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Physical Review A

Origin of spectral holes in pump-probe studies of homogeneously broadened lines

Robert W. Boyd, Shaul Mukamel

Published Apr 1, 1984• 102 citations
Publisher

Abstract

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 (${T}_{1}$) matrix.

DOI
10.1103/physreva.29.1973

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Nonlinear susceptibilities of molecular aggregates: Enhancement of χ ( 3 ) by size

Francis C. Spano, Shaul Mukamel

We investigate the third-order nonlinear-optical susceptibility ${\ensuremath{\chi}}^{(3)}$ in molecular aggregates and its scaling with aggregate size. The aggregate is modeled as a collection of N-interacting, homogeneously broadened two-level systems, where the interaction includes both the static dipole-dipole coupling and superradiant coupling. Our expression for ${\ensuremath{\chi}}^{(3)}$, derived using Liouville-space Green-function techniques, contains contributions from excitons as well as biexcitons. The scaling of ${\ensuremath{\chi}}^{(3)}$ with aggregate size and its dependence on the laser detuning and homogeneous and inhomogeneous dephasing rates are analyzed. ${\ensuremath{\chi}}^{(3)}$ contains terms that scale as \ensuremath{\sim}${N}^{2}$ and \ensuremath{\sim}N(N-1). However, under off-resonant conditions these terms interfere destructively, resulting in a linear \ensuremath{\sim}N scaling, which is precisely what is expected for N monomers. Detailed calculations of phase-conjugate degenerate four-wave mixing and pump-probe spectroscopy are performed. In the first case we predict the existence of a narrow dephasing-induced resonance superposed on the broader superradiant exciton line shape. Pump-probe spectroscopy shows a series of biexciton absorption lines, some of which are superradiantly broadened.

Physical Review A•Nov 1, 1989•292 citations
The theory of pump-probe experiments is examined using the nonlinear optical susceptibility \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(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 Im\ensuremathχ(3)\mathrm{Im}{\ensuremath{\chi}}^{(3)}Im\ensuremathχ(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 \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3) that is valid away from the impact limit and contains a realistic population relaxation (T1{T}_{1}T1​) matrix.
We report the experimental investigation of spectral hole burning and four-wave mixing at the interface of glass and a dense cesium vapor using two tunable diode lasers in a pump-probe experiment. Narrow spectral features were observed in the center of the strongly self-broadened D2{\mathrm{D}}_{2}D2​ line at Cs number density from 9\ifmmode\times\else\texttimes\fi{}1015{10}^{15}1015 to 2.8\ifmmode\times\else\texttimes\fi{}1016{10}^{16}1016${\mathrm{ncm}}^{\mathrm{\ensuremath{-}}3}$. The origin of the resonances is associated with the coherent scattering of laser beams at a self-induced population grating, and their spectral widths reflect the relaxation rate of the population difference. A simple asymptotic expression is used to fit the spectral line shapes and to infer a population relaxation time of 3.7(2) ns, which is almost an order of magnitude shorter than natural lifetime of the excited state.
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.
A theory of line shape for a Doppler broadened probe in the presence of a strong infrared pump is presented. The optical analog of Bloch equations has been developed; the T1, T2 relaxations are introduced phenomenologically in these equations. Doppler broadening of both the pump and signal transitions are taken into account to obtain the absorption coefficient for the signal. This leads to a Doppler broadened Gaussian function weighted by a pump induced Lorentzian function whose position and linewidth are controlled by the detuning and power respectively of the pump radiation. The absorption coefficient is further modulated by a Lorentzian function containing the fourth power of the radiation. The off-resonance pumping leads to two symmetrically located ‘‘Lamb dips’’ or ‘‘peaks’’ around the Gaussian pedestal for co- and counter propagating laser radiations. A calculation of the line shape of the central sub-Doppler feature for the on-resonance pumping is in good agreement with the observed variation of linewidth with pressure. Numerical results are presented graphically to demonstrate the pump and signal power dependence of the central hole or peak for the bent and cascade-type double resonances. The results obtained confirm the expected behavior based on level populations.
The response of three- and four-level homogeneous laser systems to an amplitude modulated field in the presence of an incoherent pump is treated using a rate equation model. A spectral hole due to the finite bandwidth of the population oscillations is observed with a modulated field at either the pump or the laser frequency. The spectral width of the hole broadens at a rate proportional to the sum of the dimensionless intensities of the pump and laser fields. The power levels required to broaden the 1/T1 hole in various laser systems is presented. The experimental implications of this result are discussed.
We present a theoretical study of phase-conjugate (PC) emission by backward degenerate four-wave mixing (DFWM) processes in a resonant gas medium. In this study, high-intensity (pump) saturation and Doppler broadening are simultaneously taken into account. We develop a general formalism to deal with the saturation effects in the case of a three-level atom (with degenerate frequencies) and obtain analytical expressions of the PC field amplitude (after integration over the velocity distribution) under the following conditions: (1) the probe beam and one of the pump beams have intensities weak enough to be treated at the lowest perturbation order, whereas the other pump intensity is arbitrarily high; (2) the probe crosses the standing pump wave at grazing incidence. It is shown that the saturation induced by the forward pump (copropagating with the probe beam) leads to entirely different behavior of the nonlinear susceptibility from that induced by the backward pump (counterpropagating with the probe beam). This saturation anisotropy is specific to the inhomogeneous broadening of the gas medium, which is also responsible for the frequency splitting experimentally observed in the intensity line shape of the PC emission for large pump intensities. At saturation, this splitting is theoretically shown to be proportional to the Rabi frequency, with a proportionality coefficient critically dependent on the relaxation processes; in the case of a single-relaxation model, the splitting is equal to 1.5 times the Rabi frequency. The PC intensity line shape is demonstrated to be of dispersive origin in the case of backward saturation, whereas for forward saturation, narrow structures are predicted in both real and imaginary parts of the nonlinear susceptibility, also leading to the intensity line-shape splitting. One also finds that the reflectivity of PC mirrors based on resonant DFWM strongly depends on the origin of the saturation. For a saturating forward pump, the PC emission intensity tends toward zero with increasing pump intensities, and it reaches a finite (nonzero) limit in the case of backward saturation. Most of these predictions have been observed experimentally [ BlochD., Phys. Rev. Lett.49, 719 ( 1982)].
We investigate the third-order nonlinear-optical susceptibility ${\ensuremath{\chi}}^{(3)}$ in molecular aggregates and its scaling with aggregate size. The aggregate is modeled as a collection of N-interacting, homogeneously broadened two-level systems, where the interaction includes both the static dipole-dipole coupling and superradiant coupling. Our expression for ${\ensuremath{\chi}}^{(3)}$, derived using Liouville-space Green-function techniques, contains contributions from excitons as well as biexcitons. The scaling of ${\ensuremath{\chi}}^{(3)}$ with aggregate size and its dependence on the laser detuning and homogeneous and inhomogeneous dephasing rates are analyzed. ${\ensuremath{\chi}}^{(3)}$ contains terms that scale as \ensuremath{\sim}N2{N}^{2}N2 and \ensuremath{\sim}N(N-1). However, under off-resonant conditions these terms interfere destructively, resulting in a linear \ensuremath{\sim}N scaling, which is precisely what is expected for N monomers. Detailed calculations of phase-conjugate degenerate four-wave mixing and pump-probe spectroscopy are performed. In the first case we predict the existence of a narrow dephasing-induced resonance superposed on the broader superradiant exciton line shape. Pump-probe spectroscopy shows a series of biexciton absorption lines, some of which are superradiantly broadened.