lecture 5. self-amplified spontaneous emission. flash and

Lecture 5. Self-amplified spontaneous emission. FLASH and the European XFEL in Hamburg

X-Ray Free Electron Lasers

Igor Zagorodnov

Deutsches Elektronen Synchrotron

TU Darmstadt, Fachbereich 1826. June 2017

PD Dr. Igor Zagorodnov | X-Ray Free Electron Lasers. Lecture 5 | 26. June 2017 | Seite 2

Contents

� Motivation

� Shot noise in electron beam

� Current modulation from shot noise

� FEL start up from shot noise

� Statistical properties of SASE radiation

� FEL facilities

� Outlook


Motivation

Electrons produce spontaneous undulators radiation

How to obtain a useful external field ?

SASE

A. Kondratenko, E. Saldin, Part. Accelerators 10, 207 (1980)

R.Bonifacio et al, Opt. Comm.50, 373 (1984)


MotivationLow -energy undulator test line (LEUTL), USA, 530nmSASE FEL operation in the visible and near-ultraviolet range wasaccomplished in 2001 at the low-energy undulator testline LEUTL at ArgonneNational Laboratory near Chicago, USA


MotivationTESLA Test Facility (TTF), Hamburg

In 2001 a successful SASE experiment was carried out at DESY in Hamburg at the vacuum-ultraviolet (VUV) wavelength of 109 nm.


Shot-noise in electron beam

Fluctuations of the electron beam current density serve as the input signal in the SAS EFEL

Laser pulse Spectrum

~ω ρω∆

ω ω∆[ . ]t a u

( )P t ( )P ω



The electron beam current (at the undulator entrance) consists from electrons randomly arriving at time tk

1

( ) ( )N

kk

I t e t tδ=

= −∑The electron beam averaged over an ensemble of bunches

( ) ( )I t eNF t≡The electron beam profile function can be, for example,

[0, ]1

( ) ( )r TF t tT

χ=2

221( )

2T

t

gT

F t e σ

πσ

−=

[0, ]1, 0 ,

( )0, otherwiseT

t Ttχ

≤ ≤=



In frequency domain

1 1

( ) ( ) ( ) kN N

i ti t i tk

k k

I I t e d e e t t d e e ωω ωω ω δ ω∞ ∞

= =−∞ −∞

= = − =∑ ∑∫ ∫It follows from central limit theorem that the real and imaginary parts are normally distributed

The probability density distribution of spectral power is an exponential distribution

( )2

221, Re ( ), or Im ( )

2x

x

x

p x e x I x Iσ ω ωπσ

−= = =

( ) 21, , ( )

x

p x e x x Iλ λ ωλ

−= = =


Shot-noise in electron beamFirst-order correlation function

( )

'* 2

1 1

' '2 2

1

( ) ( ') k n

k k n

N Ni t i t

k n

N Ni t i t i t

k k n

I I e e

e e e e e

ω ω

ω ω ω ω

ω ω −

= =

− −

= ≠

= =

= +

∑∑

∑ ∑

1

1( ) ( ) ( ) k

Ni ti t i t

kk

F F t e d t t e d eN

ωω ωω ω δ ω∞ ∞

=−∞ −∞

= = − =∑∫ ∫

* 2 2 *( ) ( ') ( ') ( 1) ( ) ( ')I I e NF e N N F Fω ω ω ω ω ω= − + −


Shot-noise in electron beamFirst-order correlation function

* 2 2 *( ) ( ') ( ') ( 1) ( ) ( ')I I e NF e N N F Fω ω ω ω ω ω= − + −2 2

2( )T

gF eω σ

ω−

= ( ) ( )( )

sin 0.5( ) sinc 0.5

0.5rT

F TT

ωω ω

ω= =

* 2( ) ( ') ( ')I I e NFω ω ω ω≈ − 2 2( )I e Nω ≈

*( ) ( ') 1, for 1TNF Fω ω ωσ<< >>

Averaged spectral current density (“white noise”)


