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$: SLAC Summer School on Electron and Photon Beams · PDF fileSLAC Summer School on Electron and ... Match the beam to the radiation size and minimize diffraction SSSEPB, July 22-26,$
SLAC Summer School onElectron and Photon Beams

Tor RaubenheimerLecture #3: High Gain FEL’s

2

Outline

• Synchrotron radiation• Bending magnets• Wigglers and undulators• Inverse Compton scattering• Free Electron Lasers• FEL Oscillators• High gain X-ray FEL’s• Coherence and Seeding

SSSEPB, July 22-26, 2013

Lecture #1

Lecture #2

Lecture #3

3

Types of FEL’s


zz

xx

High Gain Free Electron Laser Principle

• Due to sustained interaction, some electrons lose energy, while others gain energy modulation at 1

• e losing energy slow down, and e gaining energy catch up density modulation at 1 (microbunching)

• Microbunched beam radiates coherently at 1, enhancing the process exponential growth of radiation power

uu

ee11

xx--rayray

• Electrons slip behind EM wave by 1 per undulator period ( u)

+ + +

+ + +

KK//

vvxxEExx < < 00 vvxxEExx > 0…> 0…

+

SSSEPB, July 22-26, 2013 4

5

Early LCLS Gain Measurements


6

1-D Wave Equation

Allowing the field to change slowly (>> u) along the undulator, the wave equation in 1-D is

whereand

In the 1-D approximation, we ignored the self-fields (space charge) from the beam and diffraction effects of the radiation as well as the matching of the beam and radiation fields.To solve this, we drop the 2nd derivatives because the field is assumed to be changing slowly where J1 is the component of thecurrent at the radiation wavelength

tJtzE

tczx

x 02

2

22

2

),(1

)](exp[),(),( tkzitzEtzEx )cos(),( zkKJtzJ uzx

1

~0

~

4),(1 JKctzE

tcz r

7

1-D FEL Equations

Now we have the pendulum equations for the motion of the particles within the radiation bucket which depends on a varying electric field but the field and current equations depend on the particle distribution

2N+2 equations


~

10

~

1

_

1

~

22

~

4

)exp(2

)exp(

... 1 where2

JcKEkz

iN

JJ

imc

EeKdz

d

Nnkdz

d

ru

n

N

nz

nr

n

nun

#2, p. 23

8

1-D Solution (I)

Introduce dimensionless variables:

then neglect and linearize in terms of collective variablesa


3/12

222

~

222

2/1][

41

2][

/ˆ and 2ˆ

KJJK

kII

Emc

JJeKa

zkz

rA

u

)exp(21

ˆ

)exp(ˆˆ

ˆˆ

n

nn

nn

iaz

cciazd

dzd

d

)exp(ˆ)exp(

nn

n

iP

ibCompex field

Bunching parameter

Collective momentum iPzd

db

bzd

da

ˆ

ˆ

ˆa

zddP

and

R. Bonifacio, C. Pellegrini, and L.M. Narducci, “Collective instabilities and high-gain regime in a free-electron laser,” Opt. Commun., 50, 373 (1984).

9

1-D Solution (II)

This has a general solution of the form withhas three roots: one oscillatory, one damped and one growing.

The equation can be futher generalized to include the detuning and energy spread

where is the fractional detuning ( - r)/ r and is the width of a uniform energy spread in units of

The high gain FEL differs from the small signal gain in that the highest gain occurs on resonance


)ˆ exp(~ zia 13

10

Evolution of the FEL Microbunching

FEL bucket grows rapidly asthe beam begins to bunch andradiate coherently. After a ¼ synchrotron oscillation the bunching begins to decreaseagain and the FEL saturates

SSSEPB, July 22-26, 2013 From P. Schmuser et al.

11

Gain Length and Lethargy

After some initial mode competition, the exponentially mode will grow.The 1-D power gain length is given in terms of

At the beginning, the growingmode is competing with theoscillatory and the dampedmodes field is roughlydivided between modes andit takes 1 ~ 2 gain lengths to get started


34

uG0L

12

Gain versus Detuning

In the low gain FEL the maximum gain arises when the FEL is slightly detuned and the beam energy is higher than the amplified signal.

