Rigorous simulations of a helical core fiber by the use of transformation optics formalism

Maciej Napiorkowski; Waclaw Urbanczyk

doi:10.1364/OE.22.023108

1. Introduction

Spiral coiling of a step-index fiber gives rise to two interesting features. The first one is a circular birefringence, which can be exploited in optical metrology, for example in fiber-optic current sensors based on the Faraday Effect [1]. The second one is an increase in the loss of higher-order modes caused by fiber bending, which leads to single-mode propagation, even for high normalized frequencies [2]. Single-mode large mode area (LMA) fibers obtained in this way have been studied for applications in high-power fiber lasers [3] to reduce power density and avoid nonlinear effects leading to a decrease in beam quality and fiber degradation.

Obtaining single mode fibers by coiling becomes increasingly difficult as the diameter of the fiber core increases. To suppress higher-order modes in larger cores, it is necessary either to use a core with lower numerical aperture, which is much more susceptible to fabrication and environmental perturbations, or to reduce the bending radius, which can induce stress sufficient to damage the coiled fiber. These limitations may be overcome by the use of helical-core fibers in which high curvature of the core can be obtained without introducing significant stress. This is possible because coiling of the fiber core is performed during the fabrication process by spinning a preform of the step-index fiber with an offset core [2]. First application of the helical-core fibers to high-power fiber lasers [4] led to increased interest in rigorous modeling of such structures, which is especially important because the pitch of the helical core cannot be modified after the fiber is fabricated. Another method of obtaining LMA fibers with helical structural elements has been proposed recently in [5,6]. In this case, a single mode operation is obtained thanks to coupling of higher-order modes propagating in the central straight core to the leaky modes propagating in the helical side cores.

To determine circular birefringence of the fundamental mode in helical-core fibers, simple models based on geometrical considerations [7,8] and coupled mode equations [9] have been proposed so far. Accurate simulations of the confinement loss and intensity profiles of guided modes, which are crucial in modeling helical-core LMA fibers, proved to be much more difficult. The simplest method of the estimation of the confinement loss in helical-core fibers uses approximate semi-analytical formulas developed by Marcuse [10,11]. This approach is based on equivalence between the helical-core fiber and the fiber bent in-plane, in which a bending radius is determined by the helix pitch and the core offset [11]. Marcuse’s bend loss model cannot be used to calculate losses in LMA fibers because it uses approximations valid only for fibers which are single-mode before bending or coiling. As it was shown in [12] the loss obtained for a multimode fiber with the Marcuse model can be overestimated by orders of magnitude. Attempts to improve the Marcuse model resulted in more precise semi-analytical bend loss formulas [12,13], including those designated especially to calculate the loss in helical-core fibers [14].

Properties of helical-core fibers were also studied using numerical methods. In [15] a finite difference method was used to model mode deformation in helical-core LMA fibers, which can significantly reduce the effective mode area of the guided modes [16,17]. In [15] the curvature of the core was represented as equivalent refractive index distribution frequently used to model in-plane bending. The observed five-fold compression of the fundamental mode area agrees qualitatively with experimental results but the approximations used in this method disregard polarization effects arising in real helical-core fibers. To conclude, none of the currently used approximate methods, neither semi-analytical nor numerical, can be used for rigorous vectorial modeling of the helical-core fibers.

Recently, a new full vectorial method of simulations based on the transformation optics formalism was used to rigorously model twisted optical fibers [18]. In this method, a geometry of the twisted fiber is represented as an equivalent change in permittivity and permeability in the straight fiber, which can be analyzed as a two-dimensional structure. The transformation method has already been used successfully to model propagation characteristics of twisted microstructured fibers [19,20]. In [20] the calculated polarization-dependent resonant coupling between circularly polarized fundamental and cladding modes predicted by numerical simulations was verified experimentally. Furthermore, in [21] the method was used to analyze an electrostatic field distribution in a helicoidal weakly twisted structure.

In this work we use the transformation optics formalism to perform first rigorous simulations of the helical-core fiber which in many cases lead to the results significantly different from those obtained previously with the use of the approximate methods [8,14,15]. The analysis of the fundamental modes characteristics shows that circular birefringence in a helical-core LMA fiber can be significantly modified by twist-induced mode deformation. Furthermore, the polarization and field distribution in the first-order modes calculated versus the helix pitch disagree with approximate results obtained using the methods based on an equivalent in-plane bending model [15]. Finally, the calculated loss of the fundamental and the first-order mode is substantially different from that predicted in [14], which results in different single-mode operation range.