Current modulation from shot-noiseWe consider a rectangular averaged current

[0, ]1

( ) ( )r TF t tT

χ= [0, ]1, 0 ,

( )0, otherwiseT

t Ttχ

≤ ≤=

( ) ( )rI t eNF t=

( )( ) sinc 0.5rF Tω ω=

( )( ) ( ) sinc 0.5rI eNF eN Tω ω ω= =


Current modulation from shot-noise

2 2

0 0 0

2

0

1 1 1( ) ( ) ( )

1 1( )

T

r

S d I t dt I dT T

F dT

ω ω ω ωπ

ω ωπ

∞ ∞

∞

= = =

=

∫ ∫ ∫

∫

Spectral power density of averaged current

Parseval's theorem

( ) ( ) ( )2

2~ sinc 0.5 0, for 1I

S T TT

ωω ω ω

π= ≈ >>



( ) ( ) 22

( )shot

IIS

T T

ωωω

π π≡ =

We are interested in an averaged spectral power density of shot noise, which by analogy can be written as

2 2( )I e Nω ≈

( ) 22

0( )shot shot

I e N eIS S

T T

ωω

π π π≡ = = =



The amplification takes place in bandwidth ∆ω and we can replace the power of the current in this bandwidth by power of the averaged current with fluctuations at amplitude

1,shot 0 0shotav

b b

SI ej j

A A I

ω ωπ

∆ ∆≡ = =ɶ

2

0

( )av shot shotI S d Sω ω ω∞

= ≈ ∆∫

22 2

0 0 0

1 1( ) ( ) ( )

T

avI I t dt I d S dT T

ω ω ω ωπ

∞ ∞= = =∫ ∫ ∫

0shot

eIS

π=

Let us introduce an averaged current

The current density reads

0

0b

IA

j=


FEL start up from shot-noise

01

[ ]( )

4x zr

cK JJdE z j

dz

µγ

= −ɶ ɶ

2 , 1,2,...n u nd

k n Ndz

ψ η= =

High -gain FEL model with space-charge

2 2 2

( )[ ]( )

2nin z n

xe r r e

d eEeK JJE e

dz m c m c

ψη ψγ γ

== − ℜ −ɶ

( ) ( )( )0

0 1

( ) sgnN

zz n n m n m

m

jE

Nψ π ψ ψ ψ ψ

ωε == − − − −∑

1 01

2m

Ni

z zm

j j eN

ψ−

== ∑ɶ



23 2

ˆ ˆ2 0x x xx

E E Ei iEη η

′′′ ′′ ′+ + − =

ΓΓ Γ

ɶ ɶ ɶɶ

3( )

1

( , ) ( ) j zx j

j

E z c eα ηη η

==∑ɶ 0 0

0 02

γ γ ω ωηγ ω− −= ≈

1

1 2 3 22 2 2

31 2 3

(0)1 1 1

(0)

(0)

x

x

x

Ec

c E

c E

α α α

α α α

′= ′′

ɶ

ɶ

ɶ

11

2

3

(0)

(0)

(0)

x

x

x

Ec

c E

c E

−

′= ′′

A

ɶ

ɶ

ɶ

ˆηηρ

=

At time t=0



01

[ ](0) (0)

4x zr

cK JJE j

µγ

′ = −ɶ ɶ 01

[ ](0) (0)

4x zr

cK JJE j

µγ

′′ ′= −ɶ ɶ

1 0 01 1

2 22n n

N Ni i

z z n u z nn n

j ij e k ij eN N

ψ ψψ η− −

= =′ ′= − = −∑ ∑ɶ

1 01

2n

Ni

z zn

j j eN

ψ−

== ∑ɶ 2 , 1,2,...n u n

dk n N

dzψ η= =

1 0 1(0) 2 (0)z u zj i k jη′ = −ɶ ɶ0(0) , 1,2,...n n Nη η≡ =

00 1

[ ](0) 2 (0)

4x u zr

cK JJE i k j

µηγ

′′ =ɶ ɶ



11 1 0

2 1

03

(0) 0[ ]

