Download Flux, Intensity, Brilliance and all those extremely

1
Flux, emittance, and brilliance
Electromagnetic radiation transports energy, because there is energy in electric and magnetic
fields. For an electromagnetic plane wave the energy density [J/m3] is
u  0E2 ,
(1)
where  0  8.8541878  10 12 As/Vm is the vacuum permittivity, and E is the electric field
[V/m]. Intensity is defined as the average energy per unit time and unit area. For a harmonic
wave in vacuum the intensity is
I
1
 0 cE 02 ,
2
(2)
where c  2.99792458  10 8 m/s, is the speed of light in vacuum and E0 is the amplitude of the
wave [V/m]. Intensity then gets the unit of W/m2, as expected. If you don’t feel comfortable
with this, you must consult your Electricity and Optics books. It is covered e. g., in Hecht:
Optics , chapter 3.3, and you can find a summary in Swedish in elektromagnetiska_vagor.doc.
In a quantum mechanical description the intensity is related to number of photons per time
and area. The energy density is related to the number of photons, n ph , and their frequency,  :
u
n phh
V
,
(3)
where V is the volume of a beam with cross section A, and h  6.626076  1034 Js is Planck’s
constant. The intensity ((average) energy per unit time and unit area) is then
I
n phh
(4)
tA
The classical equation (2) and the quantum mechanical equation (4) appear very different. As
long as we are doing experiments in the ‘linear regime’, where excitations are made with onephoton-at-a-time, it is convenient to count the number of photons, as opposed to considering
the E-field amplitude. Thus, equation (4) is a more convenient starting point, and an obvious
question is what is required from a radiation source, to get a large intensity. In addition, many
experiments require monochromatic radiation. Therefore, the spectral purity of the source is
an important parameter. One has therefore introduced a number of concepts which are
relevant for experiments. On most of these concepts there is no consensus, so one has to be
careful and look at the definition in each specific case.
The flux of a source is normally defined:

n ph
t  0.1%BW
(5)
2
‘Flux’ does take the spectral purity into account, since it explicitly measures the number of
photons/second in a 0.1% bandwidth (BW), e. g, at a nominal photon energy of 1000 eV, only
photons/second within the 999.5-1000.5 eV band contributes to  .
The usefulness of a source is also critically dependent on its size and angular distribution.
Since the size boundary is not abrupt, but the intensity can be considered to be a distribution,
the spatial extensions are often given as the standard deviation of this distribution,  x and  y
in the horizontal and vertical direction, respectively. Similarly, the angular distribution of the
flux is characterized by the standard deviation of the intensity around a nominal direction,  x
and  y , with respect to the horizontal or vertical plane, respectively. Note that these measures
contain an implicit assumption of a statistical distribution of the intensity. In addition since
the two dimensions are separated the solid angle defined by  x y gets the unit [rad2] rather
than [sterad]. This is convenient as long as small angles are considered.
The emittance in the horizontal and vertical directions is defined:
 x   x x
 y   y y
(6)
It is immediately realized that small emittance is desirable for a radiation source. As intensity
is inversely proportional to the cross section area, a small spatial extension implies large
intensity. Also one can realize that the source quality is higher if all the photons go in the
same direction, as opposed to being isotropically distributed. No focussing optics is needed to
increase intensity if we have a hypothetical point source which radiates in one direction, only.
In fact, the intensity cannot be increased. At the other extreme: it is difficult to design an
efficient optical system if the source is large and radiates in all directions.
These notions are quantified in Liouville’s theorem, which states that the emittance can only
be conserved or made larger in any optical system. We will come back to this in the optics
section. Here, it suffices to understand that a small emittance is important; this motivates the
introduction of brilliance, (flux/emittance) which is a central concept:
B
n ph
t   x   y  0.1% BW
(7)
A common unit for brilliance is
B 
photons
s  mm  mrad 2  0.1% BW
2
This is the common unit in which various radiation sources are compared. We shall see that
the synchrotron provides outstanding brilliance for most of the electromagnetic spectrum.
3
Accelerating charged particles radiate
To see how charged particles radiate we first go back to the classical description. For
synchrotron radiation we shall see that the relativistic transformations are essential. There is,
however, no quantum mechanics needed.
Electromagnetic radiation is emitted when charged particles are accelerating. Using
Maxwell’s equations it can be shown (strictly in Attwood 2.2 and 2.3, handwavingly in Hecht:
‘Optics’ 3.4 and in Stralningskallor.doc) that the power per unit angle from a charge at
dp
momentum change
is
dt
2
dp
e
sin 2 
dP
dt

