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Physical Review A

Conical lens for atom focusing

B. Dubetsky, P. R. Berman

Published Sep 1, 1998• 18 citations
Publisher

Abstract

Light reflected from a conical mirror produces an intensity gradient that can be used to focus atoms to a single spot. Within the Raman-Nath approximation the position of the focal plane, peak atomic density, and spot size have been determined for a range of field strengths. Theoretical estimates of the field strength needed to focus atoms using a conical lens are found to be three to four orders of magnitude lower than that needed for focusing atoms using standing-wave fields.

DOI
10.1103/physreva.58.2413

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Light reflected from a conical mirror produces an intensity gradient that can be used to focus atoms to a single spot. Within the Raman-Nath approximation the position of the focal plane, peak atomic density, and spot size have been determined for a range of field strengths. Theoretical estimates of the field strength needed to focus atoms using a conical lens are found to be three to four orders of magnitude lower than that needed for focusing atoms using standing-wave fields.
We propose a lens for atoms with reduced chromatic aberrations and calculate its focal length and spot size. In our scheme a two-level atom interacts with a near-resonant standing light wave formed by two running waves of slightly different wave vectors, and a far-detuned running wave propagating perpendicularly to the standing wave. We show that within the Raman-Nath approximation and for an adiabatically slow atom-light interaction, the phase acquired by the atom is independent of the incident atomic velocity.
The focusing of atoms after interacting with both far-detuned and resonant standing wave fields in the thin-lens regime is considered. Exact quantum expressions for the Fourier components of the density (that include all spherical aberration) are used to study the focusing numerically. The following lens parameters and density profiles are calculated as functions of the pulsed field area $\ensuremath{\theta}:$ the position of the focal plane, peak atomic density, atomic density pattern at the focus, focal spot size, depth of focus, and background density. The lens parameters are compared to asymptotic, analytical results derived from a scalar diffraction theory for which spherical aberration is small but non-negligible $(\ensuremath{\theta}\ensuremath{\gg}1).$ Within the diffraction theory analytical expressions show that the focused atoms in the far-detuned case have an approximately constant background density $0.5(1\ensuremath{-}0.635{\ensuremath{\theta}}^{\ensuremath{-}1/2})$ while the peak density behaves as $3.83{\ensuremath{\theta}}^{1/2},$ the focal distance as ${\ensuremath{\theta}}^{\ensuremath{-}1}(1+1.27{\ensuremath{\theta}}^{\ensuremath{-}1/2})$ ${L}_{T}/2\ensuremath{\pi},$ the depth of focus as ${0.304L}_{T}{\ensuremath{\theta}}^{\ensuremath{-}3/2},$ and the focal spot size $0.0592\ensuremath{\lambda}{\ensuremath{\theta}}^{\ensuremath{-}3/4},$ where LT{L}_{T}LT​ is the Talbot distance and \ensuremath{\lambda} is the wavelength of the light. Focusing by the resonant standing wave field leads to a new effect, a Rabi-like oscillation of the atom density. For the far-detuned lens, chromatic aberration is studied quantitatively with the exact Fourier results. Similarly, the degradation of the focus that results from angular divergence in beams or thermal velocity distributions in traps is studied quantitatively with the exact Fourier method and understood analytically using the asymptotic results. Overall, we show that strong thin-lens focusing is possible with modest laser powers and with currently achievable atomic beam characteristics.
A near-resonant standing laser light field is used to focus a diverging beam of sodium atoms onto the atomic-beam axis. The investigations presented here distinguish between channeling of atoms with trajectories close to the axis and imaging of the atomic source. For laser detunings close to resonance, acceleration of the atoms in the transverse direction occurs. The amount of acceleration changes as either the intensity or the detuning of the standing wave is varied, thereby altering the position at which the atoms are imaged. This results in a variable focal-length lens for atoms. The experimental results are in good agreement with a theoretical model based on a continued-fraction solution for the standing-wave light forces.
The focusing of an atomic beam using near-resonant laser light is considered. Path-integral techniques are employed to transform the problem into a standard diffraction integral. This approach is general and allows us to deal with thick laser lenses. Starting from the basic form of the potential energy for an atom in a laser beam, we derive the propagation kernel for the atomic wave function for the particular case of a TEM01*, or doughnut, mode laser beam. Both the full three-dimensional propagation kernel and its paraxial approximation are discussed. We show that the paraxial case can be obtained from the three-dimensional case by a stationary-phase approximation of the propagation equation. Numerical results for the focusing of a Gaussian atomic beam are presented. These results show that spot diameters on the order of 20 Å are obtainable for many reasonable choices of laser and atomic beam parameters and that for most of these cases the thin-lens approximation is not valid. The effects of the lowest-order aberrations are also briefly discussed. Spherical aberration is found to contribute significantly to the focal spot diameter, at least for the doughnut mode laser beam considered here.