Outline
1. Introduction
1I t d ti
2. Light source properties vs electron beam
performance
3 Stochasticity
3.
St h ti it off photoemission
h ti i4. Distribution of circulating electron beam
5. Approach to coherent X-rays

Ring Accelerator Physics
for Non Accelerator Specialists
XFEL Research & Development Division
RIKEN SPring-8
SP i 8 CCentert
Hitoshi Tanaka

Cheiron School 2012@SPring-8

1

1 Introduction (1)1.

2

1 Introduction (2)1.

Even any member in an experimentexperiment-related
related division orBL scientists, who are not the accelerator specialists,should understand the relation between the SRS
properties and electron beam performances,
to keep the present experimental condition,
to improve the condition as much as possible,
andto challenge new experiments by using
the advanced SRSR.
Cheiron School 2012@SPring-8

Cheiron School 2012@SPring-8

3

You need high quality SR
for better experiments
For this purpose what
you should do?

SampleSample

Cheiron School 2012@SPring-84

1 Introduction (3)1.

OutlineMutual understanding through communication

1. Introduction
1I t d ti
2. Light source properties vs electronbeam performance
3 Stochasticity
3.
St h ti it off photoemission
h ti i4. Distribution of circulating electron beam
5. Extra: Approach to coherent X-rays

Experimentalusers

Beamlineengineers& scientists

Lightg source
Engineers &Physicists
Cheiron School 2012@SPring-85

Cheiron School 2012@SPring-8

6

2. Light
g source p
properties
pvs electron
beam performance(1)2. Light
g source p
properties
pvs electron
beam performance(2)Main devices to supply SR to users are“ d l“undulators”,” whichhi h are iinstalledll d iin magnet-freefstraight sections in 3rd-generation SR sources.

In an undulator, radiation from a single electron
at each undulation interferes with each other.
Spectral narrowing
by resonant condition
for a single electron isλλ1= u (1+K2/2+γ2θ2)2γ2where K and γ are a
deflecting parameterand relative energy.

Deflecting
g angleg ψ=γ-1K

Cheiron School 2012@SPring-87

Cheiron School 2012@SPring-88

2. Light
g source p
properties
pvs electron
beam performance(3)2. Light
g source p
properties
pvs electron
beam performance(4)Q: Why does the radiation concentrate within the γ cone ?
A: The electron can run after the emitted light with thealmost light speed only in this limited angle.
retarded time
R(t’)
( )
t = t’+ cpresent time
ΔtTime squeezing factor κ(t(t’)=)Δt’κ(t’) small → large acceleration →strong radiation

Q: Why does the radiation concentrate within the γ cone ?
A: The electron runs after the emitted light with thealmost light speed only in this limited angle.

κ(t' ) =rr
Δt= 1 − n(t' )⋅ β(t' )Δt'1⎛ 11⎛ 1⎞κ(t' ) ≈ ⎜ 2 + θ 2 ⎟2⎝γ⎠θ <1/1/γ → κ~1/1/γ2

κ(t’)

Intense radiation
is emitted in thisnarrow angle
gWidth~2/2/γ2⎞κ(t' ) ≈ ⎜ 2 + θ 2 ⎟2⎝γ⎠

θCheiron School 2012@SPring-8

9Cheiron School 2012@SPring-8

10

2. Light
g source p
properties
pvs electron
beam performance(5)2. Light
g source p
properties
pvs electron
beam performance(6)If electron beam were point-like without spatial
divergence and , all electrons could have noangular divergence, radiations were coherent !

In order to obtain the same resonance condition
over most of circulating electrons distributed,distributed
sufficiently small beam emittance is required.

Real radiation properties are obtained by convoluting
radiations from all N electrons
In the real world Nelectrons are
distributed in aphase space, neverdegenerate on thesame point.

Cheiron School 2012@SPring-8

11Interferencezone

Large emittance

Small emittance~2/γ

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2. Light
g source p
properties
pvs electron
beam performance(7)2. Light
g source p
properties
pvs electron
beam performance(8)In order to obtain the same resonance condition
over most of circulating electrons distributed,distributed
sufficiently small beam energy spread is required.

Ib⋅Nu2⋅γ2 ⋅Fn(K, δ) ,B∝
σxσx’σyσy’ (~εx ⋅ εy)λu(1+K2/2+γ2θ2)
λ1=2(γ+Δγ)2

ΛnΔλ1 ~ 2λ1 <Δγ/γ>
γγ
Spectral broadening by theenergy spread is much less thanintrinsic broadening ~1/Nu.

