THE HAWKING-UNRUH TEMPERATURE\\ AND DAMPING IN A LINEAR FOCUSING CHANNEL
KIRK T.~McDONALD
The Hawking-Unruh effective temperature, $\hbar a^\star \over 2 \pi ck$, due to quantum fluctuations in the radiation of an accelerated charged-particle beam can be used to show that transverse oscillations of the beam in a practical linear focusing channel damp to the quantum-mechanical limit. A comparison is made between this behavior and that of beams in a wiggler.
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\begin{document}

\title{THE HAWKING-UNRUH TEMPERATURE\\
 AND DAMPING IN A LINEAR FOCUSING CHANNEL}

\author{KIRK T.~McDONALD}
\address{Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
\\
{\tt mcdonald@puphep.princeton.edu
\\
http://puhep1.princeton.edu/$\tilde{\phantom{a}}$mcdonald/accel/}
\\
(January 30, 1998)
}


\maketitle

\begin{abstract}
The Hawking-Unruh effective temperature, $\hbar a^\star \over 2 \pi ck$,  due to
quantum fluctuations in the radiation of an accelerated charged-particle beam
can be used to show that transverse oscillations of the beam in a
practical linear focusing channel damp to the quantum-mechanical limit.  
A comparison is made between this behavior and that of beams in a wiggler.
\end{abstract}

\section{Introduction}

Many of the effects of quantum fluctuations on the behavior of charged
particles can be summarized concisely by an effective temperature first
introduced in gravitational fields by Hawking \cite{Hawking}, and
applied to accelerated particles (with the neglect of gravity) by 
Unruh \cite{Unruh}.

Hawking argued that the effect of the strong gravitational field of a black
hole on the quantum fluctuations of the surrounding space is to cause 
the black hole to radiate with a temperature
\begin{equation}
T = {\hbar g \over 2\pi ck},
\label{eq1}
\end{equation}
where $g$ is the acceleration due to gravity at the surface of the black hole,
$c$ is the speed of light, and $k$ is Boltzmann's constant.  Shortly
thereafter, Unruh argued that an accelerated observer should become excited
by quantum fluctuations to a temperature
\begin{equation}
T = {\hbar a^\star \over 2\pi ck},
\label{eq2}
\end{equation}
where $a^\star$ is the acceleration of the observer in its instantaneous rest
frame.  

In a series of papers, Bell and co-workers \cite{Bell}, have noted that
electron storage rings provide a demonstration of the utility of the 
Hawking-Unruh temperature (\ref{eq2}), with emphasis on the question of the 
incomplete polarization of the electrons due to quantum fluctuations of
synchrotron radiation.  The author has commented on how the Hawking-Unruh
temperature can be used to characterize quickly the limits on damping of
the phase volume of beams in electron storage rings \cite{McDonald87}, leading 
to well-known results of Sands \cite{Sands}.

\section{Quantum Analysis of a Linear Focusing Channel}

Recently, Chen, Huang and Ruth have discussed radiation damping in a linear
focusing channel \cite{Huang1,Huang2,Huang3},
finding that in such devices the beam can be 
damped to the quantum mechanical limit set by the uncertainty principle.
I show here how this result follows very quickly from an application of
the Hawking-Unruh temperature.

A linear focusing channel is a beam-transport system that confines the
motion of a charged particle along a straight central ray via a potential that
is quadratic in the transverse spatial coordinates.   This potential can
be characterized by a spring constant $k$, and hence the frequency $\omega$
of transverse oscillations (as observed in the laboratory frame) of a
particle of mass $m$ and Lorentz factor $\gamma$ is
\begin{equation}
\omega = \sqrt{k \over \gamma m}.
\label{eq3}
\end{equation}
If the amplitude of the oscillation in transverse coordinate $x$ is called
$x_0$, then the amplitude $a_0$ of the corresponding transverse acceleration is
\begin{equation}
a_0 = x_0 \omega^2 = {k x_0 \over \gamma m}.
\label{eq4}
\end{equation}

To apply the Hawking-Unruh temperature, we consider the motion
in the instantaneous rest frame of the particle.  Supposing the transverse
oscillations are small, the instantaneous rest frame is very nearly the
frame in which the particle has no longitudinal motion.  Quantities measured
in the instantaneous rest frame will by denoted with the superscript $\star$.
Thus, in the instantaneous rest frame the amplitude of the transverse 
acceleration as measured is
\begin{equation}
a^\star_0 = \gamma^2 a_0 = {\gamma k x_0 \over m},
\label{eq5}
\end{equation}
the frequency of the oscillation is
\begin{equation}
\omega^\star = \gamma \omega,
\label{eq6}
\end{equation}
and hence the transverse spring constant of the focusing channel appears as
\begin{equation}
k^\star = m \omega^{\star 2} = \gamma k.
\label{eq7}
\end{equation}

