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Contents

Section 12.1 - The Ray Model of Light

Section 12.2 - Images by Reflection

Section 12.3 - Images, Quantitatively

Section 12.4 - Refraction

Section 12.5 - Wave Optics

Section 12.1 - The Ray Model of Light

Section 12.2 - Images by Reflection

Section 12.3 - Images, Quantitatively

Section 12.4 - Refraction

Section 12.5 - Wave Optics

Ads for one Macintosh computer bragged that it could do an arithmetic calculation in less time than it took for the light to get from the screen to your eye. We find this impressive because of the contrast between the speed of light and the speeds at which we interact with physical objects in our environment. Perhaps it shouldn't surprise us, then, that Newton succeeded so well in explaining the motion of objects, but was far less successful with the study of light.

The climax of our study of electricity and magnetism was discovery that light is an electromagnetic wave. Knowing this, however, is not the same as knowing everything about eyes and telescopes. In fact, the full description of light as a wave can be rather cumbersome. We will instead spend most of our treatment of optics making use of a simpler model of light, the ray model, which does a fine job in most practical situations. Not only that, but we will even backtrack a little and start with a discussion of basic ideas about light and vision that predated the discovery of electromagnetic waves.

Despite its title, this chapter is far from your first look at light. That familiarity might seem like an advantage, but most people have never thought carefully about light and vision. Even smart people who have thought hard about vision have come up with incorrect ideas. The ancient Greeks, Arabs and Chinese had theories of light and vision, all of which were mostly wrong, and all of which were accepted for thousands of years.

One thing the ancients did get right is that there is a distinction between objects that emit light and objects that don't. When you see a leaf in the forest, it's because three different objects are doing their jobs: the leaf, the eye, and the sun. But luminous objects like the sun, a flame, or the filament of a light bulb can be seen by the eye without the presence of a third object. Emission of light is often, but not always, associated with heat. In modern times, we are familiar with a variety of objects that glow without being heated, including fluorescent lights and glow-in-the-dark toys.

How do we see luminous objects? The Greek philosophers Pythagoras (b. ca. 560 BC) and Empedocles of Acragas (b. ca. 492 BC), who unfortunately were very influential, claimed that when you looked at a candle flame, the flame and your eye were both sending out some kind of mysterious stuff, and when your eye's stuff collided with the candle's stuff, the candle would become evident to your sense of sight.

Bizarre as the Greek “collision of stuff theory” might seem, it had a couple of good features. It explained why both the candle and your eye had to be present for your sense of sight to function. The theory could also easily be expanded to explain how we see nonluminous objects. If a leaf, for instance, happened to be present at the site of the collision between your eye's stuff and the candle's stuff, then the leaf would be stimulated to express its green nature, allowing you to perceive it as green.

Modern people might feel uneasy about this theory, since it suggests that greenness exists only for our seeing convenience, implying a human precedence over natural phenomena. Nowadays, people would expect the cause and effect relationship in vision to be the other way around, with the leaf doing something to our eye rather than our eye doing something to the leaf. But how can you tell? The most common way of distinguishing cause from effect is to determine which happened first, but the process of seeing seems to occur too quickly to determine the order in which things happened. Certainly there is no obvious time lag between the moment when you move your head and the moment when your reflection in the mirror moves.

Today, photography provides the simplest experimental evidence that nothing has to be emitted from your eye and hit the leaf in order to make it “greenify.” A camera can take a picture of a leaf even if there are no eyes anywhere nearby. Since the leaf appears green regardless of whether it is being sensed by a camera, your eye, or an insect's eye, it seems to make more sense to say that the leaf's greenness is the cause, and something happening in the camera or eye is the effect.

Another issue that few people have considered is whether a candle's flame simply affects your eye directly, or whether it sends out light which then gets into your eye. Again, the rapidity of the effect makes it difficult to tell what's happening. If someone throws a rock at you, you can see the rock on its way to your body, and you can tell that the person affected you by sending a material substance your way, rather than just harming you directly with an arm motion, which would be known as “action at a distance.” It is not easy to do a similar observation to see whether there is some “stuff” that travels from the candle to your eye, or whether it is a case of action at a distance.

Newtonian physics includes both action at a distance (e.g., the earth's gravitational force on a falling object) and contact forces such as the normal force, which only allow distant objects to exert forces on each other by shooting some substance across the space between them (e.g., a garden hose spraying out water that exerts a force on a bush).

One piece of evidence that the candle sends out stuff that travels to your eye is that as in figure a, intervening transparent substances can make the candle appear to be in the wrong location, suggesting that light is a thing that can be bumped off course. Many people would dismiss this kind of observation as an optical illusion, however. (Some optical illusions are purely neurological or psychological effects, although some others, including this one, turn out to be caused by the behavior of light itself.)

A more convincing way to decide in which category light belongs is to find out if it takes time to get from the candle to your eye; in Newtonian physics, action at a distance is supposed to be instantaneous. The fact that we speak casually today of “the speed of light” implies that at some point in history, somebody succeeded in showing that light did not travel infinitely fast. Galileo tried, and failed, to detect a finite speed for light, by arranging with a person in a distant tower to signal back and forth with lanterns. Galileo uncovered his lantern, and when the other person saw the light, he uncovered his lantern. Galileo was unable to measure any time lag that was significant compared to the limitations of human reflexes.

The first person to prove that light's speed was finite, and to determine it numerically, was Ole Roemer, in a series of measurements around the year 1675. Roemer observed Io, one of Jupiter's moons, over a period of several years. Since Io presumably took the same amount of time to complete each orbit of Jupiter, it could be thought of as a very distant, very accurate clock. A practical and accurate pendulum clock had recently been invented, so Roemer could check whether the ratio of the two clocks' cycles, about 42.5 hours to 1 orbit, stayed exactly constant or changed a little. If the process of seeing the distant moon was instantaneous, there would be no reason for the two to get out of step. Even if the speed of light was finite, you might expect that the result would be only to offset one cycle relative to the other. The earth does not, however, stay at a constant distance from Jupiter and its moons. Since the distance is changing gradually due to the two planets' orbital motions, a finite speed of light would make the “Io clock” appear to run faster as the planets drew near each other, and more slowly as their separation increased. Roemer did find a variation in the apparent speed of Io's orbits, which caused Io's eclipses by Jupiter (the moments when Io passed in front of or behind Jupiter) to occur about 7 minutes early when the earth was closest to Jupiter, and 7 minutes late when it was farthest. Based on these measurements, Roemer estimated the speed of light to be approximately \(2\times10^8\) m/s, which is in the right ballpark compared to modern measurements of \(3\times10^8\) m/s. (I'm not sure whether the fairly large experimental error was mainly due to imprecise knowledge of the radius of the earth's orbit or limitations in the reliability of pendulum clocks.)

Many people are confused by the relationship between sound and light. Although we use different organs to sense them, there are some similarities. For instance, both light and sound are typically emitted in all directions by their sources. Musicians even use visual metaphors like “tone color,” or “a bright timbre” to describe sound. One way to see that they are clearly different phenomena is to note their very different velocities. Sure, both are pretty fast compared to a flying arrow or a galloping horse, but as we have seen, the speed of light is so great as to appear instantaneous in most situations. The speed of sound, however, can easily be observed just by watching a group of schoolchildren a hundred feet away as they clap their hands to a song. There is an obvious delay between when you see their palms come together and when you hear the clap.

The fundamental distinction between sound and light is that sound is an oscillation in air pressure, so it requires air (or some other medium such as water) in which to travel. Today, we know that outer space is a vacuum, so the fact that we get light from the sun, moon and stars clearly shows that air is not necessary for the propagation of light.

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If you observe thunder and lightning, you can tell how far away the storm is. Do you need to know the speed of sound, of light, or of both?

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When phenomena like X-rays and cosmic rays were first discovered, suggest a way one could have tested whether they were forms of light.

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Why did Roemer only need to know the radius of the earth's orbit, not Jupiter's, in order to find the speed of light?

The reason why the sun feels warm on your skin is that the sunlight is being absorbed, and the light energy is being transformed into heat energy. The same happens with artificial light, so the net result of leaving a light turned on is to heat the room. It doesn't matter whether the source of the light is hot, like the sun, a flame, or an incandescent light bulb, or cool, like a fluorescent bulb. (If your house has electric heat, then there is absolutely no point in fastidiously turning off lights in the winter; the lights will help to heat the house at the same dollar rate as the electric heater.)

This process of heating by absorption is entirely different from heating by thermal conduction, as when an electric stove heats spaghetti sauce through a pan. Heat can only be conducted through matter, but there is vacuum between us and the sun, or between us and the filament of an incandescent bulb. Also, heat conduction can only transfer heat energy from a hotter object to a colder one, but a cool fluorescent bulb is perfectly capable of heating something that had already started out being warmer than the bulb itself.

Not all the light energy that hits an object is transformed into heat. Some is reflected, and this leads us to the question of how we see nonluminous objects. If you ask the average person how we see a light bulb, the most likely answer is “The light bulb makes light, which hits our eyes.” But if you ask how we see a book, they are likely to say “The bulb lights up the room, and that lets me see the book.” All mention of light actually entering our eyes has mysteriously disappeared.

Most people would disagree if you told them that light was reflected from the book to the eye, because they think of reflection as something that mirrors do, not something that a book does. They associate reflection with the formation of a reflected image, which does not seem to appear in a piece of paper.

Imagine that you are looking at your reflection in a nice smooth piece of aluminum foil, fresh off the roll. You perceive a face, not a piece of metal. Perhaps you also see the bright reflection of a lamp over your shoulder behind you. Now imagine that the foil is just a little bit less smooth. The different parts of the image are now a little bit out of alignment with each other. Your brain can still recognize a face and a lamp, but it's a little scrambled, like a Picasso painting. Now suppose you use a piece of aluminum foil that has been crumpled up and then flattened out again. The parts of the image are so scrambled that you cannot recognize an image. Instead, your brain tells you you're looking at a rough, silvery surface.

Mirror-like reflection at a specific angle is known as specular reflection, and random reflection in many directions is called diffuse reflection. Diffuse reflection is how we see nonluminous objects. Specular reflection only allows us to see images of objects other than the one doing the reflecting. In top part of figure d, imagine that the rays of light are coming from the sun. If you are looking down at the reflecting surface, there is no way for your eye-brain system to tell that the rays are not really coming from a sun down below you.

Figure f shows another example of how we can't avoid the conclusion that light bounces off of things other than mirrors. The lamp is one I have in my house. It has a bright bulb, housed in a completely opaque bowl-shaped metal shade. The only way light can get out of the lamp is by going up out of the top of the bowl. The fact that I can read a book in the position shown in the figure means that light must be bouncing off of the ceiling, then bouncing off of the book, then finally getting to my eye.

This is where the shortcomings of the Greek theory of vision become glaringly obvious. In the Greek theory, the light from the bulb and my mysterious “eye rays” are both supposed to go to the book, where they collide, allowing me to see the book. But we now have a total of four objects: lamp, eye, book, and ceiling. Where does the ceiling come in? Does it also send out its own mysterious “ceiling rays,” contributing to a three-way collision at the book? That would just be too bizarre to believe!

The differences among white, black, and the various shades of gray in between is a matter of what percentage of the light they absorb and what percentage they reflect. That's why light-colored clothing is more comfortable in the summer, and light-colored upholstery in a car stays cooler that dark upholstery.

