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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.

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- 1. Advantages of Finite- Element Method (FEM) for Nano-Optics Simulations
- 2. 2 Typical simulations in nano-optics Compute solution to Maxwellβs equations in frequency-domain: π» Γ π(π«)β1 π» Γ π π«, π β π2 π π« π π«, π = iΟπimp π«, π π π« , π(π«): Spatially dependent permeability and permittivity of the setup. πimp (π«): Current impressed by an external source (plane-wave illumination, laser beam, dipole emitter, fiber mode, etc.) π: Harmonic frequency of the source term. π π«, π : Time-harmonic electric field strength. The time-dependent field is given as π π«, π‘ = β{π π«, π β πβπππ‘}.
- 3. 3 Example: Add-drop multiplexer (integrated optics) In-coupling waveguide mode, i.e. πimp(π«, π) Drop port Add port Through port Wave guide: π1 = 12.1π0 Substrate: π2 = 2.3π0 Electric field intensity π π«, π 2 fulfilling π» Γ π(π«)β1 π» Γ π π«, π β π2 π π« π π«, π = iΟπimp π«, π
- 4. 4 How to solve Maxwellβs equations? ο§ RCWA (rigorous coupled wave analysis): The geometry is discretized into individual layers. The diffraction of incident plane waves at the structure is calculated. [Wikipedia] ο§ 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] ο§ FEM (finite element method): The geometry is discretized into variable shapes like triangles, tetrahedrons, prisms (solution: next slide). Various methods are used to solve Maxwellβs 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 Maxwellβs 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 ο§ Description of FEM software JCMsuite ο§ Free trial download of JCMsuite ο§ Getting started with JCMsuite ο§ Benchmark of rigorous methods for electromagnetic field simulation ο§ Benchmark of quasinormal mode solvers

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