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ANALYSIS OF CHERENKOV FREE-ELECTRON LASER DRIVEN BY A
FLAT ELECTRON BEAM
V. Kumar# and Yashvir Kalkal, Accelerator and Beam Physics Laboratory, Materials and Advanced
Accelerator Science Division, Raja Ramanna Centre for Advanced Technology, Indore, India
Abstract
A thin dielectric slab at the top of an ideal conductor
supports surface electromagnetic waves, which can be
amplified by an electron beam co-propagating in the close
vicinity of the dielectric surface. Under suitable
conditions, powerful coherent terahertz radiation can be
produced using this device, which is called as Cherenkov
Free-Electron Laser (CFEL). In this paper, we present an
analysis of CFEL driven by a two dimensional flat
electron beam travelling close and parallel to the dielectric
surface. The existence of the surface mode is explained
and derived in terms of the singularity of the reflectivity of
this system. A formula for small signal gain is derived by
analysing the residue at the singularity. Analysis is also
extended to understand the behaviour of the system at
saturation.
ELECTROMAGNETIC FIELD
CALCULATIONS
We start the analysis by writing down the
electromagnetic field due to the continuous electron beam.
The schematic of CFEL setup is shown in Fig. 1. We
consider a sheet electron beam travelling with a speed v
along the z axis, at a height h above the dielectric slab
having dielectric constant , thickness d and length L. The
dielectric slab is placed over a conducting surface and the
system is assumed to have translational invariance in the y
direction.
INTRODUCTION
In a Cherenkov Free-electron Laser (CFEL) [1],
coherent electromagnetic radiation is produced due to the
interaction of an electron beam with the co-propagating
surface electromagnetic mode supported by the dielectric
slab placed over an ideal conductor. Over the last several
decades, many theoretical investigations have been made
to study the mechanism of CFEL [1-4]. In all the previous
analyses, the size of the electron beam is taken to be either
very large or infinite, which does not seem to be
appropriate. This is because the supported modes which
interact with the electron beam are evanescent in the
direction perpendicular to the dielectric surface and are
confined very close to the dielectric surface. In this paper,
we present a surface mode analysis of CFEL by
considering a thin sheet electron beam, propagating very
close to the dielectric surface. The approach followed is
similar to Ref. 5 for the analysis of Smith-Purcell freeelectron lasers (SPFELs). Analysis has been made for
small signal gain and saturation behaviour of the system.
In the next section, we discuss the electromagnetic field
due to the sheet electron beam, and its reflection by the
dielectric surface. Reflection coefficient has a singularity,
which gives rise to the surface mode. Analysis of the
singularity and calculation of residues are discussed in the
following section. This is followed by the calculation for
the small-signal gain and efficiency at saturation by setting
up coupled Maxwell-Lorentz equations in the next section.
Finally, we present some conclusions.
_________________
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[email protected]
Figure 1: Schematic of Cherenkov FEL using a sheet
electron beam.
The dielectric slab supports an evanescent surface mode
at a resonant frequency 𝜔 as discussed in the next section.
The electron beam interacts with this surface mode and
develops the strongest Fourier component in current
around the frequency 𝜔. We can then write the surface
current density as 𝐾(𝑧, 𝑡)𝑒 𝑖(𝑘0𝑧−𝜔𝑡) + 𝑐. 𝑐., where c.c.
represents complex conjugate, 𝐾(𝑧, 𝑡) = (𝐼 ⁄𝛥𝑦)〈𝑒 −𝑖𝜓 〉
and 𝜓 = 𝑘0 𝑧 − 𝜔𝑡 is the electron phase, 𝑘0 = 𝜔⁄𝑐𝛽, c is
the speed of light,  = v/c, I is the electron current, Δy is
the electron beam width in y direction and 〈… 〉 denotes
averaging over the number of electrons distributed over
one wavelength of evanescent mode. We assume slow
variation in surface current density of the type 𝑒 𝜇𝑧 and
look for the possible solutions of μ. If the real part of μ is
positive, then there will be an enhanced bunching at 𝜔.
