USPAS course on - LEPP
Transcript of USPAS course on - LEPP
USPAS course on
Recirculated and Energy Recovered Linacs
Ivan Bazarov, Cornell University
Geoff Krafft, JLAB
Synchrotron Radiation Effect on the Beam
January 26, 2008 USPAS’08 R & ER Linacs 2
Contents
• Coherent (CSR) vs. incoherent (ISR) radiation
• ISR: Energy spread
– Dipole magnets
– Insertion devices
• Emittance growth
– Dipole magnets
– Insertion devices
– Examples of calculations
• Coherent synchrotron radiation
– Radiative component of the electric field
– Beam energy dependence
– Shielding effect
January 26, 2008 USPAS’08 R & ER Linacs 3
Quantum excitation
Quantum excitation in accelerator physics refers to
diffusion of phase space (momentum) of charged particles
(mainly e+/e–) due to recoil from emitted photons.
Because radiated power scales as ∝ γ4 and critical photon
energy (divides synchrotron radiation spectral power into
two equal halves) as ∝ γ3 , the effect becomes important at
high energies (typically ≥ 3 GeV) in electron accelerators
unless bunches are extremely short.
January 26, 2008 USPAS’08 R & ER Linacs 4
ISR vs. CSR
Incoherent synchrotron radiation (ISR): λ << σz. The
radiation power scales linearly with the number of
electrons).
Coherent synchrotron radiation (CSR): σz ≤ λ. When
radiation wavelength becomes comparable with the bunch
length (or density modulation size), the radiated power
becomes quadratic with peak current. The effect can be
important at all energies if the bunch is sufficiently short.
=
I = I0×Ne
=I = I0×Ne
2
January 26, 2008 USPAS’08 R & ER Linacs 5
Circular traj. in the “rest” frame
Lorentz back-transform 4-vector ),( kcr
ω
January 26, 2008 USPAS’08 R & ER Linacs 6
Recoil due to photon emission
hν
ecB
Eβρ =
ecB
hνβρ −
Photon emission takes place in forward direction within a
very small cone (~ 1/γ opening angle). Therefore, to 1st
order, photon removes momentum in the direction of
propagation of electron, leaving position and divergence
of the electron intact at the point of emission.
January 26, 2008 USPAS’08 R & ER Linacs 7
Energy spread
Synchrotron radiation is a stochastic process. Probability
distribution of the number of photons emitted by a single electron is
described by Poisson distribution, and by Gaussian distribution in
the approximation of large number of photons.
If emitting (on average) Nph photons
with energy Eph, random walk growth
of energy spread from its mean is
If photons are emitted with spectral
distribution Nph(Eph), then one has to
integrate:
22
phphE EN=σ
random walk
∫= phphphE dEENE )(22σ
January 26, 2008 USPAS’08 R & ER Linacs 8
Spectrum of SR from bends
Photon emission (primarily) takes place in deflecting
magnetic field (dipole bend magnets, undulators and
wigglers). Spectrum of synchrotron radiation from bends
is well known (per unit deflecting angle):
ργω
ω
ω
ω
ωγ
α
ψ
/2
3
9
4
3c
Se
I
d
dN
c
c
ph
≡
∆=
January 26, 2008 USPAS’08 R & ER Linacs 9
Energy spread from bends
∫=ρ
γπ
σ γ
dsmccCE
7422 )(332
55h
Sands’ radiation constant for e–:
For constant bending radius ρ and total bend angle Θ (Θ =
2π for a ring) energy spread becomes:
Radiated energy loss:
3
5
32 GeV
m1086.8
)(3
4 −⋅==mc
rC cπ
γ
πρ
σ
2)m(
1)GeV(106.2
22
5510
2
2 Θ⋅= −
EE
E
πργγ
2
4 Θ=
ECE
πργ
2)m(
)GeV(0886.0)MeV(
44 Θ=
EE
January 26, 2008 USPAS’08 R & ER Linacs 10
Energy spread from undulator
222 )( γεσ phphphphE NdEENE ≈= ∫
)1(
22
21
2
K
hc
p +=
γ
λε γ
)1)(cm(
)GeV(950)eV(
2
21
22
K
E
p +=
λε γPhoton in fundamental
Radiated energy / e–
Naively, one can estimate
22
222
3
4
mc
LKErE
p
uc
λ
πγ = )m(
)cm(
)GeV(725)eV(
22
222
u
p
LKE
Eλ
γ =
)m()cm(
)1(763.0
2
212
u
p
ph LKKE
Nλε γ
γ +=≈
)m()1)(cm(
)GeV(107
2
2133
22213
2
2
u
p
E LK
KE
E +⋅≈ −
λ
σ
January 26, 2008 USPAS’08 R & ER Linacs 11
Energy spread from planar undulator
)m()cm(
)()GeV(108.4
33
22213
2
2
u
p
E LKFKE
E λ
σ −⋅≈
12 )4.033.11(2.1)( −+++≈ KKKKFwith
In reality, undulator spectrum is
more complicated with
harmonic content for K ≥ 1 and
Doppler red shift for off-axis
emission.
