1 Radiation from Accelerated Charge

Home , Larmor formula

Physics 139 Relativity

Relativity Notes 2001

G. F. SMOOT

Oce 398 Le Conte

DepartmentofPhysics,

University of California, Berkeley, USA 94720

Notes to b e found at

http://aether.lbl.gov/www/classes/p139/homework/homework.html

1 Radiation From Accelerated Charge

1.1 Intro duction

You have learned ab out radiation from an accelerated charge in your classical

electromagnetism course. We review this and treat it according to the prescriptions

of Sp ecial Relativity to nd the relativistically correct treatment.

Radiation from a relativistic accelerated charge is imp ortant in:

1 particle and accelerator physics { at very high energies 1 radiation losses,

e.g. synchrotron radiation, are a dominant factor in accelerator design and op eration

and radiative pro cesses are a signi cant factor in particle interactions.

2 astrophysics { the brightest sources from the greatest distances are usually

relativistically b eamed.

3 Condensed matter physics and biophysics use relativistically b eamed radiation as

a signi cant to ol. An example we will consider is the Advanced Light Source ALS

at the Lawrence Berkeley Lab oratory.Now free electron lasers are now a regular to ol.

We will need to use relativistic transformations to determine the radiation and

power emitted by a particle moving at relativistic sp eeds.

Lets lo ok at the concept of relativistic b eaming to get an idea b efore wego

into the details which require a fair amount of mathematics. 1

Radiation from an accelerated relativistic particle can b e greatly enhanced.

Part of this e ect is due to the ab erration of angles and part due to the Doppler

e ect.

1.2 Doppler E ect

From time dilation we are used to the notion that a moving clo ck or system op erating

at frequency in its rest frame will app ear to b e slower to a reference system.

t = t 1

0 0

so that if the p erio d in the rest frame is t =1= , then

= = 2

The factor leads to the relativistic transverse Doppler shift. The frequency shift one

would observe for a clo ck or system moving transversely to the line of sight.

Thus the time b etween wave p eaks crests or pulses is

3 t = t =

Observer

in observers frame

u -

If the sources is moving at an angle to the observer's line of sight, then the

di erence in arrival times, t , of successive pulses or crests is

v d

=t1 cos t = t

c c

1 v

1 cos 4 =

obs

which when inverted or multiplied by c yields:

= 1 cos 5 =

obs obs

1 cos c

obs

1.3 Radiation by an Accelerated Charge Near Rest

In 1897 Larmor derived the formula for the radiation by an accelerated charged

particle. He found for the p ower and angular distribution:

2 2

2q dP q

2 2

P = ~a ~a = jaj sin 6

3 3

3c d 4c

where is the angle to the direction of acceleration. It is our task to nd the

relativistically consistent and correct version of these formulae.

We can rederive the Larmor formula for your education. We consider the

electric eld to b e a real physical entity that p oints radially backtoacharge at rest.

If wegointo a moving frame, the electric eld lines will continue to p oint radially

back to the instanteous p osition of the charge. The transformation of the electric eld

works out precisely that way. The Lorentz-Fitzgerald contraction along the direction

of motion causes an increase by the factor of the transverse comp onent of the eld.

Gauss's law continues to hold in that an integral over a closed surface, suchasa

sphere, gives the net charge within. 3

Nowifacharge is diverted from uniform motion, then by our earlier arguments

ab out causality, the electric eld lines out at radius R can not b e e ected by that

change from uniform motion until a time t = R=c later. In fact we exp ect that the

electic eld lines will change at the sp eed of light since light is an electromagnetic

phenomenon. Thus at a time t after a brief t = disturbance change from one

state of uniform motion to another - also called acceleration there is a critical radius

R = ct. Inside of radius R c the electric eld lines p oint radially to the new

instanteous p osition of the charge and outside of radius R + c the electric eld lines

p oint radially to the virtual instanteous p osition of the undisturb ed charge. The

virtual instanteous p osition is where the charge would have b een had it not b een

disturb ed. There is a near discontinuity in the eld lines where they must makea

jaunt nearly p erp endicular to radial. Nearly means that the angle b etween the eld

line and p erp endicular to radial is of order c =v t where v is the velo citychange due

to the disturbance.

