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Longitudinal Beam Physics on UMER

Patrick O’Shea and the UMER TeamJune 23, 2008

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Outline

• Importance of longitudinal structure• Some gun physics• Launching waves• Speed of sound• Beam transport

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Longitudinal (time)structure

Real electron beam distributions are not named after famous people!

time

current

Real beam

Gaussian beam

Real beam = Asymmetric Bactrian Camel with Attendant Obelisk Distribution

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Problem StatementProblem:

Real beams have unwanted velocity (energy) or density modulations

Reasons:

• Drive laser fluctuations in photoinjectors

• Over-focusing or under-focusing in longitudinal focusing systems

Technique:

• Introduce perturbation “deliberately” in an intense beam and study its evolution.

•UMER provides the platform for such experiments

Real Beam Pulse

Cur

rent

(A)

Ideal Beam Pulse

Time

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Sources of problems

bends

Cavities and structures

electron source, low energy injector/linac,

Where HowWhat

Bunch structure can excite coherent synchrotron radiation

Coherent Synchrotron Radiation (CSR)

Bunch structure can excite high frequency modes

Wakefields/Higher OM

Space charge converts density modulation to energy modulations, and causes time dependant defocusing

Space charge (self) fields

Cau

sal c

onne

ctio

n

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Two Types of Longitudinal Problems on UMER

• Longitudinal effects in the gun and space charge driven instabilities

• Longitudinal effects in in multi-turn transport: evolution of perturbations

These are relevant to other accelerators

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1. Longitudinal structure and space charge instabilities in the gun

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Space-charge driven longitudinal beam breakupD.H Dowell et al. Phys Plasmas, 4, 3369 (1997)

0.5 nC

1.0 nC

2.0 nC

3 nC

4 nC

5 nC

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φ = 0z = 0

φ = +Voz = DEAC A

Child- Langmuir Limit Revisited

CA

Conventional Child-Langmuir Analysis:Uniform 1-D current distributionSteady statePulse length τp >> Electron transit time T

Real Photoinjector Analysis:non-uniform 3-D time-dependant current distributiontransient statePulse length τp < Electron transit time T

e-

C

A

e-

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Cathode – Anode Transit Time (T) in guns

2

0

2D mTeV

=

Non-relativistic

0

D A / C gap 2.5 cm in UMERV Gun voltage 10 kV

= == =

For UMER gun T ≈ 1 ns

f 2f

A

1mc 1T

eE

γ −γ

=

Relativistic

f

A

relativistic factor at gun exit 3 for 1 MeVE = average applied electric field 30 MVγ = ≈

≈

For Relativistic gun T ≈ .15 ns

For a typical RF photoinjector τp ≈ 0.01 ns

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Charge per Pulse vs Drive Laser IntensityExperimental data

1.Ágúst Valfells, D. W. Feldman, M. Virgo, P. G. O'Shea, and Y. Y. Lau, “Effects of pulse-length and emitter area on virtual cathode formation in electron guns”, Phys. Plasmas 9, 2377 (2002)

5-kV gunTota

l Cha

rge

in C

urre

nt P

ulse

[nC

]

Relative Laser Intensity

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Critical Current (Jcrit) in short pulse modefor onset of space charge instabilities

T= τt pX /2

3

31 142

− −=

t

CRIT CLt

XJ J

X

X

2

2

31 142

− −=

t

CRIT CLt

XQ Q

X

1<<tX3

4≈CRIT CL

t

J JX

34

≈CRIT CLQ Q

1.Ágúst Valfells, D. W. Feldman, M. Virgo, P. G. O'Shea, and Y. Y. Lau, “Effects of pulse-length and emitter area on virtual cathode formation in electron guns”, Phys. Plasmas 9, 2377 (2002)

Short pulse mode

Critical current density:

Critical charge density:

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UMER is a unique test bed for studying the evolution of current perturbations

Thermionic only, 100ns pulseThermionic + Photoemission ( 5ns pulse)

Drive Laser

Heater Photocathode

Electron Beam

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UMER Drive Laser Setup

Nd:YAG Laser

KTP/BBO Crystals

Electron Gun

UV (355nm) LaserFWHM = 5nsE-beam

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Time (ns)

