Emittance Diagnostics 

Emittance, what is it ?

Emittance, why  measuring it ?

How to measure it ?

What can go wrong ?

CAS Beam diagnosticDourdan,  2.6.2008 Hans Braun / CERN

Emittance, what is it ?

particle position x

particle angle x’

ε = Area in x, x’ plane occupied by beam particlesdivided by π

x

Beam envelope along beamline

x

x’

x

x’

x

x’

Along a beamline the orientation and aspect ratio of beam ellipse in x, x’ plane varies, but area πε remains constant

z

εβ )()( zzw = withdescribed is z along widthBeam

w

2w

converging beam

beam waist divergingbeam

waist)-(anti maximum or (waist) minimum has size beam for diverging. is beam 0 for

,converging is beam 0 for .correlated are x'and x stronghow describes

divergence half beam the is

widthhalf beam the is

dindependen are parameters these ofonly therefore

1relation theby connected are ,, parameters norientatio ellipse three the

parameters Twiss or Snyder-Courant called,, , parameters 4by described is norientatio its and ellipse Beam

2

0

''2 22

=<>

+=

++=

ααα

αγε

βε

βαγ

γαββαγε

γαβε

xxxx

Transport of single particle described with matrix algebra

( ) ( )( ) ( )

( ) ( )( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛

−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⋅⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛

LkLkkLkkLkM

LM

xxL

LkLkkLkkLkL

xx

MMMxx

Mxx

QuadrupoleDrift

A

BB

BBCABC

cossinsin1cos

101

'101

cossinsin1cos

101

''' 1

1

1

1

1

1

2

2

x

z

drift quadrupole drift

LA LB LC

⎟⎟⎠

⎞⎜⎜⎝

⎛'1

1

xx

⎟⎟⎠

⎞⎜⎜⎝

⎛'2

2

xx

generic names of matrix elements ⎟⎟⎠

⎞⎜⎜⎝

⎛=

'' scsc

M

1

2

Transport of Twiss parameters β, α, γ of beam ellipse

x

z

drift quadrupole drift

LA LB LC

εγεαεβ

εγαββ

γαβ

γαβ

12

1122

2

12

112

2

1

1

1

22

22

2

2

2

1

1

2

2

2

|2

'''2'''''

2

''''

ssccw

sscc

ssccssscsccc

sscc

xx

scsc

xx

+−=

⋅+−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⋅

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−+−

−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⇒⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛

1

2

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

1

1

1

γαβ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

2

2

2

γαβ

So far we implicitly assumed that beam occupies x, x’ space area with a sharp boundary

∫∞

∞−

= ')',()( dxxxxf ρ

particle density in x, x’ space transverse beam profile

projection on x axis

2 wbeamwidth

surface πε

βε=w

In reality beam density in x, x’ space is rarely a constant on a well defined area but often like this

∫∞

∞−

= ')',()( dxxxxf ρ

particle density in x, x’ space transverse beam profile

projection on x axis

beamwidth ?

surface πε ?

Commonly used definitions of emittance

εrms , 1 sigma r.m.s. emittance ⇔

ε2rms , 2 sigma r.m.s. emittance ⇔

ε90% , 90% emittance ⇔

βε=RMSw

2βε

=RMSw

Definition of wRMS

∫

∫

∫

∫

∞+

∞−

∞+

∞−

∞+

∞−

+∞

∞−

=

−=

dxxf

dxxfxx

dxxf

dxxfxxw

COG

COG

RMS

)(

)(

)(

)()( 2

Definition of w90

%90)(

)(90

90 =

∫

∫∞+

∞−

+

−

dxxf

dxxfW

W

βε=90w

Relations between different emittance definitions

Always ε2rms = 4 εrms

For Gauss beam distribution only ε90% = 2.7055 εrms

wrms = σXw90% =1.6449 σX

εrms , 68.3% of beam ε2rms , 95.5% of beam ε90% , 90% of beam

Slice emittance

important for free electron lasers !

x

z

x

x’

x

x’

slice A slice B

εΑ εΒ

x2’

Emittance is only constant in beamlines without acceleration

x1’Px

P1

Px

P2

acceleration

x

x’

x

x’

normalised emittance preserved with acceleration !