1/Nu=0.05

B: Brilliance (phs/sec/mm2/mrad2/100mA)
Ib: Beam current (mA)Nu: Undulator period number
σx,σy: Horizontal
H it l andd verticalti l bbeam sizesi((m))σx’,σy’: Horizontal and vertical angular divergence (rad)δ: Beam energy spread
K :Deflection parameter
p13

Cheiron School 2012@SPring-8

Nu:undulator period number

Cheiron School 2012@SPring-8

2. Light
g source pproperties
pvs electron
beam performance(9)η x2δ2 , Photonσx= σp2+βxεx+ηx2δ2 , σx’’= σpp’2+γγxεx+η'σy= σp2+βyεy +ηy2δ2, σy’= σp’2+γyεy +η’y2δ2 ,1 x,y21+αdβx,y-dβdηx,yd, η'x,y=, γx,y=.
αx,y=βx,ydsds

ElectronemittanceElectronenergyspread

14Outline1. Introduction
1I t d ti
2. Light source properties vs electron beam
performance
3 Stochasticity
3.
St h ti it off photoemission
h ti i4. Distribution of circulating electron beam
5. Extra: Approach to coherent X-rays

εx,εy: Horizontal and vertical emittance (m•rad)
βx,y : Horizontal and vertical betatoron functions at ID
ηx,y : Horizontal and vertical dispersion functions at ID
σp,σσp’’: Spatial and angular divergence of photon beam
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Cheiron School 2012@SPring-816

3 Stochasticity of photoemission(1)3.

3 Stochasticity of photoemission(2)3.

The photo-emission process is not continuous, in otherwords, quantized process. So, the emission position, thenumber of the emission photons, and the emission photon
energy have fluctuations.

A relativistic electron accelerated in a magnetic field willradiate electromagnetic energy at a rate which is
proportional to the square of the accelerating force.

This stochasticity (random fluctuation) causes finite
spread of the circulating electron beam in the 6D
phase space. The density distribution is generally
Gaussian due to the central limit theorem.

Cheiron School 2012@SPring-8classical electron radius<Averaged radiation power>
re c
re c
2E4 .
E2F⊥2 = 2
P= 323(m0c )
3 (m0c2)3 ρ2<Averaged radiation energy per turn> Averaged properties
TLdl
are smooth !
U=∫Pdt = ∫P c ,88 5E4(GeV)
88.5E
for the case with the constant ρ.
U(keV)=ρ(m)17

3 Stochasticity of photoemission(3)3.

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3 Stochasticity of photoemission(4)
3.
~2keV

In the quantized radiation process, the integrated parameter,
energy loss per unit time also fluctuates. Magnitude of thefluctuation can be defined by a mean square.

~30keV<Mean square of energy loss fluctuation per unit time>55
3 hc γ3 .
Uu ,uc =c24 32 ρT
uc: critical photon energy ~ 3.2<u><Nu2> =

ρ=~40m<Averaged photo
photo-emission
emission rate>

uc represents the photon energy at the peak brilliance of<N>(phs/s) ~ 3.2P/ uc
Cheiron School 2012@SPring-8

the BM radiation
radiation.
19

Cheiron School 2012@SPring-8

20

3 Stochasticity of photoemission(5)3.

3 Stochasticity of photoemission(6)3.

Let’s estimate the parameters!

SPring-8 parameter is the next example.

<Case-1C1E
E=11GGeV,V ρ=5m>
5<Case-2C2E
E=88GGeV,V ρ=40m>
40

88.5×14U
1.7×107×108= 17.7,
17 7 P(keV/s)=T=2π×5/c= 15
3 6.856.85×10
10-19×2.998×10
2.998 108×1957
19573= 0.440 44
uc (keV)=(k V)
2555 1.7×10
1 7 108 × 0.44=0.99×10
0 44=0 99 108<N 2> (keV
<Nu(k V2/s)/ )=24 3

88.5×84
U=11.1×101×1010
9 1 103, P(keV/s)== 9.1×1040
T=2π×40/c
3 6.856.85×10
10-19×2.998×10
2.998 108×15656
156563= 29.629 6
uc (keV)=(k V)240
55 1.1×10
1 1 1010 × 29.6=4.31×10
29 6=4 31 1011
<N 2> (keV
<Nu(k V2/s)/ )=24 3<N>(photons/s) ~ 3.2×1.7×108 / 0.44 = 1.2×109<N>(photons/s) ~ 3.2×1.1×1010 / 29.6 = 1.2×109

