Spectral Properties of Synchrotron Radiation

US Particle Accelerator School

January 18-22, 2010

- Motivation

- Conceptual View of SR Emission

- Angular Spectral Power Density

- Practical Applications

- Visible Light

- Undulator Radiation - revisited

Spectral Properties of Synchrotron Radiation

“The Physics of Synchrotron Radiation” A. Hofmann“Synchrotron Radiation” H. Wiedemann

USPAS January 18, 2010 Synchrotron Radiation Properties

Motivation for Understanding Field Pattern

- photon beam frequency spectrum

- photon beam angular distribution (vertical)

- total photon beam power

- power in a given bandwidth

- photon flux in a given bandwidth

- photon brightness in a given bandwidth

- photon beam coherence, polarization, etc

For engineering, SR science applications and diagnostic purposes

we need to know...


radiation emission in particle system

Synchrotron Radiation Basics

radiation emission from a storage ring


Intensity of two modes: s and p

radiation emission in laboratory system

SR Basics (cont’d)

infrared to x-ray spectrum!

(ordinary heat to passage-through-matter)


First light - GE synchrotron

SR Basics (cont’d)

A Billion dollar user machine


Angular Spectral Power Density Functions

23222

32

222

23

12

122

3,

ps

cc

s KF

ps ,,

2

ss

c

s FFP

dd

Pd

power density power density

s-mode polarization p-mode polarization

Can we derive these equations, Prof. Schwinger?

Surely, you’re joking...


23222

31

22222

23

12

122

3,

pp

cc

s KF

1/ and the Critical Frequency

observer1/electron

photon

dt

6)sin(

3xxx r

2

3 3cc

GeVkeVEc 3@7

33

4

rd

ct

x=1/

dt=10-19 sec!

c

1/

1/

fc=1019 Hz!


Derivation of Angular Spectral Power Density

ret

dVtr

ttV )(

)'()(

r

ret

dVtr

tt )(

)'()(

JA

dt

dV

AE

AB

Start with Lienard-Wiechert Potentials

scalar potential from charge

vector potential from current

electric field

magnetic field

b(t’)

n(t’) r(t’)

observer

particle

p 11

p 14

Poynting Vector (power flux)o

BES


Electro-magnetic field (cont’d)

dt

dV

AE

AB

After a Jacksonian derivation

ret

ret

‘difficulty evaluating the above equations’

‘advantageous to calculate Fourier transforms’

p 17,20

31

)(

4

1t

βn

ββnn

Ercop

)(

c

EnB

)(t

Prof. Hofmann


dtet ti

p )(

2

1~EE )(

dterc

ctrti

o

)/)'('(

34

1

2

1~

pp

βn

ββnnE

-1)(

dtec

i ctrti

o

)/)'('(

4

1

2

~

pp

βnnE )(

'0,,'4

1

2

~ )/)'('( dtetcr

ei ctrti

o

co

pp

)(E

Fourier Transform Field Equations

Still looks bad but n and r are approximately constant, except |r| in phase term

Integrate by parts

Decompose vector into components

p 35

p 36

p 61,64,65

z

y

ot’

observer

p 58

relativistic approximation


Small Angle and Relativistic Approximations

c

tt

c

trt o )'sin(cos

')'(

'r

21cos

2

6

')'sin(

3t

tt ooo

r

b

co 22

11

b

2

32

2

22

6

'

2

1'

)'('

r

tct

c

trt

')6

'

2

)1('(exp(0,,

4

1

2

~2

32

2

22

dttct

itcr

eio

co

r

pp

)(E

'0,,'4

1

2

~ )/)'('( dtetcr

ei ctrti

o

co

pp

)(E

p 60-65*

p 65


du

uuu

cr

e

ccco

31

4

3sin

4

3

2

~ 322

3/23/1

2/3

p

)(Ex

Change Variables and Integrate

2/322

3/2

22

2/31

21

22

3~

p

ccco

Kcr

e)(Ex

2/322

3/1

22

2/31

21

22

3~

p

ccco

Kcr

ei)(Ey

')6

'

2

)1('(exp(0,,

4

1

2

~2

32

2

22

dttct

itcr

eio

co

r

pp

)(E

du

uu

cr

ie

ccco

31

4

3cos

4

3

2

~ 322

3/23/2

2/3

2

p

)(Ey

p 66

p 67


.

