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Journal of the Physical Society of Japan

Theory of the Kapitza-Dirac Effect

Hiroshi Ezawa, Hiroshi Namaizawa

Published Feb 1, 1969• 13 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.26.458

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Classical theory of scattering of an electron beam by a laser standing wave

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A classical theory of scattering of an electron beam by a laser standing wave (the Kapitza-Dirac effect) is presented. The prediction of the theory is in good agreement with all the experimental results which were reported independently by Bartell et al., Schwarz et al., Takeda et al., and Pfeiffer. This shows that the interaction of a free electron with coherent radiation is basically a classical process.

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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.

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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.

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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
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 Kapitza-Dirac effect is the diffraction of a well-collimated particle beam by a standing wave of light. Why is this interesting? Comparing this situation to the introductory physics textbook example of diffraction of a laser beam by a grating, the particle beam plays the role of the incoming wave and the standing light wave the role of the material grating, highlighting particle-wave duality. Apart from representing such a beautiful example of particle-wave duality, the diffracted particle beams are coherent. This allows the construction of matter interferometers and explains why the Kapitza-Dirac effect is one of the workhorses in the field of atom optics. Atom optics concerns the manipulation of atomic waves in ways analogous to the manipulation of light waves with optical elements. The excitement and activity in this new field of physics stems in part from the realization that the shorter de Broglie wavelengths of matter waves allow ultimate sensitivities for diffractive and interferometric experiments that in principle would far exceed their optical analogues. Not only is the Kapitza-Dirac effect an important enabling tool for this field of physics, but diffraction peaks have never been observed for electrons, for which it was originally proposed in 1933. Why has this not been observed? What is the relation between the interaction of laser light with electrons and the interaction of laser light with atoms, or in other words what is the relation between the ponderomotive potential and the lightshift potential? Would it be possible to build interferometers using the Kapitza-Dirac effect for other particles? These questions will be addressed in this paper.
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 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.
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 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.