Analytic models for beam propagation and far-field patterns in slab and bow-tie X-ray lasers

Simplified analytic models for beam propagation in slab and bow-tie x-ray lasers yield convenient expressions that provide both a framework for guidance in computer modeling and useful approximates for experimenters. In unrefracted bow-tie lasers, the laser shape in conjunction with the nearly-exponential weighting of rays according to their length produces a small effective aperture for the signal. We develop an analytic expression for the aperture and the properties of the far-field signal. Similarly, we develop the view that the far-field pattern of refractive slab lasers is the result of effective apertures that are created by the interplay of refraction and exponential amplification. We present expressions for the size of this aperture as a function of laser parameters as well as for the intensity and position of the far-field lineout. This analysis also yields conditions for the refraction limit in slab lasers and an estimate for the signal loss due to refraction