## Abstract

In this study, we show that transformation optics formalism can be used to rigorously model a wide range of twisted anisotropic fibers, which could only be analyzed using perturbative methods. If the material anisotropy of fibers has an intrinsic origin or is induced by axially or helically symmetric physical factors, then they can be transformed into a form usable in rigorous two-dimensional (2D) modeling. We demonstrate the effectiveness of the proposed approach in 2D modeling of the propagation characteristics of first-order eigenmodes in twisted and spun fibers with high linear birefringence. We derive the equivalent electric permittivity tensors for such fibers in the helical coordinate system and study the evolution of the first-order modes toward vortex modes with increasing twist rate. The obtained results confirm that the proposed method can reveal phenomena that cannot be predicted by analytical approaches.

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

## 1. Introduction

Helical twists provide an additional degree of freedom in shaping the propagation characteristics of optical fibers. If a fiber is twisted at room temperature, it undergoes elastic deformation, which induces shear stress and consequently additional circular birefringence [1]. However, the use of such fibers is limited to relatively small twist rates, as silica fibers twisted at room temperature typically break for a twist period shorter than 1 cm. Twist-induced stress can be greatly reduced in spun fibers fabricated by spinning the preform during the drawing process, where the fiber undergoes inelastic deformation above the glass transition temperature [2]. For such a fabrication method, single-millimeter twist periods can be obtained [3]. Most often, the preform spinning method is used to reduce birefringence, which arises from technological imperfections in nearly cylindrically symmetric fibers [2]. Conversely, spinning the fiber with an offset core forms a helical core, giving rise to many interesting wave phenomena. Helical core fibers and twisted linearly birefringent fibers fabricated by preform spinning show much higher circular birefringence than that in the case of elastic deformation and therefore can be used in current sensors based on the Faraday effect [3,4]. Furthermore, an increase in the loss of higher-order modes caused by the curvature of the helical core leads to single-mode guidance, even for large normalized frequencies [4]. More recently, spun microstructured fibers have been used to preserve the chirality of orbital angular momentum (OAM) modes [5], generate broadband circularly polarized supercontinuum [6] and guide modes in a partially open ring of holes, which can be used in displacement sensors [7]. New wave phenomena related to the twist-induced coupling between the core and cladding modes were observed in fiber structures, in which the helix pitch was reduced to tens of micrometers through the local twisting of the already drawn fiber in a miniature heat zone [8]. Twist-induced coupling in conventional and microstructured fibers can be used in sensors [8–10], filters [8,9,11], couplers [8,9], polarizers [8,9,12], and optical vortex generators [13,14]. The coupling effects in twisted fibers can also be used to obtain single-mode guidance in a fiber with a very large central core surrounded by helical side cores [15] or hollow-core photonic crystal fibers [16]. Furthermore, twisted microstructured fibers can guide modes localized around the symmetry axis even in the absence of a core [17]. The generation of arbitrary polarization and OAM states has also been recently studied in twisted birefringent fibers [18].

The theoretical analysis of wave phenomena present in spun and twisted fibers can be performed using various perturbative approaches [13,14,19–22] or rigorous full-vector modeling methods [23–25]. Rigorous modeling of twisted fibers based on the finite element method (FEM) [23,24] and full-vector ﬁnite-element beam propagation method [25] were used to analyze structures of complex geometry and revealed the existence of wave phenomena that could not be predicted by simplified perturbative approaches [12,26]. The rigorous methods [23–25] are based on transformation optics formalism, in which any change of the coordinate system can be represented as an equivalent change in the permittivity and permeability [23]. If the geometry of a twisted fiber is represented in helicoidal [23,24] or helical polar [27] coordinates, it can be analyzed as a two-dimensional (2D) structure. However, the equivalent two-dimensional transformation of material properties in this coordinate system, which is also necessary for 2D modeling, was derived in [23,24,27] only for isotropic materials. The necessary formulas for the anisotropic case have been proposed in [28,29] and used in [30] to set up the principle of invisibility cloaking. In some cases, a 2D representation of twisted anisotropic materials can be obtained, as for cylindrical perfectly matched layers in helicoidal coordinates [24], but the transformed permittivity of anisotropic materials generally depends on all spatial coordinates. Therefore, simple cases of linearly birefringent fibers twisted at room temperature with additional circular birefringence induced by shear stress were analyzed using only perturbative methods, which already predicted interesting phenomena, such as the robust separation of optical vortex modes [22].

