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Equations are derived for determining magneto-optic coefficients in multilayer systems with arbitrary directions of their magnetizations. The equations are cast in a matrix form that is suitable for numerical simulations. This establishes a framework for calculating the Kerr and Faraday effects for a multilayer system in much the same way as has been applied previously to the bulk. Numeri-0 cal Kerr results are presented for the following systems: bulk Fe, an overlayer of 50 A of Fe on Au, and for an Fe/Au superlattice for different directions of the magnetization and different angles of incidence.

Self-amplified spontaneous emission in a free-electron laser has been proposed for the generation of very high brightness coherent x-rays. This process involves passing a high-energy, high-charge, short-pulse, low-energy-spread, and low-emittance electron beam through the periodic magnetic field of a long series of high-quality undulator magnets. The radiation produced grows exponentially in intensity until it reaches a saturation point. We report on the demonstration of self-amplified spontaneous emission gain, exponential growth, and saturation at visible (530 nanometers) and ultraviolet (385 nanometers) wavelengths. Good agreement between theory and simulation indicates that scaling to much shorter wavelengths may be possible. These results confirm the physics behind the self-amplified spontaneous emission process and forward the development of an operational x-ray free-electron laser.

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The reflection-transmission problem is considered for light from a boundary between two magnetic media with arbitrary direction of the magnetization within each medium. Explicit formulas are derived for the magneto-optic coefficients.