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Q ui ck Ti m e ™ an d a T I FF ( U nc om p r es se d) de co m pr e ss or ar e n ee de d t o s ee t h is pi ct u re .
RF EFFECTS IN PHOTO-INJECTORS
Massimo Ferrario
INFN-LNF
solenoid
Zb
B
z
E
Ez
z
z
E

Z1

Z2

Z


C
Madison, June 28 - July 2
• Trapping a plane wave in a box
• Longitudinal RF effects
• Transverse RF effects
The solutions to the electromagnetic wave equation in free space are
transversely polarized waves (the electric field is transverse to the propagation
vector) that have phase velocity c, the speed of light. These properties are
problematic from the viewpoint of charged particle acceleration, because in
order for a charged particle to absorb energy from an applied electric force, the
motion of the charged particle must have a component parallel to the electric
field:
 
dU
 q v E
dt
If the motion of the particle in an accelerating wave is rectilinear in the zdirection, the electric
 field must be rotated to have a longitudinal component in
order for acceleration to occur. This can be accomplished by using a smoothwalled waveguide.
Let us consider a charge co-propagating in the z direction with a
wave having a longitudinal component Ez
v z  c
Ez,t   Eze
the particle experiences an accelerating voltage

z  ct
Vacc  Re
Vacc  Re


itkz
L
i t kz o 
Eze 
dz
o
Ee
L
 z

i  kz o 
 c

z
dz
o
The energy gain depends on the spatial pattern of the field and on
the phase relation (phase slippage)

The wave is syncronous when


k
 c
Vacc  Re

L
Ez z e i o dz
o
Interaction with a plane wave: particle at rest
y
Ex  cBy
x
B
H

 0
E
E
z

Non relativistic approx:

x

e
Ex  v z By coskz  t 
mo
e
ÝzÝ
v x By coskz  t 
mo
ÝxÝ 
z
Interaction with a plane wave: particle/wave co-propagating
y
Ex  cBy
x
B
H

 1
E
E
E' (Ex 0,0)
z
 


E    E '  v  B'  Ex  E'x


v


B   B'  2  E '  By   E'x   Ex


c
c
c
a
-1
b
  1

1
2
 1
1
2 2

F e Ex  cBy cost  kz
FE
E'
using
kz  kct  t
-1
FB=FE
 e1   E
x
cos 1   t 
  
eEx
 2 cos 2 
2
2 
Interaction with a plane wave with an angle
Wave
front
x
z
Ez  Eo sin  e 

i t k z cos  y sin 
vz 

kz


k cos

c
c
cos
Not yet suitable

  1   cos  
t

Plane wave reflected by a perfectly conducting plane
xy
E+
E-
H+
H-
'

 
zz
'



In the plane xz the field is given by the superposition of the incident and reflected
wave
Ex,z,t   E x o ,z o ,t o e itik  E x o ,z o ,t o e itik '
  z cos  x sin 
'  z cos '  x sin  '
 E and H // only
Boundary conditions require that
E,H
H//

E
n
H
E //
x
Ez 0,z   E sin e ikzcos  E sin ' e ikzcos '  0
  
E   E
z
Taking into account the boundary conditions the longitudinal
component of the field becomes
Ez x,z,t   E sin  e
itikz cos x sin 
 E sin  e
itikz cos x sin 
ikx sin
ikx sin
e

e
 E sin  e itikz cos 2i
2i
 2iE sin  sinkx sin  e itikz cos
x-SW
pattern
z-TW
pattern
xy
2d
EEz
Notice that
Ez a,z,t   0
Not only on the conductor:
a  nd

d

2 sin
E+
d
E+
d/ 2
Ez
E-
- 1
z
We can confine the plane wave with a second parallel conducting
plane located where Ez = 0 so that boundary condition are fulfilled
Ez a,z,t   0
x
n
k sin  
a

n= 1
n= 2
n= 3
z


nx  itkz z
nx 
Ez x,z,t   ReiA sin
e
sint  kz z
 A sin
 a 
 a 


A  2E sin
kz  k cos 
This is a guided plane wave also called TMn mode

n indicates the number of half 
wavelength between
the two plates
Not all angles  are possible for a given distance a between the two
plates
d

