Chapter Twenty Nine Notes:
Reflection and Refraction
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The Big Idea:
When waves interact with matter, they can
be Reflected, Transmitted, or a combination of both. Waves that
are transmitted can be Refracted.
Reflection occurs when light bounces off objects. How much
reflection depends upon how even the surface is. If the surface is
rough, the light scatters. If the surface is smooth and flat, the light
will bounce off it at equal angles. That is why a flat mirror reflects a
good likeness of the object being reflected.
Refraction is the bending of a wave when it enters a medium where
it's speed is different. The refraction of light when it passes from a
fast medium to a slow medium bends the light ray toward the
normal to the boundary between the two media. The amount of
bending depends on the indices of refraction of the two media and
is described quantitatively by Snell's Law.
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Waves are a means by which energy travels. Many different particles
move in waves. The waves on an ocean are physical waves caused
mainly by wind. Light is an electromagnetic wave caused by excited
electrons. The movement of a wave is complicated, but both
electromagnetic and physical waves use similar ways to describe the
motion.
Sound waves can echo back from a cliff, and light waves are reflected
from the surface of a pond. We use the word reflection, normally
applied only to light waves in ordinary speech, to describe any such
case of a wave rebounding from a barrier. Figure (a) shows a circular
water wave being reflected from a straight wall. In this chapter, we
will concentrate mainly on reflection of waves that move in one
dimension, as in figure (b), on the next page.
Wave reflection does not surprise us. After all, a
material object such as a rubber ball would bounce
back in the same way. But waves are not objects, and
there are some surprises in store. (a) Circular water waves
are reflected from a
boundary on the left. PSSC
Physics.
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First, only part of the wave is usually
reflected. Looking out through a window, we
see light waves that passed through it, but a
person standing outside would also be able
to see her reflection in the glass. A light wave
that strikes the glass is partly reflected and
partly transmitted (passed) by the glass. The
energy of the original wave is split between
the two. This is different from the behavior of
the rubber ball, which must go one way or
the other, not both.
Second, consider what you see if you are
swimming underwater and you look up at the
surface. You see your own reflection. This is
utterly counterintuitive, since we would
expect the light waves to burst forth to
freedom in the wide-open air. A material
projectile shot up toward the surface would
never rebound from the water-air boundary!
(b) A wave on a coil
spring,
initially
traveling to the left, is
reflected from the
fixed
end.
PSSC
Physics.
What is it about the difference between two media that
causes waves to be partly reflected at the boundary
between them? Is it their density? Their chemical
composition? Ultimately all that matters is the speed of
the wave in the two media. A wave is partially reflected
and partially transmitted at the boundary between media
in which it has different speeds. For example, the speed
of light waves in window glass is about 30% less than in
air, which explains why windows always make
reflections. Figures (c) and (d) show examples of wave
pulses being reflected at the boundary between two coil
springs of different weights, in which the wave speed is
different.
Reflections such as (a) and (b), where a wave
encounters a massive fixed object, can usually be
understood on the same basis as cases like (c) and (d)
later in his section, where two media meet. Example (b),
for instance, is like a more extreme version of example
(c). If the heavy coil spring in (c) was made heavier and
heavier, it would end up acting like the fixed wall to
which the light spring in (b) has been attached.
(c) A wave in
the
lighter
spring, where
the
wave
speed
is
greater,
travels to the
left and is
then
partly
reflected and
partly
transmitted at
the boundary
(d) A wave
moving
to
the right in
the heavier
spring
is
partly
reflected at
the boundary
with
the
lighter
spring. The
reflection is
uninverted.
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Objects can be seen by the light they emit, or, more often, by the
light they reflect. Reflected light obeys the law of reflection, that the
angle of reflection equals the angle of incidence.
For objects such as mirrors, with surfaces so smooth that any hills or
valleys on the surface are smaller than the wavelength of light, the
law of reflection applies on a large scale. All the light travelling in
one direction and reflecting from the mirror is reflected in one
direction; reflection from such objects is known as specular
reflection.
Most objects exhibit diffuse reflection, with light being reflected in
all directions. All objects obey the law of reflection on a microscopic
level, but if the irregularities on the surface of an object are larger
than the wavelength of light, which is usually the case, the light
reflects off in all directions.
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Plane mirrors
A plane mirror is simply a mirror with a flat surface; all of us use
plane mirrors every day, so we've got plenty of experience with
them. Images produced by plane mirrors have a number of
properties, including:
1. the image produced is upright
2. the image is the same size as the object (i.e., the magnification is m = 1)
3. the image is the same distance from the mirror as the object appears to be (i.e., the
image distance = the object distance)
4. the image is a virtual image, as opposed to a real image, because the light rays do
not actually pass through the image. This also implies that an image could not be
focused on a screen placed at the location where the image is.