2. Modeling method

Propagation characteristics of a helical-core fiber were modeled using a finite element method (FEM). As suggested in [18] the calculations were performed not in the Cartesian but in helicoidal coordinate system in which the helical-core fiber is invariant with respect to one of the coordinates. Helicoidal coordinates (ξ₁, ξ₂, ξ₃) are related to Cartesian coordinates (x,y,z) by the following equations [18]:

\begin{array}{l} x = ξ_{1} \cos (A ξ_{3}) + ξ_{2} \sin (A ξ_{3}) \\ y = - ξ_{1} \sin (A ξ_{3}) + ξ_{2} \cos (A ξ_{3}) \\ z = ξ_{3} \end{array},

where A is a twist rate related to the pitch Λ of the helical-core fiber in the following way:

| A | = 2 π Λ^{- 1},

where the positive value of A corresponds to the left-handed helix.

Instead of using different form of Maxwell’s equation in the non-Cartesian coordinate system, we represent the change of coordinates used in the simulations as an equivalent change of permittivity ε and permeability µ tensors [22]:

ε^{i' j'} = {| \det (J^{i'}_{i}) |}^{- 1} J^{i'}_{i} J^{j'}_{j} ε^{i j},

μ^{i' j'} = {| \det (J^{i'}_{i}) |}^{- 1} J^{i'}_{i} J^{j'}_{j} μ^{i j},

where J^i’_i is the Jacobian transformation matrix defined in the following way:

J^{i'}_{i} = \frac{\partial x^{i'}}{\partial x^{i}} .

In case of materials isotropic in Cartesian coordinates (ε and µ are scalars), the equivalent ε’ and µ’ in the helicoidal coordinate system are invariant with respect to ξ₃ [18]:

ε' = (\begin{matrix} 1 + A^{2} ξ_{2}^{2} & - A^{2} ξ_{1} ξ_{2} & - A ξ_{2} \\ - A^{2} ξ_{1} ξ_{2} & 1 + A^{2} ξ_{1}^{2} & A ξ_{1} \\ - A ξ_{2} & A ξ_{1} & 1 \end{matrix}) ε,

μ' = (\begin{matrix} 1 + A^{2} ξ_{2}^{2} & - A^{2} ξ_{1} ξ_{2} & - A ξ_{2} \\ - A^{2} ξ_{1} ξ_{2} & 1 + A^{2} ξ_{1}^{2} & A ξ_{1} \\ - A ξ_{2} & A ξ_{1} & 1 \end{matrix}) μ .

Because in the helicoidal coordinates the geometry of the fiber and the equivalent ε’ and µ’ tensors depend only on ξ₁ and ξ₂, we can represent a helical-core fiber by a two-dimensional model. Such approach conserves full information about electromagnetic phenomena and significantly reduces memory consumption in comparison with 3D models. In case of arbitrary anisotropic materials, the equivalent ε’ and µ’ may depend on ξ₃ and therefore the proposed method does not necessarily lead to the two-dimensional model. Despite this fact, it is possible to define twisted cylindrical Perfectly Matched Layers (PML) absorbing boundary conditions, which can be used to model rigorously the propagation loss in twisted optical fibers [19].

In our work we used the transformation optics formalism to calculate the loss, the effective refractive index, the polarization structure and intensity profiles for the fundamental and the first-order modes in the helical-core fiber.

3. Helical-core fiber structure

The simulations were performed for the helical-core fiber with a cross-section shown in Fig. 1.

Fig. 1 Cross-section of the analyzed helical-core fiber. Arrow shows the twist direction.

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To easily compare our rigorous results with those obtained previously by approximate methods, we modeled the fiber of the same geometry as analyzed in [14]. The simulations were conducted for three different core diameters d = 20 µm, 40 µm, 60 µm and three core offsets Q = 100 µm, 200 µm, 300 µm. In most of the simulations, we assumed that the refractive index of the cladding is equal to the refractive index of silica glass n_cl = 1.44976 at λ = 1053 nm. Numerical aperture NA = (n_co² - n_cl²)^1/2 = 0.1 was obtained by setting the core refractive index n_co to 1.45321. In a few series of calculations performed for wavelengths different than λ = 1053 nm, material dispersion was not taken into account to better demonstrate changes in propagation characteristics caused by the fiber twist.

