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Journal of Physics B Atomic Molecular and Optical Physics

The theory of Faraday rotation in an autoionising resonance

J P Connerade

Published Sep 28, 1988• 10 citations
Publisher

Abstract

Expressions for magneto-optical rotation in an isolated autoionising resonance embedded in a flat continuum are obtained by applying the Kramers-Kronig relations to the Fano profile formula for photoabsorption. By considering rotation per unit optical depth in a Beutler-Fano profile, optimum conditions for detecting the rotation are deduced, and are shown to depend on both the width and the shape index. The far-wing approximation turns out to be inapplicable in the presence of a continuum background, but large rotation can occur in the transmission window of a narrow resonance of low q.

DOI
10.1088/0953-4075/21/18/001

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Expressions for magneto-optical rotation in an isolated autoionising resonance embedded in a flat continuum are obtained by applying the Kramers-Kronig relations to the Fano profile formula for photoabsorption. By considering rotation per unit optical depth in a Beutler-Fano profile, optimum conditions for detecting the rotation are deduced, and are shown to depend on both the width and the shape index. The far-wing approximation turns out to be inapplicable in the presence of a continuum background, but large rotation can occur in the transmission window of a narrow resonance of low q.
New results, both theoretical and experimental, are reported for magneto-optical rotation (MOR) in an autoionizing line. These include the first experimental demonstration of MOR in a Beutler-Fano profile and the first simulations of MOR patterns based on a previously proposed theory. In essence, the method constitutes the first determination of the variation of refractive index within an autoionizing resonance. The potential of such measurements for the determination of atomic properties is discussed.
The electromagnetic (EM) radiation gain for the induced photorecombination of an electron or ion (atom) into the autoionising state of the atom (negative ion) is considered in terms of the rotating-wave approximation, taking account of free-free transitions. Analytical expressions for the radiation gain coefficient have been derived for long and short times of action of the EM radiation. It has been shown that taking account of the induced direct transitions between the continua results in the 'renormalisation' of natural and ionisation widths of the autoionising state and, under certain conditions, essentially changes the character of the radiation gain; in particular, a typical asymmetric Fano-type profile appears.
We know that the Fano profile is obtained for single-photon absorption to the ionization continuum in the presence of an autoionizing state. We have shown that, in the case of (1+1)-photon absorption to an autoionizing state embedded in the ionization continuum, the absorption profile changes radically to a mirror image of the Fano profile for the value of q=\ifmmode\pm\else\textpm\fi{}1, q being the Fano q parameter. Moreover, for higher values of q, the (1+1)-photon absorption profile exhibits a broad minimum, while a distinct interference pattern is obtained in the profile of single-photon absorption. The absorption line shape has been found to oscillate between the Fano and a modified Fano profile for the autoionization through odd and even number of steps, respectively, via resonances.
Abstract The Kramers–Kronig relations are a pivotal foundation of linear optics and atomic physics, embedding a physical connection between the real and imaginary components of any causal response function. A mathematically equivalent, but simpler, approach instead utilises the Hilbert transform. In a previous study, the Hilbert transform was applied to absorption spectra in order to infer the sole refractive index of an atomic medium in the absence of an external magnetic field. The presence of a magnetic field causes the medium to become birefringent and dichroic, and therefore it is instead characterised by two refractive indices. In this study, we apply the same Hilbert transform technique to independently measure both refractive indices of a birefringent atomic medium, leading to an indirect measurement of atomic magneto-optical rotation. Key to this measurement is the insight that inputting specific light polarisations into an atomic medium induces absorption associated with only one of the refractive indices. We show this is true in two configurations, commonly referred to in literature as the Faraday and Voigt geometries, which differ by the magnetic field orientation with respect to the light wavevector. For both cases, we measure the two refractive indices independently for a Rb thermal vapour in a 0.6 T magnetic field, finding excellent agreement with theory. This study further emphasises the application of the Hilbert transform to the field of quantum and atomic optics in the linear regime.
Results of a fairly straightforward experiment on resonant magneto-optical rotation by rubidium-87 atoms revealed strong time-dependence of the polarization plane of light emerging from atomic vapors following a sudden irradiation with a laser beam. The rotation of the plane appears as a not direct consequence of the influence of the magnetic field on atoms. Reported measurements conducted using a vapor cell without any buffer gas or an anti-relaxation wall coating show that transmitted light has initially the same (linear) polarization as the incident one. Rotation of the polarization plane caused by an axial magnetic field develops in time scales similar to the pace of establishing the optical pumping/relaxation equilibrium in the atomic ensemble. The traditional passive Faraday rotation picture providing working description for the resonant magneto-optical effects in steady-state conditions does not explain the observed sequence of evolution of the polarization. The picture has to be augmented with analysis of time dependence in the spatial distribution of relevant quantum states within the atomic ensemble as well as longitudinal and transversal intensity profiles of the light beam. Isolation of external effects caused by the optical pumping and relaxation processes from consequences of internal atomic coherences should uncover more details of mechanisms of the latter in the presence of magnetic field. Plentitude of data available from time-dependent polarization measurements provides strong incentives for further development and tests of comprehensive models of coherent interactions of atomic ensembles with radiation and possibly a base for additional advance in understanding and applications of these phenomena.