uspas course on recirculated and energy recovered linacs
TRANSCRIPT
USPAS 2005 Recirculated and Energy Recovered Linacs 1&+(66/(33&+(66/(33
USPAS Course on
Recirculated and Energy Recovered Linacs
I. V. Bazarov
Cornell University
G. A. Krafft and L. Merminga
Jefferson Lab
Emittance and energy spread growth due to synchrotron radiation
USPAS 2005 Recirculated and Energy Recovered Linacs 2&+(66/(33&+(66/(33
Contents
• Coherent vs. incoherent emission
• Energy spread
- dipole magnets
- insertion devices
• Emittance growth
- dipole magnets
- H-fuction
- examples of calculation
- insertion devices
USPAS 2005 Recirculated and Energy Recovered Linacs 3&+(66/(33&+(66/(33
Quantum excitation
‘Quantum excitation’ in accelerator physics refers to diffusion of
phase space (momentum) of 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).
Here we consider incoherent synchrotron radiation primarily (λ <<
σz, so that the radiation power scales linearly with the number of
electrons). When radiation wavelength becomes comparable with
the bunch length (or density modulation size), radiated power
becomes quadratic with peak current. This coherent synchrotron
radiation (CSR) effects can be important at all energies when bunch
length becomes is sufficiently short.
USPAS 2005 Recirculated and Energy Recovered Linacs 4&+(66/(33&+(66/(33
Circular trajectory in the “rest” frame
Lorentz back-transform 4-vector ),( kc&
ω
USPAS 2005 Recirculated and Energy Recovered Linacs 5&+(66/(33&+(66/(33
Coherent radiation from a bend magnet
∞→
=
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
USPAS 2005 Recirculated and Energy Recovered Linacs 6&+(66/(33&+(66/(33
CSR scaling, longitudinal and transverse effect
ρ
l3SR γ
ρλ ∝
2
4
SR ργ
NP ∝3432
2
CSR
1
lNP
ρ∝
l≥CSR
λ
1 E-1 5
1 E-1 3
1 E-1 1
1 E-0 9
1 E-0 7
1 E-0 5
1 E-0 3
1 E-0 5 1 E-0 4 1 E-0 3 1 E-0 2 1 E-0 1 1 E+0 0 1 E+0 1 1 E+0 2 1 E+0 3
wav e le ngth [mm]
P [
W/m
ra
d/
σl
λcutoff
CSR enhanced spectrum
( )λSRIN ∗
Longitudinal Effect
)(ct
E
∂∂
s
tail
head
Transverse Effect
Dispersion ≠ 0
emittance
growth+ +- -
USPAS 2005 Recirculated and Energy Recovered Linacs 7&+(66/(33&+(66/(33
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.
USPAS 2005 Recirculated and Energy Recovered Linacs 8&+(66/(33&+(66/(33
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
USPAS 2005 Recirculated and Energy Recovered Linacs 9&+(66/(33&+(66/(33
Spectrum of synchrotron radiation 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
ωω
ωω
γα
ψ
/2
3
9
4
3c
Se
I
d
dN
c
c
ph
≡
∆=
USPAS 2005 Recirculated and Energy Recovered Linacs 10&+(66/(33&+(66/(33
Energy spread from bends
∫=ρ
γπ
σ γds
mccCE
7422 )(332
55!
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
πρσ
2)m(
1)GeV(106.2
22
5510
2
2 Θ⋅= −
EE
E
πργγ2
4 Θ=
ECE
πργ2)m(
)GeV(0886.0)MeV(
44 Θ=
EE
USPAS 2005 Recirculated and Energy Recovered Linacs 11&+(66/(33&+(66/(33
Energy spread from planar undulator
222 )( γεσ phphphphE NdEENE ≈= ∫
)1(
22
21
2
K
hc
p +=
γλ
ε 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
λπ
γ = )m()cm(
)GeV(725)eV(
22
222
u
p
LKE
Eλγ =
)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 +⋅≈ −
λσ
USPAS 2005 Recirculated and Energy Recovered Linacs 12&+(66/(33&+(66/(33
Energy spread from planar undulator (contd.)
)m()cm(
)()GeV(108.4
33
22213
2
2
u
p
E LKFKE
E λσ −⋅≈
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
Nph
∆ω(a
.u.)
USPAS 2005 Recirculated and Energy Recovered Linacs 13&+(66/(33&+(66/(33
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
2sH
E
Ea x
ph
x =δ here22 2
xxxxxxxxH ηβηηαηγ ′+′+=
USPAS 2005 Recirculated and Energy Recovered Linacs 14&+(66/(33&+(66/(33
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
!
πρε
2)m(
)m()GeV(103.1rad)-m(
22
55
10 Θ⋅= − HE
x
USPAS 2005 Recirculated and Energy Recovered Linacs 15&+(66/(33&+(66/(33
H-function
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
USPAS 2005 Recirculated and Energy Recovered Linacs 16&+(66/(33&+(66/(33
Example of triple bend achromat
mm 6.3=H
mm 1.9=H
mm 66.0≈−GC
H
4×3°-bends
USPAS 2005 Recirculated and Energy Recovered Linacs 17&+(66/(33&+(66/(33
Emittance and energy spread in ¼ CESR
energy = 5 GeV
large dispersion
section for bunch
compression
USPAS 2005 Recirculated and Energy Recovered Linacs 18&+(66/(33&+(66/(33
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
USPAS 2005 Recirculated and Energy Recovered Linacs 19&+(66/(33&+(66/(33
Emittance growth in undulator (contd.)
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
βγη
βγη
βγβ
ββρη
βρη
βρβ
KkK
LK
k
kL
kH
p
u
p
pu
p
Unless undulator is placed in high dispersion region,
contribution to emittance remains small.