4. Simulation results

In the first step we calculated in the helical coordinate system (ξ₁, ξ₂ ξ₃) the effective refractive indices n’_eff of the fundamental modes in three helical-core fibers with different core diameters d = 20 µm, 40 µm, 60 µm and the core offset Q = 100 µm. In Fig. 2 we show the calculated n’_eff versus the twist rate 1/Λ, which represents a number of twists for one millimeter length of the fiber.

Fig. 2 Effective indices n’_eff of the fundamental modes calculated versus the twist rate 1/Λ for the helical-core fiber with core diameters d = 60 µm (blue), 40 µm (red), 20 µm (green) and core offset Q = 100 µm, λ = 1053 nm. Fundamental modes have either left-handed circular (solid line) or right-handed circular (dotted line) polarization.

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As it is shown in Fig. 2, the effective indices n’_eff calculated in the helicoidal coordinate system rise rapidly with the twist rate 1/Λ. This is a purely geometrical effect and reflects the fact that the light travels a longer path along the spiral core.

To calculate the birefringence, the values of effective indices n’_eff must be first transformed from the helicoidal to Cartesian coordinate system using the corrections, which depend on the azimuthal modal number m and the mode polarization. We define the left-handed circular polarization as having a clockwise temporal rotation of an electric vector, looking in the direction of the source. If the mode and the helical core have the same handedness, the correction is equal to -mλ/Λ for HE modes and + mλ/Λ for EH modes; if the handedness is opposite, the sign of both corrections is reversed. In the previous publications, these corrections were derived from the coupled mode equations in helicoidal coordinates [5] or the modification of a mode angular momentum (spin) operator [23]. In this work, we point to the fact that the corrections can be also obtained by analyzing the field evolution in the hybrid modes caused by a rotation of the Cartesian coordinate system around the fiber symmetry axis (ξ₃). In the Cartesian coordinate system, the electric field evolution of the mode guided in the fiber with centrally located core can be represented as:

\vec{E} (r, θ, z, t) = \vec{E} (r, θ) \exp (- j β z) \exp (j ω t),

where

\vec{E} (r, θ)

is the electric field distribution, β is the mode propagation constant and ω is the mode frequency. Let us consider a HE mode characterized by the azimuthal mode number m propagating in the untwisted fiber with centrally located core. For such a mode, an angular modulation of the radial E_r and azimuthal E_θ components of the electric field can be expressed as E_r(r)sin(mθ) and E_θ(r)cos(mθ), respectively. If we observe such a HE mode in the helicoidal coordinate system rotating around the fiber axis with the twist rate A = 2π/Λ, there will be an additional modulation of E_r and E_θ components, which depends on z coordinate:

E_{r} (r) \sin (m (θ - A z)) = - j \frac{E_{r} (r)}{2} [\exp (j m (θ - A z)) - \exp (- j m (θ - A z))],

E_{θ} (r) \cos (m (θ - A z)) = \frac{E_{θ} (r)}{2} [\exp (j m (θ - A z)) + \exp (- j m (θ - A z))],

where r is the distance from the center of the core.

By grouping the exponential factors in the above equations, one can separate two circularly polarized modes, respectively the right-handed circularly polarized (RHC) mode:

\frac{1}{2} [\begin{matrix} - j E_{r} (r) \\ E_{θ} (r) \end{matrix}] \exp (j m θ) \exp (- j m A z) = {\vec{E}}_{R H C} (r, θ) \exp (- j m A z)

and the left-handed circularly polarized (LHC) mode:

\frac{1}{2} [\begin{matrix} j E_{r} (r) \\ E_{θ} (r) \end{matrix}] \exp (- j m θ) \exp (j m A z) = {\vec{E}}_{L H C} (r, θ) \exp (j m A z) .

