1. Introduction

Optical vortices, first used in optical tweezers [1], have found many applications in metrology [2], communication [3], fiber lasers [4], and quantum optics [5]. The demand for compact, integrated devices that utilize optical vortices has created interest in the direct generation of vortex beams in optical fibers. The most well-known methods for in-fiber generation of optical vortices use resonant coupling in long-period gratings [6,7] or selective couplers [8]. Alternative, non-resonant solutions employ fibers with special dispersion characteristics [9], apodized gratings [10] and photonic lanterns [11]. There are also many methods utilizing twisted optical fibers of helicoidal symmetry, which lifts the degeneracy of spatial modes of the same order and different polarizations [12–16]. It has also been demonstrated that beams with arbitrary polarization and orbital angular momentum (M) can be generated in twisted microstructured fibers with periodic inversion of the twist rate [17,18]. However, the operation range of resonant methods is limited to no more than tens of nanometers, whereas broadband non-resonant methods require the use of sophisticated fibers, which are difficult to fabricate. Moreover, although photonic lanterns allow the generation of vortex modes over a very broad spectral range (at least 550 nm according to [11]), they require precise control over the phase of the multiple linearly polarized (LP) modes excited at the input.

It was recently shown that the broadband generation of optical vortices could be achieved in gradually twisted commercially available highly birefringent (HB) fibers with stress-applying parts (SAPs) [19]. In such a fiber, the LP₁₁ eigenmodes of the non-twisted section can be quasi-adiabatically transformed into the eigenmodes of the twisted fiber. As the twist rate increases, these eigenmodes evolve into circularly polarized HE₂₁^± phase vortices and linearly polarized TE₀₁ and TM₀₁ polarization vortices [20,21]. In this study, we propose an effective method for modeling the transformation of modes in gradually twisted birefringent fibers and conduct a detailed numerical analysis of the conversion process in HB fibers of different geometries, starting from initially excited LP modes and progressing to high-quality vortex modes with increasing twist rate. This includes an analysis of the gradual evolution of the spin and orbital angular momenta as functions of the propagation distance, the crosstalk between the targeted and other vortex modes, and the relationship between the minimum conversion length and the twist profile for different HB fibers.

The propagation of electromagnetic waves in optical fibers with a geometry that varies along the propagation distance can be accurately modeled using beam propagation methods (BPMs) [22] or mode-matching methods (MMMs) [23–25]. The formulation of the finite-element BPM method valid for twisted anisotropic fibers was presented in [22]. However, BPMs have a high computational cost, especially in the case of a long propagation distance, and are only used to model fibers with constant and periodically inverted twist rates [17,18,22]. The method presented in this paper is based on MMMs in which the fiber is divided into multiple sections of constant geometry. Typically, eigenmodes are calculated for each section of a fiber and used to determine the transmission and reflection of electromagnetic waves at the interface between adjacent sections [23–25]. In general, a large number of local core, cladding, and radiation modes of the analyzed fiber must be rigorously calculated to obtain accurate results [23–25]. This leads to a high computational cost. In some cases, the modeling method can be simplified without loss of accuracy, as in the case of the eigenmode propagation method (EPM), which was recently used for the analysis of twisted microstructured fibers with periodic inversion of the twist direction [18]. It was shown in [18] that high accuracy could be obtained with a much smaller number of rigorously calculated eigenmodes because, after the twist inversion, any mode can be represented as a superposition of a small number of modes having the same absolute value of the orbital angular momentum |M|. Furthermore, the EPM was shown to be more numerically stable than BPMs for a fiber with a periodically changing twist rate [18].

The method used in this study allows for the modeling of the mode conversion in a gradually twisted HB fiber with a centrally-located cylindrical core and a uniform linear birefringence distribution, which is a good approximation of HB fibers with SAPs twisted at high temperatures [26,27]. A significant advantage of our approach is that it does not require a rigorous calculation of the eigenmode structure in every section of the fiber that has a different twist rate. We demonstrate that for the twisted HB fibers with SAPs, the coupled-mode approach in helicoidal coordinates can be used to obtain the solutions for local eigenmodes in successive fiber sections and that the results obtained with this approach are nearly identical to the results of rigorous numerical simulations performed using the finite element method (FEM) and the transformation optics formalism [21,28,29]. Furthermore, the core eigenmodes of the analyzed HB fibers can be accurately represented as a superposition of eigenmodes of the isotropic fiber from the corresponding spatial mode group (circularly polarized hybrid HE₂₁^± modes, and linearly polarized TE₀₁ and TM₀₁ modes in the case of the first-order eigenmodes of the twisted HB fiber). As a result, the number of modes required to model the conversion process accurately is greatly reduced, in a manner similar to that described in [18]. Furthermore, the proposed representation allows for a straightforward calculation of the orbital angular momenta and spin (related to polarization) of the local eigenmodes and their superpositions.

