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 (LP₁₁) versus helix pitch in twisted and spun fibers with linear birefringence induced by stress-applying parts (SAPs) and show that the LP₁₁ 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

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

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 {x₁, x₂, x₃}. The sample direction of the principal axes at a point specified by the coordinates (x'₀, y'₀, 0) is shown in Fig. 1(a). For z = 0, the axes of the Cartesian and helicoidal systems overlap.

 

Fig. 1. Principal axes {x₁, x₂} (blue) related to the helical intrinsic anisotropy at points specified by helicoidal coordinates (x’ = x'₀, y’ = y'₀, z), which corresponds to a helix characterized by the twist rate A in Cartesian coordinates. For z = 0, the axes of Cartesian ({x, y}, black) and helicoidal ({x’, y'}, red) coordinate systems overlap (a). For z ≠ 0, the orientation of principal axes in the helicoidal coordinates remains constant because of the helical symmetry of intrinsic anisotropy, which rotates with respect to Cartesian coordinates (b).

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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

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_xz and ɛ_yz. For isotropic materials, the last term vanishes because 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 B₂O₃ and pure silica cladding.

 

Fig. 2. Single period of a twisted highly birefringent fiber with stress-applying parts (SAPs).

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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]

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]:

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−ɛ₁p₄₄), if we approximate (1−2ɛ₁p₄₄) with (1−ɛ₁p₄₄)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−ɛ₁p₄₄)^1/2 ≈ 1−ɛ₁p₄₄/2. 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 ɛ₁p₄₄λ/Λ (where 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 (Δn_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 HE₂₁^± modes, but the effective refractive indices of the TE₀₁ and TM₀₁ modes differ by Δn_TE-TM. If the fiber is twisted at room temperature, then the additional stress-induced optical activity modifies the effective refractive indices in the Cartesian coordinates n_eff of HE_|J|N^± and EH_|J|N^± modes by a factor proportional to the total angular momentum J [21]:

The perturbative analysis of higher-order modes with |M| = 1 in twisted anisotropic fibers is the most complicated because in addition to Δn_l and Δn_c, which must be taken into account for all eigenmodes, there is additional splitting of the effective refractive indices of the TE₀₁ and TM₀₁ modes represented by Δn_TE-TM. Therefore, in [22] the modes characterized by |M| = 1 were analyzed only in two cases, which justified specific simplifications. The first considered case was a fiber with a large linear birefringence Δn_l and negligible Δn_c, which correspond to very small twist rates. In the second case, Δn_l and Δn_c were much larger than Δn_TE-TM, 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.

In the following paragraphs, we reveal the phenomena that occur when Δn_l, Δn_c, and Δn_TE-TM 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 Δn_TE-TM is much smaller than Δn_l and Δn_c 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 Δn_l = 1×10⁻⁴ and 4×10⁻⁴, which are typical values in commercially available HB fibers with SAPs [33]. To illustrate the effect of Δn_TE-TM, which scales approximately with the squared refractive index contrast between the core and cladding δn², we analyzed low- and high-contrast fibers. A high-contrast fiber has a core diameter d = 4 µm doped with 20 mol% of GeO₂, which results in δn = 2.87×10⁻² and Δn_TE-TM = 1.35×10⁻⁴. for λ = 1000 nm. A low-contrast fiber has a core diameter d = 8.03 µm doped with 5 mol% of GeO₂ doping, which for λ = 1000 nm leads to δn = 7.17×10⁻³ and Δn_TE-TM = 8.51×10⁻⁶. 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_cut-off of 1510 nm. We compare the results obtained for spun anisotropic fibers, for which the equivalent permittivity tensor 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.

 

Fig. 3. Intensity distribution (color map) and transverse electric field (arrows) of LP₁₁ modes in a linearly birefringent (Δn_l = 1×10⁻⁴) non-twisted low-contrast fiber with Δn_l>>Δn_TE-TM = 8.51×10⁻⁶ (a) and in a high-contrast fiber with Δn_l ≈ Δn_TE-TM = 1.35×10⁻⁴ (b). For high twist rates (Δn_c>>Δn_l), first-order modes in high- and low-contrast fibers resemble the hybrid modes (HE₂₁^±, TE₀₁, and TM₀₁) of isotropic fibers (c). Field profiles were obtained at λ = 1000 nm and are ordered in each group from the mode with the smallest n_eff on the left side to the mode with the largest n_eff on the right side.

