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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


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.


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


Circular traj. in the “rest” frame

Lorentz back-transform 4-vector ),( kcr

ω


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.


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


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

≡

∆=


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


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


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.)


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 ′+′+=


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


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


Example of TBA

mm 6.3=H

mm 1.9=H

mm 66.0≈−GC

H

4×3°-bends


Example: emittance and energy

spread in ¼ CESR

Energy = 5 GeV

large dispersion

section for bunch

compression

5 GeV ring (768 m)

FODO lattice


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


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


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


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+ +- -


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−=− '


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


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


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


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).


Shielding effect on CSR wake

σz = 0.3 mm, ρ = 10 m

Agoh, Yokoya, PRST-AB 7, 054403 (2004)



Simple beam trajectory: a short drift followed by circular trajectory

1.2 m


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.


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)


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.

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