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Physical Review A

Atomic vortex beams in focal regions

L. E. Helseth

Published Jan 14, 2004• 8 citations
Publisher

Abstract

We discuss some classes of atomic lenses capable of generating atomic vortex beams. For certain lens classes one may associate a topological charge with the beam in the focal region, and it is shown that an energy of the order $\ensuremath{\Elzxh}/\ensuremath{\tau}$ is needed to alter the topological charge by $\ensuremath{\Delta}n=\ifmmode\pm\else\textpm\fi{}1,$ where $\ensuremath{\tau}$ is the atom interaction time with the lens. We also find that the vortex can be manipulated by the properties of the atomic lens. In particular, it is suggested that the circulation varies locally in space.

DOI
10.1103/physreva.69.015601

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We discuss some classes of atomic lenses capable of generating atomic vortex beams. For certain lens classes one may associate a topological charge with the beam in the focal region, and it is shown that an energy of the order \ensuremath\Elzxh/\ensuremathτ\ensuremath{\Elzxh}/\ensuremath{\tau}\ensuremath\Elzxh/\ensuremathτ is needed to alter the topological charge by \ensuremathΔn=\ifmmode±\else\textpm\fi1,\ensuremath{\Delta}n=\ifmmode\pm\else\textpm\fi{}1,\ensuremathΔn=\ifmmode±\else\textpm\fi1, where \ensuremathτ\ensuremath{\tau}\ensuremathτ is the atom interaction time with the lens. We also find that the vortex can be manipulated by the properties of the atomic lens. In particular, it is suggested that the circulation varies locally in space.
In the field of atom optics, there has been much interests in developing components such as lenses, mirrors, beam splitters and waveguides to manipulate neutral atoms. Especially many clever methods to focus neutral atoms have been developed. The lenses have been constructed using the static electrical and magnetic fields, Fresnel zone plates, and radiation pressure forces. Before the laser cooling and trapping techniques were invented, the electrostatic fields could not be used for manipulating neutral atoms because of its weak interaction energy with atoms. After the cold atoms were available, it has been possible to manipulate atoms with weak electrostatic interaction energy. We developed an atom lens system using electrostatic fields.
Recently, the focusing of an atomic beam using a laser has been considered by Balykin and Letokhov.1 Their calculations explicitly assume the thin lens approximation. Here, using path integral techniques, we derive the focal length and lowest-order aberrations of the laser atomic beam lens as a function of both laser and atomic beam parameters without making the thin lens approximation. Starting from the basic form of the potential energy for an atom in a laser beam, the propagation kernel is derived for the TEM(01)* or donut mode beam. From this result the focal length and lowest-order aberrations can be found for any atomic beam profile. We consider in particular a Gaussian atomic beam. Both the full 3-D propagation kernel and its paraxial approximation are discussed. We show how the paraxial case can be obtained from the 3-D case via a stationary phase approximation of the propagation equation.
We report the generation of a crossed, focused, optical vortex beam by using a pair of hybrid holograms, which combine the vortex phase and lens phase onto a spatial light modulator. We study the intensity distributions of the vortex beam in free propagation space, and the relationship of its dark spot size with the incident Gaussian beam’s waist, the lens’s focal length, and its orbital angular momentum. Our results show that the crossed, focused, vortex beam’s dark spot size can be as small as 16.3 μm and adjustable by the quantum number of the orbital angular momentum, and can be used to increase the density of trapped molecules. Furthermore, we calculate the optical potential of the blue-detuned, crossed vortex beam for MgF molecules. It is applicable to cool and trap neutral molecules by intensity-gradient-induced Sisyphus cooling, as the intensity gradient of such vortex beam is extremely high near the focal point.
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 design and implementation of a multiplexed spiral phase mask in an experimental optical tweezer setup are presented. This diffractive optical element allows the generation of multiple concentric vortex beams with independent topological charges. The generalization of the phase mask for multiple concentric vortices is also shown. The design for a phase mask of two multiplexed vortices with different topological charges is developed. We experimentally show the transfer of angular momentum to the optically trapped microparticles by enabling orbiting dynamics around the optical axis independently within each vortex. The angular velocity of the confined particles versus the optical power in the focal region is also discussed for different combinations of topological charges.
We propose a new method to generate a focused hollow laser beam by using an azimuthally distributed 2π-phase plate and a convergent thin lens,and calculate the intensity distribution of the focused hollow beam in free propagation space.The relationship between the waist w0 of the incident collimated Gaussian beam and the dark spot size of the focused hollow beam at the focal point,and the relationship between the focal lenght f of the thin lens and the dark spot size are studied respectively.The optical potential of the blue-detuned focused hollow beam for^85 Rb atoms is calculated.Our study shows that when the larger waist w of the incident Gaussian beam and the shorter focal length f of the lens are chosen,we can obtain an extremely small dark spot size of the focused hollow beam,which can be used to form an atomic lens with a resolution of sereval angstroms.