Frequency bandwidth optimization of photothermal technique for thermal conductivity depth profiling

Abstract

The frequency bandwidth of photothermal technique for thermal conductivity depth profiling has been studied, which is based on an iterative numerical algorithm for inverse calculation of inhomogeneous thermal conductivity depth profiles of solid samples. This inversion algorithm should require a sufficient frequency bandwidth since the numerical experiments have demonstrated that systematic errors will occur in the inversion from insufficient data. According to the thermal wave propagation property, at low frequency range, the thermal wave can penetrate into deep region of the sample, but it will depress spatial resolution because of its long thermal wavelength. Nevertheless, lack of low frequency the detected information will cause the derivation of the depth profiling in the deep part. On the other hand, with high frequency, the thermal wave only can penetrate into shallow regions near the surface of the sample, although the spatial resolution will be intensified because of its short thermal wavelength. However, lack of high frequency information will induce the depth profiling derivation near the surface layer. Theoretical analyses have indicated that there is a high frequency fhigh, at which or over the amplitude of the surface temperature tends to be proportional inversely to square root of frequency f, but the phase tends to a constant 0.25π. Besides, there is a low frequency flow, at which or below the amplitude will tend to be inversely proportional to f and the phase will tend to a constant 0.5π under the heat-insulation boundary condition at the rear surface. With the frequency bandwidth (flow,fhigh) all the necessary information is essentially included for reconstructing the depth profiling. Practically, the numerical experiments have also demonstrated the frequency bandwidth can be less than (flow,fhigh) based on the investigated conditions, which will be described in detail in the paper.