In the instantaneous rest frame, the charge particle finds itself in a
bath of radiation of characteristic temperature given by eq.~(\ref{eq2})
with acceleration $a^\star$ given by eq.~(\ref{eq5}).  This bath can be
regarded as the effect of quantum fluctuations, which excite transverse
oscillations (having two degrees of freedom) to characteristic energy 
$U^\star$ (as measured in the instantaneous rest frame) given by
\begin{equation}
U^\star = kT = {\hbar a^\star_0 \over 2 \pi c} = {\hbar \gamma k x_0 \over 2 \pi
m c}.
\label{eq8}
\end{equation}
The energy of transverse oscillation can also be written in terms of
the (invariant) transverse amplitude $x_0$ as
\begin{equation}
U^\star = {k^\star x_0^{\star 2}\over 2} = {\gamma k x_0^2 \over 2}.
\label{eq9}
\end{equation}
Hence, the amplitude of excitation of the transverse oscillations is
\begin{equation}
x_0 = {\hbar \over \pi m c} = {\lambdabar_C \over \pi},
\label{eq10}
\end{equation}
where $\lambdabar_C$ is the (reduced) Compton wavelength of the particle.

The amplitude (\ref{eq10}) must, however, be compared to the amplitude
of the zero-point oscillations of the system, considered as a quantum
oscillator:
\begin{equation}
x_{0,{\rm zero\ point}} 
= \sqrt{\hbar \over \gamma m \omega} = \sqrt{\lambdabar_C \lambdabar \over
\gamma},
\label{eq11}
\end{equation}
where $\lambdabar = c/\omega$ is the laboratory (reduced) wavelength of
the transverse oscillation as measured along the beam axis.
In practical laboratory devices, we will have $\lambdabar \gg \gamma
\lambdabar_C$.  Hence,
the excitation of the transverse oscillations by fluctuations in the
radiation of the oscillating charge, as are described by the Hawking-Unruh
temperature, is negligible compared to the zero-point fluctuations of the
transverse oscillations.  In this sense, we can say along with Huang, Chen
and Ruth that the radiation does not excite the transverse oscillations, and
those oscillations will damp to the quantum-mechanical limit.  

In futuristic
devices, for which $\gamma > \lambdabar/\lambdabar_C$, \ie, when
\begin{equation}
\gamma > {mc^2 \over k \lambdabar_C},
\label{eq11a}
\end{equation}
quantum excitations of oscillations in a linear focusing channel would become
important.  When (\ref{eq11a}) holds, the transverse oscillations
would be relativistic even when their amplitude is only a Compton wavelength.
The strength of the transverse fields in the channel would 
then exceed the QED critical field strength (in the average rest frame),
\begin{equation}
E_{\rm crit} = {m^2c^3 \over e\hbar} = 1.6 \times 10^{16}\ \mbox{V/cm} 
= 3.3 \times 10^{13}\ \mbox{Gauss},
\label{eq6a}
\end{equation}
and the beam energy would be rapidly dissipated by pair creation.

Another way of viewing a practical linear focusing channel is that
its Hawking-Unruh excitation energy,
(\ref{eq8}), is small compared to the zero-point energy, $\hbar \omega^\star/2
= \gamma \hbar \omega/2$ of transverse oscillations.

The quantum-mechanical limit for transverse motion can, of course, also be 
deduced from the uncertainty principle:
\begin{equation}
\sigma_x \sigma_{p_x} \gsim \hbar,
\label{eq12}
\end{equation}
which leads to a minimum normalized emittance of
\begin{equation}
\epsilon_N = {\sigma_x \sigma_{p_x} \over mc} \approx \lambdabar_C,
\label{eq13}
\end{equation}
corresponding to geometric emittance of
\begin{equation}
\epsilon_x = {\epsilon_N \over \gamma \beta_z} \approx 
{\lambdabar_C \over \gamma}.
\label{eq14}
\end{equation}

\section{Semiclassical Analysis}

In a quantum analysis of a linear focusing channel, we found that the
transverse oscillations can damp to the limit set by the
uncertainty principle.  Hence, in a classical analysis we would expect
the damping to be able to proceed until the transverse amplitude was zero.

Indeed, a simple analysis confirms this.  Transform to the longitudinal rest
frame, in which the particle's motion is purely transverse.  The particle
has nonzero kinetic energy in this frame, but its average momentum is zero.
The radiation due to the transverse oscillation is reflection symmetric about 
the transverse plane in this frame, so the radiation carries away energy but 
not momentum.  With time, all of the energy would be radiated away, and the
particle would come to rest.   The transverse oscillations will have damped
to zero without affecting the longitudinal motion.

If we add the concept of photons to the preceding analysis, we can say that
the radiated photons carry away momentum along the direction of emission, but
the radiation pattern is symmetric, so the averaged radiated momentum is
zero.  Again, the radiation carries away energy, now in the form of photons.

