Calculation of Reflection Matrix Elements of a Grating for Growing Evanescent Waves

Reflection matrix elements of a grating play an important role in the study of Smith-Purcell (SP) free-electron lasers (FELs). Especially, the matrix element R00, which couples the incident co-propagating evanescent wave to the reflected co-propagating evanescent wave, is important for evaluation of the gain of an SP-FEL system [1]. We present a calculation of R00for rectangular grating and study its frequency dependence for a given phase velocity of incident wave. For the numerical calculation, we use the modal expansion method and extend it to include waves having slowly varying amplitude. The singularity of R00is studied in some detail and we find that it is possible to get a simple formula for the location of the singularity if we choose the eigenmodes of the groove as a basis set as done by Andrews et al. [2].