-50 0 50 100 150

Nor

mal

ized

Cur

rent

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Laser induced perturbation

Space Charge Instabilities in the UMER Gun

Jayakar Charles Tobin Thangaraj

Perturbation below the critical current density

Time (ns)

0 20 40 60 80 100 120

Nor

mal

ized

cur

rent

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

5.8 ns

Perturbation splitting into sub pulses

At large laser power

Perturbation above the critical current density

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Perturbation apparent

No perturbation

Relative Laser IntensityTota

l Cha

rge

in C

urre

nt P

ulse

[nC

]

Charge per pulse vs laser intensityOnset of longitudinal instability

Experimental data

1.Ágúst Valfells, D. W. Feldman, M. Virgo, P. G. O'Shea, and Y. Y. Lau, “Effects of pulse-length and emitter area on virtual cathode formation in electron guns”, Phys. Plasmas 9, 2377 (2002)

34

≈CRIT CLQ Q

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Normalized Pulse Length [τp/Ttransit]

Critical Current Density vs. pulse lengthSimulation

Nor

mal

ized

Cur

rent

Den

sity

(Jcr

it/JC

L)2

3

31 142

− −=

t

CRIT CLt

XJ J

X

T= τt pX /

1.Ágúst Valfells, D. W. Feldman, M. Virgo, P. G. O'Shea, and Y. Y. Lau, “Effects of pulse-length and emitter area on virtual cathode formation in electron guns”, Phys. Plasmas 9, 2377 (2002)

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2. Evolution of longitudinal structure in beam transport

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Time (ns)

-50 0 50 100 150

Nor

mal

ized

Cur

rent

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Laser induced perturbation

Experimental Results of Laser-induced Space Charge waves

Near gun

UMER Top View

Time (ns)

-50 0 50 100 150

Nor

mal

ized

Cur

rent

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Slow waveFast wave

At s = 5.75 m

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Evolution of Multiple Pulses

Bergoz (62.6 cm)BPM 0 (82.6 cm)

BPM 2 (258 cm)

BPM 1 (194 cm)

BPM 3 (323 cm)

BPM 4 (386 cm)

BPM 5 (450 cm)

BPM 6 (514 cm)

BPM 7 (578 cm)

BPM 8 (642 cm)

BPM 9 (706 cm)

BPM 10 (770 cm)

BPM 11 (834 cm)

BPM 12 (898 cm)

Modulation observed to disappear,

return, then start to disappear again

as beam travels through UMER

John Harris,PhD dissertation 2005https://drum.umd.edu/dspace/handle/1903/2906

Gun

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Theory of Space Charge Waves(1-D Cold fluid model)

( ) ( )

( ) ( )

( ) ( )

0 1

0 1

0 1

Space charge line density ,

Velocity v , v v

,

−

−

−

Λ = Λ + Λ

= +

= +

i t kz

i t kz

i t kz

z t e

z t e

Current I z t I I e

ω

ω

ω

Beam2a 2b

z

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Derivation of Sound Speed

( ) ( )

( ) ( )

( ) ( )

0 1

0 1

0 1

,

v , v v

,

i t kz

i t kz

i t kz

z t e

z t e

I z t I I e

ω

ω

ω

−

−

−

⎧Λ = Λ + Λ⎪⎪ = +⎨⎪

= +⎪⎩

Definition of perturbation

( )

0 30

v0

v vv s

z tq E

z t mγ

∂ Λ⎧ ∂Λ+ =⎪⎪ ∂ ∂

⎨∂ ∂⎪ + =⎪ ∂ ∂⎩

ContinuityequationMomentum equation

(one-dimension cold-fluid model)