Consider: a charge moving with velo city v c abruptly, at time t =0,is

decelerated at a constant rate a until it comes to rest.

J -

At t =0, x= 0 and at t = , x = v =2.

Now consider elds at a time t .At a distance r>ct , the eld will b e

f f

that of a uniformly moving charge, emanating from the \virtual present p osition" the

p oint where the particle would have b een, x = v t , if it had continued unaccelerated.

0 f

At a distance r

f o

There is a transition region which is nearly a spherical shell v cA

particular eld line L de nes a cone of angle, , inside, which contains a certain

ux. Its continuation L de nes another cone whichmust contain the same ux by

reason of Gauss's law relating the eld ux and the enclosed charge. Thus = and

L is parallel to L.

Consider the p ortion connecting these two regimes.

insert gure

The radial comp onent of the electric eld, E must b e the same in the shell as

just outside of it on either side Gauss's law.

q q q

= = 7 E =

2 2

r ct r ct

f f 4

By the geometry of the situation

E v t sin

o f

= 8

E c

q v t sin v t sin qv sin

o f o f o

E = E = = : 9

2 3

c c ct c t

f f

Now ct = r , and a = v =t , so that

f o f

q asin

E = 10

c r

The signi cance of this result is that E / 1=r while E / 1=r .At a large distance

the tangential electric eld E will dominate.

From our general knowledge of varying vacuum elds we know that there will

~ ~ ~

B of strength equal to E and p erp endicular b oth to E and ~r . b e a comp onentof

The energy density energy p er unit volume in the transition layer is

2 2 2 2 2

Energy E B E q a sin

u = = + = = 11

4 2

Volume 8 8 4 4c r

The volume of the shell is it area 4r times its thickness c and the average of

sin =2=3,

Z Z Z Z

2 1 1 1

1 1 1

2 2 2 2

< sin > = sin dcos d = sin dcos = 1 x dx

4 2 2

0 1 1 1

1 2 1

3 1

x x j = 12 =

2 3 3

so that the energy in the transition layer is

2 2

2 q a

: E =

3 c

The radiated p ower is then the energy p er unit time:

2 2

2 q a

P = : 13

3 c

which is precisely the formula derived by Larmor in 1897.

1.4 Radiation from Circular Orbit

Supp ose that ~a ? B , giving motion in a circle and that 1:

insert gure: Reference frame S Lab oratory and reference frame S

which is the electron instantaneous rest frame

two column 5

The lab oratory reference frame S has B p erp endicular to the plane of the

circular orbit B = B = 0, and B = B and E =0

x y z

m v

F =ej~vBj= 14

0 2;3

By transformation law F = F

column 2 Transform elds:

E = E B 15

or more precisely

~ ~ ~

E = E B

= 0 + B e^

x z y

= Be^ 16

Thus

~ ~

F =eE =+ qBe^ = m ~a 17

y o

Thus

a = 18

The p ower emitted is

2 2 2 2 4 2

2 2 q a q B

= 19 P =

3 2 3

3 c 3 m c

This is the correct relativistic form! b ecause P is energy p er unit time and eachis

the 0-comp onentofa4-vector!!

To make a relativistic generalization we employ the concept of covariance.

The Poynting vector is the 0-0 comp onent of the electromagnetic stress tensor

and the radiated electromagnetic energy is the 0-comp onent of a Lorentz 4-vector.

Time is the 0-comp onent of a Lorentz 4-vector. So the ratio energy p er time is an

invariant.

Can one nd a Lorentz invariant that reduces to Larmor's formula as ! 0?

orrect relativistic formula! Is it unique? Yes, if we require it to If so, it will b e the c

~ ~

involve only and d =dt and not higher p owers.

How is one to construct it? Non-relativistically,

dv 1 dp

= 20

dt m dt

So the Larmor p ower is

e d~p d~p 2

21 P =

2 3

3 m dt dt c

o 6

To get an invariant, exp erience tells us to substitute d for dt, i.e. d~p =dt ! d~p =d ,

and add the fourth comp onent:

! !

