Biomedical Engineering Reference
In-Depth Information
90
o
.Thus,for
and thus the angle for which
y
i
¼ y
c
is when (
n
i
/
n
t
)sin
y
c
¼
1 or, rather,
y
t
¼
all values of
y
i
> y
c
, the light is totally reflected at the boundary. This phenomenon is
what is used for the creation of optical fibers, in which the core of the fiber (where the
light is supposed to propagate) has an index of refraction slightly higher than the clad-
ding surrounding it, so the light launched into the fiber will totally internally reflect
allowing for minimally attenuated propagation down the fiber.
17.2.2 Light Interaction with Participating Media: The Role of Absorption
and Scattering
When light is incident on tissue either from a laser or from other light transmitting
devices, tissue acts as a participating medium by reflecting, absorbing, scattering, and
transmitting various portions of the incident wave of radiation. Ideally, an electromag-
netic analysis of the light distribution in the tissue would be performed. Unfortunately,
this can be quite cumbersome, and a reliable database of electrical properties of
biological tissues would be required. An alternate practical approach to the problem is
to use transport theory that starts with the construction of the differential equation for
propagation of the light intensity. In the following section, the equation for light intensity
distribution in a purely absorbing medium will be derived. With this motivation, the next
section introduces the general equation of transfer for a medium that scatters as well as
absorbs the light.
The Case of Pure Absorption
In order to describe the distribution and transport of laser energy in a nontransparent
participating medium, the medium can be viewed as having two coexisting “phases”: a
material phase for all the masses of the system and a photon phase for the electromag-
netic radiation. Figure 17.5 shows the material phase as circles and the photon phase
as curved arrows that strike the material phase. The energy balance equation for
the material phase is introduced and discussed in the thermodynamic descriptions in
Sections 17.3 and 17.5. The energy balance equation for the photon phase is discussed
following.
Consider an infinitesimal volume of the material under irradiation (Figure 17.6). The rate
of change of radiative energy,
)
, with time is the difference between the incoming and
outgoing radiative fluxes in the element minus the rate of energy absorption by the material
phase. The difference between the incoming and outgoing fluxes is, in the limit, the nega-
tive of the divergence of the radiative flux. Therefore, denoting the energy absorbed by
the material phase, which is the laser heat source term, by
(
rad
U
Q
L
and the radiative flux by
~
q
ð
rad
Þ
the governing differential equation for energy rate balance in the photon phase can
be written as
@
U
ð
rad
Þ
=@
t
¼r
q
ð
rad
Þ
Q
L
ð
17
:
25
Þ
Any possible photon emission or scattering by the material phase has been ignored.