In this study, we use transformation optics formalism to derive a formula that allows for rigorous 2D modeling of a wide range of twisted anisotropic fibers, in which the material anisotropy is axially symmetric or has helical symmetry and rotates with the fiber structure. The first condition is satisfied if the anisotropy is induced by axially symmetrical external factors, such as radial stress, hydrostatic pressure, elastic twist, or axial elongation. The second condition is fulfilled if the fiber birefringence has an intrinsic origin, i.e., if it is related to the fiber itself, as in the case of material anisotropy generated by stress-inducing elements located in the cladding of the twisted fiber, or if it is induced by helical external factors characterized by the same helix period as the fiber, as in the case of helical microbending [31]. Furthermore, we demonstrate the effectiveness of the proposed approach in the rigorous modeling of the propagation characteristics of first-order eigenmodes in twisted fibers with high linear birefringence, which were previously analyzed using perturbative methods [22]. In particular, using FEM, we numerically study the evolution of first-order modes (LP_{11}) versus helix pitch in twisted and spun fibers with linear birefringence induced by stress-applying parts (SAPs) and show that the LP_{11} modes are gradually transformed into vortex modes with increasing twist rate. The obtained results reveal that rigorous modeling predicts new phenomena, even in the case of relatively simple material anisotropy.

## 2. Transformation optics and material anisotropy

Transformation optics formalism is based on the fact that in electromagnetism, a change in the coordinate system from {*x*, *y*, *z*} to {*x*’, *y*’, *z*'} is equivalent to a change in material properties [23,24,27–29]. The permittivity and permeability tensors in the new coordinate system {*x*’, *y*’, *z*'} are given by

**ɛ**and

**µ**represent the material properties in {

*x*,

*y*,

*z*} (usually Cartesian) coordinates and

**J**

^{−1}is the inverse of the Jacobian matrix:

**J**

^{−T}is the transposition of

**J**

^{−1}. Transformation optics can be used in numerical simulations to reduce the dimensionality of some problems by choosing a coordinate system in which the geometry of the waveguide and its material properties become independent of one of the coordinates. In particular, the structure of twisted fibers can be simplified in helicoidal coordinates {

*x*’,

*y*’,

*z*'}, which are related to Cartesian coordinates by [23]

*A = 2π/Λ*is the twist rate expressed in [rad/m], which describes the rotation of the

*x’*and

*y*’ axes of the helicoidal coordinates system around the

*z*axis with respect to the Cartesian

*x*and

*y*axes by the angle −

*Az*, where

*z = Λ*corresponds to the full rotation angle. All lines parallel to the

*z*axis in helical coordinates are helices in the Cartesian coordinate system with a pitch distance equal to

*Λ*, which are left-handed for positive

*A*. The inverse of the Jacobian matrix

**J**

^{−1}used in this case is given by

*R*(

*Az*) is the rotation matrix and det(

**J**

^{−1}) = 1.

If the material properties of the twisted fiber are isotropic and therefore can be represented by scalars *ɛ* and *µ*, then according to Eqs. (1) and (4), the equivalent **ɛ**’ in helicoidal coordinates {*x*’, *y*’, *z*} can be expressed as [23]

In this study, *ɛ* and **ɛ** (used for anisotropic materials) indicate values expressed in Cartesian coordinates, and **ɛ**’ is a corresponding value expressed in helical coordinates. In the following analysis, the same arguments (*x*’, *y*’, *z*) are used for *ɛ*, **ɛ**, and **ɛ**’ only to indicate the position in space. Equation (5) is *z*-independent if *ɛ*(*x*’, *y*’, *z*) can be expressed as *ɛ*(*x*’, *y*’), i.e., if its value in the Cartesian coordinate is a constant at a helix specified by a given *x*’ and *y*’. Such a condition is satisfied if the twist rate *A*, which defines the helicoidal coordinates in Eq. (3), is the same as that for the fiber twist. A similar relation holds for the scalar magnetic permeability *µ*.

For anisotropic fibers, the magnetic permeability remains scalar *µ*, whereas the electric permittivity is represented as a second-rank tensor **ɛ** defined with respect to the orientation of the Cartesian axes {*x*, *y*, *z}*:

In this case, the equivalent electric permittivity tensors in the helical coordinate **ɛ**’ cannot be reduced to Eq. (5), but it is expressed as

**ɛ**(

*x*’,

*y*’,

*z*) is taken at a point with coordinates (

*x*’,

*y*’,

*z*) in the helicoidal reference system, but its components are defined with respect to the orientation of the Cartesian axes, as in Eq. (6). In general, owing to the presence of the rotation matrices

*R*(

*Az*) in Eq. (7), which do not vanish as in the case of scalar

*ɛ*(

*x*’,

*y*’,

*z*), the obtained equivalent tensor

**ɛ**'(

*x*’,

*y*’,

*z*) is not

*z*-independent. In particular,

**ɛ**'(

*x*’,

*y*’,

*z*) depends on all spatial coordinates even if

**ɛ**(

*x*’,

*y*’,

*z*) can be represented by

**ɛ**(

*x*’,

*y*’). That is, all of its components are constant along the helix defined by constant values of

*x*’ and

*y*’, which is a sufficient condition for

*z*-independent

**ɛ**’ in the case of isotropic

*ɛ*. Therefore, to date, transformation optics formalism has only been used to analyze isotropic twisted fibers, for which Eq. (5) holds.