d 16 2 m02 0c3
2
,
(8)
where e is the charge of the particle,  is the angle between the direction of acceleration and
the direction of the emitted radiation, and r is the distance between the charge and the
position where the intensity is measured. This is the classical donut shape-angular
distribution. The intensity reaches maximum in the direction perpendicular to acceleration,
and zero along the acceleration direction. The intensity is proportional to the square of the
acceleration, and radially it is proportional to inverse square of the distance, just like a
spherical wave. From this expression one can derive (basically integrating over angles and
distance) the total power emitted from an accelerating charge (see Attwood):
2
dp
e
dt
P
6m02 0c3
2
(9)
The last two equations work fine as long as the speed of the charge is small relative to the
speed of light. At higher speeds the general treatment becomes very complex, and it was
developed in detail by Julian Schwinger: (On the Classical Radiation of Accelerated Electrons
J. Schwinger Phys. Rev. 75, 1912-1925 (1949) [View Page Images or PDF (1622 kB) [Order
Document]]). Here we will point out some important features and make a simplified derivation.
Total power
The main problem for the early accelerator developers was that the particles emit radiation
during the acceleration associated with the bending of the beam path. Had this energy loss
been described by equation (9) it would not have been a big problem. The total radiated power
is, however, much larger. The main reason is time dilation:
dt  d ,
where we introduce the relativistic factor
(10)
4
1

2

v
1 2
c
E
m0c 2
(11)
The essence is that time intervals in the lab system, dt , correspond to much smaller time
intervals, d , in the system of the particle. The difference is huge, as the energy of the
particles, E , in the synchrotrons considered here is several GeV, giving 1000<  <5000).
dp
Correspondingly, as the particle experiences the momentum change
in its system we
d
dp
measure 
in the lab system. Plugging this into eq. (9) we get:
dt
2
dp
e
dt
P
2
6m0  0c3
2
2
(12)
For centripetal acceleration in a bend with radius R, we have
dp pv

dt
R
(13)
We use this equation, and the fact that
E  ( pc)2  (m0c2 )2  pc
(14)
at the considered high energies. Plugging (13) and (14) into (12), and assuming that v  c we
get the useful expressions:
4
e 2 2 vE
e 2c  E 
e 2c


P


4
6m02 0c 3 Rc
60 R 2  m0c 2 
60 R 2
2
(15)
Thus, the radiated power is proportional to the fourth power of the particle energy. In the
classical case it would have been only to the square of the energy. A tacit assumption in the
‘derivation’ of the total power expression is that the energy of the particle is constant. This is
obviously impossible because the particle radiates part of its energy. It is a safe approximation
if the radiated energy is small compared to the initial energy of the beam. The energy lost per
revolution in a circular orbit is
E  P
2R e2 4

