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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

Excitation of transitions between atomic or molecular energy levels by monochromatic laser radiation - II. Excitation under conditions of strictly inhomogeneous line broadening and mixed homogeneous and inhomogeneous broadening

Robert A. L. Smith

Published Jun 30, 1978• 12 citations
Publisher

Abstract

Abstract The development with time of the excitation of a transition between two atomic or molecular energy levels under the influence of monochromatic laser radiation is examined under conditions of strictly inhomogeneous line broadening due to such causes as doppler shift arising from translational velocity. The ratio of the number of molecules, N2, in an excited state to the total number, N, is calculated for various ratios of the intensity parameter β ═ E0μ/ħ to the half line-width ∆, where E0 is the amplitude of the electric field in the incident radiation and is the dipole moment matrix element for the transition. Excitation functions obtained in a previous paper (I) for various values of the ratio γ/β, where γ is the half line-width in the absence of inhomo­geneous broadening, are used to obtain the variation of N2/N with time under conditions of mixed broadening for various values of the ratio γ/∆ and γ/β, when the exciting radiation is in exact resonance with the central frequency of the transition.

DOI
10.1098/rspa.1978.0117

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Abstract The development with time of the excitation of a transition between two atomic or molecular energy levels under the influence of monochromatic laser radiation is examined under conditions of strictly homogeneous line-broadening. The ratio of the number of atoms or molecules, N2, in an excited state to the total number, N, is calculated for various ratios of the intensity parameter β ═ E0μ/ħ to the half line-width γ. E0 is the amplitude of the electric field in the incident radiation and the dipole moment matrix element for the transition. Exact expressions for N2/N as a function of time are given for the situation when the frequency ω/2π of the radiation is exactly equal to that of the transition ω0/2π. Numerical calculations are made for the situation when the ‘detuning’ δ ═ ω — ω0 is not equal to zero. Approximate expressions are also derived in terms of δ for use in a later paper which deals with mixed homogeneous and inhomogeneous line-broadening.

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Physical Review A•Jan 1, 1979•1 citation
Abstract The development with time of the excitation of a transition between two atomic or molecular energy levels under the influence of monochromatic laser radiation is examined under conditions of strictly inhomogeneous line broadening due to such causes as doppler shift arising from translational velocity. The ratio of the number of molecules, N2, in an excited state to the total number, N, is calculated for various ratios of the intensity parameter β ═ E0μ/ħ to the half line-width ∆, where E0 is the amplitude of the electric field in the incident radiation and is the dipole moment matrix element for the transition. Excitation functions obtained in a previous paper (I) for various values of the ratio γ/β, where γ is the half line-width in the absence of inhomo­geneous broadening, are used to obtain the variation of N2/N with time under conditions of mixed broadening for various values of the ratio γ/∆ and γ/β, when the exciting radiation is in exact resonance with the central frequency of the transition.
Abstract The development with time of the excitation of a transition between two atomic or molecular energy levels under the influence of monochromatic laser radiation is examined under conditions of strictly homogeneous line-broadening. The ratio of the number of atoms or molecules, N2, in an excited state to the total number, N, is calculated for various ratios of the intensity parameter β ═ E0μ/ħ to the half line-width γ. E0 is the amplitude of the electric field in the incident radiation and the dipole moment matrix element for the transition. Exact expressions for N2/N as a function of time are given for the situation when the frequency ω/2π of the radiation is exactly equal to that of the transition ω0/2π. Numerical calculations are made for the situation when the ‘detuning’ δ ═ ω — ω0 is not equal to zero. Approximate expressions are also derived in terms of δ for use in a later paper which deals with mixed homogeneous and inhomogeneous line-broadening.
Abstract The treatment of transient excitation of transitions between two atomic or molecular energy levels by monochromatic laser radiation under conditions in which phase-destroying collisions predominate is treated by means of numerical and approximate analytical solutions of the equations giving the rate of change of a modified density matrix. The analytical solutions are then used to obtain the transient values of the population of the excited state when Doppler broadening is taken into account. Numerical calculations are carried out for the ratio γ'/γ = 10, where 2γ is the damping constant for diagonal and γ' for non-diagonal elements of the density matrix. This extends the treatm ent given in two previous papers for the situation when γ' = γ, corresponding to very low pressures for which phase-destroying collisions are relatively unimportant.
Abstract The effect of detuning from the centre of the Doppler-broadened profile of a transition is studied by use of the rate of change of a modified density matrix. The variation of population density of the two energy levels involved in the transition is obtained under transient conditions. Two values of the ratio of the damping parameters 2γ and γ' for the on-diagonal and off-diagonal matrix elements are considered; γ' = γ corresponds to very low gas pressures and y' = 10y to conditions for which phase-destroying collisions predominate. While these considerations apply both in the visible and infra-red spectral regions, and equally to atoms and molecules, they are at present of special interest in connection with excitation of transitions of importance in laser photochemistry.
The effect of detuning from the centre of the Doppler-broadened profile of a transition is studied by use of the rate of change of a modified density matrix. The variation of population density of the two energy levels involved in the transition is obtained under transient conditions. Two values of the ratio of the damping parameters 2y and T' for the on-diagonal and off-diagonal matrix elements are considered; y' = y corresponds to very low gas pressures and ' = 10y to conditions for which phase-destroying collisions predominate. While these considerations apply both in the visible and infra-red spectral regions, and equally to atoms and molecules, they are at present of special interest in connection with excitation of transitions of importance in laser photochemistry.
The effect of collisions on the resonance fluorescence spectrum of two-level atoms excited by monochromatic resonance radiation is studied. Analysis is performed for systems where the Doppler broadening is small compared to the collision frequency (high buffer gas pressures) and for the general case where collisions arbitrarily change the phase of the radiation-induced dipole moment (from completely interrupted to completely unaffected phase of the dipole moment). Both at a relatively low and at a high excitation radiation intensity, the resonance fluorescence spectrum is shown to depend on whether the two-level system is closed or open. This is especially true for the narrow unshifted Rayleigh scattering line. It is shown that, although the absorption line is homogeneously broadened, the resonance fluorescence spectrum exhibits a clearly pronounced anisotropy. In a direction close to the direction of propagation of the excitation radiation, the Rayleigh scattering line is maximally narrowed. Under certain conditions (that can easily be created in experiments), the width of this line is proportional to the diffusion coefficient of atoms interacting with the radiation.
Using the exact solution for a sine-driven two-level system we have calculated the nonlinear absorption coefficient of a gas of two-level molecules with homogeneous (hard collisions, spontaneous decay) and inhomogeneous (Doppler effect) broadening mechanisms. We have studied the dependence of the form of the absorption lines on relaxation parameters, on the strength of the electric field, and for various ratios between the homogeneous and inhomogeneous linewidths. We include considerations about saturation of the medium caused by the external field that gives rise to the so-called power broadening of the spectral lines. The dependence of the linewidth on the saturation parameters and on the relaxation parameters is determined. These parameters appear in the ratio of the purely homogeneous and purely inhomogeneous linewidths. Asymptotic behaviors of this linewidth for the homogeneous limit and for the inhomogeneous limit are also determined.