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Physical Review A

Controllable blazed grating for electrons using Kapitza-Dirac diffraction with multiple-harmonic standing waves

Pierre Vidil, Bénoît Chalopin

Published Dec 11, 2015• 7 citations
Publisher

Abstract

We demonstrate the possibility to produce a blazed grating for electron waves using Kapitza-Dirac diffraction on a standing wave created by several harmonics of a fundamental wavelength. We show that an optimized multicolor standing wave can lead to strongly asymmetric diffraction patterns. Using a bichromatic standing wave, we predict a diffraction probability larger than 50% in a specific order. This order can be chosen with the spatial relative phase between the fundamental and second harmonic. Since the Kapitza-Dirac diffraction grating is experimentally more feasible than Bragg diffraction, this could be an interesting alternative to be used in electron interferometry.

DOI
10.1103/physreva.92.062117

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We demonstrate the possibility to produce a blazed grating for electron waves using Kapitza-Dirac diffraction on a standing wave created by several harmonics of a fundamental wavelength. We show that an optimized multicolor standing wave can lead to strongly asymmetric diffraction patterns. Using a bichromatic standing wave, we predict a diffraction probability larger than 50% in a specific order. This order can be chosen with the spatial relative phase between the fundamental and second harmonic. Since the Kapitza-Dirac diffraction grating is experimentally more feasible than Bragg diffraction, this could be an interesting alternative to be used in electron interferometry.
Coherent scattering of an electron beam by the Kapitza-Dirac effect from a standing laser wave which comprises two frequency components is studied. To this end, the Schr\"odinger equation is solved numerically with a suitable ponderomotive potential. Besides, an analytical solution for electron diffraction in the asymptotic domain of large field amplitudes is obtained and a mathematical model in reduced dimensionality for the scattering amplitude in the Bragg regime is presented. We demonstrate distinct interference signatures and relative phase effects when the standing wave contains a fundamental frequency and its second harmonic. The influence of the relative field intensities on the Rabi oscillation dynamics is also discussed.
We report on the possibility of diffracting electrons from light waves traveling inside a dielectric medium. We show that, in the frame of reference which moves with the group velocity of light, the traveling wave acts as a stationary diffraction grating from which electrons can diffract, similar to the conventional Kapitza–Dirac effect. To characterize the Kapitza–Dirac effect with traveling light waves , we make use of the Hamiltonian Analogy between electron optics and quantum mechanics and apply the Helmholtz–Kirchhoff theory of diffraction. Introduction and background. The Kapitza-Dirac effect, i.e. the diffraction of electrons from a periodic light wave, has been predicted in the 1930's. Only now, with the advent of sufficiently intense lasers, it became possible to observe the effect experimentally. Commonly a standing wave is used to create the diffraction grating, but also other configurations, involving counter propagating waves of different frequencies have been considered in the past. In this paper, we address the question whether it is possible to observe the effect with a single light beam, travelling with a reduced group velocity within a dielectric medium. Main results. We confirm that the Kapitza-Dirac effect can be observed with a single beam of light, provided the longitudinal velocity of the electrons is commensurate with the reduced group velocity. Using Lorentz transformations between a laboratory and reference frame associated with the travelling light field, we show that in this frame of reference the light acts as a stationary diffraction grating for electrons. In contrast to a standing wave, a single, travelling light wave can give rise to inelastic diffraction when electrons exchanges energy with the light field. Wider implications. Diffraction phenomena occur for all kinds of waves and represent a signature of fundamental interference processes. Our extension of the standard Kapitza-Dirac effect to the case of a single light beam enlarges the scope to observe such effects, and might lead to simplified schemes in interferometry and microscopy. Figure. Geometry of the electron diffraction in the laboratory (a) and wave (b) frames of reference. Electrons overtake the slowed light and experience diffraction for specific angles (c). The "butterfly wings" represent the regions where the Kapitza-Dirac effect with traveling waves occurs (c).
Abstract In their famous 1927 experiment, Davisson and Germer observed1 the diffraction of electrons by a periodic material structure, so showing that electrons can behave like waves. Shortly afterwards, Kapitza2 and Dirac3 predicted that electrons should also be diffracted by a standing light wave4. This Kapitza–Dirac effect is analogous to the diffraction of light by a grating, but with the roles of the wave and matter reversed. The electron and the light grating interact extremely weakly, via the ‘ponderomotive potential’5, so attempts to measure the Kapitza–Dirac effect had to wait for the development of the laser. The idea6 that the underlying interaction with light is resonantly enhanced for electrons in an atom led to the observation7 that atoms could be diffracted by a standing wave of light. Deflection of electrons by high-intensity laser light, which is also a consequence of the Kapitza–Dirac effect, has also been demonstrated8. But the coherent interference that characterizes wave diffraction has not hitherto been observed9, 10. Here we report the diffraction of free electrons from a standing light wave—a realization of the Kapitza–Dirac effect as originally proposed.
We theoretically demonstrate the feasibility of producing an electron spin beam splitter using Kapitza-Dirac diffraction on bichromatic standing waves which are created by the fundamental frequency and its third harmonic. The relativistic electron in the Bragg regime absorbs three photons with a frequency of $\ensuremath{\omega}$ and emits a photon with a frequency of $3\ensuremath{\omega}$; we introduce this as the four-photon Kapitza-Dirac effect. In this four-photon Kapitza-Dirac effect distinct spin effects arise in different polarizations of the third harmonic laser beam. It is shown that the shape of the Rabi oscillation between the initial and scattered states is changed and finds two unequal peaks for all polarizations of laser beams. For the case of circular polarization of the fundamental and third harmonic, despite Rabi oscillation, the spin down electron beam in 0.23 fs intervals maintains its momentum and spin. Also we find that if a suitable combination of a linearly polarized fundamental laser beam and a third harmonic with circular polarization in the condition $\stackrel{P\vec}{p}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{A}$ term is involved, it results in a spin-dependent oscillation that preserves the spin state of the initial electron.
Similar to the optical diffraction of light passing through a material grating, the Kapitza-Dirac effect occurs when an electron is diffracted by a standing light wave. In its original description, the effect is time independent. Here, we extended the Kapitza-Dirac effect to the time domain. By tracking the spatiotemporal evolution of a pulsed electron wave packet diffracted by a 60-femtosecond (where one femtosecond = 10 −15 seconds) standing wave pulse in a pump-probe scheme, we observed time-dependent diffraction patterns. The fringe spacing in the observed pattern differs from that generated by the conventional Kapitza-Dirac effect. By exploiting this time-resolved diffraction scheme, we can access the time evolution of the phase properties of a free electron and potentially image ionic potentials and electronic decoherences.