Back in the lab frame, we view the photons as carrying away a small amount of
longitudinal
momentum on average, as a result of the Lorentz transformation of the
energy radiated in the longitudinal rest frame.  
This momentum, however, is only that part of the particle's longitudinal
momentum associated with its transverse oscillation; the longitudinal
velocity of the particle is unaffected.

On average, the photons carry away no transverse momentum in the lab frame, 
and the average momentum of the
radiated photons is therefore parallel to the beam axis in lab frame.
However, there is no need to argue that the momentum of individual radiated
photons is parallel to the beam axis, nor to imply that the matter of
the focusing channel absorbs transverse momentum in a manner than affects
the kinematics of the radiation process \cite{Huang2}.

\section{Comparison of a Linear Focusing Channel to a Wiggler}

A comparison with the behavior of particle beams in a wiggler is instructive.
Here the transverse confinement of the beam motion is provided by a series
of alternating transverse magnetic fields.  This has the notable effect that
even if a particle enters the wiggle parallel to the beam axis, transverse
oscillations will result whose amplitude is independent of the initial
transverse coordinate.  

In contrast, a particle that enters a linear focusing channel parallel to and 
along the axis undergoes no oscillation, no matter what is the particle's
longitudinal momentum.

We thereby see that radiation damping cannot reduce the
oscillations in a wiggler to zero unless the longitudinal momentum falls to 
zero also, since the wiggler continually re-excites
transverse oscillations for any particle with nonzero kinetic energy.

Another difference between a wiggler and a linear focusing channel can be seen 
by going to the longitudinal rest frame.  In the case of the wiggler, the
alternating magnetic fields in the laboratory transform to fields that are
very much like a plane wave propagating against the direction of the
laboratory motion of the beam.  The radiation induced by this effective
plane wave is not symmetric with respect to the transverse plane, but
results in a net kick of the particle into the backward direction.

Viewed in the lab frame, we find that along with the damping of their transverse
oscillations, the particles' longitudinal momenta are
significantly reduced.  To maintain the initial longitudinal momentum,
the beam must be reaccelerated. The momentum (and energy)
added back into the beam then increases the amplitude of the transverse
oscillations, and the damping cannot continue beyond some limit.

In contrast, in a linear focusing channel, the transverse damping proceeds
without significant reduction in the longitudinal momentum of the particle,
and the transverse oscillations can damp to the quantum limit without the
need of adding energy back into the beam.

%xxx


\section*{Acknowledgments}

I wish to thank Pisin Chen, Ron Ruth and Max Zolotorev for conversations
on radiation damping.
This work was supported in part by
DoE grant DE-FG02-91ER40671.

%\section*{Appendix}

\section*{References}

\begin{thebibliography}{99}

\bibitem{Hawking}
S.W.~Hawking,
{\sl Black Hole Explosions},
 Nature {\bf 248}, 30-31 (1974);
{\sl Particle Creation by Black Holes},
Comm.\ Math.-Phys.\ {\bf 43}, 199-220 (1975).

\bibitem{Unruh}
W.G.~Unruh,
{\sl Notes on Black Hole Evaporation},
Phys. Rev. D {\bf 14}, 870-892 (1976); 
{\sl Particle Detectors and Black Hole Evaporation},
Ann.\ N.Y.\ Acad.\ Sci.\ {\bf 302}, 186-190 (1977).

\bibitem{Bell}
J.S.~Bell and J.M.~Leinaas,
{\sl Electrons as Accelerated Thermometers},
Nuc.\ Phys.\ {\bf B212}, 131-150 (1983);
{\sl The Unruh Effect and Quantum Fluctuations of Electrons in Storage Rings},
%Nuc. Phys. 
{\bf B284}, 488-508 (1987);
J.S.~Bell, R.J.~Hughes and J.M.~Leinaas,
{\sl The Unruh Effect in Extended Thermometers},
Z.\ Phys. {\bf C28}, 75-80 (1985);
W.G.~Unruh,
{\sl Acceleration Radiation for Orbiting Electrons}
(hep-th/9804158, 23 Apr 98);
J.M.~Leinaas,
{\sl Accelerated Electrons and the Unruh Effect}
(hep-th/9804179, 28 Apr 1998).

\bibitem{McDonald87}
K.T.~McDonald,
{\sl The Hawking-Unruh Temperature and Quantum Fluctuations in Particle
Accelerators},
Proceedings of the 1987 IEEE Particle Accelerator Conference,
E.R.~Lindstrom and L.S.~Taylor, eds. (Washington, D.C., Mar.~16-19, 1987)
pp.~1196-1197.

\bibitem{Sands}
M.~Sands, 
{\sl The Physics of Electron Storage Rings},
SLAC-121, (1970); 
also in Proc.\ 1969 Int.\ School of Physics, `Enrico Fermi', 
ed. by B.~Touschek (Academic Press, 1971), p. 257.

