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USPAS Accelerator Physics June 2013

Accelerator Physics

Synchrotron Radiation

A. Bogacz, G. A. Krafft, and T. Zolkin

Jefferson Lab

Colorado State University

Lecture 8


Synchrotron Radiation

'

'

'

'

ct ct z

x x

y y

z ct z

Accelerated particles emit electromagnetic radiation. Emission

from very high energy particles has unique properties for a

radiation source. As such radiation was first observed at one of

the earliest electron synchrotrons, radiation from high energy

particles (mainly electrons) is known generically as synchrotron

radiation by the accelerator and HENP communities.

The radiation is highly collimated in the beam direction

From relativity


Lorentz invariance of wave phase implies kμ = (ω/c,kx,ky,kz) is a

Lorentz 4-vector

z

x x

y y

z z

k c

k k

k k

k c k

2 2 2 2

sin sin cos/ / /

/ / 1 cos /

/ / 1 cos /

x y x y z

z

z

k k k k k

c c c

c c k c

c c k c


sin

sin1 cos

Therefore all radiation with θ' < π / 2, which is roughly ½ of the

photon emission for dipole emission from a transverse

acceleration in the beam frame, is Lorentz transformed into an

angle less than 1/γ. Because of the strong Doppler shift of the

photon energy, higher for θ → 0, most of the energy in the

photons is within a cone of angular extent 1/γ around the beam

direction.


Larmor’s Formula

2 2

22

3 2 3

0 0

1 1

6 6

q eP t a p

c m c

For a particle executing non-relativistic motion, the total power

emitted in electromagnetic radiation is (Larmor, verified later)

Lienard’s relativistic generalization: Note both dE and dt are the

fourth component of relativistic 4-vectors when one is dealing

with photon emission. Therefore, their ratio must be an Lorentz

invariant. The invariant that reduces to Larmor’s formula in the

non-relativistic limit is

2

06

due duP

c d d


2 2

6 2

06

eP t

c

For acceleration along a line, second term is zero and first term

for the radiation reaction is small compared to the acceleration

as long as gradient less than 1014 MV/m. Technically

impossible.

For transverse bend acceleration

2

4 4

2

06

e cP t

2

ˆc

r


Fractional Energy Loss

23 4

06

eE

3 34

3

e

beam

rE

E

For one turn with isomagnetic bending fields

re is the classical electron radius: 2.82

10-13 cm


Radiation Power Distribution

2

5/32

0 /

3

8cc

dP eK x dx

d

Consulting your favorite Classical E&M text (Jackson,

Schwinger, Landau and Lifshitz Classical Theory of Fields)


Critical Frequency

Critical (angular) frequency is

33

2c

c

Energy scaling of critical frequency is understood from 1/γ

emission cone and fact that 1– β ~ 1/(2 γ2)

A B

1/γ

Atc

32Bt

c c c

3 32Bt t

c c c


Photon Number

2 2

4

5/32 2

0 00 0

3

8 6c

dP e e cP d K x dxd

d

1dn dP

d d

0

0

8

15 3c

dnd

d

dnd

d

2

0

5 5 1

4 1372 3 2 3

c en n

c


Insertion Devices (ID)

Often periodic magnetic field magnets are placed in beam path

of high energy storage rings. The radiation generated by

electrons passing through such insertion devices has unique

properties.

Field of the insertion device magnet

0ˆ, , cos 2 / IDB x y z B z y B z B z

Vector potential for magnet (1 dimensional approximation)