U(keV)=

Cheiron School 2012@SPring-8

21Outline
1. Introduction
1I t d ti
2. Light source properties vs electron beamperformance
3 Stochasticity
3.
St h ti it off photoemission
h ti i4. Distribution of circulating electron beam
5. Extra: Approach to coherent X-rays

U(keV)=Cheiron School 2012@SPring-822

4. Distribution of circulatingg electronbeam(1)6D-phase space volume of a single electron, εs6 comprisesof canonical variables ((x,, px(x’),
p ( ), y, py(y’),
py(y ), t,, ps(ΔE/E)).
p ())
In an ideal case, 6D-phase space volume can be written byεth productthed t off areas off theth threethorthogonalthl 2D spaces sz.
z=x,y,sTransverse motion
Longitudinal motion

εs6=εsxεsyεss
These are invariants ofthe motion for aconservative system.
systemCheiron School 2012@SPring-823

Cheiron School 2012@SPring-824

4. Distribution of circulatingg electronbeam(2)

4. Distribution of circulatingg electronbeam(3)You can understand the relation between the 2D
phase spacepp
and the emittance byy using
g a simplepharmonic oscillator.

We need potential wells to stabilize threegoscillation modes to keepp electron
orthogonal
beam in a storage ring.

x = A・cos(ωt+φ0)
dx=x’= -A・ω sin(ωt +φ0)dt
1 dx
ω dt = -A・sin(ωt+φ0)

1xx’
ω
Ax

Action=A, Angle=φ= ωt +φ0 ,i l area
Invariant= x 2+(x’/ω)2=A2= circle
= εsx
πωtt

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4. Distribution of circulatingg electronbeam(4)
SR light properties reflects the 3×2D phasespace distribution of circulating electrons
electrons.

Quadrupole magnets generate theadequate potential wells for two
transversal oscillation modes
modes, which
are called betatron oscillations in thehorizontal and vertical planes
planes.
RF acceleration electric field
generates the adequate potential wellfor longitudinal oscillation mode,
which is called a synchrotron
yoscillation.
Cheiron School 2012@SPring-826

4. Distribution of circulatingg electronbeam(5)Width of the Gaussian distribution of N
circulating electrons is determined by theWe use the following three ensemble-averaged
emittances
itt
t express beam
tobdidistribution
t ib ti iin ththreeorthogonal phase spaces.

dynamical equilibrium between theradiation excitation and damping.

<εs6> = < εsx > < εsy > < εss >=εx εy εs,εx : horizontal emittance (m rad)εy : vertical emittance (m rad)εs : longitudinal emittance (m rad)Cheiron School 2012@SPring-827

Cheiron School 2012@SPring-828

4. Distribution of circulatingg electronbeam(6)4. Distribution of circulatingg electronbeam(7)Remember the invariant of a harmonic oscillator.

Energy spread σΔΕ

d< A2>
dε= dt z
dtz = x,y,s.

lim<AΔA> = -2
Δt→0 2Δt

Equilibrium condition:(A ΔA)2 - A2> lim <AΔA> lim <ΔA2>
d< A2> lim <(A+ΔA)+ Δt→0 Δt=0= Δt→0
= Δt→0 2 ΔtdtΔtDamping term
Averaged energy dissipation

Excitation term
Quantum effect

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4. Distribution of circulatingg electronbeam(8)τ=(z-z(z z0)/cz

Bunch length στreference electron z=z0στ has two components;
dilation factorστ1= time spread by energy spreadαστ2= time spreadpby
y stochastic p
photoemission
ω ΔΕ/Ε

στ1 >>στ2angular frequency
of energy oscillation

10-4-3αστ = ω σΔΕ/Ε 10104= ~10 psec
@ SPring-8

Cheiron School 2012@SPring-8

τ=t-t
=t t0

τεE×T ,Uτε =lim <ΔA2>
55= <Nu2> =Δt→0Δt
24 3

Uu .
cTSince σΔΕ is not the emittance,emittance phase average factor
2>,should be considered, σ 2 = < A2ΔΕσΔΕ=29< A2> ,55
96 3

E×uc ,σΔΕ/Ε=55
96 3ucE.