2 sin
If a would be
sin 
==>
a
c
then

c
2a

2
Implying a normal incidence and
 no propagation
2
Cut-Off condition:Only
waves with < c = 2a can propagate



sin 
1
c
The phase velocity is given by


c
vz  


kz k cos cos 
c

1    c 
2
c
1  sin 2 
> c if  < c
Im if  > c
The previous relation is identical to

  ckz    c2
2
2

=kzc

That is a dispersion relation
c
kz
We must slow down the wave propagation
In order to slow down the waves we have to load the cavity
by introducing some periodic obstacle into it
d
z

Ez (z)  E0 Im

n
 2 n    
an expi
z 
d


With this general form of the solution, the field can be viewed as the
sum ofmany wave components, which are termed spatial harmonics,
having different longitudinal wave-numbers kz,n  2 n  / d ,
and thus different phase velocities .
v ,n   / kz,n

Beam Dinamycs in Photo-Injectors:
RF effects
Micro-Bunch Production with rf photo-injectors
Luca Serafini (INFN - Milan)
Workshop on 2nd Generation Plasma Accelerators,
Kardamyli, Greece, June 1995
Longitudinal RF effects
solenoid
Zb
B
z
E
Ez
z
z
z exit  N  1 / 2
E

Z1

Z2

Z


C

2

Ez  Eo coskz sint   o 
Eo peak accelerating field
o phase as the particle lives the cathode surface

Let us try to compute the energy gain for a particle with =0
N 1 / 2 / 2
T  eEo
 coskzsint   dz
o
0
1
kz  kct 

t
Impulsive Approximation to Electron Capture:
assuming a constant electric field nearby the cathode surface (z~0)
Ez  Eo sin 
The longitudinal equation of motion becomes
d
eEo sin


 //  
dt
mc
assuming

1
1   //2
d // eEo sin 

1   //2
dt
mc
eEo sin t
 // 
eEo sin t 2
mc 1  

mc



giving


2


eEo sin t
mc


z
1


1



eEo sin  
mc






2

eEo
Introducing the dimensionless vector potential amplitude 
2mc 2 k
and

1  2kct sin  
2
We obtain
˜z 
2 sin 
1

2 sin 
Wich has an asymptotic behavior

1
˜z 
kct 
2 sin 
t 
 
 1
˜z  kz
Defining the phase slippage as
     o  t  kz
 ˜z 
  t  
 kc 
It has an asymptotic behavior given by

1
     o 
2 sin 
Which has a minimum at = /2 that is also the phase of maximum
acceleration and it is small if > 1/2
( ex: n= 2.856
 GHz, Eo = 100 MV/m ==> =1.63 )
Assuming that the photo-electrons are emitted from the cathode
directly at the speed of light, at a phase equal to the asymptotic
phase, the energy gain is simply given by
N 1 / 2
T  eEo
 cos˜zsin˜z   dz
0
  1
T
 1  2
2
mc
N 1 / 2
 cos˜zsin˜z   d˜z
0
  1   N  1 / 2 sin    cos 
where the non relativistic part of the motion has been considered
equivalent
to a new definition of the injection phase

Phase Compression







Defining
o
o
o as the phase separation between two
different photo-electrons emitted at two launching pahse, their
asimptotical distance will be given by:

               o
d

 o   o
d
By defining the relative change in phase distance between the two
electrons
d 



 
 1
 o
d
cot 
 1
2 sin 

The final length of the emitted electron bunch will be given by:
cot 
Lbunch  1 
Llaser
2 sin 
The injection phase o which corresponds to an asymptotic phase
 treshold below which we have Phase Compression
= /2 is the
and above which Phase Expansion.
For injection phases close to o = 0 the Compression can produce
bunching even down to a factor 4
Energy Spread
At the gun exit we have
  1   N  1 / 2sin  cos 
substituting
    

