11 Application of Accelerators and Storage Rings
691
with v x (z) from Eq. (11.63). Combining Eqs. (11.71), (11.72), and (11.74) and
neglecting derivatives of SVA quantities compared to fast terms yields
∂
∂z +
1
c
∂
∂t
ˆ
E x (z, t) = −
cμ 0 K
2γ r
ˆ
j 1 (z, t) cos (k u z) exp (ik u z) exp
ik l ˆ
z sin (2k u z)
.
On the right hand side appears the product of three functions that are z periodic in
λ u respectively λ u /2, but for the SVA approach we are only interested in variations
large compared to the undulator period. Therefore we average this product along
one undulator period and get
∂
∂z
+
1
c
∂
∂t
ˆ
E x (z, t) = −
cμ 0 ˆ
K
4γ r
ˆ
j 1 (z, t) .
(11.75)
Again the longitudinal oscillation is regarded by the modified undulator parameter ˆ
K, Eq. (11.67).
Periodic Approach
The periodic approach is applicable if the initial bunch and electromagnetic stimulation are sufficiently long in time, or more precisely if time variations are slowly
compared to coherence time L coh /c, with L coh the coherence length defined below
in Eq. (11.85). The time dependency in Eq. (11.75) is neglected ˆ
E x (z, t) → ˜
E x (z),
ˆ
j 1 (z, t) → ˜
j 1 (z), j 0 (z − vt) → j 0 and only particles in one micro-period are
considered. We choose N electrons with start phases ψ ν in the range 0 ≤ ψ < 2π.
Then the modulation amplitude follows from Fourier series expansion
˜
j 1 = 2j 0
N
ν=1
exp (−iψ ν ) /N.
(11.76)
The sum
N
ν=1
exp (−iψ ν ) /N is called bunching factor.
Coupled First-Order Equations
Combining Eqs. (11.68), (11.69) and (11.75) one obtains a set of coupled first-order
equations
d
dz ψ ν = 2k u η ν
d
dz η ν = −
e ˆ
K
2m e c 2 γ 2 Re
˜
E x exp (iψ ν )
d
dz
˜
E x = −
μ 0 c ˆ
K
4γ
˜
j 1
(11.77)
which, together with Eq. (11.76), describe the evolution of the phases ψ n and
energy deviations η n of the N electrons, as well as the growth of ˜
E x (z) and ˜
j 1 (z).
Longitudinal Coulomb forces (“space charge forces”) are of minor importance in
short-wavelength FELs [14] and are neglected here and in Eq. (11.78) below.
691
with v x (z) from Eq. (11.63). Combining Eqs. (11.71), (11.72), and (11.74) and
neglecting derivatives of SVA quantities compared to fast terms yields
∂
∂z +
1
c
∂
∂t
ˆ
E x (z, t) = −
cμ 0 K
2γ r
ˆ
j 1 (z, t) cos (k u z) exp (ik u z) exp
ik l ˆ
z sin (2k u z)
.
On the right hand side appears the product of three functions that are z periodic in
λ u respectively λ u /2, but for the SVA approach we are only interested in variations
large compared to the undulator period. Therefore we average this product along
one undulator period and get
∂
∂z
+
1
c
∂
∂t
ˆ
E x (z, t) = −
cμ 0 ˆ
K
4γ r
ˆ
j 1 (z, t) .
(11.75)
Again the longitudinal oscillation is regarded by the modified undulator parameter ˆ
K, Eq. (11.67).
Periodic Approach
The periodic approach is applicable if the initial bunch and electromagnetic stimulation are sufficiently long in time, or more precisely if time variations are slowly
compared to coherence time L coh /c, with L coh the coherence length defined below
in Eq. (11.85). The time dependency in Eq. (11.75) is neglected ˆ
E x (z, t) → ˜
E x (z),
ˆ
j 1 (z, t) → ˜
j 1 (z), j 0 (z − vt) → j 0 and only particles in one micro-period are
considered. We choose N electrons with start phases ψ ν in the range 0 ≤ ψ < 2π.
Then the modulation amplitude follows from Fourier series expansion
˜
j 1 = 2j 0
N
ν=1
exp (−iψ ν ) /N.
(11.76)
The sum
N
ν=1
exp (−iψ ν ) /N is called bunching factor.
Coupled First-Order Equations
Combining Eqs. (11.68), (11.69) and (11.75) one obtains a set of coupled first-order
equations
d
dz ψ ν = 2k u η ν
d
dz η ν = −
e ˆ
K
2m e c 2 γ 2 Re
˜
E x exp (iψ ν )
d
dz
˜
E x = −
μ 0 c ˆ
K
4γ
˜
j 1
(11.77)
which, together with Eq. (11.76), describe the evolution of the phases ψ n and
energy deviations η n of the N electrons, as well as the growth of ˜
E x (z) and ˜
j 1 (z).
Longitudinal Coulomb forces (“space charge forces”) are of minor importance in
short-wavelength FELs [14] and are neglected here and in Eq. (11.78) below.