Biomedical Engineering Reference
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Solution: (a) The focal length of the lens was
given as 22.5 mm, yielding F = f / D 2. 8 . (b) The
FOV is determined from the sensor array dimen-
sions, not the lens aperture. The sensor size was
given as 19.2 mm horizontally by 10.8 mm verti-
cally. The horizontal angular FOV is found by cal-
culating 2 arctan(19. 2mm/2(22. 5 mm)) = 0. 807
rad = 46. 2 . The vertical angular FOV is found
by calculating 2 arctan(10. 8 mm/2(22. 5 mm)) =
0. 471 rad = 27. 0 . (c) The maximum distance for
s o was found previously to be 300 m. From Figure
1.2 , it is obvious that the horizontal FOV is what
determines how much of the fence will be imaged.
At a range of s o
the lens and the geometry of the lens that would
be required to take into account the exact nature
of the propagation of light. This intentional mis-
match is due to the much higher cost of produc-
ing a geometrically perfect lens. For example, a
common form of monochromatic aberration is
spherical aberration, which occurs when the lens
is manufactured with a radius of curvature that
matches a sphere; it is much cheaper to fabricate
this type of lens than what is called an aspheric
lens, which more closely matches the physics
related to the propagation of light. Spherical
aberration causes incorrect focus, but the effect
is negligible near the center of the lens. A typical
spherical lens made from crown glass exhibits
spherical aberration such that only 43% of the
center lens area (i.e., 67% of the lens diameter)
can be used if objectional misfocus due to spheri-
cal aberration is to be avoided.
Chromatic aberration is primarily due to the
unavoidable fact that the refractive index of any
material, including lens glass, is wavelength
dependent. Therefore, a single lens will exhibit
slightly different focal lengths for different
wavelengths of light. The fact that monochro-
matic aberrations are also wavelength depend-
ent means that even differences in monochromatic
aberrations due to differences in wavelength can
be considered contributors to chromatic aberra-
tion. Chromatic aberration appears in a color
image as fringes of inappropriate color along
edges that separate bright and dark regions of
the image. Even monochrome images can suffer
from degradation due to chromatic aberration,
since a typical monochrome image using inco-
herent light is formed from light intensity that
spans many wavelengths. A compound lens
made of two materials (e.g., crown glass and
flint glass), called an achromatic lens , can correct
for a considerable amount of chromatic aberra-
tion over a certain range of wavelengths. Better
= 300 m and an angular FOV of
0.807 rad, the horizontal distance of the fence that
will be imaged is (2)(300) tan(0. 807/2) 256m .
This same answer could also be found using
Eq. (1.2) .
1.2.1.5 Aberrations
No optical system is perfect, and the imperfec-
tions result in what are called aberrations [7] .
Most aberrations are due to imperfections in
lenses and are usually categorized as either
monochromatic or chromatic. Up to this point
in Section 1.2 , the discussion has centered on
aspects of an optical system that must be con-
sidered one wavelength at a time. 4 This is how
monochromatic aberrations must be treated.
A chromatic aberration, on the other hand, is
a function of multiple wavelengths. See addi-
tional references such as Smith [8] for more
information on aberrations.
Common forms of monochromatic aberra-
tions include spherical, coma, astigmatism, field
curvature, defocus, barrel distortion, and pin-
cushion distortion. Monochromatic aberrations
are primarily due to either an unintended imper-
fection (which is usually caught and eliminated
in the lens manufacturing stage) or an inten-
tional mismatch between the actual geometry of
4 Depending on the needs of the application, a designer can make individual calculations at many wavelengths,
two calculations at the longest and shortest wavelengths, or one calculation at the approximate midpoint of the
range of wavelengths.
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