Thermal gratings and phase in high-order, transient-grating spectroscopy

Thermal gratings are a well known feature in one-dimensional (i.e., single excitation) transient-grating spectroscopy. This paper presents theory and experiments for thermal gratings in multiple dimensions (i.e., with many excitations). The theory of thermal gratings is extended to an arbitrary number of dimensions using an incoherent Hilbert-space formalism. Interference between Hilbert-space pathways makes it impossible for a thermal grating to propagate across multiple time intervals. The only surviving signal is a hybrid--a population grating between excitations and a thermal grating between the final excitation and the probe. This theory is tested on auramine O in methanol (1D) and in an ionic liquid (3-butyl-1-methylimidazolium hexafluorophosphate) (1D and 2D). In methanol, the ground-state recovery and thermal-grating signals are well separated in time; in the ionic liquid, they are not. Using the results of the theory, accurate subtraction of the thermal-grating signal is possible, extending the useful time range of the experiments. Both the comparison to the theory and the subtraction of the thermal-grating signal are dependent on accurate measurements of the time-dependent phase in these systems. Models are proposed to account for the time-dependent phase. Beer's law is generalized to multidimensional grating spectroscopy. This law provides conventions for consistently comparing the absolute phases and magnitudes between grating and nongrating experiments and between experiments of differing dimensionality.