Introduction to Prisms
Functional
Properties
There
are four functional properties of a prism: image transposition, deviation,
displacement, and dispersion.
- Image transposition
is the inversion of an image’s orientation in one meridian or the
reversion of an image’s orientation in two meridians.
- Deviation is the
change in the direction of propagating light.
- Displacement is the
shift in the position of an optical centerline without changing its
direction of propagation.
- Dispersion is the
deviation of different wavelengths or polarizations of light into
different angles of propagation.
Any
one prism is usually designed to perform just one or two of the four possible
functions.
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The
common denominator shared by all prisms is polyhedral form. In other words, all
prisms tend to be solid chunks of glass or crystal with polygonal faces. Sometimes they also have one or more curved
faces.
Refraction and Reflection
Two
principles of operation explain the performance of a prism: refraction and
reflection. Refraction is the deviation of light as it passes across the
boundary between media of different refractive indices, such as glass and air.
For example, a lens deviates and focuses transmitted light with its refractive
power. Reflection is the deviation of light when it rebounds from a reflective
surface while a curved mirror focuses light with its reflective power.
Because
the surfaces of most prisms are flat, they do not form images as do lenses or
curved mirrors; nevertheless, engineers sometimes use the word image to
identify the output of a prism. A
dispersive prism uses refraction to spread white light into a rainbow. Each
color is refracted into its own unique direction. The size of the wedge angle
controls the dispersive power of the prism. The glass from which the prism is
made also controls its dispersive power.
Special
dispersive prisms have been designed to separate light of orthogonal
polarizations. These prisms employ crystalline materials whose index of
refraction is a function of orientation with the crystal lattice.
Total
Internal Reflection
Reflection
is usually understood to be a property of metallic surfaces, but bare glass
also can reflect light. Light can be
trapped inside a solid glass body if it approaches the glass-air boundary at angles
greater than the critical angle. This is called total internal reflection. It is total because 100% of the light
experiences reflection; none of it escapes. It is internal because the light must
already be inside the glass itself. The reflective efficiency of total internal
reflection exceeds that of any coating.
Prisms
that transpose images tend to use total internal reflection. Production is
simplified because additional coating procedures are eliminated. Total internal
reflection has limitations. The critical angle is the angle of incidence that
must be exceeded in order to realize total internal reflection. This angle is a
function of the refractive index of the glass and the medium in which it is
immersed (usually air). This limits the field of view, or the range of angles,
for which the light will experience total internal reflection. Prisms of low-index
glass have smaller fields of view than prisms of high-index glass. Low-index
prisms are the most commonly used because their fields of view are large enough
for most applications, while the glass is robust and available.
Aberrations
Prisms
can introduce aberrations into an optical system. Aberrations arise when the trajectories of
some rays in a bundle of light do not follow those predicted by paraxial imaging
theory. In cases where a bundle of light in a prism has a significant angle of
convergence or divergence, a designer must compensate for prismatic aberrations
by adjusting the design of other components in the system.
Collimated
beams of light rarely pick up aberrations in a prism. An important exception to
this rule of thumb can be found in a wedge. A wedged prism will always
introduce aberrations because optical path lengths through the wedge are not
equal. Sometimes, however, the aberration introduced by a wedge is desirable.
For instance, a collimated monochromatic beam can be reshaped with a wedge’s
astigmatic power.
Summary
Prisms
offer unique properties for use by the optical systems engineer. With their transposing, deviating and
displacing powers, prisms are used to orient images with precise control. With
their dispersive power, prisms separate the wavelengths or polarizations in a
beam of light. Total internal reflection
provides a convenient and efficient way to fold systems without requiring
separate mirrors or sophisticated coatings.
Aberrations introduced by prisms sometimes require special attention,
but by using collimated beams in prisms, designers can avoid problems.
Right-Angle Prisms
The
right-angle prism is one of the simplest and most versatile prisms. Its name
derives from the size of its apex angle: 90°. Because the legs of a standard
right-angle prism are of equal length, its cross-sectional view displays the
form of an isosceles right triangle (Figure G-1).
Figure G-1
Engineers
commonly make use of a right-angle prism’s total internal reflection. Total
internal reflection is the terminology used to describe the total reflection of
light from a surface of bare glass. Light will experience this kind of reflection
only if it approaches the surface from inside the glass at an angle that is
greater than a special angle of incidence called the critical angle. When light has experienced total internal
reflection at a surface of glass, it is completely trapped inside the glass at
that point. The efficiency of total internal reflection is so high that no
coating can equal its reflectivity.
