STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

Equations (A.5) and (A.6) are sometimes referred to as the Helmholtz equations.
Assume solutions of the following form, which can be used to describe either
traveling or standing waves
E(x, y, z) = E(x, y)e ±ikz
H(x, y, z) = H(x, y)e ±ikz
185
(A.7)
(A.8)
Note that the Laplacian operator can be separated into a transverse and longitudinal
component and rewritten as
∇ 2 = ∇ 2 t + ∂2
∂z
(A.9)
Using Eq. (A.9) and the form of the solutions in Eq. (A.7) and Eq. (A.8), the
Helmholtz equation reduces to a 2D wave equation
where
2
∇t + γ 2
⎧
⎪⎨
⎪⎩
γ 2 ≡ ɛoµoω 2 − k 2
⎫
E
⎪⎬
= 0 (A.10)
H
⎪⎭
(A.11)
It is useful to rewrite the Maxwell equations in terms of components parallel and
transverse to the z-axis. The transverse components of E and H can then be
expressed in terms of Ez and Hz, thereby simplifying the problem considerably
Et = i
γ2
k∇t
Ez − µ0ω ˆz × ∇t
Hz
Ht = i
γ 2
k∇t
Hz + ɛ0ω ˆz × ∇t
Ez
(A.12)
(A.13)