We can write the surface current density as 𝐾0 𝑒 𝑖(𝛼0𝑧−𝜔𝑡) +
𝑐. 𝑐., where 𝐾0 is independent of z and t, and 𝛼0 = 𝑘0 − 𝑖𝜇.
Here the sheet beam acts as a source of electromagnetic
field, which is incident on the dielectric slab. By solving
the Maxwell equations due to the above current density,
one finds the electromagnetic field as:
𝐻𝑦𝐼 (𝑥, 𝑧) =
1
𝜃(𝑥)𝐾(𝑧)exp[−𝜃(𝑥)𝛤0 𝑥] ,
2
(1)
(𝛼02
2 ⁄ 2 )1⁄2
where 𝛤0 =
−𝜔 𝑐
, 𝜃(𝑥) = 1 for 𝑥 > 0, and
𝜃(𝑥) = −1 for 𝑥 < 0. The electromagnetic field has 𝐻polarisation, which means 𝐻𝑥𝐼 = 𝐻𝑧𝐼 = 𝐸𝑦𝐼 = 0 and 𝐸𝑧 is
given by the expression 𝐸𝑧𝐼 = (𝑖 ⁄𝜖0 𝜔 )(𝜕𝐻𝑦𝐼 ⁄𝜕𝑥 −
𝛿(𝑥)𝐾), where (x) is the Dirac delta function.
Now, due to the dielectric slab, the incident
electromagnetic field is reflected back towards the sheet
beam. The reflected and the incident electromagnetic
fields couple through reflection coefficient R of the
dielectric slab, i.e. 𝐴𝑅 = 𝑅𝐴𝐼 . Here, AI and AR are the
amplitudes of incident and reflected waves. Once we know
the reflection coefficient for the system, we can find out
the amplitude of reflected wave for a given incident
electromagnetic wave. By adding the contributions of
incident and reflected electromagnetic field, one finds the
expression of total electric field experienced by the
electron beam at x = 0 as:
𝐸𝑧 =
𝑖 𝐼𝑍0
( 𝑅𝑒 −2𝛤0 ℎ − 1)⟨𝑒 −𝑖𝜓 ⟩ .
2𝛽𝛾𝛥𝑦
(2)
Here, 𝑍0 = 1⁄𝜖0 𝑐 = 377 𝛺 is the characteristic
impedance of free space, 𝜖0 is the permittivity of free
space and  is the relativistic Lorentz factor. The total
longitudinal electric field is given by 𝐸𝑧 𝑒 𝑖𝜓 + 𝑐. 𝑐.. Here, R
is a function of frequency and the growth rate parameter μ.
In the next section, we discuss the behaviour of reflection
coefficient around the frequency of the surface mode.
SINGULARITY IN REFLECTIVITY
We first discuss the reflection coefficient of the
dielectric slab placed over the metallic surface, for the
incident electromagnetic wave generated by the beam. By
solving the Maxwell equations with appropriate boundary
conditions, following formula is obtained [6].
𝑅=
1 + 𝑎 𝑡𝑎𝑛[𝛼0 𝑏]
,
1 − 𝑎 𝑡𝑎𝑛[𝛼0 𝑏]
(3)
where 𝑎 = (𝛾⁄𝜖 )√𝜖𝛽 2 − 1 and 𝑏 = 𝑑√𝜖𝛽 2 − 1. Note
that for 𝜇 = 0, R is singular at 𝑘0 = (1⁄𝑏) tan−1 (1⁄𝑎).
This is same as the dispersion relation for a single slab
CFEL. This is due to the fact that the condition for a
system to support slow wave of wave vector 𝑘0 = 𝜔⁄𝑣 is
equivalent to the requirement that reflection coefficient is
singular for this combination of  and k0. Due to the
singularity in R, the dielectric slab supports the evanescent
wave on its own i.e., without any incident wave. In order
to study the nature of singularity in R at 𝜇 = 0, we
perform Laurent series expansion of R as a function of .
By expanding the numerator and denominator on right
hand side of Eq. (3) in terms of μ, we get the following
expression for R:
𝑅=
2 −
𝑖𝑏
𝑎
𝑖𝑏
𝑎
(1 + 𝑎2 )𝜇 −
(1 + 𝑎2 )𝜇 +
𝑏2
𝑎2
𝑏2
𝑎2
(1 + 𝑎2 )𝜇 2 + 𝑂(𝜇 3 ) . .