(0) 1 (0)4

2(0)

x

x zr

ux

EccK JJ

c E j

i kc E

µγ

η

− −

′= = −

′′

A A

ɶ

ɶ ɶ

ɶ

11 1

2

3

(0)

(0) 0

0(0)

x in

x

x

E Ec

c E

c E

− −

′= = ′′

A A

ɶ

ɶ

ɶ

Start up from current modulation at t=0

Start up from seed field at t=0



On resonance energy

0r

r

γ γηγ−= ≡ 3

0xx

EiE

′′′− =

Γ

ɶɶ

zxE Aeα=ɶ 3 3iα = Γ

( )1 3 2iα = + Γ

Γ

Imα

Reα

( )2 3 2iα = − Γ3 iα = − Γ

1α2α

3

1

j zx j

j

E c eα

==∑ɶ

*

11 2 32 2 2

1 2 3

1 1 11 1

3 3α α α

α α α

−

= =

*A A



*11

1 *0 02 1 1 2

*3 3

0[ ] [ ]1

1 (0) (0)4 3 4

0z z

r r

ccK JJ cK JJ

c j j

c

αµ µ α

γ γα

−

= − = −

A ɶ ɶ

11

2

3

1

0 13

0 1

inin

EcE

c

c

− = =

A

Start up from current modulation

Start up from seed field

0 0, 1, 0 0

[ ] [ ](0)

4 4in shot z shotr r

cK JJ cK JJ eE j j

I

µ µ ωγ γ π

∆= =Γ Γ

ɶ

12ω ρω∆ ≈



zΓ

b

P

Wρ

SASE vs. seeded FEL

SASE

seeded

beam energybW − gain parameterΓ −


Statistical properties of SASE radiationInterference

CoherenceCoherence is a property of waves that enables interference.

Temporal coherence is the measure of correlation between the wave and itself delayed. it characterizes how well a wave can interfere with itself at a different time. The delay over which the phase or amplitude wanders by a significant amount is defined as the coherence time


Statistical properties of SASE radiation


Statistical properties of SASE radiationCoherence time

1

1 1~ ~cohτ

ω ω ρ∆

The time-averaged intensity (blue) detected at the output of an interferometer plotted as a function of delay. The interference envelope gives the degree of coherence



c cohI

N lce

=

Typical length of one spike

cohl

b b

coh c

L TM

l τ= =

Coherence length

1~coh c

cl cτ

ρω=

Number of cooperative electrons

[µm]s

Laserpuls[GW]P

Number of spikes (longitudinal modes)


6M = 2.6M =


1~ 2λ ρλ∆Spikes in spectrum

V. Ayvazyan et al, Eur. Phys.Journ. D 20, 149 (2002)

Spectrum

long bunch (~100fs) short bunch (~40fs)

( )S ω ( )S ω


Statistical properties of SASE radiationFluctuations of SASE pulse energy

radiation energyradU Pdt= −∫rad

rad

Uu

U= 1u = ( )22 u uσ = − 2M σ −=


Statistical properties of SASE radiationFluctuations of SASE pulse energy (linear regime)

1

( )( )

M MMu

MM u

p u eM

−−=

Γrad

rad

Uu

U= 1

0

( ) z tz t e dt∞

− −Γ = ∫Gamma distribution, M – number of modes


Statistical properties of SASE radiationFluctuations of SASE pulse energy (after saturation, 13 nm, FLASH)


0 10 20 30 400

0.5

1

1.5

2


b

P

Wρ

g

z

L

3 3 lnsatc

g

LN

L= +

SASE with

cN

electrons on coherence length

Saturation length (SASE)


electrons

radiation


Longitudinal profile with large statical fluctuations

Transverse profile is coherent

Coherence


FEL facilitiesTESLA Test Facility ( until 2002)



Three undulator modules. Total length 15m


FEL facilitiesTESLA Test Facility II ( 2002-2006)

From 2003 on, TTF1 was expandedto TTF2, an FEL user facility for thespectral range of soft x-rays, includinga new tunnel and a new experimentalhall (in the foreground). In April 2006,the facility was renamed FLASH: FELin Hamburg


FEL facilitiesFLASH ( from 2006)


Phase space linearizationrollover compression vs. linearized compression

~ 1.5 kA

~2.5 kAQ=1 nC

Q=0.5 nC


Phase space linearization

In accelerator modules the energy of the electrons is increased from 5 MeV (gun) to 1200 MeV (undulator).

