## Abstract

Using orbital angular momentum (OAM) as a spatial information channel attracts a lot of attention due to its infinite multiplexing capability. The research on OAM carrying fibers is intensively studied and ring-core fibers are promising candidates for them. At the same time, generating OAM modes in those fibers are also important topic. Here, the evolutions of polarization states including OAM of light in helically twisted ring-core photonic crystal fibers (PCFs) are investigated for generating OAM states. The degeneracy of some of the mode sets is spontaneously broken, and the birefringence cause a geometric phase (GP) in the twisted PCFs while preserving the vectorial nature of the modes. It is demonstrated that an arbitrary polarization and OAM state can be generated by using the GP in uniformly twisted and twisted PCFs with periodical inversion. In the ring-core PCF presented in this paper, the degeneracy of HE_{31} and EH_{31} mode sets are broken, and OAM light with the topological charge of 2 and 4 can be generated. The wavelength dependence is very small compared with that of OAM generators based on long-period gratings (LPGs). Furthermore, by properly setting the period of the inversion, OAM light with different topological charges can be simultaneously generated with the same fiber structure. These results indicate that the proposed approach is effective for the generation of OAM modes with the conventional fiber modes.

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

## 1. Introduction

Recently, helically twisted PCFs have attracted a lot of attentions for manipulating the spatial state of the light. Since the first report that connects the coupling between core and cladding modes with OAM [1], a lot of work have been done, such as an optical activity and a circular dichroism [2,3], higher-order OAM mode generation [4] at a wavelength near the dip, and a circular polarized (CP) mode filtering in off-axis core twisted PCFs [5]. The nature of the “twisted” modes have also been intensively studied such as, role of symmetry [6] and scaling effect [7]. These studies are for PCFs with rotational symmetry, and basically, there are no birefringence originated from the cross-sectional structure of the fiber (linear birefringence).

In [8,9], the polarization effects in twisted birefringent PCFs (TB-PCFs) have been investigated and shown that various polarization states can be generated. In [9], it was pointed out that the polarization evolution in the TB-PCFs can be explained by a GP [10], and by accumulating the GP based on periodical inversion of the twisting, arbitrary polarization state on a Poincare sphere can be generated for the fundamental mode of TB-PCFs. For higher-order modes, which is more important in terms of OAM applications, the preliminary studies were conducted and the possibility of generating OAM modes in TB-PCFs was discussed. However, in [9], the generation of OAM modes was only examined by observing the vortex of far-field interference patterns (FFIPs) from TB-PCF mixed with the fundamental mode and the quality of the OAM mode was not clearly evaluated. Also, when using periodically inverted twisting structure, the modal power calculated by BPM for helical waveguides [11–13] was oscillated, showing the unstable behavior of the analysis method (this is also shown in this paper, later). Furthermore, the investigated rotation angles for the inversion were only 45 and 90 degrees, considering the symmetry of the modes, and optimum angles for generating OAM modes were not discussed. Therefore, for discussing efficient OAM generation in the twisted PCFs, more stable analysis method as well as proper evaluation method of OAM mode quality is desired.

In TB-PCFs, since the linear birefringence is introduced by breaking the rotational symmetry of the cross-sectional structure, the GP can be easily generated. However, the modes are similar to linearly polarized (LP) modes rather than fiber vector modes (HE and EH modes). Because the OAM modes are traditionally defined as the sum of HE or EH modes [14], it is useful if there is a fiber having vector modes with birefringence for the generation of OAM modes based on GP. Ring core PCFs with circular cladding geometry [15,16] have been intensively studied for OAM fiber and their vector mode sets exhibit good characteristics for OAM mode transmission.