2 2

dp~ dp~ dE d~p

= c 22

d d d d

Notice that

EdE = c pdp 23

dE pc mv c

2 2 2

dp = dp : 24 = dp =

2 2

c E mc

So that

1 dp~ d~p dp

2 2 2

j j = j j 25

c d d d

Giving

2 3

d~p e dp 2

2 2

4 5

j j 26 P =

2 3

3 m c d d

It is p ossible to write this in manyways. One wayis

2 e

_ _

2 2 6

~ ~ ~

27 j j j j P =

3 c

1.5 Power and Angular Distribution Summary

We can calculate these in a consistentwayby using these formula as correct in the

rest primed frame of the electron and transform the accelerations forces, angles,

frequencies, etc. into the lab oratory frame. What we need is to show that p oweis

a Lorentz invariant P = P for any emitter that emits with front-back symmetry

zero net momentum in its instantaneous rest frame. To do this we make use of the

invariance of ~a u~ which is zero for all systems.

du~ 1 d 1 d

~a u~ = u~ = u u = c =0

d 2 d 2d

This is a consequence invariance of the sp eed of light and four-vector velo city.

In the zero net radiation momentum in instanteous rest frame casea ~ ~a = ~a ~a

since in the rest frame a =0.Thus the p ower can b e evaluated in any frame can b e

found by computing the acceleration in that frame and squaring it.

2 2

2q 2q

0 0 02 02

P = ~a ~a = a + a

? k

3 3

3c 3c

2 2 2 4

a + a 28

? k

3c 7

where a is the acceleration p erp endicular to the motion of the charged particle and

a is the acceleration comp onent parallel to the charge particle motion. In the last

3 0 2 0

= a and a = a line wehave made use of the transformation of accelerations a

evaluated in the instantaneous rest frame primed of the electron. Note that there

is a factor of di erence in the transformation of accelerations p erp endicular and

parallel to the direction of motion. This translates into a di erence b etween and

in the p erp endicular and parallel cases.

We get a similar expression for the angular distribution:

2 2 2

a + a

dP q

2 0

= sin 29

3 4

d 4c 1 cos

We are making use of the conversion

1 dP dP

4 4 0

d d 1 cos

Evaluation for p erp endicular and parallel cases yields:

2 2 2 2

q a sin cos dP 1

= 1

3 4 2 2

d 4c 1 cos 1 cos

2 2 2 2 4 4

4q a 1 2 cos2 +

? 8

! 30

3 2 2 6

c 1 +

2 2

2 2 2 2 2

4q a

q a sin

k k

= ! 31

3 6 3 2 2 6

d 4c 1 cos c 1 +

Note the large p owers of 8 and 10 which shows the seriousness of the

relativistic e ects. Before we follow this up in detail, we review radiation near the

rest frame of the emitting particle.

1.5.1 Case I: acceleration parallel to motion

_ _

~ ~ ~ ~

Consider k ; acceleration parallel to motion = 0. Recalling that

2 2

=11= , then

; =

_ _

= ; =

3 2 6

2 e _

32 P =

3 c

2 e dp

P = 1

2 3

3 m d c

o 8

e dp 2 1 dt

= d =

2 3 2

3 m c dt

2 e d

2 2

= m c p = m c

o o

2 3

3 m c dt

e 2 _

= 33

3 c

where the conversion makes use of the relations

= 1

2 _ d 1

dt 2

_ _

= = 34

_ c d E

v dt m c

1 1 dE dE

= 35 =

m c vdt m c dx

o o

2 2

P 2 e =m c dE

dE =dt 3 m c dx

So that the p ower radiated compared to the energy change p er unit distance is

2 e dE

P = 37

2 3

3 m c dx

Nowwe can compare the radiated p ower with the acceleration p ower

2 2

P 2 e =m c dE =dx dE

= 38

dE =dt 3 m c dE =dt dx

Note dE =dx=dE =dt dt=cdt when 1. The ratio of p owers, radiated to

acceleration, is negligible unless energy gain in 2:8 10 cm is of order of the rest

mass - i.e. for an electron 0:511 MeV.

