# Dispersion Equation for Smith-Purcell FEL

## TL;DR

This paper derives a dispersion equation for Smith-Purcell free-electron lasers considering small groove depth, revealing that excitation regimes depend on beam height rather than current, with growth rate proportional to the square root of current.

## Contribution

It provides a new dispersion equation for Smith-Purcell FELs accounting for small groove depth, clarifying excitation regime dependencies and growth rate behavior.

## Key findings

- Regimes depend on beam height, not current
- Growth rate scales with square root of beam current
- Dispersion equation derived for small groove depth

## Abstract

For the grating, which has depth of grooves as a small parameter, the dispersion equation of the Smith-Purcell instability was obtained. It was found that the condition of the Thompson or the Raman regimes of excitation does not depend on beam current but depends on the height of the beam above grating surface. The growth rate of instability in both cases is proportional to the square root of the electron beam current.

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Source: https://tomesphere.com/paper/1702.00242