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Journal of the Optical Society of America B

Theory of heterodyne pump–probe experiments with femtosecond pulses

Antonio Mecozzi, Jesper Mørk

Published Nov 1, 1996• 46 citations
Publisher

Abstract

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.

DOI
10.1364/josab.13.002437

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Journal of the Optical Society of America B•Jan 1, 1987•0 citations
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.
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.
We demonstrate that the transient coherent nonlinearity (coherent artifact) affecting the pump-probe response of semiconductor optical amplifiers can be experimentally separated from the incoherent transient. The technique is based on measuring the mirror component of the coherent artifact which is a background-free four--wave mixing signal at a different frequency with respect to the transmitted probe in a heterodyne detection scheme. Measurements on amplifiers of different length reveal strong deviations from the commonly expected symmetric shape of the coherent artifact in case of long waveguides.
We show that in order to describe completely the coherent coupling between the pump and probe pulses in a pump– probe measurement of transient absorption, the influence of induced phase gratings must be included. The importance of phase gratings is demonstrated experimentally for the case of a bleachable dye and analyzed in terms of transient four-wave mixing. These results are relevant to the interpretation of pump–probe measurements on all time scales performed with pulses from a single laser, particularly when the pulse duration is comparable with the material response time.
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.
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.