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Journal of Physics B Atomic and Molecular Physics

The coherence properties of scattered light in a level crossing experiment

Alan V. Durrant

Published Jan 1, 1972• 7 citations
Publisher

Abstract

Existing theories of level crossing and modulation effects in the light scattered from optically thin vapours are restricted to cases where the light from different atoms is either completely incoherent (lateral scattering) or completely coherent (forward scattering). In the latter case the resonances are Doppler broadened equivalents of the well known Lorentzian forms predicted for lateral scattering. The more general case of partial coherence of the scattered light, which might be expected at small scattering angles, has not previously been considered. In this paper the degree of coherence of the scattered light has been formulated in terms of an autocorrelation function. For scattering from an ideal gas the autocorrelation function may be split into two terms representing complete incoherence and complete coherence. The predicted resonances for scattering into an arbitrary direction are mixtures of the Lorentzian and Doppler broadened forms, but the latter have a strong angular dependence that effectively restricted their observation to the small angular range associated with diffraction from the active volume of the vapour.

DOI
10.1088/0022-3700/5/1/022

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The intensity of polarized light reflected from an absorbing vapour has been calculated for optical radio-frequency double resonance and level-crossing. The reflection coefficient was obtained from the refractive index of the vapour, using the equations of classical electromagnetism. The result obtained by integration over a range of frequencies corresponding to broad-band illumination shows that all the familiar magnetic resonance effects are to be expected in the reflected light, except that the resonance curves should be Doppler- and pressure-broadened. A study of the zero-field level-crossing curve, as a function of vapour pressure, should enable one to follow the gradual transition from diffuse to coherent scattering, and to test whether the simple theory of selective reflection is adequate.

