Accelerator Physics Topic IX Wigglers, Undulators, and FELs

J. J. Bisognano

Topic Nine: Wiggler to FELs

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UW Spring 2008

Accelerator Physics

Accelerator PhysicsTopic IX

Wigglers, Undulators, and FELs

Joseph Bisognano

Engineering Physics &

Synchrotron Radiation Center

University of Wisconsin

J. J. Bisognano


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Bending Magnet Radiation

CERN School 1998

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Wiggler or Undulator (Insertion Devices)

CERN School 1998

More flux or higher brightness

Wigglers: high field, broad spectrum

Undulators: low field, interference peaked spectrum

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Insertion Devices

CERN School 1998

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Light Source

CERN School 1998

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Ideal ID Field Pattern(infinite pole tips in x)

0divBby )sin()sinh(~

)cos()cosh(~

)cos()cosh()cosh(

)cos()sinh(),(

)cos()()2cos()(),(

potentialscalar Magnetic

0

skzkBB

skzkBB

skzkB

B

skzkAzs

LaplaceMaxwell

skzfzfzs

uus

uuz

uugz

uu

us

u

u

Gap and period go hand in hand

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Gap Dependence of Magnetic Field

CERN School 1998

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Equation of Motion of Electrons in IDsNeglecting vertical motion, we have

)(

)(

sBme

xs

sBme

sx

z

z

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First Order Solution

)cos(4

~)(

)sin(2

~)('

1);cos(~

"

';,

2

2

skcm

Besx

skcm

Besx

withskcmBe

x

cxxcssx

Assume

uu

uu

u

Since there is a Bs, one can get a vertical force; i.e., focusing

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Basic Parameters

1

1

2

~1

KWiggler

ceinterferenKUndulator

anglenaturalKK

mcBe

AngleMaximum

u

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Second OrderEnergy Conservation says that if x is moving it’s at the expense of longitudinal energy

)2sin(8

*)(

)cos()(

)2cos(4

*)(

)sin()(

]2/1[2

11*

2

2

2

2

22

tk

Kctts

tk

Ktx

tcK

cts

tKc

tx

K

uu

uu

u

u

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In Beam Frame

Beam frame coordinates t and frequency

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Lorentz Transforms and Radiating

frame lab the into back go to need weNow

ck or c/frequency a

withoscillated it c, almost going stillsit' Since

/period withwiggler

shortened seeselectron frame, beam In

ubeam

,

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Photon Frequency in LabExpect a “blue” shift since waves get pushed together as beam is moving toward observer

Use fact that energy of photon is hf, momentum is hf/c

knobstuninggiveandK

Kck

ck

ulab

ulab

beamlab

)2/1(2

)cos1(

)cos1(

2222

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Undulator SpectrumSince train is of finite length (N cycles), there is a width to spectrum, but it is very narrow, order 1/N

If one includes that motion is really not perfectly sinusoidal (remember the figure 8 and energy modulation) but that it does repeat in time, there is harmonic generation

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Cern SchoolHigher harmonics add to reach of an undulator

Require care in phase errors of undulator periodic fields

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Fundamental power/total power=1/(1+K2/2)1/2Cern School

R Walker

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Spontaneous Emission

w

w

ww

w

N

N

mc

eBK

Kc

11

12

)2/1/()/2(2

undulator)(planar radiation sSpontaneou

2

220

Note that for higher frequency, you need higher energy or shorter undulator period

Shorter undulator period implies smaller gap

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R Walker, CERN School

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Brightness/Brilliance

L

L

FluxB

AreaPhaseSpaceFlux

Brightness

RRxx

RRxx

zzxx

2

4

2/

)2(

/

22

22

2

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Physics of FELs• An electron beam moving on a linear trajectory

will have no net energy coupling to a co-moving E&M wave, just “jiggled”

• In a wiggler (really undulator), an electron beam develops a transverse oscillation, as we’ve just seen

• If the oscillation stays in phase with the fields, there can be a net exchange of beam energy to the wave; i.e., the electron beam acts to amplify the electromagnetic wave

