Biomedical Engineering Reference
In-Depth Information
f
3
x
10
19
3x10
17
3x10
14
3x10
11
3x10
8
3x10
6
V
I
S
Very low
frequency (VLF)
radio waves
Radio
"medium"
waves
Radio
"long"
waves
U
V
Micro-
waves
γ
rays
X-rays
IR
λ
1
μ
m
1 pm
1 mm
1 mm
1 m
1 km
FIGURE 17.2
The electromagnetic spectrum.
For simplicity, consider the plane monochromatic (single wavelength) waves propagat-
ing in free space in the O
z
direction; then the general solution to the wave equation for
the electric field can be written as
E
x
¼
e
x
cos (
o
t
kz
þ d
Þ
ð
17
:
12
Þ
x
E
y
¼
e
y
cos (
o
t
kz
þ d
Þ
ð
17
:
13
Þ
y
in which
d
y
are arbitrary phase angles. This solution can be described completely by
means of two waves (e-field in the X-Z plane and e-field in the Y-Z plane). If these waves
are observed at a particular value of “z”—say, “z
o
”—they take the oscillatory form
E
x
d
x
and
þ d
0
x
Þ
d
0
x
¼ d
¼
e
x
cos (
o
t
kz
o
ð
17
:
14
Þ
x
þ d
0
y
Þ
d
0
y
¼ d
y
E
y
¼
e
y
cos (
o
t
kz
o
ð
17
:
15
Þ
and the top of each vector appears to oscillate sinusoidally with time along a line. E
x
is said
to be linearly polarized in the direction O
x
, and E
y
is said to be linearly polarized in the
direction O
y
.
It can be seen from Eq. (17.15) that if the light is fully polarized, it can be completely
characterized as a 2 by 1 matrix in terms of its amplitude and phase (e
x
exp(j
d' ) and e
y
exp(j
d')). This vectoral representation is known as the Jones vector. When the optical system
design includes the propagation of the light through nonscattering, and thus nondepolariz-
ing, medium such as a lens or clear biological sample, the medium can be completely char-
acterized by a 2 by 2 Jones matrix. Therefore, the output of the propagation of the polarized
light can be modeled as a multiplication of the Jones matrix of the optical system and the
input light vector. A system that yields both polarized and partially polarized light, such
as that obtained from tissue scattering, can be characterized using 4 by 4 matrices known
as Mueller matrices and a 4 by 1 matrix known as the Stokes vector.
Returning back to Eq. (17.15), the tip of the polarized light vector is the vector sum of E
x
and E
y
which, in general, is an ellipse as depicted in Figure 17.3 whose Cartesian equation
in the X-Y plane at z
¼
z
o
is
E
x
2
c
x
2
E
y
2
c
y
2
sin
2
=
þ
=
þ
2(E
x
E
y
=
e
x
e
y
Þ
cos
d ¼
d
ð
17
:
16
Þ