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Dealing with light in terms of rays is known as geometrical optics,
for good reason: there is a lot of geometry involved. It's relatively
straight-forward geometry, all based on similar triangles, but we
should review that for a plane mirror.
Consider an object placed a certain distance in front of a mirror, as
shown in the diagram. To figure out where the image of this object
is located, a ray diagram can be used. In a ray diagram, rays of light
are drawn from the object to the mirror, along with the rays
that reflect off the mirror. The image will be found where the
reflected rays intersect. Note that the reflected rays obey the
law of reflection. What you notice is that the reflected rays
diverge from the mirror; they must be extended back to find
the place where they intersect, and that's where the image is.
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Analyzing this a little further, it's easy to see that the height
of the image is the same as the height of the object. Using
the similar triangles ABC and EDC, it can also be seen that the
distance from the object to the mirror is the same as the
distance from the image to the mirror.
Concave and Convex Mirrors
Concave and Convex Mirrors
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It was mentioned earlier in this lesson that light reflects off surfaces
in a very predictable manner - in accordance with the law of
reflection. Once a normal to the surface at the point of incidence is
drawn, the angle of incidence can then be determined. The light ray
will then reflect in such a manner that the angle of incidence is equal
to the angle of reflection. This predictability concerning the
reflection of light is applicable to the reflection of light off of level
(horizontal) surfaces, vertical surfaces, angled surfaces, and even
curved surfaces. As long as the normal (perpendicular line to the
surface) can be drawn at the point of incidence, the angle of
incidence can be measured and the direction of the reflected ray can
be determined. A series of incident rays and their corresponding
reflected rays are depicted in the diagram below. Each ray strikes a
surface with a different orientation; yet each ray reflects in
accordance with the law of reflection.
The Law of Reflection is Always Observed
(regardless of the orientation of the surface)
In physics class, the behavior of light is often studied by
observing its reflection off of plane (flat) mirrors. Mirrors
are typically smooth surfaces, even at the microscopic levels.
As such, they offer each individual ray of light the same surface
orientation. But quite obviously, mirrors are not the only type
s of objects which light reflects off of. Most objects which reflect
light are not smooth at the microscopic level. Your clothing,
the walls of most rooms, most flooring, skin, and even paper are all rough
when viewed at the microscopic level. The picture at the right depicts a
highly magnified, microscopic view of the surface of a sheet of paper.
Reflection off of smooth surfaces such as mirrors or a calm body of
water leads to a type of reflection known as specular reflection. Reflection
off of rough surfaces such as clothing, paper, and the asphalt roadway
leads to a type of reflection known as diffuse reflection. Whether the
surface is microscopically rough or smooth has a tremendous impact upon
the subsequent reflection of a beam of light. The diagram below depicts two
beams of light incident upon a rough and a smooth surface.
A light beam can be thought of as a bundle of individual light rays which are
traveling parallel to each other. Each individual light ray of the bundle follows the
law of reflection. If the bundle of light rays is incident upon a smooth surface,
then the light rays reflect and remain concentrated in a bundle upon leaving the
surface. On the other hand, if the surface is microscopically rough, the light rays
will reflect and diffuse in many different directions.
Why Does a Rough Surface Diffuses A Beam of Light?
For each type of reflection, each individual ray follows the law of reflection.
However, the roughness of the material means that each individual ray meets a
surface which has a different orientation. The normal line at the point of
incidence is different for different rays. Subsequently, when the individual rays
reflect off the rough surface according to the law of reflection, they scatter in
different directions. The result is that the rays of light are incident upon the
surface in a concentrated bundle and are diffused upon reflection. The diagram
below depicts this principle. Five incident rays (labeled A, B, C, D, and E)
approach a surface. The normal line (approximated) at each point of incidence is
shown in black and labeled with an N. In each case, the law of reflection is
followed, resulting in five reflected rays (labeled A,, B,, C,, D,, and E,).
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Like any wave, a sound wave doesn't just stop when it reaches the end
of the medium or when it encounters an obstacle in its path. Rather, a
sound wave will undergo certain behaviors when it encounters the end
of the medium or an obstacle. Possible behaviors include reflection off
the obstacle, diffraction around the obstacle, and transmission
(accompanied by refraction) into the obstacle or new medium . In this
part of Chapter 29, we will investigate behaviors which have already
been discussed in a previous Chapter and apply them towards the
reflection, diffraction, and refraction of sound waves.
