| Summary: | Approved for public release; distribution is unlimited === Of the many Free Electron Lasers (FELs) in the world today, most are big and expensive. This is true across the wavelength spectrum, from long to very short. In contrast, the FEL facility in progress at NPS, which will initially operate at long wavelengths and at electron energies of only a few MeV, is inexpensive and smaller. However, longer wavelengths lead to more diffraction, which may result in beam spread and interaction with the undulator surfaces. Anticipating the possibility, in this thesis we analyze mathematically the free space Hermite-Gaussian modes of the optical beam, and then compare them to the Hybrid modes, where waveguide plates control diffraction along one axis, allowing free space diffraction along the other axis. We continue the analysis of the relativistic electron beam, co-propagating with the optical wave in the Hybrid Mode, to define new operating condition for the FEL.
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