1.5.2 Case I I: acceleration p erp endicular to motion

_ _

~ ~ ~ ~

Centrip etal acceleration: ? . ~a = c and ~v = c .

insert gure / diagram to showvector directions

Then in the relation to nd the rate of change of the energy-momentum four-

vector

dE d~p 1

j j

c dt dt 9

E=dt 0; since F ? ~v , so that no work is b eing done on the particle. Then

2 e d~p

P = j j 39

3 m c d

and

d~p

j j = !j~pj 40

where ! = c= is the orbital angular frequency of an orbit with radius . One can

derive this relationship

dp ds vdt

= d = = = !dt

= !p

Thus

! =

Nowwe can moveontothepower loss rate

d = p = m c

2 e

2 2 2

! j~pj P =

2 3

3 m c

2 e c

= mc 41

2 3

3 m c

2 e c

4 4

42 P =

The energy gain for a particle p er turn in an accelerator is

E =2 43

The radiation loss is

4 e

3 4

E = 44

In practical units

E=1 GeV

E =1 MeV = 8:85 10 45

=1 meter

The p ower radiated by a bunch of electrons

Power=1 watt = 10 [E =1 MeV turn ][J=1 amp] 46 10

provided the radiation is incoherent.

Aside: How to get these practical unit relations:

4 e

3 4

E =

Start by putting =1.If is not very near to 1, then one gets negligible radiation

power.

E in GeV E

= =

m c 0:511 MeV

2 2 2 2

e m c m c e 0:511 MeV

o o

= = r =2:810 cm

m c in cm

and the conversion from in cm to m is = 100 . So that

cm m

13 4

2:8 10 0:511 [E in GeV] 4

MeV E =

4 4

3 100 5:11 10 in m

[E in GeV]

= 8:85 10 MeV

in m

[E in GeV]

keV 47 = 88:5

in m

Giving the conversion used ab ove.

E turn

Power = Numb er of electrons

Electron turn sec

V E J 2J

= E =

2 eV e

Power in kW = 88:5[E in GeV] J in amps=R in m

= 26:5[E in GeV] B in teslasJ in amps

E in MeV

= V in MV 48

Now Some Numb ers and History E.O. Lawrence invented the cyclotron

and the rst was built here at Berkeley. Later his colleague Edwin McMillen and

indep endently in the Soviet Union by V.I. Veksler invented the idea of phase stability

which made the synchrotron p ossible.

Synchrotron radiation was rst observed in a lab oratory in 1947. That

lab oratory was in Berkeley.

Early Synchrotrons:

First synchrotron was op erated with 8 MeV electrons in 1946 byGoward and

Barnes in Wo olwich Arsenal, UK. In 1947 GE labs op erated an electron synchrotron

at 70 MeV. So on after there were many op erating. 11

Table 1: Parameters for Sample Accelerators

Accelerator LBL Cornell LHC ALS FermiLab SSC Elo

Max. Energy GeV 0.3 10 45 1-2 1000 20,000 10

Particle e e e e p-p p-p p

BTesla 0.33 0.135 1.248 4.4 6.6 7.7

Radius m 1 100 4249 1000 11.7 km 50 km

Bending R m 4.01 10.1 km

Beam Current ma 400 73 100

Single Bunch ma 1.6 0.00167

E-gain/turn MeV 0.05 10.5 350 1 5.26

E-loss/turn MeV 0.001 8.8 0.112 0.001 18

Synchrotron Power 45 kW 9.1 kW 1.8 MW

RF Power kW 16000 300 1600 61000

RF MHz 713.94 352 500 53.1 374.74 412

Harmonic 1800 31324 328 1113 103,680 146500

Beam lifetime hrs 14 4 24 48

Fill time 30 min 2.1 min 40 min 4 hrs

An early synchrotron at Berkeley had a radius of ab out 1 meter and a

maximum energy of ab out 0.3 GeV. The synchrotron radiation E 1keV/turn

max

could b e noticed. The acceleration voltage was only a few keV/turn.

At big electron synchrotron was built at Cornell and op erated at 10 GeV .

The radius was ab out 100 meters. It encloses a fo otball eld. The magnetic eld was

B = 3.3 kG 0.33 Tesla. The accelerator voltage was ab out 10.5 MeV/turn and the

synchrotron losses were E 8:8 MeV/turn.

rad

LBL Advanced Light Source is designed to provide synchrotron radiation as a

to ol for research.