When a wave reaches the boundary between one medium another
medium, a portion of the wave undergoes reflection and a
portion of the wave undergoes transmission across the
boundary. As discussed in the previous part of the chapter,
the amount of reflection is dependent upon the dissimilarity of the
two medium. For this reason, acoustically minded builders of
auditoriums and concert halls avoid the use of hard, smooth materials
in the construction of their inside halls. A hard material such as
concrete is as dissimilar as can be to the air through which the sound
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moves; subsequently, most of the sound wave is reflected by the
walls and little is absorbed. Walls and ceilings of concert halls are
made softer materials such as fiberglass and acoustic tiles. These
materials are more similar to air than concrete and thus have a
greater ability to absorb sound. This gives the room more pleasing
acoustic properties.
Reflection of sound waves off of surfaces can lead to one of two
phenomenon - an echo or a reverberation. A reverberation often
occurs in a small room with height, width, and length dimensions of
approximately 17 meters or less. Why the magical 17 meters? The
affect of a particular sound wave upon the brain endures for more
than a tiny fraction of a second; the human brain keeps a sound in
memory for up to 0.1 seconds. If a reflected sound wave reaches the
ear within 0.1 seconds of the initial sound, then it seems to the
person that the sound is prolonged. The reception of multiple
reflections off of walls and ceilings within 0.1 seconds of each other
causes reverberations - the prolonging of a sound. Since sound
waves travel at about 340 m/s at room temperature, it will take
approximately 0.1 s for a sound to travel the length of a 17 meter
room and back, thus causing a reverberation (recall that, t = v/d =
(340 m/s)/(34 m) = 0.1 s). This is why reverberations is common in
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rooms with dimensions of approximately 17 meters or less. Perhaps
you have observed reverberations when talking in an empty room,
when honking the horn while driving through a highway tunnel or
underpass, or when singing in the shower. In auditoriums and
concert halls, reverberations occasionally occur and lead to the
displeasing garbling of a sound.
Reflection of sound waves also lead to echoes. Echoes are different
than reverberations. Echoes occur when a reflected sound wave
reaches the ear more than 0.1 seconds after the original sound wave
was heard. If the elapsed time between the arrival of the two sound
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waves is more than 0.1 seconds, then the sensation of the first
sound will have died out . In this case, the arrival of the second
sound wave will be perceived as a second sound rather than the
prolonging of the first sound. There will be an echo instead of a
reverberation.
Reflection of sound waves off of surfaces is also affected by the
shape of the surface. As mentioned earlier, flat or plane surfaces
reflect sound waves in such a way that the angle at which the wave
approaches the surface equals the angle at which the wave leaves
the surface. Reflection of sound waves off of curved surfaces leads
to a more interesting phenomenon. Curved surfaces with a parabolic
shape have the habit of focusing sound waves to a point. Sound
waves reflecting off of parabolic surfaces concentrate all their
energy to a single point in space; at that point, the sound is
amplified. Perhaps you have seen a museum exhibit which utilizes a
parabolic-shaped disk to collect a large amount of sound and focus
it at a focal point. If you place your ear at the focal point, you can
hear even the faintest whisper of a friend standing across the room.
Parabolic-shaped satellite disks use this same principle of reflection
to gather large amounts of electromagnetic waves and focus it at a
point (where the receptor is located).
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Boundary Behavior for Waves on a Rope
Suppose that there is a thin rope attached to a thick rope, with each
rope held at opposite ends by people. And suppose that a pulse is
introduced by the person holding the end of the thin rope. If this is
the case, there will be an incident pulse traveling in the less dense
medium (thin rope) towards the boundary with a more dense medium
(thick rope).
Upon reaching the boundary, two behaviors will occur.
A portion of the energy carried by the incident pulse is reflected and
returns towards the left end of the thin rope. The disturbance which
returns to the left after bouncing off the boundary is known as the
reflected pulse.
A portion of the energy carried by the incident pulse is transmitted
into the thick rope. The disturbance which continues moving to the
right is known as the transmitted pulse.
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These two behaviors - reflection and transmission - were first
introduced in the beginning of the chapter, and an earlier chapter. It
was mentioned that the passage of the energy from the incident
medium into the transmitted medium was accompanied by a change
in speed and wavelength. In the case of a pulse crossing the
boundary from a less dense medium into a more dense medium, the
speed and the wavelength are both decreased. On the other hand, if
a pulse crosses the boundary from a more dense medium into a less
dense medium, the speed and the wavelength are both increased.