\bibitem{Huang1}
Z.~Huang, P.~Chen and R.D.~Ruth,
{\sl Radiation Reaction in a Continuous Focusing Channel},
Phys.\ Rev.\ Lett.\ {\bf 74}, 1759-1762 (1995).

\bibitem{Huang2}
Z.~Huang, P.~Chen and R.D.~Ruth,
{\sl Radiation and Radiation Reactions in Continuous Focusing Channels},
in {\sl Advanced Accelerator Concepts}, AIP. Conf. Proc. {\bf 335},
646-658 (1995).

\bibitem{Huang3}
Z.~Huang and R.D.~Ruth,
{\sl Effects of Focusing on Radiation Damping and Quantum Excitation in 
Electron Storage Rings},
Phys.\ Rev.\ Lett.\ {\bf 80}, 2318 (1998). 

%rrr


\end{thebibliography}

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\def\ecrit{{E_{\rm crit}}}
%%%



Here we expand slightly on the results of Bell {\it et al.}, and encourage the
reader to consult the literature for more detailed understanding.
\medskip
\centerline{\bf Applicability of the Idea}
\medskip
When an accelerated charge radiates, the discrete energy and momentum of the
radiated photons induce fluctuations on the motion of the charge.  The insight
of Unruh is that for uniform linear acceleration (in the absense of the 
fluctuations), the fluctuations would excite any internal degrees of freedom
of the charge to the temperature stated above.  His argument is very general
({\it i.e.}, thermodynamic) in that it does not depend on the details of
the accelerating force, nor of the nature of the accelerated particle.  The
idea of an effective temperature is strictly applicable only for uniform
linear acceleration, but should be approximately correct for other 
accelerations, such as that due to uniform circular motion.

A charged particle whose motion is confined by the focusing system of a 
particle accelerator exhibits transverse and longitudinal oscillations
about its ideal path.  These oscillations are excited by the quantum
fluctuations of the particle's radiation, and thus provide an excellent physical
example of the viewpoint of Unruh.

Further, the particles take on a thermal distribution of energies when
viewed in the average rest frame of a bunch, which transforms to the
observed energy spread in the laboratory.  While classical synchrotron
radiation would eventually polarize the spin-$1 \over 2$ particles completely,
the thermal fluctuations oppose this, reducing the maximum beam polarization.

It is suggestive to compare the excitation energy $U^\star = kT$, as would be
observed in the particle's rest frame, to the rest energy $mc^2$ when 
the acceleration is due to laboratory electromagnetic fields $E$ and $B$.  
Noting that $a^\star = eE^\star/m$ we find
$${U^\star \over mc^2} = {\hbar eE^\star \over 2\pi m^2c^3} = {\left[ 
E_\parallel + \gamma\left( E_\perp + \beta B_\perp \right)\right] \over 2\pi
\ecrit},$$
where the particle's laboratory momentum is $\gamma\beta mc$, and
$$\ecrit \equiv {m^2c^3 \over e\hbar}.$$
For an electron,
$$\ecrit = 1.3 \times 10^{16}\,{\rm volts/cm} =
4.4 \times 10^{13}\,{\rm gauss}.$$
($\ecrit$ is the field strength at which spontaneous pair production becomes
highly probable, {\it i.e.}, the field whose voltage drop across a
Compton wavelength is the particle's rest energy.)  We might expect that the
fluctuations become noticeable when $U^\star \sim 0.1$ eV, and hence comparable
to any other thermal effects in the system, such as the particle-source
temperature.

For linear accelerators $E_\parallel \sim 10^6$ volts/cm at best, so $U^\star
< 10^{-5}$ eV.  The effect of quantum fluctuations is of course negligible
because the radiation itself is of little importance in a linear accelerator.

For an electron storage ring such as LEP, $\gamma \sim 10^5$, and $B_\perp \sim
10^3$ gauss, so that $U^\star \sim 0.2$ eV.  For the SSC proton storage ring,
$\gamma \sim 2\times 10^4$, while $B_\perp \sim 6 \times 10^4$ gauss, so that
$U^\star \sim 2$ eV.  As is well known, in essentially all electron storage
rings, and in future proton rings, the effect of quantum fluctuations is
quite important.