We used the proposed method to analyze the conversion of LP₁₁ modes to vector vortex modes in gradually twisted HB fibers, although it is valid for modes of any order. We examined the effects of the refractive index contrast, birefringence, and twist rate profiles of HB fibers on the intensity, transverse electric field, and ellipticity angle distributions for the first-order eigenmodes and their superpositions generated due to crosstalk, which can be significant if the changes in the twist rate along the propagation distance occur too rapidly. The results of our simulations imply the possibility of obtaining vector vortex modes from LP₁₁ modes with less than −30 dB of crosstalk using a few centimeters of gradually twisted HB fiber in a spectral range spanning hundreds of nanometers. Furthermore, the conversion length can be further reduced at the cost of limiting the desired operating wavelength range or by targeting a specific output vortex mode.

2. Numerical method

2.1. Accurate eigenmode analysis based on a coupled-mode approach

In general, the field profiles and propagation constants of modes in twisted fibers can be rigorously determined using numerical simulations based on a formalism using FEM and transformation optics [21,28,29]. We show that these parameters can also be accurately obtained in the birefringent twisted fibers with stress-applying elements using a simple numerical model based on the coupled-mode theory expressed in helicoidal coordinates. The field profiles and propagation constants of the eigenmodes of twisted birefringent fiber obtained using the proposed approach agree very well with the results of the rigorous numerical simulations.

Existing perturbative methods use certain approximations, such as disregarding twist-induced circular birefringence for small twist rates or splitting of the TE/TM mode propagation constants for high twist rates, to analytically describe the first-order eigenmodes of twisted anisotropic fibers [20]. Although such methods can accurately predict the eigenmode structure and provide useful analytical formulas within the limits of low and high twist rates, they are not accurate in a range of twist rates for which the linear birefringence and the twist-induced circular birefringence have comparable magnitudes [21].

The method proposed in this study produces an accurate representation of the eigenmodes in the analyzed twisted HB fiber for any twist rate and any mode order by numerically solving an eigenvalue problem defined by 4×4 or 2 × 2 matrices. As an input, the method requires the rigorously calculated propagation constants and field distributions of the eigenmodes of a non-twisted isotropic fiber. This approach is orders of magnitude faster than rigorous numerical simulations performed with FEM and yields practically identical results. Similar to [20], the effect of a perturbation represented by linear birefringence on the modes of a twisted isotropic fiber is analyzed in a helicoidal coordinate system, in which the higher order eigenmodes are linearly polarized TE and TM modes or circularly polarized HE^± and EH^± hybrid modes modes created by a linear superposition of hybrid modes of a non-twisted isotropic fiber:

We analyzed only those HB fibers in which the residual torsional stress could be disregarded owing to the high temperatures used in the twisting process. In such fibers, the anisotropy originates primarily from the difference in thermal expansion of the SAPs doped with B₂O₃ and pure silica cladding [26,27]. Therefore, the stress-induced material birefringence in the core and surrounding cladding can be considered uniform in each cross-section. The permittivity tensor for such fibers has helicoidal symmetry (i.e., it rotates with the twisted fiber). For z = 0, this tensor can be expressed as:

The field profiles and propagation constants of the first-order modes in twisted HB fibers can be accurately obtained by numerically solving the following 4×4 eigenvalue problem:

For simplicity, we normalize the cross product of the local eigenmodes to ${({\textrm{e}_k^A} )^\ast }\textrm{e}_l^A = {\delta _{kl}}$.

For mode groups characterized by |M|≥2, the eigenvalue problem can be solved analytically because it is described by two 2×2 matrices. This is related to the fact that according to Eq. (6), the coupling between the higher-order HE^± and EH^± modes is non-zero only in the pairs of modes with the same sign of the total angular momentum:

Moreover, the eigenvalue problem for modes with M = 0, including the fundamental modes, can be formulated as follows:

Consequently, the field profiles of non-first-order eigenmodes of the twisted HB fiber can be accurately described as a superposition of only two hybrid modes of the twisted isotropic fiber ($\mathrm{HE}_{|M+1|, L}^{\pm} / \mathrm{EH}_{|M-1|, L}^{\pm}$ characterized by the same sign of M for |M|≥2, and $\mathrm{HE}_{1, L}^{\pm} / \mathrm{HE}_{1, L}^{\mp}$ for M = 0). In Fig. 1 we show the flow chart of the calculations of the propagation constants and field distributions of modes of different orders in the twisted HB fibers using the proposed method.

 

Fig. 1. Block diagram showing the steps in the calculations of the propagation constants and field distributions for modes of different orders in the twisted HB fiber.

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The prosed methods is accurate and computationally very efficient because it does not require rigorous twisted FEM calculations. This made it possible to efficiently model the conversion of the LP₁₁ modes to vortex modes in gradually twisted HB fibers using the approach presented in the next section.