We have already seen that the physiological sensation of loudness relates to the sound's intensity (power per unit area), but is not directly proportional to it. If sound A has an intensity of 1 \(\text{nW}/\text{m}^2\), sound B is 10 \(\text{nW}/\text{m}^2\), and sound C is 100 \(\text{nW}/\text{m}^2\), then the increase in loudness from B to C is perceived to be the same as the increase from A to B, not ten times greater. That is, the sensation of loudness is logarithmic.

The same is true for the brightness of light. Brightness is related to power per unit area, but the psychological relationship is a logarithmic one rather than a proportionality. For doing physics, it's the power per unit area that we're interested in. The relevant unit is \(\text{W}/\text{m}^2\). One way to determine the brightness of light is to measure the increase in temperature of a black object exposed to the light. The light energy is being converted to heat energy, and the amount of heat energy absorbed in a given amount of time can be related to the power absorbed, using the known heat capacity of the object. More practical devices for measuring light intensity, such as the light meters built into some cameras, are based on the conversion of light into electrical energy, but these meters have to be calibrated somehow against heat measurements.

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The curtains in a room are drawn, but a small gap lets light through, illuminating a spot on the floor. It may or may not also be possible to see the beam of sunshine crossing the room, depending on the conditions. What's going on?

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Laser beams are made of light. In science fiction movies, laser beams are often shown as bright lines shooting out of a laser gun on a spaceship. Why is this scientifically incorrect?

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A documentary film-maker went to Harvard's 1987 graduation ceremony and asked the graduates, on camera, to explain the cause of the seasons. Only two out of 23 were able to give a correct explanation, but you now have all the information needed to figure it out for yourself, assuming you didn't already know. The figure shows the earth in its winter and summer positions relative to the sun. Hint: Consider the units used to measure the brightness of light, and recall that the sun is lower in the sky in winter, so its rays are coming in at a shallower angle.

Note how I've been casually diagramming the motion of light with pictures showing light rays as lines on the page. More formally, this is known as the ray model of light. The ray model of light seems natural once we convince ourselves that light travels through space, and observe phenomena like sunbeams coming through holes in clouds. Having already been introduced to the concept of light as an electromagnetic wave, you know that the ray model is not the ultimate truth about light, but the ray model is simpler, and in any case science always deals with models of reality, not the ultimate nature of reality. The following table summarizes three models of light.

The ray model is a generic one. By using it we can discuss the path taken by the light, without committing ourselves to any specific description of what it is that is moving along that path. We will use the nice simple ray model for most of our treatment of optics, and with it we can analyze a great many devices and phenomena. Not until section 12.5 will we concern ourselves specifically with wave optics, although in the intervening chapters I will sometimes analyze the same phenomenon using both the ray model and the wave model.

Note that the statements about the applicability of the various models are only rough guides. For instance, wave interference effects are often detectable, if small, when light passes around an obstacle that is quite a bit bigger than a wavelength. Also, the criterion for when we need the particle model really has more to do with energy scales than distance scales, although the two turn out to be related.

The alert reader may have noticed that the wave model is
required at scales smaller than a wavelength of light (on
the order of a micrometer for visible light), and the
particle model is demanded on the atomic scale or lower (a
typical atom being a nanometer or so in size). This implies
that at the smallest scales we need *both* the wave
model and the particle model. They appear incompatible, so
how can we simultaneously use both? The answer is that they
are not as incompatible as they seem. Light is both a wave
and a particle, but a full understanding of this apparently
nonsensical statement is a topic for section 13.2.

Without even knowing how to use the ray model to calculate anything numerically, we can learn a great deal by drawing ray diagrams. For instance, if you want to understand how eyeglasses help you to see in focus, a ray diagram is the right place to start. Many students under-utilize ray diagrams in optics and instead rely on rote memorization or plugging into formulas. The trouble with memorization and plug-ins is that they can obscure what's really going on, and it is easy to get them wrong. Often the best plan is to do a ray diagram first, then do a numerical calculation, then check that your numerical results are in reasonable agreement with what you expected from the ray diagram.

Figure j shows some guidelines for using ray diagrams effectively. The light rays bend when they pass out through the surface of the water (a phenomenon that we'll discuss in more detail later). The rays appear to have come from a point above the goldfish's actual location, an effect that is familiar to people who have tried spear-fishing.

- A stream of light is not really confined to a finite number of narrow lines. We just draw it that way. In j/1, it has been necessary to choose a finite number of rays to draw (five), rather than the theoretically infinite number of rays that will diverge from that point.
- There is a tendency to conceptualize rays incorrectly as objects. In his Optics, Newton goes out of his way to caution the reader against this, saying that some people “consider ... the refraction of ... rays to be the bending or breaking of them in their passing out of one medium into another.” But a ray is a record of the path traveled by light, not a physical thing that can be bent or broken.
- In theory, rays may continue infinitely far into the past and future, but we need to draw lines of finite length. In j/1, a judicious choice has been made as to where to begin and end the rays. There is no point in continuing the rays any farther than shown, because nothing new and exciting is going to happen to them. There is also no good reason to start them earlier, before being reflected by the fish, because the direction of the diffusely reflected rays is random anyway, and unrelated to the direction of the original, incoming ray.
- When representing diffuse reflection in a ray diagram, many students have a mental block against drawing many rays fanning out from the same point. Often, as in example j/2, the problem is the misconception that light can only be reflected in one direction from one point.
- Another difficulty associated with diffuse reflection, example j/3, is the tendency to think that in addition to drawing many rays coming out of one point, we should also be drawing many rays coming from many points. In j/1, drawing many rays coming out of one point gives useful information, telling us, for instance, that the fish can be seen from any angle. Drawing many sets of rays, as in j/3, does not give us any more useful information, and just clutters up the picture in this example. The only reason to draw sets of rays fanning out from more than one point would be if different things were happening to the different sets.

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Suppose an intelligent tool-using fish is spear-hunting for humans. Draw a ray diagram to show how the fish has to correct its aim. Note that although the rays are now passing from the air to the water, the same rules apply: the rays are closer to being perpendicular to the surface when they are in the water, and rays that hit the air-water interface at a shallow angle are bent the most.

To change the motion of a material object, we use a force. Is there any way to exert a force on a beam of light? Experiments show that electric and magnetic fields do not deflect light beams, so apparently light has no electric charge. Light also has no mass, so until the twentieth century it was believed to be immune to gravity as well. Einstein predicted that light beams would be very slightly deflected by strong gravitational fields, and he was proved correct by observations of rays of starlight that came close to the sun, but obviously that's not what makes mirrors and lenses work!

If we investigate how light is reflected by a mirror, we will find that the process is horrifically complex, but the final result is surprisingly simple. What actually happens is that the light is made of electric and magnetic fields, and these fields accelerate the electrons in the mirror. Energy from the light beam is momentarily transformed into extra kinetic energy of the electrons, but because the electrons are accelerating they re-radiate more light, converting their kinetic energy back into light energy. We might expect this to result in a very chaotic situation, but amazingly enough, the electrons move together to produce a new, reflected beam of light, which obeys two simple rules:

- The angle of the reflected ray is the same as that of the incident ray.
- The reflected ray lies in the plane containing the incident ray and the normal (perpendicular) line. This plane is known as the plane of incidence.

The two angles can be defined either with respect to the normal, like angles B and C in the figure, or with respect to the reflecting surface, like angles A and D. There is a convention of several hundred years' standing that one measures the angles with respect to the normal, but the rule about equal angles can logically be stated either as B=C or as A=D.

The phenomenon of reflection occurs only at the boundary between two media, just like the change in the speed of light that passes from one medium to another. As we have seen in section 6.2, this is the way all waves behave.

Most people are surprised by the fact that light can be reflected back from a less dense medium. For instance, if you are diving and you look up at the surface of the water, you will see a reflection of yourself.

Each of these diagrams is supposed to show two different rays being reflected from the same point on the same mirror. Which are correct, and which are incorrect?

(answer in the back of the PDF version of the book)

The fact that specular reflection displays equal angles of incidence and reflection means that there is a symmetry: if the ray had come in from the right instead of the left in the figure above, the angles would have looked exactly the same. This is not just a pointless detail about specular reflection. It's a manifestation of a very deep and important fact about nature, which is that the laws of physics do not distinguish between past and future. Cannonballs and planets have trajectories that are equally natural in reverse, and so do light rays. This type of symmetry is called time-reversal symmetry.

Typically, time-reversal symmetry is a characteristic of any process that does not involve heat. For instance, the planets do not experience any friction as they travel through empty space, so there is no frictional heating. We should thus expect the time-reversed versions of their orbits to obey the laws of physics, which they do. In contrast, a book sliding across a table does generate heat from friction as it slows down, and it is therefore not surprising that this type of motion does not appear to obey time-reversal symmetry. A book lying still on a flat table is never observed to spontaneously start sliding, sucking up heat energy and transforming it into kinetic energy.

Similarly, the only situation we've observed so far where light does not obey time-reversal symmetry is absorption, which involves heat. Your skin absorbs visible light from the sun and heats up, but we never observe people's skin to glow, converting heat energy into visible light. People's skin does glow in infrared light, but that doesn't mean the situation is symmetric. Even if you absorb infrared, you don't emit visible light, because your skin isn't hot enough to glow in the visible spectrum.

These apparent heat-related asymmetries are not actual asymmetries in the laws of physics. The interested reader may wish to learn more about this from optional chapter 5 on thermodynamics.

A number of techniques can be used for creating artificial visual scenes in computer graphics. Figure l shows such a scene, which was created by the brute-force technique of simply constructing a very detailed ray diagram on a computer. This technique requires a great deal of computation, and is therefore too slow to be used for video games and computer-animated movies. One trick for speeding up the computation is to exploit the reversibility of light rays. If one was to trace every ray emitted by every illuminated surface, only a tiny fraction of those would actually end up passing into the virtual “camera,” and therefore almost all of the computational effort would be wasted. One can instead start a ray at the camera, trace it backward in time, and see where it would have come from. With this technique, there is no wasted effort.

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If a light ray has a velocity vector with components \(c_x\) and \(c_y\), what will happen when it is reflected from a surface that lies along the \(y\) axis? Make sure your answer does not imply a change in the ray's speed.

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Generalizing your reasoning from discussion question A, what will happen to the velocity components of a light ray that hits a corner, as shown in the figure, and undergoes two reflections?

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Three pieces of sheet metal arranged perpendicularly as shown in the figure form what is known as a radar corner. Let's assume that the radar corner is large compared to the wavelength of the radar waves, so that the ray model makes sense. If the radar corner is bathed in radar rays, at least some of them will undergo three reflections. Making a further generalization of your reasoning from the two preceding discussion questions, what will happen to the three velocity components of such a ray? What would the radar corner be useful for?

\myoptionalsubsection[2]{The principle of least time for reflection}

We had to choose between an unwieldy explanation of reflection at the atomic level and a simpler geometric description that was not as fundamental. There is a third approach to describing the interaction of light and matter which is very deep and beautiful. Emphasized by the twentieth-century physicist Richard Feynman, it is called the principle of least time, or Fermat's principle.

Let's start with the motion of light that is not interacting with matter at all. In a vacuum, a light ray moves in a straight line. This can be rephrased as follows: of all the conceivable paths light could follow from P to Q, the only one that is physically possible is the path that takes the least time.

