Femtosecond spectroscopy from the perspective of a global multidimensional response function

At the microscopic level, multidimensional response functions, such as the nonlinear optical susceptibility or the time-ordered response function, are commonly used tools in nonlinear optical spectroscopy for determining the nonlinear polarization resulting from an arbitrary excitation. In this Account, we point out that the approach successfully developed for the nonlinear polarization can also be used in the case of a directly observable macroscopic quantity. This observable can be, for example, the electric field radiated in a nonlinear mixing experiment, the rate of fluorescence resulting from one- or two-photon absorption, or the rate of a photochemical reaction. For each of these physical processes, perturbation theory can be used to expand the measured quantity in a power series of the exciting field, and an appropriate global response function can be introduced for each order of perturbation. At order n, the multidimensional response function will depend on n variables (either time or frequency) and have the same general properties as the nonlinear susceptibility resulting, for example, from time invariance or causality. The global response function is introduced in this Account in close analogy with the nonlinear susceptibility or the time-ordered microscopic response. We discuss various applications of the global response function formalism. For example, it can be shown that in the weak field limit, a stationary signal induced in a time-invariant system is independent of the spectral phase of the exciting field. Although this result had been demonstrated previously, the global response function enables its derivation in a more general way because no specific microscopic model is needed. Multidimensional spectroscopy is obviously ideally suited to measure the global multidimensional response function. It is shown that the second (or third)-order response can be exactly measured with 2D (or 3D) spectroscopy by taking into account the exact shape of the exciting pulses. In the case of a 2D measurement of the third-order response, a particular projection of the complete 3D response function is actually measured. This projection can be related to a mixed time and frequency representation of the response function when the pulses are assumed to be infinitely short. We thus show that the global response function is a useful tool for deriving general results and that it should help in designing future experimental schemes for femtosecond spectroscopy.