2.2. Modeling of the mode conversion process

In general, an accurate calculation of the transmission and reflection of electromagnetic waves at the interface between adjacent locally-invariant sections of optical fiber with a structure that varies along the direction of propagation requires the use of a large number of local modes (including radiation modes) in each section [23,24]. In this work, by local modes we mean the modes calculated for each section of the gradually twisted HB fiber, assuming that the parameters of the fiber and the twist rate A are constant over the section length. According to Eqs. (7–10), in the analyzed twisted HB fibers with centrally localized cylindrical core, propagation of modes of different orders can be analyzed separately. Although we focus on the twist-induced evolution of the first-order eigenmodes of a twisted HB fiber, which have the most complex structure of all higher-order modes [20], the proposed approach can also be applied to the evolution analysis of the modes of other orders.

One of the consequences of Eq. (8) is that the forward-propagating first-order eigenmodes of the analyzed HB fiber for any twist rate can be accurately represented as superpositions of the forward-propagating first-order eigenmodes of the twisted isotropic fiber. As a result, the first-order eigenmodes of the twisted isotropic fiber are sufficient to satisfy the field continuity relation; therefore, there is no reflection at the interface between the adjacent fiber sections characterized by twist rates A_i and A_i+1. Consequently, the transmission of any of the first-order modes or their superposition is described as follows:

To describe propagation along a fiber composed of N sections with nonzero lengths dz_i, we have to consider the phase changes introduced by the difference in propagation constants of the respective eigenmodes β_k(A_i). The relation between the input and output fields after propagation through multiple sections is expressed as:

Using Eq. (13), the transmission matrix T_i,1 is calculated, corresponding to twist rates changing from 0 to A_i along the fiber length, which allows analysis of the evolution of the field distribution propagating in a twisted anisotropic fiber with a varying twist rate. The fields obtained using Eq. (13) are expressed in the basis of the local eigenmodes of the twisted HB fiber, which depend on the twist rate, but can also be represented as a superposition of the modes of the twisted isotropic fiber using the coefficients $a_{km}^A$ introduced in Eq. (8):

The proposed approach can be considered as a simplified version of the EPM used recently for the analysis of twisted microstructured fibers, in which the eigenmodes cannot be represented as a superposition of the modes of the twisted isotropic fiber [18]. The propagation method formulated in this section does not require rigorous simulations to obtain the structure of the eigenmodes and can be easily integrated with the coupled-mode approach outlined in the previous section. Furthermore, as described in the following section, the proposed representation allows for a straightforward analysis of the evolution of spin angular momentum (σ) and orbital angular momentum (M) along the direction of propagation.

2.3. Angular momenta of first-order eigenmodes and their superpositions

The method described in the previous sections allows the expression of the first-order eigenmodes of the HB twisted fiber and their z-dependent superpositions in the basis of the first-order modes of the twisted isotropic fiber, Eq. (8). A significant advantage of using such basis is the possibility of a straightforward analysis of the evolution of spin angular momentum (σ) and orbital angular momentum (M) along the direction of propagation.

According to [31], for paraxial beams with field components E_z equal to zero, as in Eq. (4), the angular momentum vectors are directed along the z-axis, and their values (M and σ) are expressed by

Moreover, because ɛ as expressed in Eq. (3) is almost equal to unity matrix multiplied by a scalar ɛ, we may use the free space approximation of Eq. (16), in which ɛ is removed from the numerator and denominator. Using rigorous simulations, we verified that this approximation did not lead to significant inaccuracy, as the calculated absolute difference in the values of both M and σ for the eigenmodes of the analyzed fibers was less than 2×10⁻⁴. If a superposition of first-order local modes of the twisted HB fibers, which can be represented as a superposition of the first-order modes of the twisted isotropic fiber defined in Eq. (4):

The total angular momentum is determined only by the contributions of the HE₂₁^± eigenmodes characterized by J = ±2, whereas the spin and orbital angular momenta are also affected by the non-zero contributions of the TE₀₁ and TM₀₁ modes because their superpositions generally have non-zero values of σ = −M.

3. Evolution of first-order modes in twisted birefringent fibers

Using the proposed approach, we analyzed the evolution of LP₁₁ modes to vortex modes induced by gradual twisting in two HB fibers with different refractive index contrasts. We identified the reasons for the observed differences in the mode evolution and established guidelines for obtaining a quasi-adiabatic conversion of LP₁₁ modes to vector vortex modes for propagation distances of a few centimeters.

