Mesh points:

Mesh points (sometimes called nodal points) are not used in the Surface Charge Method to find the electrostatic field.

Mesh points are however be used by the program for two other purposes.

1. Tracing rays

Mesh points can be used, by choice, for tracing rays (ie integrating trajectories). In the ‘mesh method’ the potentials and field components are calculated and stored at mesh points in the vicinity of the ray. This is done during the evolution of the ray. When the program needs a potential or field at a point on the ray it looks for the values at nearby mesh points and interpolates between them. If a mesh does not yet exist and the program cannot find the required value then the program creates the mesh point, calculates the potential or field components and stores the information. The mesh points that are created in this way form an incomplete array (which is a square array in CPO-2D and a cubic array in CPO-3D).

The potentials are stored in one array of mesh points, and the field components in a separate array. Both arrays have the same spacing and origin. The potentials and/or fields at the mesh points are used at the time of creation of the points (and again later if there are further rays passing through the same region) to calculate the fields needed for integrating the ray. The mesh technique is therefore particularly useful when a large number of neighbouring rays is required.

The spacing of the mesh points is specified by the User

2. Space-charge calculations

The second use of mesh points is in space-charge calculations. The space through which the beam passes is notionally divided into an array of square or cubic cells, each of which can hold a space-charge. As a ray passes through a cell it deposits a charge there equal to the product of the current and the time spent traversing the cell.

The space-charge cells are created only where they are needed, in the volume traversed by the rays. The total charges in the cells (each with its weighted centre-of-gravity) are used to calculate space-charge potentials and fields.

The space-charge cells are completely independent of the ray cells (if these are used), and so the mesh spacing of the space-charge cells does not have to have the same as the mesh spacing used for the ray mesh points when rays are traced by the mesh method.

But cells do not have to be used. Another method of assigning space-charge is also available -the ‘ray tube’ method. This is usually a more accurate method, particularly for beams that are long and narrow. In this method the charge associated with each step of a ray is put in a narrow tube that surrounds the step.