exact theory of otr and cotr
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7/21/2019 Exact Theory of OTR and COTR
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A.Nause, A.Gover
Tel Aviv University. Internal Report.
Exact Theory of Optical Transition Radiation
(OTR) and Coherent OTR
September 18, 2010
1 Introduction
Transition radiation (TR) is the electromagnetic radiation emitted by a
charged particle when it hits a conducting or dielectric plate or foil. The
wide frequency band radiation emitted on both sides of foil originates from
the Fourier components of the terminated (or suddenly appearing) current
of the charge particle in either side of the foil, as well as from the currents
induced on the foil by the charge particle.
The first detailed theory of TR was published by Ginzburg and Frank [1].
They calculated the Coulomb electrostatic field component in the frequency
domain of an electron of velocity v propagates perpendicularly to the screen.
This field component on point r on the screen is:
Er e x x, k lr,B ~re i 1
where This fieldwas assumed to be reflected from the screen and diffracted
towards an observation point in the far field.
Based on this model the far field spectral radiation intensity of TR from
a single electron can be calculated:
dP . e
2
fiii
fj 2 fj 2
d O d ~
=
16 7f
3,4
V ~
[ 11 ;
+
l1 i) ~
,-4]2
2
this expression is usually used to calculate the TR far field emission pattern
of a charged particle beam by single convolution with the electron beam
spatial and angular distribution function.
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This procedure is not sufficient if one needs to find the TR field of a beam
in the near field (the screen position) or its optical image. This is especially
a problem if the emission from the electrons in the beam is phase correlated.
In this case, an exact diffraction integral expression is required, including the
radiation field phase. Shkvarunets and Fiorito [2]presented a more complete
vector diffraction model based on Love s field equivalence Theorem, but it
was not employed for calculation of the near field of partially coherent field
distribution. Geloni et al [3] presented a diffraction theory that included
imaging of an optically pre-modulated beam. This was carried out in the
paraxial diffraction approximation.
In the present article we present an exact vectorial diffraction theory of
TR from a single electron incident vertically on a conductive screen. The
source of the diffraction integral is the current of the electron itself and its
image charge. Since the complex field solution is exact at any distance, it
replicates the Coulomb field of the electron on the screen (1) in the reactive
near field range and is valid in the Fraunhofer far field and the Fresnel near
field zones as well. We can then employ it to calculate coherent end partially
coherent TR radiation from an electron beam.
Optical Transition Radiation (OTR) is used extensively as diagnostics of
the charge distribution across the cross-section of electron beams [4-5]. If the
electrons in a caustic beam hit the screen at random times, their radiative
emission is uncorrelated and the imaged screen OTR radiation intensity at
any frequency bandwidth replicates the beam current density distribution.
However, if there is temporal (incidence phase) correlation between the elec-
trons in the beam, the beam current contains spatial and temporal frequency
Fourier components, in excess to the Fourier components of the individual
electrons. The emitted Transition radiation then contains in its spectrum
these Fourier components on top of the wide band Fourier components of
the single electron.
One kind of correlated-electrons coherent emission is connected to the
electron pulse shape and duration. Electron beam micropulses of picosec-
ond duration, carry current with Fourier frequency components in the range
of TeraHertz. The Coherent Transition Radiation (CTR) emitted by such
beam-rnicropulses is measured with fast THz detectors, and is used exten-
sively for diagnosis of the beam pulse shape and duration [6].
Another kind of correlated-electrons partially coherent TR emission effect
which were observed on OTR diagnostic screens, .. in the optical frequency
regime by a camera takes place if the electron beam current (density) ismod-
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ulated at optical frequencies. It was found unexpectedly [LCLSl that such
modulation takes place due to random energy modulation by Coulomb collec-
tive interaction micro-dynamics along a beam transport line which turns into
current modulation after passage through energy dispersive elements. This
Coherent Optical Transition Radiation (COTR) effect has been measured in
many laboratories [7-8]. It is usually a disruptive effect, which disenables
OTR screens as diagnostics means, because the partial transverse coherence
of the emitted OTR radiation produces speckled image of the beam.
2 Emission from a line source
The electric field created by a current density J(r) can be calculated as
E = -iwJl
Ge( r, r )J(r ) d
3
r ,
where Ge(r , r ) is the Green function
(3)
eiklr-r l
Ge(r,
r =
I
+
k
2
Ir_ r l
4
If we define a line charge J(r ) = e J z ) o x/ )o y /), and define R
=
Jp2
+ z -
ZI)2
we can find the Green function G(r, z/). by writing the
electric field in the transverse and the axial directions
E
= z e
z
+ E p e - p we
can calculate the transverse electric field as
v v I 1 1
Ep =
-zwJl
1(z )Gp(p , z, z ) dz ,
where in this case the transverse Green function is simply
5
1 [ o
e
ikR
G
p
=
k2
[)p [)zIi )
6 )
resulting with the exact expression for the transverse Green function. This
solution can be use in order to estimate radiation in the reactive near zone:
ikR 3 3
G
p =
-p(z - Z l e
R
[1
+
k ~ - kRI)2 1 7 )
In most practical problems, we are interested in the range k R =
27r~
1. In this range, the last 2 terms of the exact solution can be neglected.
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3 Transverse Green Function in the Fernel
and Fraunhofer Limits
Estimation of the Green function in the Fresnel and Fraunhofer limits will
be performed in order to simplify these results. The Fernel limit is usually
defined for a planar source, therefore, we need to modify the condition. De-
riving these limits are based on the expansion of the R term to the second
order in ~ [? ]
Longitudinal Fresnel Near Zone Limit
Where we defined r = J(z 2
+
p2 ). Now, defining
cose
= ~ and using the
known expansion (1
+
E 4
=
1
+
~ E - k E 2 we write:
z 1Z,2
R ~ r[ l - - cosB
+ --
sin
2
B] (9 )
r 2 r2
This result will now be substituted into the Green function phase. In the
denominator we will substitute the first order
R ~ r .
