The free electron laser (FEL) is proposed to meet the Navy's need for a speed-of-light high energy laser weapon capable of engaging a variety of targets including anti-ship cruise missiles, small boats, and theater ballistic missiles. A key attribute of FELs is good optical beam quality; in other words, they operate in only a few of the lowest-order transverse Gaussian modes. For weapons applications, a good mode quality is desired because it delivers the highest intensity on target ensuring a high level of lethality. A few higher-order modes can arise from the interaction of the electron beam with the optical beam, or from misalignments of the electron beam or resonator mirrors. High intensity on FEL optics can lead to mirror distortion due to heating and insufficient cooling of the mirror substrate. Mirror distortions, including astigmatism, can cause higher-order modes to appear affecting FEL performance. Therefore, it is important to quantify these higher-order modes because doing so uniquely identifies the optical field and may allow for corrective optics to single out the best modes for FEL lethality. This thesis will review free electron laser theory, and for the first time develop analytical solutions to quantify Hermite-Gaussian higher-order modes, develop a diagnostic for modal analysis, and determine the tolerance limits on mirror distortions.
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