In this paper, the evolutions of polarization states including OAM in helically twisted ring-core PCFs with triangular lattice cladding are investigated. The degeneracy of some of the mode sets is spontaneously broken and the linear birefringence cause the GP in the twisted PCFs, while preserving the vectorial nature of the modes. It is demonstrated that an arbitrary polarization and OAM state can be generated by using the GP in uniformly twisted PCFs and twisted PCFs with periodical inversion. To simply analyze the modal evolution in the twisted fibers, an EPM based on modal basis transformation is newly formulated to avoid unstable behavior associated with the helical BPM. The results obtained by the EPM agree very well with those obtained by the BPM, showing the usefulness of the method. To visualize the polarization state of the higher-order mode group in PCFs, a higher-order Stokes parameter (or higher-order Poincare sphere (HO-PS)) [17,18] is introduced. By using HO Stokes parameters, the polarization evolution of the HO mode group can be tracked, and the quality of the OAM mode can be easily evaluated. In the ring-core PCF presented in this paper, the degeneracy of HE_{31} and EH_{31} mode sets are broken with preserving their vectorial nature, and OAM light with the topological charge of 2 and 4 can be generated by using uniform twisting or periodically inverted twisting. The wavelength dependence is very small compared with that of OAM generators based on helical LPGs [19–21]. Furthermore, for the twisted PCFs with periodical inversion, by properly setting the period of the inversion, OAM light with different topological charges can be simultaneously generated with the same fiber structure. These results indicate that the proposed approach is effective for the generation of OAM modes with the conventional fiber modes, and the ring-core geometry is advantageous for connecting with ring-core OAM fibers [15,16].

## 2. Theoretical methods

#### 2.1 Transformation optics formalism

In the theoretical analysis of helically twisted PCFs, guided mode analysis based on the finite-element method (FEM) formulated for helical systems [22] have been used. In this FEM, the concept of transformation optics (TO) is used for the formulation, namely, the helically twisted structure is replaced with a straight waveguide with equivalent relative permittivity, **ε**_{r}, and permeability, **μ**_{r}, tensors. For the twisted waveguides, left CP and right CP (LCP and RCP) modes are obtained from this guided mode analysis.

We consider a helically twisted ring-core PCF shown in Fig. 1. The origin is at the center of the fiber. The cladding is composed of hexagonally arranged air holes with diameter *d* and pitch Λ. There are six rings of air holes, and air-holes of first ring are replaced with SiO_{2} and an air-hole with the diameter of *d*_{c} is placed at the center, resulting in ring-type core. The computational window is a circle, and the outer region is a cylindrical perfectly matched layer (CPML) region with a thickness *d*_{PML}. The fiber is helically twisted along the *z* direction with a twisting rate of α [rad/m]. The propagation direction is *z*, and *xy* is the transverse plane. The refractive indexes of silica and air are taken as 1.45 and 1.0.

By formulating the problem with the TO, in the modeling coordinate, the relative permittivity **ε**_{r} and permeability **μ**_{r}, are modified due to the coordinate transformation, and is given by

Here, we assume an ordinary optical isotropic medium with a refractive index of *n* and a relative permeability of 1 in the original coordinate. ** T** is a metric tensor and is given by

Here, $\varphi=\theta+\alpha z$, where $\theta $ and $\varphi $ are the polar angle in the original and the twisted coordinates, respectively, and *r* is the distance from the origin. The parameter s is related to CPML.Detailed formulation can be found in [11,22]. By solving vector wave equation under the permittivity and permeability given by (1) and (2), one can obtain the propagation constant and the mode field of the twisted modes.

#### 2.2 Eigenmode propagation method based on the basis transformation

Although a lot of information can be extracted from the twisted mode, it is difficult to treat longitudinal evolution of the spatial state of the light in the waveguide with arbitrary input conditions. For example, in the most experiment, the input light is a guided mode of non-twisted fiber and it is not twisted mode. Therefore, a BPM analysis [11–13] is more suitable for investigating the longitudinal behavior of spatial state of the light in the twisted waveguides. It was reported that calculated optical spectra of twisted fibers are in excellent agreement with the measured results [11–13,23]. One drawback of the BPM is its computational cost. If the length of the fiber is long, more computational time is required. Furthermore, as shown later, numerically unstable behavior is seen for the analysis of periodically inverted twisting structure. Therefore, alternative analysis method for the beam propagation analysis is useful.

Here, we propose an EPM based on the basis transformation. As shown later, in some cases, the non-twisted modes can be regarded as the superposition of twisted modes. Since the twisted modes are diagonalized in the twisted waveguides, it is easy to calculate their propagation characteristics.

