Thermally corrected solutions of the one-dimensional wave equation for the laser-induced ultrasound

Abstract

When thermal and stress confinement conditions are satisfied, the propagation of laser-induced ultrasound (LIU) is governed by an inhomogeneous wave equation for pressure, and the laser temporal profile can be modeled by a Dirac delta distribution. If these conditions are not fulfilled, the coupled differential equations for temperature and pressure must be solved, considering a laser pulse with finite time-width. Here, an exact solution of the boundary value problem for the 1D-wave equation is obtained in both the frequency domain and the time domain. Since highly absorbent optical materials are out of the validity range of the confinement conditions, these are used as a numerical and experimental model. It is shown that the impulse-response model correctly predicts the time of flight of photoacoustic waves. However, when considering a laser pulse with finite time-width, the resultant theoretical amplitude of the LIU signal decays rapidly to zero, which is not observed in the experiment. To overcome this, we propose a thermal correction on the LIU source, defined typically as the optical penetration length, which imposes a redefinition of stress and thermal confinement in just one statement. Additionally, with the aim of comparing the corrected-theoretical results with the acquired electrical signals, the sensor and the oscilloscope were modeled as an RC circuit in parallel. It was found that the amplitude of the electrical signal was proportional to the difference of the LIU amplitudes at the faces of the sensor. It is demonstrated that even though the sensor impulse response is modeled as a Dirac delta distribution, this difference strongly affects the shape of the LIU electrical signals, hiding relevant information of the acoustic waves.