The remaining discussion is restricted to beams in storage rings (= transverse
particle accelerators).
\medskip
\centerline{\bf Beam-Energy Spread}
\medskip
An immediate application of the excitation energy $U^\star$ is to the 
beam-energy spread.  In the average rest frame of a bunch of particles, the
distribution of energies is approximately thermal, with characteristic
kinetic energy $U^\star$, and momentum $p^\star = \sqrt{2mU^\star}$.  The
spread in laboratory energies is then given by
$$U_{\rm lab}  \approx \gamma (mc^2 + U^\star \pm \beta p^\star c) 
\approx U_0 \left( 1 \pm \gamma \sqrt{\lambda_C \over \pi\rho} \right),$$
where $U_0 = \gamma mc^2$ is the nominal beam energy,
$\rho = U_0/eB_\perp$ is the radius of curvature of the central
orbit, and $\lambda_C = \hbar/mc$ is the Compton wavelength. 
Writing this as
$$\left( {\delta U \over U_0}\right)^2 \approx {\gamma^2\lambda_C \over 
\pi\rho},$$
we obtain the standard result, as given by equation (5.48) of the review by
Sands.$^6$
\medskip
\centerline{\bf Beam Height}
\medskip
The quantum fluctuations of synchrotron radiation drive the oscillations of
particles about the bunch center, and set lower limits on the transverse and
longitudinal beam size.  If we associate a harmonic oscillator with each
component of the motion about the bunch center, then each oscillator will
be excited to amplitudes whose corresponding energy is $U^\star = kT^\star$.

For example, consider the vertical betatron oscillations which determine the 
beam height.  The frequency of these oscillations is $\omega = \nu_z \omega_0
= \nu_z c/R$, where $\nu_z$ is the vertical betatron number, and $R = L/2\pi$
is the mean radius of the storage ring.  In the average rest frame of a bunch
the oscillation frequency appears to be $\omega^\star = \gamma\omega$, and
the spring constant in this frame is given by $k^\star = m\omega^{\star 2} =
\gamma^2 m\omega^2$.  The typical amplitude of oscillation in this frame is
then
$${1 \over 2}k^\star z^{\star2} \approx U^\star = {\hbar a^\star \over 
2\pi c} = {\hbar\gamma^2 a \over 2\pi c} = {\hbar\gamma^2 c \over 2\pi\rho},$$
noting that in uniform circular motion the acceleration is transverse.
For the vertical oscillation
the lab frame amplitude $z$ is the same as $z^\star$.  Combining the above
we find
$$z^2 = {\lambda_C R^2 \over \pi\nu_z^2 \rho},$$
which reproduces the standard result, such as equation (5.107) of Sands.$^6$

An analogous argument is given in ref. 5 to derive the beam height in a weakly 
focused storage ring.
\medskip
\centerline{\bf Bunch Length and Beam Width}
\medskip
A similar analysis can be given for oscillations in the plane of the orbit.
However, radial and longitudinal excursions are also directly coupled to energy
excursions, which proves to be the stronger effect.  As the present method
finds the standard result for the beam-energy spread, the usual results for
bunch length and beam width follow at once.  [In ref. 6, use equations (5.64)
and (5.93) to yield expressions (5.65) and (5.95).]
\medskip
\centerline{\bf Beam Polarization}
\medskip
Sokolov and Ternov$^7$ predicted that quantum fluctuations in synchrotron
radiation limit the transverse polarization of the beam to 92\%.  In the absense
of quantum fluctuations the polarization should reach 100\% after long times.
Bell and Leinaas$^3$ realized that the thermal character
of the fluctuations provides an alternate view of the depolarizing mechanism.
In ref. 5 they provide a detailed justification that the
thermodynamic arguments are fully equivalent to the original QED calculation
of Sokolov and Ternov.  In the process they find that for circular motion
in a weakly focused ring (betatron), the effective temperature due to 
quantum fluctuations is
$$kT = {13 \over 96} \sqrt{3} {\hbar a^\star \over c},$$
which is about 1.5 times Unruh's result for linear acceleration.
\medskip
\centerline{\bf Radiation Spectrum}
\medskip
Because of the quantum fluctuations the motion of the particles departs from
the central orbit, and a classical calculation of the synchrotron-radiation
spectrum is incorrect in principle.  The deviations become significant only
when the characteristic energy of the radiation approaches the beam energy,
{\it i.e.}, when $\gamma B_\perp/\ecrit \sim 1$, and the prominent effect is
the cutoff at the high-energy end of the spectrum.

In the regime where the quantum corrections to the radiation spectrum are 
small the author has given an estimate of their size.$^8$  For this we
imagine the accelerated charge is surrounded (in its rest frame) by a
bath of photons with a Planck spectrum of temperature 
$kT = \hbar a^\star/2\pi c$.  The correction to the classical spectrum is
considered to arise from the Thomson scattering of these virtual photons off
the charged particle.  
In the lab frame the spectral correction is proportional to the Lorentz 
transform of the Planck
spectrum, whose peak photon energy is then $2\gamma kT = \hbar\gamma^3 c/
\pi\rho$, essentially the same as that of the classical spectrum.  
On integrating over energy, the
total rate of the correction term is the classical (Larmor) rate times
$${\alpha \over 60\pi} \left( {\gamma B_\perp \over \ecrit} \right)^2,$$
which is indeed very small at present storage rings.
\medskip
\centerline{\bf Acknowledgements}
\medskip
I would like to thank Ian Affleck and Heinrich Mitter for several discussions
on this topic.  This work was supported in part by the U.S. Department of Energy
under contract DOE-AC02-76ER-03072.
\medskip
\hbox to \hsize{\hss\vrule width2.5cm height1pt \hss}
\medskip
%\ninepoint
$^{1}$S.W. Hawking, `Black-Hole Explosions,' Nature {\bf 248}, 30 (1974).

