Modelling Tools

Computer Simulation Technology (CST) [121] is a high-frequency electromagnetic software that has been frequently used to understand the properties of dual-grating structures. It uses both a finite element method (FEM) solver and adaptive meshing to generate solutions to Maxwell's Equations in a variety of structures with different boundary conditions. CST includes a full 3D electromagnetic simulation tool and various static solvers

43

and is widely used in electromagnetic design and analysis. CST simulation suite is comprised of 7 packages to meet the various demands of users, as follows:

(1) CST Microwave studio: it is used mainly for the fast and accurate 3D simulation of high frequency devices such as filters, couplers, planar and multi-layer structures. (2) CST EM studio: it is used mainly for the design and analysis of static and low

frequency electromagnetic applications such as motors, sensors, actuators, et al. (3) CST Particle studio: it is used mainly in the analysis of charged particle dynamics

subject to static and dynamic fields. Applications include electron guns, cathode ray tubes, magnetrons, and wake fields.

(4) CST Cable studio: it is used mainly for the simulation of signal integrity and electromagnetic compatibility (EMC) / electromagnetic interference (EMI) analysis of cable harnesses.

(5) CST PCB studio: it is used mainly for the simulation of signal integrity and EMC/EMI on printed circuit boards (PCB).

(6) CST Mphysics studio: it is used mainly for thermal and mechanical stress analysis. (7) CST Design studio: it is a versatile tool that facilitates 3D electromagnetic/circuit

co-simulation and synthesis.

In the following studies, CST microwave studio and particle studio are used for simulation of our DLAs. CST microwave studio includes various solvers such as the eigenmode solver, time domain solver, frequency domain solver, integral equation solver, multilayer solver, asymptotic solver. It supports hexahedral and tetrahedral mesh settings along with a fully automated meshing procedure specific to the desired frequency range, dielectrics and metallic edges, etc. An adaptive meshing process can refine the mesh settings in the calculation region according to the desired accuracy of the solutions by repeating the simulations. CST particle studio includes particle tracking solver and PIC solver. PIC solver calculates the particle movement in the total electromagnetic field due to the mutual coupling of the external field with the space charge field due to the particles. The algorithm of a PIC simulation is shown in Figure 34. At the starting point of the cycle, the program calculates the charge density and current density on the grid (uniform charged cloud) from the given initial particle quantities such as position and velocity of the particles after performing some weighting. At the second step of the cycle, the program calculates the electromagnetic field from Maxwell’s equations. At the third step of the cycle, the program interpolates the fields from the grid and particles by waiting again to get the force on the particles. The last step is to solve the equations of motion to get the new particle quantities.

44

Figure 34. CST PIC algorithm [121].

CST Microwave studio with time domain solver is usually used to compute the electromagnetic fields in dual-grating structures. It should be noted here that the time domain solver for CST is based on the finite integration technique [122] which is similar but not the same as FDTD method. Open boundary conditions (equal to a fully absorbing boundary) are used at both laser input and output faces along the y direction. Periodic boundary conditions are used on both the surfaces in the z direction, to represent the periodic characteristics of this structure. The electromagnetic fields are uniform along the x direction, so magnetic boundary conditions are used at both outer surfaces in the x direction. There are two symmetry planes for a four-period dual-grating structure: magnetic (𝐻t= 0) at the YZ plane,

and electric (𝐸t= 0) at the XY plane. Furthermore, a hexahedral mesh type is used, as it

matches the geometry of the structures. For a dual-grating structure, the mesh size is chosen to be much smaller than the operating laser wavelength, to increase accuracy. The mesh density is determined by three parameters: lines per wavelength (LPW), lower mesh limit (LML), and mesh line ratio limit (MLRL). LPW determines the minimum number of mesh lines to be used across, for the minimum wavelength set by the simulation. The default value for LPW is 10, and LPW higher than 10 will increase the accuracy of the result as well as the calculation time. LML, which defines a maximum distance 𝐿_max between two mesh lines, is given as follows:

𝐿max = 𝐷min/√3LML, (46)

where 𝐷_min represents the smallest face diagonal of the bounding box of the calculation domain. MLRL stands for the minimum ratio between two mesh lines (the absolute smallest mesh step). The calculation time depends not only on the absolute number of mesh cells, but also on the distance between mesh lines, since the smallest time step is determined by the smallest distance in a mesh.

45

The electromagnetic fields generated in the Microwave studio are then loaded into the same dual-grating structure to interact with the injected particles in the vacuum channel, through Particle studio with PIC solver. The energy gain and accelerating gradient can be calculated by analysing the phase space of the particles.

VSim [123] is a PIC code combined with a self-consistent electromagnetic field solver based on the FDTD numerical method. It allows more freedom to create the desired simulation by using the Python language. It can also compute electromagnetic fields between 1, 2, or 3 dimensions. It can be used for simulations in the time domain as well as for particle dynamics. Similar to CST Microwave studio, VSim can take a user-specified dielectric design with boundary conditions and compute the electromagnetic fields involved. For a dual-grating structure illuminated by a laser pulse with a Gaussian envelope, the periodic boundary conditions are applied along the electron channel in the z direction, magnetic boundary conditions are used in the x direction, and perfectly matched layers (PMLs) are employed along the laser propagation direction (y-axis) to absorb the transmitted light. It then tracks and characterizes loaded particles travelling through these fields, as well as the self-fields generated by those particles in the vacuum channel of the dual-grating structure. To solve for the fields as a function of time, VSim takes into account initial conditions of the electromagnetic fields, particle positions and velocities, and the user-defined mesh parameters. The algorithm is the same as that of CST PIC solver which can be seen in Figure 34. In each cell of the mesh, the magnetic field on the boundary of the cell is used to update the electric field at the centre of each cell over one half time-step, according to Faraday's law. During the next half time-step, the electric field at the centre of the cell is used to update the magnetic field on the boundary according to Ampere's law. The particle positions and velocities are then updated using the Biot-Savart law. The new particle positions/velocities are then used to update the electric and magnetic fields. VSim therefore is well-suited for modelling DLA structures and performing intensive PIC simulations to obtain all the relevant physical characteristics of DLA structures.

In addition, the mesh setup at convergence for CST and VSim are found to be different. CST automatically generates the mesh sizes by setting up mesh density parameters (LPW, LM and MLRL). As described previously, LPW, LM and MLRL are set to 80, 80, and 50 for convergence. For VSim, we found that dx (the mesh size along x direction) and dy (the mesh size along y direction) is set to period/200 and period/100 in order for convenience. These mesh setup will be used for the following simulations.

In the following simulations, CST and VSim code have been used to perform the optimization studies, and results from both codes are shown for comparison.

46