A comparison between Hankel and Fourier methods for photothermal radiometry analysis

Abstract

Photothermal radiometry has recently been investigated for use in the multidimensional thermal characterization of anisotropic samples. In application, there are two principal thermal models available for such characterization: a Cartesian model for the heat equation, which requires the application of three Fourier transforms to arrive at a solution (dubbed the Fourier technique), and a cylindrical model for the heat equation, which requires the application of a Hankel transform and a single Fourier transform (dubbed the Hankel technique). The Fourier technique allows for three-dimensional characterization, while the Hankel technique is expected to greatly reduce the computational time required. As these models can be very computationally expensive, the potential to reduce this cost is of great interest. In this work, these multidimensional models are presented after which they are compared for accuracy, computational time, and assumption limitations. It was found that both the Fourier and Hankel techniques could accurately arrive at desired thermal properties, but that the Hankel Technique reduced the computational time by between 100× and 250× depending upon mesh spacings. Accuracy limitations were found as the eccentricity of the heating laser was increased with a less than 13% error being induced from a beam with a 3–1 axis ratio. The Hankel technique shows ideal application in computationally expensive models which employ a relatively circular beam shape.