v
3 0 R
(16)
where we use that v  c . In convenient units we get
E(MeV )  8.85 10 2
E 4 (GeV)
R(m)
(17)
5
In practice storage rings also include straight sections, with total length, L, where the charged
particles do not radiate. This makes the average power less than what is given by equation
(15), but the energy loss per revolution is not influenced. Obviously it does, however,
influence the average power since the electron since the total path length becomes 2R  L .
As long as we do not discuss free electron lasers, the charged particles radiate incoherently.
Therefore it suffices to consider one electron at a time. To get to the properties of the
synchrotron facility one simply adds the contribution from each electron. The average power
emitted by a storage ring with beam current, I, is then:
P(kW )  8.85  10 2
E 4 (GeV )
I (mA)
L
R
( m)
2
(18)
Angular distribution
We have seen that relativity (essentially time dilation) helps us to get higher flux (eq. 5). Here
we shall see that relativity also helps to decrease emittance, and hence to increase the
brilliance in another way.
The angular dependence of the intensity from an accelerating charge is given by equation (8).
If the charge is moving at high speed, this distribution must be transformed from the rest
system of the electron to the lab system. This transformation shows that almost all radiation is
emitted in a narrow cone, in the forward direction. Here we will not transform the whole
distribution, but consider only a special case where a photon is emitted perpendicular to the
direction of the particle (it can be trivially generalized to all emission angles). Thus, the
electron is moving in the x-direction with speed v  c , sending out radiation in the y-direction
with velocity u y '  c . In a Galilean transformation we would have expected the radiation to
have the speed utot  u y '2 v 2  c 2 , and an angle of 45 degrees to the x-axis. We know that
this cannot happen, and the crucial point is again: time dilation. Time intervals in the rest
system of the particle are smaller than in the lab system. Therefore the perpendicular speed is
reduced, and leaves only the ‘inertial component’.
y’
S’
vc
S
y
u' y  c

x’
x
6
Doing the Lorentz transformation of velocities explicitly according to Physics Handbook, we
find the velocities in the lab system:
u ' x v
vc
vu' x
1 2
c
1
1
1
uy 
u'y  c
 1  vu'x

2
c
ux 
(19)
(20)
We see that the perpendicular component is killed in the transformation (remember that  is a
big number). Combining the two equations we get the angle of the radiation in the lab system:

uy
ux

1
(21)

From this equation we know only, that if the radiation is emitted perpendicular to the velocity
1
in the rest system it will have the angle
in the lab system. Considering equation (5) one

realizes that all radiation with forward components the will be folded into a cone with this
maximum angle. Radiation emitted with backward components will have larger angles. The
concentration in the forward direction is, however obvious, and even if this is not strictly true
1
it is often said that the radiation is concentrated within a cone with a top angle . This

concentration automatically leads to lower emittance, and better usefulness of the beam. It
also gives a key to understanding the energy spectrum of synchrotron radiation.
Energy Distribution (Attwood 5.2)
The angular distribution gives the picture of a searchlight sweeping by the observer. One can
realize that this gives rise to a broad energy distribution, just because the time it takes to
sweep by the observer is short.

B
A
R

The observer starts to detect photons at when the electron is at A, these photons will reach the
detector after the photon transit time r. The observer stops detecting photons when the
7
electron is a B, this point is reached after a time e. The pulse width 2 is the difference of
these two times.
2   e   r 
R  2 2 R  sin  R R R  1 1 

     
v
c
v c   v c 
(22)
This is a very short time, and should not be confused with the time associated with the bunch
length (as is done e. g. in Hecht: Optics), which is much longer. Here we are still talking one
electron. To give it a more convenient form we introduce the common relativistic parameter
v
c

(23)
When   1 one can get the following useful relation:
2 
1
1
1


2
1   1    21   
1 
(24)
Plugging (23) and (24) into (22) we get:
2 
R1 1
R
1     R3
  