The above discussion was limited to the behavior of a wave on a
rope. But what if the wave is a light wave traveling in a threedimensional medium? For example, what would happen if a light
wave is traveling through air and reaches the boundary with a glass
surface? How can the reflection and transmission behavior of a light
wave be described? First, the light wave behaves like the wave on the
rope: a portion of the wave is transmitted into the new medium
(glass) and a portion of the wave reflects off the air-glass boundary.
Second, the same wave property changes which were observed for
the wave on the rope are also observed for the light wave passing
from air into glass; there is a change in speed and wavelength of the
wave as it crosses the air-glass boundary. When passing from air
into glass, both the speed and the wavelength decrease.
Finally, and most importantly, the light is observed to
change directions as it crosses the boundary separating
the air and the glass. This bending of the path of light is
known as refraction. A one-word synonym for refraction
is bending. The transmitted wave experiences this
refraction at the boundary. As seen in the diagram at the
right, each individual wavefront is bent only along the
boundary. Once the wavefront has passed across the
boundary, it travels in a straight line. For this reason, refraction is called a
boundary behavior. A ray is drawn perpendicular to the wavefronts; this ray
represents the direction which the light wave is traveling. Observe that the ray
is a straight line inside of each of the two media, but bends at the boundary.
Again, refraction is a boundary behavior.
The Ray Model of Light
In this unit, we will rely heavily on the use of rays to represent the direction in
which light is moving. While we often think of light behaving as a wave, we will
still find it useful to represent its movement through a medium using a line
segment with an arrowhead (i.e., a ray) to depict the refraction of light. The ray
is constructed in a direction perpendicular to the wavefronts of the light wave;
this accurately depicts the light wave's direction. In this sense, we are viewing
light as behaving as a stream of particles which head in the direction of the ray.
The idea that the path of light can be represented by a ray is known as the ray
model of light.
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Refraction of waves involves a change in the direction of waves as
they pass from one medium to another. Refraction, or bending of
the path of the waves, is accompanied by a change in speed and
wavelength of the waves. So if the medium (and its properties) are
changed, the speed of the waves are changed. Thus, waves passing
from one medium to another will undergo refraction. Refraction of
sound waves is most evident in situations in which the sound wave
passes through a medium with gradually varying properties. For
example, sound waves are known to refract when traveling over
water. Even though the sound wave is not exactly changing media,
it is traveling through a medium with varying properties; thus, the
wave will encounter refraction and
change its direction. Since water has a
moderating affect upon the temperature
of air, the air directly above the water tends to be cooler than the air
far above the water. Sound waves travel slower in cooler air than
they do in warmer air. For this reason, the portion of the wavefront
directly above the water is slowed down, while the portion of the
wavefronts far above the water speeds ahead. Subsequently, the
direction of the wave changes, refracting downwards towards the
directly above the water is slowed down, while the portion of the
wavefronts far above the water speeds ahead. Subsequently, the
direction of the wave changes, refracting downwards towards the
water. This is depicted in the diagram above and below.
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As light travels through a given medium, it travels in a straight line.
However, when light passes from one medium into a second medium,
the light path bends. Refraction takes place. The refraction occurs
only at the boundary. Once the light has crossed the boundary
between the two media, it continues to travel in a straight line. Only
now, the direction of that line is different than it was in the former
medium. If when sighting at an object, light from that object changes
media on the way to your eye, a visual distortion is likely to occur.
This visual distortion is witnessed if you look at a pencil submerged
in a glass half-filled with water. As you sight through the side of the
glass at the portion of the pencil located above the water's surface,
light travels directly from the pencil to your eye. Since this light does
not change medium, it will not refract. (Actually, there is a change of
medium from air to glass and back into air. Because the glass is so
thin and because the light starts and finished in air, the refraction
into and out of the glass causes little deviation in the light's original
direction.) As you sight at the portion of the pencil which was
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submerged in the water, light travels from water to air (or from
water to glass to air). This light ray changes medium and
subsequently undergoes refraction. As a result, the image of the
pencil appears to be broken. Furthermore, the portion of the
pencil which is submerged in water appears to be wider than the
portion of the pencil which is not submerged. These visual
distortions are explained by the refraction of light.
But why does light refract? What is the cause of such behavior?