We consider *N*-mode case: there are *N* twisted and non-twisted guided modes. In the twisted waveguide, the complex modal amplitude vector ** a** after the propagation distance of

*z*is given by

**is the transmission matrix and the subscript “**

*T*_{T}” denotes the twisted system. β

_{i}is the propagation constant of the i-th twisted guided mode. If the twisted modes can be regarded as the superposition of the non-twisted guided modes, they can be expressed as

_{C}” denotes the non-twisted (Cartesian) system.

*T*_{TC}is a translation matrix between twisted and non-twisted modes. <

*T*

_{i}|

*C*

_{j}> is a modal overlap between i-th twisted mode,

*T*

_{i}, and j-th non-twisted mode,

*C*

_{j}. This is analogous to the Clebsch-Gordan coefficient in quantum mechanics [24], which is used for the basis transformation of angular momentum of holes in semiconductors. By inserting (6) into (4), output Cartesian modal amplitude for Cartesian mode input is given by

The evaluation of (9) is considerably easy and fast. It should be noted that the EPM can be used only when the non-twisted modes can be regarded as the superposition of twisted modes. If there are continuous radiation modes, which is difficult to evaluate by the guided mode analysis, the method should not be used, and the BPM analysis is necessary. It should be noted that the EPM can be viewed as a simpler version of mode matching method [25]. The formulated EPM is simpler since we do not treat backward reflection.

#### 2.3 Higher-order Stokes parameters

To visualize and evaluate the various polarization states for HO mode group of the fibers, we employ HO Stokes parameters and HO-PS [17]. In this paper, we use a traditional definition for OAM modes [14], namely,

*p*is the radial order of the mode.

*l*and σ are topological charge and spin angular momentum, respectively. The superscript e (o) denotes even (odd) HE or EH modes. For the fiber modes with the azimuthal mode order larger than 1, there are four modes in the mode group (two HE and two EH modes). Therefore, to express the polarization states of the mode group, two Poincare spheres are necessary for sgn(

*l*) = sgn(σ) and sgn(

*l*) ≠ sgn(σ). The points on this HO-PS is expressed by HO Stokes parameters and given by

*l*) = sgn(σ) and sgn(

*l*) ≠ sgn(σ). Details of these symbols and equations are summarized in Appendix. The poles of HO-PS (

*S*

_{3}= ±1) correspond to the OAM modes given by (10). By using the HO Stokes parameters, the quality of OAM modes can be easily evaluated.

## 3. Modal characteristics of twisted ring-core PCFs

#### 3.1 Basic characteristics

We consider a ring-core PCF shown in Fig. 1. *d* = 1.6 μm and Λ = 2.0 μm. The wavelength of the light is λ = 1.55 μm, unless otherwise noted. Here, the basic modal characteristics of the non-twisted fiber is investigated. Table 1 shows the effective refractive indexes of the ring-core PCF with *d*_{c} = 2.4 μm calculated by full-vector FEM. The mode No is sorted from the mode with larger effective index, *n*_{eff}. The mode label is corresponding vector mode expression (HE, EH, TE, and TM). Other than HE_{11}, four modes make one mode group. In the mode group, there are two sets of modes. For example, in the mode group 3, there are two HE_{31} and two EH_{11} modes. We call these two HE_{31} modes as a mode set. For a mode set, the mode with larger (smaller) refractive index is denoted as HE_{31-1} (HE_{31-2}). It should be noted that for the mode group 4, since the field distributions are like LP modes rather than vector modes, we used LP notations for the modes. The confinement losses of these modes are sufficiently small. There are 28 guided modes in this ring-core PCF (the imaginary part of *n*_{eff} < 10^{−8}).

Since the fiber has a rotational symmetry, the fiber is basically non-birefringent. Table 2 shows the effective refractive index difference, Δ*n*_{eff}, of each mode set. As shown in the table, the mode set birefringence is basically zero, however, it is on the order of 10^{−3} for the mode sets of TE-TM, HE_{31}, and EH_{31}. We call these birefringences as “spontaneously broken degeneracy” (SBD) in this paper. It is well known that TE and TM modes are decoupled in the fiber modes. For HE_{31} and EH_{31} modes, the cladding three-fold rotational symmetry causes the birefringence. For the application of OAM fiber for long-distance transmission, the SBD is not preferable for stable OAM mode propagation, and therefore, ring-core PCFs with circular arrangement cladding were proposed [15,16]. However, in this paper, the SBD is used for the generation of OAM modes by twisting the fiber. The left panel of Fig. 2 shows *S*_{z} (*z*-component of the Poynting vector) and vector field distributions of HE_{31} and EH_{31} mode sets. The right panel of Fig. 2 shows Δ*n*_{eff} of HE_{31} and EH_{31} mode sets as a function of *d*_{c}. The mode set birefringence on the order of 10^{−3} can be obtained. In this paper, *d*_{c} is set to 2.4 μm, hereafter.