  v c  c
2 c
(25)
Using Heisenberg’s uncertainty principle
E   

2
(26)
we get an energy spread
E 
2c 3
R
(27)
To relate this energy spread to the magnetic field rather than the radius, we use the expression
for the Lorentz force:
m0
v2
 evB
R
and get (again using that v  c )
(28)
8
E 
2eB 2
m0
(29)
With reasonable values,   3000 and B=1T , the energy spread is around 1 keV!
An exact expression for the intensity distribution over photon energies and emission angle can
be derived. The mathematical description of this distribution is, however, quite complex, and
involves modified Bessel functions (see e. g. Jackson, Classical Electrodynamics (Wiley NY
(1998)). The on-axis energy distribution is fully characterized by one important parameter: the
critical energy. At this energy, half the radiated power is at higher energies, and half is at
lower (see Fig. 5.7 in Attwood). The expression for the critical energy is almost identical with
equation (29)!
Ec 
3eB 2
2m0
(30)
In practical units this becomes:
Ec (keV )  0.6650 E 2 (GeV ) B(T )
(31)
Thus, by controlling two parameters, the energy of the electron beam and the magnetic field
one can fully determine the energy distribution. The on-axis flux is given by Attwood, eq. 5.6:
d 3 FB
d
dd
 0
 1.33  1013 Ee2 (GeV ) I ( A) H 2 (

E
)
Ec
(32)
in the units
[photons/s/mrad2/0.1%BW]
Note (Attwood Fig 5.7) that the angle-integrated energy distribution of intensity is different
from the on-axis distribution. This difference is accounted for by the exact expression, but it
can also be intuitively understood from our simple derivation: Off-axis, the ‘observed speed’
of the particle (which is the crucial variable in eq. (22)) is achieved by a projection on the
plane. Lower speed, implies longer sweeping-by times, and hence lower energy. Therefore,
lower energies are emphasized as the as a function of angle in the vertical plane. In other
words, the on-axis distribution emphasizes the higher energies. Or: The energy spectrum at
larger angles emphasizes lower energies. Or: the higher energies you choose to look at, the
smaller the deviation from the axis. The standard deviation of the angular distribution is
related to the energy
570  Ec 
 ' y mrad  
 
 E
Polarization
0.43
9
It can be realized from symmetry arguments that the on-axis radiation must be linearly
polarized in the horizontal plane. This is true classically, because as the orbiting electron is
observed its acceleration has components only in the horizontal plane. Although radiation in
other directions is folded into the forward direction in the relativistic transformation, off-plane
components from above and below the plane tend to cancel in the plane.
Off-axis these components do not cancel. Now, the orbiting electron is observed at angle, and
consequently, there are acceleration components in more than one direction perpendicular to
the beam direction, hence the polarization state becomes more complicated.
The exact expression also contains the angular distribution, and it can be shown (e. g. in
Jackson) that, integrated over all energies:
I ( ) 
1
5
(1 
 1
2
 2   2 


5 2
)
7 1  2
2

where  is the angle from the horizontal plane. As expected, the intensity decreases rapidly as
the angle increases. Of utmost importance is that the first term in the second bracket (1) of the
equation corresponds to radiation with the polarization vector in the horizontal plane, whereas
5 2
the second term (
) corresponds to radiation with the polarization vector
7 1  2
2
perpendicular to the horizontal plane. As expected, we have horizontally plane-polarized
radiation on axis, where the second term is zero. The second term increases with angle to
unity at large angles; thus the vertically polarized radiation has a maximum at some angle,
and at large angles the vertical and horizontal components have the same magnitude!
In general, the radiation is elliptically polarized. At large angles the radiation is circularly
polarized! And: The handedness above and below the plane is the opposite!
This (of course) also comes out of the exact master equation. One can also get a feeling for
this by considering the classical electron, orbiting in the horizontal plane: Radiation in the
vertical direction is circularly polarized, and the handedness of polarization is opposite, above
and below the plane. In the relativistic transformation this radiation is now folded in the
(almost) forward direction.
Therefore one can conveniently select the polarization state of the radiation by selecting the
emission angle.
Time structure
10
The time structure of synchrotron radiation is set by technical demands. It is impossible to
make a continuous current in the ring, so the electrons must come in bunches.
Correspondingly, the radiation comes in flashes.
In modern synchrotrons it is found that optimum beam conditions are achieved when the
bunches are around 20-100 ps long, and separated by around 2 ns.
The separation between bunches can be made larger, e.g. various bunch fill patterns can be
designed for specific experiments. It is not easy, however, to make the times shorter.
We will come back to time structure in the context of free electron lasers.