To distinguish from normalised emittance εN , ε is quoted as “geometric emittance” !

εγβε relrelN =

cmPPP

relrelγβεε == with2

112

ε1

ε2

Commonly used units for emittance

mm·mrad, m·rad, μm, m, nm

1 mm·mrad=10-6 m·rad=1μm=10-6 m=103nm

Often a π is added to the unit to indicate that the numerical value describesa surface in x, x’ space divided by π, i.e. 1 π·mm·mrad

The units for normalised emittance are the same as for geometric emittance

Due to the various definitions it is recommended to always mentionthe emittance definition used when reporting measurement values !

Emittance, why measuring it ?

The emittance tells if a beam fits in the vacuum chamber or not

( )zazzw <= εβ )()(

LHC IR region

aperture a(z)

( ) yy z εβ×10

Emittance is one of key parameters for overall performance of an accelerator

o Luminosity of colliders for particle physics

o Brightness of synchrotron radiation sources

o Wavelength range of free electron lasers

o Resolution of fixed target experiments

Therefore emittance measurement is essential to guide tune-up of accelerator !

Measurement of emittance is closely linked with measurement of Twiss parameter β, α, γ .

o Provide initial beam conditions for a beamline, thus allowing to compute optimum setting of focusing magnets in a beamline.Important to minimise beam losses in transfer line !

o Measurement of Twiss parameters at injection in synchrotron or storage ring allows to “match” Twiss parameters to circulating beam. This helps to minimise injection losses and to avoid emittance growth due to filamentation of unmatched beam.

o Measurement of emittance as function of time in synchrotrons and storage rings allows to identify, understand and possibly mitigate mechanism of emittance growth.

o Measurement of emittance at different locations in a beam line or in a linear accelerator allows to verify and/or improve beam optics model of this line. Moreover, it allows to identify, understand and possibly mitigate mechanism of emittance growth.

Emittance, how to measure it ?

Methods based on transverse beam profile measurements

Slit and pepperpot methods

Methods based on Schottky signal analysis⇒ F. Caspers Lecture, June 3rd

Direct methods to measure transverse profiles and beam divergence⇒ see literature list

Emittance measurement in transfer line or linac

Twiss parameter β,α,γ are a priori not known, they have to bedetermined together with ε.

Method A x

z

wA

profile monitors

wB

wC

Reference point whereβ, α, γ will be determined

La Lb Lc

εγεαεβ

εγεαεβ

22

222

2

101

2

LLw

Ls'c'sc

ssccw

+−=⇒

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛+−=

drift for ,

see Enrico Bravin’s lecture

( )ε

αεαε

βεβεαεγεβεεαγβεαεγεβε

γεαεβε

γεαεβε

εγεαεβ

εγεαεβ

εγεαεβ

===−⋅⇒=−⋅=−⋅

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⋅⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

⇒⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⋅

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⇓

+−=

+−=

+−=

−

,,)()(

212121

212121

2

2

2

22222

2

2

21

2

2

2

2

2

2

2

2

2

22

22

22

C

B

A

CC

BB

AA

CC

BB

AA

C

B

A

CCC

BBB

AAA

www

LLLLLL

LLLLLL

www

LLw

LLw

LLw

notationMatrix in rewritten be can

Derivation of emittance and Twiss parameters from beam width measurements

Emittance measurements often require inversion of 3x3 Matrices.Many Math and spreadsheet programs and programming languageshave functions for this.

If you insist to code it yourself you can use :

The inverse of a 3x3 matrix:

| a11 a12 a13 |-1 | a33a22-a32a23 a32a13-a33a12 a23a12-a22a13 || a21 a22 a23 | = 1/DET * | a31a23-a33a21 a33a11-a31a13 a21a13-a23a11 || a31 a32 a33 | | a32a21-a31a22 a32a11-a31a12 a22a11-a21a12 |

with DET = a11(a33a22-a32a23)-a21(a33a12-a32a13)+a31(a23a12-a22a13)

Emittance measurement in transfer line or linac, Method B

x

z

wA, wB, wC

profile monitorReference point whereβ, α, γ will be determined

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛+−=

)()()()(

101

22221

1211222

magmag

magmag

ImImImImL

s'c'sc

ssccw , εγεαεβ

Adjustable magnetic lens with settings A,B,C(quadrupole magnet, solenoid, system of quadrupole magnets…)

