As we showed in the previous section (Topic
24. Thin Lenses, part 3. Images using positive lenses), a point source of light
at the focal point of a positive lens produces an image which is a parallel
beam of light. This configuration is used in spotlights, lighthouses, and
similar applications. In order to minimize the size of the device, the focal
length of the lens must be as short as possible, since this sets the distance
between the lens and the lamp. This short focal length is usually implemented
in a very large lens so as to capture as much of the output of the lamp as
possible. The resulting lens is large and very heavy.
Since the imaging power of a lens is
produced by refraction at its edges, the center of the lens makes almost no
contribution. A Fresnel lens is designed using this idea. The interior glass of
the lens is removed, and the curved surface of the lens is approximated by a
series of curved segments. The effect is to preserve most of the focusing power
of the lens while at the same time saving a considerable amount on its size and
weight.
The design of a Fresnel lens usually
starts from a plano-convex lens as shown below in the left panel. The lens is
then cut into horizontal sections and each section is then aligned so that the
curved edges on the right hand side of the lens are aligned. Then the extra
material on the back side of the lens is removed.
The
resulting lens has the same refractive performance as the original, since the
shapes of both the front and the rear surfaces have not changed. However, it
has much less material and is therefore much lighter.
Although
it looks like the modified Fresnel should have the same performance as the
original, this is not quite true. A ray through the center of the Fresnel lens,
for example, reaches a point on the image side much faster than a ray through
the center of the original lens, because a larger fraction of its path is in
air. Thus a packet of rays which travel through the Fresnel lens are separated
in time compared to the same packet of rays passing through the original lens.
This same effect is true for rays that travel through other parts of the lens.
This results in subtle changes to the image that are usually not important for
spotlights and lighthouses, but can be significant whenever the phase of the
light is important.
To
build a spotlight or a lighthouse, a Fresnel lens is usually combined with a
concave mirror. The concave mirror is
placed behind the lamp with the lamp at its center of curvature. Every ray that
leaves the lamp traveling backwards strikes the mirror perpendicular to its
surface in this case and is therefore reflected back on itself. These reflected
rays therefore travel in the same directions as the rays that emerge from the
front of the lamp. The lamp is placed at
the focal point of the Fresnel lens, so that all of these rays emerge from the
lens in a parallel beam. This configuration uses most of the light that is
emitted by the lamp, although the rays that do not strike either the mirror or
the lens are wasted.
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