In this work, we show that transformation optics formalism can be used for the rigorous modeling of twisted anisotropic fibers if the material anisotropy has helical symmetry and rotates with the fiber structure or is axially symmetric. For example, we consider lossless media, for which **ɛ**(*x*, *y*, *z*) is a real symmetric tensor. In this case, **ɛ**(*x*, *y*, *z*) at each point in the fiber cross section can be represented as a diagonal matrix in a locally defined principal reference system {*x _{1}*,

*x*,

_{2}*x*}. The sample direction of the principal axes at a point specified by the coordinates (

_{3}*x'*,

_{0}*y'*, 0) is shown in Fig. 1(a). For

_{0}*z*= 0, the axes of the Cartesian and helicoidal systems overlap.

If the principal axes of **ɛ**(*x*’, *y*’, *z*) rotate with the twisted fiber, as in the case of intrinsic anisotropy or external physical factors of helical symmetry, then their orientation is fixed in the helicoidal coordinate system {*x*’, *y*’, *z*}, as shown in Fig. 1(b). In this case, the rotation of the principal axes of the permittivity tensor **ɛ**(*x*’, *y*’, *z*) observed in the Cartesian coordinates can be expressed as

**ɛ**(

*x*’,

*y*’, 0) is the anisotropic permittivity tensor at

*z*= 0, for which the axes of the Cartesian coordinate system and helicoidal coordinate systems overlap, as shown in Fig. 1(a). The condition given by Eq. (8) is also satisfied for axially symmetrical external factors, which do not change upon rotation around the

*z*-axis, but does not hold for anisotropy induced by external factors of directional nature, such as bending or lateral stress, which change orientation in the helical coordinates. If the above relation is substituted into Eq. (7), then the

*z*-dependent rotation matrices vanish, and one obtains

The equivalent permittivity tensor **ɛ**’ for anisotropic fibers is a sum of three distinct *z*-independent parts: the tensor **ɛ**(*x*’, *y*’, 0), which describes the anisotropy in Cartesian coordinates (*x*’ = *x* and *y*’ = *y* for *z *= 0); the twist-induced correction depending on a single component of permittivity tensor *ɛ _{zz}*(

*x*’,

*y*’, 0), which is analogous to the permittivity change in isotropic materials; and the correction of permittivity in the transverse plane related to

*ɛ*and

_{xz}*ɛ*. For isotropic materials, the last term vanishes because

_{yz}*ɛ*(

_{xz}*x*’,

*y*’, 0) =

*ɛ*(

_{yz}*x*’,

*y*’, 0) = 0, and Eq. (9) is reduced to Eq. (5) because

*ɛ*(

_{zz}*x*’,

*y*’, 0) =

*ɛ*(

*x*’,

*y*’, 0).

Of note, the derived formula for the equivalent permittivity tensor **ɛ**’ is general and can be used to represent a twisted anisotropic fiber as a 2D structure in helical coordinates, as long as Eq. (8) is satisfied. It allows for a rigorous 2D modeling of a wide range of effects that can impact the propagation characteristics of twisted fibers through induced material anisotropy. This includes anisotropy induced by the difference in thermal expansion coefficients between pure silica and doped structural elements of the fiber or by axially symmetrical external physical factors acting on the twisted fibers (e.g., hydrostatic pressure, axial elongation, or radial stress). Structural mechanical simulations, which are beyond the scope of this study, are required to determine the stress tensor and permittivity tensor **ɛ**(*x*’, *y*’, *z*) for a specific fiber load. Nonetheless, if the resulting tensor **ɛ**(*x*’, *y*’, *z*) is invariant in the helical coordinates, i.e., satisfying Eq. (8), then it is possible to use the general form of the permittivity tensors represented by Eq. (9) to model the propagation characteristics of the fibers in 2D.

## 3. Modeling of spun and twisted highly birefringent fibers

The proposed approach can be used for the rigorous modeling of twisted highly birefringent (HB) fibers with linear birefringence induced by SAPs, which have the form of a double helix rotating around a centrally located core (Fig. 2). The perturbative analysis performed in [22] for higher-order modes in various types of anisotropic fibers twisted at room temperature showed that higher-order modes in such fibers are non-degenerate optical vortices, which can be used in OAM multiplexing. In the following sections, we report the results of rigorous numerical simulations of similar structures showing some new phenomena that cannot be predicted by perturbative methods [22]. Moreover, we analyzed and compared twisted and spun fibers. Twisted fibers are fibers twisted at room temperature, which undergo elastic deformation inducing shear stress and consequently have additional birefringence that is not directly related to the presence of SAPs. By contrast, spun fibers are fibers twisted at high temperatures during the drawing process by preform spinning. In such fibers, there is only a small amount of residual torsional stress, which has a significant effect only in low-birefringent fibers [32]. In spun HB fibers, the residual torsional stress can be disregarded, and it can be assumed that the anisotropy in such fibers primarily originates from the difference in thermal expansion of the SAPs doped with B_{2}O_{3} and pure silica cladding.

