1. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
DEVELOPMENT OF THE THEORY OF MEASUREMENT OF ELECTRON BEAM ABSOLUTE ENERGY BY “RESONANCE ABSORPTION” METHOD R.A. Melikian Yerevan Physics Institute, Yerevan, Armenia
Abstract
We consider in detail some theoretical aspects of the absolute energy measurement of an electron beam
by means of the “Resonance Absorption” method. We carefully analyse the issues of the accuracy of the
electron beam absolute energy accuracy at certain electrons distribution of energies and angles, the laser
beam parameters, and the choice of the magnet length with the edge effects taken into account.
1. Essence of the method
The possibility of precise measurement of absolute energy of the electron beam in range of a few hundred
GeV by "Resonance Absorption of Photons in a Magnetic Field" method was offered in [1, 2]. In this
report we consider more correctly and in detail the measurement accuracy problem of electron beam
energy with Gaussian energy distribution and with various angle distributions of electrons. Moreover,
we consider the issues concerning the laser beam parameters, the admissible limits of the magnet length,
as well as the choice of the magnet length with the edge effects taken into account.
The energy of electrons can be determined from the following condition of the resonant absorption of the
laser photons by electrons at transitions between the quantum energy levels in a magnetic field:
(1a)
(1) or
R.A.Melikian,YerPhI, , Zeuthen

2. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
where
is the laser frequency,
is the relativistic factor of electron,
is the component of
electron velocity along the axis of the magnetic field (along z-axis),
is the magnetic field,
is the incidence angle of photons with respect to the z –axis,
is the injection angle of the
electrons in the magnetic field with respect to the z –axis,
is the electron energy.
From the resonance condition (1a) the
-factor of electron can be determined:
(2)
Let us find the energy of the electron for case when
and the spatial angle
around the
-axis
is limited to the interval
(Fig.1).
Fig.1
The dependence of electron energy
on the angle
found from (2) for the fixed values of
and
is illustrated on Fig.2.
Fig.2
R.A.Melikian,YerPhI, , Zeuthen

3. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
From the Fig.2 and the relations (1), (2) it follows that:
1. The energy of the electron reaches minimum for the angles
and is equal to
(3)
2. Angles
and
should satisfy the restriction:
(4)
3. Energy
of the electron is less sensitive to changes of the angle
near
Namely, this
behavior of
will be used to determine of the absolute energy of the electron beam.
In reality, the electron beam has some spread over energy:
We assume that the electron beam
has Gaussian energy distribution (Fig.3) and spatial angle
around the
-axis is limited to the
interval
(Fig.4).
Fig.3
Fig.4
Since the absorption of photons by electrons is possible only in the energy interval
we shall consider the dependence
only in this strip, and find the corresponding quantity
R.A.Melikian,YerPhI, , Zeuthen

4. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
A. It is clear, that for the quantities
when curve
lies outside of the interval
(where there are no electrons), the absorption of photons is absent and
(Fig.5).
(Fig.5)
B. With increase of
the intensity of absorption
grows and reaches its maximum
when curve
passes through the point
for the some
(Fig.6) .
(Fig.6)
R.A.Melikian,YerPhI, , Zeuthen

5. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
Thus,
we can determine the value of
corresponding to the absorption intensity
and
calculate the electron beam energy
, using the resonant condition (1a) for the point
(Fig.6):
(5)
whence we find or
(6)
Comparing the resonant conditions
(7)
and
(8)
written for points
and
where
we see that .
Using (5) and (7) we can estimate the value of
(9)
R.A.Melikian,YerPhI, , Zeuthen

6. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
C. Determination of electron beam energy for coaxial distribution of electrons over angles.
When for the some
the curve
passes through the point
then only the half of the number
of electrons
with energies
and with angles
will absorb photons (Fig.7).
It means that for
the absorption intensity of photons will be
(Fig.7).
(Fig.7)
Thus, we can determine the value of
corresponding to the value of the absorption intensity
and calculate the electron beam energy
using the resonant condition (1a) for the point
(Fig.7):
(10)
From (10) we obtain
(11)
From (11) it follows, that for interesting for us energies of electrons
one always has
and we can calculate the quantity of
with the accuracy
by means of the simple formula
(12) or
(12a)
R.A.Melikian,YerPhI, , Zeuthen

7. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
Accuracy of measurement of electron beam energy
According to the formula (12) the accuracy of determination
depends on the accuracy of the
measurement
i.e.
(13)
On the other hand, since
(14)
it follows that the accuracy of determination of the energy
depends on the degree of stability of the
laser frequency
and of the magnetic field
as well as the accuracy of their measurement.
Thus, since the value of
can be determined only with accuracy
the
dependence
on the diagram (Fig.7) will be represented not as some line, but as some "strip" with
the width
(Fig. 8).
Fig. 8
R.A.Melikian,YerPhI, , Zeuthen

8. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
Measurement of electron beam energy at misalignment of electron beam with respect to the z-axis
Since the angle spread of the electron beam on ILC for high energies is very small
it is
difficult to practically provide the coaxiality of the e-beam with respect to the z-axis. Therefore, we shall
consider measurement of electron beam energy when its angles of incidence are limited to the interval
(Fig.9).
We assume that the distribution of the number of electrons over angles has the
shape represented on Fig.10.
Fig.9
Fig.10
For the angles
we shall find the interval of values
for which the resonant absorption of
photons is possible. On Fig.11 the dependence
is illustrated the for the some
when the curve
lies above the strip
Then, the electrons with angles within the interval
appear outside of the strip
and therefore
Fig.11
R.A.Melikian,YerPhI, , Zeuthen

9. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
B. On Fig.12 the dependence
is illustrated for the some
when the curve
passes over bottom of the strip
In this case the maximum quantity of electrons can
be absorbed and therefore
(Fig.13).
Fig.12
Fig.13
Thus, as a result of the measurement
we can determine
and, using the resonant condition (1a)
(15)
for the points (Fig.12)
and
we find or
(16)
R.A.Melikian,YerPhI, , Zeuthen

10. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
On Fig.14 the dependence
is illustrated (according to (2)) for some
when the curve
passes through the midpoint of the strip
Then, only the half of the
electron number
will contribute in the process of absorption and consequently the intensity of
photon absorption will be
Fig.14
Fig.15
Thus, as a result of the measurement
we can determine
Using the resonant condition (1а)
and
(17)
for the points (Fig.14)
and
we find or
(18)
R.A.Melikian,YerPhI, , Zeuthen

11. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
From (15) and (17) we find the relations:
(19) and
(20)
where (see Fig.14)
(21)
From (15) and (17) we also find ( with accuracy
) the approximate relation:
(22)
Substitution of the values
and
from (21), (22) into the equation (20) yields an algebraic
equation for one unknown variable
The exact solution of this equation is rather cumbersome and
will not be presented here.
Fig.15
Thus, at misalignment of the electron beam with respect to the z-axis, the measuring of
and
with
accuracy
allow us to determine the energy of distribution centre of electron beam with accuracy
R.A.Melikian,YerPhI, , Zeuthen

12. then the relation (23) can be written as:
Choice of the magnet length, with the edge effects taken into account
For determination of the magnet length
we use the classical formula for the growth of the electron
energy [1]:
(23)
where
is the laser intensity parameter,
is the amplitude of electric field of the wave. If
is the length of the formation of photon absorption with the energy
during time
and taking into account that
(24)
then the relation (23) can be written as:
(25)
Since only the integer numbers of photons can be absorbed from (25) for given
we can
choose the magnet length
within the limits:
(26)
The condition (26) has a practical meaning and allows us to choose the length of the magnet
using
the homogeneous part of the magnetic field and exclude the influence of the edge effects.
Fig.16
R.A.Melikian,YerPhI, , Zeuthen

13. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
Choice of intensity and diameter of the laser beam
The laser intensity, necessary for the absorption of photons by electrons on the length
can be
determined from the formula (25)
(27)
Let us note that for a given intensity
the amplitude of the electric field
of an electromagnetic
wave can be determined according to the formula (24) and does not depend on the presence of electrons.
Let us consider restrictions on the diameter of the laser beam and limits of its admissible values.
It is
known,
that, due to diffraction, the light beam of diameter
diverges in the limits:
where
is the length of wave.
The area of electrons interaction with the laser beam in the magnetic field is schematically shown on
Fig.1. If
is the length of interaction of the electron and
the light beames, then from Fig.1 we have
(28)
Absorption of photons can occur if the following
condition is satisfied
(29)
Fig.17
From the diagrams considered above for
(Fig. 6, Fig. 12) it is clear that in these cases we have
and
can be chosen in a wide interval of values.
R.A.Melikian,YerPhI, , Zeuthen

14. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
Feasibility of measurement of an electron beam energy by a method "Resonant Absorption of Photons
in a Magnetic Field " is much easier than acceleration of electrons by the same method.
Firstly, for measurement of a electron energy the absorption of one or a few photons on length of a
magnet is sufficient , while for acceleration of electrons on appreciable energy
the number
of the absorbed photons should be very great. Therefore for acceleration of electrons the necessary laser
intensity is much more than for measurement of electron beam energy.
Second, for measurement of electron energy it is necessary to use a constant and homogeneous magnetic
field while for electron acceleration it is necessary to use a complex magnetic field profile [3].
Summary
1. The possibility of the measurement of the electron beam absolute energy with accuracy for
Gaussian for distributions over energy for various distributions over angles of electrons is shown.
2. Admissible limits of the length of a magnet allow the choice of using the homogeneous area of the
magnetic field, effectively excluding the influence of the edge effects.
3. Restrictions on the diameter of the laser beam and its admissible limits are considered.
R.A.Melikian,YerPhI, , Zeuthen

15. R.A.Melikian,YerPhI, 09-11.04.2008, Zeuthen
References
1. R.A. Melikian. The Possibility of Precise Measurement of Absolute Energy of the Electron Beam by
Means of “Resonance Absorption” Method Meeting on Beam Energy Measurement. June 06-08,
2007, Zeuthen.
2. D.P. Barber, R.A. Melikian. On the Possibility of Precise Measuring of Electron Beam Energy Using
Resonance Absorption of Laser Wave by Electrons in Static Magnetic Field. Proc. 7th EPAC, 2000,
Vienna.
3. S.V. Shchelkunov, et al., The LACARA Vacuum Laser Accelerator Experiment: Beam Positioning
and Alignment in a Strong Magnetic Field.
Advanced Accelerator Concepts Workshop, 2006.
R.A.Melikian,YerPhI, , Zeuthen