In the helicoidal coordinate system, the factor exp( ± jmAz) lifts the degeneracy of the propagation constant β of the circularly polarized modes:

\begin{matrix} {\vec{E}}_{R H C} (r, θ, z, t) = {\vec{E}}_{R H C} (r, θ) \exp (- j m A z) \exp (- j β z) \exp (j ω t) = \\ = {\vec{E}}_{R H C} (r, θ) \exp (- j (β + m A) z) \exp (j ω t) = \\ = {\vec{E}}_{R H C} (r, θ) \exp (- j β_{R H C} z) \exp (j ω t), \end{matrix}

\begin{matrix} {\vec{E}}_{L H C} (r, θ, z, t) = {\vec{E}}_{L H C} (r, θ) \exp (j m A z) \exp (- j β z) \exp (j ω t) = \\ = {\vec{E}}_{L H C} (r, θ) \exp (- j (β - m A) z) \exp (j ω t) = \\ = {\vec{E}}_{L H C} (r, θ) \exp (- j β_{L H C} z) \exp (j ω t) . \end{matrix}

Consequently, the effective refractive indices n’_eff of the circularly polarized modes in the helicoidal coordinate system are given by:

n'_{e f f} = (β \pm m A) \frac{λ}{2 π} = n_{e f f} \pm m \frac{λ}{Λ} .

In case of the left-handed helical core fiber characterized by A > 0, n’_eff of the right-handed HE modes is raised and n’_eff of the left-handed HE modes is lowered by a factor of mλ/Λ. Analogous reasoning for EH modes results in the reversed relationship. The consideration outlined above leads to the conclusion that rotation of the coordinate systems induces an additional circular birefringence equal to 2mλ/Λ, which must be subtracted to determine the real circular birefringence observed in the Cartesian coordinate system. The corrected values of birefringence are then compared with the circular birefringence obtained from an approximate formula presented in [7], which is based on geometrical considerations neglecting the core diameter [8]:

B = \frac{2 λ (S - Λ)}{Λ S} = \frac{2 λ (\sqrt{{(2 π Q)}^{2} + Λ^{2}} - Λ)}{Λ \sqrt{{(2 π Q)}^{2} + Λ^{2}}},

where S is the arc length of the core for one pitch. Circular birefringence obtained from Eq. (16) and the results of numerical simulations for the fibers with different core diameters are compared in Fig. 3.

Fig. 3 Comparison between circular birefringence in Cartesian coordinates calculated numerically using transformation optics formalism (color lines) and obtained from Eq. (16) (black line) as a function of the twist rate 1/Λ. Simulation parameters: Q = 100 µm, λ = 1053 nm.

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The circular birefringence calculated numerically using the transformation optics formalism is higher than the birefringence obtained from Eq. (16) and the observed differences rise with the core diameter and the twist rate 1/Λ. This discrepancy is a result of mode deformation caused by the curvature of the helical core, which is not taken into consideration when deriving Eq. (16). The Eq. (16) is based on a purely geometrical model, in which the position of the center of the fundamental mode follows the helix defined solely by the core offset Q and the pitch Λ. In reality, especially in case of large core fibers, the fundamental mode is significantly displaced in the direction of the bent. Consequently, the distance between the center of the fundamental mode and the center of the fiber becomes greater than the core offset Q. As a result, the fundamental mode passes a longer path S per one helix turn, which leads to a greater circular birefringence according to Eq. (16). One can verify this explanation by calculating, with the use of Eq. (16), the effective mode offset Q_eff which results in the same circular birefringence as obtained from the numerical simulations (B_s):

Q_{e f f} = \sqrt{\frac{{(\frac{2 B_{s} Λ}{2 B_{s} - λ Λ})}^{2} - Λ^{2}}{4 π^{2}} .}

In Fig. 4 we show the calculated value of Q_eff as a function of a pitch rate and different core diameter. The values of Q_eff obtained in this way match the distance between the center of the fundamental mode and the center of the fiber. This confirms the fact that the observed effect of a circular birefringent increase versus the core diameter is related only to the fundamental mode displacement.

Fig. 4 Calculated effective mode offset Q_eff versus the twist rate 1/Λ related to the displacement of the fundamental mode in a helical-core fiber. Insets show intensity profiles calculated for 1/Λ = 0.2 mm⁻¹. Simulation parameters: d₁ = 60 µm (blue), d₂ = 40 µm (red), d₃ = 20 µm (green), Q = 100 µm and λ = 1053 nm.

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We have also analyzed the spectral dependence of circular birefringence in helical core fibers. A comparison of circular birefringences calculated versus the wavelength for different core diameters in the helical core fiber with the offset Q = 100 µm is shown in Fig. 5. As predicted by Eq. (16), the birefringence increases against the wavelength but in real fibers the growth is greater than that foreseen by Eq. (16). This effect is caused by the twist-induced fundamental mode displacement.