2

~2

2 0

2

0

22

02

c

r

dd

Ud

dd

Pd

p

p

E

Flux, Energy and Power Density

p 41,51

Return to Poynting’s vector for power flux

dtd

Ud

rcoo

2

2

1

2EBES (where U is the radiated energy)

1. Solve for radiated energy U into a unit solid angle

dEc

rdttE

c

rSdtr

d

dU

2

0

22

0

22 )()( (by Parseval’s theorem)

2. Differentiate with repect to to find angular spectral energy density

2

0

2

)(2

Ec

r

dd

dU

(factor of 2 from positive frequencies only) p 52

p 51

3. Note that power=energy/time: UP o

p

2 (time interval is one turn)

Angular Spectral Power Density p 52

.

2

~2

2 0

2

0

22

02

c

r

dd

Ud

dd

Pd

p

p

E

Angular Spectral Power Density

Angular Spectral Power Density

where from before

2/322

3/2

22

2/31

21

22

3~

p

ccco

Kcr

e)(Ex

2/322

3/1

22

2/31

21

22

3~

p

ccco

Kcr

ei)(Ey

we still have to square these expressions

there are two polarizations

.~~

2

2 22

0

2222

rpps

yx EEr

dd

Pd

dd

Pd

dd

Pdp 68

.~~

2

2 22

0

2222

rpps

yx EEr

dd

Pd

dd

Pd

dd

Pd

The Fs and Fp Functions

.,,,2

ps s

c

sss

c

s FP

FFP

dd

Pd

.12

122

3,

.12

122

3,

23222

31

22222

23

23222

32

222

23

p

p

p

s

cc

s

cc

s

KF

KF

p 83

where .3

22

42

00

r

ccmrPs is the total radiated power for one particle

and we define

plot as a function of normalize frequency and normalized angle...

p 84


The Angular Spectral Power Density Functions

23222

32

222

23

12

122

3,

ps

cc

s KF

ps ,,

2

ss

c

s FFP

dd

Pd

power density power density

s-mode p-mode

visible

x-ray

1/

23222

31

22222

23

12

122

3,

pp

cc

s KF

Integral Over Vertical Angle

sum

))()((16

393/23/5

c

ccc

KzdzKS

s

p

))()((16

393/23/5

c

ccc

KzdzKS

p

p

p 89,90where

cc

s

ccc

s SP

SSP

ddd

Pd

d

dP

ps

0

2

Sands

c

zdzKScc

p

)(

8

393/5

universal dipole flux curve

Integral Over Frequency

sum

p 96

.

132

15

2

.1

1

32

21

2

.,,

2722

22

2522

0 0

2

p

p

p

s

ps

s

s

ss

c

s

P

d

dP

P

d

dP

dFFP

ddd

Pd

d

dP


)(,2

)(,2

22

2

2

22

22

22

2

222

22

p

p

p

ps

p

p

s

ss

s

s

dFS

ddd

pd

ddd

pd

dFS

ddd

pd

ddd

pd

s

s

s

s

RMS Opening Angle as a Function of Frequency

22 RMS

3/1

2 449.0

r

at long wavelengths

p 94

p 96

mrm

nm2

8

550449.0

3/1

2

(opening angle of green light)

Total Integrals – A Reality Check

ps ,,

2

ss

c

s FFP

dd

Pd

p

2

0 0

.1, cs dFdd

sPdddd

Pd

2

)(

)(5.88/

4

m

GeVEcdsPdtPU sso

r

for I=200ma, E=3GeV, r=7.5m, P=200kW, U=1MeV/turn

total power

.3

22

42

00

r

ccmrPs


Power Through a Diagnostic Beam Line

total mirror SiO2 windows >420nm UV Filtersigma

pi250 W 5 W 1mW <1mW

Cold Finger for Mirror Protection

sigma pi


)cos()( zkBzB uoy

)',0),'cos()'( ctt

k

KtR u

u

u bb

),0),'sin()'( b

b t

Kt u

u

)0,0),'cos()'( t

cKKt u

uu

bb

)

'cos,sinsin),'cos(cossin)'(

p

u

up

up

r

ctt

kr

Krtr

bqq

bq

Weak Field Undulator:

b(t’)

n(t’) r(t’)

observer

particle

Undulator Radiation - revisited

origin

R(t’)