First, we consider the spun HB fiber in which the helical shape of SAPs leads to a stress distribution with helical symmetry. In this case, the induced intrinsic anisotropy satisfies Eq. (8). In our analysis, we disregarded the shear stress and the stress generated by the doped core, which were much lower than those induced by SAPs [33]. In such a case, the stress-induced material birefringence in the core and surrounding cladding, which has a dominating influence on the propagation properties of the core modes, can be considered uniform, and the permittivity tensor can be expressed as [34]

*ɛ*

_{1}(

*x*,

*y*, 0) =

*n*

^{2}(

*x*,

*y*, 0) represents the distribution of the refractive index averaged with respect to polarization (in the step index, fiber

*n*takes only two values:

*n*

_{cor}_{e}and

*n*), and δ

_{clad}*ɛ*≈ 2

*n*Δ

_{core}*n*is proportional to the stress-induced linear birefringence Δ

_{l}*n*. Inserting Eq. (10) into Eq. (9) and using the relation

_{l}**ɛ**(

*x*,

*y*, 0) =

**(**

*ɛ**x*’,

*y*’, 0), we obtain the following form of the equivalent permittivity tensor

**ɛ**’ in the helicoidal coordinates:

The case of HB fibers twisted at room temperature is more complex. Such a twist is an elastic deformation inducing shear stress, which adds to the tensile stress generated by the SAPs and can take comparable values. Therefore, the permittivity tensor in a twisted fiber contains an additional term, which represents the effect of the shear stress [1]:

*p*

_{44}is the elasto-optic coefficient (

*p*

_{44}≈ −0.075 for silica glass). After substituting Eq. (12) into Eq. (9), we obtain the following expression for the equivalent permittivity tensor

**ɛ**’ in the HB fibers twisted at room temperature:

The above equation becomes identical to the equivalent permittivity tensor **ɛ**’ given by Eq. (10) for the spun fibers with the twist rate *A* multiplied by the correction coefficient (1−*ɛ _{1}p_{44}*), if we approximate (1−2

*ɛ*) with (1−ɛ

_{1}p_{44}*p*

_{1}_{44})

^{2}. It should be noted that the propagation characteristics of the optical fiber are related to its refractive index, which is determined equally by

**ɛ**’ given by Eq. (13), and

**µ**’, which is not affected by the stress and transforms in the same way as in the isotropic case (Eq. (5)). Therefore, the stress-related correction to the twist rate

*A*for the refractive index is given by (1−

*ɛ*)

_{1}p_{44}^{1/2}≈ 1−

*ɛ*/2. Consequently, the twist-induced circular birefringence of the fundamental modes in the Cartesian reference system (the difference in principal refractive indices for left- and right-handed circularly polarized fundamental modes in the twisted fiber) is given by

_{1}p_{44}*ɛ*(where

_{1}p_{44}λ/Λ*Λ =*2π/

*A*), which agrees with the analytically predicted and experimentally verified results [1].

## 4. Structure of first-order modes in spun and twisted birefringent fibers

In this section, we demonstrate the effectiveness of the proposed approach in the rigorous modeling of the polarization structure of first-order modes with azimuthal number |*M|* = 1 in twisted and spun HB fibers with linear birefringence induced by SAPs. We model the effect of twist and linear birefringence (Δ*n _{l}*) on higher-order modes with azimuthal number (absolute value of OAM

*M*) |

*M|*= 1 and radial number

*N*= 1. The modes characterized by |

*M|*= 1 have the most complex structure out of the higher-order modes. In principle, in an isotropic fiber with circular symmetry, the higher-order mode group with |

*M|*> 1 and

*N*= 1 is composed of two pairs of exactly degenerate modes HE

_{(|M|+1)1}and EH

_{(|M|−1)1}. These modes can be equivalently presented as pairs of circularly polarized optical vortices HE

_{|J|N}

^{±}and EH

_{|J|N}

^{±}, where ± corresponds to the sign of the total angular momentum

*J*=

*M+σ*and

*σ*is the spin angular momentum equal to +1 for right-handed circular polarization or to −1 for left-handed circular polarization

*.*For |

*M|*= 1 and

*N*= 1, there are two exactly degenerate circularly polarized HE

_{21}

^{±}modes, but the effective refractive indices of the TE

_{01}and TM

_{01}modes differ by Δ

*n*If the fiber is twisted at room temperature, then the additional stress-induced optical activity modifies the effective refractive indices in the Cartesian coordinates

_{TE-TM}.*n*of HE

_{eff}_{|J|N}

^{±}and EH

_{|J|N}

^{±}modes by a factor proportional to the total angular momentum

*J*[21]:

*n*

_{eff}_{0}represents the mode effective index of the non-twisted fiber. This stress-related effect lifts the degeneracy between the HE

_{|J|N}

^{±}and EH

_{|J|N}

^{±}modes of opposite handedness. In the helicoidal coordinate system, used both in the perturbative analysis [22] and rigorous modeling, there is an additional correction to the effective refractive index

*n’*given by which is a purely geometrical effect unrelated to stress. The effects described by Eqs. (14) and (15) lead to twist-induced circular birefringence in the helicoidal coordinate system Δ

_{eff}*n*, which is equal to 2|

_{c}*J*|(1−

*ɛ*/2)

_{1}p_{44}*λ/Λ*for circularly polarized HE

_{|J|N}

^{±}and EH

_{|J|N}

^{±}modes and 0 for the TE

_{01}and TM

_{01}modes.