The low-contrast fiber we used was an idealized version of the fiber used as a mode converter in [19], with a core of diameter d = 6.19 µm doped with 6.1 mol% of GeO₂, resulting in a refractive index contrast δn = 8.75 × 10⁻³ for λ = 1000 nm. The high-contrast fiber had a core of diameter d = 3.41 µm doped with 20 mol% of GeO₂, resulting in a refractive index contrast δn = 2.87 × 10⁻² for λ = 1000 nm. Both fibers had the same linear birefringence Δn_l = 2.51 × 10⁻⁴ and the same normalized frequency (V = 3.10, λ = 1000 nm), ensuring two-mode propagation in the range from 800 to 1300 nm.

First, we characterized the evolution of the first-order eigenmodes versus twist rate as a function of the effective refractive indices in helicoidal coordinates (n’_eff= λβ’/(2π)) and of two other parameters that allow the analysis of similarities between the eigenmodes and the vector vortex modes, that is, the average ellipticity angle and the intensity distribution. It should be noted that in the paraxial approximation, the spin angular momentum σ determined by Eq. (20) is related to the intensity-weighted average of the ellipticity angle ϑ_av in the following way [32]:

 

Fig. 2. Intensity profile (a) and intensity along symmetry axes (b) of a sample eigenmode. Annularity coefficient Γ = [max(I_⊥)/max(I_||)] is defined as the ratio of maximal intensity along the mode’s symmetry axes indicated by red and black dots; Δx and Δy denote the distance from the center of the core.

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The behavior of n’_eff, ϑ_av, and Γ as functions of the twist rate expressed as the number of rotations per millimeter length (1/Λ = A/(2π) [1/mm]) is displayed in Fig. 3, for the first-order eigenmodes in the low-contrast and high-contrast fibers at λ = 1000 nm. We refer to the eigenmodes, which in the limit of high twist rates evolve toward TE₀₁, TM₀₁, and HE₂₁^± modes, as quasi-TE₀₁, quasi-TM₀₁, and quasi-HE₂₁^± modes, respectively. We have verified that the effective indices n’_eff obtained by the coupled-mode approach presented in the previous sections and the rigorous simulations with FEM were almost identical, differing by less than 4 × 10⁻⁷ for the low-contrast fiber and by less than 2 × 10⁻⁶ for the high-contrast fiber. Furthermore, the absolute value of the difference between the coefficients a_km describing the decomposition of the eigenmodes of the twisted HB fiber in the basis of the modes of isotropic twisted fiber (0≤|a_km |≤1) (see Eq. (8)) was less than 0.01 for the low-contrast fiber and less than 0.005 for the high-contrast fiber. As a result, the annularity coefficient Γ obtained using the perturbation method and FEM differed by less than 0.03 and 0.01, respectively for the low- and high-contrast fiber. Corresponding differences for the average mode ellipticity ϑ_av approached 1.5° and 3° in a narrow twist rate range. Such a good agreement validates the use of a simplified, much faster, coupled-mode approach in the analysis of mode evolution in gradually twisted HB fibers.

 

Fig. 3. Effective refractive index n’_eff in the helicoidal coordinate system (a, d), annularity coefficient Γ (b, e), and intensity-weighted average ellipticity angle ϑ_av (c, f) for the quasi-HE₂₁⁺ (red), quasi-TE₀₁ (green), quasi-TM₀₁ (blue), and quasi-HE₂₁⁻ (black) modes in the idealized low-contrast (a, b, c) and high-contrast (d, e, f) fibers at λ = 1000 nm. The results were obtained using the coupled-mode theory (Eq. (7)) (solid) and FEM (dotted). The majority of the curves overlap.

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For the non-twisted, low-contrast fiber, the linear birefringence Δn_l = 2.51×10⁻⁴ was much greater than the difference between the effective refractive indices of the TE₀₁ and TM₀₁ modes in the isotropic fiber (Δn_TE-TM = 1.13×10⁻⁵ at λ = 1000 nm), which resulted in two pairs of nearly degenerate LP₁₁ modes for 1/Λ = 0, as depicted in Fig. 3(a). Moreover, the evolution of Γ was nearly identical for all the eigenmodes, as depicted in Fig. 3(b), and the polarization in the pairs of fast and slow modes differed only by the sign of ϑ_av, as depicted in Fig. 3(c) [21]. In the high-contrast non-twisted fiber, the linear birefringence Δn_l = 2.51×10⁻⁴ was of the same order of magnitude as Δn_TE-TM = 1.21×10⁻⁴ at λ = 1000 nm, which lifted the degeneracy of the LP₁₁ modes, as depicted in Fig. 3(d). The lack of mode degeneracy led to a distinct evolution of the annularity coefficient Γ and the average ellipticity angle ϑ_av for all of the first-order eigenmodes [21], as depicted in Fig. 3(e),(f).