( )ε

αεαε

βεβεαεγεβεεαγβεαεγεβε

γεαεβε

γεαεβε

εγεαεβ

εγεαεβ

εγεαεβ

===−⋅⇒=−⋅=−⋅

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⋅⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

⇒⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⋅

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⇓

+−=

+−=

+−=

−

,,)()(

222

222

2

2

2

22222

2

2

21

22

22

22

22

22

22

2

2

2

222

222

222

C

B

A

CCCC

BBBB

AAAA

CCCC

BBBB

AAAA

C

B

A

CCCCC

BBBBB

AAAAA

www

ssccssccsscc

ssccssccsscc

www

ssccw

ssccw

ssccw

notationMatrix in rewritten be can

Derivation of emittance and Twiss parameters from beam width measurements

To determine ε, β, α at a reference point in a beamline one needs

at least three w measurements with different transfer matrices between the

reference point and the w measurements location.

Different transfer matrices can be achieved with different profile monitor locations,different focusing magnet settings or combinations of both.

Once β, α at one reference point is determined the values of β, α at every pointin the beamline can be calculated.

Three w measurements are in principle enough to determine ε, β, αIn practice better results are obtained with more measurements. However, with more than three measurements the problem is over-determined.

χ2 formalism gives the best estimate of ε, β, α for a set of n measurements wi , i=1-n with transfer matrix elements ci, si .

( )

∑∑∑∑

∑∑∑∑

∑∑∑∑

∑

====

====

====

=

=+−⇒=∂∂

=+−⇒=∂∂

=+−⇒=∂∂

−+−=

+−

n

iii

n

ii

n

iii

n

iii

n

iiii

n

iii

n

iii

n

iii

n

iii

n

iii

n

iii

n

ii

n

iiiiii

iiii

i

wcsscsc

wscscscsc

wcscscc

wsscc

sscc

w

1

22

1

4

1

3

1

22

1

2

1

3

1

22

1

3

1

22

1

22

1

3

1

4

1

2222

22

20

20

,20

2

2

γεαεβεγεχ

γεαεβεαεχ

γεαεβεβεχ

χ

γεαεβεχ

γεαεβε

γεαεβεγεαεβε

2

2

2

2

2

minimum for Conditions

minimises whichvalues, , , of set Find

known not priori a , , but , widthbeam half Predicted

widthbeam half Measured

εαεα

εβεβεαεγεβε

γε

αε

βε

γε

αε

βε

===−⋅

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⋅

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−

−

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⋅

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−

−

∑

∑

∑

∑∑∑

∑∑∑

∑∑∑

∑

∑

∑

∑∑∑

∑∑∑

∑∑∑

=

=

=

−

===

===

===

=

=

=

===

===

===

,,)(

2

2

2

2

2

2

2

1

22

1

2

1

221

1

4

1

3

1

22

1

3

1

22

1

3

1

22

1

3

1

4

1

22

1

2

1

22

1

4

1

3

1

22

1

3

1

22

1

3

1

22

1

3

1

4

n

iii

n

iiii

n

iii

n

ii

n

iii

n

iii

n

iii

n

iii

n

iii

n

iii

n

iii

n

ii

n

iii

n

iiii

n

iii

n

ii

n

iii

n

iii

n

iii

n

iii

n

iii

n

iii

n

iii

n

ii

wc

wsc

wc

sscsc

scscsc

scscc

wc

wsc

wc

sscsc

scscsc

scscc

notationmatrix in rewritten be can conditions These

χ2 formalism to determine ε, β, α

Girder 104 April 2007

Emittance measurement insertion in CTF3 linac

quadrupoletriplet

OTR screensteering

magnet

beam direction

4.5 m

Example of ε measurement in CTF3 linac

x

z

v

profile monitorMoveable slit

L

x

x’

v/L

From width and position of slit image mean beam angle and divergence of sliceat position u is readily computed.

By moving slit across the beam complete distribution in x, x’ space is reconstructed.