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The structure of the eigenmodes in spun and twisted linearly birefringent fibers for small twist rates, for which Δn_l is much larger than Δn_c, resembles the LP₁₁ 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 LP₁₁ eigenmodes. If Δn_TE-TM ≪ Δn_l, as in the case of a low-contrast fiber, then the LP₁₁ 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 Δn_TE-TM ≈ Δn_l, 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 (HE₂₁^±, TE₀₁, and TM₀₁) and pure LP₁₁ modes obtained for Δn_TE-TM ≪ Δn_l, 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 HE₂₁ and TM₀₁ modes, for which the refractive index difference is much smaller than that for the HE₂₁ and TE₀₁ modes.

 

Fig. 4. Intensity-weighted average of the ellipticity angle ϑ (a, b), local ellipticity angle ϑ(r) (c, d), and effective refractive index in the helicoidal coordinate system n’_eff (e, f) for quasi-HE₂₁⁺ (solid, red), quasi-TE (dashed, red), quasi-TM (solid, green), and quasi-HE₂₁⁻ (dashed, green) modes. Radial profile of the intensity distribution of the first-order modes is shown in (c, d) as a black dotted line. Simulations were performed at λ = 1000 nm for the twisted isotropic fibers with a core diameter d = 8.03 µm doped with 5 mol% of GeO₂, (δn = 7.17×10⁻³) (a, c, e) and with core diameter d = 4 µm doped with 20 mol% of GeO₂, (δn = 2.87×10⁻²) (b, d, f).

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For high twist rates, for which Δn_c is much larger than Δn_l and Δn_TE-TM, 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 HE₂₁^± modes and linearly polarized TE₀₁ and TM₀₁ 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-HE₂₁^± 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:

 

Fig. 5. Annularity coefficient Γ = [max(I_⊥)/max(I_||)] is defined as a ratio of the maximal intensity along the axis perpendicular to the intensity maxima (red marks on red line) to the maximal intensity along the axis parallel to the intensity maxima (black marks on the black line) Δx and Δy denote distance from the core center.

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In the previously unexplored range of moderate twist rates [22], the linearly polarized LP₁₁ 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 LP₁₁ 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 LP₁₁ mode with a line of zero intensity, separating the maxima to 1 for an optical vortex with a perfectly annular shape.

 

Fig. 6. Effective refractive index in the helicoidal coordinate system n’_eff (a), annularity coefficient Γ (b), and averaged ellipticity angle ϑ (c) for the quasi-HE₂₁⁺ (solid, red for the twisted fiber, black for the spun fiber), quasi-TE (dashed, red for the twisted fiber, black for the spun fiber), quasi-TM (solid, green for the twisted fiber, gray for the spun fiber), and quasi-HE₂₁⁻ (dashed, green for the twisted fiber, gray for the spun fiber) modes. Simulations were performed at λ = 1000 nm for a core doped with 5 mol% of GeO₂, (δn = 7.17×10⁻³), core diameter d = 8.03 µm, and linear birefringence Δn_l = 1×10⁻⁴.

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Fig. 7. Effective refractive index in the helicoidal coordinate system n’_eff (a), annularity coefficient Γ (b), and averaged ellipticity angle ϑ (c) for quasi-HE₂₁⁺ (solid, red for the twisted fiber, black for the spun fiber), quasi-TE (dashed, red for the twisted fiber, black for the spun fiber), quasi-TM (solid, green for the twisted fiber, gray for the spun fiber), and quasi-HE₂₁⁻ (dashed, green for the twisted fiber, gray for the spun fiber) modes. Simulations were performed at λ = 1000 nm for a core doped with 5 mol% of GeO₂, (δn = 7.17×10⁻³), core diameter d = 8.03 µm, and linear birefringence Δn_l = 4×10⁻⁴.

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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 Δn_l = 1×10⁻⁴ and in Fig. 7 for Δn_l = 4×10⁻⁴_.