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Oscillators and SASEs• If one puts beam/wiggler into optical resonantor, there is a

feedback loop that generates an oscillator and a laser

• If the wiggler is long enough, the energy modulation of the electron beam can generate “microbunches” which can radiate coherently, generation self-amplified spontaneous emission (SASE) from the Schottky noise on the beam, lasing without mirrors from a beam instability

• Or one can “seed” the beam with an energy modulation induced by an external laser

• Sources are tunable (beam energy or wiggler field) and coherent

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Basic FEL Configuration•

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Jlab FEL

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Spontaneous Emission

w

w

ww

w0

N

N

mceB

K

Kc

wiggler)(helical radiation sSpontaneou

11

12

)1/()/2(2

2

22

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FEL Dynamics I

)sinˆcosˆ(

21

1

)sinˆcosˆ(

2

2

0

0

skyskxK

K

s

skyskxBB

Consider

ww

wwww

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FEL Dynamics II

)2

11(

;)(

sin

Emc

e

Eˆ

)cos(ˆ)sin(ˆ(EE

radiation polarized circularly

2

2

00

0

000

K

k

k

cswithtskk

mc

KeE

sB

tksytksx

Suppose

ww

w

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FEL Dynamics III

period radcc

cc

c in photon

c

time in period one through moves Electron

K

0 if exchangeenergy net get llWe'

www

w

w

ww0

0

00

0

0

22

1//

/,

/

)1/()]1/(2[

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Another Pendulum Equation

0sin

2

2

sin2

)1(2

2

202

2

2

r

w

w

w

r

r

wr

mc

KeE

Then

K

Let

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CERN School

Gain only when energy of beam doesn’t quite match “ideal” energy

If wiggler is two long, process reverses, unless wiggler is “tapered”

η

φ

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FEL parameters

•

eadenergy spr smallneed1/N is height Bucket

mKVNe

NfNG

VNmcG

field inenergy lossenergy electron

G

W

w

ww

E

,

16)/(4

)4()4(4

)/()(

22

223

3

822 2

0

Need high beam density

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SYLee Text

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SYLee Text

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SYLee Text

Looks like derivative of undulator power spectrum: fluctuation-dissipation or Madey’s theorem

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High Gain Regime

So far, we haven’t included how the increasing electromagnetic wave affects the continued electron motion

Also, there is a density variation developing

Also, at high enough frequencies there are no good mirrors to make an optical resonator

“High Gain” regime, really an instability saves the day, and points to X-ray lasers

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Basic Principle: Coherent Synchrotron Radiation

If we can get “microbunching” of electron beam, strong enhancement over incoherent

synchrotron radiation

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High Gain FEL to the Rescue: Basic Feedback Loop

• Electron beam responds to co-traveling electromagnetic wave in a wiggler/undulator

– Electrons radiate by stimulated emission in wiggler– Electrons move relative to each other: density variations at wavelength of

radiation

• Density variations radiate coherently in wiggler/undulator• Electromagnetic field is enhanced, with changes to both its amplitude and

phase• Electron move relative to each other in response to to co-traveling

electromagnetic other: density variations grow at wavelength of radiation

• Genuine instability with exponential growth of both the density variation and the electromagnetic radiation

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Further Details

• Can send beam through a dispersive compressor where the microbunching through energy variation is enhanced, “optical klystron”

• Generates higher harmonics

• Since Schottky (shot) noise is “noisy,” can instead seed with laser

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Zhirong Huang, SLAC

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Max Cornacchia,SLAC

LCLS Project Overview

BESAC, Feb. 26-27, 2001

LCLS layout

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(fs)

The SASE radiation is powerful, but noisy!

Solution: Impose a strong coherent modulation with an external laser source

A SASE FEL amplifies random electron density modulations

Spectrum From a SASE FEL

Graves

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Bill Graves

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Brookhaven Laser Seeding Demonstration

Buncher

e-

Laser

800 nm

Modulator

266 nm

output

Radiator

•Suppressed SASE noise

•Amplified coherent signal

•Narrowed bandwidth

•Shifted wavelength

High Gain Harmonic Generation (HGHG)

SASE x105

HGHG

L.H. Yu et al., Phys. Rev. Lett. 91, 74801 (2003).

