USPAS Course on Recirculated and Energy Recovered Linacs

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


Contents

• Coherent vs. incoherent emission

• Energy spread

- dipole magnets

- insertion devices

• Emittance growth

- dipole magnets

- H-fuction

- examples of calculation

- insertion devices


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.


Circular trajectory in the “rest” frame

Lorentz back-transform 4-vector ),( kc&

ω


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


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+ +- -


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 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

≡

∆=


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


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


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.)


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 ′+′+=


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


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


Example of triple bend achromat

mm 6.3=H

mm 1.9=H

mm 66.0≈−GC

H

4×3°-bends


Emittance and energy spread in ¼ CESR

energy = 5 GeV

large dispersion

section for bunch

compression


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


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.

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