Heat Conduction in Lenses

We consider several heat conduction problems for glass lenses with different boundary conditions. The problems dealt with in Sections sec:1 to sec:3 are motivated by the problem of an airborne digital camera that is initially too cold and must be heated up to reach the required image quality. The problem is how to distribute the heat to the different lenses in the system in order to reach acceptable operating conditions as quickly as possible. The problem of Section sec:4 concerns a space borne laser altimeter for planetary exploration. Will a coating used to absorb unwanted parts of the solar spectrum lead to unacceptable heating? In this paper, we present analytic solutions for idealized cases that help in understanding the essence of the problems qualitatively and quantitatively, without having to resort to finite element computations. The use of dimensionless quantities greatly simplifies the picture by reducing the number of relevant parameters. The methods used are classical: elementary real analysis and special functions. However, the boundary conditions dictated by our applications are not usually considered in classical works on the heat equation, so that the analytic solutions given here seem to be new. We will also show how energy conservation leads to interesting sum formulae in connection with Bessel functions. The other side of the story, to determine the deterioration of image quality by given (inhomogeneous) temperature distributions in the optical system, is not dealt with here.