#### 3.2 Twisted modes of ring-core PCFs

Figure 3 shows *n*_{eff} of the twisted ring-core PCF calculated by full-vector FEM for helical waveguides [22]. *n*_{eff} of the mode set is split with the twisting and shows circular birefringence. Since each twisted mode is a mixture of non-twisted modes, it is difficult to label the mode, like HE or EH. Therefore, we just refer the twisted mode with the mode No as in Table 1. For example, the mode No of the twisted modes in the mode group 3 are 7, 8, 9, and 10. To investigate the modal mixture of the twisted modes, we calculate modal overlap coefficient < *T*_{i}|*C*_{j}> given by (8). For the mode group 3 and 5 with α = 3.14 rad/mm,

From these results, following conclusion can be extracted. First, the twisted mode can be viewed as the superposition of the mode set of non-twisted modes in this case. For example, twisted mode 7 are the mixture of non-twisted HE_{31} mode set. Second, if the mode set birefringence is large, the power ratio of the modes is not 50:50. If the mode set is degenerate, the power ratio is 50:50. Figure 4 shows the modal overlap power as a function of α for (left) HE_{31} and (right) EH_{31} mode sets. The modal mixture is increased for large twisting rate. There results indicate that if the non-twisted vector mode is launched in the twisted fiber, the phase retardation is caused for the mode set via geometric phase [9]. The retardation leads to non-zero *S*_{3} component of the HO Stokes parameters, resulting in the generation of OAM modes. If there is no mode-set birefringence, *S*_{3} component is not generated, and the polarization state go round and round on the equator of HO-PS. It should be noted that too large twisting rate make the higher-order modes leaky, and we use α = 3.14 rad/mm, hereafter.

## 4. Arbitrary polarization state generation based on the geometric phase

#### 4.1 Uniformly twisted ring-core PCFs

Here, the polarization evolution in the uniformly twisted ring-core PCFs are analyzed. The left panel of Fig. 5 shows modal powers in twisted ring-core PCF with α = 3.14 rad/mm calculated by the EPM. The input mode is HE_{31-1}. The modal power conversion between HE_{31-1} and HE_{31-2} modes can be seen and no mode conversions occur for EH_{11} mode set. This is because as shown in (12a), HE_{31-1} mode can be decomposed into the twisted mode 7 and 8, in which no EH_{11} mode is included. The right panel of Fig. 5 shows the same one calculated by helical BPM. The modal conversion behavior agrees very well with that calculated by EPM. For the BPM, slightly oscillating behavior can be seen in Poynting vector of whole cross-section of the fiber (black dashed line). This is prominent for the periodically inverted fibers, as shown later. The left panel of Fig. 6 shows the HO Stokes parameters for HE_{31-1} mode input. A periodical CP component (*S*_{3}) cab be seen. This result is consistent to the fundamental mode case of TB-PCF [9]. The right panel of Fig. 6 shows the *S*_{z} and phase of the electric field at the propagation distance of 150 μm (red dots in the left panel), where *S*_{3} is maximum. A helical phase distribution with the topological charge of 2 is clearly seen. The left panel of Fig. 7 shows modal powers in twisted ring-core PCF with α = 3.14 rad/mm calculated by the EPM for the EH_{31-1} mode input. As in Fig. 5, the modal power conversion between EH_{31-1} and EH_{31-2} modes can be seen, and the results obtained by EPM agree well with those of BPM (the right panel of Fig. 7). The left panel of Fig. 8 shows the HO Stokes parameters and similar results are obtained. The right panel of Fig. 8 shows the *S*_{z} and phase of the electric field at the propagation distance of 210 μm (red dots in the left panel). A helical phase distribution with the topological charge of 4 is clearly seen. However, the value of *S*_{3} is -0.63 and the quality of the OAM beam is not sufficient.

