Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Bragg Scattering of Free Electrons Using the Kapitza-Dirac Effect

Daniel L. Freimund, Herman Batelaan

Published Dec 30, 2002• 88 citations
Publisher

Abstract

Bragg scattering has been observed for free electrons using a standing wave of light. Both the rocking curve and the angular electron distribution have been measured. The results of a numerical simulation to the Schrödinger equation are consistent with our experimental data. Unlike the diffraction regime which uses thin crystals, the Bragg regime requires the use of thick crystals. We point out several applications in atom optics that could be realized in electron optics.

DOI
10.1103/physrevlett.89.283602

Related papers

Electron matter optics and the Kapitza -Dirac effect

Daniel L. Freimund

The diffraction of electrons by a standing wave of light was first proposed by Kapitza and Dirac in 1933. We observed this effect for the first time in September 2001 by showing separate electron diffraction peaks. It was first observed for a focused laser beam, where the standing wave of light serves as a thin light crystal. Later, the effect was also observed with an unfocused laser beam, where the standing wave serves as a thick light crystal, where scattering can only occur at the Bragg angle. The theory behind the effect has been formulated within both atomic optics and electron optics communities. It involves the ponderomotive potential that an electron experiences within the electromagnetic field of two counter-propagating laser beams. The derivations of the differential equations given by both communities have identical results with their own benefits when making numerical calculations. These theoretical calculations fit well to our data, which falls between the limits of the diffraction and Bragg regimes. To analyze the data collected in these experiments, the statistical behavior of the arrival of electrons, governed by the Poisson distribution, was taken into account. The Gaussian nature of the laser beam and its consequences on focusing needed to be addressed, along with how far from diffraction limited the laser beam actually was. There may be a possibility of new effects arising from the interaction of electrons and light. The effect is like KDE, but modified to include two counter-propagating lasers of different frequency, the consequence of the longitudinal velocity of the electrons, and the role that the polarization of the laser light plays.

•Jan 1, 2003•1 citation

Diffraction of Electrons by Standing Electromagnetic Waves: The Kapitza-Dirac Effect

Michael Martin Nieto

The Bragg diffraction of electrons off of a standing electromagnetic wave, first predicted by Kapitza and Dirac, is discussed. The original derivation of Kapitza and Dirac is reviewed. This is followed by a more general derivation, which uses quantum mechanical, Born approximation, scattering theory. A review of the recent experiment results is given. Because the Kapitza-Dirac effect reverses the usual roles of “particles” and “waves” in Bragg diffraction, the discussion emphasizes the physical insight that this phenomenon gives into light and the particle-wave duality of quantum mechanics.

American Journal of Physics•Feb 1, 1969•7 citations

Observation of the Kapitza-Dirac effect

Dan Freimund, Kayvan Aflatooni, Herman Batelaan

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.

APS•May 1, 2001•0 citations

Spin-dependent two-photon Bragg scattering in the Kapitza-Dirac effect

Sven Ahrens, Zhenfeng Liang, Tilen Čadež +1 more

We present the possibility of spin-dependent Kapitza-Dirac scattering based on a two-photon interaction only. The interaction scheme is inspired from a Compton scattering process, for which we explicitly show the mathematical correspondence to the spin dynamics of an electron diffraction process in a standing light wave. The spin effect has the advantage that it already appears in a Bragg scattering setup with arbitrary low field amplitudes, for which we have estimated the diffraction count rate in a realistic experimental setup at available x-ray free-electron laser facilities.

arXiv:2001.10177•Physical Review A•Sep 9, 2020•11 citations

Illuminating the Kapitza-Dirac Effect with Electron Matter Optics [Colloquium]

Herman Batelaan

The observation of the Kapitza-Dirac effect raises conceptual, theoretical, and experimental questions. The Kapitza-Dirac effect is often described as diffraction of free electrons from a standing wave of light or stimulated Compton scattering. However, for the two-color Kapitza-Dirac effect these two interpretations appear to lead to paradoxical conclusions. The discussion of this paradox deepens our understanding of both of these versions of the Kapitza-Dirac effect.

•Jan 1, 2007•0 citations

Colloquium: Illuminating the Kapitza-Dirac effect with electron matter optics

Herman Batelaan

The observation of the Kapitza-Dirac effect raises conceptual, theoretical, and experimental questions. The Kapitza-Dirac effect is often described as diffraction of free electrons from a standing wave of light or stimulated Compton scattering. However, for the two-color Kapitza-Dirac effect these two interpretations appear to lead to paradoxical conclusions. The discussion of this paradox deepens our understanding of both of these versions of the Kapitza-Dirac effect.

Reviews of Modern Physics•Jul 13, 2007•127 citations
Bragg scattering has been observed for free electrons using a standing wave of light. Both the rocking curve and the angular electron distribution have been measured. The results of a numerical simulation to the Schrödinger equation are consistent with our experimental data. Unlike the diffraction regime which uses thin crystals, the Bragg regime requires the use of thick crystals. We point out several applications in atom optics that could be realized in electron optics.
The diffraction of electrons by a standing wave of light was first proposed by Kapitza and Dirac in 1933. We observed this effect for the first time in September 2001 by showing separate electron diffraction peaks. It was first observed for a focused laser beam, where the standing wave of light serves as a thin light crystal. Later, the effect was also observed with an unfocused laser beam, where the standing wave serves as a thick light crystal, where scattering can only occur at the Bragg angle. The theory behind the effect has been formulated within both atomic optics and electron optics communities. It involves the ponderomotive potential that an electron experiences within the electromagnetic field of two counter-propagating laser beams. The derivations of the differential equations given by both communities have identical results with their own benefits when making numerical calculations. These theoretical calculations fit well to our data, which falls between the limits of the diffraction and Bragg regimes. To analyze the data collected in these experiments, the statistical behavior of the arrival of electrons, governed by the Poisson distribution, was taken into account. The Gaussian nature of the laser beam and its consequences on focusing needed to be addressed, along with how far from diffraction limited the laser beam actually was. There may be a possibility of new effects arising from the interaction of electrons and light. The effect is like KDE, but modified to include two counter-propagating lasers of different frequency, the consequence of the longitudinal velocity of the electrons, and the role that the polarization of the laser light plays.
The Bragg diffraction of electrons off of a standing electromagnetic wave, first predicted by Kapitza and Dirac, is discussed. The original derivation of Kapitza and Dirac is reviewed. This is followed by a more general derivation, which uses quantum mechanical, Born approximation, scattering theory. A review of the recent experiment results is given. Because the Kapitza-Dirac effect reverses the usual roles of “particles” and “waves” in Bragg diffraction, the discussion emphasizes the physical insight that this phenomenon gives into light and the particle-wave duality of quantum mechanics.
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 present the possibility of spin-dependent Kapitza-Dirac scattering based on a two-photon interaction only. The interaction scheme is inspired from a Compton scattering process, for which we explicitly show the mathematical correspondence to the spin dynamics of an electron diffraction process in a standing light wave. The spin effect has the advantage that it already appears in a Bragg scattering setup with arbitrary low field amplitudes, for which we have estimated the diffraction count rate in a realistic experimental setup at available x-ray free-electron laser facilities.