Greg Hammett, May 20, 2010:

In looking over my 2005 paper with Nevins recently, I noticed that we didn't seem to define the factor $d_\Par({\bf k})$ in Eq. (5) very well. As I recall in my previous discussions with Andris Dimits, the parallel electric field in PIC codes is typically calculated with a 3 point stencil of the form:

E_i = -(phi_{i+1}-phi_{i-1})/(2*dz)

In which case $d_\Par({\bf k}) = sin(k_\Par \Delta z) / (k_\Par \Delta z)$, where $\Delta z$ is the parallel grid spacing. For well-resolved modes with $k_\Par \Delta z \ll 1$, just replace $d_\Par$ with 1.