#### 4.2 Twisted ring-core PCFs with periodical inversion

It was found that in [9], if the direction of twisting is periodically inverted at proper distance, at which excited *S*_{3} is maximum, the GP is accumulated, and perfect LCP or RCP can be generated for the fundamental mode of TB-PCF. The concept is expanded to, general, HO mode case. The inset in Fig. 9 shows the schematic of the fiber with periodically inversion. Λ_{inv} is the inversion period. First, the fiber is twisted in α > 0 direction, and it is inverted (α < 0) after Λ_{inv} propagation. In [9], the inversion period corresponded to 45- or 90-degree rotation, considering the symmetry of the mode. However, as shown in Fig. 6 and 8, the period for maximum *S*_{3} excitation for HO mode case seems to be insensitive to the symmetry (27 and 38 degrees for HE_{31} and EH_{31} modes). Therefore, the distance for the maximum *S*_{3} excitation cannot be inferred by the symmetry of the structure and the propagation analysis is necessary.

It should be noted that although the fabrication of this structure is challenging at this stage, multiple splicing of inverted sections may be one of the fabrication methods. Or, recently reported 3D printing technology for the fiber-based devices [26], where a twisted PCF is really fabricated, may also be a candidate.

Based on the results shown in Fig. 6 and 8, Λ_{inv} is set to 150 and 210 μm for HE_{31} and EH_{31} mode inputs. When calculating the propagation characteristics of the periodically inverted structure with *N*_{p} sets of inverted and non-inverted sections by the EPM, Eq. (9) is modified as

*z*= 2

*N*

_{p}Λ

_{inv}.

Figure 9 shows the modal power calculated by (left) EPM and (right) BPM for HE_{31-1} mode input. Figure 10 shows the same one for EH_{31} mode input. For short propagation distance, both EPM and BPM results agree well. However, the modal powers obtained by BPM are highly oscillated for longer distance propagation, while the powers obtained by EPM are very stable, showing the usefulness of EPM for periodically inverted fibers. The left panel of Fig. 11 shows HO Stokes parameters for HE_{31-1} mode input. Due to the accumulation of the GP, *S*_{3} is also accumulated and takes the value from -1 to 1. The center and right panel of Fig. 11 shows the BPM-calculated *S*_{z} and phase of the electric field at the propagation distance of 200 and 600 μm (red dots in the left panel). The helical phase front with the opposite helicity can be seen for *z* = 200 and 600 μm, corresponding to *l* = ±2. The left panel of Fig. 12 shows the same one for EH_{31} mode input. Accumulated *S*_{3} excitation can also be seen. The right panel shows the BPM-calculated *S*_{z} and phase of the electric field at the propagation distance of 400 μm (red dot in the left panel). A helical phase distribution with the topological charge of 4 is seen, and unlike the case of Fig. 8, the value of *S*_{3} is -1 and the quality of the OAM beam is improved.

Figures 13 show HO-PS plot of the HO Stokes parameters shown in Figs. 6, 8, 12, and 13. As stated in 2.3, two spheres are necessary to express the polarization state of HO modes [17]. However, there are almost no modal powers for another mode set, we only plot one of the spheres containing majority of the modal power. In case of uniform twisting, the trajectory is a circle as in the single mode case [9]. For periodically inverted structure, the trajectory is very complex and covers whole Poincare sphere, showing the possibility for generating arbitrary polarization state for HO mode.

In the above example, Λ_{inv} is set to the distance, at which *S*_{3} excitation is maximum in case of uniformly twisted fibers. Of course, the distance can be changed to arbitrary values. The left panel of Fig. 14 shows HO Stokes parameters for HE_{31-1} mode input for Λ_{inv} = 210 μm, which is the same value for EH_{31} mode case. Although the polarization evolution is very complex compared with that of Fig. 11, there are multiple distances, at which *S*_{3} = ±1. The right panels show BPM-calculated *S*_{z} and phase of the electric field at the propagation distance of 500 μm (red dot in the left panel) for HE_{31} and EH_{31} mode inputs. Helical phase distributions with the topological charges of 2 and 4 can be seen, and *S*_{3} = +1 and -0.94 for HE_{31} and EH_{31} modes. Therefore, almost perfect OAM modes with different topological charges can be generated with the same fiber structure.

