Page 168 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Waveguides
wave will be deflected from its straight path by the waveguide walls
along the curve. For a straight segment we can simplify the analysis by
considering simple harmonic propagation along the axis of the
waveguide, thereby “separating” the variables somewhat. Thus we start
the analysis by assuming that a wave of the form (4.10) that propagates in
the x-direction through an isotropic medium:

(
E t() = E yz,( )exp [ i – ωt – kx)]
t
(4.60)
(
H t() = H yz,( )exp [ i – ωt – kx)]
t
Note that E yz,( ) and H yz,( ) are real values. For this case, harmonicity
t
t
implies that ∂∂t⁄ ≡ i – ω and axial propagation of a planar distribution
,
⁄
,
⁄
,
implies that ∇ ≡ ( ik ∂∂y ∂ ∂z) = ( ik ∇ t , ) so that
⁄
,
⁄
∇ ≡ ( ∂∂y ∂ ∂z) . We now check to see what this assumption induces
t
by inserting (4.60) into the four Maxwell equations (4.1a)-(4.1d) as cor-
rected for dielectric materials:
∇ × E + ike × E = – iωB t gives ∇ × E = 0 ,
t
t
x
t
t
t
ω
e × E = – ----B (4.61a)
x t t
k
∇ × H + ike × H = – iωD t gives ∇ × H = 0 ,
t
t
t
t
x
t
ω
e × H = – ----D (4.61b)
x t t
k
ε  ∂E ty + ∂E tz = ε∇ •( E ) = 0 implies that ∇ • D = 0 (4.61c)
∂  y z ∂  t t t t
µ  ∂H ty + ∂H tz = µ∇ • H ) = 0 implies that ∇ • B = 0 (4.61d)
(
∂  y z ∂  t t t t
µ
ε
Additionally, we have assumed that and vary so slightly with space
that the term involving its gradient can be dropped. We see that the mag-
netic induction is perpendicular to the electric field, and the magnetic
field is perpendicular to the electric displacement. Furthermore, we see
that the transverse field distributions E and H are rotation free and
t t
Semiconductors for Micro and Nanosystem Technology 165