INTRODUCTION TO HIGH

BRIGHTNESS ELECTRON BEAMS

Wai-Keung Lau [ NSRRC Winter School on Free Electron Lasers 2019

Jan. 14th – 18th, 2019

Single Particle Dynamics in EM Fields

Relativistic dynamics

Applied EM fields are usually expressed as expanded series called paraxial

approximation

Complicated EM field distributions can only be solved by numerical methods.

Examples of simulation codes are: POISSON/SUPERFISH

High Frequency Structure Simulator (HFSS) -- http://www.ansoft.com/products/hf/hfss/

CST Microwave Studio -- http://www.cst.com/Content/Products/MWS/Overview.aspx

BvEqvmdt

d

dt

pd

0

The Lorentz factor is in general not a constant of time

Lorentz force law

What is a Charged Particle Beam?

an “ordered flow” of charged particles

all particles are moving

along the same trajectory

for a perfect beam

a random distribution

of charges

something in between

(real world)

x’

x

(xi,xi’) x’

x

a single particle in trace-space

a system of particles in trace-space

Trace Space Area Definition of Emittance

Trace space area definition of emittance

The trace-space area Ax is related to the phase-space area in x-px plane by

We can therefore useful to define normalized emittance that is independent of particle acceleration such that

However, beams with quite different distributions in trace space may have the same area!!

xdxdAxx

many authors identify the emittance as trace-space area divided by (unit: [m-rad])

[ m-rad]

xx

z

x dxdpmc

dxdpp

A

11this integral can be a

constant as long as

Liouville’s theorem

applies.

n

Let f (x,x’) be the distribution function such that . N is the total number of particles. Beam parameters can be defined accordingly as:

Nxdxdxxf ,

xdxdxxfxdxdxxfxxxx

xdxdxxfxdxdxxfxx

xdxdxxfxdxdxxfxx

xdxdxxfxdxdxxfxx

xdxdxxfxdxdxxxfx

xx

x

x

,/,

,/,

,/,

,/,

,/,

2

2

averaged beam size

averaged beam divergence

rms beam size

rms beam divergence

beam correlation

root-mean square of a set of n values is defined as:

22

2

2

1

1nrms xxx

nx

Define rms emittance as

If x and x’ are not correlated (at beam waist where the beam is neither converging nor diverging)

RMS emittance gives quantitative information on beam quality

RMS emittance gives more weight to the particles in the outer region of the trace-space area. Therefore, remove some of the outer particles will significantly improve RMS emittance without too much degradation of beam intensity.

det~ 222222 xxxxx xxxx

xxx xx ~~~

2

2

xxx

xxx

-matrix of beam distribution

Effective Emittance In a system that all forces (space charge and external forces) acting on the particles are

linear (i.e. proportional to particle displacement x from the beam axis), it is still useful to

assume an elliptical shape for the area occupied by the beam in trace space such that

We are able to define an emittance as

The relation between and the corresponding RMS quantities

are given by

mmx xxA

xmmx

Axx

xthxx ~,~,~ xmm xx ,,

xx

thm

m

xx

xx

~4

~2

~2

x

x’

Beam Brightness

Definition of beam brightness:

It is useful know that total beam current that can be confined within a 4-D trace space volume V4. We can define average brightness as

If any particle distribution whose boundary in 4-D trace space is defined by a hyperellipsoid

one finds and average brightness is

RMS brightness is then defined as:

dsd

dI

d

JB

4V

IB dsdV4

1

2

2

22

2

2

yx

yb

b

yxa

a

x

yxdsd 2/2

yx

IB

2

2

(note: 2/2 is left out)