The perturbative analysis of higher-order modes with |*M|* = 1 in twisted anisotropic fibers is the most complicated because in addition to Δ*n _{l}* and Δ

*n*, which must be taken into account for all eigenmodes, there is additional splitting of the effective refractive indices of the TE

_{c}_{01}and TM

_{01}modes represented by Δ

*n*. Therefore, in [22] the modes characterized by |

_{TE-TM}*M|*= 1 were analyzed only in two cases, which justified specific simplifications. The first considered case was a fiber with a large linear birefringence Δ

*n*and negligible Δ

_{l}*n*, which correspond to very small twist rates. In the second case, Δ

_{c}*n*and Δ

_{l}*n*were much larger than Δ

_{c}*n*, which was disregarded. By contrast, in the approach based on transformation optics formalism, any fiber with helically symmetric anisotropy can be modeled with equal ease.

_{TE-TM}In the following paragraphs, we reveal the phenomena that occur when Δ*n _{l}*, Δ

*n*, and Δ

_{c}*n*are of the same order of magnitude, and none can be omitted. Furthermore, we show that even in the case of a fiber in which Δ

_{TE-TM}*n*is much smaller than Δ

_{TE-TM}*n*and Δ

_{l}*n*and in twisted fibers without linear birefringence, there are phenomena that could not be predicted by perturbative methods. We discuss the effect of twist on first-order modes in fibers with linear birefringence Δ

_{c}*n*= 1×10

_{l}^{−4}and 4×10

^{−4}, which are typical values in commercially available HB fibers with SAPs [33]. To illustrate the effect of Δ

*n*, which scales approximately with the squared refractive index contrast between the core and cladding

_{TE-TM}*δn*, we analyzed low- and high-contrast fibers. A high-contrast fiber has a core diameter

^{2}*d*= 4 µm doped with 20 mol% of GeO

_{2}, which results in

*δn*= 2.87×10

^{−2}and Δ

*n*= 1.35×10

_{TE-TM}^{−4}. for

*λ*= 1000 nm. A low-contrast fiber has a core diameter

*d*= 8.03 µm doped with 5 mol% of GeO

_{2}doping, which for

*λ*= 1000 nm leads to

*δn*= 7.17×10

^{−3}and Δ

*n*= 8.51×10

_{TE-TM}^{−6}. Both cores have the same normalized frequency

*V*= 3.64, ensuring a two-mode propagation at a wavelength

*λ*of 1000 nm and cut-off wavelength

*λ*of 1510 nm. We compare the results obtained for spun anisotropic fibers, for which the equivalent permittivity tensor

_{cut-off}**ɛ**’ is given by Eq. (11) and twisted anisotropic fibers with

**ɛ**’ given by Eq. (13). Essentially, silica fibers twisted at room temperature typically break for helix pitches

*Λ*shorter than 1 cm, which limits their use to relatively small twist rates. In this study, however, we present the simulation results obtained in a greater range of twist rates (with

*Λ*as small as 0.5 mm) to emphasize some of the effects related to twist-induced stress, which are very subtle for

*Λ*> 1 cm.

The structure of the eigenmodes in spun and twisted linearly birefringent fibers for small twist rates, for which Δ*n _{l}* is much larger than Δ

*n*, resembles the LP

_{c}_{11}modes of non-twisted anisotropic fibers, as predicted in [22]. In a non-twisted linearly birefringent fiber, first-order hybrid modes with an annular intensity distribution are coupled and form linearly polarized LP

_{11}eigenmodes. If Δ

*n*≪ Δ

_{TE-TM}*n*, as in the case of a low-contrast fiber, then the LP

_{l}_{11}modes have a sharp intensity minimum between two intensity maxima and their transverse electric fields are oriented either perpendicularly or parallel to the anisotropy axis, as shown in Fig. 3(a). When Δ

*n*≈ Δ

_{TE-TM}*n*, as in the case of a high-contrast fiber, the eigenmodes of the isotropic fiber are not fully coupled and resemble a transition state between the hybrid modes (HE

_{l}_{21}

^{±}, TE

_{01}, and TM

_{01}) and pure LP

_{11}modes obtained for Δ

*n*≪ Δ

_{TE-TM}*n*, as shown in Fig. 3(b). In this case, we observed two pairs of modes with distinct intensity profiles. Modes that resemble those observed in a low-contrast fiber are obtained owing to the coupling between the HE

_{l}_{21}and TM

_{01}modes, for which the refractive index difference is much smaller than that for the HE

_{21}and TE

_{01}modes.