Proceedings of the Physical Society•Jun 1, 1967•12 citations
Existing theories of level crossing and modulation effects in the light scattered from optically thin vapours are restricted to cases where the light from different atoms is either completely incoherent (lateral scattering) or completely coherent (forward scattering). In the latter case the resonances are Doppler broadened equivalents of the well known Lorentzian forms predicted for lateral scattering. The more general case of partial coherence of the scattered light, which might be expected at small scattering angles, has not previously been considered. In this paper the degree of coherence of the scattered light has been formulated in terms of an autocorrelation function. For scattering from an ideal gas the autocorrelation function may be split into two terms representing complete incoherence and complete coherence. The predicted resonances for scattering into an arbitrary direction are mixtures of the Lorentzian and Doppler broadened forms, but the latter have a strong angular dependence that effectively restricted their observation to the small angular range associated with diffraction from the active volume of the vapour.
Abstract A theoretical and experimental study has been made of the intensity of forward scattered light in conditions of radio-frequency double resonance and level-crossing. The main conclusions are (i) that the magnetic resonance and level-crossing effects which are familiar in the laterally scattered light are to be found also in the forward scattered light, but (ii) that the resonance curves are Doppler-broadened, and the strength of the oscillatory field necessary to generate them is correspondingly increased, and (iii) that coherence narrowing is much more pronounced in forward than in lateral scattering. The differences derive from the coherence between the radiation from different atoms which is present in forward, but not in lateral scattering, and which is implicitly recognized in the classical theory of the propagation of light in polarizable media. The equations of electromagnetism form the basis of the present analysis. In the presence of the oscillatory magnetic field the polarization tensor is complex and time-dependent. The time dependence leads to the result that the number of eigenwaves necessary to describe propagated waves in the medium is four, as against the usual two. An arbitrary wave is resolved into a superposition of the four eigenwaves, all of which have different propagation constants. The analysis is very general, and is applicable to experiments with lasers. The components of the polarization tensor are derived by finding first the polarizability of individual atoms from a semi-classical theory of their interaction with light and magnetic fields. The macroscopic polarization, obtained by summing over individual atoms, is then given by folding the individual polarizabilities with the Doppler distribution of resonance frequencies. In the particular case studied in detail (the inter-combination resonance line, 2537 A, in mercury), the Doppler width is much greater than the radiation width, and the imaginary (absorptive) part of the polarizability is Gaussian. Expressions are derived in a ‘weak scattering’ approximation for the intensity of the forward scattered light. They are magnetic resonance functions which are found to be the Doppler-broadened equivalents of the familiar functions, based on Lorentzians, characteristic of scattering by a single atom. In the case of double resonance the forward scattered light is strongly modulated, as is the laterally scattered light. To investigate coherence narrowing, the zero-field level-crossing has been studied with the restriction of ‘weak scattering’ removed. The result is obtained that the width of levelcrossing curves is inversely proportional to the vapour density for magnetic fields such that the Zeeman splitting is small in relation to the Doppler width. This analytical result has been verified experimentally.
The semi-classical method for calculating the observable properties of light scattered from an optically thin atomic vapour is placed in the context of the well known input-output relations for random signals transmitted through a linear system. The autocorrelation of the scattered field amplitude, the statistic from which the mean intensity spectral density and intensity fluctuations may be deduced, is expressed in terms of the autocorrelation of the incident optical field and the autocorrelation of the scattering system; these latter are quantities that can be readily measured or calculated. An appendix contains a classical and a semi-classical calculation of the scattering system autocorrelation function for a vapour of atoms optically excited from a J=0 level under the conditions of a level crossing experiment. In the course of this calculation the relationship between the matrix element of the electric dipole moment operator and the corresponding classical quantity is carefully examined.
Existing theories of optical pumping do not take into account the relative phase of the optical field which monitors the resonances. In experiments on excited atoms the optical phase relations have a profound effect on magnetic resonance curves. Signals in the forward-scattered light show Doppler broadening, modified by strong coherence narrowing; a result quite different from that for the laterally scattered light. The question is raised as to whether similar effects are to be expected for resonances in the ground states. The problem is analysed in terms of the classical theory of the propagation of light in polarizable media. The (complex) polarizability of the sample under conditions of optical pumping and magnetic resonance is calculated to first order in perturbation theory. An expression is obtained similar to that found for resonance in the excited states, but different in so far as the factor which describes the magnetic resonance (the density matrix for the ground states) is independent of the factor which describes the Doppler effect (the plasma dispersion function). Since the intensity of the transmitted light is a function of the imaginary part of the polarizability, the conclusion is that the ground-state resonance curves should not be Doppler-broadened, which is in agreement with the known facts.
A theory of resonant light scattering from two level atoms in a gas is developed for an arbitrarily strong pump field. The Doppler effect and two body foreign gas collisions are considered. Elastic, inelastic (upward and downward), and the velocity changing collisions are retained in the general formulation. The impact approximation is not made, and the expressions for all (field dependent) collision rates are generated internally. Particular attention is paid to the lower state interaction, and a more complete parameterization of the theory is achieved. The general theory is non-Markovian and the quantum regression theorem is not used; it reduces to existing theories in special cases. It is shown that the coherent scattering is not a general property of the system if the lower state interaction is present; the "coherent" component acquires a small, but nonzero, width in the strong field region. This width could provide information about some new collision parameters.
The resonant scattering of monochromatic light in gases is analyzed. Explicit expressions for the differential cross section are obtained in the limits (Doppler width) \ensuremath{\gg} (natural width) \ensuremath{\gg} (collision width) and (collision width) \ensuremath{\gg} (natural width) \ensuremath{\gg} (Doppler width). In the former limit, the frequency distribution of scattered light is narrowed in the forward and backward directions, and has the full absorption width for right-angle scattering. In the collision-dominated limit, the cross section separates into a coherent and a fluorescent component. The motion of the atoms broadens the $\ensuremath{\delta}$ function distribution characterizing coherent scattering from a stationary target. The interference between light scattered by different atoms is assessed.
The intensity of polarized light reflected from an absorbing vapour has been calculated for optical radio-frequency double resonance and level-crossing. The reflection coefficient was obtained from the refractive index of the vapour, using the equations of classical electromagnetism. The result obtained by integration over a range of frequencies corresponding to broad-band illumination shows that all the familiar magnetic resonance effects are to be expected in the reflected light, except that the resonance curves should be Doppler- and pressure-broadened. A study of the zero-field level-crossing curve, as a function of vapour pressure, should enable one to follow the gradual transition from diffuse to coherent scattering, and to test whether the simple theory of selective reflection is adequate.