#### 4.3 Wavelength dependence

In the conventional OAM generators based on LPG [18–20], the exact phase matching is necessary for efficient mode conversion, leading to strong wavelength dependence. In contrast to LPG, the wavelength dependence of the proposed method is considerably small because the conversion mechanism is totally different. The left and right panels of Fig. 15 show the *S*_{3} of uniformly twisted fiber for HE_{31} and EH_{31} mode input with α = 3.14 rad/mm for various wavelengths. Although for longer distance propagation, the wavelength dependence can be seen, it is almost negligible for short propagation distance. The left panel of Fig. 16 show the *S*_{3} of periodically inverted twisted fiber for HE_{31} mode input with α = 3.14 rad/mm and Λ_{inv} = 150 μm. At the first distances where *S*_{3} ≈ ±1, there almost no wavelength dependence. The right panel of Fig. 16 shows the same one for EH_{31} mode input with Λ_{inv} = 210 μm. Similar characteristics are observed. There results indicate that the proposed method has very large bandwidth compared with that of LPG in terms of OAM mode generation.

#### 4.4 Hole diameter dependence

Here, we show the fabrication tolerance of *S*_{3} excitations. The left panel of Fig. 17 shows the *S*_{3} of uniformly twisted fiber for HE_{31} mode input with α = 3.14 rad/mm for various values of hole diameters. Δ*d* = +0.1 μm means that the diameters of all the holes are increased by 0.1 μm. Similar to the wavelength dependence, the variation is very small for short propagation distance. The right panel shows the same one for periodically inverted twisted fiber with α = 3.14 rad/mm and Λ_{inv} = 150 μm. Again, the dependence is small for short propagation distance.

## 5. Conclusion

The complex polarization evolution of HO modes in twisted ring-core PCFs are analyzed based on simple numerical method (EPM) formulated for this problem. The results obtained by the method agree very well with those obtained by rigorous BPM analysis. The complex polarization behavior is visualized by using HO Stokes parameters and HO-PS, which enables us to evaluate the quality of OAM mode easy. Using the SBD modes of ring-core PCFs, it is possible to generate arbitrary polarization state (including OAM modes) through the GP, existing in uniformly twisted or periodically inverted PCFs. The results indicate that the proposed fiber is useful for generating OAM modes.

## Appendix

## A.1. Circular polarization basis expression of fiber vector modes, higher-order Stokes parameters, and the higher-order Poincare sphere

Here, the formulation of HO Stokes parameters are summarized. The linear combination of the even and odd fiber modes with the azimuthal order of *v* can be expressed by circular polarization basis as

**is the modal electric field,**

*E**f*(

*r*) is the radial modal field distribution function for each basis (the detail can be found in [14]), and φ is the azimuthal angle. The radial order

*p*is omitted for simplicity.

**and**

*L***are LCP and RCP bases. For weakly guiding fiber,**

*R***component can be neglected, and**

*z***or**

*L***component is dominant. Namely, for HE modes and for EH modes. Here,**

*R***and**

*L***(defined for source-point-of-view) are given by**

*R**l*=

*v*-1 for HE modes and

*l*=

*v*+1 for EH modes.

**and**

*x***are modal bases in Cartesian coordinate. Upper (lower) sign corresponds to HE (EH) mode. We redefine above basis for the topological charge**

*y**l*as [17]

Arbitrary state with the topological charge *l*, |ψ_{l}>, can be expressed by the linear combination of these bases as

HO Stokes parameters are defined as (11) from the above modal amplitude of |*R *> and |*L *> bases [17]. The HO Stokes parameters can be mapped on HO-PS and the bases for *S*_{1} are given by

The bases for *S*_{2} are given by

Finally, the bases for *S*_{3} are |*R *> and |*L*>.

## Funding

Japan Society for the Promotion of Science (20K20988).

## Acknowledgement

This work was partly supported by SEI Group CSR Foundation.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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