(1 + 𝑎2 )𝜇 2 + 𝑂(𝜇 3 ) . .
. (4)
Here, we have division of two infinite series. By
performing the required algebra and keeping the terms of
the order of 1⁄µ and µ0 in the Laurent series, we obtain
the following simple expression for reflection coefficient
𝑖𝜒
+ 𝜒1 ,
(5)
µ
where 𝜒 = 2𝑎 ⁄(1 + 𝑎2 )𝑏 and 𝜒1 = (1 − 𝑎2 )⁄(1 + 𝑎2 ).
In the next section, we use above formula for R to set the
coupled Maxwell-Lorentz equations for a CFEL.
𝑅= −
CALCULATIONS OF CFEL GAIN
We now set up the Maxwell-Lorentz equation for a
CFEL and make an analysis for small signal gain. This
approach is familiar in case of the conventional FELs and
SP-FELs, and it turns out to be very useful for the detailed
analysis there [5,7]. By using Eq. (2) and Eq. (5), we
obtain the following expression for amplitude of the
electromagnetic field:
𝐸𝑧 =
𝑖 𝐼𝑍0
𝜒
( (−𝑖 + 𝜒1 )𝑒 −2𝛤0ℎ − 1) ⟨𝑒 −𝑖𝜓 ⟩.
2𝛽𝛾𝛥𝑦
𝜇
(6)
The above expression has two parts. The first part is a
component of surface mode and the remaining terms are
independent of growth rate and are identified as spacecharge terms. Replacing μ by 𝑑 ⁄𝑑𝑧, we obtain a steady
state differential equation of electric field for the surface
mode as:
𝑑𝐸
𝐼𝑍0 𝜒 −2𝛤 ℎ −𝑖𝜓
=
𝑒 0 ⟨𝑒 ⟩ .
𝑑𝑧 2𝛽𝛾𝛥𝑦
(7)
Also, we can derive the following equation of motion
for the electron in the presence of above field:
𝑑𝛾𝑖
𝑒𝐸 𝑖𝜓
=
𝑒 𝑖 + 𝑐. 𝑐.,
𝑑𝑧 𝑚𝑐 2
(8)
𝛾𝑖 − 𝛾𝑝
𝑑𝜓𝑖
𝜔
= 2 3(
).
𝑑𝑧
𝑐𝛾 𝛽
𝛾𝑝
(9)
Here, the subscript i denotes the ith particle, and 𝛾𝑝 =
1/√1 − 𝑣𝑝2 /𝑐 2 , where vp is the phase velocity of the
surface mode. We define the following dimensionless
variables:
𝜁 = 𝑧 ⁄𝐿 ,
𝜂𝑖 =
𝑘0 𝐿
(𝛾 − 𝛾𝑝 ) ,
𝛽2𝛾 3 𝑖
(10)
(11)
ℇ=
8𝜋𝑘0 𝐿2
𝐸,
𝐼𝐴 𝑍0 𝛽 2 𝛾 3
𝒥 = 4𝜋
(12)
𝐼 𝜒 𝑘0 𝐿3 −2𝛤 ℎ
𝑒 0 .
𝐼𝐴 𝛥𝑦 𝛽 3 𝛾 4
(13)
Here 𝜁 is the dimensionless distance which varies from 0
to 1, 𝜂𝑖 is the normalised energy detuning of ith electron,
The dimensionless field amplitude is given by ℇ, 𝒥 is the
dimensionless beam current and 𝐼𝐴 = 17.04 𝑘𝐴 is the
Alfven current. In terms of these dimensionless variables,
the coupled Maxwell-Lorentz equations assume the form:
𝑑ℇ
= 𝒥⟨𝑒 −𝑖𝜓 ⟩ ,
𝑑𝜁
(14)
𝑑𝜂𝑖
ℇ
= 𝑒 𝑖𝜓𝑖 + 𝑐. 𝑐.,
𝑑𝜁
2
(15)
𝑑𝜓𝑖
= 𝜂𝑖 .