20

14s

g IEz c tπε

∂Λ ∂⎛ ⎞= − +⎜ ⎟∂ ∂⎝ ⎠2ln bg

a=Maxwell’s equation and

boundary conditions

( )2 2 20v 0sk C kω − − =Dispersion equation

Phase velocity of fast/slow waves

0

0v v

v v

+

+

= +

= =

=

−s

s

s

f C

Ck

kω

ω

M. Reiser, Theory and Design of Charged Particle Beams, Chapter 6, (2008)

Cs = Sound speed

For UMER Cs ≈ 106 m/s

5 4

=so o

egIor Cmvπε γ

05

0 04Λ

=sqg

Cmπε γ

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Evolution of Space-Charge WavesFast (Forward) and Slow (Backward) Waves

( )

( )

0

0 0

01

1

0

,2

v

v

2 v,

v

v

s

s

s

s

s

zh tC

zh tC

zh zh

z

t t

t

CCC

z t

η

η

⎡ ⎤ΛΛ = +⎢ ⎥

⎣ ⎦⎡ ⎤

=

⎛ ⎞−⎜ ⎟+⎝ ⎠

⎛ ⎞−

⎛ ⎞−⎜ ⎟−⎝

⎜

⎠

⎛ ⎞− ⎟+⎝ ⎠

⎜ ⎟− +⎢ ⎥⎣ −⎝ ⎦⎠

( ) ( )( ) ( )( ) ( ) ( )

1 0

1 0

1 0

v 0, v0,0,

p

p

p

t tI t I t

t t

δ

η

η δ

⎧⎪⎪⎨⎪⎪⎩

=

=

Λ = − Λ

Definition of perturbation

Algebraic equations of

Assume pure density perturbation 0δ =

Red = slow (backward) waveBlue = fast (forward) wave)

h function is a wave that depends of the initial conditions etc.

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Evolution of Space-Charge Waves

Fast waveSlow

wave

Fast wave

Slow wave

Line Charge Density – (Λ1) Velocity – (v1)

( )

( )

0

0 0

01

1

0

,2

v

v

2 v,

v

v

s

s

s

s

s

zh tC

zh tC

zh zh

z

t t

t

CCC

z t

η

η

⎡ ⎤ΛΛ = +⎢ ⎥

⎣ ⎦⎡ ⎤

=

⎛ ⎞−⎜ ⎟+⎝ ⎠

⎛ ⎞−

⎛ ⎞−⎜ ⎟−⎝

⎜

⎠

⎛ ⎞− ⎟+⎝ ⎠

⎜ ⎟− +⎢ ⎥⎣ −⎝ ⎦⎠

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Space Charge Wave Transport Experimental Results from UMER

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Time (ns)

-50 0 50 100 150

Nor

mal

ized

Cur

rent

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Laser induced perturbation

Experimental Results of Laser-induced Space Charge waves

Near gun

UMER Top View

Time (ns)

-50 0 50 100 150

Nor

mal

ized

Cur

rent

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Slow waveFast wave

At s = 5.75 mCs (exp) = 1.54*106 m/s

Cs (theory) = 1.30*106 m/s

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Multiturn observation of waves in UMER

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Experimental observation of waves in UMERMultiturn observation of waves in UMER

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Time (ns)

-50 0 50 100 150

Nor

mal

ized

Cur

rent

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Laser induced perturbation

Time (ns)

-50 0 50 100 150

Nor

mal

ized

Cur

rent

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Slow waveFast wave

Time (ns)

0 20 40 60 80 100 120

Nor

mal

ized

cur

rent

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

5.8 ns

Perturbation splitting into sub pulses

UV (355nm) FWHM= 5nsNd:YAG Laser

Heater Photocathode

Electron Beam

Space Charge Waves & Instabilities

At s= 5.75 m

At large laser power: explosive splitting in gun

At low laser power: gentle splitting in transport

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Space Charge Converts Density Modulation into Energy Modulations

Q = 0.16 nC

Experiment at Brookhaven DUV FEL after perturbed beam is accelerated to 75 MeV

Jonathan Neumann, Dissertation 2005https://drum.umd.edu/dspace/handle/1903/2437

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Is the inverse Humpty-Dumpty Effect possible ?

Illustration by John Tenniel From Through the Looking-Glass.

Humpty Dumpty sat on a wall.Humpty Dumpty had a great fall.All the king's horses and all the king's menCouldn't put Humpty together again.