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Ultra-Relativistic Effects of Laser
Beam and Electron Interactions
Basic equations, exact solutions and modelling
IOP Series in Coherent Sources, Quantum Fundamentals, and Applications
About the Editor
F J Duarte is a laser physicist based in Western New York, USA. His career has
expandedthreecontinentswhilecontributingintheacademic,industrialanddefense
sectors. Duarte is editor/author of 15 laser optics books and sole author of three
books: Tunable Laser Optics, Quantum Optics for Engineers, and Fundamentals of
Quantum Entanglement. Duarte has made original contributions in the fields of
coherentimaging,directedenergy,high-powertunablelasers,lasermetrology,liquid
and solid-state organic gain media, narrow-linewidth tunable laser oscillators,
organic semiconductor coherent emission, N-slit quantum interferometry, polar-
ization rotation, quantum entanglement, and space-to-space secure interferometric
communications. He is also the author of the generalized multiple-prism grating
dispersion theory and pioneered the use of Dirac’s quantum notation in N-slit
interferometry and classical optics. His contributions have found applications
in numerous fields, including astronomical instrumentation, dispersive optics,
femtosecond laser microscopy, geodesics, gravitational lensing, heat transfer, laser
isotope separation, laser medicine, laser pulse compression, laser spectroscopy,
mathematical transforms, nonlinear optics, polarization optics, and tunable diode
laserdesign.DuartewaselectedFellowoftheAustralianInstituteofPhysicsin1987
and Fellow of the Optical Society of America in 1993. He has received various
recognitions, including the Paul F Foreman Engineering Excellence Award and the
David Richardson Medal from the Optical Society.
Coherent Sources, Quantum Fundamentals, and Applications
Since its discovery, the laser, has found innumerable applications from astronomy
to…zoology.Subsequently,wehavealsobecomefamiliarwithadditionalsourcesof
coherentradiationsuchasthefreeelectronlaser,opticalparametricoscillators,and
coherent interferometric emitters. The aim of this book Series in Coherent Sources,
Quantum Fundamentals, and Applications is to explore and explain the physics
and technology of widely applied sources of coherent radiation and to match them
withutilitarianandcutting-edgescientificapplications.Coherentsourcesofinterest
are those that offer advantages in particular emission characteristics areas such as
broadtunability,highspectralcoherence,highenergy,orhighpower.Anadditional
area of inclusion are those coherent sources capable of high performance in the
miniaturized realm. Understanding of quantum fundamentals can lead to new and
better coherent sources and unimagined scientific and technological applications.
Applicationareasofinterestincludetheindustrial,commercial,andmedicalsectors.
Also,particularattentionisgiventoscientificapplicationswithabrightfuturesuch
as coherent spectroscopy, astronomy, biophotonics, space communications, space
interferometry, quantum entanglement, and quantum interference.
Publishing benefits
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animation and video, and incorporating supplementary files in their work.
Do you have an idea of a book that you’d like to explore?
For further information and details of submitting book proposals, see iopscience.
org/books or contact Ashley Gasque at [email protected].
Ultra-Relativistic Effects of Laser
Beam and Electron Interactions
Basic equations, exact solutions and modelling
Alexandru Popa
National Institute for Laser, Plasma and Radiation Physics, Magurele City, Romania
IOP Publishing, Bristol, UK
ªIOPPublishingLtd2021
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AlexandruPopahasassertedhisrighttobeidentifiedastheauthorofthisworkinaccordance
withsections77and78oftheCopyright,DesignsandPatentsAct1988.
ISBN 978-0-7503-3984-1(ebook)
ISBN 978-0-7503-3981-0(print)
ISBN 978-0-7503-3982-7(myPrint)
ISBN 978-0-7503-3983-4(mobi)
DOI 10.1088/978-0-7503-3984-1
Version:20211101
IOPebooks
BritishLibraryCataloguing-in-PublicationData:Acataloguerecordforthisbookisavailable
fromtheBritishLibrary.
PublishedbyIOPPublishing,whollyownedbyTheInstituteofPhysics,London
IOPPublishing,TempleCircus,TempleWay,Bristol,BS16HG,UK
USOffice:IOPPublishing,Inc.,190NorthIndependenceMallWest,Suite601,Philadelphia,
PA19106,USA
Contents
Preface xi
Author biography xii
Symbols xiii
1 Introduction 1-1
References 1-2
2 Exact solutions of the basic equations 2-1
2.1 Initial hypotheses 2-1
2.2 Solution of the classical equation of electron motion 2-4
2.2.1 Equation of the electron motion 2-4
2.2.2 Calculation of velocities and accelerations 2-6
2.2.3 Calculation of the electron coordinates 2-8
2.3 Solution of the Klein–Gordon equation 2-9
2.3.1 Relation necessary for our analysis 2-9
2.3.2 Solving the Klein–Gordon equation 2-10
2.4 Solution of the Dirac equation 2-11
2.4.1 Dirac equation 2-11
2.4.2 Solution of the system equivalent to the Dirac equation 2-13
2.4.3 Verification of the solution of the Dirac equation 2-17
2.4.4 Determination of the conditions of validity of the 2-20
classical models
References 2-22
3 Modelling ultra-relativistic interactions in electron plasmas 3-1
3.1 Initial hypotheses 3-1
3.2 Phase effect 3-2
3.2.1 Bijective dependence between phase η and time t 3-2
3.2.2 The decreasing ofΔη when I increases 3-3
L
3.3 Effect of strong electron acceleration in the ultra-relativistic regime 3-4
3.3.1 Variation of electron velocity with phase η 3-4
3.3.2 Physical interpretation of the effect of variation ofΔη with I 3-6
L
3.4 Electromagnetic field generated by the electron motion 3-8
3.4.1 Calculation of the components of the electromagnetic field 3-8
3.4.2 Periodic and aperiodic domains 3-12
vii
Ultra-RelativisticEffectsofLaserBeamandElectronInteractions
3.4.3 Modelling the electromagnetic field in the periodic domain 3-13
3.4.4 Modelling the electromagnetic field in the aperiodic domain 3-15
3.5 Very intense pulses having very large frequency spectra 3-18
3.5.1 Algorithm of our calculations 3-18
3.5.2 Generation of extremely bright pulses 3-19
3.5.3 Calculation of the frequency spectrum and comparison 3-21
with the experimental data
References 3-23
4 Modelling interactions between laser beams and 4-1
ultra-relativistic electron beams
4.1 Initial hypotheses 4-1
4.2 Solution of the equation of electron motion in the S′ system 4-2
4.3 Solution of the Klein–Gordon equation in the system S 4-7
4.4 Solution of the Dirac equation in theS′ system 4-9
4.5 Relations for the linearly polarized laser field 4-11
4.6 Comparison with experimental results from the literature 4-12
4.6.1 Preliminary observations 4-12
4.6.2 Results from SLAC experiments 4-12
4.6.3 Results from CERN experiments 4-13
4.6.4 Concordance between our solution and the experimental 4-14
data from the literature
4.7 General conditions for the validity of classical equations in the 4-15
S system
References 4-16
5 Modelling the radiation damping effect in ultra-relativistic 5-1
interactions
5.1 Initial hypotheses 5-1
5.2 Expressions for damping force and damping energy 5-3
5.2.1 Conventions for writing the four-vectors and the 5-3
electromagnetic field tensor
5.2.2 Components of the radiation damping force and the 5-4
damping energy
5.3 Radiation damping parameters calculated in the S′ system 5-7
5.3.1 Components of the external force and kinetic energy of the 5-7
electron in the S′ system
viii
