Brief explanation of some advantages of the Finite-Element Method (FEM) for simulations in photonics and nano-optics. Benchmarks against FDTD, RCWA and other FEM solvers show the exceptionally short computation times of the higher-order FEM method implemented in JCMsuite.

1. Advantages of Finite-
Element Method (FEM) for
Nano-Optics Simulations

2. 2
Typical simulations in nano-optics
Compute solution to Maxwell‘s equations in frequency-domain:
𝛻 × 𝜇(𝐫)−1 𝛻 × 𝐄 𝐫, 𝜔 − 𝜔2 𝜖 𝐫 𝐄 𝐫, 𝜔 = iω𝐉imp 𝐫, 𝜔
𝜇 𝐫 , 𝜖(𝐫): Spatially dependent permeability and permittivity of the setup.
𝐉imp
(𝐫): Current impressed by an external source (plane-wave
illumination, laser beam, dipole emitter, fiber mode, etc.)
𝜔: Harmonic frequency of the source term.
𝐄 𝐫, 𝜔 : Time-harmonic electric field strength. The time-dependent
field is given as 𝐄 𝐫, 𝑡 = ℜ{𝐄 𝐫, 𝜔 ⋅ 𝑒−𝑖𝜔𝑡}.

3. 3
Example: Add-drop multiplexer (integrated optics)
In-coupling
waveguide mode,
i.e. 𝐉imp(𝐫, 𝜔)
Drop port
Add port
Through port
Wave guide: 𝜖1 = 12.1𝜖0
Substrate: 𝜖2 = 2.3𝜖0
Electric field intensity 𝐄 𝐫, 𝜔 2
fulfilling
𝛻 × 𝜇(𝐫)−1
𝛻 × 𝐄 𝐫, 𝜔 − 𝜔2
𝜖 𝐫 𝐄 𝐫, 𝜔 = iω𝐉imp
𝐫, 𝜔

4. 4
How to solve Maxwell’s equations?
 RCWA (rigorous coupled wave analysis):
The geometry is discretized into
individual layers. The diffraction of
incident plane waves at the structure is
calculated. [Wikipedia]
 FDTD (finite difference time-domain
method): The geometry is discretized
into uniform patches (squares, cubes).
The equations are solved in a time and
space discrete manner. [Wikipedia]
 FEM (finite element method): The
geometry is discretized into variable
shapes like triangles, tetrahedrons,
prisms (solution: next slide).
Various methods are used to solve Maxwell’s equations rigorously, e.g.:
FDTD

5. 5
1. Choose computational domain with
appropriate boundary condition and
sub-divide the geometry into patches
2. Expand the electric/magnetic fields with
local ansatz functions which are defined
on the triangles/tetrahedrons/prisms
etc. and plug into weak formulation of
Maxwell‘s equations
3. Solve sparse matrix equation with fast
numerics
FEM Recipe

6. 6
The flexibility of FEM
Due to its flexible geometry
discretization, FEM can be applied to
various geometries and complex
shapes.

7. 7
Examples of shape discretization
Cavity for sensing applications Helix nanoantenna
Circular grating resonator
for optical switching
Smooth vs. rough surface

8. 8
The efficiency of FEM
The precision of the FEM solution
can be locally adapted. This leads to
highly accurate results at short
computation times.

9. 9
Hp-Finite Element Method
FEM numerical parameters: h - triangle size, p - polynomial order
Suitable non-uniform combination of p and h refinements
leads to superior convergence [Babuska, 1992]

10. 10
Comparison of convergence speed
FEM
RCWA
FDTD
Comparison:
FEM vs. FDTD vs. RCWA
Benchmark Problem:
Rigorous Mask Simulation for
Lithography
FEM faster and more accurate
by orders of magnitude
[Benchmark of FEM, Waveguide
and FDTD Algorithms for Rigorous
Mask Simulation.
Proc. SPIE 5992, 368, 2005.]

11. 11
Comparison of convergence speed (2)
Comparison: The FEM solver of JCMsuite is compared to other
commercial and non-commercial FEM-solvers (see reference for a
detailed description)
Benchmark problem: Complex eigenfrequencies of the modes of a
2D plasmonic crystal composed of a periodic array of metallic
squares in air.
[Quasinormal mode solvers
for resonators
with dispersive materials.
JOSA A 36, 686, 2019.]

12. 12
Resources
 Description of FEM software
JCMsuite
 Free trial download of JCMsuite
 Getting started with JCMsuite
 Benchmark of rigorous methods for
electromagnetic field simulation
 Benchmark of quasinormal mode
solvers