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Journal of Physics B Atomic and Molecular Physics

The time development of adiabatic two-photon absorption. II. Rate equation regime

K A Eagles, Carlos R. Stroud, L. Allen

Published Jul 14, 1982• 9 citations
Publisher

Abstract

The time dependence of the two-photon absorption process is experimentally investigated in the intermediate intensity rate equation regime, where incoherent relaxation plays an important role, as a function of laser linewidth and Doppler width. The saturation effects observed are in excellent agreement with the predictions of Allen and Stroud (1982).

DOI
10.1088/0022-3700/15/13/018

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The time dependence of the two-photon absorption process is experimentally investigated in the intermediate intensity rate equation regime, where incoherent relaxation plays an important role, as a function of laser linewidth and Doppler width. The saturation effects observed are in excellent agreement with the predictions of Allen and Stroud (1982).
The time dependence of the two-photon absorption process is experimentally investigated in the low intensity regime as a function of laser linewidth, Doppler width, polarisation and delay between pulses, using 5 ns pulses at two different frequencies. Excellent agreement is found between experiment and the theoretical predictions of Allen and Stroud. The experiment arguments and expands earlier work on the dependence of linewidth in the two-photon process. The lineshape of the Hansch-type laser producing the pulses is shown to be best described by a Gaussian rather than a Lorentzian or as the result of an Ornstein-Uhlenbeck process.
An unconventional perturbation treatment of the theory of two-photon absorption in a four-level atomic system close to two-photon resonance is developed on the basis of the optical Bloch equations. The results for the longtime populations of the intermediate and upper states are valid in low- and medium-intensity regimes. In both regimes the upper state population shows a lorentzian resonance centred on zero two-photon detuning with a width equal to the natural linewidth of the upper state. The intermediate state population shows a similar behaviour in the intermediate regime only, with no significant dependence on two-photon detuning in the low-intensity regime. However, in both regimes the population of the virtual intermediate state is at least comparable to that of the resonant upper state. The perturbation theory results have been checked by direct numerical calculations based on the Bloch equations. Numerical calculations confirm the presence of a high-intensity regime characterized by power shifts, broadening and saturation effects.
The coherent interaction of two smoothly varying, near-resonant, two-photon pulses with a three-level system can be described by "two-photon damped Bloch equations" which are analogous to those for a one-photon transition in a two-level system except for the presence of a two-photon coupling and a frequency shift. These equations are solved for the cases ${\ensuremath{\gamma}}_{1}$, ${\ensuremath{\gamma}}_{2}\ensuremath{\ll}\ensuremath{\Omega}$, ${\ensuremath{\gamma}}_{1}={\ensuremath{\gamma}}_{2}$, and $\frac{{\ensuremath{\gamma}}_{2}{k}^{2}{\ensuremath{\epsilon}}^{4}}{{\ensuremath{\Omega}}^{2}}$, ${\ensuremath{\gamma}}_{1}\ensuremath{\ll}\ensuremath{\Omega}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase relaxation widths, respectively, and $\ensuremath{\Omega}$ is the Rabi frequency. The leading contribution to the refractive index is intensity dependent, caused by the level shifts inherent in multiphoton processes; it includes a relaxation dependent part which is important at times shorter than ${\ensuremath{\gamma}}_{1}^{\ensuremath{-}1}$. The second-order contributions depend on the square of the intensity and the time-integrated square of the intensity. The latter contribution, which is relaxation dependent, causes line asymmetry at the long-wavelength wing; it consists of a term proportional to ${\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$ and only important at early times and a term proportional to $2{\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$.
We study three ways in which two-photon ionization, at high light intensities and near an intermediate-state resonance, may be described. We calculate the instantaneous ionization rate, the total ion count, and a saturable rate constant. We find that the total ion count, which is the experimentally interesting quantity in most cases, is very closely modelled by our saturable rate constant. The instantaneous ionization rate bears no useful relation to the total ion count in many cases. In comparing these three descriptions of ionization we have included the effects of detuning from resonance, finite light bandwidth, finite intermediate-state lifetime, and light intensity; we have varied all of these parameters independently over several orders of magnitude. Our conclusions are in good agreement with, and extend the findings of, Ackerhalt and Shore