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 - I. Excitation under conditions of strictly homogeneous line broadening
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 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 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 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.
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.
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.
We have studied the excitation spectrum arising from the optical mixing of the frequencies of a strong bichromatic field interacting with a three-level atom, where the two initially populated modes ωa and ωb are equal to the two atomic transition frequencies, respectively. In the limit of high photon densities, the excitation spectrum near the frequency ω = ωa – 2ωb has been calculated as a function of the parameter η = Ωa2/Ωb2, where Ωa and Ωb are the Rabi frequencies of the two laser fields, respectively. For [Formula: see text] and for weak fields for which [Formula: see text], the spectral function describes a Lorentzian line peaked at the frequency ω = ωa – 2ωb and has a width of the order of γ0, where γ0/2 is the natural width for a two-level atom. When Ωb2 > γ02 and [Formula: see text], the band at ω = ωa – 2ωb splits into two bands described by two Lorentzian lines peaked at ω = ωa – 2ωb ± Ωb/√2 and have spectral widths of the order of 3γ0/4. The ratio of the height of the band ω = ωa – 2ωb to the height ω = ωa – 2ωb ± Ωb/2 is 3:2. The probability amplitudes for both bands take large negative values indicating that optical amplification of the signal field may be expected to occur at these frequencies. When Ωa = Ωb = Ω, η = 1, and for Ω2 < γ02, the spectral function describes a single band at ω = ωa – 2ωb while for Ω2 > γ02, the single band splits into five pairs of bands which are separated from the frequency ω = ωa – 2ωb by frequency shifts which are equal to: ± Ω/√2, [Formula: see text], ± Ω, ± Ω√2, and ± Ω√3, respectively, and have spectral widths of the order of 3γ0/4. For [Formula: see text] and for laser fields for which Ωa2 > γ02 and Ωb2 > γ02, the spectral function consists of three pairs of bands. The probability amplitudes for these bands vary linearly with η and may take large values for [Formula: see text]. A complete discussion of the excitation spectrum as well as a graphical representation of the derived results has been given.
The rate of absorption and emission of radiation is calculated for two laser fields interacting with a homogeneously broadened, three-level molecular system. The time-dependent Schrödinger equation is used in the density matrix formalism and the laser fields are treated classically. The laser fields, at frequencies ωpand ωs, are off resonance by δpand δs, respectively, such that|\delta_{p}|, |\delta_{s}| \ll \omega_{p}, \omega_{s}. Analytic expressions are derived for the imaginary part of the electric susceptibility,\chi"(\omega_{p})and\chi"(\omega_{s}). The present work differs from previous calculations in that the laser fields may have arbitrary values of field strength and each field may be on or off resonance. A discussion is presented of the properties of the solutions for\chi"(\omega_{p})and\chi"(\omega_{s}), for both strong and weak laser fields, based on computer evaluation of the general analytic expressions. It is shown that the solutions may be divided into single-photon and two-photon, or Raman, contributions. Conditions for optimizingdI(\omega_{s})/dzand the photon conversion efficiency are derived. These results may be applied to analyzing the pulsed, optically pumped submillimeter laser and to studying two-photon absorption in a gas.