Propagation of Electromagnetic Radiation through a Resonant Medium

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Abstract
The propagation of electromagnetic radiation through a resonant medium is studied. The slowly varying envelope of the electromagnetic field is assumed to have a slowly varying phase in addition to a slowly varying amplitude. The nonlinear coupled equations of motion are solved exactly under rather general assumptions. Three classes of solutions are obtained; two of them reduce to the solutions of Crisp and of McCall and Hahn when the slowly varying phase is zero. Because of the gauge invariance of the equations of motion, there is a term proportional to the vector potential in addition to the terms of its space and time derivatives in the equation describing the evolution in time of the electromagnetic field. Hence, a characteristic frequency exists in the infrared region below which the wave number becomes imaginary.