More rigorous treatment gives
ω/ω1
Np
h∆
ω(a
.u.)
January 26, 2008 USPAS’08 R & ER Linacs 12
Emittance growth
Consider motion:
where
As discussed earlier, emission of a photon leads to:
changing the phase space ellipse
E
Exx x
∆+= ηβ
E
Exx x
∆′+′=′ ηβ
)()(
si
xxxesax
ψβ β=
E
Exx
E
Exx
ph
x
ph
x
ηδδ
ηδδ
β
β
′+′==′
+==
0
0
E
Ex
E
Ex
ph
x
ph
x
ηδ
ηδ
β
β
′−=′
−=
222 2 ββββ βαγ xxxxa xxxx′+′+=
)(2
2
2 sHE
Ea x
ph
x =δ here22 2 xxxxxxxxH ηβηηαηγ ′+′+=
January 26, 2008 USPAS’08 R & ER Linacs 13
Emittance growth in bend
( )xx
s
si
xxx aesa x ββσ ψ 22)(2
2
1)( ==
phphphphxx dEsHENEdscE
a )()(2
1
2
1 2
2
2 &∫ ∫=∆=ε
In bends:
∫=3
5
22
364
)(55
ργ
πε γ HdsmccC
x
h
πρε
2)m(
)m()GeV(103.1rad)-m(
22
55
10 Θ⋅= − HE
x
January 26, 2008 USPAS’08 R & ER Linacs 14
H-function (curly H)
As we have seen, lattice function H in dipoles (1/ρ ≠ 0) matters
for low emittance.
In the simplest achromatic cell (two identical dipole magnets with
lens in between), dispersion is defined in the bends. One can show
that an optimum Twiss parameters (α, β) exist that minimize
Such optimized double bend achromat is known as a Chasman
Green lattice, and H is given by3
154
1ρθ=H
H
θ θ
January 26, 2008 USPAS’08 R & ER Linacs 15
Example of TBA
mm 6.3=H
mm 1.9=H
mm 66.0≈−GC
H
4×3°-bends
January 26, 2008 USPAS’08 R & ER Linacs 16
Example: emittance and energy
spread in ¼ CESR
Energy = 5 GeV
large dispersion
section for bunch
compression
5 GeV ring (768 m)
FODO lattice
January 26, 2008 USPAS’08 R & ER Linacs 17
Emittance growth in undulator
HE
Ex 2
2
2
1 σε ≈
with energy spread σE/E calculated earlier.
∫ +′+′=uL
u
dsL
H0
22 )2(1
γηηαηηβ
For sinusoidal undulator field
Differential equation for dispersion
ppp kskBsB λπ /2 with cos)( 0 ==
sk pcos11
0ρρη ==′′
Kk p
γρ =0with
January 26, 2008 USPAS’08 R & ER Linacs 18
0
0
2)cos1(
1)( η
ρη +−= sk
ks p
p
For an undulator with beam waist (β∗) located at its center
skk
s p
p
sin1
)(0ρ
η =′
+++≈
++++≈
∗∗
∗
∗∗∗
∗
2
0
22
22
0
2*
2
2
2
222
00
2
2
0
22
0
2*
2
2
0
2
82
121
2
2
1182
121
2
β
γη
β
γη
βγ
β
ββ
ρη
β
ρη
βρ
β
KkK
LK
k
kL
kH
p
u
p
pu
p
Unless undulator is placed in high dispersion region,
contribution to emittance remains small.
Emittance growth in undulator
January 26, 2008 USPAS’08 R & ER Linacs 19
Coherent radiation from a bend
∞→
=
Ω−+=
Ω ∫ NdzzSc
zif
dd
IdfNNN
dd
Idfor ,)(exp)( ,)]()1([
2
0
22 ωω
ωω
ω
rms
( )222 /4exp)( λσπω −=f
Form factor
Gaussian Uniform
)/2(sinc)( 2 λπω lf =
half length
January 26, 2008 USPAS’08 R & ER Linacs 20
CSR: longitudinal and transverse e
ρ
l3SR γ
ρλ ∝
2
4
SR ρ
γNP ∝
3432
2
CSR
1
lNP
ρ∝
l≥CSR
λ
1E-15
1E-13
1E-11
1E-09
1E-07
1E-05
1E-03
1E-05 1E-04 1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+03
wavelength [mm]
P [W
/mra
d/0.