Conditions for good resolution: v >> s

u

u

Emittance measurement with moveable slit

Intensity at u, x’ in x, x’-space is mapped to position u+x’·L on screen

g

g

Phase space reconstruction at SPARCcourtesy of Massimo Ferrario / LNF 

X’

X

Single shot emittance measurement for 50 MeV proton beam at LINAC II / CERN

quadrupoles for pointto parallel imaging

fast deflector magnetsSEM grid

profile monitor Slit

fast sweeping magnet

time resolved profile monitor

quadrupoles forpoint to parallel imaging

slit (fixed position)

time

beam

posi

tion

on p

rofil

e m

onito

r

Single shot emittance measurement for 50 MeV proton beam at LINAC II / CERN

Pepperpot

similar to moving slit method

+ allows measurement in both planes simultaneously- pepperpot dimensions have to be well adapted to beam conditions

hole spacing << beam widthshole image >> hole diameterhole image < hole spacing

Where to use what ?

quadrupole scan for beamlines where linear beam optics is valid,i.e. where space charge forces are small and energy spread is not too large

Slit / pepper-pot for low energy beams, where beam can be stopped in slit / pepperpot mask

high energyhigh space charge forces

large energy spread

slit / pepperpot ‐ + +

quadrupole scan + ‐ ‐

drift with several monitors + ‐ +

profile monitor with variable Z-position(CCD camera & screen)

pepperpot

switchable lens f=25-100 mm

beam generation

online data acquisition and analysis systemThe "ultimate" emittance measurement set-up

beam

Slice emittance measurement

off crest acceleration

time slice

ATF at BNL

deflecting RF cavity

Slice emittance set-up at FLASH / DESY

Emittance measurement in storage ring or synchrotron

Beam profile measurement at one location of ring is sufficient to determine ε

)'2)('2(2)(

'''2'''''

2

)()()(

)()()(

'''2'''''

2

)()()(

)()()(

),(.

),(

22

22

22

22

scscsz

ssccssscsccc

sscc

zzz

zzz

ssccssscsccc

sscc

CzCzCz

zzz

CzzMz

s'c'sc

CzzM

++−−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−+−

−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⋅

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−+−

−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+++

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⇒

+

⎟⎟⎠

⎞⎜⎜⎝

⎛=+

β

γαβ

γαβ

γαβ

γαβ

gives rEigenvecto the for Solving

. matrix of rEigenvecto is Therefore

nceCircumfere ithperiodic w be must functions Twissposition. and parameters magnet from calculated be can

positon monitor profile at starting revolution ring one formatrix transfer particle single is

)()(

2

zwzw

βεεβ =⇒= halfwidth beam Measured

Complication in storage ring

Emittance determination requires profile monitors at location with

For measurement of β(z) and D(z) see Jörg Wenniger’s lecture.

For measurement of see Mario Ferianis’s lecture

( ) ( ) ( )

( ) ( )2

2

22

PPzDz

PPzDzzw

Δ>>

Δ+=

εβ

εβ

PPΔ

Longitudinal emittance

In synchrotron or storage rings with Qs << 1

If intensity dependend synchrotron tune shift is negligible it is sufficient

to measure either or and the synchrotron oscillation frequency ωs

usingI

PPz

PPzL

Δ⋅−

Δ=

22ε

22

PPzL

Δ=ε

2z2

PPΔ

222

1

PP

cz

S

Δ⎟⎟⎠

⎞⎜⎜⎝

⎛−

=ω

γα

Spectrometer magnetY-plane deflection

RF deflector cavityX-plane deflection

X-Y image monitor

beam

x

y

bunch centered on zero cresting deflection

x’

t

A

( )

xt wLA

w

tLAx

ω

ω1sin

≈

=

YYP

P

Y

wD

w

PPDy

1=

Δ=

Δ

wY

wX

Longitudinal emittance measurement for Linacs I

YPP

Xt

wD

wLA

βε

βεω

>>

>>

Δ2

2

)(

)(For good accuracy

required

Spectrometer magnetY-plane deflection

Profile monitor with preservation of time information like OTR or Cerenkov radiator

+ observation with streak camera

beam

Longitudinal emittance measurement for Linacs II

x

y

wY

YYP

P

Y

wD

w

PPDy

1=

Δ=

Δ

wt

streak direction

Pitfalls with emittance measurements

Beam width determination

Space charge

Chromatic effects

Calibration errors