yx

IB

~~~

Thermal Emittance of a Beam from Cathode For a beam from a thermionic cathode at temperature T, rms thermal velocity spread

is related to rms beam divergence such that

Assume Maxwellian velocity distribution from a round cathode with radius rs,

where T is the temperature of the cathode, then

On the other hand, for a beam emitted from the cathode

Thermal emittance of such beam from the cathode is given by

0, /~~ vvx thx

Tk

vvvmfvvvf

B

zyx

zyx2

exp,,

222

0

2/1

~~

m

Tkvv B

yx

2/~~sryx

0

2/1

, 2~

v

m

Tk

r

B

syx

2/1

00

0

3

0

2

2

2

1

2

~~

drrn

drrnrx

a

a

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X

Y=exp(-X.2)

2/1

22~

mc

Tkr B

sn

the total Coulomb force acting on q by a thin volume dV

of charge Q is independent of r !!

Planar Diode with Space Charge – Child-Langmuir Law

0

2

22

dx

d

constxJ x

02

2 xexm

2/12/1

0

2

2 1

/2

me

J

dx

d

C

me

J

dx

d

2/1

2/1

0

2

/2

4

xm

eJ4/12/1

0

4/3 22

3

4

3/4

0

d

xVx 2

2/3

0

2/1

0

2

9

4

d

V

m

eJ

2

2

2/3

06 /1033.2 mAd

VJ

x

e-

=0

=V0

cathode

C=0 under the boundary conditions: =0 and

d/dx=0 at x=0; the condition d/dx=~x1/3=0 at

x=0 implies that the special case electric field

at cathode surface is null (a steady state

solution).

Emission of electrons

from cathodes:

1. Thermionic

emission

2. Secondary

emission

3. Photo-emission

with

Child-Langmuir law

Electrostatic repulsion forces between

electrons tend to diverge the beam

The current density required in the

electron beam is normally far greater

than the emission density of the cathode

Optimum angle for parallel beam is

referred to as Pierce electrodes

Conical diode is needed for convergent

flow

Defocusing effect of anode aperture has

to be considered a simple diode gun 140 kV Electron Gun System for TLS

r

z focusing continuous beam

For a given focusing channel, the

size of beam waist is in general

determined by beam emittance,

space charge forces etc..

beam expansion due

to space charge

propagation of a continuous beam and a bunched beam in drift space

change of particle

distribution in phase

space due to space

charge (transverse

beam size, bunch

length, divergence,

energy spread etc..)

A Cylindrical Beam in a Strong Magnetic Field

Particles enter the drift tube with kinetic energy qb.

A new potential will be setup in the drift tube which will reduce the K.E. of

the particles according to energy conservation law.

As the potential is strong enough, K.E. of particles completely convert into

potential energy. There exists a beam current limit!!

b

0

(z)

rqrqTrT eb

E-BEAM

- + b

DRIFT TUBE

STRONG MAGNETIC FIELD

a b CATHODE

a

b

c

Ie ln21

40

0

electric fields of a uniformly moving charged particle

v = 0 v < c v = c

1/

Er 2q(z-ct)/r 0ˆ BzEeF

v c

B 2q(z-ct)/r

16

Paraxial Approximation of Axisymmetric

Electric and Magnetic Fields General assumptions:

Consider only static electric and magnetic fields at this point.

That is, no time-varying fields and displacement currents are excluded.

Electric and magnetic fields for axisymmetric lenses in a beam focusing system do not usually have azimuthal components (i.e. ).

Other field components have no azimuthal dependence

In paraxial approx., fields are calculated at small radii from the system axis with the assumption that the field vectors make small angles with the axis

0

0

B

E

0

rz EE

zr EE zr BB

0

0

E

B

0

rz BB

0;0 BE

17

The axial and radial fields in paraxial

approximation are:

4

44

2

22

3

33

644,

162,

z

Er

z

ErEzrE

z

Er

z

ErzrE

z

r

4

44

2

22

3

33

644,

162,

z

Br

z

BrBzrB

z

Br

z

BrzrB

z

r

(1)

(2)

(3)

(4)

18

Derive linear equations of particle motion in which only terms up to first order in r and r’=dr/dz are considered

Assumptions: ◦ Cylindrical beam; self-fields are neglected ◦ Particle trajectories remain close to the axis. That is, r <<

b. And b is the solenoid or radii of electrodes that produce the electric and magnetic fields. This also implies that the slopes of the particle trajectories remain small (i.e., or ).

◦ Azimuthal velocity must remains very small compared to the axial velocity (i.e., ). Thus, in this linear approximation, we have

zr

vzr

vrrvz 2/12222

center line

focusing solenoid

B fields

r z-axis

particle trajectory

b

1r

19

