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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