And why is there this one exception to the refraction of light? An
analogy of marching soldiers is often used to address this
question. In fact, it is not uncommon that the analogy be
illustrated in a Physics class with a student demonstration. A
group of students forms a straight line (shoulder to shoulder) and
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connect themselves to their nearest neighbor using meter sticks. A
strip of masking tape divides the room into two media. In one of the
media (on one side of the tape), students walk at a normal pace. In
the other media (or on the other side of the tape), students walk
very slowly using baby steps. The group of students walk forward
together in a straight line towards the diagonal strip of masking
tape. The students maintain the line as they approach the masking
tape. When an individual student reaches the tape, that student
abruptly changes the pace of her/his walk. The group of students
continue walking until all students in the line have entered into the
second medium. The diagram below represents the line of students
approaching the boundary (the masking tape) between the two
medium. On the diagram, an arrow is used to show the general
direction of travel for the group of students in both medium.
Observe that the direction of the students changes at the
"boundary."
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The broken pencil phenomenon occurs during your everyday
spear-fishing outing. Fortunately for the fish, light refracts as it
travels from the fish in the water to the eyes of the hunter. The
refraction occurs at the water-air boundary. Due to this bending of
the path of light, a fish appears to be at a location where it isn't. A
visual distortion occurs. Subsequently, the hunter launches the spear
at the location where the fish is thought to be and misses the fish.
Of course, the fish are never concerned about such hunters; they
know that light refracts at the boundary and that the location where
the hunter is sighting is not the same location as the actual fish.
How did the fish get so smart and learn all this? They live in schools.
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Now any fish who has done his/her physics homework knows that
the amount of refraction which occurs is dependent upon the angle
at which the light approaches the boundary. We will investigate this
aspect of refraction in great detail later. For now, it is sufficient to
say that as the hunter with the spear sights more perpendicular to
the water, the amount of refraction decreases. The most successful
hunters are those who sight perpendicular to the water. And the
smartest fish are those who head for the deep when they spot
hunters who sight in this direction.
Since refraction of light occurs when it crosses the boundary, visual
distortions often occur. These distortions occur when light changes
medium as it travels from the object to our eyes.
There are many effects of
refraction. a. The apparent
depth of a glass block is
less than the real depth.
b. The fish appears to be
nearer than it actually is.
c. The full glass mug
appears to hold more root
beer than it actually does.
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It has been mentioned in our discussion that the refraction or
bending of light occurs at the boundary between two materials;
and once a light wave has crossed the boundary it travels in a
straight line. The discussion has presumed that the medium is a
uniform medium. A uniform medium is a medium whose optical
density is everywhere the same within the medium. A uniform
medium is the same everywhere from its top boundary to its
bottom boundary and from its left boundary to its right boundary.
But not every medium is a uniform medium, and the fact that air
can sometimes form a nonuniform medium leads to an interesting
refraction phenomenon - the formation of mirages.
A mirage is an optical phenomenon which creates the illusion of
water and results from the refraction of light through a nonuniform
medium. Mirages are most commonly observed on sunny days
when driving down a roadway. As you drive down the roadway,
there appears to be a puddle of water on the road several yards
(maybe one-hundred yards) in front of the car. Of course, when
you arrive at the perceived location of the puddle, you recognize
that the puddle is not there. Instead, the puddle of water appears
to be another one-hundred yards in front of you. You could carefully match the
perceived location of the water to a roadside object; but when you arrive at
that object, the puddle of water is still not on the roadway. The appearance of
the water is simply an illusion.
Mirages occur on sunny days. The role of the sun is to heat the roadway to
high temperatures. This heated roadway in turn heats the surrounding air,
keeping the air just above the roadway at higher temperatures than that day's
average air temperature. Hot air tends to be less optically dense than cooler
air. As such, a nonuniform medium has been created by the heating of the
roadway and the air just above it. While light will travel in a straight line
through a uniform medium, it will refract when traveling through a nonuniform
medium. If a driver looks down at the roadway at a very low angle (that is, at a
position nearly one-hundred yards away), light from objects above the
roadway will follow a curved path to the driver's eye as shown in the diagram
below.
Light which is traveling downward into this less optically dense air begins to
speed up. Though there isn't a distinct boundary between two media, there is
a change in speed of a light wave. As expected, a change in speed is
accompanied by a change in direction. If there were a distinct boundary
between two media, then there would be a bending of this light ray away from
the normal. For this light ray to bend away from the normal (towards the
boundary), the ray would begin to bend more parallel to the roadway and then
bend upwards towards the cooler air. As such, a person in a car sighting
downward at the roadway will see an object located above the roadway.