For high twist rates, for which Δ*n _{c}* is much larger than Δ

*n*and Δ

_{l}*n*, the perturbative model [22] predicts that the effect of linear anisotropy vanishes and the LP modes are transformed into hybrid modes of an isotropic fiber, i.e., circularly polarized HE

_{TE-TM}_{21}

^{±}modes and linearly polarized TE

_{01}and TM

_{01}modes, as shown in Fig. 3(c). A rigorous modeling confirmed this prediction only for spun fibers. In twisted linearly birefringent fibers, the effect of linear birefringence will be canceled for short

*Λ*, and the resulting mode structure will be similar to that obtained for twisted isotropic fibers, which is different from that predicted by perturbative methods [21,22]. In a twisted isotropic fiber, the eigenmodes have intensity profiles identical to those obtained in spun fibers (Fig. 3(c)), but they are generally elliptically polarized and are referred to as quasi-TE, quasi-TM, and quasi-HE

_{21}

^{±}modes. Their polarization structure cannot be represented by a linear superposition of the first-order modes of the unperturbed fiber.

The differences between perturbative and rigorous solutions increase with the twist rate and can be observed in the relation between the intensity*-*weighted average of the ellipticity angle ϑ of the first-order modes defined as:

*S*is the axial component of the Poynting vector, and the twist rate, which is shown in Fig. 4(a) for the low-contrast fiber and in Fig. 4(b) for the high-contrast fiber. The obtained values of the averaged ellipticity angle differ from the perturbatively predicted ±45° for the quasi-HE

_{z}_{21}

^{±}mode and 0° for the quasi-TE and quasi-TM modes. This effect can be associated with the radial dependence of the local ellipticity angle

*ϑ*(

*r*) of first-order modes, as shown in Fig. 4(c) and Fig. 4(d) and related to the shear stress, which increases with the distance from the fiber axis and the twist rate. The radial dependence of the ellipticity angle of the quasi-HE

_{21}

^{±}mode is similar in both analyzed fibers. Therefore, it has a greater effect on the intensity

*-*weighted average of the ellipticity angle ϑ for the low-contrast fiber in which the mode has greater radius. Moreover, for the quasi-HE

_{21}

^{±}mode, there is a step change in

*ϑ*(

*r*) observed at the interface between the core and cladding, which is more pronounced for the high-contrast core, for which the change in the material refractive index and elasto-optic correction of the relative permittivity (see Eq. (12)) at the interface is greater than that for the low-contrast fiber. In addition, there is an abrupt change in the averaged ellipticity angle

*ϑ*of the quasi-TE and quasi-TM modes of the low-contrast fiber related to the resonant coupling observed at 1/

*Λ*≈ 1 [1/mm], Fig. 4(a). This coupling is observed when a small variation of

*n’*of the quasi-TE and quasi-TM modes induced by the elasto-optic effect (not predicted by the perturbative methods), which is proportional to (1/

_{eff}*Λ*)

^{2}, matches the refractive index difference between the TE

_{01}and TM

_{01}modes Δ

*n*, as shown in Fig. 4(e). For the high-contrast fiber, a similar effect is not observed in the analyzed twist rate range, due to the much larger Δ

_{TE-TM}*n*and smaller change in

_{TE-TM}*n’*, as shown in Fig. 4(f). In this case, phase matching between the TE

_{eff}_{01}and TM

_{01}modes occurs at 1/

*Λ*≈ 8 [1/mm].

In the previously unexplored range of moderate twist rates [22], the linearly polarized LP_{11} modes of HB fibers gradually evolved toward the spun or twisted isotropic fiber modes. The intensity distribution of the eigenmodes changes with increasing twist from a shape characterized by two distinct maxima along the slow or fast axis (characteristic of LP_{11} modes in non-twisted anisotropic fibers; Fig. 3(a)) toward the annular shape characteristic of optical vortices (Fig. 3(c)). To describe the evolution of the mode shape versus twist rate, we introduce a parameter called the annularity coefficient Γ, which is defined as the ratio of maximal intensity along the axis perpendicular to the intensity maxima to the maximal intensity along the axis parallel to the intensity maxima Γ = [max(I_{⊥})/max(I_{||})], as shown in Fig. 5. This parameter changes from 0 for the LP_{11} mode with a line of zero intensity, separating the maxima to 1 for an optical vortex with a perfectly annular shape.

The relation between the twist rate expressed as a number of rotations per millimeter length (1/*Λ = A*/(2π) [1/mm]) and the effective refractive index *n’ _{eff}* in the helicoidal coordinate system, the annularity coefficient Γ, and intensity

*-*weighted average of the ellipticity angle ϑ of the first-order eigenmodes in the low-contrast core of the spun and twisted fibers is presented in Fig. 6 for Δ

*n*= 1×10

_{l}^{−4}and in Fig. 7 for Δ

*n*= 4×10

_{l}^{−4}

_{.}

The difference between the effective refractive indices of the TE_{01} and TM_{01} modes in the isotropic low-contrast core is equal to Δ*n _{TE-TM}* = 8.51×10

