Resonances in Mie scattering by an inhomogeneous atomic cloud

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. Introduction. – Mie theory is the well-known solution of Maxwell’s equations for the scattering of electromagnetic radiation by spherical objects [1, 2]. Via calculation of the electric and magnetic fields inside and outside the object the theory predicts the total optical cross section, which determines the amount of scattered light, and the form factor, which characterizes the far-field radiation pattern [3, 4]. Simple solutions are available in regimes where the object size R differs very much from the radiation wavelength λ, or when the refractive index m is close to unity. For example, for small phase-shifts |m−1|R/λ≪ 1 in optically dilute media |m−1| ≪ 1, one enters the Rayleigh-Debye-Gans regime whereas for small particles and small phase-shifts, one obtains Rayleigh scattering by point-like objects. For objects whose size is of the order of the radiation wavelength (e.g. water droplets in the atmosphere or in emulsions), Mie’s full theory has to be used to find the scattering pattern. Mie scattering differs from Rayleigh scattering in several respects. While the intensity of Rayleigh-scattered radiation scales with the object size as R and is identical in forward and backward direction, the intensity of Mie-scattered radiation is roughly independent of wavelength, and it is larger in forward than in backward direction. The greater the particle size, the more light is Mie-scattered into forward direction. The hallmark of Mie scattering, however, are the Mie resonances: those are sets of parameters (size, refraction index, wavelength), where Mie scattering is particularly strong or particularly weak. The sharpness of some Mie resonances make them useful for measuring unknown parameters such as particles’ size. Recently, a series of papers demonstrated how collectivities of point-like Rayleigh-scattering particles may cooperate [5–10] in scattering radiation into the forward direction and the relationship to Mie scattering was pointed out [11, 12]. Here, we show that the theory of collective scattering by smooth distributions of point-like scatterers is equivalent to Mie scattering by demonstrating that the premisses of both models are identical. Hence, we may apply the Mie scattering technique to atomic clouds as long as their granularity, as well as collisions and nonresonant atomic interactions, can be neglected. In Mie theory, boundary conditions of the scattering object assume a fundamental role: these are generally sharp since dielectric spheres typically have homogeneous densities, while atomic clouds, in general, have parabolic or Gaussian density distributions and smooth boundaries. Within the framework of the Mie theory, we compare calculations for homogeneous and parabolic densities, and identify the impact of sharp or smooth boundaries on the occurrence and shape of Mie resonances. Our study reveal that some resonances will persist, although not the