Optical lenses are used in a wide range of applications to manipulate incoming light or in general electromagnetic radiation. Due to the refractive index of the lens, the radiation can be focused or dispersed depending on the function of the lens. Three-dimensional (3D) simulations of the interaction of electromagnetic radiation and lenses, which are typically made of dielectric materials can be used in the design process of lenses and lens systems or optimization without the need to construct costly prototypes and perform lengthy experiments. In the following, the process of a 3D optical lens simulation is described in detail.

#### Dielectric Materials

Dielectrics are materials that can be polarized when applying an electric field to them. Usually, they are used as electric insulators, e.g, in capacitors. In PICLas, dielectric materials can be used in electrostatic or electrodynamic simulations. In the first case, Poisson's equation that describes the electric potential can be solved under consideration of dielectric materials. In the latter, Maxwell's equations that consider the interaction of electromangetic fields have to be applied. In PICLas, higher-order discontinuous Galerkin methods (DGSEM) are utilized for both Poisson's and Maxwell's equations in combination with dielectric materials. These methods combine classical features that are typically attributed to finite-volume and finite-element methods, namely numerical surface fluxes between adjacent simulation elements and polynomial basis functions that represent the solution within them. For more details on the theory and numerical implementation, see Pfeiffer et al. (electrostatic) and Copplestone (electrodynamic) in the list of references.

#### 3D Simulation Setup

In the simulation setup, a cuboid domain of 1.3 m x 0.4 m x 0.4 m with unstructured elements is used. The rotationally symmetric optical lens with constant dielectric properties and a diameter of 20 cm is placed inside the domain. In front of the lens, an electromagnetic dipole is positioned that emits a continuous signal with constant frequency of 80 Ghz. The high-order Maxwell solver of PICLas is used for this setup due to the excellent scalability and performance on HPC systems. A total of 168 856 elements are employed within the simulation and the polynomial degree *N* of the solution within each element is varied between 7 - 11 (for each simulation, a global polynomial degree is selected), hence, giving a spatial order of approximation *O(N+1)*, while the temporal order is fixed at 4.

As the simulation uses an explicit time integration scheme, the highest velocity of the system, the speed of light, must be resolved in each grid cell. The criterion that enforces this is the so-called CFL condition, which limits the distance that information may travel in a given time step. Therefore, due to the very small elements near the lens corners, the time step is effectively dictated by the smallest elements in the system and essentially the required computational costs of the simulation is increased when tiny geometric features have to be resolved.

Zoom on the lens near the center of the domain showing the size and distribution of the dielectric mesh elements. The red region indicates the dielectric material, outside the permittivity of free space is used.

The image on the right shows the electric field strength in and around the lens. The lens has a strong effect on incoming electromagnetic radiation, which changes the direction of travel and a clear reduction of the wavelength inside the lens is visible due to the dielectric material. This shows that the spatial resolution must be increased inside the dielectric region to guarantee that the decreased wavelengths therein are adequately resolved.

#### Outlook

Due to numerical stability, the smallest mesh cell dictates the time step in a scheme with explicit time integration. These small elements are usually encountered in complex three-dimensional applications where small geometrical features are to be resolved. This can lead to heavy computational demands when solving large systems in combination with high frequencies (small wavelengths). When the local effect of small geometrical features is negligible, implicit methods can mitigate the computational costs by increasing the global time step because these methods lift the stability restriction with respect to the time step. On the other hand, a single implicit time step requires more computational resources as compared with an explicit one. Therefore, the speed-up is only achieved if the time step is increased correspondingly. Additionally, a large time step can be used as a starting point, which may then be successively increased until the desired solution quality is achieved instead of beginning with the tiny time step of the explicit simulation. PICLas offers such a fully implicit method for solving Maxwell's equations and has already been appliedĀ successfully to technical applications, see Ortwein (2019).

More information about the underlying theory and modelling can be found here:

- Pfeiffer, M., Hindenlang, F., Binder, T., Copplestone, S.M., Munz, C.D. and Fasoulas, S., (2019). A particle-in-cell solver based on a high-order hybridizable discontinuous Galerkin spectral element method on unstructured curved meshes.
*Computer Methods in Applied Mechanics and Engineering*,, pp.149-166.*349* - Copplestone, S.M., (2019). Particle-based Numerical Methods for the Simulation of Electromagnetic Plasma Interactions.
*Verlag Dr. Hut*. - Ortwein, P., (2019). Implicit Time Integration Strategies for a Particle-in-Cell Solver.
*Verlag Dr. Hut*.