Of course, this is not a usual event. When was the last time that you looked
downward at a surface and saw an object above the surface? While not a
usual event, it does happen. For instance, suppose you place a mirror on the
floor and look downward at the floor; you will see objects located above the
floor due to the reflection of light by the mirror. Even a glass window placed
on the floor will reflect light from objects above the
floor. If you look downward at the glass window at
a low enough angle, then you will see objects
located above the floor. Or suppose that you are
standing on the shore of a calm pond and look
downward at the water; you might see objects
above the pond due to the reflection of light by the
water.
Photograph of Mount Moran in the
Grand Teton National Park in Wyoming
- taken by Becky Henderson
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So when you experience this sunny day phenomenon, your mind
must quickly make sense of how you can look downward at the
roadway and see an object located above the road. In the process of
making sense of this event, your mind draws upon past experiences.
Searching the database of stored experiences, your mind is
interested in an explanation of why the eye can sight downward at a
surface and see an object which is located above the surface. In the
process of searching, it comes up with three possible explanations
based upon past experiences. Your mind subtly ponders these three
options.
◦ There is a mirror on the road. Someone must have for some reason placed
a mirror on the road. The mirror is reflecting light and that is why I see an
image of the oncoming truck when I look downward at the road.
◦ There is a glass window on the road. My gosh, do you believe it! Someone
has left a glass window on the road. The glass window is reflecting light
and that is why I see an image of the oncoming truck when I look
downward at the road.
◦ There is water on the road. It must have rained last night and there is a
puddle of water left on the road. The water is reflecting light and that is
why I see an image of the oncoming truck when I look downward at the
road.
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Of the three possible explanations of the image of the truck, only
one makes a lot of sense to the mind - there is water on the road.
After all, while both glass windows and mirrors can reflect light,
nowhere in your mind's database of past experiences is there an
account of a mirror or glass window being seen on a roadway. Yet
there are plenty of times that a water puddle has been observed to
be present on a roadway. Smart person that you are, you then
conclude that there is a puddle of water on the road which is
causing you to see objects located above the road when you sight
downward at the road. The illusion is complete.
A driver might see a mirage
on a hot day. The “wet”
street is actually dry!
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When you watch the sun set,
you see the sun for several
minutes after it has actually
sunken below the horizon.
This is because light is
refracted
by
the
earth’s
atmosphere as shown in the
figure. Since the density of the
atmosphere changes gradually,
the refracted ray bends gradually to produce a curved path. The
same thing occurs at sunrise, so our daylight is about 5 minutes
longer because of atmospheric refraction.
When the sun (or moon) is near the horizon, the rays from the
lower edge are bent more than the rays from the upper edge. This
produces a shortening of the vertical diameter, and makes the sun
(or moon) look oval instead of round, as in the figure.
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Newton's experiments illustrated the dispersion of sunlight into
a spectrum (and recombination into white light). Sunlight
consists of a mixture of light with different wavelengths. A
dispersive medium is one in which different wavelengths of
light have slightly different indices of refraction. For example,
crown glass is a dispersive medium since the index of refraction
for violet light in crown glass is higher than for red light. This is
responsible for chromatic aberration. (Manufacturers of optical
glass customarily specify the refractive index of a material for
yellow sodium light, the D line.)
Light passing through a rectangular prism can experience
lateral displacement. In a prism with non-parallel sides, the
displacement is described by the angle of deviation between the
ray incident to the prism and the ray emerging from it.
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One of nature's most splendid masterpieces is the rainbow. A
rainbow is an excellent demonstration of the dispersion of light
and one more piece of evidence that visible light is composed of a
spectrum of wavelengths, each associated with a distinct color.
To view a rainbow, your back must be to the sun as you look at
an approximately 40 degree angle above the ground into a region
of the atmosphere with suspended droplets of water or even a
light mist. Each individual droplet of water acts as a tiny prism
which both disperses the light and reflects it back to your eye. As
you sight into the sky, wavelengths of light associated with a
specific color arrive at your eye from the collection of droplets.
The net effect of the vast array of droplets is that a circular arc of
ROYGBIV is seen across the sky. Exactly how do the droplets of
water disperse and reflect the light? And why does the pattern
always appear as ROYGBIV from top to bottom? These are the
questions which we will seek to understand on this section of The
chapter. To understand these questions, we will need to draw
upon our understanding of refraction, internal reflection and
dispersion.