The intensity, transverse electric field, and ellipticity angle distributions for the first-order eigenmodes of the high-contrast fiber for the chosen twist rates at λ = 1000 nm are depicted in Fig. 4. The overlap coefficients between the presented fields calculated using the perturbation method and the FEM results differ from 1 by less than 2×10⁻⁴. The relationship between the initial LP₁₁ modes in the non-twisted fiber section and the modes they transformed into for high twist rates was not related to their polarization or symmetry, but only to the order of the values of n’_eff for 1/Λ = 0 [19]. Whereas all the eigenmodes had almost perfectly ring shapes for 1/Λ = 1 [1/mm], only the quasi-HE₂₁^± modes had a uniform polarization structure similar to the corresponding hybrid modes. Furthermore, Γ = 1 can be obtained not only for the single hybrid mode but also for the superposition of TE₀₁ and TM₀₁ modes, having nonzero ϑ_av, as seen for the quasi-TM₀₁ mode obtained for 1/Λ = 1 [1/mm], Fig. 4. Consequently, the polarization state of the eigenmodes represented by ϑ_av, or equivalently by σ, was a better indicator of similarity between the eigenmodes of the twisted HB fiber and TE₀₁, TM₀₁, and HE₂₁^± hybrid modes than the annularity coefficient Γ because, according to Eqs. (20) and (23), the value of ϑ_av equals ±45° or 0° only for pure HE₂₁^± or TE₀₁, TM₀₁ modes. For those twist rates that correspond with a twist-induced circular birefringence Δn_c = 4λ/Λ >> Δn_TE-TM, the relation between ϑ_av and 1/Λ for quasi-HE₂₁^± modes can be expressed as [21]:

 

Fig. 4. Intensity, transverse electric field and ellipticity angle distributions calculated for different twist rates 1/Λ for all of the first-order modes in the high-contrast fiber at λ = 1000 nm.

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In order to analyze the effect of crosstalk more directly, it is convenient to express the local electric field E(z,A) in the fiber cross-section as a superposition of the local first-order eigenmodes e_k:

We can identify the ranges of twist rate in which there is a possibility of significant crosstalk related to the term (e^A_k)*δe^A_m/δΛ⁻¹ by expressing it through the elements ${({\textrm{e}_k^{{A_{i + 1}}}} )^\ast }\textrm{e}_m^{{A_i}}$ of the T_i+1,i transfer matrix defined in Eq. (11):

 

Fig. 5. Sum of the moduli of (e^A_k)·δe^A_m/δΛ⁻¹ which can be seen as the upper limit of |dc_k/dΛ⁻¹|, with respect to 1/Λ for different eigenmodes [quasi-HE₂₁⁺ (red), quasi-TE₀₁ (green), quasi-TM₀₁ (blue), and quasi-HE₂₁⁻ (black)] for the low-contrast (a) and the high-contrast (b) fiber at λ = 1000 nm.

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In general, to reduce the crosstalk, the increase in the twist rate dΛ⁻¹/dz with the propagation distance should be the slowest at the beginning of the fiber. Thus, the small value of dΛ⁻¹/dz acts to reduce the detrimental effects of the large value of δe^A_m/δΛ⁻¹ and the small value of β_m − β_k in Eq. (26). Unfortunately, an implicit relation between the values of δe^A_m/δΛ⁻¹, the differences in β of the eigenmodes, and the parameters of the analyzed fibers cannot be expressed analytically in a concise way because of the complexity of the eigenvalue problem given by Eq. (7). However, we verified numerically that the reduction in Δn_l increases the value of δe^A_m/δΛ⁻¹ and decreases the value of β_m − β_k at the beginning of the twisted fiber. Therefore, it was expected that the energy exchange between the eigenmodes would be smaller in high-contrast fibers than in low-contrast fibers of the same length. Any combination of the fiber parameters and mode type would lead to different optimal variations of dΛ⁻¹/dz. In order to reduce complexity and establish the more general relations described in the latter part of this section, we limited our investigation to twist rate profiles that corresponded with 1/Λ ∝ zⁿ.

We describe the efficiency of the conversion from the first-order LP₁₁ mode to the first-order eigenmode of the twisted HB fiber in terms of the crosstalk µ, which is defined as the ratio of the power transferred to other eigenmodes to the power remaining in the targeted eigenmode:

In Fig. 6, we show the relation between the exponent n in the formula for the twist rate 1/Λ = (z/l)ⁿ/Λ_end and the minimum fiber length l needed to obtain −30 dB crosstalk at the output of the twisted fiber section, which can be considered as a requirement for quasi-adiabatic mode conversion. We show the relation for wavelengths λ = 700 nm and λ = 1000 nm. The results were obtained for 1/Λ_end = 1.75 [1/mm] for which, in the wavelength range 700–1000 nm, the intensity profiles of the eigenmodes are almost perfectly circularly symmetric (Γ > 0.99). Moreover, the intensity-averaged ellipticity angle ϑ_av differs from perfectly circular or linear polarization by no more than 3°. As shown in the following discussion, quasi-adiabatic conversion could be obtained in both analyzed fibers for all of the first-order modes in a broad spectral range, and the crosstalk could be arbitrarily reduced by increasing the fiber length. For both fibers, the conversion length depends on the mode type, wavelength, and exponent coefficient n, but it is systematically shorter (by approximately two orders of magnitude) for the high-contrast fiber with greater differences in β. The conversion length could be further reduced by increasing Δn_l, as it would lead to greater differences in β at the cost of obtaining the eigenmodes, which are less similar to the vector vortex modes HE₂₁^±, TE₀₁, TM₀₁ in terms of polarization. However, this effect could be compensated for by increasing 1/Λ_end.