1%] B
W]
σl
λcutoff
CSR enhanced spectrum
( )λSRIN ∗
Longitudinal Effect
)(ct
E
∂
∂
s
tail
head
Transverse Effect
Dispersion ≠ 0
emittance
growth+ +- -
January 26, 2008 USPAS’08 R & ER Linacs 21
Lienard-Wiechert formula: two e–’s
32322)1(
]')[(
)1()(
β
β
β
β
γ ′⋅−
×′−×+
′⋅−
′−= rr
rrvv
rr
rrr
nL
ann
c
e
nL
nePE
β ′r
nr
Coulomb term radiative term
L – distance between P and P′P′ – position of (red) particle at retarded time t′
cLtt /' =−
LLsss
β−=− '
January 26, 2008 USPAS’08 R & ER Linacs 22
Energy loss in a circle
+−
−−+
+−
+−=⋅
32
2
3222
2
2
2
)8/1(2
)1(8/
)8/1(
8/31)(
u
u
uu
u
R
cePEnec
β
β
βγ
ββ
rr
24)1('
3Ru
Russ +−=− β 'φφ −=u
PP′
1<<u
Problem: There is a singularity
when u → 0
Solution: subtract work due to
Coulomb force
22 )'()(
ss
neePEne
sc−
⋅=⋅γ
ββ
rrrr
January 26, 2008 USPAS’08 R & ER Linacs 23
Energy loss in a circle
−+−
+−
+−+
+−
−−=≡
2232
2
2232
2
2
2
)124/(
1
)8/1(
8/311
)8/1(2
)1(8/)(
ββ
β
γβ
ββ
uu
u
uu
u
R
eEeK
CSRCSR
Sagan, EPAC’06, p 2829
)(ct
E
∂
∂
s
tail
head
CSR wake Energy spread
for Gaussian
January 26, 2008 USPAS’08 R & ER Linacs 24
Shielding effect
Radiation at high frequency can be
limited by critical wavelength (low
energy), or by the bunch form factor.
Low frequency content is cut-off by
conducting vacuum chamber (c.f.
waveguide modes).
Schwinger predicted for uniform
bunch of length l reduction in total
CSR power by a factor of
l
h
R
l
23
3/1
infinite conducting plates
h
January 26, 2008 USPAS’08 R & ER Linacs 25
Simulation of shielding effect
…
…
–
+
+
+
+
–
–
h
Simple geometry of two parallel plates is simulated through adding
‘infinite’ number of image charges of alternating polarity, and direct
application of Lienard-Wiechert formula (still need to solve for the
retarded time).
January 26, 2008 USPAS’08 R & ER Linacs 26
Shielding effect on CSR wake
σz = 0.3 mm, ρ = 10 m
Agoh, Yokoya, PRST-AB 7, 054403 (2004)
January 26, 2008 USPAS’08 R & ER Linacs 27
Shielding effect on CSR wake
Simple beam trajectory: a short drift followed by circular trajectory
1.2 m
January 26, 2008 USPAS’08 R & ER Linacs 28
Shielding effect on CSR wake
Simple beam trajectory: a short drift followed by circular trajectory
1.2 m
January 26, 2008 USPAS’08 R & ER Linacs 29
Beam energy effect on CSR
Interesting feature of CSR is its independence from beam energy
(e.g. radiated power of coherent fraction). However, at low beam
energies (important for the injector), CSR can be substantially
weaker, while Coulomb term dominates.
January 26, 2008 USPAS’08 R & ER Linacs 30
CSR simulation tools
• 1D, high gamma approximation: elegant
• 1D, correct at low energy, shielding: BMAD
• 1D, correct at low energy, shielding, space
charge: GPT (Cornell version)
• Self-consistent: Tredi3D, CSRTrack (one of
the routines)
• …
3/1
<<
zz
xR
σσ
σ(1D criterion)
January 26, 2008 USPAS’08 R & ER Linacs 31
Summary
• Practical tools for evaluating degradation of
emittance and energy spread due to synchrotron
radiation presented
• ISR is typically negligible for low and medium beam
energies (< 3 GeV), can dominate beam line design
and higher energies
• CSR is important whenever the bunch is short (bunch
compressors), except at very low energies (injector)
when the space charge dominates.