There are countless paths by which light rays
from the sun can pass through a drop. Each
path is characterized by this bending towards
and away from the normal. One path of great
significance in the discussion of rainbows is
the path in which light refracts into the
droplet, internally reflects, and then refracts
out of the droplet. The diagram at the right
depicts such a path. A light ray from the sun enters the droplet with
a slight downward trajectory. Upon refracting twice and reflecting
once, the light ray is dispersed and bent downward towards an
observer on earth's surface. Other entry locations into the droplet
may result in similar paths or even in light continuing through the
droplet and out the opposite side without significant internal
reflection. But for the entry location shown in the diagram at the
right, there is an optimal concentration of light exiting the airborne
droplet at an angle towards the ground. As in the case of the
refraction of light through prisms with nonparallel sides, the
refraction of light at two boundaries of the droplet results in the
dispersion of light into a spectrum of colors. The shorter wavelength

blue and violet light refract a slightly greater amount than the
longer wavelength red light. Since the boundaries are not parallel to
each other, the double refraction results in a distinct separation of
the sunlight into its component colors.
The angle of deviation between the incoming
light rays from the sun and the refracted
rays directed to the observer's eyes is
approximately 42 degrees for the red light.
Because of the tendency of shorter wavelength
blue light to refract more than red light, its angle of deviation from
the original sun rays is approximately 40 degrees. As shown in the
diagram, the red light refracts out of the droplet at a steeper angle
toward an observer on the ground. There are a multitude of paths by
which the original ray can pass through a droplet and and
subsequently angle towards the ground. Some of the paths are
dependent upon which part of the droplet the incident rays contact.
Other paths are dependent upon the location of the sun in the sky
and the subsequent trajectory of the incoming rays towards the
droplet. Yet the greatest concentration of outgoing rays is found at
these 40-42 degree angles of deviation. At these angles, the

dispersed light is bright enough to result in a rainbow display in the
sky. Now that we understand the path of light through an individual
droplet, we can approach the topic of how the rainbow forms.
A rainbow is most often viewed as a circular arc in the sky. An
observer on the ground observes a half-circle of color with red
being the color perceived on the outside or top of the bow. Those
who are fortunate enough to have seen a rainbow from an airplane
in the sky may know that a rainbow can actually be a complete
circle. Observers on the ground only view the top half of the circle
since the bottom half of the circular arc is prevented by the presence
of the ground (and the rather obvious fact that suspended water
droplets aren't present below ground). Yet observers in an airborne
plane can often look both upward and downward to view the
complete circular bow.

Often a secondary bow , with colors reversed, can be seen arching at
a greater angle around the primary bow. The secondary bow is
formed by similar circumstances and is the result of double
refraction within the raindrops, as illustrated in the figure. Because
most of the light is refracted out the back during the extra
reflection, the secondary bow is much dimmer.
29.12 Total Internal Reflection
A common Physics lab is to sight through the long side of an isosceles triangle at
a pin or other object held behind the opposite face. When done so, an unusual
observation - a discrepant event - is observed. The diagram on the left below
depicts the physical situation. A ray of light entered the face of the triangular block
at a right angle to the boundary. This ray of light passes across the boundary
without refraction since it was incident along the normal. The ray of light then
travels in a straight line through the glass until it reaches the second boundary.
Now instead of transmitting across this boundary, all of the light seems to reflect
off the boundary and transmit out the opposite face of the isosceles triangle. This
discrepant event bothers many as they spend several minutes looking for the light
to refract through the second boundary. Then finally, to their amazement, they
looked through the third face of the block and clearly see the ray. What
happened? Why did light not refract through the second face?
The phenomenon observed in this part of the lab is known as total
internal reflection. Total internal reflection, or TIR as it is intimately
called, is the reflection of the total amount of incident light at the
boundary between two medium.
To understand total internal reflection, we will begin with a thought
experiment. Suppose that a laser beam is submerged in a tank of water
(don't do this at home) and pointed upwards towards water-air boundary.
Then suppose that the angle at which the beam is directed upwards is
slowly altered, beginning with small angles of incidence and proceeding
towards larger and larger angles of incidence. What would be observed in
such an experiment? If we understand the principles of boundary
behavior, we would expect that we would observe both reflection and
refraction. And indeed, that is what is observed (mostly). But that's not
the only observation which we could make. We would also observe that
the intensity of the reflected and refracted rays do not remain constant.
At angle of incidence close to 0 degrees, most of the light energy is
transmitted across the boundary and very little of it is reflected. As the
angle is increased to greater and greater angles, we would begin to
observe less refraction and more reflection. That is, as the angle of
incidence is increased, the brightness of the refracted ray decreases and
the brightness of the reflected ray increases. Finally, we would observe
that the angles of the reflection and refraction are not equal. Since the light
waves would refract away from the normal (a case of the SFA principle of
refraction), the angle of refraction would be greater than the angle of incidence.