0ˆ, , sin 2 /2

IDID

BA x y z A z x A z z


Electron Orbit

Field Strength Parameter

Uniformity in x-direction means that canonical momentum in

the x-direction is conserved

0

2

IDeBK

mc

v 1

cos 2 /v 2

x IDID

z z

Kx z dz z

v sin 2 /x ID

eA z Kz c z

m


Average Velocity

zz xz

2

2

11

Energy conservation gives that γ is a constant of the motion

Average longitudinal velocity in the insertion device is

2

2

2

22*

2

11

Kz

Average rest frame has

2/11

12

2

2*

2*

K


Relativistic Kinematics

In average rest frame the insertion device is Lorentz contracted,

and so its wavelength is

*** / ID

The sinusoidal wiggling motion emits with angular frequency

** /2 c

Lorentz transformation formulas for the wave vector of the

emitted radiation

***

*

*

***

cos

sinsin

cossin

cos1

kk

kkk

kkk

kk

z

yy

xx


ID (or FEL) Resonance Condition

cos1

coscos

*

*

*

**

k

kz

Angle transforms as

Wave vector in lab frame has

cos1

2

cos1 *

*

**

*

ID

ckk

In the forward direction cos θ = 1

2

*2 21 / 2

2 2

ID IDe K


Power Emitted Lab Frame

2224 *2

0

2 1

6 2ID

e KP c

Larmor/Lienard calculation in the lab frame yields

Total energy radiated after one passage of the insertion device

22 2 * 2

0

26 ID

eE NK


Power Emitted Beam Frame

22* 2

*

0

2 2 1

6 2

eP cK

Larmor/Lienard calculation in the beam frame yields

Total energy of each photon is ħ2πc/λ*, therefore the total

number of photons radiated after one passage of the insertion

device

2

3

2NKN


Spectral Distribution in Beam Frame

* 2*4 2 2 *

* 2

0

sin32

dP e ck a

d

Begin with power distribution in beam frame: dipole radiation

pattern (single harmonic only when K<<1; replace γ* by γ, β* by β)

Number distribution in terms of wave number

*2 *2

2

* *24

y zk kdNNK

d k

Solid angle transformation

*

22 1 cos

dd


Number distribution in beam frame

22 2 2 2

2

44

sin sin cos cos

4 1 cos

dNNK

d

Number distribution as a function of normalized lab-frame energy

Energy is simply

2 1ˆ 1 cos 1 cosID

cE E

2

2 2

2 3 2

ˆ1

ˆ 4

dN ENK

dE


Average Energy

Limits of integration

1 1ˆ ˆcos 1 cos 1 1 1

E E

Average energy is also analytically calculable

20 max

0

ˆˆ

2 /2ˆ

ˆ

ID

dNE dE

EdEE c

dNdE

dE


dN/dEγ

Eγ 0 βγ2Eund (1+β)βγ2Eund

θ = π

sin θ = 1/ γ

θ = 0

Number Spectrum

2 /und IDE c


Conventions on Fourier Transforms

1

2

i t

i t

f f t e dt

f t f e d

Results on Dirac delta functions

For the time dimensions

1

1

2

i t

i t

t e dt

t e d


3

3

3

3 3

3

1

2

1

2

ik x

ik x

ik x

f k f x e d x

f x f k e d k

x x e d k

For the three spatial dimensions


Green Function for Wave Equation

Solution to inhomogeneous wave equation

2 2 2 2

2 2 2 2 2

1, ; ,

4

G x t x tx y z c t

x x t t

Will pick out the solution with causal boundary conditions, i. e.

This choice leads automatically to the so-called Retarded Green

Function

, ; , 0 G x t x t t t


In general

because there are two possible signs of the frequency for each

value of the wave vector. To solve the homogeneous wave

equation it is necessary that

i.e., there is no dispersion in free space.

3

, ; , 0

, ; ,

i k x t i k x t

G x t x t t t

G x t x t

A k e B k e d k t t

k k c


Continuity of G implies

2

3

2

2

2

, ; ,1 4

4

2

t

i k x t i k x t

ik x i t

G x t x tx x

c t

i A k e i B k e d k

c x x

cA k e e

i

Integrate the inhomogeneous equation between and

i t i tA k e B k e

t t t t


23

2

, ; ,

1

2

2 2

i k x x t t i k x x t t

ik x xi t t

G x t x t

ce e d k

i

t t

c e ce dk

x x

/0

ik x xi t te

e dk t tx x

x x c t t

x x

Called retarded because the influence at time t is due to the

source evaluated at the retarded time

/t t x x c


Retarded Solutions for Fields

2 2 2 2

2 2 2 2 2

0

2 2 2 2

02 2 2 2 2

1

1

x y z c t

A Jx y z c t

3

0

30

,1, /

4

,, /

4

x tx t d x dt x x c t t

x x

J x tA x t d x dt x x c t t

x x

Tip: Leave the delta function in it’s integral form to do derivations.

