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arxiv: 2604.12014 · v1 · submitted 2026-04-13 · ⚛️ physics.optics · cs.NA· math.NA

And Yet Another FEM-Based Mode Solver for Dielectric Waveguides

Ergun Simsek This is my paper

Pith reviewed 2026-05-10 14:54 UTC · model grok-4.3

classification ⚛️ physics.optics cs.NAmath.NA
keywords dielectric waveguidesfinite element methodmode solverhybrid modesspurious modesopen sourceintegrated photonics
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The pith

A mixed Nedelec-Lagrange finite element discretization of Maxwell's equations yields an accurate open-source mode solver for dielectric waveguides that matches commercial results to within 0.05 percent error.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a full-vector finite element method that solves for the modes supported by dielectric waveguides. It discretizes Maxwell's curl equations in the frequency domain by assigning edge elements to the transverse field components and nodal elements to the longitudinal component. This combination is intended to model hybrid modes correctly while eliminating the non-physical spurious solutions that often appear in other formulations. The implementation is released in both MATLAB and Python with an emphasis on reproducibility and cloud compatibility. Validation on standard waveguide geometries shows close agreement with a commercial solver, and mesh-refinement studies illustrate the expected accuracy gains versus added computational cost.

Core claim

The mixed Nedelec-Lagrange discretization of Maxwell's curl equations in the frequency domain enables accurate computation of hybrid modes in dielectric waveguides while suppressing spurious modes, as demonstrated by implementation in MATLAB and Python and validation against COMSOL Multiphysics with relative errors below 0.05 percent.

What carries the argument

The mixed Nedelec-Lagrange finite element discretization, in which edge elements approximate the transverse electric field components and nodal elements approximate the longitudinal component.

If this is right

  • Researchers can obtain reliable effective indices and field profiles for hybrid modes without commercial software licenses.
  • The open implementations allow direct inspection and modification of the discretization for teaching or custom extensions.
  • Convergence rates with mesh refinement provide a quantitative guide for balancing accuracy against run time in practical calculations.
  • Cloud-compatible code lowers the barrier for users without local high-performance computing resources.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same discretization strategy could be tested on waveguides with material loss or gain to check whether the suppression of spurious modes remains effective.
  • Extension to three-dimensional or bent waveguide geometries would require only changes to the mesh generator while retaining the core element types.
  • The code base offers a concrete starting point for adding support for nonlinear or anisotropic materials in future versions.

Load-bearing premise

The combination of edge and nodal elements prevents spurious solutions while still capturing the hybrid character of the waveguide modes.

What would settle it

Running the solver on a rectangular dielectric waveguide and finding either relative errors larger than 0.05 percent against a reference commercial solution or the appearance of non-physical modes with imaginary propagation constants.

Figures

Figures reproduced from arXiv: 2604.12014 by Ergun Simsek.

Figure 1
Figure 1. Figure 1: Normalized field components |Ex|, |Ey|, |Ez| of the first three modes W = 1.0 µm and height H = 0.6 µm. The upper cladding thickness is hclad = 2.6 µm, and the buried oxide (BOX) thickness is hbox = 2.0 µm. The total simulation window width and height are wsim = 6.0 µm and hsim = 4.6 µm both in our solver and COMLSOL Multiphysics. The operating wavelength is again λ = 1.55 µm. The refractive index contrast… view at source ↗
Figure 2
Figure 2. Figure 2: Mesh quality (points per wavelength) vs. (left) error and (right) computation time. Both figures [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

We present a full-vector finite element method (FEM) mode solver for dielectric waveguides based on a mixed Nedelec-Lagrange discretization of Maxwell's curl equations in the frequency domain. The formulation combines edge elements for transverse field components with nodal elements for the longitudinal component, enabling accurate modeling of hybrid modes while effectively suppressing spurious solutions. The solver is implemented in both MATLAB and Python with an emphasis on reproducibility, computational efficiency, and accessibility, including compatibility with cloud-based platforms. Numerical validation is performed on representative waveguide structures, demonstrating excellent agreement with COMSOL Multiphysics, with relative errors below 0.05%. Convergence studies confirm the expected accuracy trends with mesh refinement, while highlighting the trade-off between computational cost and precision. The proposed implementation provides a flexible and reliable open-source tool for integrated photonics research and education.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript describes a full-vector finite element method (FEM) mode solver for dielectric waveguides based on a mixed Nedelec-Lagrange discretization of Maxwell's curl equations in the frequency domain. It provides implementations in MATLAB and Python, validates the solver on representative waveguide structures with relative errors below 0.05% compared to COMSOL Multiphysics, and includes convergence studies with mesh refinement.

Significance. If the validation holds, the paper contributes an open-source, reproducible tool for modeling dielectric waveguides that could aid research and education in integrated photonics. The mixed element approach is standard for suppressing spurious modes in such problems, and the emphasis on accessibility and cloud compatibility adds practical value. The low reported errors and convergence behavior align with expectations for this discretization.

minor comments (3)
  1. The claim of 'excellent agreement' with relative errors below 0.05% would be strengthened by specifying the waveguide structures and modes considered in the validation.
  2. More details on the boundary conditions and how the eigenvalue problem is solved would improve clarity for readers implementing similar solvers.
  3. The paper could benefit from additional references to prior open-source FEM waveguide solvers for context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. The report does not contain any specific major comments or points requiring clarification or rebuttal.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper implements a standard mixed Nedelec-Lagrange FEM discretization of the frequency-domain curl-curl Maxwell equations for dielectric waveguide modes. This rests on well-established finite-element theory for edge and nodal elements, with no derivation that reduces to fitted inputs, self-defined quantities, or self-citation chains. Validation consists of direct numerical comparison to COMSOL on representative structures (relative error <0.05%), which is an external benchmark rather than a constructed prediction. No load-bearing uniqueness theorems, ansatzes smuggled via citation, or renaming of known results appear in the provided text. The method is self-contained against external references and mesh-refinement studies.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on established numerical methods and standard electromagnetic theory without introducing new fitted parameters or postulated entities.

axioms (2)
  • domain assumption Maxwell's curl equations hold in the frequency domain for linear dielectric media
    The formulation is based directly on these equations as described in the abstract.
  • standard math Nedelec edge elements and Lagrange nodal elements satisfy the required continuity and approximation properties for the vector and scalar fields
    Standard finite element theory is invoked to justify the mixed discretization.

pith-pipeline@v0.9.0 · 5433 in / 1225 out tokens · 40317 ms · 2026-05-10T14:54:23.330576+00:00 · methodology

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Reference graph

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