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The Journal of Chemical Physics

Calculation of vibrational level populations in multiphoton absorption processes

Jon T. Hougen

Published Aug 1, 1976• 18 citations
Publisher

Abstract

A simple procedure is described for computing steady-state excited vibrational state populations for multiphoton absorption processes in many-level systems. The procedure utilizes techniques already in the literature, i.e., expansion of the density matrix as a Fourier series in the frequency of the monochromatic radiation field, and introduction of linewidths and thermal relaxation by the addition of empirical terms to the density matrix time-development equation. Numerical computations can be carried out quickly on idealized systems, using commercially available band matrix inversion routines. A sample calculation for a 10-level, nine-photon process is presented.

DOI
10.1063/1.433181

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•May 20, 1991•0 citations
A simple procedure is described for computing steady-state excited vibrational state populations for multiphoton absorption processes in many-level systems. The procedure utilizes techniques already in the literature, i.e., expansion of the density matrix as a Fourier series in the frequency of the monochromatic radiation field, and introduction of linewidths and thermal relaxation by the addition of empirical terms to the density matrix time-development equation. Numerical computations can be carried out quickly on idealized systems, using commercially available band matrix inversion routines. A sample calculation for a 10-level, nine-photon process is presented.
We show that a class of multiphoton-absorption problems has exact solutions. In these problems an NNN-level atom or molecule absorbs energy from near-resonance monochromatic laser fields, and one asks for the time-dependent populations of the energy levels of the absorber. We find that the equal-Rabi and the harmonic-Rabi NNN-level systems studied numerically in the preceding paper are associated with Chebyshev and Hermite polynomials. Using the exact solutions we find exact expressions for the time averages of occupation probabilities in terms of sums of squares of polynomials. In addition we find the first approximate analytic expression for the long-period population oscillations observed in numerical solutions; and we show that a steady-state population inversion between the highest and lowest levels may be maintained by multiphoton pumping.
Multiphoton excitation and dissociation of large molecules under collisionless conditions is discussed in terms of an intercontinuum random coupling model. The mathematical approach described in a previous paper is used to obtain the general solution for a system of consecutively coupled discrete states, quasicontinuous manifolds, and continuous (dissociative) manifolds of molecular levels (eigenstates of the total molecular Hamiltonian), where the radiative coupling matrix elements are assumed to be given as a linear combination of smoothly varying and randomly varying (over level indices in the molecular manifolds) components. In the range of discrete molecular levels the time evolution is coherent and described in terms of the optical Bloch equation. In the quasicontinuous and continuous ranges the time evolution may be described in terms of Markoffian kinetic equations for the number of photons absorbed by the molecule, provided that the intramolecular vibrational relaxation widths associated with the optically active molecular modes is much larger than the Rabi frequency associated with the excitation of these modes. The kinetic evolution itself consists of direct multiphoton excitation processes (simultaneous transitions from the upper discrete levels to all higher energy molecular manifolds) resulting from the smooth component in the radiative coupling, and a consecutive excitation process described by the Pauli master equation with rates given by the golden rule expression. The interaction which enters into the golden rule expression is the variance in the radiative coupling. The direct excitation component contributes a negligible part of the overall excitation even if the random and smooth radiative coupling components are comparable. The resulting incoherent time evolution of the multiphoton excitation process is consistent with available experimental results. Coherent effects in the time evolution are expected for higher radiation field intensities, where the Rabi frequency becomes comparable to the intramolecular vibrational relaxation rate.
A theoretical analysis is made of the action of high-intensity bichromatic pump radiation on a system of vibrational–rotational molecular levels. The effective Hamiltonian approximation is used to reduce the problem to the solution of kinetic equations for the density matrix with a periodic external perturbation. These can be solved by adopting a basis of quasienergy states. The Raman susceptibility profile is obtained and investigated for an isolated vibrational resonance. It is shown that any change in the polarizability under excitation can be determined from the dependence of the population difference on the pump and scattered radiation intensities.
Using a semiclassical, effective-state approach to laser pulse propagation in a dilute gas of polyatomic molecules, a numerical model of multiphoton excitation which includes quasi- continuum transitions is used to calculate the effects of coherent propagation on multi-photon absorption. The initial laser field is assumed to be a quasimonochromatic Gaussian envelope which is nearly resonant with a portion of the lower vibrational states of the molecule. The molecular model consists of a single ground state and several bands of discrete states which are successively connected via electric dipole transitions. This model correctly mimics some of the major characteristics of multiphoton absorption, such as power threshold, fluence dependent absorption, and complete population inversion at high fluences. The model also allows for the calculation of the polarization induced in the medium by the electric field, and the subsequent influence that the polarization has on the field. As the pulse propagates, the Rabi oscillations induced in the lower level states become encoded onto the pulse, creating a series of peaks and troughs similar to the behavior of pulse propagation in two-level systems. Another coherent propagation effect which can change the excitation process is coherent sideband generation. The temporal distortions and the generation of new frequencies results in changes in the population distribution among the states, as well as changes in the total population transferred to the quasi-continuum. The pulse reshaping occurs while the total fluence of the pulse is decreasing due to energy loss to the quasi-continuum. The effective increase in total population transfer due to propagation effects is offset by the decrease in fluence. The resulting transfer of energy from the pulse to the molecule depends strongly on the initial characteristic of the pulse. These processes play an important role in any system where the efficient transfer of laser energy to the medium is required, such as laser-induced chemistry and laser isotope separation.
Results are reported on coherent monochromatic multiphoton excitation in many-level systems, which are representative for some of the basic mechanisms for atomic and molecular multiphoton processes. Numerical solutions are discussed that use the Floquet and quasi-resonant approximations in the framework of the URIMIR program package. The excitation schemes include direct three-photon excitation, two-photon excitation with diagonal coupling, Göppert-Mayer-type two-photon processes, multiphoton excitation with off-resonant inter-mediates, and practically irreversible coherent excitation into dense spectral structures. Several interesting phenomena are observed, such as nonlinear line shifts and broadenings of multi-photon resonances of relevance for multiphoton spectroscopy and almost constant intermediate population inversions, potentially useful for laser design. The accurate numerical results are compared with approximate solutions from perturbation theory, and with simple analytical solutions from Rabi-type formulae.
A general program has been developed to calculate multiphoton processes in atomic hydrogen to arbitrarily high order in perturbation theory. This program can be used to calculate quantities in intense field multiphoton processes, such as the generalized cross section of multiphoton absorption, the a.c. Stark effect (shift and the broadening of the level due to the field interaction), and the parameters in nonlinear optics such as the nonlinear index of refraction and the nonlinear susceptibility. Effective parametrization of high-order susceptibilities has been achieved over a wide range of frequencies, utilizing concepts from quantum defect theory to express the results of lengthy calculations in a compact form that also permits extrapolation across thresholds and resonance poles. Significant progress has been made in direct numerical solution of the time-dependent Schroedinger equation. In addition, contributions have been made in: understanding the role of spatial dimensionality in the solution of model systems (e.G., the one-dimensional delta-function potential); the development of a general R-matrix code for perturbative computations of multiphoton processes in many-electron atoms; and multiphoton ionization with two commensurate laser frequencies.