Fig. 5 Comparison of circular birefringence calculated versus 1/Λ for λ = 1053 nm (solid lines) and λ = 500 nm (dotted lines) (a) and versus the wavelength for 1/Λ = 0.15 mm⁻¹ (b). Simulation parameters d₁ = 60 µm (blue), d₂ = 40 µm (red), Q = 100 µm; black lines represent birefringence claculated using Eq. (15).

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In the next step, we have analyzed a behavior of the first-order modes guided in helical-core fibers. The results obtained by the rigorous method differ significantly from the predictions based on approximate methods. In [15] the mode intensity profiles in a helical core fiber are obtained using equivalence between the helical-core fiber and the fiber bent in-plane as postulated by Marcuse [11]. Such assumption implies that eigenmodes of a helical-core fiber are linearly polarized LP modes guided in a fiber bent in-plane. Therefore, using this simplified approach, it is not possible to model the experimentally observed circular polarization in the first order modes [7,8].

Complex polarization effects observed in the helical-core fiber make it impossible to represent its higher order eigenmodes as either linearly polarized LP modes or HE/EH hybrid modes guided in standard step-index fibers with a circular core. Because amplitude distribution in the first-order modes of a weakly twisted helical-core fiber resembles the distorted hybrid modes HE^e₂₁, HE^o₂₁,TE₀₁ and TM₀₁, in the later part of this work we will refer to them respectively as: HE₂₁’ + , HE₂₁’-, TE₀₁’- and TM₀₁’ + . The calculated n’_eff of the first-order modes versus the twist rate 1/Λ is presented in Fig. 6.

Fig. 6 Effective refractive indices n’_eff calculated in the helicoidal coordinate system versus the twist rate 1/Λ for four first-order modes of a helical-core fiber (a) and the difference between respective n’_eff and the average value n’_av of effective indices of four modes (b). Simulation parameters: d = 20 µm, Q = 100 µm, λ = 1053 nm.

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The effective refractive indices n’_eff of the first-order modes rise with the twist rate in a way similar to n’_eff of the fundamental modes but in this case the difference n’_eff-n’_av becomes a strongly nonlinear function of the twist rate for 1/Λ greater than 0.1 mm⁻¹. This is caused by significant mode deformation, which leads to the separation of two first-order mode pairs with intensity profiles resembling those of linearly polarized LP₁₁ modes. The first pair is composed of HE₂₁’- and TM₀₁’ + modes, while the second pair of HE₂₁’ + and TE₀₁’- modes. The intensity profiles in the modes making one pair become more similar with the rising twist rate 1/Λ, while the field structure gradually evolves towards the circular polarization with opposite handedness. Modes marked with ‘-‘ have the same handedness as the helical core, while those marked with ‘ + ‘ have the opposite handedness.

The values of n_eff for all the first order modes transformed to the Cartesian coordinate system are presented in Fig. 7.To determine n_eff of higher-order modes in Cartesiancoordinates, one must use the correction ± mλ/Λ represented by Eq. (15). In the analyzed case, the corrections should be applied only to HE₂₁’ + and HE₂₁’- modes characterized by the azimuthal mode number m = 2. The effective index n_eff of the TM₀₁’ + and TE₀₁’- modes with m = 0 is not modified by the transformation between the helical and Cartesian coordinate system. Twist-induced changes in the effective indices n_eff of individual first-order modes are accompanied by evolution of the intensity profiles presented in Fig. 8.

Fig. 7 Difference between n_eff of individual modes and the average value n_av of effective indices of four modes calculated in the Cartesian coordinate system versus the twist rate 1/Λ. In graph (a) we present the simulation results for the full analyzed range, while in graph (b) only for a weakly twisted fiber. Simulation parameters: d = 20 µm, Q = 100 µm, λ = 1053 nm.

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Fig. 8 Evolution of the intensity profiles of the first-order modes in a helical-core fiber (d = 20 µm, Q = 100 µm) versus twist rate 1/Λ. Wavelength λ = 1053 nm.

