Bent optical waveguide finite element analysis with a 3D envelope Maxwell model

Abstract: With the goal of accurately extracting the optical field losses in a three-dimensional (3D), circularly coiled waveguide (e.g., bent optical fiber), this effort presents the numerical methodologies that are implemented for an envelope Maxwell model that propagates electromagnetic fields as an entirely boundary value problem. Our unique modeling approach includes an ultraweak variational formulation of the envelope Maxwell model in the curved geometry of the bending, which is discretized by the discontinuous Petrov-Galerkin (DPG) method, which permits residual-driven mesh and polynomial-order adaptivity. This also, then, requires a unique approach for constructing perfectly matched layers (PMLs) as absorbing boundary conditions in both the direction of optical field propagation and in the tangential directions, where unguided energy escapes the waveguide. Our coiled waveguide modeling technology extracts the mode confinement losses from the propagation of the coherent optical field through the bent waveguide. We verify our simulations against the semi-analytical results from the analogous bent slab waveguide problem, and we successfully demonstrate stable convergence to loss values for the 3D coiled optical fiber problem, which has never been done previously for our specific modeling approach.

• The paper introduces a robust 3D envelope Maxwell model using DPG discretization to capture bend-induced confinement loss in optical waveguides.
• It validates the method against analytical benchmarks and semi-analytic Bessel solutions across various fiber geometries and bend radii.
• The study demonstrates effective mesh adaptivity and PML integration to resolve full 3D guided and radiative field interactions in bent fibers.

Finite Element Analysis of Bent Optical Waveguides via 3D Envelope Maxwell Models

Introduction and Context

The modeling and quantification of bend-induced confinement loss in coiled optical waveguides is critical for designing advanced photonic devices, particularly for large-mode-area (LMA) fiber lasers and amplifiers where differential mode suppression and thermal management are essential. The paper "Bent optical waveguide finite element analysis with a 3D envelope Maxwell model" (2604.00155) introduces a high-fidelity computational framework for directly solving the full vectorial Maxwell boundary value problem (BVP) in physically realistic, fully three-dimensional, toroidal fiber geometries. The approach leverages an envelope-based reformulation of Maxwell's equations coupled with a discontinuous Petrov-Galerkin (DPG) discretization and residual-driven $hp$-adaptivity.

The authors target the robust extraction of mode confinement loss—distinguishing guided versus radiative optical energy—in bent fibers, explicitly modeling the interaction of guided and radiative field components with all relevant material interfaces. Notably, the loss mechanism associated with geometric bending is captured without resorting to modal decomposition or paraxial approximations, enabling the assessment of arbitrary fiber cross-sections and the accommodation of real-world refractive index distributions.

Physical and Mathematical Model

The problem geometry is that of a step-index optical fiber segment spliced into a toroidal (circularly bent) extension, with the bend radius $r_0$ substantially exceeding the fiber radius $a$, as depicted in (Figure 1).

Figure 1: Schematic of a straight fiber spliced to a bent fiber segment, illustrating the transition from a perfectly guiding regime to a geometry supporting confinement loss due to curvature.

The physical fiber consists of a high-index glass core, lower-index glass cladding, and an outer polymer coating, with their characteristic dimensions and refractive index step structure illustrated in (Figure 2).

Figure 2: Cross-sectional step-index fiber structure with core, cladding, and coating regions and the associated refractive index profile.

Maxwell's equations are cast in a time-harmonic, full-vectorial form with complex-valued, spatially varying permittivity and permeability. Crucially, the field ansatz employs a full-envelope shift in the $\theta$ angular coordinate—$$E(x, r, \theta) = e^{-ik r_0 \theta} \mathcal{E}(x, r, \theta)$$—where the envelope model reduces oscillatory behavior for improved numerical tractability in long-domain simulations. This ansatz leads to modified curl and material tensors that naturally encode the geometric effects of curvature.

To handle the unbounded nature of physical domains (where radiative modes escape radially and longitudinally), specialized perfectly matched layers (PMLs) are developed for both the longitudinal ($\theta$) and transverse (radial $r$ and $\rho$) coordinates. The envelope Maxwell system is then formulated as an ultraweak variational problem and discretized using the DPG finite element method. The DPG approach equips the scheme with robust residual-based adaptivity and a posteriori error control, essential given the multiscale nature of bent fiber physics.