What about reflection? If light is going to go from one point to another, being reflected on the way, the quickest path is indeed the one with equal angles of incidence and reflection. If the starting and ending points are equally far from the reflecting surface, o, it's not hard to convince yourself that this is true, just based on symmetry. There is also a tricky and simple proof, shown in figure p, for the more general case where the points are at different distances from the surface.

Not only does the principle of least time work for light in a vacuum and light undergoing reflection, we will also see in a later chapter that it works for the bending of light when it passes from one medium into another.

Although it is beautiful that the entire ray model of light can be reduced to one simple rule, the principle of least time, it may seem a little spooky to speak as if the ray of light is intelligent, and has carefully planned ahead to find the shortest route to its destination. How does it know in advance where it's going? What if we moved the mirror while the light was en route, so conditions along its planned path were not what it “expected?” The answer is that the principle of least time is really a shortcut for finding certain results of the wave model of light, which is the topic of the last chapter of this book.

There are a couple of subtle points about the principle of least time. First, the path does not have to be the quickest of all possible paths; it only needs to be quicker than any path that differs infinitesimally from it. In figure p, for instance, light could get from A to B either by the reflected path AQB or simply by going straight from A to B. Although AQB is not the shortest possible path, it cannot be shortened by changing it infinitesimally, e.g., by moving Q a little to the right or left. On the other hand, path APB is physically impossible, because it is possible to improve on it by moving point P infinitesimally to the right.

It's not quite right to call this
the principle of *least* time. In figure q, for
example, the four physically possible paths by which a ray
can return to the center consist of two shortest-time paths
and two longest-time paths. Strictly speaking, we should
refer to the *principle of least or greatest time*, but
most physicists omit the niceties, and assume that other
physicists understand that both maxima and minima are possible.

Infants are always fascinated by the antics of the Baby in
the Mirror. Now if you want to know something about mirror
images that most people don't understand, try this. First
bring this page closer and closer to your eyes, until you
can no longer focus on it without straining. Then go in the
bathroom and see how close you can get your face to the
surface of the mirror before you can no longer easily focus
on the image of your own eyes. You will find that the
shortest comfortable eye-mirror distance is much less than
the shortest comfortable eye-paper distance. This demonstrates
that the image of your face in the mirror acts as if it had
depth and existed in the space *behind* the mirror. If
the image was like a flat picture in a book, then you
wouldn't be able to focus on it from such a short distance.

In this chapter we will study the images formed by flat and curved mirrors on a qualitative, conceptual basis. Although this type of image is not as commonly encountered in everyday life as images formed by lenses, images formed by reflection are simpler to understand, so we discuss them first. In section 12.3 we will turn to a more mathematical treatment of images made by reflection. Surprisingly, the same equations can also be applied to lenses, which are the topic of section 12.4.

We can understand a mirror image using a ray diagram. Figure a shows several light rays, 1, that originated by diffuse reflection at the person's nose. They bounce off the mirror, producing new rays, 2. To anyone whose eye is in the right position to get one of these rays, they appear to have come from a behind the mirror, 3, where they would have originated from a single point. This point is where the tip of the image-person's nose appears to be. A similar analysis applies to every other point on the person's face, so it looks as though there was an entire face behind the mirror. The customary way of describing the situation requires some explanation:

**Customary description in physics:**There is an image of the face behind the mirror.**Translation:**The pattern of rays coming from the mirror is exactly the same as it would be if there were a face behind the mirror. Nothing is really behind the mirror.

This is referred to as a *virtual* image, because the
rays do not actually cross at the point behind the mirror.
They only appear to have originated there.

Imagine that the person in figure a moves his face down quite a bit --- a couple of feet in real life, or a few inches on this scale drawing. The mirror stays where it is. Draw a new ray diagram. Will there still be an image? If so, where is it visible from?

(answer in the back of the PDF version of the book)The geometry of specular reflection tells us that rays 1 and 2 are at equal angles to the normal (the imaginary perpendicular line piercing the mirror at the point of reflection). This means that ray 2's imaginary continuation, 3, forms the same angle with the mirror as ray 1. Since each ray of type 3 forms the same angles with the mirror as its partner of type 1, we see that the distance of the image from the mirror is the same as that of the actual face from the mirror, and it lies directly across from it. The image therefore appears to be the same size as the actual face.

Figure c shows an old-fashioned device called a praxinoscope, which displays an animated picture when spun. The removable strip of paper with the pictures printed on it has twice the radius of the inner circle made of flat mirrors, so each picture's virtual image is at the center. As the wheel spins, each picture's image is replaced by the next.

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The figure shows an object that is off to one side of a mirror. Draw a ray diagram. Is an image formed? If so, where is it, and from which directions would it be visible?

An image in a flat mirror is a pretechnological example: even animals can look at their reflections in a calm pond. We now pass to our first nontrivial example of the manipulation of an image by technology: an image in a curved mirror. Before we dive in, let's consider why this is an important example. If it was just a question of memorizing a bunch of facts about curved mirrors, then you would rightly rebel against an effort to spoil the beauty of your liberally educated brain by force-feeding you technological trivia. The reason this is an important example is not that curved mirrors are so important in and of themselves, but that the results we derive for curved bowl-shaped mirrors turn out to be true for a large class of other optical devices, including mirrors that bulge outward rather than inward, and lenses as well. A microscope or a telescope is simply a combination of lenses or mirrors or both. What you're really learning about here is the basic building block of all optical devices from movie projectors to octopus eyes.

Because the mirror in figure d is curved, it bends the
rays back closer together than a flat mirror would: we
describe it as *converging*. Note that the
term refers to what it does to the light rays, not to the
physical shape of the mirror's surface . (The surface itself
would be described as *concave*. The term is not all
that hard to remember, because the hollowed-out interior of
the mirror is like a cave.) It is surprising but true that
all the rays like 3 really do converge on a point, forming
a good image. We will not prove this fact, but it is true
for any mirror whose curvature is gentle enough and that is
symmetric with respect to rotation about the perpendicular
line passing through its center (not asymmetric like a
potato chip). The old-fashioned method of making mirrors and
lenses is by grinding them in grit by hand, and this
automatically tends to produce an almost perfect spherical surface.

Bending a ray like 2 inward implies bending its imaginary
continuation 3 outward, in the same way that raising one
end of a seesaw causes the other end to go down. The image
therefore forms deeper behind the mirror. This doesn't just
show that there is extra distance between the image-nose and
the mirror; it also implies that the image itself is bigger
from front to back. It has been *magnified* in the front-to-back direction.

It is easy to prove that the same magnification also applies to the image's other dimensions. Consider a point like E in figure e. The trick is that out of all the rays diffusely reflected by E, we pick the one that happens to head for the mirror's center, C. The equal-angle property of specular reflection plus a little straightforward geometry easily leads us to the conclusion that triangles ABC and CDE are the same shape, with ABC being simply a scaled-up version of CDE. The magnification of depth equals the ratio BC/CD, and the up-down magnification is AB/DE. A repetition of the same proof shows that the magnification in the third dimension (out of the page) is also the same. This means that the image-head is simply a larger version of the real one, without any distortion. The scaling factor is called the magnification, \(M\). The image in the figure is magnified by a factor \(M=1.9\).

Note that we did not explicitly specify whether the mirror was a sphere, a paraboloid, or some other shape. However, we assumed that a focused image would be formed, which would not necessarily be true, for instance, for a mirror that was asymmetric or very deeply curved.

If we start by placing an object very close to the mirror,
g/1, and then move it farther and farther away, the image at
first behaves as we would expect from our everyday
experience with flat mirrors, receding deeper and deeper
behind the mirror. At a certain point, however, a dramatic
change occurs. When the object is more than a certain
distance from the mirror, g/2, the image appears upside-down
and in *front* of the mirror.

Here's what's happened. The mirror bends light rays inward, but when the object is very close to it, as in g/1, the rays coming from a given point on the object are too strongly diverging (spreading) for the mirror to bring them back together. On reflection, the rays are still diverging, just not as strongly diverging. But when the object is sufficiently far away, g/2, the mirror is only intercepting the rays that came out in a narrow cone, and it is able to bend these enough so that they will reconverge.

Note that the rays shown in the figure, which both
originated at the same point on the object, reunite when
they cross. The point where they cross is the image of the
point on the original object. This type of image is called a
*real image*, in contradistinction to the virtual
images we've studied before.

Definition: A real image is one where rays actually cross. A virtual image is a point from which rays only appear to have come.

The use of the word “real” is perhaps unfortunate. It sounds as though we are saying the image was an actual material object, which of course it is not.

The distinction between a real image and a virtual image is an important one, because a real image can be projected onto a screen or photographic film. If a piece of paper is inserted in figure g/2 at the location of the image, the image will be visible on the paper (provided the object is bright and the room is dark). Your eye uses a lens to make a real image on the retina.

Sketch another copy of the face in figure g/1, even farther from the mirror, and draw a ray diagram. What has happened to the location of the image?

(answer in the back of the PDF version of the book)If you are wearing glasses right now, then the light rays from the page are being manipulated first by your glasses and then by the lens of your eye. You might think that it would be extremely difficult to analyze this, but in fact it is quite easy. In any series of optical elements (mirrors or lenses or both), each element works on the rays furnished by the previous element in exactly the same manner as if the image formed by the previous element was an actual object.

Figure h shows an example involving only mirrors. The Newtonian telescope, invented by Isaac Newton, consists of a large curved mirror, plus a second, flat mirror that brings the light out of the tube. (In very large telescopes, there may be enough room to put a camera or even a person inside the tube, in which case the second mirror is not needed.) The tube of the telescope is not vital; it is mainly a structural element, although it can also be helpful for blocking out stray light. The lens has been removed from the front of the camera body, and is not needed for this setup. Note that the two sample rays have been drawn parallel, because an astronomical telescope is used for viewing objects that are extremely far away. These two “parallel” lines actually meet at a certain point, say a crater on the moon, so they can't actually be perfectly parallel, but they are parallel for all practical purposes since we would have to follow them upward for a quarter of a million miles to get to the point where they intersect.

The large curved mirror by itself would form an image \(\text{I}\), but the small flat mirror creates an image of the image, \(\text{I}'\). The relationship between \(\text{I}\) and \(\text{I}'\) is exactly the same as it would be if \(\text{I}\) was an actual object rather than an image: \(\text{I}\) and \(\text{I}'\) are at equal distances from the plane of the mirror, and the line between them is perpendicular to the plane of the mirror.

One surprising wrinkle is that whereas a flat mirror used by itself forms a virtual image of an object that is real, here the mirror is forming a real image of virtual image \(\text{I}\). This shows how pointless it would be to try to memorize lists of facts about what kinds of images are formed by various optical elements under various circumstances. You are better off simply drawing a ray diagram.

Although the main point here was to give an example of an
image of an image, figure i also shows an interesting case where we
need to make the distinction between *magnification*
and *angular magnification*.
\(\text{I}\)f you are looking at the moon through this telescope, then
the images \(\text{I}\) and \(\text{I}'\) are much *smaller* than the actual
moon. Otherwise, for example, image \(\text{I}\) would not fit inside
the telescope! However, these images are very close to your
eye compared to the actual moon. The small size of the image
has been more than compensated for by the shorter distance.
The important thing here is the amount of *angle*
within your field of view that the image covers, and it is
this angle that has been increased. The factor by which it
is increased is called the *angular magnification*, \(M_a\).

◊

Locate the images formed by two perpendicular mirrors, as in the figure. What happens if the mirrors are not perfectly perpendicular?

