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theatrical spotlights, in camera viewfi nder focusers, and
also as novelty wide-angle lenses for automobile rear
windows. Think of taking a set of nested concentric
circular cookie-cutters and slicing a lens into a series
of rings. Then shave off the bulk of the glass in the rear,
leaving only the front curvature. Cement the resulting
rings to a thin fl at sheet, and the result is like a Fresnel
lens (which is actually molded). This is a really thin
lens!
Using the Thin Lens Approximation and simple
geometrical rules, it is possible to quickly draw the
principle light paths in a lens system. For each lens, a
light ray down the system axis (a perpendicular through
the lens center) travels straight on. A ray parallel to
the axis, emerging from an off-axis point of the object
as if it came from infi nity, is bent by the lens to pass
through the rear focal point, and a ray passing from the
same object point through the front focal point (at an
angle) emerges from the lens parallel to the axis (note
the symmetry!). Where the latter two rays cross is the
point where the original point of the object is imaged.
In cases where the image rays converge to a focus, the
image is termed real.
If, in leaving the lens, the rays only diverge, then
an image can only be seen by the use of an additional
lens, say that in your eye, and the image is called vir-
tual. Convex lenses and concave mirrors can give real
or virtual images, depending on whether the object is
closer or farther than one focal length away from the
optic. Concave lenses and convex mirrors yield only
virtual images.
Snell’s Law, which describes the refraction of light,
along with the law of refl ection (the angle of incidence
equals the angle of refl ection), can be applied at each
point of a boundary surface to predict the path of light
rays as they pass through. By doing this step by step at
all points (or a sample) of a surface of known shape,
one can follow a ray of light through a system of any
complexity, and in reasonable detail. This process is
called ray tracing, and until the advent of computers
was carried out by hand. There some complications to
this process, however.
First, the index of refraction of any real transparent
material varies with the color (wavelength or frequency)
of light. This effect is called dispersion, and explains
why prisms are able to spread white light out into the
visible spectrum. In general the index is greatest in
the blue and diminishes continuously into the red and
infrared. Mirrors do not suffer from this problem. This
effect causes any lens to send blue rays to a different
focus point than green or red ones. The result is called
chromatic aberration, and results in color fringes sur-
rounding images of objects with sharp edges, such as
the Moon. It was discovered in the middle 1700s that
this problem could be removed by sandwiching together
two lenses of different indices of refraction, one convex
and the other concave, carefully choosing their focal
lengths, to make the dispersion of the second approxi-
mately cancel that of the fi rst. These pairs are called
achromatic lenses. All modern lenses for cameras and
telescopes are achromats.
Second, every lens or mirror has a natural limit to
resolution caused by the wave nature of light, called
the diffraction limit. Waves bend around any edge,
straight, curved or jagged. This is termed diffraction.
Light does this and this results in fuzzy shadow edges
(visible under careful examination) and also fuzziness
in the whole image. The larger the opening of the lens
or mirror, the less important this is and the sharper the
image. The longer the wavelength of light (the redder
it is) the worse the effect. This diffraction limit can not
be evaded in conventional optics.
The diffraction effect can actually be put to use,
however. A simple round pinhole carefully made in an
opaque sheet will give a real image on almost any size
surface behind, at almost any distance. What you gain
in areal coverage and depth of focus you sacrifi ce in
speed and exposure time. In conventional optics (lenses
or mirrors)the aperture is wide and concentrates a lot
of light. In pinhole cameras, just a tiny amount of light
passes through. A conventional “fast” lens might have
a focal length to diameter ratio (“f/stop” or “speed”) of
f/1.2 or f/1.8. The speed of a pinhole is usually f/150
or more.
Aside: A more effi cient way than a pinhole to use dif-
fraction to manipulate light and form an image is a sec-
ond invention of Fresnel: the zone plate. Fresnel found
an exact formula to compute the widths of transparent
gaps between and widths of concentric opaque rings to
form a lens based on diffraction. A zone plate looks just
like a bullseye, but it is a mathematical construct.
There are also distortions of image shape due to lens
or mirror shape. If the surfaces are spherical then off-
axis rays do not focus at the same distance as on-axis
ones. Spherical aberration, along with comatic aber-
ration (images at the edge of the fi eld of view spread
out into fan “tails” like comets), barrel distortion and
pincushion distortion (rectangular objects have “swol-
len” or “collapsed” images, respectively) along with
chromatic aberration all have to be reduced to make
an optical system produce high quality images. With
the use of ray tracing in modern computers all of these
problems can be solved.
Compound lenses have been designed for many dif-
ferent purposes. Perhaps the two greatest challenges are
to fi nd excellent wide angle lenses, and to fi nd zoom
lenses which maintain focus and image quality, along
with maximum speed at every magnifi cation. Again
computers have allowed many different solutions to
these problems.