Exact Solutions for the Temperature Rise in a Laser-Heated Slab

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Abstract
The mathematical treatment of laser machining involves the solution of the equation of heat conduction under conditions which generally impose the use of either approximate or numerical methods. Therefore, it seemed of interest to investigate the heating of an infinitely extended slab of finite thickness for various configurations of the laser beam for which exact solutions can be found. In particular, solutions are given (i) for beams with a Gaussian profile and for pencil‐shaped beams (intensity a radial step function) decaying exponentially, and for their limits of very fast (surface source) and very slow (radial heat flow only) decay, and (ii) for beams with radially constant intensity (axial heat flow). For the latter, the case of the slab resting on a substrate is considered by means of a modified image method for planar sources, enabling us to find all solutions by the superpositon of the well‐known result for point sources.

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