1.6 Synchrotron Radiation Basics

Consider the non-relativistic case of a charged particle in a circular orbit caused bya

magnetic eld. That particle will radiate electromagnetic waves at a frequency given

by the orbit frequency or the Lamor frequency

qB qB

= ! =

L L

mc 2mc

due to the acceleration of b ending in the magnetic eld. As the particle's energy is

increased relativistic e ects will b ecome imp ortant. For the same orbit the particle

will b oth b egin to radiate more energy and at more frequencies - which are at the 12

orbit frequency and its harmonics. The p eak p ower will b e emitted at a frequency

whichisat times the orbit frequency. In the next sections we will understand

this.

1.6.1 Synchrotron Emitted Power

To nd the total emitted p ower we can use the Lamor 1897 formula

2 2

2 q q 2

2 4 2 2 2

P = j~a j = a + a

emitted o

? k

3 3

3 c 3 c

where ~a is the particle acceleration in its instantaneous rest frame and the right hand

side of the equation uses the acceleration transform law from the particle rest frame.

= q~v E

and since E =0 wehave = constant.

d dP

~ ~

= m ~v=q~v B F =

dt dt

With = constant,

d~v

m = q~v B

Thus

dv q dv

=0; = ~v B

dt dt m

We can conclude jv j = constant and jv j = constant. Wehave uniform circular

motion of the pro jected motion on the normal plane. That is a simple helical motion

around the uniform magnetic eld.

The frequency of rotation or gyration is

; ! a = ! v ! =

? B ? B

m c

Note that the gyration orbit frequency is the Lamor frequency divided by .

We can nowevaluate the transformation of the Lamor formula for the p ower

radiated since we know a = ! v and a =0

? B ?

2 2 4 2

2 2 q q qB q B 2

4 2 2 4 2 2 2

! v = v =

P =

B ? ? ?

3 3 2

3c 3c m c 3 m c

2 2 2 2

= r c B

o ?

2 2 2 2 2

c U =2 c U sin =2

T B T B

2 2 2

where r = e =m c is the classical radius of the electron, =8r =3 is the Thomson

o e T

crossection, U = B =8 is the energy density of the magnetic eld, and is the helix

pitch angle angle of the gyrating particle with resp ect the magnetic eld lines. This

is the relativistically correct form that wesaw previously. 13

1.6.2 Synchrotron Radiation Frequency Sp ectrum

First we consider the frequency distribution of a mono energetic distribution, i.e. we

consider the radiation from a particle at an energy E corresp onding to . When the

particle's energy increases as grows larger the ab erration of angles moves most of

the radiated p ower into a cone of half angle 1= in the instantaneous direction

of motion of the particle. Thus an observer will see a pulse of radiation whenever the

particle's instantaneous velo citysweeps past his direction. This will happ en once p er

orbit. This pulse will b e narrow b oth b ecause the ab erration of angles and b ecause of

the time dilation and Doppler e ect. Since the relativistic particle is moving towards

the receiver observer, the received pulse is sharp ened compressed in time bya

factor of order . The time compression go es at

2 2

=1 cos 1 + =2 !

2 2

1 1= ! 1 =2 and =2 where the limit comes for 1 since =

=2.

Thus the observer will see a pulse every orbit with width of the pulse

separation. Fourier theory tells us that the signal will app ear at the orbit frequency

and its harmonics and that the p ower will p eak at a frequency which is near

where = ! =2 = qB =m c.

B L e

For a magnetic eld B =10 Gauss, whichisatypical value in the Galaxy

and manypowerful radio galaxies, = 28 Hz. The electrons that pro duce emission

3 4

at radio frequencies of a few GHZ therefore have Lorentz factors 10 10 . The

spacing b etween successive harmonics is = = , for very high this spacing is so

B L

narrow as to negligible for all but the highest frequency resolution observations. In

astrophysical sources, this is often blurred and smo othed byvariations in the electron

energy a p ower law sp ectrum and byvariations in the magnetic eld intensity and

direction.

