STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

187
Ez(z) = A sin kz + B cos(kz) (A.16)
The boundary conditions require that the tangential components of E must vanish
at z = 0 and z = d. These conditions are satisfied when
Ez(ρ, φ, z) = Eo cos
pπz
ψ(ρ, φ) (p = 0, 1, 2, ...) (A.17)
d
where ψ(ρ, φ) describes the azimuthal dependence of the fields. It follows from Eq.
(A.14) and Eq. (A.15) that the transverse components of the electric and magnetic
field are given by
Et = − pπ
sin
dγ2 Ht = iɛoω
cos
γ2 pπz
d
pπz
d
∇tψ(ρ, φ) (A.18)
[ˆz × ∇tψ(ρ, φ)] (A.19)
The problem can be simplified further by taking advantage of the azimuthal sym-
metry in the problem and writing
ψ(ρ, φ) = ψ(ρ)e ±imφ
where m is an integer. Substituting Eq. (A.20) into Eq. (A.10) yields
(A.20)
d2ψ 1 dψ
+
dρ2 ρ dρ +
γ 2 − m2
ρ2
ψ = 0 (A.21)
which is Bessel’s equation and whose solutions are Bessel functions of order m and
denoted as Jm(γρ). One of the solutions diverges for ρ = 0 which is physically
unacceptable and so this solution is disregarded.
as
From Eq. (A.17), Eq. (A.20) and the solution to Eq. (A.21), Ez can be written