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arXiv:physics/9806019 · Physical Review A

High-resolution amplitude and phase gratings in atom optics

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

Published Dec 1, 1998• 22 citations
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Abstract

An atom-field geometry is chosen in which an atomic beam traverses a field interaction zone consisting of three fields, one having frequency $\ensuremath{\Omega}=2\ensuremath{\pi}c/\ensuremath{\lambda}$ propagating in the $\mathbf{z\ifmmode \hat{}\else \^{}\fi{}}$ direction and the other two having frequencies $\ensuremath{\Omega}+{\ensuremath{\delta}}_{1}$ and $\ensuremath{\Omega}+{\ensuremath{\delta}}_{2}$ propagating in the $\ensuremath{-}\mathbf{z\ifmmode \hat{}\else \^{}\fi{}}$ direction. For ${n}_{1}{\ensuremath{\delta}}_{1}{+n}_{2}{\ensuremath{\delta}}_{2}=0$ and $|{\ensuremath{\delta}}_{1}|T,|{\ensuremath{\delta}}_{2}|T\ensuremath{\gg}1,$ where ${n}_{1}$ and ${n}_{2}$ are positive integers and T is the pulse duration in the atomic rest frame, the atom-field interaction results in the creation of atom amplitude and phase gratings having period $\ensuremath{\lambda}/[{2(n}_{1}{+n}_{2})].$ In this manner, one can use optical fields having wavelength \ensuremath{\lambda} to produce atom gratings having periodicity much less than \ensuremath{\lambda}.

DOI
10.1103/physreva.58.4801

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The interaction of an atomic beam with three spatially separated linearly polarized standing-wave fields is considered, taking into account the magnetic degeneracy and hyperfine splitting of the atomic levels. A spatial modulation of the ground-state density matrix occurs, and is not washed out as a result of spontaneous decay from the upper states. The interaction with fields separated by a distance L results in a focusing of harmonics at distances L'=(m/n)L, where m and n are integers; the period of the matter gratings is \ensuremath{\lambda}/2n, where \ensuremath{\lambda} is the wavelength of the radiation field. For a beam having angular divergence \ensuremath{\theta}, one can observe these higher harmonics independently from each other if L\ensuremath{\gg}\ensuremath{\lambda}/\ensuremath{\theta}. The gratings can be probed using a third standing-wave laser field in the focal plane of the harmonic under investigation. General expressions that allow one to calculate the probe absorption for atoms having arbitrary fine and hyperfine structure are derived. Explicit calculations for the alkali metals are presented. It is predicted that spatial modulations having periods 30--500 nm and modulation depths 2.5--50 % can be produced in this manner. The focusing of the spatial harmonics is interpreted in terms of a shadow effect.

