We consider the full set of electromagnetic fields with azimuthal symmetry. Many of the ideas for the electromagnetic field representation and solution can trace roots to Langdon [10] and Eastwood [7]. Maxwell's equations in integral form are written:

We define a new set of field variables which encapsulate the mesh metrics:

In Eqs. (1.2), the line integrals are along cell sides, and the surface integrals are over surfaces formed by cell faces, where fields are as defined on the Yee mesh [11]. The constituency equations for the integral-form variables become

where and are coupling matrices with the dimensionality
of capacitance and inductance, respectively. On a general mesh,
the couplings can involve fields from one or more neighboring
cells; the general mesh is beyond the scope of the present
paper. On a non-uniform orthogonal Yee mesh
in cylindrical **z**-**r**
coordinates, the capacitances in OOPIC are written:

Similarly, the inductance elements for a non-uniform orthogonal
Yee mesh in cylindrical **z-r** coordinates in OOPIC are:

The Maxwell curl equations can be written in terms of these variables and discretized. The transverse magnetic (TM) set becomes:

where over
the cell volume. The indices refer to locations on
the Yee mesh, and are ordered in **z, r** for a right-handed
cylindrical coordinate system. The corresponding transverse
electric (TE) set can be written:

The TM and TE field equations are advanced in time using a leap frog advance [1]. The source terms in Eqs. (6) and (7), I, are the currents resulting from charged particle motion. The field algorithms are completed by initial conditions and boundary conditions.

Fri Mar 24 02:14:14 WET 1995