II order expansion
around <>
sin      sin    cos  
 2
cos      cos    sin  
 2
 1
2 
  1    N  1 / 2 sin   cos  1   
 2


  N  1 / 2cos   sin  
2
2
sin 
cos 
We are interested in the phase


giving
2
 1
2 
      1   N  1 / 21    
 2


and
  1   N  1 / 2
1
       1 2
2
So that
Or in terms of the rms quantity

  
 
2
     k z
 
 
2
 k z
Electron phase, energy at gun exit (2nd iris): simulations vs. analytical
Exercise: show that the Longitudinal emittance
is given by
where
In particular for a Gaussian distribution show that
The fields that accelerate the charged particles in a radio-frequency
linac cavity also give rise to transverse components of the Lorentz
force that deflect the particles.
Transverse RF effects
Assuming cylindrical symmetry, transverse electromagnetic
fields in an accelerating structure can be obtained by a linear
expansion near the axis of the Maxwell equations:
r
 D   e
Er  
1
r

Ez
z
0
D
 H 
 Je
t
r

B 
1
rc 2

0
˜rd˜r  
r0
Ez
t
r Ez
2 z
˜rd˜r 
r0
r0
r Ez
2c 2 t
r0

The radial component of the Lorentz force becomes

Fr  q Er  v0 B
qr Ez
  
2  z
qr dEz

2 dz
qr Ez
  
2  z

1 Ez

r0 c t
0 Ez

c t
r0


r0 


r0 
(v0  c ).
r0
Thus in the ultra-relativistic limit , the net radial Lorentz force is
simply proportional to the total z-derivative of the accelerating
field.
Considering as a longitudinal component
Ez r,z,t   Ez z  sint   o   Eo coskz sint   o 
r dEz z 
Er r,z,t   
sint   o 
2 dz
r k
B r,z,t  
Ez z cost   o 
2c

Tc
Tnc
Symmetric structure
If we integrate the radial forces through an isolated electromagnetic
structure with constant velocity and radial offset we obtain for the
full momentum transfer
er
pr 
2mc 2


Tnc
Tc
dEz
er
dt 
E T  Ez T c 
2  z  nc 
dt
2mc
er
E sin  2n   Eo sin   0
2  o
2mc
This result explicitly shows that the sum of all inward and outward
impulses applied within the structure cancel for an ultra-relativistic

particle
traversing a cylindrically symmetric structure.
But in the last half cell
Anti-symmetric structure
Tc
er
pr 
2mc 2

Texit
Tc
z
Texit
dEz
er
dt 
E sin   kr sin 
2 o
dt
2mc
This represents a defocussing kick when 0 < < 

r 
pr
2
 10 mrad @ S  band, 100 MV / m
Thus it usually will be necessary to focus the beam immediately
after leaving the cavity
Transverse RF emittance
Rewriting pr in Cartesian coordinates:
It gives the phase space distribution: a collection of lines with
different slopes corresponding to different 
The normalized transverse emittance is:
By inserting px we obtain
and assuming that  is small
writing
so that
sin      sin    cos  
sin
2

 2
2
sin 
    sin    2cos  sin  
one obtains from:
2
 2
2

cos 2   sin 2 

It has a minimum for <>=/2
 
Away from the minimum we have

2

 


2
For a Gaussian beam distriution
x, y, z  
1 x 2 y 2 z 2
 
 2
2
2 
z
  x
1
2 
3/2
 x2 z
e
The rms quantitie are
 2  k 2 z2

 4  k 4 z4
And the emittance becomes:

rf
 xmin

k 3 x 2  z2
2




Exercise: show that for a uniform distribution in a cylinder of
radius a and length L, the relevant moments for the distribution
are:
And the transverse emittance is:
Second order transverse focusing in electromagnetic
accelerating fields
Tc
Tnc
Symmetric structure
if one relaxes the assumptions of constant velocity and offset from the
design orbit and examines the effects of the alternating gradient forces,
one finds that they give rise to a second order secular focusing force
2
x  0
sinkpz x  0
1.50
1.00
x
0.50