Designers
use the right-angle prism in one of two orientations. The first orientation is called the single
mirror or leg-hypotenuse-leg orientation. The second orientation is called the
double mirror or hypotenuse-leg-leg-hypotenuse orientation. For both
orientations, incoming light must travel parallel to the plane that includes
the right-angle vertex.
In the
single mirror, or leg-hypotenuse-leg orientation, the prism acts like a single
mirror. Light enters the prism through one of its legs, reflects off its
hypotenuse by total internal reflection, and then exits through its second leg.
The centerline of the incoming light must be perpendicular to the entrance face
(Figure G-2).
As in
the case of a flat mirror angled at 45° to the incoming light, the prism in
this orientation inverts the image while deflecting its direction of
propagation by 90°.
Figure G-2
In the
double mirror or hypotenuse-leg-leg-hypotenuse orientation, the prism acts like
two mirrors. Light enters the prism through its hypotenuse, reflects at its
first and second legs by total internal reflection, and then exits back through
its hypotenuse (Figure G-3).
Figure G-3
When
used in the double mirror orientation, the right angle prism can be called a retroreflector
because an incoming beam of light is reflected back upon itself.
In
retroreflective mode, as long as the incoming light remains parallel to the
plane that contains the vertex angle, the alignment of the prism within that
plane is not critical; exact retroreflection will still occur (Figure G-4).The
dimension that controls the accuracy of the retroreflection is the right angle
at the vertex of the prism. The outgoing beam will be inclined to the incoming
beam by an amount equal to twice the deviation
of the
vertex angle from 90°. For example, if a right angle prism were manufactured
with a tolerance of ±1 minute of arc, then the incoming and outgoing beams
could cross each other with an inclination of no more than ±2 minutes of arc.
Figure G-4
The
retroreflective capability of a right-angle prism is limited to action in the
plane that includes its right-angle vertex. If retroreflective action is
required for randomly oriented light, then the designer must use a corner cube retroreflector.
Dove Prisms
As the
dove prism is rotated about its own long axis, the orientation of its image
rotates at twice the angular displacement.
Thus, an image can be rotated through 180° by rotating the dove prism
through only 90° (Figure G-5).
Figure G-5
Relating
the orientation of a dove prism’s image to its object can be confusing. Its
image will always be an inversion of its object. Rotation cannot substitute for
a second, orthogonal inversion nor can it reverse or “undo” an inversion. The dove prism’s shape is unique because of
its oblong profile (Figure G-6).
Figure G-6
Nevertheless,
the dove prism is simply one section of a right-angle prism (Figure G-7).
Figure G-7
A
collimated beam of light directed into the prism through one inclined face will
be refracted toward the base where total internal reflection inverts the beam
and directs it out through the second inclined face (Figure G-8).
Figure G-8
Because
the two inclined faces are symmetrically angled with respect to the base, the
output beam travels the same trajectory as the input beam; there is no
deviation or displacement of the beam. Engineers
use dove prisms to invert an image or to provide continuous control of the
orientation of an inverted image. Limitations are related to its size (it must
be rather long compared to its aperture) and to its aberrational effects upon
beams that are converging or diverging. Collimated
beams are preferred because they do not experience aberration as they pass
through the prism.
BK7 Roof Prisms
Engineers
gave the roof prism its name because of a roof-like structure that allows the
prism to invert an image in two meridians: left-right and up-down. It is also
known as an Amici prism.
A roof
prism can appear to have a rather complex geometry, but it is simply a modified
right-angle prism in which the “roof ” replaces the hypotenuse (Figure G-9).
Figure G-9
In its
typical orientation the roof prism receives an input beam through one leg and
reflects the output beam through the other leg (Figure G-10).
Figure G-10
The
beam experiences only two reflections in the roof, but each reflection is a
compound reflection because each section of the roof is tilted in both the
original plane of incidence and its orthogonal plane. The double compound
reflection reverts the image or inverts the image in two
orthogonal meridians: up-down and left-right. The final image is said to be a reverted
copy
of the input.
A
reverted image is called an erect image when it has been transposed to the same
orientation as when viewed with the unaided eye. The terminology of an
“erected” image comes from experience with a simple lens. When an observer views a distant scene
through a lens with positive power, such as a bi-convex lens, the image will be
upside down and inverted left to right (Figure G-11).
Figure G-11
Some
additional optical elements must be used to erect the image to invert it in
both the vertical and horizontal meridians. A second positive lens can be used
to form a Keplerian telescope; in more complex systems, an erecting prism, such
as a roof prism, can be used (Figure G-12).