 

Fig. 6. Minimal length of gradually twisted fiber required to obtain −30 dB crosstalk versus the exponent n in the relation 1/Λ = (z/l)ⁿ/Λ_end, 1/Λ_end = 1.75 1/mm, for quasi-HE₂₁⁺ (red), quasi-TE₀₁ (green), quasi-TM₀₁ (blue), and quasi-HE₂₁⁻ (black) modes in low-contrast (a) and high-contrast (b) fiber. The simulations were conducted for λ=700 nm (solid) and 1000 nm (dashed).

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As shown in Fig. 6, in the low n range, the required conversion length rapidly decreased with n, owing to the reduction of dΛ⁻¹/dz in the initial section of the fiber, which compensated for the high δe^A_m/δΛ⁻¹ shown in Fig. 5. For values of n lower than the optimal value, the exchange of energy between local modes occurred almost exclusively in the pairs of modes with the smallest differences in their propagation constants β, that is, in pairs of slow (quasi-HE₂₁⁺ /quasi-TE₀₁) and fast (quasi-TM₀₁/quasi-HE₂₁⁻) modes. As a result, the required conversion length was almost identical for the modes of a single pair. For values of n greater than the optimal value, the term dΛ⁻¹/dz = n(zⁿ⁻¹/lⁿ)/Λ_end becomes sufficiently high (especially for a larger propagation distance z) to increase the value of de^A_m/dz to the level needed for significant exchange of energy between the modes, even for the small values of δe^A_m/δΛ⁻¹ that occur for large z. As a consequence, significant crosstalk can arise between quasi-TE₀₁ and quasi-TM₀₁ modes for high twist rates (at greater propagation distances), for which the difference between β decreases, as shown in Fig. 3.

In Fig. 7(a),(b) we show the minimal lengths l required to ensure values of −30 dB and −40 dB for the crosstalk versus wavelength, in the range from 400 to 1000 nm, for all of the first-order modes in low-contrast and high-contrast fibers with twist rate profiles given by 1/Λ = (z/l)³ /1.75 [1/mm]. The exponent n = 3 was chosen because it is close to the optimum for high-contrast fiber and can be obtained experimentally [19].

 

Fig. 7. Minimal length of gradually twisted HB fiber ensuring −30 dB, and −40 dB crosstalk, versus wavelength, for quasi-HE₂₁⁺ (red), quasi-TE₀₁ (green), quasi-TM₀₁ (blue), and quasi-HE₂₁⁻ (black) modes in low-contrast (a) and high-contrast (b) fibers with twist rate profiles given by 1/Λ = (z/l)³/1.75 [1/mm]. The inverse of the difference between β (c) in pairs of modes (quasi-TM₀₁ and quasi-HE₂₁⁻) (solid) and (quasi-HE₂₁⁺ and quasi-TE₀₁) (dashed) in weakly twisted (1/Λ < 0.01 [1/mm]) low-contrast (red, left y axis) and high-contrast (blue, right y axis) fiber, which corresponds to the shape of respective curves shown in (a,b).

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The shapes of the curves in Fig. 7(a),(b) showing the spectral dependence of the minimal conversion length are similar to the curve showing the inverse of the difference between the propagation constants in pairs of quasi-HE₂₁⁺ /quasi-TE₀₁ and quasi-TM₀₁/quasi-HE₂₁⁻ modes in weakly twisted (1/Λ < 0.01 [1/mm]) HB fibers (Fig. 7(c)). This effect is related to the fact that for n = 3, the energy is exchanged primarily between the modes of a single pair in the weakly twisted initial fiber section. In the low-contrast fiber, the difference between the values of β was similar in both pairs of modes, leading to similar conversion lengths. In the high-contrast fiber, the modes in the quasi-TM₀₁/quasi-HE₂₁⁻ pair were much less separated in the long wavelength range and therefore required much greater conversion length than the modes in the quasi-HE₂₁⁺/quasi-TE₀₁ pair. Consequently, the required conversion length could be significantly reduced if only selected first-order vortex modes were generated. The results obtained for the quasi-HE₂₁⁺ and quasi-HE₂₁⁻ modes can be exchanged if the handedness of the twisted fiber is reversed. However, regardless of the handedness, in the analyzed fibers, the conversion length required to ensure a given level of crosstalk will be the greatest for the quasi-TM₀₁ mode, which is always placed in a pair of less separated modes.

