Coherent synchrotron radiation: theory and simulations .
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Transcript of Coherent synchrotron radiation: theory and simulations .
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1 Sasha Novokhatski October 4, 2011
Coherent synchrotron radiation:theory and simulations.
Sasha Novokhatski
ICFA Beam Dynamics Mini Workshop on Low Emittance Rings 2011
October 3-5, 2011 Heraklion, Crete, Greece
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CSR images
LCLS: CSR light on the YAG screen, when the electron bunch is bent down
CSR
Incoherent light
Radiation from a horizontal bend
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Theory of CSR
In 1949 J.Schwinger published a paper “On the Classical radiation of Accelerated Electrons” (Phys. Rev. v. 75, Num 12, 1920 ), where he presented his approximation for the spectrum of the synchrotron radiation for the circular trajectory of an electron
42/3 2
05/32 2
/
34
cc
e EP K dmc
introducing the critical frequency3 3
0 2 2
3 32 2c
E c Emc mc
He verified that his approximation gives the total power
42
0 20
23
e EP t P dmc
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I had a chance to check E^4
Measurement at PEP-II HER during the energy scan in 2008
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SR spectrum
For small frequencies power is independent of the electron energy
1/31/ 6 2 2
3
0
3
4eP
c
For high frequencies power exponentially decreases
1/ 21/ 2 42
02
3 3 exp4 2 c c c
e EPmc
c
CSR-impedance ?
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Critical frequency and power loss
We know that a bunch of electrons with a finite length may coherently excite electromagnetic fields if their frequencies less than a bunch (critical) frequency
bc
Comparing the critical frequencies we may introduce an equivalent bunch length for SR fields
3
2
32c b SR
Emc
Now we rewrite the formula for the SR power
4/3 4 /342 2 2
1/30 02 2
2 2 3 2 33 3 2 2SE SE
e E Q QP tmc
Schiff, Nordvick, Saxon, Murphy, Derbenev, …
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7
24 3 3
2 201/3 2/ 33 2 2
1 1,3,..
sin 32, ,3 3 33
j
pp
nj j j
n j
j
nZ c
K h K Kh n n nn
jh
CSR shielding in 1954
Murphy, Warnock, …
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It can be a very strong shielding
hyperbolic sine
s
122 12
inh( )pp
FS
hP
P
h
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CSR simulations
• R. Li. Nucl. Instrum. Meth. Phys. Res. A, 429, 310, 1998.• G. Bassi et al., Nuc. Instrum. Methods Phys. Res. A, 557,
pp. 189–204 (2006).• M. Borland, Phys. Rev. ST Accel. Beams 4, 070701
(2001).• G.V. Stupakov and I. A. Kotelnikov, Phys. Rev. ST Accel.
Beams, 12, 104401 (2009).• T.Agoh and K.Yokoya, Phys. Rev. ST Accel. Beams, 7,
054403 (2004).• …
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Time-domain presentation
A bunch and radiation are moving together for a long time. How do they separate? A chamber wall cuts the field?
How does the bunch field change when the bunch has been rotated in a magnetic field?
A bunch retarding causes the field radiation?
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Recently a new method was developed at SLAC
• The method is based on an implicit scheme for solving the electromagnetic equations, Maxwell’s equations.
• This algorithm is free of frequency dispersion effects which means that all propagating waves will have their natural phase velocity, completely independent of simulation parameters like mesh size or time step.
• Other known methods, usually explicit, have “mesh driven” dispersion and because of this they need a much smaller mesh size which slows down calculations and can sometimes cause unstable solutions.
• An implicit scheme is a self-consistent method that allows us to calculate fields of much shorter bunches.
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First results: Field dynamics in a magnet
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Movie showChamber wall
Chamber wall
Initial direction
v
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Movie showChamber wall
Chamber wall
Initial direction
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Bunch self-field remakes itself moving in a magnetic field
V
The upper field lines take the position of the lower lines
The white box shows the bunch location
The red arrow shows the bunch velocity vector
Green arrows show field line directions
The lower field lines take the place of the upper lines
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The picture becomes clear if we decompose the field
= +
a field of a moving dipole
a field of a bunch moving straight in initial direction
Decomposition of the field of a bunch moving in a magnetic field into two fields:
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Detailed plot of a dipole field
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Comparison with a classical synchrotron radiation
3
23
(=6)
- region in front of a particle
equivalent bunch length
for a bending radius
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An absolute dipole electric field in time
When a dipole is created an electric field appears between a real bunch and a virtual bunch. This field increases in value and reaches a maximum value when the bunches are completely separated and then it goes down as the bunches move apart leaving fields only around the bunches.
The white oval shows the real bunch contour.
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Electrical forces inside a bunch
Bunch shape collinear force
eF J E
ti
me
transverse force
| |JF E JJ
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Electrical forces inside a bunch
Bunch shape collinear force
eF J E
The transverse force is the well known space-charge force, which probably is compensated by a magnetic force in the ultra-relativistic case.
The collinear force is responsible for an energy gain or an energy loss. The particles, which are in the center, in front and at the end of the bunch are accelerating, whereas the particles at the boundaries are decelerating. The total effect is deceleration and the bunch loses energy, however the bunch gets an additional energy spread in the transverse direction.
transverse force
| |JF E JJ
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A new bunch field
A long way to a steady-state regime
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Coherent edge radiation
Bunch trajectory
Initial beam direction
Some fields propagate along
initial beam direction
Image of the magnetic field
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Magnetic field plots
Magnified
Initial beam direction
Current beam direction
Bunch field
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Images of radiation (transverse magnetic field)
Bunch field
Synchrotronradiation
Edgeradiation
Z=0.73 m Z=1.0 m
very similar to the images, which we have seen on the YAG screen after the dump magnets, which bend down the beam at LCLS.
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0 50 100 150 200 250 300 350 400
-5000
-4000
-3000
-2000
-1000
0
1000
2000
Signal
cT/h
Hx
0 50 100 150 200 250 300 350 4000
50
100
150
200
250
300
350
0
500
1000
1500
2000
2500
3000
3500
Spectrumradiation bunch
wavelength/hra
diat
ion
bunc
h
Signal and spectrum of edge radiation
First peak in spectrum corresponds to and is a little bit less than the bunch wavelength = 2πσ
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PEP-II LER pumping chamber
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PEP-II LER dipole (a model)X beam position and magnetic field
0 100 200 300 400 500 600
-1.5
-1
-0.5
0
0.5
1
1.5
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
teta X magnetic field
distance [mm]
X-p
ositi
on [m
m],
X' [
degr
ee]
mag
netic
fiel
dn [G
auss
]
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Beam and chamber fields
Beam direction
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Magnified by 10 times
Beam direction
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Magnified by 100 times
Beam direction
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Magnified by 1000 times
Beam direction
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Beam power loss
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Beam energy loss
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Spectrum of a 1mm bunch
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Energy loss along a bunch
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Next step: combine CSR and Fokker-Planck calcultions.
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The author would like to thank
Mike Sullivan and R. Clive Field
for help and valuable comments;
Franz-Josef Decker, Paul J. Emma and Yunhai Cai
for support and interest in this work;
Physicists of the SLAC Beam Physics Department
for useful discussions.
Acknowledgments