Theoretical model for round continuous-wave transient temperature distribution in round optically transmissive materials using the reverse thermal wave transform

This paper is a continuation of my work in describing the applications to which the Reverse Thermal Wave Transform may be applied. As the title implies, this paper addresses the application of the Reverse Thermal Wave Transform to Transmissive optical components. A number of researchers have asked about the time temperature history of imposed laser beams in transmissive crystals. Very often the crystals are being used to shift the wavelength of the laser beam to a shorter wavelength. The problems that are described in this paper are quite generic to any transmissive material problem. The equations are not limited to lasers only, however. The same equations and concepts apply when used with Synchrotron and X-Ray radiation sources. Not to put too fine a point on it, the equations are also applicable to those problems that are associated with Microwave radiation sources as well. In this lecture we will also point out some additional equations and concepts which have not been described before relative to transient heat transfer in flat plates. Heretofore, when using the equations shown by Carslaw and Jaeger the expectation was that the temperature shown in the illuminated area was, in fact, the average temperature throughout the irradiated area. As I show in this paper this is only true for the case where the irradiating source flux density is spread over a radius that is equal to, or greater than, R o equal to or greater than 6 (root)(alpha) (tau) . When the radius is R o less than 6 (root)(alpha) (tau) there is a temperature gradient from the center of the component out to the edge of the laser beam, X- Ray beam, et al. There is, then, the gradient from the edge of the laser beam out to the edge of the component. These are two different gradients that are computed very differently. Since Carslaw and Jaeger were only dealing with the one dimensional semi-infinite plate models, this detail failed to materialize in their equations. We will use a set of examples complete with nodal maps and graphical representations to describe how the temperature gradient results from the Reverse Thermal Wave Transform.