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Adaptive Finite Element Method for Simulation of Optical Nano Structures
Jan Pomplun, Sven Burger, Lin Zschiedrich, Frank Schmidt
TL;DR
The paper develops a finite element treatment of time-harmonic Maxwell equations for nano-optical simulation, including bounded structures with unbounded exteriors. It constructs vectorial finite-element spaces, uses PML-based transparent boundaries, and applies the method to leaky modes in hollow-core photonic crystal fibers.
Problem
Nano-optical simulations require formulations that account for bounded waveguides together with infinite, possibly inhomogeneous exteriors and radiation losses.
Method
The method discretizes time-harmonic Maxwell equations with local vectorial polynomial basis functions and represents unbounded exteriors through PML-based transparent boundary conditions.
Results
The approach computes leaky propagating modes of hollow-core photonic crystal fibers, while the resulting stiffness matrix has O(N) nonzeros and practical linear computational scaling with unknowns.
Takeaways & Limitations
Finite elements provide a sparse computational framework for modeling radiation losses in photonic crystal fibers with exterior domains included.
Takeaways & Limitations
The exterior-field treatment uses the Silver-Müller radiation condition, and the incoming field must satisfy Maxwell’s equations near the boundary.
Abstract
from arXiv · showhide
We discuss realization, properties and performance of the adaptive finite element approach to the design of nano-photonic components. Central issues are the construction of vectorial finite elements and the embedding of bounded components into the unbounded and possibly heterogeneous exterior. We apply the finite element method to the optimization of the design of a hollow core photonic crystal fiber. Thereby we look at the convergence of the method and discuss automatic and adaptive grid refinement and the performance of higher order elements.
1 Introduction
The paper presents finite elements for solving time-harmonic Maxwell equations in nano-optical simulations. It frames the method across several problem classes and outlines the paper’s organization around their formulations and discretization.
- The finite element method computes steady-state electromagnetic fields at a fixed frequency ω.
- The paper considers propagation mode, resonance, and scattering problems as representative nano-optical simulation tasks.
2 Problem Classes
The paper organizes time-harmonic Maxwell problems into propagation, resonance, and scattering classes. These classes share a curl-curl operator but differ in boundary conditions, unknowns, and resulting algebraic problems.
- Propagation mode problems describe guided light fields in waveguides, resonance problems describe resonator eigenmodes, and scattering problems describe light interacting with obstacles.
- All three classes use Maxwell’s eigenvalue equations formulated as a second-order curl-curl equation for the electric field.
- The finite element method discretizes the common differential operator while incorporating class-specific boundary conditions and unknown quantities.
- Discretized scattering problems produce linear systems, whereas propagation mode and resonance problems produce eigenvalue problems for the assembled finite element matrix.
3 Propagation mode problems
Propagation modes are formulated for waveguide geometries invariant along one direction and characterized by a propagation constant or effective refractive index. The method includes the infinite exterior to represent bounded and leaky modes, using PML-based transparent boundaries.
- The electric and magnetic fields are decomposed into transversal and longitudinal components before deriving the propagation-mode problem.
- The propagation-mode formulation is a generalized eigenvalue problem for propagation constant kz and electric propagation mode Epm(x, y).
- The effective refractive index neff is introduced as the eigenvalue used in the numerical analysis of propagation modes.
- Including the infinite exterior allows computation of bounded and leaky modes, with leaky modes representing radiation losses; PML realizes transparent boundary conditions and supports inhomogeneous exteriors.
4 Resonance problems
Resonance problems seek electromagnetic eigenmodes and frequencies in cavities coupled to their exterior. Their formulation combines interior and exterior Maxwell equations with boundary and radiation conditions, while preserving Maxwell’s structure in discretization.
- A resonance problem seeks tuples (ω, E) satisfying Maxwell’s equation in the cavity interior and exterior.
- The resonance formulation imposes a boundary condition on Γ together with the Silver-Müller radiation condition at infinity.
- Gradient fields lie in the curl operator’s kernel and generate many zero-frequency solutions, so finite-element ansatz functions must preserve this structure discretely.
5 Scattering problems
Scattering problems combine an interior Maxwell problem with an exterior radiation problem, requiring boundary conditions that represent strictly outgoing scattered fields. The FEM uses transparent boundary conditions to truncate the unbounded exterior.
- The scattering formulation specifies Maxwell’s equations separately in the interior and exterior domains, with a boundary condition at Γ.
- The Silver-Müller radiation condition enforces that Eout is a strictly outward-radiating solution.
- For inhomogeneous exteriors, the radiation condition requires generalization, while the FEM imposes transparent boundary conditions on Γ.
- The weak formulation uses vector-valued test functions Φ ∈ H(curl, Ω) after multiplying Maxwell’s equation and integrating over Ω.
6 Weak formulation of Maxwell’s equations
The weak formulation converts Maxwell’s equations into a variational problem over H(curl, Ω). Boundary treatment requires Neumann data obtained from a Dirichlet-to-Neumann operator, and the formulation remains exact.
- Partial integration produces the weak formulation of Maxwell’s equations.
- The weak problem requires Neumann data on Γ for the electric field.
