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

Laser and Particle Beams

Peierls’ dielectric response for clarification of interacting electron with laser beams

Heinrich Hora

Published May 3, 2013• 0 citations
Publisher

Abstract

Abstract The formulation of the momentum of electromagnetic radiation or photons in media is a historical question. After Peierls discussed all these options, a crucial new value for the dominating dispersion factor σ was derived for non-ionized media with the consequence that an opto-acoustic coupling exists. After the experimental confirmation of the Kapitza-Dirac effect the generalization of the effect for quantum modulation of electron beams by crossing laser beams is ascertained. The evaluation of repeated experiments of this quantum modulation is very precisely confirming by measurements that the Peierls factor σ = 1/5 is correct.

DOI
10.1017/s0263034613000128

Related papers

Kapitza-Dirac effect with lasers and non-resonant interaction for quantum modulation of electron beams (Schwarz-Hora effect)

Heinrich Hora, Peter H. Handel

The initial Kapitza-Dirac effect for crossing energetic electron beams with laser beams had been generalized for quantum modulation of the electrons. The recent developments provide advanced insights with the initial experimental facts, where consequences are leading to results about non-resonant laser interaction with condensed media including nonlinearities with applications to quantum 1/f noise theory in electronic devices.

Applied Physics Letters•Apr 8, 2013•3 citations

Generalised scattering model for the quantum modulation of electrons

Handel, P. H.

Since the first experiments of the quantum modulation of electron beams by lasers with a crystal as medium in the interaction area ~d1warz-Hora effect) were reported, extensive theoretical studies were performed.A very detaile/ rLproduction of these experiments was achieved (by Andrick and Langhans, further by Weingartshofer et al) where the medium in the interaction area was a gas.The initial clarification of the experiments with crystal as a quantum phenomenon is using a threshold defining a correspondence principle for electromagnetic interaction.Recent experiments by Weingartshofer et al independently confirm this correspondence principle.For two other experiments the ranges of quantum and classical regimes are elaborated to explain the basically differing character of observations.The discussion of the coherence for matter waves is followed up from the beginning of the Schwarz-Hora effect including developments reviewed by Hawkes and indicating further generalizations by the Hanbury-Brown/ Twiss effect.A consequent generalization of the scattering theory of the quantum modulation is presented based on a Green's function solution which results in the fact that far-field superpositions only lead to interferences.The problems of the re-emission of photons from quantum modulated electron beams is discussed and experiments on indirect observations of the phenomena in microwave amplification and in the 1/f noise Handel effect are reported.In the last case. . the self-interaction of electrons are involved in the Handel effect.A further second-quantised derivation and the coherent state Handel effect is presented.

•Mar 18, 2022•0 citations

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

Investigation of the Kapitza-Dirac effect in the relativistic regime

Sven Ahrens

Quantum mechanical diffraction is of particular interest, because it contradicts our everyday life experience. \nThis theoretical work considers the diffraction of electrons at standing waves of light, referred \nto as the Kapitza-Dirac effect. The work focuses on a special version of a Kapitza-Dirac effect in which \nthe electron interacts with three photons. The particular property of this 3-photon Kapitza-Dirac effect \nis, that the electron spin is rotated. \nThis work considers different relativistic and non-relativistic quantum mechanical wave equations \nwhich are described in momentum space. On one hand, the quantum dynamics of the diffracted electrons \nis solved numerically in momentum space and the properties of the 3-photon Kapitza-Dirac effect \nare investigated in detail. On the other hand, the quantum dynamics is solved via time-dependent \nperturbation theory and is compared with the numerical results. \nIn contrast to the originally proposed Kapitza-Dirac effect with two interacting photons, the number \nof absorbed and emitted photons by the electron is not equal for the 3-photon Kapitza-Dirac effect. \nTherefore, the diffraction process only appears for relativistic electron momenta in laser propagation \ndirection. Furthermore, a very high field strength of the laser beam is required for driving the Kapitza- \nDirac effect with a measurable diffraction probability. The electron spin is rotated along the axis of the \nmagnetic field of the laser beam, when it undergoes the diffraction process. The rotation angle of the \nspin rotation depends on the electron momentum component in laser polarization direction. Therefore, \nthe probability for flipping the electron spin can be tuned by choosing the electron momentum in \nthe direction of the laser polarization. An experimental investigation may by established by utilizing \nfuture X-ray laser facilities.