Figure G-12
A roof
prism will introduce aberrations into an image if the beams of light that form
the image converge or diverge as they pass through the prism. Image-forming elements
that complete the system must be designed to compensate for these aberrations.
If the beams at the prism are collimated, then no special compensation is required.
Penta Prisms
The
penta prism deviates a beam through 90° in a way that preserves the orientation
of the input (Figure G-13). Only two of
the inclined sides are used to reflect light; manufacturers create the fifth
side, or face, by removing a corner to reduce the weight and size of the prism.
Figure G-13
Most
people familiar with penta prisms have gained their knowledge from single lens
reflex (SLR) cameras. Over the years this design has come to dominate the
market for the highest-quality 35mm cameras. The prism is commonly housed in an
extension of the camera’s body directly above the lens.
A
penta prism in an SLR serves to fold the path of the viewfinding system. A
photographer views the camera’s focusing screen by looking through the eyepiece
of the viewfinder and the penta prism (Figure G-14).
Figure G-14
In
some cameras, extra facets are polished into the sides of the penta prism.
Designers use the extra facets to bring images of the exposure control panels
into the viewfinder to reduce the weight of the prism or to transpose the image
created by the camera’s objective lens.
A
penta prism has five sides when viewed in cross-section. It is a modification
of the right-angle prism (Figure G-15).
Figure G-15
Engineers
specify a penta prism rather than a double mirror because of several advantages
found in the prism’s construction. One of those advantages is stability. Glass has
a low coefficient of expansion and alignment of the reflecting surfaces of the
prism will remain constant over a long time even when exposed to environmental changes
and shocks.
A
second advantage is in simplicity and ease of assembly. A penta prism can be placed into a single
mount whereas two mirrors would require two separate mountings and a more
complex procedure for assembly.
Still
another advantage to the prism lies in its size. A penta prism can be easily
manufactured with small dimensions and then simply placed into an assembly. A miniature mirror system might require
special design for mountings and special tools for assembly.
The
geometry of reflection inherent in a penta prism deserves special note. The 90°
deviation, which is imparted to the incoming beam, remains constant for different
angles of incidence. This invariance means that alignment of a penta prism is
not critical in terms of the deviation of a centerline through 90°.
Although
this invariance to alignment is shared by other prisms oriented for two
coplanar reflections, the penta prism is unique in its ability to deviate the
centerline by precisely 90°. The manufacturing tolerance of the 45° angle between
the two reflecting faces determines the accuracy of the centerline’s 90°
deviation.
An
application of the above principle can be found in laser scanning systems where
a rotating penta prism is substituted for a rotating polygon mirror or
holographic disc. The penta prism takes an input laser beam and images it as a
line scan in the focal plane of a scan lens.
Rhombic Prisms
Rhombic,
rhombus, and rhomboidal are synonymous when referring to a prism that works
like a simple periscope. The root of the terminology is in the Greek word
rhomboid, meaning “non-equilateral parallelogram,” and that is exactly the
shape of a rhombic prism when viewed in cross-section (Figure G-17).
Figure G-17
Designers
will specify a rhombic prism when they need to displace an optical centerline
without changing its direction.
This
prism also features an image that remains in the same orientation as the object
(Figure G-18).
Figure G-18
In the
most common form of a rhombic prism, the reflecting facets are cut at 45° to
the entrance and exit faces. Total internal reflection can be used with this geometry.
Note that only a small portion of the entrance and exit faces are actually used
for input and output beams. Because it controls the size of the reflecting
facets, the thickness of the slab determines the aperture of the prism.
Rhombic
prisms can be found in some microscopes.
They allow variation in the spacing between eyepieces for binocular
viewing. This spacing is called the “interocular distance” because observers
adjust its size to fit the distance between their eyes. Zeiss was one of the first
to use this design feature. A rhombic prism feeds the image to each eyepiece,
and each prism is mounted on a common pivot. To adjust the interocular distance
an observer swivels the eyepieces around the pivot. Refocusing is not required because the
optical path lengths have not been changed.
Figure G-19
Figure
G-19 depicts a system in which the same image is presented to each eye. There
is no stereoscopic effect. For
stereoscopic instruments, the single incoming image is two separate images;
each image enters each rhombic prism separately, and there are two separate
pivots, one for each prism, rather than the common pivot pictured above.