◊

Locate the images formed by the periscope.

It sounds a bit odd when a scientist refers to a theory as “beautiful,” but to those in the know it makes perfect sense. One mark of a beautiful theory is that it surprises us by being simple. The mathematical theory of lenses and curved mirrors gives us just such a surprise. We expect the subject to be complex because there are so many cases: a converging mirror forming a real image, a diverging lens that makes a virtual image, and so on for a total of six possibilities. If we want to predict the location of the images in all these situations, we might expect to need six different equations, and six more for predicting magnifications. Instead, it turns out that we can use just one equation for the location of the image and one equation for its magnification, and these two equations work in all the different cases with no changes except for plus and minus signs. This is the kind of thing the physicist Eugene Wigner referred to as “the unreasonable effectiveness of mathematics.” Sometimes we can find a deeper reason for this kind of unexpected simplicity, but sometimes it almost seems as if God went out of Her way to make the secrets of universe susceptible to attack by the human thought-tool called math.

We will now derive the equation for the location of a real image formed by a converging mirror. We assume for simplicity that the mirror is spherical, but actually this isn't a restrictive assumption, because any shallow, symmetric curve can be approximated by a sphere. The shape of the mirror can be specified by giving the location of its center, C. A deeply curved mirror is a sphere with a small radius, so C is close to it, while a weakly curved mirror has C farther away. Given the point O where the object is, we wish to find the point I where the image will be formed.

To locate an image, we need to track a minimum of two rays coming from the same point. Since we have proved in the previous chapter that this type of image is not distorted, we can use an on-axis point, O, on the object, as in figure a/1. The results we derive will also hold for off-axis points, since otherwise the image would have to be distorted, which we know is not true. We let one of the rays be the one that is emitted along the axis; this ray is especially easy to trace, because it bounces straight back along the axis again. As our second ray, we choose one that strikes the mirror at a distance of 1 from the axis. “One what?” asks the astute reader. The answer is that it doesn't really matter. When a mirror has shallow curvature, all the reflected rays hit the same point, so 1 could be expressed in any units you like. It could, for instance, be 1 cm, unless your mirror is smaller than 1 cm!

The only way to find out anything mathematical about the rays is to use the sole mathematical fact we possess concerning specular reflection: the incident and reflected rays form equal angles with respect to the normal, which is shown as a dashed line. Therefore the two angles shown in figure a/2 are the same, and skipping some straightforward geometry, this leads to the visually reasonable result that the two angles in figure a/3 are related as follows:

\[\begin{equation*}
\theta_i+\theta_o = \text{constant}
\end{equation*}\]

(Note that \(\theta_i\) and \(\theta_o\), which are measured from the image and the object, not from the eye like the angles we referred to in discussing angular magnification on page 756.) For example, move O farther from the mirror. The top angle in figure a/2 is increased, so the bottom angle must increase by the same amount, causing the image point, I, to move closer to the mirror. In terms of the angles shown in figure a/3, the more distant object has resulted in a smaller angle \(\theta_o\), while the closer image corresponds to a larger \(\theta_i;\) One angle increases by the same amount that the other decreases, so their sum remains constant. These changes are summarized in figure a/4.

The sum \(\theta_i+\theta_o\) is a constant. What does this constant represent? Geometrically, we interpret it as double the angle made by the dashed radius line. Optically, it is a measure of the strength of the mirror, i.e., how strongly the mirror focuses light, and so we call it the focal angle, \(\theta_f\),

\[\begin{equation*}
\theta_i+\theta_o = \theta_f .
\end{equation*}\]

Suppose, for example, that we wish to use a quick and dirty optical test to determine how strong a particular mirror is. We can lay it on the floor as shown in figure c, and use it to make an image of a lamp mounted on the ceiling overhead, which we assume is very far away compared to the radius of curvature of the mirror, so that the mirror intercepts only a very narrow cone of rays from the lamp. This cone is so narrow that its rays are nearly parallel, and \(\theta_o\) is nearly zero. The real image can be observed on a piece of paper. By moving the paper nearer and farther, we can bring the image into focus, at which point we know the paper is located at the image point. Since \(\theta_o\approx 0\), we have \(\theta_i\approx \theta_f\), and we can then determine this mirror's focal angle either by measuring \(\theta_i\) directly with a protractor, or indirectly via trigonometry. A strong mirror will bring the rays together to form an image close to the mirror, and these rays will form a blunt-angled cone with a large \(\theta_i\) and \(\theta_f\).

\(\triangleright\) The object and image angles are the same; the angle labeled \(\theta \) in the figure equals both of them. We therefore have \(\theta_i+\theta_o=\theta =\theta_f\). Comparing figures b and c, it is indeed plausible that the angles are related by a factor of two.

At this point, we could consider our work to be done. Typically, we know the strength of the mirror, and we want to find the image location for a given object location. Given the mirror's focal angle and the object location, we can determine \(\theta_o\) by trigonometry, subtract to find \(\theta_i=\theta_f-\theta_o\), and then do more trig to find the image location.

There is, however, a shortcut that can save us from doing so much work. Figure a/3 shows two right triangles whose legs of length 1 coincide and whose acute angles are \(\theta_o\) and \(\theta_i\). These can be related by trigonometry to the object and image distances shown in figure d:

\[\begin{equation*}
\tan \theta_o = 1/d_o \tan \theta_i = 1/d_i
\end{equation*}\]

Ever since chapter 2, we've been assuming small angles. For small angles, we can use the small-angle approximation \(\tan x\approx x\) (for \(x\) in radians), giving simply

\[\begin{equation*}
\theta_o = 1/d_o \theta_i = 1/d_i .
\end{equation*}\]

We likewise define a distance called the focal length, \(f\) according to \(\theta_f=1/f\). In figure b, \(f\) is the distance from the mirror to the place where the rays cross. We can now reexpress the equation relating the object and image positions as

\[\begin{equation*}
\frac{1}{f} = \frac{1}{d_i}+\frac{1}{d_o} .
\end{equation*}\]

Figure e summarizes the interpretation of the focal
length and focal angle.^{1}

Which form is better, \(\theta_f=\theta_i+\theta_o\) or
\(1/f=1/d_i+1/d_o?\) The angular form has in its favor its
simplicity and its straightforward visual interpretation,
but there are two reasons why we might prefer the second
version. First, the numerical values of the angles depend on
what we mean by “one unit” for the distance shown as 1 in
figure a/1. Second, it is usually easier to measure
distances rather than angles, so the distance form is more
convenient for number crunching. Neither form is superior
overall, and we will often need to use both to solve any
given problem.^{2}

Suppose we need to create a parallel beam of light, as in a searchlight. Where should we place the lightbulb? A parallel beam has zero angle between its rays, so \(\theta_i=0\). To place the lightbulb correctly, however, we need to know a distance, not an angle: the distance \(d_o\) between the bulb and the mirror. The problem involves a mixture of distances and angles, so we need to get everything in terms of one or the other in order to solve it. Since the goal is to find a distance, let's figure out the image distance corresponding to the given angle \(\theta_i=0\). These are related by \(d_i=1/\theta_i\), so we have \(d_i=\infty\). (Yes, dividing by zero gives infinity. Don't be afraid of infinity. Infinity is a useful problem-solving device.) Solving the distance equation for \(d_o\), we have

\[\begin{align*}
d_o &= (1/f - 1/d_i)^{-1} \\
&= (1/f - 0)^{-1} \\
&= f
\end{align*}\]

The bulb has to be placed at a distance from the mirror equal to its focal point.

An equation like \(d_i=1/\theta_i\) really doesn't make sense in terms of units. Angles are unitless, since radians aren't really units, so the right-hand side is unitless. We can't have a left-hand side with units of distance if the right-hand side of the same equation is unitless. This is an artifact of my cavalier statement that the conical bundles of rays spread out to a distance of 1 from the axis where they strike the mirror, without specifying the units used to measure this 1. In real life, optometrists define the thing we're calling \(\theta_i=1/d_i\) as the “dioptric strength” of a lens or mirror, and measure it in units of inverse meters (\(\text{m}^{-1}\)), also known as diopters (1 D=1 \(\text{m}^{-1}\)).

We have already discussed in the previous chapter how to find the magnification of a virtual image made by a curved mirror. The result is the same for a real image, and we omit the proof, which is very similar. In our new notation, the result is \(M=d_i/d_o\). A numerical example is given in subsection 12.3.2.

The equation \(d_i=(1/f-1/d_o)^{-1}\) can easily produce a negative result, but we have been thinking of \(d_i\) as a distance, and distances can't be negative. A similar problem occurs with \(\theta_i=\theta_f-\theta_o\) for \(\theta_o>\theta_f\). What's going on here?

The interpretation of the angular equation is straightforward. As we bring the object closer and closer to the image, \(\theta_o\) gets bigger and bigger, and eventually we reach a point where \(\theta_o=\theta_f\) and \(\theta_i=0\). This large object angle represents a bundle of rays forming a cone that is very broad, so broad that the mirror can no longer bend them back so that they reconverge on the axis. The image angle \(\theta_i=0\) represents an outgoing bundle of rays that are parallel. The outgoing rays never cross, so this is not a real image, unless we want to be charitable and say that the rays cross at infinity. If we go on bringing the object even closer, we get a virtual image.

To analyze the distance equation, let's look at a graph of \(d_i\) as a function of \(d_o\). The branch on the upper right corresponds to the case of a real image. Strictly speaking, this is the only part of the graph that we've proven corresponds to reality, since we never did any geometry for other cases, such as virtual images. As discussed in the previous section, making \(d_o\) bigger causes \(d_i\) to become smaller, and vice-versa.

Letting \(d_o\) be less than \(f\) is equivalent to \(\theta_o>\theta_f:\) a virtual image is produced on the far side of the mirror. This is the first example of Wigner's “unreasonable effectiveness of mathematics” that we have encountered in optics. Even though our proof depended on the assumption that the image was real, the equation we derived turns out to be applicable to virtual images, provided that we either interpret the positive and negative signs in a certain way, or else modify the equation to have different positive and negative signs.

Interpret the three places where, in physically realistic parts of the graph, the graph approaches one of the dashed lines. [This will come more naturally if you have learned the concept of limits in a math class.]

(answer in the back of the PDF version of the book)We can even apply the equation to a flat mirror. As a sphere gets bigger and bigger, its surface is more and more gently curved. The planet Earth is so large, for example, that we cannot even perceive the curvature of its surface. To represent a flat mirror, we let the mirror's radius of curvature, and its focal length, become infinite. Dividing by infinity gives zero, so we have

\[\begin{align*}
1/d_o &= -1/d_i ,\\
\text{or}
d_o &= -d_i .
\end{align*}\]

If we interpret the minus sign as indicating a virtual image on the far side of the mirror from the object, this makes sense.

It turns out that for any of the six possible combinations of real or virtual images formed by converging or diverging lenses or mirrors, we can apply equations of the form

\[\begin{gather*}
\theta_f = \theta_i+\theta_o \\
\text{and}
\frac{1}{f} = \frac{1}{d_i}+\frac{1}{d_o} ,
\end{gather*}\]

with only a modification of plus or minus signs. There are two possible approaches here. The approach we have been using so far is the more popular approach in American textbooks: leave the equation the same, but attach interpretations to the resulting negative or positive values of the variables. The trouble with this approach is that one is then forced to memorize tables of sign conventions, e.g., that the value of \(d_i\) should be negative when the image is a virtual image formed by a converging mirror. Positive and negative signs also have to be memorized for focal lengths. Ugh! It's highly unlikely that any student has ever retained these lengthy tables in his or her mind for more than five minutes after handing in the final exam in a physics course. Of course one can always look such things up when they are needed, but the effect is to turn the whole thing into an exercise in blindly plugging numbers into formulas.