FLASH



In compressors the peak current I is increased from 1.5-50 A (gun) to 2500 A (undulator).

FLASH



FLASH

FEL radiation parameters

Wavelength Range 4.1 - 45 nm

Average Single Pulse Energy 10 - 400 µJ

Pulse Duration (FWHM) 50 - 200 fs

Peak Power (from av.) 1 - 3 GW

Average Power (5000 pulses/sec) 400 mW

Spectral Width (FWHM) 0.7 - 2 %

Average Brilliance10^17 - 10^21 photons/s/mrad2/mm2/0.1%bw


FEL facilitiesFLASH 2 ( from 2013)


FEL facilitiesFLASH 2

Photon Beam HHG SASE

Wavelength range(fundamental)

10 - 40 nm 4 - 80 nm

Average singlepulse energy

1 – 50 µJ 1 – 500 µJ

Pulse duration(FWHM)

<15 fs 10 – 200 fs

Peak power (fromav.)

1 – 5 GW 1 – 5 GW

Spectral width(FWHM)

0.1 – 1 % 0.5 – 1.5 %

Peak Brilliance*10 - 40 nm

1028 - 10311028 - 1031


FEL facilities

Intensity distrubution

for λ= 0.14 nm

radiation power ~ GW G.Gutt et al, PRL, 108, 024801 (2012)

E= 3.5-14 GeV

pulse length ~30 fs

LCLS


FEL facilitiesLCLS

P. Emma et al, Nature Photon. 4, 641(2010)

radiation power ~ GW

Pulse length ~30 fs

G.Gutt et al, PRL, 108, 024801 (2012)

λ=1.4 �


FEL facilitiesEuropean XFEL

- kürzeste Wellenlänge

- größte Brillanz



Parameter Value

SASE 1 SASE 2 SASE 3

photon energy [keV] 12.4 - 4.0 12.4 - 3.1 3.1 - 0.2

wavelength [nm] 0.1 - 0.31 0.1 - 0.4 0.4 - 6.4

peak power [GW] 24 22 100 - 135

average power [W] 72 66 300 - 800

photon beam size (FWHM) [µm] 110 110 65 - 95

photon beam divergence (FWHM) [µrad] 0.8 0.8 3 - 27

bandwidth (FWHM) [%] 0.09 0.08 0.28 - 0.73

coherence time [fs] 0.3 0.3 0.3 - 1.9

pulse duration (FWHM) [fs] 100 100 100

average brillance [x10^25, photons/(s mrad^2 mm^2 0.1% bandwidth)]

1.6 1.6 0.52 - 0.03


Linac Coherent Light Source(LCLS)

Spring-8 Angstrom Compact Laser (SACLA)

European XFEL

Standort USA Japan Deutschland

Start der Inbetriebnahme

2009 2011 2017

Beschleuniger –Technologie

normalleitend normalleitend supraleitend

Anzahl der Lichtblitze pro Sekunde

120 60 27 000

Minimale Wellenlänge

0.15 nm 0.1 nm 0.05 nm

Länge 1500 m 750 m 3400 m

FEL facilities


Outlook

� self-“seeding“

� high harmonics of laser light

Methods for improving of coherence

Monochromator


Outlook“Table-Top -FEL”

M.Fuchs et al, Nature Physics 5, 826(2009)

H.-P. Schlenvoigt et al, Nature Physics 4, 130 (2008)

� λ=740 nm

� λ=17 nmspontaneous undulator radiation with a laser plasma accelerator

lecture 5. self-amplified spontaneous emission. flash and

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