𝑑𝜁
(16)
These equations have to be solved numerically for the
complete description of the CFEL. However, there is a
regime of small-signal small-gain in CFEL, which can be
solved analytically in similar way as in conventional FELs
[7]. Once the equations are cast in dimensionless variables
with appropriate form as above, it is straightforward to get
the expression for small signal gain. Defining differential
gain as (1⁄ℇ2 )(𝑑ℇ2 ⁄𝑑𝜁 ), and following the approach
similar to given in Ref. 7, we get the following expression
for the small signal gain:
𝐺(𝛼) = 2𝒥 𝑔(𝛼),
1
(17)
𝛼
where 𝑔(𝛼) = − 3 [1 − cos 𝛼 − sin 𝛼] is the gain
𝛼
2
function, has a maximum value 𝑔𝑚𝑎𝑥 = 6.75 × 10−2 and
𝑘 𝐿
𝛼 = 20 3 (𝛾 − 𝛾𝑝 ). By substituting 𝒥 from Eq. (13) and
𝛽 𝛾
the value of 𝑔𝑚𝑎𝑥 , we obtain the final expression for small
signal gain in a single pass as:
𝐺 = 2 × 6.75 × 10−2 × 4𝜋
𝐼 𝜒 𝑘0 𝐿3 −2𝛤 ℎ
𝑒 0 .
𝐼𝐴 𝛥𝑦 𝛽 3 𝛾 4
(18)
The gain decreases exponentially with increase in electron
beam height h and increases with the surface current
density 𝐼 ⁄𝛥𝑦. The gain depends upon L and dielectric
constant ϵ, slab thickness d through the parameter χ.
There is a reduction in the gain due to diffraction of the
surface mode supported by the grating. The radiation beam
size increases due to diffraction, resulting in partial
overlap with the electron beam. The effective width of the
radiation beam needs to be taken in Eq. (18) as y.
Following a similar analysis as done for SPFEL, we
estimate that that effective value of y should be taken as
√𝜆𝐿⁄2𝛽𝑔 [8], where 𝛽𝑔 𝑐 is the group velocity. Note that
we have taken y as √2𝜋 times the rms beam width.
Next, we discuss the efficiency for power conversion in
a CFEL at saturation. We can estimate the efficiency
analytically by the argument that the maximum energy that
the electron can lose before saturation is such that it lags
the co-propagating evanescent wave by quarter a
wavelength during the transit through the dielectric slab
[9]. As per this argument, if change in velocity of the
electron is Δv due to the loss of energy, then 𝛥𝑣𝐿⁄𝑣 =
𝜋𝑣 ⁄2𝜔 . This gives us the following expression for the
efficiency 𝜂𝑒𝑓𝑓 , in the non-relativistic case:
𝛽𝜆
𝜂𝑒𝑓𝑓 =
.
(19)
2𝐿
Finally, we discuss the calculation of small signal gain
and efficiency predicted from our formula for the case of
the Dartmouth experiment [4,10] for a CFEL. The
parameters used in the calculation are listed in Table 1.
We evaluate the dispersion relation and find out that g =
0.23 for this case. The small signal gain predicted from our
calculations comes out to be 49% with an efficiency of
0.3% at saturation. Note that due to long wavelength,
diffraction effects are important and reduce the small
signal gain significantly.
Table 1. Parameters of the CFEL used in the calculation
Dielectric constant (ϵ)
Slab Thickness (d)
Length of slab (L)
Electron beam Current (I)
Electron Beam Energy
Electron beam Height (h)
Effective beam width (Δy)
Output Frequency
13.1
350 μm
0.15 m
1 mA
30 keV
35 μm
30.3 mm
0.1 THz
To summarize, we have analysed the interaction of
sheet electron beam with the surface mode supported by
the dielectric layer on a conducting surface. Our analysis is
built on an earlier successful analysis of SP-FELs [5,8,9].
We have derived a formula for the small signal gain and
efficiency at saturation for this system. Our analysis has a
scope of incorporating all realistic effects such as effect
due to three dimensional variations of the field and energy
spread of the electron beam, which makes it a very useful
approach in developing a better understanding of CFEL.
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