As you have gathered by now, there is another method which I
think is better, and which I'll use throughout the rest of
this book. In this method, all distances and angles are
*positive by definition*, and we put in positive and
negative signs in the *equations* depending on the
situation. (I thought I was the first to invent this method,
but I've been told that this is known as the European sign
convention, and that it's fairly common in Europe.) Rather
than memorizing these signs, we start with the generic equations

\[\begin{align*}
\theta_f &= \pm \theta_i \pm \theta_o \\
\frac{1}{f} &= \pm\frac{1}{d_i}\pm\frac{1}{d_o} ,
\end{align*}\]

and then determine the signs by a two-step method that depends on ray diagrams. There are really only two signs to determine, not four; the signs in the two equations match up in the way you'd expect. The method is as follows:

1. Use ray diagrams to decide whether \(\theta_o\) and \(\theta_i\) vary in the same way or in opposite ways. (In other words, decide whether making \(\theta_o\) greater results in a greater value of \(\theta_i\) or a smaller one.) Based on this, decide whether the two signs in the angle equation are the same or opposite. If the signs are opposite, go on to step 2 to determine which is positive and which is negative.

2. If the signs are opposite, we need to decide which is the positive one and which is the negative. Since the focal angle is never negative, the smaller angle must be the one with a minus sign.

In step 1, many students have trouble drawing the ray diagram correctly. For simplicity, you should always do your diagram for a point on the object that is on the axis of the mirror, and let one of your rays be the one that is emitted along the axis and reflected straight back on itself, as in the figures in subsection 12.3.1. As shown in figure a/4 in subsection 12.3.1, there are four angles involved: two at the mirror, one at the object \((\theta_o)\), and one at the image \((\theta_i)\). Make sure to draw in the normal to the mirror so that you can see the two angles at the mirror. These two angles are equal, so as you change the object position, they fan out or fan in, like opening or closing a book. Once you've drawn this effect, you should easily be able to tell whether \(\theta_o\) and \(\theta_i\) change in the same way or in opposite ways.

Although focal lengths are always positive in the method used in this book, you should be aware that diverging mirrors and lenses are assigned negative focal lengths in the other method, so if you see a lens labeled \(f=-30\) cm, you'll know what it means.

\(\triangleright\) As shown in ray diagram g/1, \(d_i\) is less than \(d_o\). The magnification, \(M= d_i/d_o\), will be less than one, i.e., the image is actually reduced rather than magnified.

Apply the method outlined above for determining the plus and minus signs. Step 1: The object is the point on the opposite wall. As an experiment, g/2, move the object closer. I did these drawings using illustration software, but if you were doing them by hand, you'd want to make the scale much larger for greater accuracy. Also, although I split figure g into two separate drawings in order to make them easier to understand, you're less likely to make a mistake if you do them on top of each other.

The two angles at the mirror fan out from the normal. Increasing \(\theta_o\) has clearly made \(\theta_i\) larger as well. (All four angles got bigger.) There must be a cancellation of the effects of changing the two terms on the right in the same way, and the only way to get such a cancellation is if the two terms in the angle equation have opposite signs:

\[\begin{equation*}
\theta_f = + \theta_i - \theta_o
\end{equation*}\]

or

\[\begin{equation*}
\theta_f = - \theta_i + \theta_o .
\end{equation*}\]

Step 2: Now which is the positive term and which is negative? Since the image angle is bigger than the object angle, the angle equation must be

\[\begin{equation*}
\theta_f = \theta_i - \theta_o ,
\end{equation*}\]

in order to give a positive result for the focal angle. The signs of the distance equation behave the same way:

\[\begin{equation*}
\frac{1}{f} = \frac{1}{d_i}-\frac{1}{d_o} .
\end{equation*}\]

Solving for \(d_i\), we find

\[\begin{align*}
d_i &= \left(\frac{1}{f}+\frac{1}{d_o}\right)^{-1}\\
&= 2.1\ \text{m} .
\end{align*}\]

The image of the store is reduced by a factor of \(2.1/7.0=0.3\), i.e., it is smaller by 70%.

In the case of a real image, there is a shortcut for step 1, the determination of the signs. In a real image, the rays cross at both the object and the image. We can therefore time-reverse the ray diagram, so that all the rays are coming from the image and reconverging at the object. Object and image swap roles. Due to this time-reversal symmetry, the object and image cannot be treated differently in any of the equations, and they must therefore have the same signs. They are both positive, since they must add up to a positive result.

\myoptionalsubsection[2]{Aberrations}

An imperfection or distortion in an image is called an aberration. An aberration can be produced by a flaw in a lens or mirror, but even with a perfect optical surface some degree of aberration is unavoidable. To see why, consider the mathematical approximation we've been making, which is that the depth of the mirror's curve is small compared to \(d_o\) and \(d_i\). Since only a flat mirror can satisfy this shallow-mirror condition perfectly, any curved mirror will deviate somewhat from the mathematical behavior we derived by assuming that condition. There are two main types of aberration in curved mirrors, and these also occur with lenses.

(1) An object on the axis of the lens or mirror may be imaged correctly, but off-axis objects may be out of focus or distorted. In a camera, this type of aberration would show up as a fuzziness or warping near the sides of the picture when the center was perfectly focused. An example of this is shown in figure i, and in that particular example, the aberration is not a sign that the equipment was of low quality or wasn't right for the job but rather an inevitable result of trying to flatten a panoramic view; in the limit of a 360-degree panorama, the problem would be similar to the problem of representing the Earth's surface on a flat map, which can't be accomplished without distortion.

(2) The image may be sharp when the object is at certain distances and blurry when it is at other distances. The blurriness occurs because the rays do not all cross at exactly the same point. If we know in advance the distance of the objects with which the mirror or lens will be used, then we can optimize the shape of the optical surface to make in-focus images in that situation. For instance, a spherical mirror will produce a perfect image of an object that is at the center of the sphere, because each ray is reflected directly onto the radius along which it was emitted. For objects at greater distances, however, the focus will be somewhat blurry. In astronomy the objects being used are always at infinity, so a spherical mirror is a poor choice for a telescope. A different shape (a parabola) is better specialized for astronomy.

One way of decreasing aberration is to use a small-diameter mirror or lens, or block most of the light with an opaque screen with a hole in it, so that only light that comes in close to the axis can get through. Either way, we are using a smaller portion of the lens or mirror whose curvature will be more shallow, thereby making the shallow-mirror (or thin-lens) approximation more accurate. Your eye does this by narrowing down the pupil to a smaller hole. In a camera, there is either an automatic or manual adjustment, and narrowing the opening is called “stopping down.” The disadvantage of stopping down is that light is wasted, so the image will be dimmer or a longer exposure must be used.

What I would suggest you take away from this discussion for the sake of your general scientific education is simply an understanding of what an aberration is, why it occurs, and how it can be reduced, not detailed facts about specific types of aberrations.

Economists normally consider free markets to be the natural
way of judging the monetary value of something, but social
scientists also use questionnaires to gauge the relative
value of privileges, disadvantages, or possessions that
cannot be bought or sold. They ask people to *imagine*
that they could trade one thing for another and ask which
they would choose. One interesting result is that the
average light-skinned person in the U.S. would rather lose
an arm than suffer the racist treatment routinely endured by
African-Americans. Even more impressive is the value of
sight. Many prospective parents can imagine without too much
fear having a deaf child, but would have a far more
difficult time coping with raising a blind one.

So great is the value attached to sight that some have imbued it with mystical aspects. Joan of Arc saw visions, and my college has a “vision statement.” Christian fundamentalists who perceive a conflict between evolution and their religion have claimed that the eye is such a perfect device that it could never have arisen through a process as helter-skelter as evolution, or that it could not have evolved because half of an eye would be useless. In fact, the structure of an eye is fundamentally dictated by physics, and it has arisen separately by evolution somewhere between eight and 40 times, depending on which biologist you ask. We humans have a version of the eye that can be traced back to the evolution of a light-sensitive “eye spot” on the head of an ancient invertebrate. A sunken pit then developed so that the eye would only receive light from one direction, allowing the organism to tell where the light was coming from. (Modern flatworms have this type of eye.) The top of the pit then became partially covered, leaving a hole, for even greater directionality (as in the nautilus). At some point the cavity became filled with jelly, and this jelly finally became a lens, resulting in the general type of eye that we share with the bony fishes and other vertebrates. Far from being a perfect device, the vertebrate eye is marred by a serious design flaw due to the lack of planning or intelligent design in evolution: the nerve cells of the retina and the blood vessels that serve them are all in front of the light-sensitive cells, blocking part of the light. Squids and other molluscs, whose eyes evolved on a separate branch of the evolutionary tree, have a more sensible arrangement, with the light-sensitive cells out in front.

The fundamental physical phenomenon at work in the eye is
that when light crosses a boundary between two media (such
as air and the eye's jelly), part of its energy is
reflected, but part passes into the new medium. In the ray
model of light, we describe the original ray as splitting
into a reflected ray and a transmitted one (the one that
gets through the boundary). Of course the reflected ray goes
in a direction that is different from that of the original
one, according to the rules of reflection we have already
studied. More surprisingly --- and this is the crucial point
for making your eye focus light --- the transmitted ray is
bent somewhat as well. This bending phenomenon is called
*refraction*. The origin of the
word is the same as that of the word “fracture,” i.e., the
ray is bent or “broken.” (Keep in mind, however, that
light rays are not physical objects that can really be
“broken.”) Refraction occurs with all waves, not just light waves.

The actual anatomy of the eye, b, is quite complex, but in essence it is very much like every other optical device based on refraction. The rays are bent when they pass through the front surface of the eye, c. Rays that enter farther from the central axis are bent more, with the result that an image is formed on the retina. There is only one slightly novel aspect of the situation. In most human-built optical devices, such as a movie projector, the light is bent as it passes into a lens, bent again as it reemerges, and then reaches a focus beyond the lens. In the eye, however, the “screen” is inside the eye, so the rays are only refracted once, on entering the jelly, and never emerge again.

A common misconception is that the “lens” of the eye is what does the focusing. All the transparent parts of the eye are made of fairly similar stuff, so the dramatic change in medium is when a ray crosses from the air into the eye (at the outside surface of the cornea). This is where nearly all the refraction takes place. The lens medium differs only slightly in its optical properties from the rest of the eye, so very little refraction occurs as light enters and exits the lens. The lens, whose shape is adjusted by muscles attached to it, is only meant for fine-tuning the focus to form images of near or far objects.

What are the rules governing refraction? The first thing to observe is that just as with reflection, the new, bent part of the ray lies in the same plane as the normal (perpendicular) and the incident ray, d.

If you try shooting a beam of light at the boundary between two substances, say water and air, you'll find that regardless of the angle at which you send in the beam, the part of the beam in the water is always closer to the normal line, e. It doesn't matter if the ray is entering the water or leaving, so refraction is symmetric with respect to time-reversal, f.