$^{2}$W.G. Unruh, `Notes on Black-Hole Evaporation,'
Phys. Rev. D {\bf 14}, 870 (1976).

$^{3}$J.S. Bell and J.M. Leinaas, `Electrons as Accelerated Thermometers,'
Nuc. Phys. {\bf B212}, 131 (1983).

$^{4}$J.S. Bell, R.J. Hughes and J.M. Leinaas, `The Unruh Effect in
Extended Thermometers,' Z. Phys. C {\bf 28}, 75 (1985).

$^{5}$J.S. Bell and J.M. Leinaas, `The Unruh Effect and Quantum 
Fluctuations of Electrons in Storage Rings,' CERN-TH.4468/86 (June, 1986).

$^{6}$

$^{7}$A.A. Sokolov and I.M. Ternov, `On Polarization and Spin Effects in the
Theory of Synchrotron Radiation,' Sov. Phys. Dokl. {\bf 8}, 1203 (1964).

$^{8}$K.T. McDonald, `Fundamental Physics During Violent Acceleration,' in
{\sl Laser Acceleration of Particles}, AIP Conf. Proc. No. 130, ed. by C. Joshi
and T. Katsouleas (Malibu, 1985), p. 23.


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\bibitem{temporary}
K.T.~McDonald and K.~Shmakov,
submitted to Phys. Rev. Lett.

\bibitem{Sprangle}
E.~Esarey \etal,
% {\sl Laser Acceleration of Electrons in Vacuum}, 5443-5455
Phys.~Rev.~E{\bf 52}, 5443 (1995);
P.~Sprangle \etal,
% {\sl Vacuum Laser Acceleration},  69-73
Opt.~Commun.~{\bf 124}, 69 (1996).

\bibitem{dressing}
K.T.~McDonald and K.~Shmakov,
submitted to Am. J. Phys.

\bibitem{McMillan}
E.M.~McMillan,
% {\sl The Origin of Cosmic Rays},   498-501
Phys.~Rev.~{\bf 79}, 498 (1950).

\bibitem{Menzel}
D.H.~Menzel and W.W.~Salisbury,
% {\sl The Origin of Cosmic Rays}, 67-71
Nucleonics, {\bf 2}, No.~4, 67 (1948).

\bibitem{Francia}
G.~Toraldo di Francia,
%{\sl Interaction of Focused Laser Radiation with a Beam of Charged Particles},
Nuovo Cim.~{\bf 37}, 1553 (1965).

\bibitem{Kibble}
T.W.B.~Kibble,
% {\sl Frequency Shift in High-Intensity Compton Scattering}, B740-753
Phys.~Rev.~{\bf 138}, B740, (1965); {\bf 150}, 1060 (1966);
% {\sl Radiative Corrections to Thomson Scattering from Laser Beams}, 627-628
%Phys.~Lett.~{\bf 20}, 627 (1966);
% {\sl Refraction of Electron Beams by Intense Electromagnetic Waves}, 1054-1056
Phys.~Rev.~Lett.~{\bf 16}, 1054 (1966);
% {\sl Mutual Refraction of Electrons and Photons}, 1060-1069
%Phys.~Rev.~{\bf 150}, 1060 (1966);
% {\sl Some Applications of Coherent States}, 299-345
{\sl Carg\`ese Lectures in Physics}, Vol.~2, ed. by M.~L\'evy (Gordon and
Breach, New York, 1968), p.~299.

\bibitem{Sarachik}
E.S.~Sarachik and G.T.~Schappert,
%{\sl Classical Theory of Scattering of Intense Laser Radiation by Free 
%Electrons}, 2738-2753
Phys.~Rev.~D {\bf 1}, 2738 (1970).

\bibitem{Kaw}
P.E.~Kaw and R.M.~Kulsrud,
% {\sl Relativistic Acceleration of Charged Particles by Superintense Laser
% Beams}, 321-328
Phys.~Fluids {\bf 16}, 321 (1973).

\bibitem{Lai}
H.M.~Lai,
%{\sl Particle Acceleration by an Intense Solitary Electromagnetic Pulse},
% 2373-2375
Phys.~Fluids {\bf 23}, 2373 (1980).

\bibitem{Bucksbaum87}
P.H.~Bucksbaum \etal,
% {\sl Scattering of Electrons by Intense Coherent Light}, 349-352
Phys.~Rev.~Lett.~{\bf 58}, 349 (1987).

\bibitem{Jonsson}
L.~J\"onsson,
%{\sl Energy Shifts Due to the Ponderomotive Potential},  1422-1425
J.~Opt.~Soc.~Am.~B{\bf 4}, 1422, (1987).