Nowwe can write the
transverse Green function in Fernels approximation:
eikr
12
G
p
= -sin((} )cos((} )_e-ikzzl+ikp~
r
10
Where kp
=
k sinB and kz
=
k cosB .
Longitudinal Fraunhofer Far Zone Limit In the Fraunhofer far zone
limit, the quadratic term of the phase is negligible. According to our deriva-
tions, the condition for this limit is:
- .
The transverse Green function in the Fraunhofer limit is then:
(11)
ikr
G = - sin((} )cos((} )~e-ikzZI
p
r
12
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4 Ginzburg s Formation Zone
Any radiation, including TR, with a known wave length is formed in a
region and not in a point. This zone is considered as the forming zone
according to Ginzburg f l . The formation zone size is determined by its typical
wavelength. The size of this formation zone is called the formation length
and is marked as
L
f. For a relativistic particle, this length is growing as
the particles energy increases. This length can be thought of as the length
in which the phase of wave emitted by a source, moving at velocity v, at
a certain point differs by
27 r
from the phase of the wave, emitted from this
source, at another point in a distance
L
f. Ginzburgs formation length value
is
13
In our Green function calculations, the integration was supposed to be
from 00 to 00but for these reasons, we can ignore the contribution of
L > Lj, since most of the radiation will be included in this region anyway,
and less calculations can be made in the numerical integration. We will
investigate this case for the exact solution with different lengths in units of
the formation length and verify the convergence of this solution.
5 Transition Radiation Picture
Transition radiation (TR), is emitted when a relativistic electron passes a
boundary between materials of different electrical properties. When this
boundary is a conductive foil, this picture can be replaced by 2 particles
moving towards each other with the same velocity, one is the electron and
the other is an image charge with opposite charge. If the foil is located at
the z =0 plane, the electron is coming from 00with velocity v, we can
define the charge density in space as
J r t)
=
J(x - xo)J(y - Yo)[-evezJ(z - v(t - to) - evezJ(z + v(t - to))]x
x { 1 -
T/(t - to ) Z
< 0 -
refl ec tion }
TJ( t - to ) Z > 0 - transmission
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Where the T J function is defined as
T J t _ t o ={ 1 t > t o }
t < t o
The frequency components of the charge density can be derived using
Fourier transform on the above expression. The result is divided into two
cases: forward TR and reflected TR:
15
J;ef(r,w)
=
-ee b(x -
xo)b(y - Yo) {
J;;r,w)
= -ec b( x - xo)b(y - Yo) {
i~z
e
v
z < 0 }
Z
z < o }
z > o
16
W
tZ
e v
W
tZ
e
v
i~z
e
v
1 7
Transverse Electric Field - Exact Solution Transverse Green function
solution was evaluated in the previous chapter. We can estimate the radiation
emitted from such electron by substitution of this solution in equation 1.
. 0 ikCR +)
v _ uaue iwto
8 8 [ 1 . ) )
e fJ
1
e, -
-8 -88m e cos j
R
dz +
k p Z _ 1
z
ikCR-:.)
(z
e
f J
+
o sin({jl)C O S 1 I
R dzJ
18
This is the exact solution, and by substituting the solution we already
obtained after performing the derivations, we can write the full solution as:
v _
iwto 1
0
p(z - z)
3i 3i
ikCR -:.) 1
Ep -
-zw f.-L e e
[_
R 3
1
+ kR - (kR I)2 )e
f J
dz +
Z
2 p Z_ Z I) 3i 3i )ikCR +)
+
o
R 3
1
+ kR - (kR )2
e
f J dz
19
As was mentioned before, in most practical cases we can use the
kR
> > 1
approximation. We derive the solution for the Green function in this approx-
imation, and the resulted field is:
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10~
N=
N=2
N=5
N=O
N=1Il
-0. 5
o
X [ m J
0. 5
x
1O ~
Figure 1: Transverse electric field amplitude for ll-:rn OTR at a distance
of Imm. Different curves are for increasing integration length in Forming
length units
1
0
p
I) , 1
I) ,
if
=
-iw,uee
iwtO
[ z - z eik(R -t) dz + p z - Z eik(R +t) dzJ
p L R3 R3
2
20
Figure 1 presents the results of OTR emission from a single electron,
in l,uTn
wavelength, at a distance of
1mm.
The Green function is solved
using the exact solution. The different curves are for different integration
limits with units of Ginzburgs formation length. A clear convergenceas the
integration length increases can be seen.
Transverse
Electric Field Far Field
The transverse Green function
in the longitudinal Fraunhofer approximation was
e
ikr
G
= -
sin(8
1
cos(8 )_e-
ikzz
(21)
P r
Using the same method as in the previous chapter, we can calculate the
transverse electric field component:
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v
t[
1 1 ]
p =
- iw f.- L e e
2W
0
0 + 0
sin(B) cos(B)
/, . : : . -k /, .: : .+k
v v
Substituting
kz = .: : .
cos(tJ) we find the far-field approximation for the
c
electric field:
(22)
p =
-i f.- L c e e iw to [ 1 1 ( )
+
1 1
0 ]
sin(B) cos(B) (23)
7 3 - cos
0
7 3
+
cos
8
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