Correlation between amplitude and phase photothermal signals based on Kramers–Kronig relations: Application to the investigation of interfacial thermal resistance in multilayers

Abstract

A general feature of frequency-domain photothermal (PT) experiments is that the signal phase follows the derivative of the log –log-amplitude. This is the consequence of the Kramers–Kronig (K–K) relations between real and imaginary parts of the thermal response of the system, represented by its thermal impedance. Simple expressions allow determining the phase from the amplitude, both theoretically and experimentally. They stem from the local Hilbert transform in differential form, instead of the traditional integral form which requires broadband information. These expressions are applied to the thermal impedance of a layered system. The measurements were performed in an earlier publication by photothermal radiometry (PTR) method to determine the interfacial thermal resistance between a 50 nm Ti layer and a steel or Si substrate. In this case, high frequency measurements are the most relevant, but they are also most prone to artifacts and background noise. It is concluded that the amplitude and phase results of Ti/steel samples are correctly linked to one another, but that the phase of Ti/Si samples violates the K–K relations at high frequency, probably due to a different signal generation mechanism than the PT one. The fact that PT phenomena are consistent with the K–K relations opens the possibility to check the self-consistency of the results obtained by PTR or by other related PT methods. It also reveals that the PT phase carries the same information as the signal amplitude, less a scaling factor that incorporates the thermal effusivity of a layer in the system.