To describe the motion of a charged particle that moves close to the beam

axis, ‘paraxial ray equation’ can be derived

This equation is not a linear differential equation and it is not very useful in

practice. However, the contribution from the last term can be neglected for a

beam without angular momentum or if the particle with constant angular

momentum, we can choose a rotating reference frame in which the last term

can be avoided. Especially when we are interested only on the evolution of

beam envelop rm along z

022 32222

22

22

rcm

pr

mc

qBrrr

022

2

22

mmmm r

mc

qBrrr

For a simple beam moving at constant energy in a field free region, its

envelop diverges at a constant rate along the beam axis z (i.e. r’m = const.).

20

Recall paraxial ray equation:

With

For pure magnetic field, particle energy conserved. Therefore,

With

Solenoid Magnetic Lenses

021 rzgrzgr

21

zg

22

2

222 c

zg L

0

02 rKr

22

22

2

2 cmc

qBK L

21

Since K2 is always positive and if the particle does not cross the axis inside the

lens field (i.e., r > 0), the lens is focusing.

For thin lens, r can be treated as a constant and taken out from the integral

Further, if f2 = f1, then

On the other hand, the particle is rotated by an angle

But for solenoids, integration of H field along z equals to NI (number of ampere

turns of the coil)

rdzKrrz

z2

1

2

12

dzBmc

qrr

z

z

2

1

2

2

2

dzBmc

q

r

r

f

z

z

2

1

2

2

2

1

2

1

z

zr Kdz

NImc

qr

20

22

Hard edge approximation:

1. The field in the region 0 < z < l is assumed to be uniform and zero outside

this region, where l is the effective length of the solenoid

2. D/l << 1

From energy conservation law,

dzzBlB

22

0

dzzBB

l

2

2

0

1(44)

effective length

23

Initial conditions at z = 0, (i.e. )

At the end of the solenoid,

where

The focal length

0,0 rrr

KzKrr

Kzrr

sin

cos

0

0

sin

cos

0

0

l

rr

rr

l

l

mc

qNI

mc

lqBKl

22

00

lr

r

f

l sin1

0

lKl

22

thin lens approximation

Image rotation:

l

r KlKdz0

(45)

(46)

(47)

24

Effects of a Lens on Trace Space Ellipse

and Beam Envelope

0

1 2

3

4

x’

x

Uniform Beam Model The beam is assumed to have a sharp boundary

The uniformity of charge and current densities assures that the

transverse electric and magnetic self-fields and the associated

forces are linear functions of transverse coordinates (see below)

This beam model allows us to extend the linear beam optics to

include the space charge forces.

constJ

constinside the boundary

0 J everywhere outside the boundary

a b

beam pipe charged particle beam

evolution of beam envelop

along the propagation

direction is exaggerated!!

• A axisymmetric laminar beam

• Particle trajectories obey

paraxial assumption that the

angle with the z-axis is small.

• The variation of beam radius

along z-axis is slow enough

that Ez and Br can be neglected.

Based on the above assumptions, we have J, and vz v are all

constant values across the beam.

Therefore, with denoting the charge density of the

beam, we obtain

Since we assumed the electric field has only a radial component

and by Gauss’ law,

vaI 2

0 /

va

I

a

IJJ z

20

2

for 0 r a

0 J for r > a

(1)

va

IrrEr 2

00 22

for 0 r a (2a)

vr

IEr

02 for r > a (2b)

the Lorentz force

exerted on an electron

in the beam has radial

component only and is

a linear function of r.

The magnetic field, on the other hand has only an azimuthal component,

is obtained by applying Ampere’s circuital law:

20

2 a

IrB

r

IB

20

for 0 r a

for r > a

(4)

2

2

ln21a

r

a

bVr s for 0 r a

r

bVr s ln2 for r > a

(5)

Integrate the electrostatic field along an arbitrary path from r = a to r = b,

IIaVs

30

44 00

2

0 (6)

by setting that = 0 at r = b, where The peak potential on the beam axis (r =

0) is thus . And

the maximum electric field at the beam

edge is

aI

a

VEa /60

2

abVV s ln210 0

The motion of a beam particle in such field is described by the radial force equation

BvEqrmrmdt

dzr

where we dropped the force term that is negligibly small and

because there is no external acceleration. zBqr r .const

Substitution for Er from the first equation of (2), B from the first equation of (3)

and with , , we have 2

00

c cvz

(8)

2

2

0

12

ca

qIrrm (9)

with rcdz

rdvr z

22

2

22 we have

(10) 3332

02 mca

qIrr

Define characteristic current (Alfve’n current) I0 as

q

mc

q

mcI

23

00

30

14

