We analyze the early-time development of the spectral transfer of optical excitation in inhomogeneously broadened systems. Previous models for the decay of the narrow-band fluorescence are generalized to include systems with asymmetric energy-dependent transfer rates. It is assumed that the energy splittings of the ions are uncorrelated. An analogous model for the early-time development of the background fluorescence is outlined. Detailed results are presented for spectral transfer over a Gaussian line shape assuming a dipole-dipole mechanism with rates that are inversely proportional to the square of the energy mismatch. Comparison is made with the treatment of Holstein et al.
The radiative-transfer rate for inhomogeneously broadened optical lines is calculated. Photon emission and absorption is supplemented by one- and two-phonon processes to achieve energy conservation. These photon-assisted radiative transfer processes are shown to be effective at low concentrations, where electrostatic and exchange processes are weak, in the intermediate or strong radiative trapping regime. Specific application is made to the case of dilute ruby, and explicit transfer rates are calculated. The coherence factors, which undermine nonradiative single-phonon-assisted energy transfer for a small energy mismatch, are not important for radiative transfer, if the photon mean free path is longer than the wavelength of the energy matching phonon. The one-phonon process is found to vary linearly with temperature, and to be independent of the energy mismatch. However, the calculated rate for the case of ruby is found to be very slow. The two-phonon-assisted radiative-transfer process rates are calculated. They dominate the one-phonon-assisted rate for all reasonable temperatures. The results of an earlier paper are shown to be related to phonon-assisted radiative transfer if one replaces the exchange coupling between transfer sites
J by $\ensuremath{\sim}\frac{{\ensuremath{\tau}}_{R}^{\ensuremath{-}1}\ensuremath{\hbar}}{(\frac{{r}_{0}}{\ensuremath{\lambda}})}$. Here ${\ensuremath{\tau}}_{R}$ is the radiative lifetime,
r0 the distance between sites, and $\ensuremath{\lambda}$ the photon wavelength involved in the radiative transfer. This replacement results in a line center to wing transfer rate equal to 1.1 \ifmmode\times\else\texttimes\fi{}
102 ${e}^{\ensuremath{-}\frac{42}{T}}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ for ruby (independent of concentration for a photon mean free path smaller than the sample size).
The time-dependent spectral redistribution in fluorescence line narrowing in an inhomogeneously broadened system is investigated by means of simple theoretical models and Monte Carlo calculations. The two-site phonon-assisted energy transfer rate is assumed to be site-symmetric and a Lorentzian function of energy mismatch. For the Monte Carlo calculations initial excitations are made at the wing, shoulder, and center of the inhomogeneously broadened Gaussian band. For the wing excitation it is argued that, because of the above Lorentzian factor, the intensity in the initial site of excitation decays with a much smaller back-transfer rate than for the center excitation, simplifying the theoretical treatment. This effect is demonstrated by the Monte Carlo study. A refinement of this model is developed for a general situation, and its predictions are compared with the Monte Carlo results.
We study the time development of spectral profile of an inhomogeneously broadened system of a low concentration of optically active random two-level impurity ions excited with a narrow-band laser pulse. The spectral transport is due to phonon-assisted spatial migration of optical excitations among the impurity ions. The basic two-site transfer rate is assumed to be site symmetric and independent of the energy levels of the initial and final sites when the thermal energy is much larger than the inhomogeneous width. A "direct transfer" model relevant to some current works is solved exactly for dipolar interaction at a low-impurity concentration. In this model transitions are allowed only between the initial site of excitation and rest of the ions. The result agrees well with a previous "exact" Monte Carlo calculation for a short time. The result is improved systematically for longer times by using a pair model, a 3-site model, and a 4-site model. In the first two models, energy transfer is allowed along one and two nearest-neighbor percolation bonds, respectively, starting from the initial site of excitation. In a 4-site model additional transfer is included between the initial ion and its second neighbor. Transitions to the background ions without back transfer are allowed in all cases. Finally, the theoretical results are compared with recent experimental data.
An inhomogeneously broadened spectral line, for a spatially random site distribution with dipole-dipole coupling to first-neighbor sites, is studied using a Monte Carlo Technique. The site-site coupling is chosen at random according to the probability density function for first-neighbor dipolar interactions $\frac{[\mathrm{exp}(\ensuremath{-}\frac{1}{{X}^{\frac{1}{2}}})]}{{X}^{\frac{3}{2}}}$; where
X is the transfer rate proportional to
r61, and
r is the spatial separation of the interacting sites. This form is appropriate for a dilute concentration of sites on a three-dimensional random lattice. We arrange the sites in a linear array, and the spectral diffusion equation (including back transfer) is solved iteratively. We show that this is an adequate description of the three dimensional system at short and intermediate times for the dilute concentrations considered in this study. A group of sites with a specific energy is excited at (time)
t=0. The time development of the excitation profile is then studied. For very short times, the decay of the initial excitation is linear in time, appropriate to the largest value of
X (near neighbor site). For longer times, but over a fairly narrow time interval, the decay proceeds as $\mathrm{exp}(\ensuremath{-}a{t}^{\frac{1}{2}})$, as calculated by Inokuti and Hirayama for dipolar coupling in the absence of back transfer. At the longest times, the decay follows the one-dimensional diffusion result, ${t}^{\ensuremath{-}\frac{1}{2}}$. The intermediate-time regime, covering most of the decay of the initial excitation profile, does not seem to follow any simple time dependence.