Don’t have to remember complicated delta-function rules


Delta Function Representation

/3

2

0

/30

2

,1,

8

, ,

8

i x x c t t

i x x c t t

x tx t d x dt d e

x x

J x tA x t d x dt d e

x x

f t t f t tt t

f x x f x xx x

Evaluation can be expedited by noting and using the symmetry of

the Green function and using relations such as


Radiation From Relativistic Electrons

/3

2

0

/30

2

3 3

/

2

0

,1,

8

, ,

8

, , v

1,

8

i x x c t t

i x x c t t

i x r t c t t

x tx t d x dt d e

x x

J x tA x t d x dt d e

x x

x t q x r t J x t q t x r t

qx t dt d e

x r t

/0

2

v ,

8

i x r t c t ttqA x t dt d e

x r t


Lienard-Weichert Potentials

0

0

0

0

/,

4

v /,

4

1,

4 ˆ1

v,

4 ˆ1

ret

ret

x r t c t tqx t dt

x r t

t x r t c t tqA x t dt

x r t

qx t

x r t n t

tqA x t

x r t n t


EM Field Radiated

/

2

0

/0

2

32

02

0

1,

8

v ,

8

ˆ ˆˆ

4 4ˆ ˆ1 1

i x r t c t t

i x r t c t t

ret

qx t dt d e

x r t

tqA x t dt d e

x r t

AE B A

t

n nq n q

Ecn nR

3

ˆ /

ret

B n E c

R

Acceleration Field Velocity Field


2

2

/

22

0

0

2

ˆ1ˆ

ˆ ˆ/ /ˆ ˆ1

ˆ 1 /,

8

v,

8

i x r t c t t

nx r t n

x r t x r t

dr dt n n dr dtd n c dn

dt dtx r t x r tx r t

n i x r t cqx t dt d e

x r t

t iqA x t dt d

t

/

/ /ˆ1

i x r t c t t

i x r t c t t i x r t c t t

ex r t

de i t n t e

dt


/

22

0

/

22 20

2

ˆˆ,

8

integrate by parts to get final result

,8 ˆ1

ˆ ˆ ˆ ˆ1 2ˆ

i x r t c t t

i x r t c t t

vel

i nq nE x t dt d e

c x r tx r t

q eE x t dt d

n x r t

n n nn

2 2

22

2 22

ˆ ˆ

ˆ

ˆ ˆ ˆ ˆ1

n n n

n

n n n n


/

22

0

/

22

0

,8 ˆ1

ˆ ˆ

ˆ ˆ1

ˆ ˆ8 ˆ1

i x r t c t t

acc

i x r t c t t

q eE x t dt d

c n x r t

n n

n n

q edt d n n

c n x r t


Larmor’s Formula Verified

0

22

2 3

0 0

22

2

2 3

0

22

3

0

For small velocities can neglect retardation

ˆ ˆ, /4

ˆ ˆ16

v sin16

v6

acc

qE x t n n R

c

dP qn n

d c

q

c

qP

c

Classical Dipole

Radiation Pattern

Angle between acceleration

and propagation directions


Relativistic Peaking

2

2

52

0

2 2 2

52

0

max

2 2 2 2

3 22 20

In far field after short acceleration

ˆ ˆ

16 ˆ1

sin

16 1 cos

1

2

For circular motions

sin cos1

16 1 cos 1 cos

n ndP t q

d c n

dP t q

d c

dP t q

d c


Spectrum Radiated by Motion

2 2

2 2

0

2

2 22

0 0

ˆ ˆ ˆ1 2 / / / 1 2

1ˆ

ˆ ˆ ˆ ˆ1

8 ˆ ˆ1 1

i R n r t R n r t R c t t i R n r

dE dPdt E H nR dt E E R dt

d d c

n n n nq

t tc c n n

e e

2 2ˆ/ / /

2

22

0 0

clearing the unprimed time integral and omega prime

integral with delta represntation

ˆ ˆ ˆ ˆ2

8 ˆ ˆ1 1

t R n r t R c t t

dt d dt d dt

n n n nq

tc c n n

2

ˆ ˆ/ /

i n r t c t i n r t c t

t

e e dt dt d


2

2 2ˆ /

23

0

22 2 2

ˆ /

3

0

ˆ ˆ

32 ˆ1

ˆ ˆ32

Factor of two difference from Jackson because he

combines positive frequency and

i n r t c t t

i t n r t c

n nd E q

e dtd d c n

d E qn n e dt

d d c

negative frequency

contributions in one positive frequency integral. I

don't like because Parseval's formula, etc. don't work!

We'll perform this calculation in new intensity regimes

of pulsed lasers.

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