In the high gain case, this differs and maximum gain arises on resonance however it takes somelength for this exponential modeto dominate


Growing mode

Damped mode

Low Gain

High Gain

13

Small Gain Limit of High Gain FEL

SSSEPB, July 22-26, 2013 From P. Schmuser et al.

14

Energy Spread and Emittance Effects

Energy spread needs to be small compared to the width of the gain curve

Tranverse emittance has two effects:

1. Increase the longitudinal slipage.

where x and y are the acceleratoroptical functions NOT velocity and Jx andJy are the particle transverse amplitudes

2. Match the beam to the radiation size and minimize diffraction


y

y

x

xz

JJK2

2

22/11

)1( 2, Kyxeff

4

15

3-D Parametrization

Many effects increase the 1-D gain length: energy spread, emittance, diffraction, finite bunch length, … These effects can be approximated as LG = LG0 (1 + ) where:


Xie, M.: Exact and variational solutions of 3D eigenmodes in high gain FELs.Nucl. Instr. Meth. A 445, 59 (2000) 92, 93

andFinally, the saturationpower is estimated toscale as:

SASE FEL Micro-Bunching Along Undulator

UCLAUCLA

S. ReicheSASE* FEL starts up from noise

log log (radiation power)(radiation power)

distancedistance

electron beam

photon beam

beam dumpbeam dumpundulatorundulator

* Self* Self--Amplified Amplified Spontaneous Spontaneous EmissionEmission

Power comes from last gainlength Lg

SSSEPB, July 22-26, 2013 16

17

SASE Pulse Evolution

Starts from noise but as the beam propagates, thespectrum narrows. However, the light is still temporally chaotic unless the correlation length iscomparable to the bunchlength

SCSSspectrum

LCLS Simulation

Initial measurements of e-beam and x-ray temporal profiles using the XTCAV

Data taken on June 4th, 2013.Beam energy 3.5GeV, 150pC.Temporal resolution is about 1.7fs rms in this test.

Preliminary results.

FEL Off FEL On


SASE Temporal Coherence

Single pass SASE FEL starts from noise

Slippage length is ~ (Lg * u)

• No phase correlation between portions

Would like to have transform limited pulsesfor many experiments Y. DingY. Ding

Z. HuangZ. Huang15 Å15 Å,2.42.4 10101111 photons,photons,IIpkpk = 2.6 kA,= 2.6 kA,

0.4 µm0.4 µm 1.2 fs1.2 fs

Simulation at 15 Å based on measured injector &linac beam & Elegant tracking, with CSR & 20 pC.

Y. DingY. DingZ. HuangZ. Huang

Simulation at 1.5 Å based on measured injector &linac beam & Elegant tracking, with CSR, at 20 pC.

1.5 Å1.5 Å,,3.63.6 10101111 photonsphotonsIIpkpk = 4.8 kA= 4.8 kA

0.4 µm0.4 µm

SIMULATED FEL PULSESSIMULATED FEL PULSES


SASE

Seeded

20

Transverse Coherence

Radiation can be decomposed into transverse modes. The TEM00 has the highest intensity on axis while other modes extend out radially. The fundemental TEM00 mode grows fastestand rapidly dominates although the othermodes can catch up once into saturation


In the LCLS

Slide 21


Linac Coherent Linac Coherent Light Source FacilityLight Source Facility

Injector at 2-km point

Existing Linac (1 km)(with modifications)

First Light April 2009, CD-4 June 2010First Light April 2009, CD-4 June 2010

New e Transfer Line (340 m)

X-ray Transport Line (200 m)

UCLAUCLA

Undulator (130 m)

Near Experiment HallNear Experiment Hall

Far Experiment HallFar Experiment Hall

LCLS Concept: Fourth Generation Workshop21 Years Ago

C. Pellegrini, A 4 to 0.1 nm FEL Based on the SLAC Linac,Workshop on Fourth Generation Light Sources, February, 1992

Herman Winick’s Study GroupClaudio Pellegrini Herman Winick

22SSSEPB, July 22-26, 2013

Engaged Bjorn Wiik andGerd Materlik duringsabbaticals at SLAC

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