Formulation and Discretization

The ultraweak variational formulation places the domain in $[L^2(\Omega;\mathbb{C}^3)]^2$ (pairs of electric and magnetic fields), with test functions in a broken $[H(\mathrm{curl}, T)]^2$ space. The DPG method introduces interface trace unknowns and employs optimal test functions generated locally, which simultaneously enhance stability and provide sharp residual-based error indicators.

The bulk computation proceeds on exact geometry meshes supporting non-affine, e.g., true toroidal mappings—a crucial feature for avoiding geometric dispersion errors that quickly dominate in high-frequency, strongly curved domains. The implementation utilizes the $r_0$0 codebase, featuring static condensation and scalable hybrid parallelism, allowing for the solution of very large 3D fiber problems.

Numerical Experiments: Convergence and Loss Extraction

Three main classes of numerical experiments are presented:

1. Verification via 2D Vacuum and Slab Waveguides

The first set of experiments considers a bent vacuum slab waveguide with perfect electric conducting (PEC) boundaries. The system admits analytic solutions via separation of variables, serving as a stringent benchmark for the DPG discretization. Uniform $r_0$1-refinements, coupled with polynomial orders $r_0$2, $r_0$3, and $r_0$4, yield DPG residuals and field errors converging at the theoretically expected rates, confirming the soundness of the formulation.

2. Confinement Loss in Open Step-Index Slab Waveguides

The second set addresses step-index slab waveguides with a radiation boundary condition implemented via a PML in the radial direction (see Figure 3).

Figure 3: Domain setup for open bent slab waveguide with marked regions for core, cladding, coating, and corresponding PMLs in radial and longitudinal directions.

Transverse electric (TE) modes are imposed as input conditions, and the simulations assess the attenuation of guided power due to curvature-induced loss. The convergence of the DPG residual is shown for several modes and bend radii. Loss exponents (twice the imaginary part of the propagation constant extracted from the integrated Poynting vector flux) computed numerically are found to match semi-analytic results from high-precision Bessel equation solvers to high accuracy:

Figure 4: Comparison between numerically extracted (via power decay) and theoretical loss exponents for the first even mode with $r_0$5, demonstrating nearly exact correspondence.

The agreement holds for a range of modes and bend radii, except at extremely low-loss regimes where numerical estimation approaches machine precision.

3. Full 3D Bent Fiber Simulations

The final, most computationally demanding experiment simulates LP$r_0$6 mode propagation in a realistic bent step-index fiber. The 3D geometry allows for a PML in both $r_0$7 and $r_0$8, and utilizes full mesh adaptivity to resolve the detailed escape of the field from the core into the cladding and coating regions.

The field distributions and mesh refinement patterns are visualized in (Figure 5) and (Figure 6):

Figure 5: Visualization of the final adapted mesh including 3D domain view, transverse input cross-section, and longitudinal slice, demonstrating anisotropic, physics-aware refinement in critical regions.

Figure 6: Irradiance profile at multiple cross sections along the fiber, capturing the progressive leakage of optical power as the mode traverses the bent region.

The loss of guided power is plotted as a function of arc length for two bend radii, showing significant differential confinement loss magnitudes (Figure 7):

Figure 7: Decay of optical power for input LP$r_0$9 mode in 3D bent fiber, with rapid attenuation for tighter bend radii compared to more gentle bending.

Implications and Future Directions

This work demonstrates, via direct simulation, that DPG-based ultraweak formulations provide a tractable and robust tool for quantifying geometric confinement loss in complex fiber systems without modal or paraxial approximations. The ability to accurately and efficiently extract the radiative loss exponents directly from 3D simulations is essential for future advances in fiber laser design, mode-filtering via coiling, and high-power amplifier TMI modeling. The methodology can accommodate further extensions, including incorporation of elasto-optic perturbations (strain-induced refractive index anisotropy) and full nonlinear gain and thermal coupling.

On the computational side, significant scalability and efficiency challenges remain as one approaches kilometer-scale fibers or nonlinear multi-physics couplings. On the theoretical front, a rigorous stability analysis for PML-truncated, curved fiber domains is highlighted as ongoing work.

Conclusion

The study establishes the viability and accuracy of full-envelope finite element Maxwell solvers for 3D bent optical fibers, including robust PML-treated open boundaries and adaptivity via DPG residuals. The direct computation of bend-induced mode loss in complex domains signals a significant methodological advance for high-power fiber device simulation and paves the way for addressing more complicated physics (such as TMI onset, strain-optic effects, and nonlinear gain), which are of profound technological interest in photonics and laser engineering.