In Fig. 8(a)-(c) we show the crosstalk versus propagation distance for a single eigenmode excited at the beginning of the gradually twisted high-contrast fiber with l = 9 cm, 1/Λ = (z/l)³/1.75 [1/mm] for λ = 400 nm, 700 nm, and 1000 nm. In addition, in Fig. 8(d)-(i), we compare the evolution of the intensity-weighted average ellipticity angle ϑ_av calculated using Eqs. (20) and (23) and the annularity coefficient Γ for the pure eigenmodes of the twisted HB fibers and the superposition of eigenmodes that arise owing to the residual power coupling between different eigenmodes when a single LP eigenmode is excited at the input. Convergence of the crosstalk simulations versus the length of fiber section with constant twist rate is shown in Fig. 9 for all wavelengths. These results prove that for the considered fiber structure and twist profile, the step of length of 10 µm is sufficient to obtain reliable results.

 

Fig. 8. Crosstalk (a, b, c), intensity-weighted average ellipticity angle ϑ_av (d, e, f), and annularity coefficient Γ (g, h, i) for the superpositions of the modes corresponding to a single LP₁₁ mode excited at the input, which evolves into quasi-HE₂₁⁺ (red), quasi-TE₀₁ (green), quasi-TM₀₁ (blue), or quasi-HE₂₁⁻ (black) mode in high-contrast fiber with l = 9 cm, 1/Λ = (z/l)³/1.75 [1/mm] obtained at λ = 400 nm (a, d, g), 700 nm (b, e, h) and 1000 nm (c, f, i). The dashed lines in (d-i), which overlap with solid lines for low crosstalk level (d,e,h), show the evolution of the pure eigenmodes.

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Fig. 9. Convergence test of the crosstalk coefficient as a function of the length of the fiber section with constant twist rate conducted for different wavelengths: 400 nm (a), 700 nm (b), and 1000 nm (c).

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The lowest crosstalk for all generated mode superpositions (less than −40 dB at the end of the fiber) was observed for λ = 700 nm. In this case, the generated output field practically consisted of one eigenmode selected by excitation of the respective LP₁₁ mode at the input to the gradually twisted HB fiber and was only residually contaminated by other eigenmodes. Therefore, the evolution of ϑ_av and Γ for such mode superposition was practically the same as for the respective individual eigenmodes. For λ = 700 nm, the average ellipticity ϑ_av differed from the pure circular or linear polarization of the vortex modes of the isotropic fiber by approximately 3°. Greater crosstalk was observed for λ = 400 nm and λ = 1000 nm; therefore, in Fig. 10, we show the intensity profiles of these more contaminated vortex modes. For λ = 400 nm, the crosstalk with other eigenmodes at the fiber end was close to −30 dB for all possible LP₁₁ excitations. As shown in Fig. 10, for such crosstalk levels there was a noticeable deviation from the perfectly annular shape. This was reflected in values of Γ = 0.87 for the superposition of the modes corresponding to quasi-TM₀₁ and quasi-HE₂₁⁻ modes and Γ = 0.91 for quasi-HE₂₁⁺ and quasi-TE₀₁ modes. Furthermore, |ϑ_av| declined to 40° for quasi-HE₂₁^± modes (which was directly related to the decrease in λ, Eq. (24)) and ϑ_av = ±14° for the quasi-TE₀₁ and quasi-TM₀₁ modes. For λ = 1000 nm, the twist-induced conversion to the quasi-HE₂₁⁺ and quasi-TE₀₁ modes was of very high purity (crosstalk lower than −50 dB); therefore, the evolution of ϑ_av and Γ for these modes along the twisted fiber was the same as for pure eigenmodes, and ϑ_av differed from pure circular or linear polarization by approximately 2° at the fiber output. However, for the quasi-TM₀₁ and quasi-HE₂₁⁻ modes, there was crosstalk of approximately −15 dB owing to a significant reduction in the propagation constant difference between these modes, as shown in Fig. 7(c). This resulted in an intensity distribution with two distinct maxima (Γ = 0.56), as shown in Fig. 10. Furthermore, owing to the high crosstalk, ϑ_av differed from the values obtained for the pure eigenmodes and was approximately −3° for the quasi-TM₀₁ mode and −37° for the quasi-HE₂₁⁻ mode. As shown in Fig. 7(b), a nearly two-fold increase in the conversion length for these modes (to l = 17 cm) was required to reduce the crosstalk to −30 dB at λ = 1000 nm.

 

Fig. 10. Intensity distributions for the superpositions of modes that arise when successive LP eigenmodes are purely excited at the input of the gradually twisted high-contrast fiber with l = 9 cm, 1/Λ = (z/l)³ /1.75 [1/mm], at λ = 400 nm and λ = 1000 nm.

