Biomedical Engineering Reference
In-Depth Information
B
(the magnetic flux density) and
D
(the electric displacement) are continuous across the
boundary. These conditions can only be true at all times and all places on the boundary if
the frequencies of all waves are the same. The relative amplitudes of the waves will now be
considered (note the values of
E
,
H
,
D
,and
B
will depend on the direction of vibration of
E
and
H
fields of the incident wave relative to the planes of incidence). In other words, they
depend on the polarization of the wave. The wave can be divided into two orthogonal
polarization states, and using Maxwell's equations, Snell's law, and the law of reflection as
well as imposing the boundary conditions, the following Fresnel equations can be derived.
The reflected and transmitted equations in the parallel configuration are
p
E
r
E
i
r
¼
¼ð
n
i
cos
y
þ
n
t
cos
y
Þ=ð
n
i
cos
y
þ
n
t
cos
y
Þ
ð
17
:
21
Þ
p
t
i
t
i
p
E
t
E
i
t
¼
¼
(2
n
i
cos
y
Þ=
(
n
i
cos
y
þ
n
t
cos
y
Þ
ð
17
:
22
Þ
p
i
t
i
The normal or perpendicular reflected and transmitted equations are
⊥
¼
E
r
E
i
r
⊥
¼
(
n
i
cos
y
n
t
cos
y
Þ=
(
n
i
cos
y
þ
n
t
cos
y
Þ
ð
17
:
23
Þ
i
t
i
t
⊥
¼
E
t
E
i
t
⊥
¼
(2
n
i
cos
y
i
Þ=
(
n
i
cos
y
i
þ
n
t
cos
y
t
Þ
ð
17
:
24
Þ
Using the preceding equations, there are two noteworthy limiting cases. The first case is as
follows. If
n
t
>
n
i
, then
y
i
> y
t
,
r
?
is always negative for all values of
y
i
, and
r
p
starts out
90
o
. This implies that the
refracted and reflected rays are normal to each other, and from Snell's law this occurs when
tan
positive at
y
i
¼
0 and decreases until it equals 0 when
y
i
þ y
t
¼
y
i
is known as the polarization angle or, more com-
monly, the Brewster angle. At this angle only the polarization with
E
normal to the plane
of incidence is reflected, and as a result, this is a useful way of polarizing a wave. Now,
as
y
i
¼
n
t
/
n
i
. This particular value of
r
p
becomes progressively negative, reaching -1.0 at 90
o
, which
implies that the surface performs as a perfect mirror at this angle. On the other hand, at
normal incidence
y
i
increases beyond
y
p
,
y
i
¼ y
r
¼ y
t
¼
0, then
t
p
¼
t
?
¼
2
n
i
/(
n
i
þ
n
t
) and
r
p
¼
-
r
?
¼
(
n
t
-
n
i
)/
(
n
i
þ
n
t
)
¼
(
n
i
-
n
t
)/(
n
i
þ
n
t
). The wave intensity, which is what can actually be measured
by a detector,
is defined proportional
to the square of
the electric field. Thus,
2
¼ð
n
t
n
i
Þ
2
¼
E
r
E
i
E
t
E
i
2
2
2
and
2
2
.
2
ð
I
=
I
i
Þ¼ð
r
Þ
¼
=ð
n
i
n
t
Þ
ð
I
t
=
I
i
Þ¼ð
t
Þ
¼
4
n
i
=ð
n
i
þ
n
t
Þ
R
p
p
The preceding expressions are useful because they represent the amount of light lost by
normal specular reflection when transmitting from one medium to another.
The second limiting case is, if
n
t
<
n
i
,then
y
t
> y
i
,
r
?
is always positive for values of
y
i
,
and
r
?
increases from its initial value at
y
i
¼
0untilitreaches
þ
1.0atwhatisknownasthe
90
o
,andfromSnell'slaw,sin
critical angle
y
c
.Atthisangle
y
i
¼ y
c
,then
y
t
¼
y
t
¼
n
i
/
n
t
sin
y
t
could be greater than
one, according to the preceding equation. This is not possible for any real value of
y
i
. Clearly for
n
i
>
n
t
(less dense to more dense medium), sin
y
t
,
