Page 267 - Electromagnetics

compared to three-dimensional fields, and this structure often allows a decomposition
into even simpler field structures.
Consider a homogeneous region of space characterized by the permittivity ˜ , perme-
ability ˜µ, and conductivity ˜σ. We assume that all sources and fields are z-invariant, and
wish to find the relationship between the various components of the frequency-domain
fields in a source-free region. It is useful to define the transverse vector component of an
arbitrary vector A as the component of A perpendicular to the axis of invariance:
A t = A − ˆ z(ˆ z · A).

For the position vector r, this component is the transverse position vector r t = ρ.For
instance we have

ρ = ˆ xx + ˆ yy, ρ = ˆρρ,
in the rectangular and cylindrical coordinate systems, respectively.
˜
˜
Because the region is source-free, the fields E and H obey the homogeneous Helmholtz
equations
˜
" #
E
2 2
(∇ + k ) = 0.
˜
H
Writing the fields in terms of rectangular components, we find that each component
must obey a homogeneous scalar Helmholtz equation. In particular, we have for the
˜
˜
axial components E z and H z ,
˜
" #
2
2
(∇ + k ) E z = 0.
˜
H z
But since the fields are independent of z we may also write
˜
" #
E z
2
2
(∇ + k ) = 0 (4.208)
˜
t
H z
2
where ∇ is the transverse Laplacian operator
t
∂ 2
2
2
∇ =∇ − ˆ z . (4.209)
t 2
∂z
In rectangular coordinates we have
∂ 2 ∂ 2
2
∇ = + ,
t 2 2
∂x ∂y
while in circular cylindrical coordinates
∂ 2 1 ∂ 1 ∂ 2
2
∇ = + + . (4.210)
t 2 2 2
∂ρ ρ ∂ρ ρ ∂φ
˜
˜
With our condition on z-independence we can relate the transverse fields E t and H t to
˜
˜
E z and H z . By Faraday’s law we have
˜
˜
∇× E(ρ,ω) =− jω ˜µH(ρ,ω)
and thus
˜ 1 ˜
H t =− ∇× E .
jω ˜µ t
© 2001 by CRC Press LLC