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Physical Review A

Limitations of pump-probe femtosecond time-resolved experiments: Time-dependent absorption and dispersion line shapes

Benjamin Fain, S. H. Lin, Vadim Khidekel

Published Apr 1, 1993• 20 citations

Abstract

In ultrashort (femtosecond) pump-probe spectroscopy, the pump pulse creates a nonstationary medium while the probe pulse propagates in such a medium. The analysis of this situation is performed in the framework of linear nonstationary optics. Constitutive equations and electrodynamics of the nonstationary media are analyzed. The propagation of the probe-pulse energy is characterized by the coarse-grained susceptibility introduced in the previous works of the authors [S. H. Lin and B. Fain, Chem. Phys. Lett. 157, 233 (1989); B. Fain, S. H. Lin, and N. Hamer, J. Chem. Phys. 91, 4485 (1989); B. Fain and S. H. Lin, Chem. Phys. 161, 515 (1992)]. The main objective of the paper is the derivation and analysis of the delay-time-dependent absorption and dispersion line shapes of the probe pulse. This line shape has a very specific form corresponding to phase-dependent Raman absorption (emission). The latter phenomenon is introduced and described in the present paper.

DOI: 10.1103/physreva.47.3222

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In ultrashort (femtosecond) pump-probe spectroscopy, the pump pulse creates a nonstationary medium while the probe pulse propagates in such a medium. The analysis of this situation is performed in the framework of linear nonstationary optics. Constitutive equations and electrodynamics of the nonstationary media are analyzed. The propagation of the probe-pulse energy is characterized by the coarse-grained susceptibility introduced in the previous works of the authors [S. H. Lin and B. Fain, Chem. Phys. Lett. 157, 233 (1989); B. Fain, S. H. Lin, and N. Hamer, J. Chem. Phys. 91, 4485 (1989); B. Fain and S. H. Lin, Chem. Phys. 161, 515 (1992)]. The main objective of the paper is the derivation and analysis of the delay-time-dependent absorption and dispersion line shapes of the probe pulse. This line shape has a very specific form corresponding to phase-dependent Raman absorption (emission). The latter phenomenon is introduced and described in the present paper.

In femtosecond laser-pulse experiments the pump pulse, with duration comparable or shorter than a typical period of intra- or intermolecular vibrations, creates a nonstationary wave packet. In this paper we use the density-matrix method to analyze creation of space–time coherences by the pump pulse and their effect on the probe pulse. Expressions for the density-matrix jumps, induced by the probe pulse, have been obtained in a general case. The material equations, determining propagation of the probe pulse, have been derived.

In the femtosecond pump-probe pulse experiments, the pump pulse with duration comparable to or shorter than a typical vibrational period creates a nonstationary wave pocket. In this paper, we use the density-matrix method to treat the dynamic behavior of the system (embedded in a heat bath) in the femtosecond pump-probe pulse experiment. The expression for the coherence of the system due to the pumping laser is derived, and the generalized master equations for the dynamical behavior of the system after the pumping pulse is over are also obtained. Some numerical calculations are performed for a two-level system to demonstrate the time-dependent behaviors of memory kernel, rate constant, and population.

Controlling stimulated emission is of importance because it competes with absorption and fluorescence under intense laser excitation. We performed resonant nonlinear optical spectroscopy measurements using femtosecond pulses shaped by π- or π/2-step phase functions and carried out calculations based on density matrix representation to elucidate the experimental results. In addition, we compared enhancements obtained when using other pulse shaping functions (chirp, third-order dispersion, and a time-delayed probe). The light transmitted through the high optical density solution was dominated by an intense stimulated emission feature that was 14 times greater for shaped pulses than for transform limited pulses. Coherent enhancement depending on the frequency, temporal, and phase characteristics of the shaped pulse is responsible for the experimental observations.

The response of a system of coherently pumped atoms to weak electromagnetic perturbing fields is discussed in general terms. The pump field is prescribed, and is assumed to propagate unattenuated through the medium which the (identical) atoms comprise, exciting a particular transition through either electric or magnetic dipole coupling. The perturbing field, which is assumed to act through electric dipole coupling, is allowed to be the (resonant) field emitted by the atoms, a weak external field absorbed by the atoms under resonant conditions, or a resonant or nonresonant wave propagating through the pumped medium. All of these cases are conveniently described within the same general framework, based on an analysis of the (single-atom) two-time dipole-moment correlation function. This function contains nonstationary components as well as stationary ones, owing to the time dependence imposed by the pump field. We are accordingly led to introduce a time-dependent linear susceptibility, which is inhomogeneous and anisotropic, due, respectively, to the variation of the pump phase within the medium and the preferred direction established by the pump-field polarization. Such disparate effects as parametric frequency conversion, emission- and absorption-line splitting, and time-dependent modulation of resonant emission and absorption functions in optical-pumping experiments emerge naturally from a single unified formalism.

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.