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Physical Review A

Coherent scattering by a spherical medium of resonant atoms

Sudhakar Prasad, Roy J. Glauber

Published Jun 20, 2011• 6 citations
Publisher

Abstract

We consider the problem of coherent resonant scattering of electromagnetic waves by a spherical medium of two-level atoms. The frequency dependence of the scattering amplitudes and cross sections reveals a complex structure of narrow peaks and dips. We relate these scattering resonances to the cooperative emission resonances characteristic of a sphere. We find the scattering to show considerable interference between the electric and magnetic multipole contributions, particularly in the lower multipole orders. This interference tends to enhance anisotropies in the differential scattering cross section even for small spheres. For such spheres, the peak values of the resonant contributions of the low-order multipoles to the total scattering cross section can increase with multipole order, in contrast to the usual decrease seen in nonresonant scattering.

DOI
10.1103/physreva.83.063821

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We consider the problem of coherent resonant scattering of electromagnetic waves by a spherical medium of two-level atoms. The frequency dependence of the scattering amplitudes and cross sections reveals a complex structure of narrow peaks and dips. We relate these scattering resonances to the cooperative emission resonances characteristic of a sphere. We find the scattering to show considerable interference between the electric and magnetic multipole contributions, particularly in the lower multipole orders. This interference tends to enhance anisotropies in the differential scattering cross section even for small spheres. For such spheres, the peak values of the resonant contributions of the low-order multipoles to the total scattering cross section can increase with multipole order, in contrast to the usual decrease seen in nonresonant scattering.
Radiation by the atoms of a resonant medium is a cooperative process in which the medium participates as a whole. In two previous papers we treated this problem for the case of a medium having slab geometry, which, under plane-wave excitation, supports coherent waves that propagate in one dimension. We extend the treatment here to the three-dimensional problem, focusing principally on the case of spherical geometry. By regarding the radiation field as a superposition of electric and magnetic multipole fields of different orders, we express it in terms of suitably defined scalar fields. The latter fields possess a sequence of exponentially decaying eigenmodes corresponding to each multipole order. We consider several examples of spherically symmetric initial excitations of a sphere. Small uniformly excited spheres, we find, tend to radiate superradiantly, while the radiation from a large sphere with an initially excited inner core exhibits temporal oscillations that result from the participation of a large number of coherently excited amplitudes in different modes. The frequency spectrum of the emitted radiation possesses a rich structure, including a frequency gap for large spheres and sharply defined and closely spaced peaks caused by the small frequency shifts and even smaller decay rates characteristic of the majority of eigenmodes.
Despite the quantum nature of the process, collective scattering by dense cold samples of two-level atoms can be interpreted classically describing the sample as a macroscopic object with a complex refractive index. We demonstrate that resonances in Mie theory can be easily observable in the cooperative scattering by tuning the frequency of the incident laser field or the atomic number. The solution of the scattering problem is obtained for spherical atomic clouds who have the parabolic density characteristic of BECs, and the cooperative radiation pressure force calculated exhibits resonances in the cloud displacement for dense clouds. At odds from uniform clouds which show a complex structure including narrow peaks, these densities show resonances, yet only under the form of quite regular and contrasted oscillations.
Despite the quantum nature of the process, collective scattering by dense cold samples of two-level atoms can be interpreted classically describing the sample as a macroscopic object with a complex refractive index. We demonstrate that Mie resonances can be easily observable in the cooperative scattering by tuning the frequency of the incident laser field or the atomic number. The solution of the scattering problem is obtained for spherical atomic clouds who have the parabolic density characteristic of BECs, and the cooperative radiation pressure force calculated exhibits resonances in the cloud displacement for dense clouds. Reminescent to uniform clouds which show a complex structure of narrow peaks, these densities show resonances, yet under the form of quite regular and contrasted oscillations.
An analysis is made of the scattering and absorption of light by a hydrogenlike atom in the field of an electromagnetic wave which mixes resonantly an excited level (n = 2) with its neighbor (n = 3). The scattering and absorption cross sections are calculated as a function of the frequency of the wave and of the intensity of the electromagnetic field. The influence of low-frequency nonresonance field on the scattering and absorption of light by an atom in the ground state is considered. It is shown that the scattering (absorption and luminescence) spectra of a hydrogenlike atom may be frequency modulated by a low-frequency amplitude modulated field.
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.
Steady electric and magnetic fields can stimulate frequency mixing of two laser waves in ensemble of free atoms. In addition to coherence conditions, the steady fields may induce additional resonance singularities essentially enhancing the cross section for scattering the sum-frequency wave. Interference between different components of the electric- and magnetic-field induced frequency summation amplitudes may cause significant effects on the efficiency of conversion. The dependence on the incident wave polarization and the atomic resonance structure is calculated analytically for the frequency mixing in atoms with a singlet structure of the ground and resonance states. Numerical estimates for the quantitative characteristics of the effect are presented for helium, alkali-earth and mercury atoms in their groundn1S0-state in the case of the two-photon resonance on excited singlet states with angular momentum 0, 1, and 2.