Wedges
The
shape of a wedge in cross-section resembles a wooden door stop. Usually a wedge
will contain a right angle, but a “right-angle wedge” is a special case (Figure
G-20).
Figure G-20
A list
of common names for a wedge includes: wedged mirror, wedged window, thin prism
and thick prism. Each name corresponds
to a different function for which this element is used.
Day/Night
Rearview Mirrors
The
most common application for a wedge is found in automobiles;
the day/night rearview mirror is a wedged mirror.
Its principle of operation is the separation of reflections
from its first and second surfaces and it can be
rotated with the flick of a switch to reduce the glare of
headlights coming from behind.
A
wedge can separate reflections from its two polished surfaces because they are
not parallel to each other. The wedge angle defines the inclination of the
first surface to the second. The angular separation between the two reflections
is twice the wedge angle. In the day/night
rearview mirror the first surface is uncoated bare glass for a dim reflection
and the second surface is coated with aluminum for a bright reflection of about
92% (Figure G-21).
Figure G-21
Wedged
Windows
Used
as a wedged window, a wedge can control the direction of back-reflected light
at each surface. The transmitted beam is deflected (Figure G-22).
Figure G-22
Applications
for wedged windows, as opposed to standard plane-parallel windows, can be found
in lasers and interferometers. These wedges are usually shallow with wedge
angles measured in minutes of arc or fractions of an angular degree.
Thin or
Thick Prisms
As a
thin or thick prism, a wedge can be used to disperse light into its constituent
colors. Although the terms thin and thick can be defined mathematically they are
used loosely. A thin prism has a small wedge angle and a thick prism has a
large wedge angle. Color analyzers and spectrographic instruments contain
prisms or diffraction gratings to bend light of different colors into different
angles for analysis.
Prismatic
wedges can also be employed in anamorphic imaging systems. In these
applications designers do not use the prism for its dispersive power, but for
its ability to change the diameter of a collimated beam with refractive power. The wedge is used to expand or contract the size
of the beam in just one meridian (Figure G-23).
Figure G-23
Cylindrical
lenses also can be used to change the size and shape of a beam. However, they
must be used in pairs and the designer must be wary of lens-induced aberrations.
The
wedge is a simple element with great versatility. It can be adapted to many
applications by choice of its wedge angle and coatings for its inclined,
polished surfaces. Sometimes it is used
for its reflective capability, sometimes for its refractive power, and
sometimes for its dispersive power.
Corner Cubes
The
marvelous retroreflective property of the corner cube has been put to use in
common products such as safety reflectors and in unique products such as the
laser ranging targets placed on the moon by the Apollo astronauts.
Principles
of Operation
A
corner cube is, geometrically speaking, cut from the corner of a cube of glass.
It has three mutually orthogonal reflecting faces and one entrance/exit face
(Figure G-24).
Figure G-24
A ray
of light entering the corner cube will experience three total internal
reflections. After the third reflection, the ray exits in exactly the opposite
direction of the original incoming ray (Figure G-25).
Figure G-25
This
retroreflective behavior is independent of the orientation between the corner
cube and the incident rays of light. It depends instead only on the accuracy of
the squareness of the corner.
Safety
Reflectors
Corner
cubes are responsible for the brilliant appearance of safety reflectors when
they are illuminated by the headlights of a car. A safety reflector is usually
a medallion of plastic whose inside surface has been molded into many small
corner cubes. When a safety reflector is illuminated with light, its corner
cubes reflect the rays of light straight back to the driver’s eye. This
retroreflection will occur for any direction in which the car approaches.
Corner
cube construction is found in the reflectors mounted on bicycles and automotive
tail lights and on highways where the shoulders, medians and signs are marked
with bright discs. Corner cubes also are stamped into sheets of painted
material to form brilliantly reflective panels used for the background of
highway signs and barriers that identify work zones along a road.
Apollo Lunar
Ranging Targets
Apollo
astronauts carried arrays of precision corner cubes to the lunar surface.
Powerful lasers are directed at the moon through telescopes on earth. The
corner cubes reflect the laser beams back to their origins. Astronomers can
make extraordinarily accurate measurements of the
lunar
distance by timing the round trip of pulses of laser light between the earth
and the moon.
The
arrays of corner cubes make measurement of the lunar distance practical because
their alignment with a telescope on earth is not critical. Not only did their retroreflective
property simplify construction and deployment of these arrays, but it also now
simplifies use of the telescopic lasers. Astronomers receive a good signal from
the lunar surface from any telescope on earth despite the constant relative
motion of earth and moon.
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