the wavelength range of 1000–2000nm. In a general ring-core fiber, the OAM modes can be formed by coherently superimposing two vector modes in the fiber, and their superposition law is given by the following formula [21]:

*l*represents the order of the OAM mode, namely the topological charge,

*m*is the number of the radial order of the mode, and ± expresses the direction of rotation. Specifically, + is the same direction as the LC polarization and the fiber twist direction. However, in a helically twisted ring-core fiber, owing to the helical channel induction of the fiber, the light energy in the ring-core is forced to travel along the helical path, so that part of the light momentum propagating in the axial direction in the original core is coupled to the transverse field component, resulting in the formation of angular momentum flow. The formed angular momentum energy flow along the azimuth angle leads to the generation of discrete OAM modes. Moreover, high-order even (odd) modes in the ring-core region would change into left (right) circular polarization modes based on the coupled-mode theory in helically twisted fibers [21,29]. Thus, the eigenstates of HPC-MOF contain OAM modes. Vector ring-core modes which carry a helical wavefront phase characterized by 2

*πl*in HPC-MOFs can be more directly denoted by the following equation:

The resonant peaks are marked in alphabetical order in Fig. 4(b). Noting that there is a small peak in Fig. 4(b) unmarked, since the couplings are weak at those peaks. There are five different topological charge OAM modes (*l *= −5, +8, +9, +10, +11) excited in the wavelength range of 1000–2000nm. Mode profiles and phase distributions of the corresponding mode are illustrated in Fig. 5. The excited OAM modes above are located at different wavelengths, which indicates that rich OAM spectra can be obtained in HPC-MOFs.

Coupling lengths *L _{c}* of the ten resonant peaks are also calculated by using the following equation [17]:

Where *β*_{1} and *β*_{2} are propagation constants of the two supermodes formed by the inner dual-core and the outer ring-core modes at the coupling wavelength, respectively. As presented in Table 1, coupling lengths show obvious differences in each generated OAM ring-core mode, which indicates that OAMs generation in the proposed fiber can be more flexible.

In-depth analysis shows that for the same even or odd dual-core supermodes, they can be coupled to different ring-core modes that carry the same OAM order. For example, peaks *i* and *j* correspond to the *EH _{91}^{-}* and

*HE*ring-core modes carrying the same OAM. In addition, they can be coupled to the same ring-core modes at different wavelengths. The peaks

_{11 1}^{+}*j*at 1084 nm and

*g*at 1296 nm correspond to the

*HE*ring-core mode and the peaks

_{11 1}^{+}*e*at 1415 nm and

*b*at 1712nm correspond to the

*HE*ring-core mode. Both the LC and RC polarization supermodes can be coupled to both the same handedness and opposite handedness polarization ring-core modes that carry OAM. However, many high-order ring-core modes and RC polarization supermodes have intersections in their effective refractive indices as presented in Figs. 4(c) and 4(d). Thus, more opposite handedness couplings occur for the RC polarization supermodes, and more peaks appear in the RC polarization supermodes; for example, peaks

_{10 1}^{+}*a*,

*b*,

*c*,

*f*,

*g*, and

*h*are the opposite couplings, but only peak

*f*belongs to the LC polarization supermodes. In addition to the significant differences in coupling occurring to the LC and RC polarization supermodes, coupling also occurs to the even and odd supermodes and exhibits impressive properties. For the same polarization supermodes, we found that all the even (odd) supermodes in inner dual-core are only coupled to the even (odd)-order ring-core modes. Therefore, the differences of coupling to the ring-core modes between the even and the odd supermodes can be expressed by the following equation:

*l*(

_{j}*l*) represents the order of

_{k}*j*(

*k*)th ring-core mode carrying even (odd) order OAM,

*l*(

_{even}*l*) represents the order of even (odd) supermodes in inner dual-core and

_{odd}*p*,

*q*are integers. It should be noted that mode order can be characterized by the azimuthal harmonic order [28]. In other words, the even supermodes can be only coupled to HE