Cheiron School 2012@SPring-830

4. Distribution of circulatingg electronbeam(9)Horizontal emittance εxExcitation: due to the discrete energy jump + energy
dispersion
Damping: due to the decrease of transverse momentum
by the photoemission + acceleration along the running
di ti
direction
Excitation
DampingωtPhase space of synchrotron((energy)) oscillation
ill ti
31

Cheiron School 2012@SPring-832

4. Distribution of circulatingg electronbeam(10)

Chasman-Green (CG) lattice is the most popularmagnet cell structure for a low emittance SR souce,
where a achromatic arc is composed of a pair ofbending magnets.

For the typical magnetic lattice structure (ChasmanGGreen:CG) based
bd storagetring,iththe hhorizontal
it lminimum emittance is written by@general ~εx min @g=

ε1@achromatx min
2Cqγ2
θb 38 15 Jx

2 bends: CG or CBA
3 bends: TBA
4 bends: QBA

εx min∝θb3

Cq: Quantumn constant 3.832×10-13 (m)θb: Deflection angle of a single bending magnet (rad)Jx: Horizontal damping partition number ~ 1
Cheiron School 2012@SPring-833

4. Distribution of circulatingg electronbeam(12)Vertical emittance εyCheiron School 2012@SPring-834

4. Distribution of circulatingg electronbeam(13)The effect of these error fields can be corrected top
0.1 % level byy the combination of beam response
analysis and skew quadrupole corrector magnets.

γ=1000/0.511∼2000
1000/0 511 2000
1/γ∼5×10-4

Principally, εy =1/1000 εx
is ppossible

Cheiron School 2012@SPring-8

θb smallerwith same
cell No.

Vertical emittance is determined by magnetic error
components
t mixing
i i theth horizontal
h it l andd verticalti l bbetatront toscillation. The main sources are vertical misalignmentsof sextupole magnets and rotational errors of quadrupolemagnets.

εy has two components;
εy1= verticalti l emittanceittbyb stochastict h ti photoemission
h ti iεy2= vertical emittance by HV coupling
Usually, εy1 << εy21-GeV storage ring

The Angular divergence
In the vertical pplane is ~ 1/γγ

4. Distribution of circulatingg electronbeam(11)35

One-dimensional diffraction limited X-ray beam is
now available
Cheiron School 2012@SPring-836

5 Approach to coherent X-rays (1)5.

Outline1. Introduction
1I t d ti
2. Light source properties vs electron beamperformance
3 Stochasticity
3.
St h ti it off photoemission
h ti i4. Distribution of circulating electron beam
5. Approach to coherent X-rays

Cheiron School 2012@SPring-837

Q: How much is a coherent
component of SR?
A: 0.1% at SPring-8

Cheiron School 2012@SPring-838

5 Approach to coherent X-rays (2)5.

5 Approach to coherent X-rays (3)5.

Q: How to estimate?

R is unity at maximum, which we call“diffraction limit”.

A: Roughly, the ratio R isσp2 σp’2
σx σy σx’ σy’

Cheiron School 2012@SPring-8

σpσp’=λ/4π
Νλuλk4πλkσp’= Νλuσx~σσpσx’’~σσp’’σy~σpσyy’~σpp’

σp =39Cheiron School 2012@SPring-840

5 Approach to coherent X-rays (4)5.
Electron beam emittance requiredfor diffraction limit at a wavelength of01
0.1nm.

5 Approach to coherent X-rays (5)5.
Electron beam emittance required is~10 pmrad

12 ~10σpσp’=λ/4π=8x10
λ/4 8 10-12
10 pm radd

The present value achieved at 3rdd GenG SR
S is ~1 nmrad

We assume N=900
N=900, λu=32mm,
λu=32mm E=8 GeV(γ=15656)

To obtain spatially coherent X-ray
X-ray, the electron beamemittance should be reduced by 2~3 orders of magnitude

σp=00.06/γ,06/γ σp’’=00.03/γ03/γ

σp=4um4um, σp’’= 2urad

Cheiron School 2012@SPring-841

5 Approach to coherent X-rays (6)5.
At present, we use “ a linear accelerator” without
no recoil by radiation + a low emittance electron
source + a bunch compressor + a long undulator,
which can realize the diffraction limited X
X-raysrays at
0.1 nm by using a SASE process.
How about the possibility of Ultra-low emittanceg ring?
gstorage

Cheiron School 2012@SPring-843

Cheiron School 2012@SPring-842