The difference between the effective refractive indices of the TE₀₁ and TM₀₁ modes in the isotropic low-contrast core is equal to Δn_TE-TM = 8.51×10⁻⁶, which is much lower than the linear birefringence Δn_l. 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

According to the above relation, the average ellipticity angle depends on the ratio of the twist-induced circular birefringence in the helicoidal coordinate system Δn_c to the linear birefringence Δn_l. Therefore, the averaged ellipticity angle ϑ of quasi-HE₂₁^± modes in a fiber with Δn_l = 1×10⁻⁴ (Fig. 6(c)) approaches ±45° (circular polarization) four times faster than in a fiber with Δn_l = 4×10⁻⁴ (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-HE₂₁^± mode when Δn_c is smaller than Δn_l. For greater twist rates, the ellipticity angle 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 TE₀₁ and TM₀₁ 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 Δn_l, 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 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 Δn_l = 4×10⁻⁴. 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.

 

Fig. 8. Effective refractive index in the helicoidal coordinate system n’_eff (a) and averaged ellipticity angle ϑ (b) for the quasi-TE (red), quasi-TM (green), quasi-HE₂₁⁺ (black), and quasi-HE₂₁⁻ (gray) modes in a twisted fiber. Simulations were performed at λ = 1000 nm for core doped with 5 mol% of GeO₂, (δn = 7.17×10⁻³), core diameter d = 8.03 µm, linear birefringence Δn_l = 0 (dotted), Δn_l = 1×10⁻⁴ (dashed), and Δn_l = 4×10⁻⁴ (solid).

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In the high-contrast fiber, the separation of TE₀₁ and TM₀₁ modes Δn_TE-_TM = 1.35×10⁻⁴ is of the same order of magnitude as the considered values of Δn_l, 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 n’_eff, annularity coefficient Γ and averaged ellipticity angle ϑ, and of the first-order modes in the high-contrast core is presented in Fig. 9 for Δn_l = 1×10⁻⁴ and in Fig. 10 Δn_l = 4×10⁻⁴.

 

Fig. 9. Effective refractive index in the helicoidal coordinate system n’_eff (a), annularity coefficient Γ (b), and averaged ellipticity angle ϑ (c) for quasi-HE₂₁⁺ (solid, red for the twisted fiber, black for the spun fiber), quasi-TE (dashed, red for the twisted fiber, black for the spun fiber), quasi-TM (solid, green for the twisted fiber, gray for the spun fiber), and quasi-HE₂₁⁻ (dashed, green for the twisted fiber, gray for the spun fiber) modes. Simulations were performed at λ = 1000 nm for a core doped with 20 mol% of GeO₂, (δn = 2.87×10⁻²), core diameter d = 4 µm, and linear birefringence Δn_l = 1×10⁻⁴.

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Fig. 10. Effective refractive index in the helicoidal coordinate system n’_eff (a), annularity coefficient Γ (b), and averaged ellipticity angle ϑ (c) for quasi-HE₂₁⁺ (solid, red for the twisted fiber, black for the spun fiber), quasi-TE (dashed, red for the twisted fiber, black for the spun fiber), quasi-TM (solid, green for the twisted fiber, gray for the spun fiber), and quasi-HE₂₁⁻ (dashed, green for the twisted fiber, gray for the spun fiber) modes. Simulations were performed at λ = 1000 nm for a core doped with 20 mol% of GeO₂, (δn = 2.87×10⁻²), core diameter d = 4 µm, and linear birefringence Δn_l = 4×10⁻⁴.

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In the high-contrast fiber and moderate twist rate range (1/Λ≤0.5 1/mm), for which Δn_c is comparable to Δn_l and Δn_TE-_TM, Eq. (17) no longer provides an accurate approximation of n’_eff, and each eigenmode is characterized by a distinct evolution of the effective refractive index n’_eff, annularity Γ, and average ellipticity angle ϑ. Furthermore, for the high-contrast fiber with Δn_l = 1×10⁻⁴ smaller than Δn_TE-_TM, we observe resonant couplings between the quasi-TE mode (red dashed line) and quasi-TM mode (green solid line) at 1/Λ ≈ 0.04 [1/mm] and the quasi-HE₂₁⁺ 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’_eff and the averaged ellipticity angle 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 Δn_TE-_TM, for which the resonant coupling between the quasi-TE and quasi-TM modes would be observed for 1/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−ɛp₄₄/2). 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 Δn_l = 1×10⁻⁴, 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 Δn_l = 0 (Fig. 4(b)).

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 HE₂₁^± modes and linearly polarized TE₀₁ and TM₀₁ 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 TE₀₁ and TM₀₁ 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.