0.00
-0.50
-1.00
-1.50
0.0
1.0
2.0
3.0
k z
4.0
5.0
6.0
sec
The approximation we will employ here assumes that the motion
can be broken down into two components, one which contains the
small amplitude fast oscillatory motion (the perturbed part of the
motion), and the other that contains the slowly varying or secular,
large amplitude variations in the trajectory.
x  xosc  xsec
The oscillatory component is analyzed by making the approximation
that the offset x=xsec is constant over an oscillation so that:
2
   0
xosc
sinkpz xsec  0
Which has a simple solution

 02
xosc  sinkp z 
2
kp
xsec
The original equation becomes

2 


x  02 sin(k p z)x   02 sin(k p z)1  sin k p z 02 x sec
k p 



 
The last step is to convert it into an averaged expression over a
period, that gives the behavior of the secular component of the
motion
xsec  x
''
x sec

2 

2
  0 sin(k p z)1  sin k p z 02  x sec
k p 



 


 
xsec
4
0
2
2kp
xsec  0
d
z
Simple Fourier decomposition of the on-axis solution
then gives the useful form

Ez (z)  E0 Im

n
 2 n    
an expi
z 
d


With this general form of the solution, the field can be viewed as the
sum of many wave
 components, which are termed spatial harmonics,
having different longitudinal wave-numbers kz,n  2 n  / d ,
and thus different phase velocities . v ,n   / kz,n

we assume the n=0 spatial harmonic is synchronous with the
relativistic particle motion, or  / kz,0  c
For a particle located at a phase  with respect to maximum
acceleration, we have




Ez  E0Im
an e i (2k0 nz n ) 


 n



q dEz
F  
2 dz
 0



 qE0 Im
ik0 nan e i (2k0 nz n ) 


 n


If the energy variation is ignorably small over a period, then one
may proceed to find the secular radial equation of motion by
averaging the lowest order oscillatory motion given by the previous
equation over a structure period.
 
osc
F
   sec
 2m0 c 2



qE  
i (2k0 nz n ) 
  0 sec
Im
ik
na
e
0
n

m0 c 2 
 n


witch has a steady-state solution

 osc




qE0 
an i (2k0 nz n ) 

  sec 1 
Im
e
2 

ik0 n
4

m
c
0

 n




Substitution of the value of  osc and averaging, we have
  
sec
F
 2m0 c 2

1  qE0 
  
  sec
2
8 m0 c 
2


2
a n2  a n
 2a n an sin(2 )

n1
This radial focusing force, like that derived from the solenoid

provides
equal focusing in both x and y, which is second order in
applied field strength . In the case of the solenoid, the net radial force
is second order due to the accompanying rotation, while in the present
case it is of second order because of the fast radial oscillatory motion
due to alternating gradient focusing

 ( ) 

2
an2  an
 2an an sin(2 )

n1
represents a sum over all spatial harmonics that contribute to the
alternating gradient force, and is equal to 1 for a pure harmonic
standingwave. Note that the synchronous harmonic (n=0) does
not contribute to the first or second order force in the ultrarelativistic limit, so that =0 for a pure forward traveling wave
(ao=1, all other an vanishing).
the alternating gradient focusing effect arises from the existence of
non-synchronous spatial harmonics
Because we have kept the energy constant, we have not kept the
effects of adiabatic damping in the equation of motion. Acceleration
can be taken into account by use of the damping term.
Because the focusing is symmetric, the resulting equations of
motion in x, y and  are all equivalent, and are of the form

   2
x   x 
  x  0
2
 
8 sin   