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4. Conclusions

In this study, we propose a rapid and accurate numerical method for modeling the process of conversion of LP modes to vortex modes in gradually twisted HB fibers. The method is valid for fibers with a centrally located cylindrical core and uniform linear birefringence in the core region, which prohibits the twist-induced coupling between the modes of different orders. Our method utilizes the fact that the core eigenmodes in such fibers can be accurately represented as a superposition of the modes of the isotropic twisted fiber, belonging to the same mode group (HE₂₁^±, TE₀₁, and TM₀₁ in the case of the first-order eigenmodes). Consequently, the coupled-mode approach in helicoidal coordinates can be used to calculate the field distributions and propagation constants of the eigenmodes of gradually twisted fibers with an accuracy nearly identical to rigorous simulations conducted using FEM and the transformation optics formalism, but orders of magnitude faster. Using our method, modeling the conversion of the LP modes to vector vortex modes and calculating the crosstalk between the eigenmodes induced by the gradual change in the twist rate is simplified to the multiplication of 4×4 matrices for LP_1,L mode groups, or 2×2 matrices for all other mode groups. Furthermore, the proposed method allows for a straightforward analysis of the evolution of the orbital angular momentum and spin (related to the mode polarization) along the direction of propagation.

Using the developed numerical approach, we analyzed the conditions for quasi-adiabatic (that is, characterized by at most −30 dB of crosstalk) conversion of LP₁₁ modes to vortex eigenmodes in gradually twisted HB fibers. We examined the effects of the refractive index contrast, birefringence, and twist rate profile on the evolution of the first-order eigenmodes and their superpositions (generated due to crosstalk) in gradually twisted fibers. The obtained results revealed that the length of the gradually twisted HB fiber required for quasi-adiabatic mode conversion depended on all of the analyzed parameters and was a function of wavelength. However, significant crosstalk was most likely to occur in the initial, weakly twisted section of the fiber, where the differences between the values of β’ for different eigenmodes were the smallest, and the derivatives of the field profiles with respect to the twist rate were the greatest. Therefore, in order to reduce the length of the gradually twisted fiber required for quasi-adiabatic conversion, the twist rate increase should not be uniform, but should be at its slowest at the beginning of the fiber. Furthermore, the minimal length needed to reduce crosstalk below −30 dB depended primarily on the minimal difference between the values of β for the eigenmodes in a weakly twisted section of the fiber, which increased with the refractive index contrast. In the analyzed low-contrast fiber with refractive index contrast δn = 8.75 × 10⁻³ at λ = 1000 nm, a gradually twisted section more than 70 cm long was needed to obtain the quasi-adiabatic mode conversion. In the high-contrast fiber with δn = 2.87 × 10⁻² at λ = 1000 nm, −30 dB crosstalk was obtained for a fiber shorter than 4 cm. Furthermore, if the twist rate profile in high-contrast fiber followed the experimentally attainable dependence 1/Λ ∝ z³ [19], it was possible to obtain crosstalk below −30 dB for all of the first-order modes in the range between λ = 400 nm and λ = 800 nm, for a propagation distance less than 10 cm. The quasi-adiabatic conversion could be obtained in a broader wavelength range (from λ = 400 nm to λ > 1000 nm), even for a smaller propagation distance, if only quasi-HE₂₁⁺ and quasi-TE₀₁ modes were targeted.

The presented results prove that efficient quasi-adiabatic conversion of LP₁₁ modes to vortex eigenmodes could be obtained in gradually twisted fibers with lengths of a few centimeters. Broadband, in-fiber mode converters exploiting this effect may find novel applications in optical communication, laser technology, and metrology. The analyzed conversion process can also be used for building fiber-based broadband sources of vortex beams for free-space applications.

In the presented analysis we did not consider the effect of core imperfections, which can have a significant impact on the mode conversion process. In particular, the mode converter demonstrated experimentally in [19] had similar birefringence, core radius, and refractive index contrast as the low-contrast fiber analyzed in this work; however, it also had a slightly elliptical shape of the core, which lifted the near-degeneracy of the even and odd pairs of LP₁₁ modes of the same polarization. As a result, the quasi-adiabatic mode conversion length in this fiber was reduced to less than 3 cm, compared to the 70 cm obtained in numerical simulations of the fiber with a perfectly cylindrical core. Moreover, the residual ellipticity of the core altered the ordering of the LP₁₁ modes with respect to the value of the effective index, compared to the fiber with a cylindrical core. Consequently, the order of transformation of the LP₁₁ modes into vortex modes was changed. Therefore, it is expected that a more detailed study, taking into account the elliptical shape of the core and real birefringence distribution (including the birefringence of circular symmetry induced by core doping, which affects the effective indices of the TM₀₁ and TE₀₁ modes), will allow us to obtain better agreement between the simulation and the experimental results.