1D-FEL Without Approximations
motivation, capabilities
1D theory  1D-solver for waves
implementation (without and with Lorentz transformation)
excitation of waves (single particle)
without self effects
one and few particles with self effects
mystery
Motivation, Capabilities
FEL codes use many approximations, as averaged equation of motion,
local periodical approximation, EM-field calculation by paraxial approximation,
one or several harmonics
these approximations are questionable for ultra-short bunches or bunches
with extreme energy modulation
it is easy to implement complete FEL effects in 1D
1D model can be used to verify approximations
capabilities of 1D model: ultra broadband (not split into harmonics, excitation
and radiation)
no local periodic approximation
complete 1D field computation
seems possible: particle = macro particle
LT method can be tested
1D Theory
EM Fields
a) undulator field
b) external wave
c) longitudinal self field
E z z   z , t  
q
 z  z 

A 
no principle problem, but neglected
d) transverse self field
J x z, t  
q
 vx,  z  z 
A 
E x , z   B y ,t
 B y , z  J x  c  2 E x ,t

E x  L  R  2
cBy   L  R  2
L and R are waves to the
left and to the right

~
RU  ct , t   R U , t 
~
LV  ct , t   L V , t 
with
~
 t R   1 J U  ct , t 
~
 t L   1 J V  ct , t 
waves to the left and to the right
Implementation
Equation of Motion
d  r   v 
 


dt p   f 
with
v 
p
c
m0
p
p2  m0 c 
2
f  q0 E  v  B 
horizontal plane, only x and z components:

~ ~
~ ~
 E x , e  v z B y , e  q0  R  L   z R  L
 Fx 
  
   q0 
~ ~


v
B
F
2

R
L
x
y
,
e
 z 


x


external
Coupled Problem
J x z, t  
q
 vx,  z  z 
A 
v
 r  


 

f
d  p  


~
1
dt  L V , t     J V  ct , t 
~
  1





R
U
,
t


J
U

ct
,
t

 



self
 binning and smoothing (on equi. mesh)
needs spatial resolution
 PDE solver (f.i. RK4)
needs time resolution


without Lorentz transformation
integration of PDE (by rk4) needs small time step for left wave
criterion: slip between source and wave < dz/c
 dt < dz/c, number of time steps ~ undulator length / photon wavelength !!!
solution a: neglect L, maxL  
maxR 
4 2
solution b: the part of the left wave, seen by the bunch, is determined by near
interaction; use J x z, t   J x z  v  , t   
solution c:
with Lorentz transformation
differenced between length scales are shrunk
huge external fields (from undulator)
same magnitude of left and right wave
it is possible, it is applicable even in 3D!
 E x ,u ' 
  c LT 

   LT 
 By ,u
 B '
1


 y ,u 
Excitation of Waves (single particle)
left and right wave
weak backward wave
L
intense fwd. wave !
R
w  6 cm
w  77 nm, 3  77 nm, 5  77 nm 
left and right wave with Lorentz transformation
less power in fwd. direction !
L
R
w  0.67 mm
higher harmonics:
m  2n  1
 mK 2 
 mK 2 



JJ  J n 
 J n 1 
2 
2 
4

2
K
4

2
K




Example: Without Self Effects
before undulator
Bu = 1T
u = 3 cm
E = 500 MeV
Nu = 10

K = 2.80
w = 77 nm
pz

same parameters,
with Lorentz transformation
frame = initial velocity
start
E/E0 = 978 = LT
pz

same parameters,
with Lorentz transformation
frame = av. velocity
start
av = 441 = LT
pz

Why are left and right waves asymmetric?
L
R
One and Few Particles with Self effects
parameters as before
particle after undulator
longitudinal momentum
left and right waves
L
left wave out of window
pz
R
pictures from solutions (a) and (b) cannot be distinguished by eye!
excitation
of waves
parameters as before, with Lorentz transformation to frame = av. velocity
particle after undulator
L
pz
R
pz
parameters as before, with Lorentz transformation to frame = av. velocity
transformation back to lab-frame
pz
direct calculation in lab frame:
2700 time steps
pz
9600 time steps
parameters as before, with Lorentz transformation to frame = av. velocity
two particles, separated by one photon wavelength
pz
left wave vs. z:
L
right wave vs. z:
R
LT^-1
pz
particle 2
-1
particle 1
 -2.8
parameters as before, with Lorentz transformation to frame = av. velocity
two particles, separated by half photon wavelength
pz
left wave vs. z:
L
right wave vs. z:
R
LT^-1
pz
particle 1
particle 2
Mystery
In the frame “av. undulator velocity” the energy loss to both
waves (left and right) is about equal.
It seems the effect from both waves to the one-particle
dynamic is similar.
In the rest frame the effect of the left wave seems negligible.
What happens if we neglect the left wave in the frame “av.
undulator velocity”?
example as before, with Lorentz transformation to frame = av. velocity
no stimulation of left wave:
L
v
 r  


 

f
d  p  


~

0
dt  L V , t  
~
  1

 R U , t    J U  ct , t 
pz
pz
LT^-1
R
for comparison: complete calculation
L
R
pz
pz
Why is L negligible in the frame “av. undulator velocity”?