^{−6}, which is much lower than the linear birefringence Δ

*n*. Therefore, the considered fiber structures are very similar to the case of intensely twisted high-anisotropic ﬁbers described in [22], for which the perturbative solution for effective refractive indices in helicoidal coordinates is given by

_{l}*n*is the scalar approximation of the effective refractive index for the first-order modes, which is equal to the effective refractive index for the TE

_{scalar}_{01}mode. If we substitute the average effective refractive index of the first-order hybrid modes for

*n*in Eq. (14), then the obtained values of

_{scalar}*n’*do not differ significantly from the rigorous results presented in Figs. (6) and (7). The differences between perturbative and rigorous solutions can be observed in the annularity coefficient Γ and averaged ellipticity angle

_{eff}*ϑ*. In the perturbative model for intensely twisted high-anisotropic ﬁbers [22], the first-order modes are assumed to be degenerate and, as a result, slow and fast linearly polarized LP

_{11}modes in non-twisted anisotropic fibers are degenerate in pairs. Therefore, any combination of two slow or two fast LP

_{11}modes, including combinations of annular intensity profiles, is also an eigenmode, and the slightest twist generates perfectly annular eigenmodes with Γ = 1. In the rigorous model, the first-order eigenmodes become annular (Γ ≈ 1) when the twist-induced circular birefringence Δ

*n*becomes much larger than the refractive index difference between the two modes of the same polarization in the non-twisted fiber. Consequently, the fast modes (green in Fig. 6(b)) for which this difference is equal to 4.0×10

_{c}^{−6}attains an annular shape slightly faster than the slow modes (red in Fig. 6(b)), for which the refractive indices differ by 4.6×10

^{−6}. Furthermore, the linearly polarized TE

_{01}and TM

_{01}modes are replaced in the perturbative model by their circularly polarized superpositions with M =

*−σ =*±1. As a result,

*ϑ*of all first-order modes approach circular polarization for high twist rates in the same way as the quasi-HE

_{21}

^{±}modes of the spun fiber in the rigorous model, for which the evolution of the ellipticity angle

*ϑ*can be described by the relation obtained in [35] for the fundamental modes. After taking into account the stress-related correction to the twist rate (1−

*ɛ*/2) this relation can be expressed as

_{1}p_{44}According to the above relation, the average ellipticity angle depends on the ratio of the twist-induced circular birefringence in the helicoidal coordinate system Δ*n _{c}* to the linear birefringence Δ

*n*. Therefore, the averaged ellipticity angle

_{l}*ϑ*of quasi-HE

_{21}

^{±}modes in a fiber with Δ

*n*= 1×10

_{l}^{−4}(Fig. 6(c)) approaches ±45° (circular polarization) four times faster than in a fiber with Δ

*n*= 4×10

_{l}^{−4}(Fig. 7(c)). The ellipticity angle

*ϑ*of the quasi-TE and quasi-TM modes increases with the twist rate in approximately the same way as for the quasi-HE

_{21}

^{±}mode when Δ

*n*is smaller than Δ

_{c}*n*. For greater twist rates, the ellipticity angle

_{l}*ϑ*of the quasi-TE and quasi-TM modes in the spun fiber monotonically approaches 0, as these modes evolve with increasing twist rate toward linearly polarized TE

_{01}and TM

_{01}modes of the spun isotropic fiber. In twisted HB fibers, the relation between the averaged ellipticity angle ϑ and twist rate is more complex, as it approaches the result obtained for twisted isotropic fibers, as shown in Fig. 4. In a fiber with greater Δ

*n*, the quasi-TE and quasi-TM modes converge to modes of twisted isotropic fibers for greater twist rates, as shown in Fig. 8. As a result, the abrupt change in

_{l}*ϑ*related to the resonant coupling between the quasi-TE and quasi-TM modes in the twisted isotropic fiber at 1/

*Λ*≈ 1 [1/mm] is much less pronounced in a fiber with Δ

*n*= 4×10

_{l}^{−4}. Furthermore, for small twist rates, the averaged ellipticity angle ϑ of the quasi-TE and quasi-TM modes in twisted linearly birefringent fibers has the opposite sign of the twisted isotropic fiber. Therefore, it will reach 0 before the effect of linear birefringence will be canceled owing to the high twist rate.

In the high-contrast fiber, the separation of TE_{01} and TM_{01} modes Δ*n _{TE}-_{TM}* = 1.35×10

^{−4}is of the same order of magnitude as the considered values of Δ

*n*, which significantly changes the effect of linear birefringence and twist on the evolution of first-order modes. The influence of twist on the effective refractive index in helicoidal coordinates

_{l}*n’*, annularity coefficient Γ and averaged ellipticity angle

_{eff}*ϑ*, and of the first-order modes in the high-contrast core is presented in Fig. 9 for Δ

*n*= 1×10

_{l}^{−4}and in Fig. 10 Δ

*n*= 4×10

_{l}^{−4}.