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To Produce Transform-Limited Hard X-ray Pulses

Use “cascaded” High Gain Harmonic Generation methods

Input

seed 0

1st stage 2nd stage …Nth

stage

Stage 1 output at

50 seeds 2nd

stage

Stage 2 output at

250 seeds 3rd

stage

…Nth stage

output at 5N0

W. Graves, MIT

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Key facility elementsKey facility elements

Photoinjecto

r

SRF

linac

SRF

linac

Bunch

compress

orEbeam

switch

Undulators

Photocathod

e laser

Bunch

compress

or

Seed laser

W. Graves, MIT

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High Harmonic Generation (HHG) Seeding

Courtesy of B. Sheehy

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HHG laser seeds at 100 eV Mod 1

100 eV 100 eV 200 eV

Rad 1

Buncher magnets

1. Initial seed is 3 nJ at 100 eV.

2. Radiator 1 amplifies the seed laser.

3. Buncher magnets control the power in each succeeding section by changing the magnitude of harmonic bunching.

1240 eV FEL seeded by 100 kW at 100 eV

2.5 GeV ebeam

1.2 GW @ 1200 eV3 m 1.5 m 2 m 22 m

Spent ebeam is dumped

Fiber link synchronization

1200 eV600 eV

3 m

Rad 2 Rad 3 Rad 4

Graves, MIT

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•Transform-limited output – longitudinal and transverse

•Many beamlines operating simultaneously

•Complete tunability from 6 – 1200 eV

•Fully tunable polarization

•Peak power and brilliance much larger than current XUV sources

•Average flux and brilliance much larger than best synchrotrons and ERLs

•Synchronization of ~10 fs to user lasers

Performance GoalsPerformance Goals

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Three Standard Operating Modes

• Single-shot—Experiments that require the highest available peak brilliance/flux and cannot be cycled rapidly.

• kHz-class experiments—often driven by pump lasers and operate from 10-1000 Hz. Requires CW SC linac.

• MHz-class experiments—includes experiments which can cycle rapidly, where time constants of interest are less than a micro-second. Also includes experiments in the energy domain needing high energy resolution and high flux. Requires CW SC linac and gun.

• All available at the same time

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Breakthrough Science

SRF Electron Injectors Superconducting

Electron Accelerator

FEL Undulatorss

Experimental AreasJacobs and Moore, SRC

Monochromators

RF Separation

Time Resolved Imaging and Coherent ScatteringTaking advantage of the short duration and variable polarization of the FEL x-ray pulse, this technique is particularly suited to study magnetization dynamics. Examples include new materials for high-speed high-density magnetic storage devices.

Resonant Inelastic X-ray ScatteringThis is a powerful technique for studies of low energy electronic and magnetic excitations in materials.

FemtochemistryAllows chemists to follow the dynamics of chemical reactions over extremely short time scales. This may enable chemists to better control reactions to create new products.

Biological SystemsComplex biological processes (for example, photosynthesis, or the transport of information from the eye to the brain) can be studied in snap shot experiments utilizing the precise time pattern and tunability of the FEL.

Exotic Materials, Clusters and NanostructuresThe FEL can be used to the selectively fabricate of atomic clusters and other nanostructures (a billionth of a meter in scale) with specifically tailored medical or material properties.The FEL can also be used to characterize the properties of these new nanoscale materials.

Ultrahigh Resolution SpectroscopyPhotoemission spectroscopy is thetool of choice to study highly correlated systems such as high Tc superconductors, now done with energy resolution in the meV range. With a pulsed FEL source energy resolution of several 10 μeV should be possible.

Bunch compressors

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Homework Problems

• In the text, the vertical focusing in an undulator is derived from a hamiltonian. From a more newtonian approach in an expansion in 1/γ show that there is vertical focusing when the expansion is carried out to second order.

• Starting at equation 5.17 of Lee, fill in the details to get to equation 5.32.

• Lee 5.1.1

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Accelerator Physics Topic IX Wigglers, Undulators, and FELs