Biomedical Engineering Reference
In-Depth Information
correction can be achieved with low-dispersion
glass (typically containing fluorite), but these
lenses are quite expensive.
In multiple-lens optical systems, aberrations of
all types can be mitigated using special combina-
tions of convex and concave lenses and specific
types of glass; this is usually described as
corrected
optics.
High-quality corrected optics, although
expensive, can achieve images very close to the
theoretical ideal. Spherical aberration, by itself,
can be minimized even in a single-lens system by
using a relatively expensive aspheric lens.
θ
θ
θ
FIGURE 1.6
Reflection (ray b) and refraction (ray c) of
incident light (ray a) encountering a boundary.
1.2.1.6 Reflection and Refraction
When light enters an optical system, it encoun-
ters a boundary between two different indices
of refraction
n
. For example, when light propa-
gates through air
(
n
1
= 1)
and enters a lens made
of crown glass
(
n
2
=
1. 5
)
, it is both refracted
and reflected5
5
at this boundary (
Figure 1.6
). For
a multiple-lens system using different types
of glass, there may be many such boundaries.
The familiar Snell's law
(sin θ
1
/ sin θ
2
=
n
2
/
n
1
)
predicts the angle of refraction; the angle of
reflection is equal to the angle of incidence
.
6
The reflectance
R
is the fraction of incident light
intensity (i.e., power) that is reflected, and the
transmittance
T
is the fraction of incident light
intensity that is refracted. Obviously,
R
+
T
= 1
.
At this point, we need to take into account the
polarization state of the incident light to deter-
mine how much of the incident light will be
reflected.
Assume the page on which the plot of
Figure
1.6
appears is the plane of incidence for the
incoming light. If the incident light is polarized
such that the electric field is perpendicular to the
plane of incidence, then
n
1
cos
θ
1
−
n
2
cos
θ
2
2
R
s
=
,
T
s
=
1
−
R
s
.
n
1
cos
θ
1
+
n
2
cos
θ
2
(1.4)
If the incident light is polarized such that the
electric field is parallel to the plane of incidence,
then
n
1
cos
θ
2
−
n
2
cos
θ
1
2
R
p
=
,
T
p
=
1
−
R
p
.
n
1
cos
θ
2
+
n
2
cos
θ
1
(1.5)
If the light is unpolarized (i.e., randomly polarized),
then a common estimate is
R
=
(
R
s
+
R
p
)/
2
.
One ramification of reflection for a vision sen-
sor designer is that the fraction of light intensity
that is reflected at the boundary never makes it
to the photodetectors. For example, for incident
5
We consider here specular reflection, where any irregularities in the boundary surface are small compared to
the wavelength (i.e., an optically smooth surface). If this is not true, diffuse reflection (i.e., scattering) occurs. We
also assume
n
2
>
n
1
. If the converse is true, then there exists a critical angle for which total internal reflection will
occur
[7]
.
6
Snell's law
is named after Willebrord Snellius (born Willebrord Snel van Royen; 1580-1626). Note that the spelling
of Snell's name has been Anglicized; the more correct Dutch spelling is Snel (or Snellius), but Snell is overwhelm-
ingly found in the literature
[22]
. It is appropriate here to mention that Snell's law and many other significant
discoveries in optics were made by Middle Eastern scientists such as Ibn Sahl (c. 940-1000) and Ibn al-Haytham
(c. 965-1039) many hundreds of years before Snel, Descartes, or Newton were born
[23]
.
