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

The forward scattering of resonance radiation, with special reference to double resonance and level-crossing experiments

Alan Corney, B P Kibble, George William Series

Published Jul 5, 1966• 87 citations
Publisher

Abstract

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.

DOI
10.1098/rspa.1966.0159

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Coherent Forward Scattering Spectroscopy (Cfs): Present Status and Future Perspectives

Gerd M. Hermann, Kay Niemax

I. INTRODUCTION A. Level-Crossing Phenomena and the Effect of Coherent Forward Scattering Coherent forward.scattering (CFS—used for the physical effect as well as for the approach of coherent forward scattering spectroscopy) has been investigated first in connection with atomic resonance radiation phenomena like double resonance and level-crossing effects.1 The very early level crossing has been measured in lateral fluorescence in a weak magnetic field.2 This “zero-field level crossing” in lateral detected resonance fluorescence is called “Hanle effect”3 and can be classically described by the model of an atomic oscillator, emitting during its Larmor precession with respect to the applied magnetic field (Figure 1).

C R C Critical Reviews in Analytical Chemistry•Jan 1, 1988•6 citations
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.
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.
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.
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.
The forward scattering of resonance radiation (2537 AA) from a 198-mercury lamp by 198-mercury vapour has been investigated as a function of magnetic field. The experiments have revealed two regions of interest; (i) in weak magnetic fields, certain features in the intensity against field (level crossing) curves shift to lower field strengths as the optical density is increased (coherence narrowing), (ii) in stronger magnetic fields, features shift to higher field values with increasing optical density. Over the range of densities investigated there is good agreement between theory and experiment.
Abstract A theory is formulated to describe the modulation which has been observed in fluorescent light from atoms subjected simultaneously to optical and radio-frequency radiation. The optical field stimulates one or more of a set of excited states of the atom, between which the radio-frequency field establishes coherence. This coherence is manifest in the fluorescent radiation. Interference between radiations of different frequency leads to modulation. General expressions are given for the intensity of the fluorescent light as a function of time. The Zeeman structure of the transition (63P1 — 6 1S0), A2537 A, in mercury is studied in detail. Modulation at frequencies, 1, 2, 3 and 4 times that of the radio-frequency field, w0, is predicted, and resonant effects at static magnetic fields, 0, 1/2, 1, 3/2, 2 and 3 times H0, the field for which w0 is the Larm or frequency. Resonances at fields other than H0 are due to excitation with light of mixed polarization. Most of the predicted effects have been found experimentally. A ‘frequency diagram’ is introduced and discussed, to represent the combined effects of static and radio-frequency magnetic fields. To each excited state belong a set of r frequencies, where r is the number of states linked by the radio-frequency perturbation. The 9 levels are drawn, as functions of H, for the states mJ = 0, ± 1, of 3P1. The resonances at fields other than H0 may be associated with intersections of frequency levels belonging to different mJ.
I. INTRODUCTION A. Level-Crossing Phenomena and the Effect of Coherent Forward Scattering Coherent forward.scattering (CFS—used for the physical effect as well as for the approach of coherent forward scattering spectroscopy) has been investigated first in connection with atomic resonance radiation phenomena like double resonance and level-crossing effects.1 The very early level crossing has been measured in lateral fluorescence in a weak magnetic field.2 This “zero-field level crossing” in lateral detected resonance fluorescence is called “Hanle effect”3 and can be classically described by the model of an atomic oscillator, emitting during its Larmor precession with respect to the applied magnetic field (Figure 1).