Numerical Simulation of the Whole Thermal Lensing Process with Z-Scan-Based Methods Using Gaussian Beams

Abstract

A general study of the diffracted far field due to thermal lens heating using Gaussian beams is presented. The numerical simulation considers the whole system, including both the optical and the thermal parameters. It is shown that the existing simplified relations found in the literature and used up to now only give the order of magnitude of the thermo-optical coefficients. More accurate, simplified formulas are derived to measure the induced thermal phase shift when working with Z-scan-based methods. Temperature estimation in absorbing media turn out to be more reliable whether using time-resolved or steady-state techniques. The extension of the calculation to the image formation in a 4f system is also addressed.

Time-Resolved cw Thermal Z-scan for Nanoparticles Scattering Evaluation in Liquid Suspension

“…The comparison of the obtained signal with the one given by Equation (5) shows that even if they have a similar profile, their amplitude can be very different. Taking into account the experimental parameters which are the same as those considered in the simulation of [ 15 ], we obtained that the measurement procedure of underestimated its value by 60%. Coming back to the OPM measurements, and taking into account the standard deviation σ over the different observations, the results that are shown in Figure 3 give an average value of the absorption: .…”

“…So, for each acquisition the absorption was measured owing to the OPM and the TL processing as explained previously. The obtained α values were not equal especially because the equation used for the fitting according to Equation (5) is derived with too many approximations, as is shown in reference [ 15 ]. Indeed, in the latter reference, based on the Helmoltz equation, the diffracted intensity profiles at finite distance were calculated numerically without any approximation considering the whole system, including the optical and the thermal parameters.…”

“…In a separate paper, we performed the simulation of the temperature distribution in the thermal lens according to Equation (1) in order to validate our numerical measurement approach. The result of the temperature distribution at various times was presented in Figure 1 in [ 15 ] and compared to the beam waist of the exciting Gaussian beam, using the same parameters as those shown in Figure 2 of reference [ 17 ].…”

“…The derivative of Equation (5) allows to optimize this signal, finding positions of the sample around where the signal is extremal. As it was shown in [ 15 ], a numerical study of the whole thermal lensing and optical processes to evaluate the accuracy related to the measurement using the simplified analytical relation (3a) and therefore (5) gave only an order of magnitude of the desired coefficients to be measured and . These relations, generally used to estimate the thermo-optical coefficients, have the advantage of existing, but a preliminary calibration of the signal must be conducted in order to obtain reliable and accurate measurements.…”

“…Already, we have experimentally demonstrated the feasibility of extracting an image of the phase shift induced by TL and applied this method to transversally map an inhomogeneous thin film doped with different concentrations of silver nanoparticles [ 14 ]. Additionally, numerical simulation considered the whole system, including both optical and thermal parameters to obtain more reliable temperature estimation in absorbing media whether in time-resolved or steady-state Z-scan techniques [ 15 ].…”