Focused-Ion-Beam (FIB) Simulation

This project simulates the motion of Ar⁺ ions through a system of electrostatic lenses and apertures. The electric field is calculated by solving the Laplace equation on a 2D grid (r-z), and ion trajectories are computed based on the resulting field.

Features

• Electrostatic potential calculation using a finite difference Laplace solver
• Electric field computation from the potential
• Ion trajectory simulation with random emission angles
• Aperture collision handling

Requirements

• Python ≥ 3.8
• NumPy
• Matplotlib
• Numba

Installation

pip install numpy matplotlib numba

Grid Configuration

The simulation space is defined in cylindrical coordinates:

Nx, Nz = 300, 800
dr, dz = 0.01, 0.01  # grid resolution in cm
r_max, z_max = Nx * dr, Nz * dz

Electrodes and Apertures (Lens Configuration)

Lenses

Defined by position z, axial length depth, inner radius r_in, outer radius r_out, and potential V.

lenses = {
    "Accelerator_Lens_1": {"z": 1.7, "depth": 0.3, "r_in": 0.2, "r_out": 1.0, "V": -30000},
    ...
}

Apertures

Defined as non-conductive barriers (e.g. PVC) with similar geometry, but no voltage. These block ions when collided with:

apertures = {
    "Aperture1": {"z": 0.7, "depth": 0.1, "r_in": 0.05, "r_out": 1.0}
}

In Depth: Simulation Steps

Laplace Solver

The Laplace equation is solved iteratively over the grid with Dirichlet boundary conditions given by the lens voltages:

phi_new[1:-1, 1:-1] = 0.25 * (phi[2:, 1:-1] + phi[:-2, 1:-1] +
                              phi[1:-1, 2:] + phi[1:-1, :-2])

Lens regions are held at constant potential throughout the iteration.

Electric Field Calculation

The electric field is calculated from the gradient of the potential:

Er = -(phi[2:, 1:-1] - phi[:-2, 1:-1]) / (2 * dr)
Ez = -(phi[1:-1, 2:] - phi[1:-1, :-2]) / (2 * dz)

Ion Trajectory Simulation

Each ion is initialized with a random angle and radial offset. The motion is integrated using the electric field:

vr += ar * dt
vz += az * dt
r += vr * dt
z += vz * dt

Ion paths terminate when:

• They leave the simulation domain
• They hit a defined aperture