\bibitem{Hagenbuch}
K.~Hagenbuch,
%{\sl Free Electron Motion in a Plane Electromagnetic Wave}, 693-696.
Am.~J.~Phys.~{\bf 45}, 693 (1979).

\bibitem{Bucksbaum90}
%For a review, see 
P.H.~Bucksbaum,
% {\sl Introduction to the High Intensity Physics of Atoms and Free Electrons},
in {\sl Atoms in Strong Fields}, 
C.~Nicholaides \etal\ eds.~(Plenum Press, 1990), p. 381.

\bibitem{Bula}
C.~Bula \etal,
%``Observation of Nonlinear Effects in Compton Scattering'',
Phys.~Rev.~Lett.~{\bf 76}, 3116 (1996).

\bibitem{Burke}
D.L.~Burke \etal,
%``Positron Production in Multiphoton Light-by-Light Scattering'',
Phys.~Rev.~Lett.~{\bf 79}, 1626 (1997).

\bibitem{Landau}
L.~Landau and E.M.~Lifshitz, 
{\sl The Classical Theory of Fields},
4th ed.~(Pergamon Press, Oxford, 1975), prob.~2, \S~47; p.~112 of the 
1941 Russian edition.

\bibitem{McMillan}
E.M.~McMillan,
% {\sl The Origin of Cosmic Rays},   498-501
Phys.~Rev.~{\bf 79}, 498 (1950).

\bibitem{Volkov} D.M.~Volkov,
Zeits.~f.~Phys.~{\bf 94}, 250 (1935).

\bibitem{Lifshitz} 
See also Secs.~40 and 101 of V.R.~Berestetskii, E.M.~Lifshitz and 
L.P.~Pitaevskii,
{\em Quantum Electrodynamics}, 2nd ed.~(Pergamon Press, Oxford, 1982).

\bibitem{Mikaelian}
K.O.~Mikaelian,
%  %%`Photoproduction of Charged Intermediate Vector Bosons,'
%  Phys.~Rev.~D {\bf 17}, 750 (1978);
%  %%`Zeroes in $\gamma + e \to W + \nu$,' 
%  Phys. Rev. D {\bf 30}, 1115 (1984).
Phys.~Rev.~D {\bf 17}, 750 (1978); {\bf 30}, 1115 (1984).

\bibitem{Ginzburg} 
I.F.~Ginzburg \etal,
%  %%`$W^\pm$ Boson Production at the $e^+e^-$, $\gamma e$ and $\gamma\gamma$
%  %%Colliding Beams,'
Nucl.~Phys.~{\bf B228}, 285 (1983).

\bibitem{Chan}
Y.W.~Chan,
%{\sl Ultra-Intense Laser Radiation as a Possible Energy Booster for
%Relativistic Charged Particle}, 305-306
Phys.~Lett.~{\bf 35A}, 305 (1971).

\bibitem{Umstadter}
%For a review, see 
D.~Umstadter \etal,
% {\sl Nonlinear Optics in Relativistic Plasmas and Laser Wake Field 
% Acceleration of Electrons}, 472-475
Science, {\bf 273}, 472 (1996).
         
\bibitem{Ostriker}
J.P.~Ostriker and J.E.~Gunn,
% {\sl On the Nature of Pulsars. I. Theory}, 1395-1417
% Ap.~J.~{\bf 157}, 1395 (1969).
% {\sl On the Motion and Radiation of\Charged Particles in Strong
% Electromagnetic Waves. I. Motion in Plane and Spherical Waves}, 523-541
Ap.~J.~{\bf 157}, 1395 (1969); {\bf 165}, 523 (1971).
% see eq.~(79) regarding angular momentum.

\bibitem{Nikishov}
A.I.~Nikishov and V.I.~Ritus,
% {\sl Quantum Processes in the Field of a Plane Electromagnetic Wave and in
% a Constant Field}, 1191-1199
Sov.~Phys.~JETP {\bf 19}, 1191 (1964);
% {\sl Effect of laser field on beta decays of nuclei}, 14-23
{\bf 58}, 14 (1983);
V.I.~Ritus,
%{\sl Effect of an Electromagnetic Field on Decays of Elementary Particles},
% 532-541
{\it ibid.}, {\bf 29}, 532 (1969).





\bibitem{Edighoffer}
J.A.~Edighoffer and R.H.~Pantell,
%{\sl Energy Exchange Between Free Electrons and Light in Vacuum}, 6120-6122
J.~Appl.~Phys.~{\bf 50}, 6120 (1979),



\bibitem{Gunn}
J.E.~Gunn and J.P.~Ostriker,
% {\sl Acceleration of High-Energy Cosmic Rays by Pulsars},   728-731
Phys.~Rev.~Lett.~{\bf 22}, 728 (1969).

