Nonlinear wake amplification by an active medium in a cylindrical waveguide using a modulated trigger b unch 3
through the stimulated emission process. Thus, the EM field
has the following generic form
ψ(T, r) =
1
2
s
J
ν
(k
s
r)ψ
s
(T )e
iω
0
T
+ c.c., (2.1)
where each field ψ(ψ∈{E
z
, E
r
, H
φ
}) depends on the radial
coordinate, r , and a variable that follows the e-beam, T ≡ t −
z/βc. Specifically, for field ψ = E
z
the index of the Bessel
function of the first kind is ν = 0 and for ψ ={E
r
, H
φ
}
the index is ν = 1. In addition, the radial wavenumber is
k
s
≡ p
s
/R, where p
s
is defined through J
0
(k
s
R) = 0 (s =
1, 2, 3,...)since E
z
(T , r = R) = 0. Finally, the dependence
of each mode on T is given by ψ
s
(T ).
Similarly to the EM fields, the current density of the
trigger bunch is assumed to be of the form
J
z
(T , r ) =
1
2
I (T )
θ(R
b
−r)
π R
2
b
e
iω
0
T
+ c.c., (2.2)
where θ(x) is the Heaviside step function and I (T ) is the
e-beam longitudinal current envelope. Assuming that the
spatially modulated bunch profile is given by f (T ) then the
current envelope is I (T ) = I
0
f (T ), where the modulated
current is I
0
= Q
b
βc/L
b
. Here, the total charge is Q
b
=
−eN
b
, where −e is the electron charge and N
b
is the number
of macro-particles that comprise the bunch; L
b
is the bunch
length.
Having in mind that the growth rate is much smaller than
the resonance frequency, the dynamics of the fields can be
separated into two major time scales: the ‘fast’ and the ‘slow’
time scales. The ‘fast’ time scale is the medium resonance
period of time 1/ω
0
whereas the ‘slow’ time scale associated
with the growth rate of the field envelopes, ψ
s
(T ),is1/ω
p
,
where ω
p
= ω
0
2N
0
μ
2
ε
0
ω
0
ω
0
is the ‘plasma frequency’ of
the active medium. Here, N
0
is the initial PID, μ = μ
12
/
√
3
is the average dipole moment and μ
12
is the dipole moment.
The parameter is the reduced Planck constant and ε
0
is the
vacuum permittivity. Thus, the dynamics of the wake and
the active medium can be described by the slowly varying
envelope approximation. Consequently, the dynamics of the
normalized EM fields derived from the Ampere and Faraday
laws read
∂
¯
E
z,s
∂τ
=−i ¯ω
0
¯
E
z,s
+
¯
k
s
2
√
ε
r
¯
E
+,s
+
¯
k
s
2
√
ε
r
¯
E
−,s
+
2
ε
r
¯
P
z,s
−
2
ε
r
¯
J
s
f, (2.3)
∂
¯
E
+,s
∂τ
=−i ¯ω
0
¯
E
+,s
−
¯
k
s
√
ε
r
Δε
−
¯
E
z,s
+
2
ε
r
Δε
−
¯
P
r,s
,
(2.4)
∂
¯
E
−,s
∂τ
=−i ¯ω
0
¯
E
−,s
−
¯
k
s
√
ε
r
Δε
+
¯
E
z,s
−
2
ε
r
Δε
+
¯
P
r,s
.
(2.5)
In our model the time T is normalized by 1/ω
A
such that
τ = T ω
A
, where ω
A
= ω
p
/(2
√
ε
r
) and, as already indicated,
ε
r
is the dielectric constant of the medium excluding the
population inversion dynamics. Also, the normalized electric
field envelopes,
¯
E
z,s
and
¯
E
r,s
, are normalized with E
0
=
1
J
1
(k
s
R)
ω
0
N
0
2ε
0
and the magnetic field envelope,
¯
H
s
,is
normalized with E
0
/(μ
0
c). In addition,
¯
E
±,s
=
¯
H
s
√
ε
r
±
¯
E
r,s
,
Δε
±
= 1 ±
1
β
√
ε
r
, ¯ω
0
= ω
0
/ω
A
and
¯
k
s
= k
s
c/ω
A
.
The expression
2
ε
r
¯
J
s
f is the normalized bunch current,
where
¯
J
s
=
I
0
π R
2
J
c
(k
s
R
b
)
J
1
(k
s
R)
√
ε
r
ω
0
μN
0
, J
c
(x) ≡ 2 J
1
(x)/x and f =
f (τ ) describes the electron bunch profile in the longitudinal
direction. In this study, the bunch injected at τ = τ
0
with a
length of
¯
L
b
= L
b
ω
A
/βc has a profile of f (τ
0
<τ <τ
1
) =
1, f (τ = τ
0
) = 1/2, f (τ = τ
1
) = 1/2 and zero otherwise,
where τ
1
= τ
0
+
¯
L
b
.
The active medium is modeled semi-classically as a
two-level system within the framework of the dipole
approximation
[22, 23]
. In addition, it is assumed that only
stimulated emission can reduce the population inversion
density and collisions of the second kind are neglected here.
The response of the active medium to the wake is through
the normalized polarization fields
¯
P
z
and
¯
P
r
,
∂
¯
P
z,s
∂τ
+
Δ ¯ω
2
¯
P
z,s
= ε
r
¯
N
¯
E
z,s
, (2.6)
∂
¯
P
r,s
∂τ
+
Δ ¯ω
2
¯
P
r,s
=
1
2
ε
r
¯
N
¯
E
+,s
−
1
2
ε
r
¯
N
¯
E
−,s
, (2.7)
where the polarization envelopes are normalized with
i
μN
0
√
ε
r
J
1
(k
s
R)
and
¯
N =
¯
N (τ ) is the normalized population
inversion density measured in units of N
0
. Also, it is
assumed for simplicity that
¯
N is radially independent. Radial
variations will be considered elsewhere.
The dynamics of the PID,
¯
N , reads
∂
¯
N
∂τ
+
¯
A
21
(
¯
N −
¯
N
e
) =−
1
4
s
[2
¯
E
∗
z,s
¯
P
z,s
+ 2
¯
E
z,s
¯
P
∗
z,s
+ (
¯
E
+,s
−
¯
E
−,s
)
¯
P
∗
r,s
+ (
¯
E
∗
+,s
−
¯
E
∗
−,s
)
¯
P
r,s
], (2.8)
where
¯
A
21
= A
21
/ω
A
is the normalized Einstein coefficient
associated with the spontaneous emission time τ
spon
=
1/
¯
A
21
and
¯
N
e
is the PID in thermal equilibrium. In this
study, we consider an active medium with a long sponta-
neous emission time compared with the order of the amplifi-
cation time of 1/ω
p
, which results in neglecting the second
term on the left-hand side of Equation (2.8) associated with
the spontaneous emission effect.
Finally, the set of equations introduced in Equations
(2.3)–(2.8) conserves energy,
∂
∂τ
¯
W
tot
= 0, (2.9)