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.