If, instead of water and air, you try another combination of substances, say plastic and gasoline, again you'll find that the ray's angle with respect to the normal is consistently smaller in one and larger in the other. Also, we find that if substance A has rays closer to normal than in B, and B has rays closer to normal than in C, then A has rays closer to normal than C. This means that we can rank-order all materials according to their refractive properties. Isaac Newton did so, including in his list many amusing substances, such as “Danzig vitriol” and “a pseudo-topazius, being a natural, pellucid, brittle, hairy stone, of a yellow color.” Several general rules can be inferred from such a list:

- Vacuum lies at one end of the list. In refraction across the interface between vacuum and any other medium, the other medium has rays closer to the normal.
- Among gases, the ray gets closer to the normal if you increase the density of the gas by pressurizing it more.
- The refractive properties of liquid mixtures and solutions vary in a smooth and systematic manner as the proportions of the mixture are changed.
- Denser substances usually, but not always, have rays closer to the normal.

The second and third rules provide us with a method for measuring the density of an unknown sample of gas, or the concentration of a solution. The latter technique is very commonly used, and the CRC Handbook of Physics and Chemistry, for instance, contains extensive tables of the refractive properties of sugar solutions, cat urine, and so on.

The numerical rule governing refraction was discovered by Snell, who must have collected experimental data something like what is shown on this graph and then attempted by trial and error to find the right equation. The equation he came up with was

\[\begin{equation*}
\frac{\sin\theta_1}{\sin\theta_2} = \text{constant} .
\end{equation*}\]

The value of the constant would depend on the combination of media used. For instance, any one of the data points in the graph would have sufficed to show that the constant was 1.3 for an air-water interface (taking air to be substance 1 and water to be substance 2).

Snell further found that if media A and B gave a constant
\(K_{AB}\) and media B and C gave a constant \(K_{BC}\),
then refraction at an interface between A and C would be
described by a constant equal to the product, \(K_{AC}=K_{AB}K_{BC}\).
This is exactly what one would expect if the constant
depended on the ratio of some number characterizing one
medium to the number characteristic of the second medium.
This number is called the
*index of refraction* of the medium, written as \(n\) in
equations. Since measuring the angles would only allow him
to determine the *ratio* of the indices of refraction
of two media, Snell had to pick some medium and define it as
having \(n=1\). He chose to define vacuum as having \(n=1\).
(The index of refraction of air at normal atmospheric
pressure is 1.0003, so for most purposes it is a good
approximation to assume that air has \(n=1\).) He also had to
decide which way to define the ratio, and he chose to define
it so that media with their rays closer to the normal would
have larger indices of refraction. This had the advantage
that denser media would typically have higher indices of
refraction, and for this reason the index of refraction is
also referred to as the optical density. Written in terms of
indices of refraction, Snell's equation becomes

\[\begin{equation*}
\frac{\sin\theta_1}{\sin\theta_2} = \frac{n_2}{n_1} ,
\end{equation*}\]

but rewriting it in the form

\[\begin{equation*}
n_1 \sin \theta_1=n_2 \sin \theta_2
\end{equation*}\]

[relationship between angles of rays at the interface between media with indices of refraction \(n_1\) and \(n_2\); angles are defined with respect to the normal] makes us less likely to get the 1's and 2's mixed up, so this the way most people remember Snell's law. A few indices of refraction are given in the back of the book.

(1) What would the graph look like for two substances with the same index of refraction?

(2) Based on the graph, when does refraction at an air-water interface change the direction of a ray most strongly?

(answer in the back of the PDF version of the book)\(\triangleright\) The tricky part is that Snell's law refers to the angles with respect to the normal. Forgetting this is a very common mistake. The beam is at an angle of \(30°\) with respect to the normal in the water. Let's refer to the air as medium 1 and the water as 2. Solving Snell's law for \(\theta_1\), we find

\[\begin{equation*}
\theta_1 = \sin^{-1}\left(\frac{n_2}{n_1}\sin\theta_2\right) .
\end{equation*}\]

As mentioned above, air has an index of refraction very close to 1, and water's is about 1.3, so we find \(\theta_1=40°\). The angle \(\alpha \) is therefore \(50°\).

What neither Snell nor Newton knew was that there is a very simple interpretation of the index of refraction. This may come as a relief to the reader who is taken aback by the complex reasoning involving proportionalities that led to its definition. Later experiments showed that the index of refraction of a medium was inversely proportional to the speed of light in that medium. Since \(c\) is defined as the speed of light in vacuum, and \(n=1\) is defined as the index of refraction of vacuum, we have

\[\begin{equation*}
n=\frac{c}{v} .
\end{equation*}\]

[\(n=\) medium's index of refraction, \(v=\) speed of light in that medium, \(c=\) speed of light in a vacuum]

Many textbooks start with this as the definition of the index of refraction, although that approach makes the quantity's name somewhat of a mystery, and leaves students wondering why \(c/v\) was used rather than \(v/c\). It should also be noted that measuring angles of refraction is a far more practical method for determining \(n\) than direct measurement of the speed of light in the substance of interest.

Why should refraction be related to the speed of light? The mechanical model shown in the figure may help to make this more plausible. Suppose medium 2 is thick, sticky mud, which slows down the car. The car's right wheel hits the mud first, causing the right side of the car to slow down. This will cause the car to turn to the right until is moves far enough forward for the left wheel to cross into the mud. After that, the two sides of the car will once again be moving at the same speed, and the car will go straight.

Of course, light isn't a car. Why should a beam of light have anything resembling a “left wheel” and “right wheel?” After all, the mechanical model would predict that a motorcycle would go straight, and a motorcycle seems like a better approximation to a ray of light than a car. The whole thing is just a model, not a description of physical reality.

However intuitively appealing the mechanical model may be, light is a wave, and we should be using wave models to describe refraction. In fact Snell's law can be derived quite simply from wave concepts. Figure j shows the refraction of a water wave. The water in the upper left part of the tank is shallower, so the speed of the waves is slower there, and their wavelengths is shorter. The reflected part of the wave is also very faintly visible.

In the close-up view on the right, the dashed lines are normals to the interface. The two marked angles on the right side are both equal to \(\theta_1\), and the two on the left to \(\theta_2\).

Trigonometry gives

\[\begin{align*}
\sin \theta_1 &= \lambda_1/h \text{and} \\
\sin \theta_2 &= \lambda_2/h .
\end{align*}\]

Eliminating \(h\) by dividing the equations, we find

\[\begin{equation*}
\frac{\sin\theta_1}{\sin\theta_2}=\frac{\lambda_1}{\lambda_2} .
\end{equation*}\]

The frequencies of the two waves must be equal or else they would get out of step, so by \(v=f\lambda \) we know that their wavelengths are proportional to their velocities. Combining \(\lambda\propto v\) with \(v\propto 1/n\) gives \(\lambda\propto 1/n\), so we find

\[\begin{equation*}
\frac{\sin\theta_1}{\sin\theta_2}=\frac{n_2}{n_1} ,
\end{equation*}\]

which is one form of Snell's law.

Ocean waves are formed by winds, typically on the open sea, and the wavefronts are perpendicular to the direction of the wind that formed them. At the beach, however, you have undoubtedly observed that waves tend come in with their wavefronts very nearly (but not exactly) parallel to the shoreline. This is because the speed of water waves in shallow water depends on depth: the shallower the water, the slower the wave. Although the change from the fast-wave region to the slow-wave region is gradual rather than abrupt, there is still refraction, and the wave motion is nearly perpendicular to the normal in the slow region.

In general, the speed of light in a medium depends both on the medium and on the wavelength of the light. Another way of saying it is that a medium's index of refraction varies with wavelength. This is why a prism can be used to split up a beam of white light into a rainbow. Each wavelength of light is refracted through a different angle.

In section 6.2 we developed an equation for the percentage of the wave energy that is transmitted and the percentage reflected at a boundary between media. This was only done in the case of waves in one dimension, however, and rather than discuss the full three dimensional generalization it will be more useful to go into some qualitative observations about what happens. First, reflection happens only at the interface between two media, and two media with the same index of refraction act as if they were a single medium. Thus, at the interface between media with the same index of refraction, there is no reflection, and the ray keeps going straight. Continuing this line of thought, it is not surprising that we observe very little reflection at an interface between media with similar indices of refraction.

The next thing to note is that it is possible to have situations where no possible angle for the refracted ray can satisfy Snell's law. Solving Snell's law for \(\theta_2\), we find

\[\begin{equation*}
\theta_2 = \sin^{-1}\left(\frac{n_1}{n_2}\sin\theta_1\right) ,
\end{equation*}\]

and if \(n_1\) is greater than \(n_2\), then there will be large
values of \(\theta_1\) for which the quantity \((n_1/n_2)\sin\theta \)
is greater than one, meaning that your calculator
will flash an error message at you when you try to take the
inverse sine. What can happen physically in such a
situation? The answer is that all the light is reflected, so
there is no refracted ray. This phenomenon is known as
*total internal reflection*,
and is used in the fiber-optic cables that nowadays carry
almost all long-distance telephone calls. The electrical
signals from your phone travel to a switching center, where
they are converted from electricity into light. From there,
the light is sent across the country in a thin transparent
fiber. The light is aimed straight into the end of the
fiber, and as long as the fiber never goes through any turns
that are too sharp, the light will always encounter the edge
of the fiber at an angle sufficiently oblique to give total
internal reflection. If the fiber-optic cable is thick
enough, one can see an image at one end of whatever the
other end is pointed at.

Alternatively, a bundle of cables can be used, since a single thick cable is too hard to bend. This technique for seeing around corners is useful for making surgery less traumatic. Instead of cutting a person wide open, a surgeon can make a small “keyhole” incision and insert a bundle of fiber-optic cable (known as an endoscope) into the body.

Since rays at sufficiently large angles with respect to the normal may be completely reflected, it is not surprising that the relative amount of reflection changes depending on the angle of incidence, and is greatest for large angles of incidence.

◊

What index of refraction should a fish have in order to be invisible to other fish?

◊

Does a surgeon using an endoscope need a source of light inside the body cavity? If so, how could this be done without inserting a light bulb through the incision?

◊

A denser sample of a gas has a higher index of refraction than a less dense sample (i.e., a sample under lower pressure), but why would it not make sense for the index of refraction of a gas to be proportional to density?

◊

The earth's atmosphere gets thinner and thinner as you go higher in altitude. If a ray of light comes from a star that is below the zenith, what will happen to it as it comes into the earth's atmosphere?

◊

Does total internal reflection occur when light in a denser medium encounters a less dense medium, or the other way around? Or can it occur in either case?

Figures n/1 and n/2 show examples of lenses forming images. There is essentially nothing for you to learn about imaging with lenses that is truly new. You already know how to construct and use ray diagrams, and you know about real and virtual images. The concept of the focal length of a lens is the same as for a curved mirror. The equations for locating images and determining magnifications are of the same form. It's really just a question of flexing your mental muscles on a few examples. The following self-checks and discussion questions will get you started.

(1) In figures n/1 and n/2, classify the images as real or virtual.

(2) Glass has an index of refraction that is greater than that of air. Consider the topmost ray in figure n/1. Explain why the ray makes a slight left turn upon entering the lens, and another left turn when it exits.

(3) If the flame in figure n/2 was moved closer to the lens, what would happen to the location of the image?

(answer in the back of the PDF version of the book)
◊

In figures n/1 and n/2, the front and back surfaces are parallel to each other at the center of the lens. What will happen to a ray that enters near the center, but not necessarily along the axis of the lens? Draw a BIG ray diagram, and show a ray that comes from off axis.