*(EH*

_{2p+1,m}*) whereas the odd supermodes can be only coupled to HE*

_{2p−1,m}*(EH*

_{2q+2,m}*) ring-core modes, which can be summed up as a modal matching rule for the supermode couplings. The coupling characteristic is shown in Fig. 6. Specifically, both the same handedness and opposite handedness couplings between dual-core supermodes and ring-core modes show agreement with the equation above.*

_{2q,m}Next, to study the effect of different twist rate on couplings between dual-core supermodes and ring-core modes occurred in HPC-MOFs, we added two sets of twist rates and the loss spectra are investigated for comparison. The resonant peaks are marked in alphabetical order in Fig. 7, and the label of the mode is changed to OAM mode using Eq. (7).

Three groups of dual-core supermode loss at different twist rates are calculated so far, and the results are presented in Fig. 7. It can be seen that more resonant peaks occur with increasing twist rate for all supermodes in inner dual-core. Specifically, more opposite handedness couplings occur in RC polarization supermodes, which means that more OAM modes are excited at different wavelengths, and the higher-order OAM modes tend to rise in the shorter wavelength range. Apparently, the RC polarization supermodes still have more peaks than the LC polarization supermodes, and also tend to be coupled to the higher-order OAM modes. The two highest order OAM ring-core modes occur at the twist rate *α* = 4398.23 rad/m, which are excited by the RC-even and RC-odd supermodes in inner dual-core, corresponding to peak *g* at 1010 nm and 1002 nm as shown in Figs. 7(c) and 7(d), respectively. They carry an OAM order as high as +14 and +13, and their mode profiles and phase distributions are shown in Fig. 8. Meanwhile, it is not difficult to find that even (odd) dual-core supermodes at different twist rates are also coupled to the even (odd)-order ring-core modes carrying corresponding order of OAM in Fig. 7, which means all the resonant peaks show good agreement with Eq. (9). In addition, with increasing twist rate, the loss spectra yield a distinct redshift, which indicates that the HPC-MOFs could have more flexible and tunable properties.

## 4. Conclusion

In conclusion, a helically twisted pig-nose-shaped microstructured optical fiber (HPC-MOF) is proposed in this study, and its OAM generation characteristics were investigated numerically. The HPC-MOF supports four supermodes via the dual-core and rich OAM modes by the ring-core. It is demonstrated that the dual-core supermodes can be coupled to the ring-core modes at different wavelengths in the range of 1000–2000nm, resulting in the generation of OAM modes. The possibility of coupling to higher-order OAM modes tends to rise at shorter wavelengths, which demonstrates the excellent performance of the HPC-MOF as a high-order OAM mode generator. Specifically, for the same even or odd supermodes, more resonant peaks occur in the RC polarization than the LC polarization supermodes, and higher-OAM mode coupling also tends to rise at the RC polarization supermodes. Simultaneously, for the same polarization supermodes, all the even (odd) dual-core supermodes can be only coupled to the even (odd) -order ring-core mode carrying corresponding order of OAM, which makes the generation of OAM modes more flexible. We presented a modal matching rule for the supermode couplings occurred in the helical fibers, and found that the results showed good agreement with the rule. Furthermore, we investigated three groups of twist rates for the loss spectra of the dual-core supermodes. The results show that with increasing twist rate, more resonant peaks occur, yielding more and higher-order OAM modes. Additionally, a redshift of the spectra is observed within the range of 1000–2000nm. The extraordinary properties and excellent performance of HPC-MOFs indicate that they can be used for OAM generator, filtering, and mode conversion devices, playing an important role in future all-fiber optical communications, mode control, and integrated photonics.

## Funding

Special Fund Project for Science and Technology Innovation Strategy of Guangdong Province (2020KTSCX032); Guangzhou Municipal Science and Technology Project (2019050001); National Natural Science Foundation of China (61935007, 61935010).

## Disclosures

The authors declare no conflicts of interest.

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