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Analyzing the simulation results presented in Fig. 7 and 8 we can identify three characteristic phases in the evolution of the first-order modes versus the twist rate. As the first one we can distinguish a weak twist rate (0<1/Λ≤0.03 mm⁻¹) for which the intensity profiles of the first-order modes and differences between their effective indices are nearly identical to those in the untwisted fiber. For the intermediate twist rate (0.03<1/Λ≤0.1 mm⁻¹), we observed small changes in n_eff-n_av against 1/Λ and gradually increasing disturbances in the modes’ intensity profiles. At first, around 1/Λ≈0.7 mm⁻¹, the circular symmetry of intensity profiles in HE₂₁’ ± modes is broken by pairs of maxima, which are oriented perpendicularly in the modes HE₂₁’ + and HE₂₁’-. Similar process occurs for the TE₀₁’- and TM₀₁’ + modes at around 1/Λ≈1 mm⁻¹. It leads to separation of two modes pairs, respectively HE₂₁’-/TM₀₁’ + with an intensity prolife resembling LP₁₁^o mode and HE₂₁’+/TE₀₁’- resembling LP₁₁^e mode. For a high twist rate (1/Λ>0.1 mm⁻¹), the deformation and displacement in the direction of the bent lead to stronger azimuthal modulation of the modes intensity profiles, which become even more similar to the intensity distribution in the LP₁₁ modes. In this regime, all the first-order modes become circularly polarized with opposite handedness in each pair. Additionally, for 1/Λ>0.16 mm⁻¹, the intensity profiles of the modes lose their plane symmetry, which is related to a helical shape of the fiber core.

For a weakly twisted fiber (1/Λ≤0.03 mm⁻¹), the difference between the effective indices n_eff of the first order modes is nearly equal to the intermodal dispersion observed in the untwisted fiber. In case of a strongly twisted fiber (1/Λ>0.1 mm⁻¹), the difference between n_eff of two circularly polarized modes in the pairs HE₂₁’-/TM₀₁’ + and HE₂₁’+/TE₀₁’- can be regarded as circular birefringence in the higher order modes. Its values are located symmetrically with respect to the circular birefringence of the fundamental mode, Fig. 9. The small dissimilarity in circular birefringence of the fundamental modes and the pairs of the first order modes HE₂₁’-/TM₀₁’ + and HE₂₁’+/TE₀₁’- is caused by differences in intensity distributions in those modes, which results in slightly different effective offsets.

Fig. 9 Difference in effective indices n_eff for the modes pairs HE₂₁’-/TM₀₁’ + (red) and HE₂₁’+/TE₀₁’- (blue) and circular birefringence of the fundamental modes calculated versus the twist rate 1/Λ. Simulation parameters d = 20 µm, Q = 100 µm, λ = 1053 nm.

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Rigorous numerical simulations allow also to analyze the twist-induced polarization changes in the first-order modes. Our numerical results confirm that in the helical-core fiber the modes HE₂₁’ ± are circularly polarized even for very small twist rates as foreseen by analytical Eqs. (8)–(11). Polarization of the TM₀₁’ + and TE₀₁’- modes changes gradually from locally linear (ellipticity angle ≈0°) for a weak twist rate to globally circular (ellipticity angle ≈ ± 45° in the whole mode cross-section) for a strongly twisted fiber. The distributions of the ellipticity angle over the surface of the TM₀₁’ + mode, which is similar in the TE₀₁’- mode, and the ellipticity angle at extreme points calculated versus the twist rate 1/Λ for the TM₀₁’ + and TE₀₁’- modes are presented in Fig. 10.

Fig. 10 Surface distribution of the ellipticity angle in TM₀₁’ + mode for 1/Λ = 0.08 mm⁻¹ (a) and ellipticity angle in the modes TE₀₁’- (blue) and TM₀₁’ + (red) calculated at the extreme points 1 (ξ₁ = 100 µm, ξ₂ = 10 µm - solid line) and 2 (ξ₁ = 110 µm, ξ₂ = 0 - dashed line) versus 1/Λ (b). Simulation parameters: d = 20 µm, Q = 100 µm, λ = 1053 nm.

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Using the rigorous transformation optics approach, we have also studied the effect of a twist rate on the confinement loss of the fundamental and the first order modes guided in the helical-core fiber. The difference in loss characteristics of the fundamental and the first-order modes determines the single-mode operation range of such fibers. In Fig. 11 the rigorously calculated loss characteristics for the fibers with the core diameter d = 60 µm and two different core offsets Q = 100 µm and 300 µm are compared to the approximate results obtained in [14] using an equivalent in-plane bent model. In the approximate bent equivalent model, the first-order mode characterized by the lowest loss is the LP₁₁^e mode [14], while in the rigorous model it is the pair of circularly polarized HE₂₁’+/TE₀₁’- modes with an intensity profile similar to the LP₁₁^e mode.

