Image Processing Reference
In-Depth Information
Appendix D: The Poynting Vector
The Poynting vector relates the movement of electromagnetic power to the
temporal variation of stored energy, that is, the magnitude and direction of
power flow It is defined to be
SEH
=× and at high frequencies, the time
averaged Poynting vector can be shown to be
S
12Re where Re
denotes the real part. This simple form assumes that
E
∝
H
, that is, that the
fields are simply proportional to each other and hence the me
d
ium is isotro-
pic. If the medium is such th
a
t its refractive index
n
< 0, then
S
is necessarily
in the direction opposite to
k
. We no
r
mally assume a plane
w
ave s
o
lution to
the wave equation and assume that
H
is perpendicular to
E
an
d
k
.
In
free
space,
HE
=
(/)[
E
×
H
*
]
= =/ Ω By convention,
E
,
H
, and
k
form
a right-handed set (Figure D.1). In some medium, the phase velocity of the
wave is given by (
, and
= (
/η
0
)
ηµε
0
(
)
120
.
0
0
r r
/ .
Let us consider the boundary conditions (Figure D.2). The quantities shown
in this figure are continuous; here
t
denotes the tangential component and
n
the normal component. If ε
eff
is for a metal, then
E
2
= , but we may have sur-
face waves and surface currents, and surface plasmons to contend with. These
boundary conditions are
c
/
µε
)
=
(
ω
/
k
)
and
HE
= (
/η , where ηηµε
)
=
(
)
rr
0
EE
t
−=
0
1
2
t
BB
n
−
=
0
1
2
n
DD
n
−
= ρ
(
surface charge)
1
2
n
s
×
ˆ
HH J
−
=
n
(surface current)
1
t
2
t
s
Another important point concerns the energy density in a medium which
is given by
1
2
1
2
W
=
εω
( ||
E
2
+
µω
( ||
H
2
0
0
Otherwise, assuming little dispersion near ω
0
we get
1
2
d
(())
ωε ω
ω
1
2
d
(())
ωµ ω
ω
W
=
||
E
2
+
||
H
2
d
d
ωω
=
ωω
=
0
0
211
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