•Jan 1, 2012•7 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

Coherent regime of Kapitza-Dirac effect with electrons

Kamila Moriová, Petr Koutenský, Neli Laštovičková Streshkova +5 more

Electron matter waves coherently diffract when passing through a periodic structure of light formed by two interfering light waves. In this so-called Kapitza-Dirac effect, the electron momentum changes due to absorption and emission of photons via stimulated Compton scattering. Until now, the effect has only been observed with low energy electrons due to the small momentum of a visible photon compared to the momentum of high energy electron leading to diffraction angles of 10^(-4) rad or smaller. We report on the observation of the Kapitza-Dirac effect in a scanning electron microscope using high energy (20 and 30 keV) electrons with de-Broglie wavelengths of 9 pm and 7 pm, respectively. The photon sidebands in the electron transverse momentum spectrum are detected in the convergent beam diffraction geometry using spatial filtering. As the coupling strength between the electrons and the light field increases, the sideband populations exhibit coherent, reversible oscillations among diffraction orders. The effect can serve as a coherent electron beam-splitter or a phase-plate in various types of electron microscopes.

quant-pharXiv:2511.14508•Physical Review Letters•Nov 19, 2025•0 citations
Abstract The formulation of the momentum of electromagnetic radiation or photons in media is a historical question. After Peierls discussed all these options, a crucial new value for the dominating dispersion factor σ was derived for non-ionized media with the consequence that an opto-acoustic coupling exists. After the experimental confirmation of the Kapitza-Dirac effect the generalization of the effect for quantum modulation of electron beams by crossing laser beams is ascertained. The evaluation of repeated experiments of this quantum modulation is very precisely confirming by measurements that the Peierls factor σ = 1/5 is correct.
Since the first experiments of the quantum modulation of electron beams by lasers with a crystal as medium in the interaction area ~d1warz-Hora effect) were reported, extensive theoretical studies were performed.A very detaile/ rLproduction of these experiments was achieved (by Andrick and Langhans, further by Weingartshofer et al) where the medium in the interaction area was a gas.The initial clarification of the experiments with crystal as a quantum phenomenon is using a threshold defining a correspondence principle for electromagnetic interaction.Recent experiments by Weingartshofer et al independently confirm this correspondence principle.For two other experiments the ranges of quantum and classical regimes are elaborated to explain the basically differing character of observations.The discussion of the coherence for matter waves is followed up from the beginning of the Schwarz-Hora effect including developments reviewed by Hawkes and indicating further generalizations by the Hanbury-Brown/ Twiss effect.A consequent generalization of the scattering theory of the quantum modulation is presented based on a Green's function solution which results in the fact that far-field superpositions only lead to interferences.The problems of the re-emission of photons from quantum modulated electron beams is discussed and experiments on indirect observations of the phenomena in microwave amplification and in the 1/f noise Handel effect are reported.In the last case. . the self-interaction of electrons are involved in the Handel effect.A further second-quantised derivation and the coherent state Handel effect is presented.
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.
Quantum mechanical diffraction is of particular interest, because it contradicts our everyday life experience. \nThis theoretical work considers the diffraction of electrons at standing waves of light, referred \nto as the Kapitza-Dirac effect. The work focuses on a special version of a Kapitza-Dirac effect in which \nthe electron interacts with three photons. The particular property of this 3-photon Kapitza-Dirac effect \nis, that the electron spin is rotated. \nThis work considers different relativistic and non-relativistic quantum mechanical wave equations \nwhich are described in momentum space. On one hand, the quantum dynamics of the diffracted electrons \nis solved numerically in momentum space and the properties of the 3-photon Kapitza-Dirac effect \nare investigated in detail. On the other hand, the quantum dynamics is solved via time-dependent \nperturbation theory and is compared with the numerical results. \nIn contrast to the originally proposed Kapitza-Dirac effect with two interacting photons, the number \nof absorbed and emitted photons by the electron is not equal for the 3-photon Kapitza-Dirac effect. \nTherefore, the diffraction process only appears for relativistic electron momenta in laser propagation \ndirection. Furthermore, a very high field strength of the laser beam is required for driving the Kapitza- \nDirac effect with a measurable diffraction probability. The electron spin is rotated along the axis of the \nmagnetic field of the laser beam, when it undergoes the diffraction process. The rotation angle of the \nspin rotation depends on the electron momentum component in laser polarization direction. Therefore, \nthe probability for flipping the electron spin can be tuned by choosing the electron momentum in \nthe direction of the laser polarization. An experimental investigation may by established by utilizing \nfuture X-ray laser facilities.
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.
Electron matter waves coherently diffract when passing through a periodic structure of light formed by two interfering light waves. In this so-called Kapitza-Dirac effect, the electron momentum changes due to absorption and emission of photons via stimulated Compton scattering. Until now, the effect has only been observed with low energy electrons due to the small momentum of a visible photon compared to the momentum of high energy electron leading to diffraction angles of 10^(-4) rad or smaller. We report on the observation of the Kapitza-Dirac effect in a scanning electron microscope using high energy (20 and 30 keV) electrons with de-Broglie wavelengths of 9 pm and 7 pm, respectively. The photon sidebands in the electron transverse momentum spectrum are detected in the convergent beam diffraction geometry using spatial filtering. As the coupling strength between the electrons and the light field increases, the sideband populations exhibit coherent, reversible oscillations among diffraction orders. The effect can serve as a coherent electron beam-splitter or a phase-plate in various types of electron microscopes.