Chapter Twenty Nine Notes: Reflection and Refraction.

Chapter Twenty Nine Notes:

Reflection and Refraction .

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.

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.

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.

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.

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.

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.

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

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 types 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,).  

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

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

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

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).

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.

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 three-dimensional 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 LightIn 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.

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 water. This is depicted in the diagram at the right.

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.

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

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

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."

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.

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.

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

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.

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!

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.

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 ReflectionA 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.

Prism’s are more efficient at reflecting light than mirrors because of total internal reflection.

TIR and the Sparkle of DiamondsRelatively 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

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.