Methods for modeling and diagnosis of nonlinear absorption in
the photothermal and photoacoustic spectrometry of homogeneous fluids
Stephen E. Bialkowski and Agnès Chartier
Department of Chemistry and Biochemistry
Utah State University
Logan, UT 84322-0300
USA
Poster: Area 6, Nonlinear phenomena and inverse problem
We have developed methods for analysis of irradiance-dependent photothermal signals in gas and liquid phase samples. The practical aspects of nonlinear signals modeling based on absorption and excited state kinetics when using Gaussian-profile excitation source lasers are addressed in this paper. The analysis is based primarily on the kinetic model for optical excitation and subsequent excited state relaxation. However, in order to obtain accurate model results, the irradiance dependent excitation laser profile and the subsequent hydrodynamic relaxation of the spatially and temporally distorted heating rate distribution resulting from nonlinear absorption and metastable state relaxation are incorporated. This heating rate is used to calculate the temperature change distribution and subsequently the optical elements needed to model the experimental photothermal interferometric, deflection, and lens signals.
Different approaches are used for pulsed and continuous laser excitation. For fast-pulsed laser excitation, excited state relaxation is slow compared to the rate for optical excitation. In this case the excited state concentration initially populated by the excitation laser is calculated based on an exponential law equation using the ratio of the integrated optical irradiance, H, to the integrated irradiance for ground state bleaching, HS
The initially excited population, N*, subsequently relaxes, perhaps through intermediate states. To obtain the thermal response, the rate of decay of the spacio-temporal excited state distribution is convoluted with the impulse-response for hydrodynamic relaxation. The resulting temperature change distribution is subsequently used to determine the relative amplitude of the of the optical element or the acoustic wave using the impulse-response for Gaussian beams. This process is facilitated by modeling the initial excited state distribution as a superposition of Gaussians with different beam waist radii. The resulting signal is a superposition of terms resulting from each Gaussian element. Equivalent considerations may be used to model results obtained using other excitation geometries, such as optical interference patterns formed in photothermal diffraction and the use of non-Gaussian-form excitation laser beams.
For continuous and chopped-continuous laser excitation, steady-state populations in the optically coupled ground and excited states are used to determine the excitation laser irradiance-dependent absorption coefficient, a , based on the bleaching irradiance, ES, through
The sample heating rate is then determined by the absorption coefficient-excitation irradiance product, a E. Sample heating by excited state absorption is incorporated as an additive term. Excited state distributions are first determined through the coupled rate equations describing excitation and relaxation. The excited state absorption heating rate is then the product of the spatially-dependent excited state concentration, the excited state absorption coefficient, and the spatially-dependent irradiance. The resulting photothermal signals are again calculated based on the theoretical response of a linear absorption system using a Gaussian laser beam. The sample heating rate is first modeled by a superposition of Gaussian sources, the appropriate optical element or photoacoustic wave amplitude is found for each source. The resulting signal is the superposition of the responses to each Gaussian source term.
Finally, the intermediate case wherein some excited state relaxation occurs during the excitation laser pulse is examined. The case where the excitation pulsed duration is less than the time scaled for hydrodynamic relaxation is examined. To obtain a solution, rate equations describing the optically-coupled ground and excited state populations are first solved, either analytically or numerically. It is possible to obtain analytical solutions in many cases since the rate equations are most often pseudo-first-order and driven by the excitation laser irradiance. This, of course, assumes that the small signal transmission is high enough to neglect pathlength-dependent irradiance changes and that the temporal profile of the laser pulse is constant. It is difficult to account for each individual heating rate resulting from the dynamic evolution of excited state populations. Instead, the evolving populations are used to first determine dynamic absorption coefficients. The total energy absorbed by the system is determined by time integration of the absorption coefficient-instantaneous irradiance product.
The presentation will describe the procedures for determining the nonlinear irradiance-dependant models which may be useful in photothermal and photoacoustic spectroscopy of chemical and biochemical substances using high power laser excitation sources. Examples of theoretical irradiance-dependent photothermal lens and photoacoustic spectroscopy signals of condensed phase bleachable organic dyes and photo-activated charge transfer compounds, and of gas phase optical saturation will be used to illustrate the utility of the approach. Finally, it is shown that nonlinear system response models can be used to determine kinetic and photophysical parameters from experimental results.