Coordinate system: p128 7.7


Radiation Geometry used in Simulator


theta

phi

ret

o rc

31

)(

4

1t

βn

ββnn

Ep

)(c

EnB

)(t

p

po

uu

p twcr

KetE 1322

22223

cos1

0),2sin(),2cos(1)(

q

qq

p

p

po

uu

p twcr

KetB 1322

22223

cos1

0),2cos(1),2sin()(

q

qq

p

p

ti

p dtet p

p )(

2

1~EE )(

u

u

uu

N

NN

EE

p

p

p

q

qq

1

1

1

322

2222)sin(

21

)2sin(),2cos(1)(

Apply Ultra-relativistic Approximations:

Fourier Transform:

22

2

11

2

q

u

Relativistic

Doppler

Frequency

Finite ID length: sin(x)/x

Observer

p132 7.21

p137 7.30=deviation

Plug Expressions into EM Field Equations

2

0

2

|)(|2

Ec

r

dd

dU

2

1

1

1

522

222222422)sin(

1

)2sin()2cos(1

u

u

uuuoou

N

NNKkccmr

dd

Pd

p

p

q

qq

p

)(,,22

wfFFPdd

PdNuuu

u

qq

ps

522

222

1

)2cos(13),(

q

q

pqs

uF

522

222

1

)2sin(3),(

q

q

pqp

uF

2

1

1

1 /

)/sin()(

p

p

u

uuN

N

NNf 22

2

1

2

q

u

i

undulator

Observer

ID length

Horizontal polarization

Vertical polarization

Form factor

p139 7.35

p 140


Re-compute power spectral density


Weak Undulator Field Patterns in s- and p-modes

)'2sin(

8',0),'cos()'(

22

2* t

k

Kctt

k

KtR u

u

uu

u

u

bb

b

)'2cos(

4,0),'sin()'(

2

2* t

Kt

Kt u

uu

u

bb

b

)'2sin(

2,0),'cos()'(

2

2

tckK

tKck

t uuu

uuu

bb

ret

o rc

31

)(

4

1t

βn

ββnn

Ep

)(

c

EnB

)(t

Effective velocityLongitudinal modulation term

p160


Strong Field Undulator – same game, carry more terms

b(t’)

n(t’) r(t’)

observer

particle

origin

R(t’)

u

u

ummum

p

u

m

m

N

NNK

r

mekiE

p

p

p

q

qq

pp

1

1

1

222*

12

*

1

*

0

3*1)sin(

1

sin2,cos2

2)(

)(,,2*2

mNmmum wfFFP

dd

Pd

qq

ps

Use Ultra-relativistic approximations and Fourier Transform…

322*

2*

1

2*22

2

1

cos*2

)2/1(

3),(

q

q

pqs

um

uu

m

K

KK

mF

322*

2

1

2*22

2

1

sin*2

)2/1(

3),(

q

q

pqp

m

uu

mKK

mF

Observer

ID length including harmonics



2

1

1

1

)sin(

)(

p

p

mu

mu

umN

N

NN

f Form factor 22*

2*

1

2

q

u

m m harmonics

p167 8.36

p167 8.38

1 mm

322*

2*

1

2*22

2

1

cos*2

)2/1(

3),(

q

q

pqs

um

uu

m

K

KK

mF

322*

2

1

2*22

2

1

sin*2

)2/1(

3),(

q

q

pqp

m

uu

mKK

mF

)()( 21 ulm

l

ulm mbJmaJ

)()()( 12122 ulmulm

l

ulm mbJmbJmaJ

22*

2*

14 q u

u

Ka

22*

**

1

cos2

q

qq

u

u

Kb



Bessel functions from ei(sint)

trajectory modulation

Found in radio, FELs, accelerator

physics, lasers, etc


The result is complicated but we already ran the simulator

- The End -


Radiated Power Field Patterns in s- and p-modes - First 3 Harmonics -

Hofmann, Chapter 8

Summary – Synchrotron Radiation Properties

- SR Emission Cone

- Spectral Angular Power Density

- Practical Applications

- Visible Light

- Undulator Radiation re-visited

.2

dd

Pd

mrm

nm2

8

550449.0

3/1

2

<1mW


Documents

Spectral Properties of Synchrotron Radiation