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Journal of the Optical Society of America B

Time-domain theory for pump–probe experiments with chirped pulses

N. Tang, Richard L. Sutherland

Published Dec 1, 1997• 34 citations
Publisher

Abstract

We present a time-domain treatment of pump–probe experiments with chirped pulses derived from the same laser source. In this theory we treat the nonlinearities as decaying either instantaneously or exponentially. Therefore the results from it cover different mechanisms originating from electronic orbitals, molecular vibrations, molecular orientations, excited-state populations, etc., as long as their decay falls into one of the two classes. We derive expressions for the transmission of the probe in each case. We divide the transmission expression from nonlinearities with exponential decay into a number of terms, each of which describes a unique situation. These include two-beam coupling in the coherent region that is not present with chirp-free pulses, two-photon absorption, and excited-state absorption. We give analytical expressions for the transmission where possible, and we give the first-order approximation and the estimated error when expression in closed form is unattainable. We discuss various aspects of these terms.

DOI
10.1364/josab.14.003412

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We present a time-domain treatment of pump–probe experiments with chirped pulses derived from the same laser source. In this theory we treat the nonlinearities as decaying either instantaneously or exponentially. Therefore the results from it cover different mechanisms originating from electronic orbitals, molecular vibrations, molecular orientations, excited-state populations, etc., as long as their decay falls into one of the two classes. We derive expressions for the transmission of the probe in each case. We divide the transmission expression from nonlinearities with exponential decay into a number of terms, each of which describes a unique situation. These include two-beam coupling in the coherent region that is not present with chirp-free pulses, two-photon absorption, and excited-state absorption. We give analytical expressions for the transmission where possible, and we give the first-order approximation and the estimated error when expression in closed form is unattainable. We discuss various aspects of these terms.
Most chirped-pulse experiments refer to a theoretical study from McGurk, Schmalz, and Flygare [J. Chem. Phys. 60, 4181 (1974)] which is well tailored to interpret the signals obtained with very fast chirped pulses, but is not sufficient to account for the signals in the case of slower chirped pulses used in spectroscopy to increase the signal-to-noise ratio. A theoretical study of the polarization of molecules subjected to a chirped pulse in a cell, uniform supersonic flow, or molecular beam is presented. Three degrees of approximation for the polarization are introduced and are compared with the numerical solution of the optical Bloch equations. These expressions enter the analytic expression of the free-induction-decay signal which is validated against experimental data on the rotational emission spectra of OCS molecules. A relation among the pulse duration, the line position in the chirped pulse, and the signal amplitude is proposed in the thermalized case. It assists in the optimization of the chirped-pulse parameters and in the estimation of the error associated with the line intensity.
We give the first analysis of heterodyne experiments that includes the propagation along the waveguide. Under the assumption that the material polarization is adiabatically eliminated we derive simple analytic expressions for the coherent coupling term, exact for Gaussian pulses and a good approximation for other shapes. We find that the measured signal does not display any phase dependence close to zero pump–probe delay. A four-wave mixing band is identified in the heterodyne signal, and a new class of experiments is proposed by analogy with the spatial case.
Recent femtosecond transmission-correlation measurements1 of large organic dye molecules revealed surprising details concerning the ultrafast relaxation dynamics of such dyes. Superposed on simple exponential decays, the observation of damped sinusoidal oscillations with a period of ~ 150 fs has been puzzling. We theoretically and numerically demonstrate that such oscillatory behavior could be due to quantum beats of closely spaced two levels, whose coherent superposition state is optically prepared and monitored by a pair of femtosecond pulses. Although studies of quantum beats have been developed in both fluorescence-type2 and nonfluorescence-type measurements, such as those based on photon echoes3 or polarization spectroscopy,4 this simple method gives a new approach to femtosecond time domain spectroscopy. The generalized perturbation theory5 of optical coherent transients is extended to the case of optical transmission-correlation or pump-probe-type experiments. The model used is a three-level system which has split levels in either the ground or optically excited states. Especially, if the ground state has sublevels, the system becomes quite similar to the case of coherent Raman beats. Both cases are shown to give quantum beats when the excitation pulses are short compared with the inverse sublevel splitting. The period and decay time of the damped sinusoid represent the separation energy and the dephasing time between the sublevels, respectively. Using third-order perturbation theory, the transmission-correlation signal STC(τ) as a function of the first-to-second pulse delay time Is derived for arbitrary-shaped excitation pulses and is given in the form of fourfold integrals with many terms.
The delayed or time-domain four-wave-mixing experiment is treated in the regime of intense near-resonant pulses. The interaction with the radiation during both pump and probe pulses is considered to all powers of the electric field amplitude. Analytical results are obtained for an effective four-level system. These include the dependence of the coherence amplitudes on the ratio of the pump-field intensities when there is a large vibrational discrepancy between ground and excited electronic states and a general solution for the unitary time development during the probe pulse. For the first time, delayed coherent anti-Stokes Raman scattering is detected from highly dilute (10-ppm) guest molecules. Illustrative examples are presented for the system of pentacene in benzoic acid at low temperature. Vibronic-free induction decay and the effect of field inhomogeneity across the beam profile are found to be essential for understanding the observed intensity and spectral distribution of the signal beam in the region of optimum pulse intensity.
The theory of pump-probe experiments is examined using the nonlinear optical susceptibility ${\ensuremath{\chi}}^{(3)}$. The results of this calculation are shown to differ from those based on a commonly employed heuristic model of pump-probe experiments which assumes that the pump beam "prepares" the system which the probe beam subsequently monitors. This model consequently accounts only for processes where the interactions with the pump field come first in time. The correct expression for the probe absorption line shape as predicted by $\mathrm{Im}{\ensuremath{\chi}}^{(3)}$ contains a sum over all the possible time orderings of the various radiative interactions. The interference terms missing from the simplified treatment are shown to account for the spectral hole in homogeneously broadened lines observed recently [L. W. Hillman et al., Opt. Commun. 45, 416 (1983)]. In addition we derive a generalized expression for ${\ensuremath{\chi}}^{(3)}$ that is valid away from the impact limit and contains a realistic population relaxation (T1{T}_{1}T1​) matrix.
It is nowadays possible to generate, in different spectral ranges, ultrabroadband light pulses with transform-limited (TL) duration corresponding to a few optical cycles. Whenever broadband pulses are employed, the dispersion experienced by the pulses causes a frequency chirp which may severely lengthen their duration and influence time-resolved measurements. While it is well known that the relative weight of ground-state versus excited-state oscillations can be controlled by the chirp in the pump pulse, the effects of probe pulse chirp have not been fully investigated. In this paper we show, both theoretically and experimentally, that a pump-probe experiment using strongly-chirped probe pulses can preserve the temporal resolution of the TL pulses.