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The Journal of Chemical Physics

Photon echo decays in optically dense media

Francis C. Spano, Warren S. Warren

Published Aug 1, 1990• 13 citations
Publisher

Abstract

In this paper we investigate the influence of optical density or superradiance on the photon echo decay using the coupled Maxwell–Bloch equations. Our theory applies to optically thick but dilute media, where the transition dipole–dipole interaction between nearby chromophores is negligible, but the far-field interaction is not. Corrections to the (normalized) echo decay in the optically thin limit, INe(tπ)=exp [−4tπ/T2] (tπ and T2 are the interpulse separation times and the coherence relaxation times, respectively), are found by expanding the polarization, excited state population and pulse electric fields in powers of ζ=αz, where α is the inverse Beer’s absorption length, and z is the coordinate along the pulse propagation axis. Numerical solutions for optically thick samples are also presented, and when ζ≫1 the normalized echo intensity is found to obey the simple analytic form: INe(tπ)=2 exp [−tπ/T1]−1, when tπ/T1<ln 2 and INe(tπ)=0 for longer times. (T1 is the population relaxation time.) In this limit the echo decay is independent of the coherence relaxation time T2, illustrating the profound effect intermolecular coupling may have on any nonlinear optical measurement. We show that at low temperatures, when T2=2T1, the echo decay rate increases as the optical density increases. However, for higher temperatures, when T2 is sufficiently shorter than 2T1, the opposite behavior results, that is, the echo decay rate decreases as the optical density increases.

