ECE 222B, Winter 2010
Solution to Homework Set 2
1. Assume that an anisotropic crystal has a permittivity tensor
~
~
ε
=
ε
xx
ˆ
x
ˆ
x
+
ε
yy
ˆ
y
ˆ
y
+
ε
zz
ˆ
z
ˆ
z
where
ε
xx
,
ε
yy
, and
ε
zz
are all real constants
>
0.
Now assume that we introduce a propagating wave of frequency
ω
.
(a) If the magnetic field of the wave takes the form
~
H
=
H
0
exp(

j
~
β
·
~
r
)
,
where
~
β
=
β
1
(ˆ
z
cos(
θ
) + ˆ
x
sin(
θ
)), what is the wave dispersion (re
lation between
β
1
and
ω
?
Is the phase velocity of the wave con
stant? Calculate
~
D
and
~
E
, and express them in terms of frequency,
wavenumber and magnetic field. Are
~
D
and
~
E
parallel to each other?
Under what conditions?
Solution:
This problem (both parts a and b) contain a rather glaring
typo. The equation for
~
H
should have read
~
H
=
H
0
exp(

j
~
β
·
~
r
)ˆ
y,
For part b, the equation for
~
D
should have read
~
D
=
D
0
exp(

j
~
β
·
~
r
)ˆ
y,
As posed, if we take the equation for
~
H
to mean any allowable direc
tion, then it will lie in a plane perpendicular to the direction of
~
beta
.
For any chosen initial polarization, the wave will then break down
into two waves, one with a ydirected
H
field, and the other with a
y
directed
D
field. Here an inconsistency occurs, since both
~
beta
and
ω
have been fixed, but cannot be the same for both waves. In the
spirit of the originally intended problem, I will give the solution for
a ydirected
H
field, and then handle the ydirected
D
field in part
b.
If
~
H
=
H
0
exp(

j
~
β
·
~
r
)ˆ
y
, then the electric field will lie in the xz
plane (since
E
and
H
must be orthogonal).
This means that the
solution for the dispersion and mode behavior will be independent of
ε
yy
. As a consequence, the mode properties will be identical to those
discussed in the handout for the uniaxial case.
(b) If the electric flux density of the wave takes the form
~
D
=
d
0
exp(

j
~
β
·
~
r
)
,
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
where
~
β
=
β
2
(ˆ
z
cos(
θ
) + ˆ
x
sin(
θ
)), what is the wave dispersion (re
lation between
β
1
and
ω
?
Is the phase velocity of the wave con
stant? Calculate
~
H
and
~
E
, and express them in terms of frequency,
wavenumber and magnetic field. Are
~
D
and
~
E
parallel to each other?
This is the end of the preview.
Sign up
to
access the rest of the document.
 '09
 Frequency, red line, group velocity, Wave propagation, phase velocity, Ferrites

Click to edit the document details