Fig. 11 Comparison of the loss calculated with rigorous (red lines) and approximate (black lines) methods [14] as a function of the helix pitch Λ. Solid lines indicate the fundamental modes, while dashed lines the first-order modes with lower loss. Simulation parameters: d = 60 µm, Q₁ = 100 µm, Q₂ = 300 µm, λ = 1053 nm.

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As it is shown in Fig. 11, the confinement loss predicted by the rigorous model is much lower than the loss obtained with the approximate method. Moreover, in the approximate model, the cut-off of the fundamental and the first order modes and the single-mode operation range are shifted towards a greater helix pitch range. In consequence, the single-mode helical-core fiber designed with the approximate model would be actually multimode. Furthermore, the approximate model predicts the increase in the single mode operation range against the core offset Q, while in the rigorous model, the difference in the loss characteristics for the fundamental and the first order modes practically does not depend on this parameter. The discrepancy between both simulation methods would most likely be smaller, if the approximate model proposed in [14] did not disregard the twist-induced mode deformation. In the single-mode operation range of the helical-core fiber characterized by d = 60 µm and Q = 100 µm, the fundamental mode is displaced by more than 20 µm from the center of the core and significantly compressed, which leads to lower propagation loss.

Additional calculations were performed to determine the single-mode operation range for the helical-core fiber with smaller core diameters d = 40 µm and 20 µm. The obtained results presented in Fig. 12 show that the reduction in core diameter widens the single-mode operation range of helical-core fibers as predicted in [14].

Fig. 12 Confinement loss characteristics calculated for the fundamental mode (solid line) and the first order modes of lower losses (modes pair HE₂₁’+/TE₀₁’- dashed line) versus the helix pitch Λ determining the single-mode operation range. Simulation parameters: d₁ = 60 µm (blue), d₂ = 40 µm (red), d₃ = 20 µm (green), Q = 100 µm, λ = 1053 nm.

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5. Summary

We are first to perform rigorous numerical simulations of a helical-core fiber using the transformation optics formalism. Basing on fully vectorial calculations, we determined the effective indices, intensity profiles, polarization and confinement losses of the fundamental and the first-order modes. The obtained results in many cases differ significantly from the predictions of approximate methods based on an equivalent in-plane fiber bent. In particular, our simulations show a significant increase in circular birefringence caused by mode deformation in LMA fibers, which was not taken into account by approximate models disregarding the size of the fiber core. This effect leads to 40% increase in circular birefringence of the fundamental mode in the fiber with 60 μm core compared to the values obtained by analytical formulas used so far.

Furthermore, using the rigorous transformation optics formalism we were able to analyze a complex twist-induced polarization evolution of the first-order modes, which could not be accounted for in the approximate methods based on the equivalent in-plane bending. Our simulations show that with a rising twist rate four hybrid modes guided by the untwisted fiber: HE₂₁^e, HE₂₁^o,TM₀₁ and TE₀₁ gradually evolve into two pairs of circularly polarized modes: HE₂₁’+/TE₀₁’- and HE₂₁’-/TM₀₁’ + with intensity profiles resembling respectively the LP₁₁^e and LP₁₁^o modes.

Finally, the most serious discrepancies between the rigorous and approximate methods were observed in the simulations of confinement losses for the fundamental and the first order modes, which define the single mode operation range. The rigorous model based on transformation formalism manifested in this case a significant advantage over the approximate models, which overestimate the propagation loss, especially in case of higher order modes.

Acknowledgment

This work was supported by Wroclaw Research Center EIT + Ltd. in the frame of the NanoMat project “Application of Nanotechnology in Advanced Materials”, within the European Funds for Regional Development, POIG, Sub-action 1.1.2.

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Rigorous simulations of a helical core fiber by the use of transformation optics formalism

Abstract

1. Introduction

2. Modeling method

3. Helical-core fiber structure

4. Simulation results

5. Summary

Acknowledgment

References and links

Cited By

Figures (12)

Equations (17)

Optics Express