## Abstract

We present a new and greatly simplified derivation of the tuning relation for a Čerenkov free-electron laser. The laser uses an electron beam to excite coherent light in a planar waveguide. The geometric (or zig-zag) theory of waveguides is used to model the set of guided modes, while properties of hte Čerenkov radiation dictate which modes are excited. The results obtained are identical to earlier research based on the formal application of Maxwell’s equations. However, the geometric approach adds physical insight into the Čerenkov free-electron laser design problem. The dependence of output wavelength on electron-beam voltage and resonator thickness is plotted. A sample laser is designed for operation from 150 to 825 *μ*m.

© 1992 Optical Society of America

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### Equations (9)

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(1)
$${\delta}_{\text{TM}}={\text{tan}}^{-1}\left\{\frac{{\left(\frac{{n}_{1}}{{n}_{2}}\right)}^{2}{\left[{({n}_{1}\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta}_{1})}^{2}-{{n}_{2}}^{2}\right]}^{1/2}}{{n}_{1}\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta}_{1}}\right\}.$$
(2)
$$2\frac{\omega d}{c}{n}_{1}\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta}_{1}-4{\delta}_{\text{TM}}=2M\pi ,$$
(3)
$$\frac{\omega d}{c}=\frac{1}{{n}_{1}\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta}_{1}}\times \left(M\pi +2\hspace{0.17em}{\text{tan}}^{-1}\left\{\frac{{n}_{1}}{\text{cos}\hspace{0.17em}{\theta}_{1}}{[{({n}_{1}\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta}_{1})}^{2}-1]}^{1/2}\right\}\right).$$
(4)
$$iq=\frac{\omega}{c}{(1-{{n}_{1}}^{2}\hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}{\theta}_{1})}^{1/2}.$$
(5)
$${\theta}_{c}={\text{cos}}^{-1}\left(\frac{1}{\beta {n}_{1}}\right)$$
(6)
$$\frac{\omega d}{c}=\frac{\beta}{{({{n}_{1}}^{2}{\beta}^{2}-1)}^{1/2}}\times \left(M\pi +2\hspace{0.17em}{\text{tan}}^{-1}\left\{{{n}_{1}}^{2}{\left[\frac{1-{\beta}^{2}}{{({{n}_{1}}^{2}{\beta}^{2}-1)}^{1/2}}\right]}^{1/2}\right\}\right).$$
(7)
$$\frac{{\sigma}_{b}}{\lambda \beta \gamma}\approx 1.$$
(8)
$$V(\text{keV})=\left[{\left(\frac{1}{1-{\beta}^{2}}\right)}^{1/2}-1\right]511.$$
(9)
$${V}_{T}(\text{keV})=\left[{\left(\frac{{{n}_{1}}^{2}}{{{n}_{1}}^{2}-1}\right)}^{1/2}-1\right]511.$$