1.7 Astrophysical Synchrotron Radiation

1.7.1 Historical Note

Although nonthermal radiation had b een observed from the Galaxy from the op ening

of radio astronomy in the pioneering work by Karl Jansky in 1933, there was no clear

evidence of its origin. In 1950 Kiep enheuer suggested that Galactic nonthermal radio

emission was synchrotron radiation and Alfv en and Herlofson prop osed that non-

thermal discrete sources were emitting synchrotron radiation. Kiep enheuer showed

that the intensity of the nonthermal Galactic radio emission can b e understo o d as the

radiation from relativistic cosmic ray electrons that move in the general interstellar

6 10

magnetic eld. He found that a eld of 10 Gauss 10 Tesla and relativistic

electrons of energy 10 eV would give ab out the observed intensity. The early 1950s

saw the development of these ideas e.g. Ginzburg et al. 1951 and following pap ers, 14

see Ginzburg 1969 that synchrotron emission was the source of non-thermal \cosmic"

radiation. This mo del was later supp orted by maps which showed that the sources of

the non-thermal comp onents were extended nebulae and external galaxies and by the

discovery that the radiation was p olarized as predicted by theory. The synchrotron

theory is widely accepted and is the basis of interpretation of all data relating to

nonthermal radio emission.

1.7.2 Context

Synchrotron radiation is a common phenonmen in astrophysics as there are almost

always plasma and magnetic elds present and energetic electrons. Because of

sto chastic scattering pro cesses, the energetic electrons tend to b e isotropically

distributed.

For an isotropic distribution of velo cities one needs to average over all angles

for a given sp eed . If is the pitch angle, the angle b etween the magnetic eld

direction and the particle velo city, then

2 2

< >= sin d

Thus

4 2

2 2 2 2 2 2

P = r c B = c U

T B

3 3

2 2

where =8=3 r is the Thomson cross section and U = B =8 is the energy

T B

density in magnetic eld.

Electrons of a given energy E = m c radiate over a wide sp ectral band,

with the distribution p eaking roughly at 16:08B =GE=GeV MHz, with

c e

a long low-p ower tail at higher frequencies, and most of the radiation in a 2:1 band

from p eak. The p eak intensityisat =0:29 =4:6B =GE=GeV MHz.

max c e

The radiation from a single electron is elliptically p olarized with the electric

vector maximum in the direction p erp endicular to the pro jection of the magnetic

eld on the plane of the sky. Explicitly the total emissivity of a single electron via

synchrotron radiation is the sum of parallel and p erp endicular p olarization

3e B sin

F x 49 j =

16 cm

0 e

where is the electron direction pitch angle to the magnetic eld B and F x

x K d is shown graphically in Figure ??.

5=2

The quantity x is the dimensionless frequency de ned as x !=! = =

c c

where ! and are the critical synchrotron frequencies. An electron accelerated by

c c

a magnetic eld B will radiate. For nonrelativistic electrons the radiation is simple

and called cyclotron radiation and its emission frequency is simply the frequency of

gyration of the electron in the magnetic eld. 15

However, for extreme relativistic 1 electrons the frequency sp ectrum is

much more complex and extends to many times the gyration frequency. This is given

the name synchrotron radiation. The cyclotron or gyration frequency ! is

! = 50

For the extreme relativistic case, ab erration of angles cause the radiation from the

electron to b e bunched and app ear as a narrow pulse con ned to a time p erio d much

shorter than the gyration time. The net result is an emission sp ectrum characterized

by a critical frequency

3 3 qB

! ! sin = sin 51

c B

2 2mc

To understand the astrophysical radiation, one must consider that cosmic ray

electrons are an ensemble of particles of di erent pitch angles and energies E .It

can generally b e assumed that the directions are fairly isotropic so that integration

over pitch angles is straightforward.

The next step is integration over electron energy sp ectrum to determine the

total synchrotron radiation sp ectrum.

If the electrons' direction of motion is random with resp ect to the magnetic

eld, and the electrons' energy sp ectrum can b e approximated as a p ower law:

dN =dE = N E , then the luminosity is given by

3 p1=2

3e 3e

p+1=2

p1=2

LN B ap; 52 I =

2 3 5

8mc 4m c

where apisaweak function of the electron energy sp ectrum see Longair, 1994, vol.