\bibitem{Kulsrud}
R.M.~Kulsrud, J.P.~Ostriker and J.E.~Gunn,
% {\sl Acceleration of Cosmic Rays in Supernova Remnants}, 636-639
Phys. Rev. Lett. {\bf 28}, 636 (1972).

\bibitem{Sengupta49}
N.D.~Sengupta,
%{\sl On the Scattering of Electromagnetic Waves by a Free Electron -- I.
% Classical Theory}, 187-198
Bull.~Math.~Soc.~(Calcutta) {\bf 41}, 187 (1949).

\bibitem{Sengupta52}
N.D.~Sengupta,
%{\sl On the Scattering of Electromagnetic Waves by a Free Electron}, 175-180
Bull.~Math.~Soc.~(Calcutta) {\bf 44}, 175 (1952).

%\begin{figure}[hbt]
%\centering \epsfig{file=accfig2.eps,width=3.25in,height=2.1in}
%\caption{
%Sketch of a waveguide structure of length $L$ and radius $R$ for acceleration 
%of electrons.  The drive electromagnetics fields have wavelength $\lambda
%\approx R$.
%}
%\label{fig2}
%\end{figure}

The waveguide is filled with a field labelled ${\bf E}_{\rm drive}$.  The
change in the fields of the electron due to the drive field will be called
the stimulated field, ${\bf E}_{\rm stim}$.  It is important to note that
even in the absence of a drive field, when an electron passes through the
waveguide structure its fields are modified by what we call the
spontaneous field, ${\bf E}_{\rm spont}$.  



%We argue that the part of the energy gain of the
%electron that is linear in the strength of the drive field is due to 
%interference between the drive field and the spontaneous field of the electron.
%Since the stimulated field is proportional to the drive field, energy gain
%due to interference between the drive and stimulated fields will be quadratic
%in the drive field, as was the case for vacuum laser acceleration.

%The above argument gives an additional perspective on why vacuum laser
%acceleration is nonlinear in the laser field strength.  There is no spontaneous
%radiation of a free electron in vacuum, and hence no interference term linear
%in the drive laser field.

Since the waveguide structure has characteristic radius of same order as
the wavelength of the radiation, the interior of the structure is in the
near zone of the fields of the electron.  The spontaneous fields of the
electron then represent modifications to the Coulomb field of the electron
that occur on distance scale $R$, the radius of the structure.  Thus,
\begin{equation}
E_{\rm spont} \approx {e \over R^2}.
\label{eq14}
\end{equation}

The stimulated field of the electron in the near zone is dominated by the
field of the dipole moment of the electron in response to the drive field.
That dipole moment, $d$, is given by
\begin{equation}
d \approx {e^2 \over m \omega^2} E_{\rm drive} 
\approx r_e \lambda^2 E_{\rm drive},
\label{eq15}
\end{equation}
where $\omega$ is the frequency of the drive wave.  The characteristic
strength of the stimulated field in the near zone is then,
\begin{equation}
E_{\rm stim} \approx {d \over R^3} \approx {r_e \over R} E_{\rm drive},
\label{eq16}
\end{equation}
noting that $R \approx \lambda$.  Thus, the stimulated fields of the electron
simply provide a small correction to the drive fields, and we neglect the
stimulated fields in further discussion.

We evaluate the field energy inside the waveguide structure just after the
electron has passed through it.  The self fields of the electron then no
longer interfere with the fields inside the structure, and we ignore the
self fields.  We have (also neglecting the stimulated fields),
\begin{eqnarray}
U_{\rm total} & \approx & L R^2 \left| {\bf E}_{\rm drive} + {\bf E}_{\rm spont}
 \right|^2
\nonumber \\
& \approx & L R^2 \biggl( E_{\rm drive}^2 +
2 {e \over R^2} E_{\rm drive} \cos\phi 
+ {e^2 \over R^4} \biggr)
\label{eq17} \\
& \approx & U_{\rm drive} + e E_{\rm drive} L \cos\phi + {Lr_e \over R^2} mc^2,
\nonumber
\end{eqnarray} 
where $\phi$ is the phase between the drive wave and the spontaneous wave of
the electron.  
%We see that the effects of the stimulated field of the electron
%are everywhere suppressed by factors of $r_e/R$, and so are negligible in
%any macroscopic waveguide structure.  Rather, the principal correction to
%the field energy is due to the interference between the drive field and the
%spontaneous field of the electron.

For the electron to extract energy from the waveguide, the phase $\phi$ 
should be chosen to be $180^\circ$.  Then the energy gain of the electron 
compensates for the loss of field energy, and is
\begin{equation}
\Delta U_e \approx e E_{\rm drive} L,
\label{eq18}
\end{equation}
in agreement with expectations.

physics0003061.extracted_bib
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M.~Sands, 
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Z.~Huang, P.~Chen and R.D.~Ruth,
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%rrr


\end{thebibliography}