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An atom-field geometry is chosen in which an atomic beam traverses a field interaction zone consisting of three fields, one having frequency \ensuremathΩ=2\ensuremathπc/\ensuremathλ\ensuremath{\Omega}=2\ensuremath{\pi}c/\ensuremath{\lambda}\ensuremathΩ=2\ensuremathπc/\ensuremathλ propagating in the z\ifmmode^\elseˆ\fi\mathbf{z\ifmmode \hat{}\else \^{}\fi{}}z\ifmmode^\elseˆ\fi direction and the other two having frequencies \ensuremathΩ+\ensuremathδ1\ensuremath{\Omega}+{\ensuremath{\delta}}_{1}\ensuremathΩ+\ensuremathδ1​ and \ensuremathΩ+\ensuremathδ2\ensuremath{\Omega}+{\ensuremath{\delta}}_{2}\ensuremathΩ+\ensuremathδ2​ propagating in the \ensuremath−z\ifmmode^\elseˆ\fi\ensuremath{-}\mathbf{z\ifmmode \hat{}\else \^{}\fi{}}\ensuremath−z\ifmmode^\elseˆ\fi direction. For n1\ensuremathδ1+n2\ensuremathδ2=0{n}_{1}{\ensuremath{\delta}}_{1}{+n}_{2}{\ensuremath{\delta}}_{2}=0n1​\ensuremathδ1​+n2​\ensuremathδ2​=0 and ∣\ensuremathδ1∣T,∣\ensuremathδ2∣T\ensuremath≫1,|{\ensuremath{\delta}}_{1}|T,|{\ensuremath{\delta}}_{2}|T\ensuremath{\gg}1,∣\ensuremathδ1​∣T,∣\ensuremathδ2​∣T\ensuremath≫1, where n1{n}_{1}n1​ and n2{n}_{2}n2​ are positive integers and T is the pulse duration in the atomic rest frame, the atom-field interaction results in the creation of atom amplitude and phase gratings having period \ensuremathλ/[2(n1+n2)].\ensuremath{\lambda}/[{2(n}_{1}{+n}_{2})].\ensuremathλ/[2(n1​+n2​)]. In this manner, one can use optical fields having wavelength \ensuremath{\lambda} to produce atom gratings having periodicity much less than \ensuremath{\lambda}.
We propose a scheme to create high-contrast, periodic atom density distributions having period $\ensuremath{\lambda}/2n$ using the Talbot effect, where $\ensuremath{\lambda}$ is the wavelength of the optical fields that scatter the atoms and n is a positive integer. This filtered Talbot lens is comprised of two standing-wave optical fields. An atomic beam propagates perpendicular to the fields. The first field, which is far detuned from the atomic transition frequency, acts as an array of lenses that focuses the atoms. The second field, positioned at the atom optical focus of the first, is resonant with the atomic transition frequency and acts as an amplitude mask, leaving unperturbed only those atoms that pass through its nodes. At distances following the interaction with the second field that are equal to an integral fraction of the Talbot length, atomic density gratings having period $\ensuremath{\lambda}/2n$ are formed.
We have studied sub-optical wavelength atomic gratings generated by the interference of atomic de Broglie waves in a time-domain interferometer. Three short optical standing- wave pulses, detuned from resonance with a cloud of approximately 100 (mu) K 85Rb atoms, act as phase gratings for atomic matter waves. Shortly after the first pulse, a modulation in the atomic density containing spatial harmonics of period (lambda) /2N appears ((lambda) is the optical wavelength, and N is any integer) and then rapidly vanishes due to the atomic thermal motion. The cloud maintains phase memory of this modulation, however, and the second pulse (applied at time T after the first one) results in a temporal sequence of atomic density gratings of periods (lambda) /2N. The appearance of a grating is referred to as an echo. The third pulse (and a subsequent traveling wave pulse) is used to detect these gratings. By this technique, we have created and detected in real time a grating with period (lambda) /4 at time 1.5T after the first pulse and observed new types of `echoes' caused by interaction of the de Broglie waves with all three phase gratings. This ability to produce sub-wavelength structures makes our techniques applicable to atomic beam lithography.
The interaction of an atomic beam with three spatially separated linearly polarized standing-wave fields is considered, taking into account the magnetic degeneracy and hyperfine splitting of the atomic levels. A spatial modulation of the ground-state density matrix occurs, and is not washed out as a result of spontaneous decay from the upper states. The interaction with fields separated by a distance L results in a focusing of harmonics at distances L'=(m/n)L, where m and n are integers; the period of the matter gratings is \ensuremath{\lambda}/2n, where \ensuremath{\lambda} is the wavelength of the radiation field. For a beam having angular divergence \ensuremath{\theta}, one can observe these higher harmonics independently from each other if L\ensuremath{\gg}\ensuremath{\lambda}/\ensuremath{\theta}. The gratings can be probed using a third standing-wave laser field in the focal plane of the harmonic under investigation. General expressions that allow one to calculate the probe absorption for atoms having arbitrary fine and hyperfine structure are derived. Explicit calculations for the alkali metals are presented. It is predicted that spatial modulations having periods 30--500 nm and modulation depths 2.5--50 % can be produced in this manner. The focusing of the spatial harmonics is interpreted in terms of a shadow effect.