- A Dirichlet-to-Neumann operator maps electric-field Dirichlet data to Neumann values while respecting the radiation condition.
- The weak formulation is stated using bilinear functionals and is an exact reformulation of Maxwell’s equations.
7 Discretization of Maxwell’s equations
The finite element discretization approximates Maxwell’s solution space with a finite-dimensional space built from local vectorial basis functions. Local support produces sparse linear systems whose solution time is practically linear in the number of unknowns.
- The computational domain is subdivided into patches, where local vectorial ansatz functions form a basis of Vh and approximate Eh by superposition.
- Testing the discrete equation with basis functions yields a linear system for the unknown coefficients ai.
- The stiffness matrix has O(N) nonzeros among O(N^2) entries because the ansatz functions have small support.
- With special solvers, computational time scales practically linearly with the number of unknowns.
8 Construction of finite elements
Finite elements are constructed locally on mesh patches with function spaces, degrees of freedom, and basis functions chosen to preserve the structure of Maxwell’s differential operators. For H(curl) fields, lowest-order edge elements use tangential edge integrals as degrees of freedom and maintain tangential continuity across patches.
- Finite-element construction chooses a patch, a local function space with desired properties, degrees of freedom, and basis functions spanning that space.
- Linear scalar elements on triangles use the three-dimensional polynomial space P 1(K) = {v = a + bx + cy}.
- Scalar nodal degrees of freedom evaluate functions at triangle nodes, producing basis functions λi with ψj(λi) = δij.
- The discrete exact sequence requires gradients of scalar finite elements to lie in the H(curl)-conforming vector space and form the curl kernel.
- Lowest-order vectorial elements extend the constant-vector space with polynomial functions whose curl remains constant.
- The construction is presented in two dimensions on triangular patches, while higher-order extensions were first constructed by Nedelec.
- Edge-element degrees of freedom are integrals of the tangential component along patch edges, and shared edges give globally continuous tangential components.
- Transparent boundary conditions are realized with PML on polygonal, star-shaped computational domains while allowing a certain class of exterior inhomogeneities.
9 Transparent boundary conditions
The method embeds a bounded finite-element domain in an unbounded, potentially heterogeneous exterior using prismatoidal coordinates and a PML-based transparent boundary treatment. The exterior is discretized semi-discretely in η and with finite elements in ξ, where analyticity makes higher-order elements advantageous.
- Coordinate construction: The exterior domain is decomposed into semi-infinite segments along straight, non-intersecting rays from the polygonal boundary to infinity.Each segment receives a local coordinate system, yielding a global distance variable ξ and generalized angular variable η.
- Coordinate construction: Local bilinear mappings transform reference elements into exterior segments, while their combination forms a mapping continuous in η.The transformed material parameters use the Jacobian and its determinant.
- Semi-discretization: Maxwell’s equations are semi-discretized in η by expanding the outgoing field in boundary trace basis functions from the interior finite-element space.The weak formulation produces a differential system for the coefficient vector as a function of ξ.
- PML realization: The PML replaces ξ with γξ, where ℜ(γ) > 0 and ℑ(γ) > 0, then truncates the transformed exterior at ξ = ρ with zero Dirichlet conditions.The resulting PML system is finally discretized in ξ using finite elements.
- PML realization: Because the exterior solution is analytic in ξ, high-order finite elements are advantageous for the ξ-discretization.The complete exterior discretization can be interpreted as a finite-element method on quadrilaterals whose quality depends on the initial rays.
- Scope: The approach permits transparent boundaries for open waveguides and inhomogeneous exteriors while retaining a finite computational domain.The PML method is used to realize the Dirichlet-to-Neumann operator for the exterior problem.
10 Application: Optimization of photonic crystal fiber design
The finite element method is applied to leaky modes in a hollow-core photonic crystal fiber, combining convergence studies, adaptive refinement, and parameter optimization. Varying geometric parameters yields a minimum reported imaginary effective refractive index of 5 · 10^-15 1/m.
- Convergence and refinement: The study evaluates relative fundamental-eigenvalue error against unknowns for uniform and adaptive refinement and several finite-element degrees.The most accurate FEM eigenvalue serves as the reference solution, and adaptive refinement shows convergence.
- Convergence and refinement: The imaginary part of the effective refractive index is another possible quantity of interest for goal-oriented grid refinement.This quantity directly relates the discretization target to radiation-loss analysis.
- Fiber optimization: The HCPCF optimization fixes six cladding rings and varies pitch Λ, hole-edge radius r, strut thickness w, and core-surround thickness t.The fiber has a hollow core formed by 19 omitted hexagonal cladding cells, with triangulation resolving its geometric features.
- Fiber optimization: Increasing the number of cladding rings decreases radiation losses, motivating the fixed six-ring configuration used for optimization.The cladding rings form a photonic crystal structure that prevents leakage from the core to the exterior.
- Fiber optimization: 5 · 10^-15 1/m is the minimum reported imaginary effective refractive index, obtained at Λ = 1597 nm, w = 38 nm, and t = 151 nm.The starting values were Λ = 1550 nm, t = 152 nm, and w = 50 nm.