In the high-contrast fiber and moderate twist rate range (1/*Λ≤*0.5 1/mm), for which Δ*n _{c}* is comparable to Δ

*n*and Δ

_{l}*n*, Eq. (17) no longer provides an accurate approximation of

_{TE}-_{TM}*n’*, and each eigenmode is characterized by a distinct evolution of the effective refractive index

_{eff}*n’*, annularity Γ, and average ellipticity angle

_{eff}*ϑ*. Furthermore, for the high-contrast fiber with Δ

*n*= 1×10

_{l}^{−4}smaller than Δ

*n*, we observe resonant couplings between the quasi-TE mode (red dashed line) and quasi-TM mode (green solid line) at 1/

_{TE}-_{TM}*Λ*≈ 0.04 [1/mm] and the quasi-HE

_{21}

^{+}mode (red solid line) at 1/

*Λ*≈ 0.065 [1/mm], which manifest as anti-crossings in Fig. 9(a) and resonant peaks in Γ and

*ϑ*in Figs. 9(b),(c). For large twist rates (1/

*Λ>*0.5 [1/mm]), the obtained results are similar to those of the low-contrast core, and Eqs. (17) and (18) derived by a perturbative approach provide a good approximation of the effective refractive index

*n’*and the averaged ellipticity angle

_{eff}*ϑ*. The influence of the elasto-optic effect on the high-contrast core in the analyzed twist rate range is less than that in the case of a low-contrast core because of the smaller mode radius and much larger Δ

*n*, for which the resonant coupling between the quasi-TE and quasi-TM modes would be observed for 1/

_{TE}-_{TM}*Λ*≈ 8 [1/mm]. Therefore, in the majority of the analyzed phenomena, the effect of shear stress can be simply described using the twist rate multiplied by (1−

*ɛp*/2). An additional effect of shear stress can be observed only in the averaged ellipticity angle of quasi-TE and quasi-TM modes in the high-contrast core with Δ

_{44}*n*= 1×10

_{l}^{−4}, which does not asymptotically approach 0 but changes sign for 1/

*Λ*≈ 1 [1/mm] to obtain the same polarization handedness as in a fiber with Δ

*n*= 0 (Fig. 4(b)).

_{l}## 4. Conclusions

We have shown that transformation optics formalism can be used to obtain 2D representations of a wide range of twisted anisotropic fibers and allow for a rigorous and efficient numerical modeling of structures, which were previously analyzed using only simplified analytical methods. The proposed approach can be used for fibers, in which the material anisotropy has an intrinsic origin or is induced by helical external factors characterized by the same helix period as the fiber. Furthermore, it can be applied to twisted fibers with anisotropy induced by axially symmetrical external physical factors, such as radial stress, hydrostatic pressure, elastic twist, or axial elongation. However, the 2D representation cannot be obtained for twisted fibers with anisotropy induced by external factors of directional nature, such as bending or lateral stress, which change orientation in the helical coordinates.

The effectiveness of the developed method was verified by modeling the structure of the first-order eigenmodes of elastically twisted highly anisotropic fibers, which were previously studied using perturbative methods in [22], and spun highly anisotropic fibers, in which the shear stress vanishes because of inelastic deformation. We analyzed a low-contrast fiber, for which the degeneration of the first-order modes assumed in [22] is a justified approximation, and high-contrast fiber, in which the difference between the refractive indices of the TE and TM modes is of the same order of magnitude as the linear birefringence and cannot be disregarded. In the low-contrast fiber, the relation between the effective refractive indices is similar to that predicted by the perturbative approach. Furthermore, our results confirm that in high twist rates, the effect of the linear birefringence vanishes, and the eigenmode structure is the same as that in twisted or spun isotropic fibers. In spun fibers, the modes evolve toward circularly polarized HE_{21}^{±} modes and linearly polarized TE_{01} and TM_{01} modes with increasing twist rate. In twisted fibers, however, the polarization structure of the eigenmodes is much more complex than that predicted by the perturbative approach. In this case, the mode ellipticity changes with the twist rate and the distance from the fiber axis because of the radial dependence of the shear stress. Furthermore, in twisted fibers, there is a subtle change in the refractive indices of the TE_{01} and TM_{01} modes, which increases quadratically with the twist rate and can lead to resonant coupling between these modes for twist rates greater than can be obtained experimentally. In a high-contrast core, the non-degeneracy of the modes leads to additional phenomena. In this case, the relation between the effective refractive index and twist rate cannot be accurately described using perturbative formulas. Furthermore, the shape and ellipticity of each of the eigenmodes change in a distinct manner with increasing twist rate.

The presented method can be applied with equal ease to simplified forms of material anisotropy and to complex anisotropy distributions, which can be obtained using structural mechanical simulations for a specific fiber load, as long as they are cylindrically or helically symmetric. It can also be used for modeling fibers with material anisotropy and form anisotropy related to the shape and position of the core, for which new wave phenomena can be expected.

## Funding

Narodowe Centrum Nauki (Maestro 8, DEC-2016/22/A/ST7/00089).

## Disclosures

The authors declare no conflicts of interests.

## 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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