And if this is the case, the angle of refraction would also be greater than the
angle of reflection (since the angles of reflection and incidence are the same).
As the angle of incidence is increased, the angle of refraction would eventually
reach a 90-degree angle. These principles are depicted in the diagram below.
The maximum possible angle of refraction is 90-degrees. If you think about it (a
practice which always helps), you recognize that if the angle of refraction were
greater than 90 degrees, then the refracted ray would lie on the incident side of
the medium - that's just not possible. So in the case of the laser beam in the
water, there is some specific value for the angle of incidence (we'll call it the
critical angle) which yields an angle of refraction of 90-degrees. This particular
value for the angle of incidence could be calculated using Snell's Law (ni =
1.33, nr = 1.000, = 90 degrees, = ???) and would be found to be 48.6 degrees.
Any angle of incidence which is greater than 48.6 degrees would not result in
refraction. Instead, when the angles of incidence is greater than 48.6 degrees
(the critical angle), all of the energy (the total energy) carried by the incident
wave to the boundary stays within the water (internal to the original medium)
and undergoes reflection off the boundary. When this happens, total internal
reflection occurs.
.
•
Total internal reflection, as the name implies: Total -- 100%. Silvered
or aluminized mirrors reflect only 90 to 95% of the incident light, and
are marred by dust and dirt; prism’s are more efficient! This is the
main reason prisms instead of mirrors are used in many optical
instruments. The figure to the right show how prisms can be used to
reflect light.
TIR and the Sparkle of Diamonds
Relatively speaking, the critical angle for the
diamond-air boundary is an extremely small
number. Of all the possible combinations of
materials which could interface to form a
boundary, the combination of diamond and air
provides one of the largest difference in the
index of refraction values. This means that there
will be a very small nr/ni ratio and subsequently a
small critical angle. This peculiarity about the
diamond-air boundary plays an important role in
the brilliance of a diamond gemstone. Having a
small critical angle, light has the tendency to
become "trapped" inside of a diamond once it
Prism’s are more
efficient at reflecting
light than mirrors
because
of
total
internal reflection.
enters. A light ray will typically undergo TIR several times before finally
refracting out of the diamond. Because the diamond-air boundary has
such a small critical angle (due to diamond's large index of refraction),
most rays approach the diamond at angles of incidence greater than
the critical angle. This gives diamond a tendency to sparkle. The effect
can be enhanced by the cutting of a diamond gemstone with a
strategically planned shape. The diagram below depicts the total
internal reflection within a diamond gemstone with a strategic and a
non-strategic cut.
Light Piping and Optical Fibers
•Total internal reflection is often demonstrated in a
Physics class through a variety of demonstrations. In
one such demonstration, a beam of laser light is
directed into a coiled plastic thing-a-ma jig. The plastic
served as a light pipe, directing the light through the
coils until it finally exits out the opposite end. Once the
light entered the plastic, it was in the more dense
medium. Every time the light approached the plastic-air
boundary, it is approaching at angles greater than the
critical angle. The two conditions necessary for TIR are
met, and all of the incident light at the plastic-air
boundary stays internal to the plastic and undergoes
reflection. And with the room lights off, every student
becomes quickly aware of the ancient truth that Physics
is better than drugs.
This
demonstration helps to illustrate the principle by which optical
.
fibers work. The use of a long strand of plastic (or other material such
as glass) to pipe light from one end of the medium to the other is the


basis for modern day use of optical fibers. Optical fibers are used in
micro-surgeries. Since total internal reflection takes place within
the fibers, no incident energy is ever lost due to the transmission of
light across the boundary. The intensity of the signal remains
constant.
Another
growing
application
of
optical
fibers
is
the
telecommunications system, as the fibers can be easily laid under
ground, and under the sea. This is a great means on transmitting
signals over long distances with minimal loss, and it is surprisingly
cheap to build, lay, and use. Both Telstra and Optus have realized
these capabilities, and are researching and laying fiber optic cables,
for use in telephone, Internet, and pay television systems.
Underwater fiber optic cables currently carry telephone and Internet
signals across the Atlantic and Pacific oceans.
The potential of the applications of optical fibers is nearly
unlimited, because of the great ability to bend the fiber, and place
it under extreme conditions, without distorting the signals being
sent through them. So the next time you pick up the phone to
speak, you may well be using an optical fiber system to do it.