*In discussion questions B-F, don't draw ultra-detailed
ray diagrams as in A.*

◊

Suppose you wanted to change the setup in figure n/1 so that the location of the actual flame in the figure would instead be occupied by an image of a flame. Where would you have to move the candle to achieve this? What about in n/2?

◊

There are three qualitatively different types of image formation that can occur with lenses, of which figures n/1 and n/2 exhaust only two. Figure out what the third possibility is. Which of the three possibilities can result in a magnification greater than one? Cf. problem 10, p. 799.

◊

Classify the examples shown in figure o according to the types of images delineated in discussion question C.

◊

In figures n/1 and n/2, the only rays drawn were those that happened to enter the lenses. Discuss this in relation to figure o.

◊

In the right-hand side of figure o, the image viewed through the lens is in focus, but the side of the rose that sticks out from behind the lens is not. Why?

\myoptionalsubsection[2]{The lensmaker's equation}

The focal length of a spherical mirror is simply \(r/2\), but we cannot expect the focal length of a lens to be given by pure geometry, since it also depends on the index of refraction of the lens. Suppose we have a lens whose front and back surfaces are both spherical. (This is no great loss of generality, since any surface with a sufficiently shallow curvature can be approximated with a sphere.) Then if the lens is immersed in a medium with an index of refraction of 1, its focal length is given approximately by

\[\begin{equation*}
f = \left[(n-1)\left|\frac{1}{r_1}\pm\frac{1}{r_2}\right|\right]^{-1} ,
\end{equation*}\]

where \(n\) is the index of refraction and \(r_1\) and \(r_2\) are the radii of curvature of the two surfaces of the lens. This is known as the lensmaker's equation. In my opinion it is not particularly worthy of memorization. The positive sign is used when both surfaces are curved outward or both are curved inward; otherwise a negative sign applies. The proof of this equation is left as an exercise to those readers who are sufficiently brave and motivated.

For most materials, we observe that the index of refraction depends slightly on wavelength, being highest at the blue end of the visible spectrum and lowest at the red. For example, white light disperses into a rainbow when it passes through a prism, q. Even when the waves involved aren't light waves, and even when refraction isn't of interest, the dependence of wave speed on wavelength is referred to as dispersion. Dispersion inside spherical raindrops is responsible for the creation of rainbows in the sky, and in an optical instrument such as the eye or a camera it is responsible for a type of aberration called chromatic aberration (subsection 12.3.3 and problem 28). As we'll see in subsection 13.3.2, dispersion causes a wave that is not a pure sine wave to have its shape distorted as it travels, and also causes the speed at which energy and information are transported by the wave to be different from what one might expect from a naive calculation. The microscopic reasons for dispersion of light in matter are discussed in optional subsection 12.4.6.

\myoptionalsubsection[2]{The principle of least time for refraction}

We have seen previously how the rules governing straight-line motion of light and reflection of light can be derived from the principle of least time. What about refraction? In the figure, it is indeed plausible that the bending of the ray serves to minimize the time required to get from a point A to point B. If the ray followed the unbent path shown with a dashed line, it would have to travel a longer distance in the medium in which its speed is slower. By bending the correct amount, it can reduce the distance it has to cover in the slower medium without going too far out of its way. It is true that Snell's law gives exactly the set of angles that minimizes the time required for light to get from one point to another. The proof of this fact is left as an exercise (problem 38, p. 804).

\myoptionalsubsection[4]{Microscopic description of refraction}

Given that the speed of light is different in different media, we've seen two different explanations (on p. 776 and in subsection 12.4.5 above) of why refraction must occur. What we haven't yet explained is why the speed of light does depend on the medium.

A good clue as to what's going on comes from the figure s. The relatively minor variation of the index of refraction within the visible spectrum was misleading. At certain specific frequencies, \(n\) exhibits wild swings in the positive and negative directions. After each such swing, we reach a new, lower plateau on the graph. These frequencies are resonances. For example, the visible part of the spectrum lies on the left-hand tail of a resonance at about \(2\times10^{15}\ \text{Hz}\), corresponding to the ultraviolet part of the spectrum. This resonance arises from the vibration of the electrons, which are bound to the nuclei as if by little springs. Because this resonance is narrow, the effect on visible-light frequencies is relatively small, but it is stronger at the blue end of the spectrum than at the red end. Near each resonance, not only does the index of refraction fluctuate wildly, but the glass becomes nearly opaque; this is because the vibration becomes very strong, causing energy to be dissipated as heat. The “staircase” effect is the same one visible in any resonance, e.g., figure k on p. 180: oscillators have a finite response for \(f \ll f_0\), but the response approaches zero for \(f \gg f_0\).

So far, we have a qualitative explanation of the frequency-variation of the loosely defined “strength” of the glass's effect on a light wave, but we haven't explained why the effect is observed as a change in speed, or why each resonance is an up-down swing rather than a single positive peak. To understand these effects in more detail, we need to consider the phase response of the oscillator. As shown in the bottom panel of figure j on p. 181, the phase response reverses itself as we pass through a resonance.

Suppose that a plane wave is normally incident on the left side of a thin sheet of glass, t/1, at \(f \ll f_0\). The light wave observed on the right side consists of a superposition of the incident wave consisting of \(\mathbf{E}_0\) and \(\mathbf{B}_0\) with a secondary wave \(\mathbf{E}^*\) and \(\mathbf{B}^*\) generated by the oscillating charges in the glass. Since the frequency is far below resonance, the response \(q\mathbf{x}\) of a vibrating charge \(q\) is in phase with the driving force \(\mathbf{E}_0\). The current is the derivative of this quantity, and therefore 90 degrees ahead of it in phase. The magnetic field generated by a sheet of current has been analyzed in subsection 11.2.1, and the result, shown in figure e on p. 666, is just what we would expect from the right-hand rule. We find, t/1, that the secondary wave is 90 degrees ahead of the incident one in phase. The incident wave still exists on the right side of the sheet, but it is superposed with the secondary one. Their addition is shown in t/2 using the complex number representation introduced in subsection 10.5.7. The superposition of the two fields lags behind the incident wave, which is the effect we would expect if the wave had traveled more slowly through the glass.

In the case \(f \gg 0\), the same analysis applies except that the phase of the secondary wave is reversed. The transmitted wave is advanced rather than retarded in phase. This explains the dip observed in figure s after each spike.

All of this is in accord with our understanding of relativity, ch. 7, in which we saw that the universal speed \(c\) was to be understood fundamentally as a conversion factor between the units used to measure time and space --- not as the speed of light. Since \(c\) isn't defined as the speed of light, it's of no fundamental importance whether light has a different speed in matter than it does in vacuum. In fact, the picture we've built up here is one in which all of our electromagnetic waves travel at \(c\); propagation at some other speed is only what appears to happen because of the superposition of the \((\mathbf{E}_0,\mathbf{B}_0)\) and \((\mathbf{E}^*,\mathbf{B}^*)\) waves, both of which move at \(c\).

But it is worrisome that at the frequencies where \(n\lt1\), the speed of the wave is greater than \(c\). According to special relativity, information is never supposed to be transmitted at speeds greater than \(c\), since this would produce situations in which a signal could be received before it was transmitted! This difficulty is resolved in subsection 13.3.2, where we show that there are two different velocities that can be defined for a wave in a dispersive medium, the phase velocity and the group velocity. The group velocity is the velocity at which information is transmitted, and it is always less than \(c\).

Electron microscopes can make images of individual atoms, but why will a visible-light microscope never be able to? Stereo speakers create the illusion of music that comes from a band arranged in your living room, but why doesn't the stereo illusion work with bass notes? Why are computer chip manufacturers investing billions of dollars in equipment to etch chips with x-rays instead of visible light?

The answers to all of these questions have to do with the subject of wave optics. So far this book has discussed the interaction of light waves with matter, and its practical applications to optical devices like mirrors, but we have used the ray model of light almost exclusively. Hardly ever have we explicitly made use of the fact that light is an electromagnetic wave. We were able to get away with the simple ray model because the chunks of matter we were discussing, such as lenses and mirrors, were thousands of times larger than a wavelength of light. We now turn to phenomena and devices that can only be understood using the wave model of light.

Figure a shows a typical problem in wave optics, enacted with water waves. It may seem surprising that we don't get a simple pattern like figure b, but the pattern would only be that simple if the wavelength was hundreds of times shorter than the distance between the gaps in the barrier and the widths of the gaps.

Wave optics is a broad subject, but this example will help us to pick out a reasonable set of restrictions to make things more manageable:

(1) We restrict ourselves to cases in which a wave travels through a uniform medium, encounters a certain area in which the medium has different properties, and then emerges on the other side into a second uniform region.

(2) We assume that the incoming wave is a nice tidy sine-wave pattern with wavefronts that are lines (or, in three dimensions, planes).

(3) In figure a we can see that the wave pattern immediately beyond the barrier is rather complex, but farther on it sorts itself out into a set of wedges separated by gaps in which the water is still. We will restrict ourselves to studying the simpler wave patterns that occur farther away, so that the main question of interest is how intense the outgoing wave is at a given angle.

The kind of phenomenon described by restriction (1) is
called *diffraction*. Diffraction
can be defined as the behavior of a wave when it encounters
an obstacle or a nonuniformity in its medium. In general,
diffraction causes a wave to bend around obstacles and make
patterns of strong and weak waves radiating out beyond the
obstacle. Understanding diffraction is the central problem
of wave optics. If you understand diffraction, even the
subset of diffraction problems that fall within restrictions
(2) and (3), the rest of wave optics is icing on the cake.

Diffraction can be used to find the structure of an unknown diffracting object: even if the object is too small to study with ordinary imaging, it may be possible to work backward from the diffraction pattern to learn about the object. The structure of a crystal, for example, can be determined from its x-ray diffraction pattern.

Diffraction can also be a bad thing. In a telescope, for example, light waves are diffracted by all the parts of the instrument. This will cause the image of a star to appear fuzzy even when the focus has been adjusted correctly. By understanding diffraction, one can learn how a telescope must be designed in order to reduce this problem --- essentially, it should have the biggest possible diameter.

There are two ways in which restriction (2) might commonly be violated. First, the light might be a mixture of wavelengths. If we simply want to observe a diffraction pattern or to use diffraction as a technique for studying the object doing the diffracting (e.g., if the object is too small to see with a microscope), then we can pass the light through a colored filter before diffracting it.

A second issue is that light from sources such as the sun or
a lightbulb does not consist of a nice neat plane wave,
except over very small regions of space. Different parts of
the wave are out of step with each other, and the wave is
referred to as *incoherent*. One way
of dealing with this is shown in figure c. After filtering
to select a certain wavelength of red light, we pass the
light through a small pinhole. The region of the light that
is intercepted by the pinhole is so small that one part of
it is not out of step with another. Beyond the pinhole,
light spreads out in a spherical wave; this is analogous to
what happens when you speak into one end of a paper towel
roll and the sound waves spread out in all directions from
the other end. By the time the spherical wave gets to the
double slit it has spread out and reduced its curvature, so
that we can now think of it as a simple plane wave.

If this seems laborious, you may be relieved to know that modern technology gives us an easier way to produce a single-wavelength, coherent beam of light: the laser.

The parts of the final image on the screen in c are called diffraction fringes. The center of each fringe is a point of maximum brightness, and halfway between two fringes is a minimum.

