exact theory of otr and cotr

7/21/2019 Exact Theory of OTR and COTR

1/8

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

14

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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

exact theory of otr and cotr

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