  Recent publications have described a method for stand-off optical detection
of explosives using resonant infra-red photothermal imaging. This technique
uses tuned lasers to selectively heat small particles of explosive lying on a
substrate surface. The presence of these heated particles is then detected
using thermal infra-red imagery. Although the method has been experimentally
demonstrated, no adequate theoretical analysis of the laser heating and
subsequent particle cooling has been developed. This paper provides the
analytical description of these processes that is necessary to understand and
optimize the operational parameters of the explosive detection system. The
differential equations for particle and substrate temperatures are derived and
solved in the Laplace transform domain. The results are used to describe
unexplained cooling phenomena measured during the experiments. A limiting
particle temperature is derived as a function of experimental parameters. The
effects of radiative and natural convection cooling of the particle and of
non-uniform particle temperature are examined and found to be negligible.
Calculations using the analytical model are compared with experimental
measurements. An analysis of thermal contact heating of the substrate is
included in the appendix.
