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arXiv:physics/9907038 · Physical Review A

Atom focusing by far-detuned and resonant standing wave fields: Thin-lens regime

J. L. Cohen, B. Dubetsky, P. R. Berman

Published Dec 1, 1999• 17 citations
arXivHTMLPDF

Abstract

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 ${L}_{T}$ 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.

DOI
10.1103/physreva.60.4886

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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 \ensuremathθ:\ensuremath{\theta}:\ensuremathθ: 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 (\ensuremathθ\ensuremath≫1).(\ensuremath{\theta}\ensuremath{\gg}1).(\ensuremathθ\ensuremath≫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\ensuremathθ\ensuremath−1/2)0.5(1\ensuremath{-}0.635{\ensuremath{\theta}}^{\ensuremath{-}1/2})0.5(1\ensuremath−0.635\ensuremathθ\ensuremath−1/2) while the peak density behaves as 3.83\ensuremathθ1/2,3.83{\ensuremath{\theta}}^{1/2},3.83\ensuremathθ1/2, the focal distance as \ensuremathθ\ensuremath−1(1+1.27\ensuremathθ\ensuremath−1/2){\ensuremath{\theta}}^{\ensuremath{-}1}(1+1.27{\ensuremath{\theta}}^{\ensuremath{-}1/2})\ensuremathθ\ensuremath−1(1+1.27\ensuremathθ\ensuremath−1/2) LT/2\ensuremathπ,{L}_{T}/2\ensuremath{\pi},LT​/2\ensuremathπ, the depth of focus as 0.304LT\ensuremathθ\ensuremath−3/2,{0.304L}_{T}{\ensuremath{\theta}}^{\ensuremath{-}3/2},0.304LT​\ensuremathθ\ensuremath−3/2, and the focal spot size 0.0592\ensuremathλ\ensuremathθ\ensuremath−3/4,0.0592\ensuremath{\lambda}{\ensuremath{\theta}}^{\ensuremath{-}3/4},0.0592\ensuremathλ\ensuremathθ\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.
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 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.
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.
This paper examines the focusing of an atomic beam by the near field of laser radiation diffracted by a small aperture. The wave properties of the atom are taken into consideration within the framework of an approximate Schrödinger equation. The possibility is demonstrated of focusing an atomic beam into a spot some 3 nm across, the atomic and laser beam characteristics being quite acceptable. The main contribution to the spot size is shown to be from spherical aberrations and diffraction effects, the chromatic aberrations of the atomic beam with a 5% velocity distribution broadening having a lesser effect.
We suggest and examine an atom optics focusing element based on the near field arising in the diffraction of laser radiation by an aperture of small size (less than an optical wavelength) in a thin conducting screen. The quasi-resonant interaction potential of the atom and the near field is found to be close to the potential of the ideal lens. Within the quasi-classical dynamics of atoms we consider the role of chromatic, spherical aberration. Taking the diffraction aberrations into account we find that it is possible to focus atoms into spot with radius 4-7 nm.
We study the optimal focusing of two-level atoms with a near-resonant standing wave light, using both classical and quantum treatments of the problem in the thin- and thick-lens regimes. It is found that the near-resonant standing wave focuses the atoms with a reduced background in comparison with far-detuned light fields. For some parameters, the quantum atomic distribution shows even better localization than the classical one. Spontaneous emission effects are included via the technique of quantum Monte Carlo wave function simulations. We investigate the extent to which nonadiabatic and spontaneous emission effects limit the achievable minimal size of the deposited structures.