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Journal of the Optical Society of America B

Atom-optical properties of a standing-wave light field

Jabez J. McClelland

Published Oct 1, 1995• 89 citations
Publisher

Abstract

The focusing of atoms to nanometer-scale dimensions by a near-resonant standing-wave light field is examined from a particle optics perspective. The classical equation of motion for atoms traveling through the lens formed by a node of the standing wave is derived and converted to a spatial trajectory equation. A paraxial solution is obtained, which results in simple expressions for the focal properties of the lens, useful for estimating its behavior. Aberrations are also discussed, and an exact numerical solution of the trajectory equation is presented. The effects on focal linewidth of angular collimation and velocity spread in the atomic beam are investigated, and it is shown that angular collimation has a much more significant effect than velocity spread, even when the velocity spread is thermal.

DOI
10.1364/josab.12.001761

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Dec 8, 2004•0 citations
The focusing of atoms to nanometer-scale dimensions by a near-resonant standing-wave light field is examined from a particle optics perspective. The classical equation of motion for atoms traveling through the lens formed by a node of the standing wave is derived and converted to a spatial trajectory equation. A paraxial solution is obtained, which results in simple expressions for the focal properties of the lens, useful for estimating its behavior. Aberrations are also discussed, and an exact numerical solution of the trajectory equation is presented. The effects on focal linewidth of angular collimation and velocity spread in the atomic beam are investigated, and it is shown that angular collimation has a much more significant effect than velocity spread, even when the velocity spread is thermal.
The use of a TEM01*-mode laser beam has been proposed as a means of focusing an atomic beam to nanometer- scale spot diameters. We have analyzed the classical trajectories of atoms through a TEM01*-mode laser beam, using methods developed for particle optics. The differential equation that describes the properties of the: first- order paraxial lens hi exactly the same form as the bell-shaped magnetic Newtonian lens that was first analyzed by Glaser for the focusing of electrons in an electron-microscope objective. We calculate the first-order properties of the lens, obtaining cardinal elements that are valid over the entire operating range of the lens including the thick and the immersion regimes. Contributions to the spot size are discussed, including four aberrations plus diffraction and atomic-beam-collimation effects. Explicit expressions for spherical chromatic, spontaneous-emission, and dipole-fluctuation aberrations are obtained. Examples are discussed for a sodium atomic beam, showing that subnanometer-diameter spots may be achieved with reasonable laser and atomic- beam parameters. Optimization of the lens is also discussed.
Atom lens generated by a laser standing wave can be used to focus an atomic beam to nanometer dimensions. Aberrations exist in atom lens owing to the influence of velocity spread, beam spread, wave nature of atoms and so on. Two theoretical models were built to describe atom lens. Spherical aberration, chromatic aberration and the aberration caused by angular spread were analyzed with semi-classical model, and diffractive aberration was analyzed using quantum model. Simulation results show that aberration originates mainly from beam spread, and diffractive aberration is larger than spherical aberration and chromatic aberration. Three solutions of optimization of experimental parameters, adding beam mask and using etching technologies for improving atom lithography quality are presented.
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.
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 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.
The two-level atom focusing in a Gaussian standing wave laser field was analyzed from the perspective of both classical mechanics and wave mechanics. The effects of source imperfection such as velocity spread and beam spread on atom focusing were analyzed by numerically integrating the classical equation of atomic motion. The ideal focal plane can be easily determined by the variation of atomic density at the minimal potential of the standing wave laser field as a function of traveling distance. In the absence of source imperfection, the contribution of diffractive aberration originating from the wave nature of the atom to broadening of feature width is larger than that of spherical aberration. Several methods for improving atom lithography experiments were presented.