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

Journal of Physics A Mathematical and General

Elastic and inelastic scattering of electrons by a standing wave of intense and coherent light

F. Ehlotzky, C. Leubner

Published Nov 1, 1975• 3 citations
Publisher

Abstract

The elastic and inelastic scattering of electrons by a standing wave of intense and coherent light are investigated by means of the Helmholtz-Kirchhoff diffraction theory. The time-averaged theories, used previously for the description of Kapitza-Dirac scattering, are shown to be inadequate.
DOI
10.1088/0305-4470/8/11/016

Related papers

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

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

Kapitza-Dirac scattering of electrons from a bichromatic standing laser wave

Matthias M. Dellweg, Carsten Müller

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.

arXiv:1503.03708•Physical Review A•Jun 2, 2015•16 citations

Theory of the Kapitza-Dirac Effect

Hiroshi Ezawa, Hiroshi Namaizawa

The Kapitza-Dirac effect is the scattering of electrons by a standing wave of very intense light, which can nowadays be realized by using laser beam. A theory is presented, in which no assumptions are necessary as to the strength of the field and to the length of the interaction time. Basic is the assumption that the laser beam should have a gentle cross-sectional profile. It allows one to separate the electron motion into two components: The one parallel to the beam is the motion in the periodic field of light, which generates “adiabatic potential” for the motion transverse to the beam. This picture can be compared to that of hydrogen molecule in which electron motion generates the adiabatic potential for nuclear vibration. In this way, it is shown that the problem can be reduced to that of one-dimensional scattering. The present theory is shown to embrace the predecessor theories of Bartell and of Fedorov.

Journal of the Physical Society of Japan•Feb 1, 1969•13 citations

Electron Scattering from a Standing Light Wave

Rodney J. Gush, Herbert P. Gush

The transition probability for electron scattering in the Kapitza-Dirac experiment is calculated using nonrelativistic theory as a function of the time of flight $t$ of the electron through the light beam, and as a function of the intensity. It is found that for intensities up to approximately ${10}^{7}$ W/${\mathrm{cm}}^{2}$ the probability equals ${sin}^{2}{a}_{0}t$, where ${a}_{0}$ is proportional to the intensity. At intensities of the order of ${10}^{9}$ W/${\mathrm{cm}}^{2}$, the dependence of the probability on the time is more complicated but it also oscillates between zero and nearly unity as the time increases. Both first- and second-order Bragg reflections are considered as well as conditions off the Bragg maximum.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 15, 1971•23 citations

Classical and quantum versions of the Kapitza-Dirac effect

Maxim A. Efremov, Mikhail Vladimirovich Fedorov

The scattering of electrons on a standing electromagnetic wave (the Kapitza-Dirac effect) is considered within the quantum-mechanical and classical descriptions of the motion of electrons. The mean scattering angle and the distribution function of the electrons after scattering are found. It is shown that assignment of the initial electronic wave function in the form of a plane wave in the quantum-mechanical picture gives rise to the characteristic parameter β = mc 2 /ħ ω 2 τ (where ω and τ are the frequency of the field and the duration of the interaction), which separates the regions of Bragg ( β ≪1) and nearly classical ( β ≫1) scattering. When β ≫1, the mean scattering angle does not depend on the choice of the description method, but the distribution functions of the electrons after scattering in the quantum and classical approaches differ significantly. When β ≪1, both the distribution function and the mean scattering angle differ. Modification of the quantum theory and assignment of the initial electronic wave function in the form of a more or less localized wave packet are apparently needed to eliminate the differences discovered. The results obtained can be used to determine the dimensions of the wave packet which characterizes the state of electrons in a beam from experiments on scattering from a standing light wave.

Journal of Experimental and Theoretical Physics•Sep 1, 1999•15 citations
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.
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.
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.
The Kapitza-Dirac effect is the scattering of electrons by a standing wave of very intense light, which can nowadays be realized by using laser beam. A theory is presented, in which no assumptions are necessary as to the strength of the field and to the length of the interaction time. Basic is the assumption that the laser beam should have a gentle cross-sectional profile. It allows one to separate the electron motion into two components: The one parallel to the beam is the motion in the periodic field of light, which generates “adiabatic potential” for the motion transverse to the beam. This picture can be compared to that of hydrogen molecule in which electron motion generates the adiabatic potential for nuclear vibration. In this way, it is shown that the problem can be reduced to that of one-dimensional scattering. The present theory is shown to embrace the predecessor theories of Bartell and of Fedorov.
The transition probability for electron scattering in the Kapitza-Dirac experiment is calculated using nonrelativistic theory as a function of the time of flight ttt of the electron through the light beam, and as a function of the intensity. It is found that for intensities up to approximately 107{10}^{7}107 W/cm2{\mathrm{cm}}^{2}cm2 the probability equals sin2a0t{sin}^{2}{a}_{0}tsin2a0​t, where a0{a}_{0}a0​ is proportional to the intensity. At intensities of the order of 109{10}^{9}109 W/cm2{\mathrm{cm}}^{2}cm2, the dependence of the probability on the time is more complicated but it also oscillates between zero and nearly unity as the time increases. Both first- and second-order Bragg reflections are considered as well as conditions off the Bragg maximum.
The scattering of electrons on a standing electromagnetic wave (the Kapitza-Dirac effect) is considered within the quantum-mechanical and classical descriptions of the motion of electrons. The mean scattering angle and the distribution function of the electrons after scattering are found. It is shown that assignment of the initial electronic wave function in the form of a plane wave in the quantum-mechanical picture gives rise to the characteristic parameter β = mc 2 /ħ ω 2 τ (where ω and τ are the frequency of the field and the duration of the interaction), which separates the regions of Bragg ( β ≪1) and nearly classical ( β ≫1) scattering. When β ≫1, the mean scattering angle does not depend on the choice of the description method, but the distribution functions of the electrons after scattering in the quantum and classical approaches differ significantly. When β ≪1, both the distribution function and the mean scattering angle differ. Modification of the quantum theory and assignment of the initial electronic wave function in the form of a more or less localized wave packet are apparently needed to eliminate the differences discovered. The results obtained can be used to determine the dimensions of the wave packet which characterizes the state of electrons in a beam from experiments on scattering from a standing light wave.