which is approx. 17 kA for electrons and 31 (A/Z) MA for ions of mass number A

and charge number Z.

(11)

Define “generalized perveance” K such that

33

0

2

I

IK

In terms of generalized perveance, the equation of motion can be expressed as

(12)

ra

Kr

2

Under the condition of laminar flow, the trajectories of all particles are similar

and scale with the factor r/a. That is, the particle at r=a will always remain at the

boundary of the beam. Thus, by setting r = a = rm, we obtain

Krr mm

(13)

(14)

022 332

2

32222

22

22

mm

n

m

mmmmr

K

rrcm

pr

mc

qBrrr

From the paraxial ray equation and considering the effects of finite emittance and

linear space charge, we obtained the beam envelop equation:

(15)

Electric field along the axis

define electron phase with respect to rf field as

then

or

acceleration of an electron in the gun is

)sin(cos 00 tkzEEz

0 kzt

00

z

dzv

dzk

00 2 1

dzk

z

center line e-

/4 /2 /2 /2

cathode

• The electron emitter is laser driven photo-cathode

• Operating mode in each cell is TM010 – like. Coupled cavity mode is -mode.

• The cavity cells are on-axis coupled through apertures

• Lengths of cells are about /2 • The first cell is shorter because

electrons move slower there.

(1)

(2)

(3)

For initial phase spread of electron 0 , spread of electron phase after acceleration is

The rf effects on electron distribution in longitudinal phase space can be derived from the approx. expression of at the end of the n+1/2 cavity:

If we write

then the longitudinal emittance by definition becomes

in terms of and

0

0

2

0

sin2

cos1

the bunch length dz = -/k

will in general be compressed during acceleration.

(11)

cossin121 n

zzz

ppp zzz

2222

zpzp zzz

22221

k

z

(12)

(13)

(14)

(15)

at the gun exit

setting to minimize transverse emittance

expand for small ,

we obtain,

for Gaussian distribution,

...!3

12

1 32

f

cossin121n

2

sin1cos

241

1 f

rf

zk

3213 zf

rf

z k

(16)

(17)

(18)

(19)

(20)

Longitudinal emittance scales as the third power of bunch length and as the square of rf frequency.

Assuming the following longitudinal E-field,

Er, B can be found from Maxwell’s equations, that is

The radial force is given by

then the equation describing radial motion is

with

0sincos tkzzEEz

z

ErE z

r

2 t

E

c

rcB z

2

cBEeF rr

rr F

mcdt

dp 1

000 cossin

2sincos

2

1cossin

2

1

tkz

dz

zdEtkz

dz

zdEtkzzE

dt

d

cerFr

Assume the transverse deflection is small, r can be regarded as constant, integrate the equation of radial motion gives the radial impulse and assume pr = 0 at t = 0; v c near cavity exit in Cartesian coordinates, Electron trajectories in trace-space is therefore are lines with different slopes Correspond to different

f

rr dtFmc

p0

1

000 cossinsincos tkztkzkrpp ffrr

sinkrpr here is the rf phase at the cavity exit

xkxpx sin

x

x’

The normalized transverse emittance is therefore and therefore the emittance is minimized when <> = 90 such that for Gaussian distribution,

222222 sinsin xkxpxp xx

rf

x

2

22424

222 sin

4

1cos

3

1xkrf

x

2

24

2

2

xkrf

x

2

22

xkrf

x

The equation of motion of a particle accelerated by a traveling-wave

where the phase of the traveling-wave is

the phase motion is

expressing in the equation of motion in , we have

tzqEdt

d

d

dmcmc

dt

d,cos0

3

zttz

2,

1

2 c

dt

d

Since <1, increases with time. The particle falls behind the initial phase on the wave

cos

211

12

0

2 mc

qE

d

d

integration

1

1

1

12sinsin

0

2

i

ii

qE

mc

at asymptotic phase ( 1), if we want to set at the rf crest for efficient acceleration, the condition is

i

ii

qE

mc

1

12sinsin

0

2

i

ii

qE

mc

1

12sin

0

2

injected beam

asymptotic beam

E

• Injected bunch: all of the electrons distributed on a single equi-potential line the ideal case

• rf nonlinearity leads to asymmetric distribution of longitudinal phase space the limitation of compression in bunch length

• Solution injection of lower energy beam !!

• flat region covered widely

• Benefit to relieve this limitation

Generally, thermionic rf gun ~ 2 MeV ;

photocathode gun ~ 4 MeV

nonlinearity of the equi-potential line

at higher injection energy is stronger

(1) γ=10, 3 ps γ=33, 335 fs,

(2) γ=5, 3 ps γ=49, 135 fs

39 Thermionic rf gun injector is suitable with

velocity bunching method for bunch

compression.

RF Bunch Compression in an RF Linac