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
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