2, page 262 for a tabulation of ap, L is the length along the line of sight through

the emitting volume, B is the magnetic eld strength, and is the frequency.

Atvery low frequencies synchrotron self-absorption is very imp ortantas

according to the principle of detailed balance, to every emission pro cess there is

a corresp onding absorption pro cess. At the lowest frequencies synchrotron self-

5=2

. absorption predicts an intensity that increases as /

The lo cal energy sp ectrum of the electrons has b een measured to b e a p ower

law to go o d approximation, for the energy intervals describing the p eak of radio

synchrotron emission at GeV energies. The index of the p ower law app ears to

increase from ab out 2.7 to 3.3 over this energy range Webb er 1983, Nishimura et

al 1991. Such an increase of the electron energy sp ectrum slop e is exp ected, as the

energy loss mechanisms for electrons increases with the square of the electron energy.

The synchrotron emission at frequency is dominated by cosmic ray electrons

1=2

of energy E 3=GHz GeV. The range of energies contributing to the radiation

intensity at a given frequency dep ends on the electron energy sp ectrum: the steep er

the electron distribution, the narrower the energy range Longair 1994. For the case 16

of most of the Galaxy, this range is of order 15 to 50. The observed steep ening of the

electrons' sp ectrum at GeV energies is used to mo del the radio emission sp ectrum at

GHz frequencies e.g. Banday&Wolfendale, 1990, Platania et al. 1998. 17

1.8 Free electron Lasers

The Free Electron laser FEL is a classical device that converts the kinetic energy

of an electron b eam into electromagnetic radiation by passing it through a transverse

p erio dic magnetic eld called the "wiggler". In contrast with conventional lasers,

the radiation of the FEL is not constrained by the discrete energy levels that x the

wavelength of emission. The wavelength of FEL radiation dep ends mainly on the

wavelength of the p erio dic magnetic eld and the energy of the electron b eam. High

p eak p owers and its large range of op erational wavelengths make it a laser of the

future. A simple schematic representation of the FEL is given in the following gure.

Akey feature is that the FEL is a true laser pro ducing coherent radiation.

Coherent radiation happ ens when the FEL is biased in the resonant condition. This

leads to an e ect where the electrons bunch more tightly so that they radiate as

a single coherent bunch. For N electrons acting indep endently, the radiation is

prop ortional to Ne . If the N electrons act coherently, as if a single particle, then

2 2

the radiation is prop ortional to N e .

One could seed the laser with an electromagnetic wave for sp eci c applications

but to have a completely tunable laser, generally the FEL op erates on the principle

of a single-pass free electron laser op erating the self-ampli ed sp ontaneous emission

SASE mo de. Electron motion through the undulator with alternating magnetic

elds forces the electrons into a sinusoidal tra jectory leading to electromagnetic

radiation which recouples to the electron bunch causing laser action through SASE.

The radiated p ower increases along the electron b eam path leading to exp onential

increase in intensity. With high enough electron current and long enough undulator

the p ower is saturated and energy oscillates b etween the electron and photon b eam.

If the resonant condition is met the energy exchange b etween the electron and photon

b eam leads to microbunching and coherent emission.

It should b e noted that the FEL do es not require any mirrors or resonating

laser cavity structure. This is a great advantage at short wavelengths where, for

example, mirrors and optics are technicaly dicult.

One can think ab out the FEL in steps: 1 What is the wavelength of light

emitted by an electron traveling down the FEL magnet structure? Once can nd this

by using the synchrotron radiation formula or by transforming to the rest frame of

the electron to nd the frequency of oscillation by the magnets and then transforming

the radiation to the lab by the Doppler formula. The approximate answer is

magnetic structure

2 What is the resonant condition? The undulator gives a resonance condition

between the electron bunch and the electromagnetic wave, when one undulator

p erio d travel length gives a time di erence b etween the electron bunch and

electromagnetic wave corresp onding to one p erio d of the electromagnetic wave. In

that situation the electrons are always going uphill against the electric eld and thus

