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accelerator cavity amplitude approximation average betatron betatron motions bucket bunch buncher C. A. Brau Chapter Coherent Radiation computed conventional lasers corresponds current density described devices dimensionless discussed distribution effects efficiency electric field electron beam electron energy electron lasers electron motions electron-beam current emittance energy spread expression focusing formula free-electron lasers frequency function harmonics IEEE illustrated in Fig index of refraction Inst integral intensity interaction J. D. Jackson laser beam laser field lasers operating longitudinal mode Madey magnetic field Meth microtron mirrors modulator Nucl obtain one-dimensional optical beam optical field optical klystron optical pulse optical resonator oscillations parameters particles peak phase plane phase space Phys quantum radius reflectance result rf linacs saturated shown in Fig small-signal gain spectrum SPIE spontaneous emission storage rings superconducting tapered wiggler trajectory transverse mode uniform wiggler velocity wave packet wavelength wiggle wiggler axis wiggler field wiggler length wiggler period