DOI
10.1063/1.459133

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•May 21, 1990•0 citations
In this paper we investigate the influence of optical density or superradiance on the photon echo decay using the coupled Maxwell–Bloch equations. Our theory applies to optically thick but dilute media, where the transition dipole–dipole interaction between nearby chromophores is negligible, but the far-field interaction is not. Corrections to the (normalized) echo decay in the optically thin limit, INe(tπ)=exp [−4tπ/T2] (tπ and T2 are the interpulse separation times and the coherence relaxation times, respectively), are found by expanding the polarization, excited state population and pulse electric fields in powers of ζ=αz, where α is the inverse Beer’s absorption length, and z is the coordinate along the pulse propagation axis. Numerical solutions for optically thick samples are also presented, and when ζ≫1 the normalized echo intensity is found to obey the simple analytic form: INe(tπ)=2 exp [−tπ/T1]−1, when tπ/T1<ln 2 and INe(tπ)=0 for longer times. (T1 is the population relaxation time.) In this limit the echo decay is independent of the coherence relaxation time T2, illustrating the profound effect intermolecular coupling may have on any nonlinear optical measurement. We show that at low temperatures, when T2=2T1, the echo decay rate increases as the optical density increases. However, for higher temperatures, when T2 is sufficiently shorter than 2T1, the opposite behavior results, that is, the echo decay rate decreases as the optical density increases.
The effects of optical density (OD) on photon echo experiments is examined both theoretically and experimentally. The theoretical treatment involves coupled Maxwell-Optical Bloch equations, and the experiments are picosecond photon echo studies of the mixed molecular crystal, pentacene in naphthalene. Under high power excitation (π/2, π) in high OD samples, the echo decays are predicted and observed to be faster than decays arising solely from the intrinsic molecular T2. The increase in the echo decay rate vanishes in the limit of low power or low OD, permitting the intrinsic molecular T2 to be measured. Previously reported concentration effects are shown to actually be optical density effects. The experimentally determined intrinsic molecular T2 of pentacene in naphthalene is shown to be independent of concentration and temperature (below ∼2 K) and T2≠2T1. It is suggested that the difference between T2 and 2T1 is due to the static multiplet level structure arising from the electronic state-phonon coupling.
We calculate the photon echo decay for a strongly coupled electron–phonon system, without making unrealistic assumptions about excitation pulse durations. We find that asymptotically, the photon echo decay is exponential, and we provide a formal expression for the decay rate 1/T2 which is independent of pulse duration. In contrast to the usual results, our expression for 1/T2 is nonperturbative in the electron–phonon interaction. We calculate the dependences of the echo amplitude on pulse power and duration, finding that the strong electron–phonon coupling modifies the simple two level system result. We also focus on the relationship between the zero phonon line shape in the homogeneous optical absorption spectrum, and the photon echo decay rate. We find that the zero phonon line shape is Lorentzian, with width 1/πT2. We show that the usual optical Bloch equation approach to analyzing time-domain experiments, which normally is only applicable if the electron–phonon coupling is weak, can be extended to the case of strong coupling. These strong coupling Bloch equations are adequate for analyzing the time dependence, but not the pulse power and duration dependences, of time-domain experiments. Our results indicate that within the Condon approximation, electron–phonon coupling cannot explain the very low temperature anomaly of recent photon echo experiments on impurities in molecular crystals.
A theoretical model for superradiant damping effect in inhomogeneously broadened media and its application to photon-echo problems are discussed. Only optically thin systems are considered, and all nonradiative relaxation mechanisms are neglected. Numerical calculations are performed for a very wide square inhomogeneous line shape and square applied pulses. It is shown that excitation of the system becomes increasingly difficult as the superradiant lifetime ${\ensuremath{\tau}}_{s}$ becomes comparable with T2∗{T}_{2}^{*}T2∗​. The energy reradiated after the end of the exciting pulse is considerable for ${\ensuremath{\tau}}_{s}\ensuremath{\approx}{T}_{2}^{*}$ and for some applied pulse areas. Two-pulse sequences produce multiple echoes as the microscopic polarizations spontaneously rephase. The first-echo amplitude maximum is found to occur for first-applied-pulse areas increasingly above $\frac{1}{2}\ensuremath{\pi}$ as ${\ensuremath{\tau}}_{s}$ decreases. For a given first-pulse area, the echo intensities increase initially as ${\ensuremath{\tau}}_{s}$ decreases (negligible damping) and finally decrease and vanish (important damping). The ratio of the second- to first-echo amplitudes is found to be proportional to $\frac{{T}_{2}^{*}}{{\ensuremath{\tau}}_{s}}$ for small superradiance levels and to decrease when ${\ensuremath{\tau}}_{s}$ approaches T2∗{T}_{2}^{*}T2∗​. An approximate estimate of this ratio is derived for the region of linearity in $\frac{{T}_{2}^{*}}{{\ensuremath{\tau}}_{s}}$.
The optical density of a resonant medium significantly affects the output signal in time resolved nonlinear optical interactions. The resonant interaction with a medium reshapes any propagating short pulse, and the mutual interaction of several such pulses, as in four-wave mixing, must be treated self-consistently. We present a theoretical framework for the proper handling of the propagation of all pulses, input as well as generated, in the small area limit. For optically thick resonant absorbers, negative time delay signals are observed, and the apparent decay rate of the induced polarization is faster than the rate observed for thin samples. We experimentally measure degenerate four-wave mixing in an atomic medium as an example, and demonstrate the quality of the theoretical model by the excellent fit to measured signals over several orders of magnitude. The improved understanding enables us to provide a simple, but surprisingly accurate, estimate for the apparent decay rate in homogeneously broadened optically thick media: If the absorption is given by \ensuremath{\alpha}, the propagation length is L, and the transverse relaxation time is T2{\mathit{T}}_{2}T2​, the apparent decay rate 2/Ta{\mathit{T}}_{\mathit{a}}Ta​ of a time resolved four-wave mixing signal is given by 2/Ta{\mathit{T}}_{\mathit{a}}Ta​=(2/T2{\mathit{T}}_{2}T2​)(1+\ensuremath{\alpha}L/2).
Using a stochastic model, we calculate the temperature dependence of the homogeneous linewidth and the two-pulse photon-echo decay rate of optically active impurities in glasses. The theory applies to the low-temperature regime T\ensuremath{\lesssim}10 K, where it is assumed that the dephasing arises from an elastic dipole interaction of the impurity with an array of thermally excited tunneling systems. Averaging over the tunneling parameters and using relaxation rates appropriate to vitreous silica, we find that the photon-echo decay rate varies as T1.1{T}^{1.1}T1.1 and the homogeneous linewidth as T1.2{T}^{1.2}T1.2 for 0.1\ensuremath{\le}T\ensuremath{\le}1 K, assuming a constant density of states for the tunneling systems. Both the photon-echo amplitude and the optical dipole-moment correlation function decay exponentially in time. The rates, however, are different, with a ratio of photon-echo dephasing time to optical-absorption dephasing time of 6.9 at T=1 K, assuming a constant density of states. The relation of our results to various experimental studies is discussed.
The phase relaxation of optically excited electronic states can be studied in fourwave mixing experiments where three excitation pulses at t = 0, t =t21, and t = T, respectively, interact via the third-order polarization of the medium to generate a nonlinear signal whose intensity depends on the dephasing time T2. For inhomogeneously broadened optical transitions the signal will be emitted as a time delayed photon echo pulse at t = T + t21 provided the pulse duration is short compared with T2. The echo amplitude decays as exp(−4t21/T2).1