adding p ower to the electromagnetic wave. That condition for very small transverse 18

movement of the electrons is that

t = =v =c = =c = =2 c

u u u

1.9 High Gain Free Electron Lasers

Motivation for high-gain FELs are as microwave sources for advanced accelerators

and ecient sources of short wavelength radation. The basic physics is that a b eam

of electrons is injected along the axis of an undulator a transverse, p erio dic ,

magnetostatic eld B z , N p erio ds. The electrons are p erio dically de ected and as

o o

a result radiate synchrotron radiation. The primary features of synchrotron radiation

are sp ontaneous emission which is incoherent: I N , in a narrow cone: 1= ,

and narrow bandwidth:

dI ! !

sinc N 53

d! d !

which p eaks at ! = ! =2= and we will see that the resonant condition is at

s s

=1 = .

s k o k

In the electron rest frame the wiggler eld lo oks like N p erio d radiation eld

with wavelength

0 0

= = =

s 0

2 2

where =1=1 . Thus the electron oscillaes N times. It pro duces a wavepacket

k k

0 0

of length N p eaked at wavelength . The sp ectrum of the radiation is the Fourier

0 0

transform of a plane wave truncated after N oscillations:

54 N I ! = sinc

In the lab oratory frame, is the Doppler upshifted wavelength

= ' 55

2 2

The exact solution is

0 s

A free electron laser has tunability via change in electron energy or the

undulator.

1 1 a

2 2

=1 ' 56

k ?

2 2

where a is he dimensionaless vector p otential of the undulator

e B

0 0

a = helical

2m c

0 19

= 0:934B per Tesla cm

0 0

e B

0 0

= planar

2 2 m c

= 0:66B per Tesla cm 57

0 0

and

= 58

1+a

1+a 59 =

1.9.1 Stimulated Emission

Inject a laser b eam with ' along the axis of the undulator. The electrons move

along curved path at v

e z

If an electron has the resonant energy E = m c

R R 0

2 2

= 1+a 60

R 0

then the relative phase b etween transverse electron and radiation oscillations remains

constant. Dep ending up on the phase the electron can give energy to the eld and

decellerate, < _ 0 stimulated emission or take energy from the eld and accelerate,

>_ 0.

An issue is that at the entrance of the undulator the electron phases are

randomly distributed. For low gain, half of electrons will accelerate and half will

decellerate. For low gain < >> . This is what is observed for the rst FEL, the

0 R

Mdey laser in 1976 op erated at 10.6 m.

But if undulator is long enough and the current is high enough, then energy

mo dulation will result in space mo dulation. There will b e \self-bunching" and it will

b e around a \right" phase for gain. Most electrons will have the same phase and the

intensity will b e prop ortional to the numb er of electrons squared. I / N . This is

collective instability of self-bunching and exp onential gain.

1.9.2 Self-Consistent Theory

To fully describ e FELs, we need a many particle, self-consistent theory that combines

relativity for the electron mechanics and tra jectories including the transverse current

J , Maxwell's equations or the sp ecial relativistic version, and an expression for the

radiation eld.

Wiggler Field

~ ~ ~

B = r A

0 0

Radiation Field

@ 1

~ ~ ~ ~ ~

A B = rA E =

c @t 20

Tra jectory Equation

dp d 1

~ ~ ~

= m v= e E + ~vB +B

0 0

dt dt c

Energy Equation

dE d

== m c =eE~v =eE v

0 ?

dt dt

The total eld on electrons from the vector p otential A

tot

~ ~ ~

A = A + A

tot 0

which is the total from the wiggler and radiation. A is p erio dic spatially either

planar or helical

ik z

A = ^ee + c:c:

for the helical eld which leads to circularly p olarized radiation:

h i

I k !t

Ae^e c:c: A =

where ! = ck = c k + k where k allows for waveguides. Let k =0: Then

? ?

1 d

m v = e E + v B

0 ? ?

dt c

e @A

tot

v rA =

tot ?

c @t

e d

= A 61

tot

c dt

d e dA

tot

= dta 62

? tot

dt mc dt

For p erfect on-axis injection 0 = 0 and

a a

tot 0

= '