◊

Why would x-rays rather than visible light be used to find the structure of a crystal? Sound waves are used to make images of fetuses in the womb. What would influence the choice of wavelength?

This chapter has “optics” in its title, so it is nominally about light, but we started out with an example involving water waves. Water waves are certainly easier to visualize, but is this a legitimate comparison? In fact the analogy works quite well, despite the fact that a light wave has a wavelength about a million times shorter. This is because diffraction effects scale uniformly. That is, if we enlarge or reduce the whole diffraction situation by the same factor, including both the wavelengths and the sizes of the obstacles the wave encounters, the result is still a valid solution.

This is unusually simple behavior! In subsection 0.2.2 we saw many examples of more complex scaling, such as the impossibility of bacteria the size of dogs, or the need for an elephant to eliminate heat through its ears because of its small surface-to-volume ratio, whereas a tiny shrew's life-style centers around conserving its body heat.

Of course water waves and light waves differ in many ways, not just in scale, but the general facts you will learn about diffraction are applicable to all waves. In some ways it might have been more appropriate to insert this chapter after section 6.2 on bounded waves, but many of the important applications are to light waves, and you would probably have found these much more difficult without any background in optics.

Another way of stating the simple scaling behavior of diffraction is that the diffraction angles we get depend only on the unitless ratio \(\lambda \)/d, where \(\lambda\) is the wavelength of the wave and \(d\) is some dimension of the diffracting objects, e.g., the center-to-center spacing between the slits in figure a. If, for instance, we scale up both \(\lambda \) and \(d\) by a factor of 37, the ratio \(\lambda /d\) will be unchanged.

The only reason we don't usually notice diffraction of light
in everyday life is that we don't normally deal with objects
that are comparable in size to a wavelength of visible
light, which is about a millionth of a meter. Does this mean
that wave optics contradicts ray optics, or that wave optics
sometimes gives wrong results? No. If you hold three fingers
out in the sunlight and cast a shadow with them, *either*
wave optics or ray optics can be used to predict the
straightforward result: a shadow pattern with two bright
lines where the light has gone through the gaps between your
fingers. Wave optics is a more general theory than ray
optics, so in any case where ray optics is valid, the two
theories will agree. This is an example of a general idea
enunciated by the physicist Niels Bohr,
called the *correspondence
principle: * when flaws in a physical theory lead to the
creation of a new and more general theory, the new theory
must still agree with the old theory within its more
restricted area of applicability. After all, a theory is
only created as a way of describing experimental observations.
If the original theory had not worked in any cases at all,
it would never have become accepted.

In the case of optics, the correspondence principle tells us that when \(\lambda /d\) is small, both the ray and the wave model of light must give approximately the same result. Suppose you spread your fingers and cast a shadow with them using a coherent light source. The quantity \(\lambda /d\) is about \(10^{-4}\), so the two models will agree very closely. (To be specific, the shadows of your fingers will be outlined by a series of light and dark fringes, but the angle subtended by a fringe will be on the order of \(10^{-4}\) radians, so they will be invisible and washed out by the natural fuzziness of the edges of sun-shadows, caused by the finite size of the sun.)

What kind of wavelength would an electromagnetic wave have to have in order to diffract dramatically around your body? Does this contradict the correspondence principle?

(answer in the back of the PDF version of the book)Returning to the example of double-slit diffraction, f,
note the strong visual impression of two overlapping sets of
concentric semicircles. This is an example of *Huygens' principle*, named after a Dutch physicist
and astronomer. (The first syllable rhymes with “boy.”)
Huygens' principle states that any wavefront can be broken
down into many small side-by-side wave peaks, g, which
then spread out as circular ripples, h, and by the
principle of superposition, the result of adding up these
sets of ripples must give the same result as allowing the
wave to propagate forward, i. In the case of sound or
light waves, which propagate in three dimensions, the
“ripples” are actually spherical rather than circular, but
we can often imagine things in two dimensions for simplicity.

In double-slit diffraction the application of Huygens' principle is visually convincing: it is as though all the sets of ripples have been blocked except for two. It is a rather surprising mathematical fact, however, that Huygens' principle gives the right result in the case of an unobstructed linear wave, h and i. A theoretically infinite number of circular wave patterns somehow conspire to add together and produce the simple linear wave motion with which we are familiar.

Since Huygens' principle is equivalent to the principle of superposition, and superposition is a property of waves, what Huygens had created was essentially the first wave theory of light. However, he imagined light as a series of pulses, like hand claps, rather than as a sinusoidal wave.

The history is interesting. Isaac Newton loved the atomic theory of matter so much that he searched enthusiastically for evidence that light was also made of tiny particles. The paths of his light particles would correspond to rays in our description; the only significant difference between a ray model and a particle model of light would occur if one could isolate individual particles and show that light had a “graininess” to it. Newton never did this, so although he thought of his model as a particle model, it is more accurate to say he was one of the builders of the ray model.

Almost all that was known about reflection and refraction of light could be interpreted equally well in terms of a particle model or a wave model, but Newton had one reason for strongly opposing Huygens' wave theory. Newton knew that waves exhibited diffraction, but diffraction of light is difficult to observe, so Newton believed that light did not exhibit diffraction, and therefore must not be a wave. Although Newton's criticisms were fair enough, the debate also took on the overtones of a nationalistic dispute between England and continental Europe, fueled by English resentment over Leibniz's supposed plagiarism of Newton's calculus. Newton wrote a book on optics, and his prestige and political prominence tended to discourage questioning of his model.

Thomas Young (1773-1829) was the person
who finally, a hundred years later, did a careful search for
wave interference effects with light and analyzed the
results correctly. He observed double-slit diffraction of
light as well as a variety of other diffraction effects, all
of which showed that light exhibited wave interference
effects, and that the wavelengths of visible light waves
were extremely short. The crowning achievement was the
demonstration by the experimentalist Heinrich Hertz and the theorist James Clerk
Maxwell that light was an
*electromagnetic* wave. Maxwell is said to have related
his discovery to his wife one starry evening and told her
that she was the only other person in the world who knew what starlight was.

Let's now analyze double-slit diffraction, k, using Huygens' principle. The most interesting question is how to compute the angles such as X and Z where the wave intensity is at a maximum, and the in-between angles like Y where it is minimized. Let's measure all our angles with respect to the vertical center line of the figure, which was the original direction of propagation of the wave.

If we assume that the width of the slits is small (on the order of the wavelength of the wave or less), then we can imagine only a single set of Huygens ripples spreading out from each one, l. White lines represent peaks, black ones troughs. The only dimension of the diffracting slits that has any effect on the geometric pattern of the overlapping ripples is then the center-to-center distance, \(d\), between the slits.

We know from our discussion of the scaling of diffraction that there must be some equation that relates an angle like \(\theta_Z\) to the ratio \(\lambda /d\),

\[\begin{equation*}
\frac{\lambda}{d} \leftrightarrow \theta_Z .
\end{equation*}\]

If the equation for \(\theta_Z\) depended on some other expression such as \(\lambda +d\) or \(\lambda^2/d\), then it would change when we scaled \(\lambda \) and \(d\) by the same factor, which would violate what we know about the scaling of diffraction.

Along the central maximum line, X, we always have positive waves coinciding with positive ones and negative waves coinciding with negative ones. (I have arbitrarily chosen to take a snapshot of the pattern at a moment when the waves emerging from the slit are experiencing a positive peak.) The superposition of the two sets of ripples therefore results in a doubling of the wave amplitude along this line. There is constructive interference. This is easy to explain, because by symmetry, each wave has had to travel an equal number of wavelengths to get from its slit to the center line, m: Because both sets of ripples have ten wavelengths to cover in order to reach the point along direction X, they will be in step when they get there.

At the point along direction Y shown in the same figure, one wave has traveled ten wavelengths, and is therefore at a positive extreme, but the other has traveled only nine and a half wavelengths, so it at a negative extreme. There is perfect cancellation, so points along this line experience no wave motion.

But the distance traveled does not have to be equal in order to get constructive interference. At the point along direction Z, one wave has gone nine wavelengths and the other ten. They are both at a positive extreme.

At a point half a wavelength below the point marked along direction X, carry out a similar analysis.

(answer in the back of the PDF version of the book)To summarize, we will have perfect constructive interference at any point where the distance to one slit differs from the distance to the other slit by an integer number of wavelengths. Perfect destructive interference will occur when the number of wavelengths of path length difference equals an integer plus a half.

Now we are ready to find the equation that predicts the angles of the maxima and minima. The waves travel different distances to get to the same point in space, n. We need to find whether the waves are in phase (in step) or out of phase at this point in order to predict whether there will be constructive interference, destructive interference, or something in between.

One of our basic assumptions in this chapter is that we will only be dealing with the diffracted wave in regions very far away from the object that diffracts it, so the triangle is long and skinny. Most real-world examples with diffraction of light, in fact, would have triangles with even skinner proportions than this one. The two long sides are therefore very nearly parallel, and we are justified in drawing the right triangle shown in figure o, labeling one leg of the right triangle as the difference in path length , \(L-L'\), and labeling the acute angle as \(\theta \). (In reality this angle is a tiny bit greater than the one labeled \(\theta \) in figure n.)

The difference in path length is related to \(d\) and \(\theta \) by the equation

\[\begin{equation*}
\frac{L-L'}{d} = \sin \theta .
\end{equation*}\]

Constructive interference will result in a maximum at angles for which \(L-L'\) is an integer number of wavelengths,

\[\begin{multline*}
L-L' = m\lambda .
\shoveright{\text{[condition for a maximum;}}\\
\text{$m$ is an integer]}
\end{multline*}\]

Here \(m\) equals 0 for the central maximum, \(-1\) for the first maximum to its left, \(+2\) for the second maximum on the right, etc. Putting all the ingredients together, we find \(m\lambda/d=\sin \theta \), or

\[\begin{multline*}
\frac{\lambda}{d} = \frac{\sin\theta}{m} .
\shoveright{\text{[condition for a maximum;}}\\
\text{$m$ is an integer]}
\end{multline*}\]

Similarly, the condition for a minimum is

\[\begin{multline*}
\frac{\lambda}{d} = \frac{\sin\theta}{m} .
\shoveright{\text{[condition for a minimum;}}\\
\text{$m$ is an integer plus $1/2$]}
\end{multline*}\]

That is, the minima are about halfway between the maxima.

As expected based on scaling, this equation relates angles to the unitless ratio \(\lambda /d\). Alternatively, we could say that we have proven the scaling property in the special case of double-slit diffraction. It was inevitable that the result would have these scaling properties, since the whole proof was geometric, and would have been equally valid when enlarged or reduced on a photocopying machine!

Counterintuitively, this means that a diffracting object with smaller dimensions produces a bigger diffraction pattern, p.

Blue light has a shorter wavelength than red. For a given double-slit spacing \(d\), the smaller value of \(\lambda /d\) for leads to smaller values of \(\sin \theta \), and therefore to a more closely spaced set of diffraction fringes, (g)

Let's also consider how the equations for double-slit diffraction relate to the correspondence principle. When the ratio \(\lambda /d\) is very small, we should recover the case of simple ray optics. Now if \(\lambda /d\) is small, \(\sin\theta \) must be small as well, and the spacing between the diffraction fringes will be small as well. Although we have not proven it, the central fringe is always the brightest, and the fringes get dimmer and dimmer as we go farther from it. For small values of \(\lambda /d\), the part of the diffraction pattern that is bright enough to be detectable covers only a small range of angles. This is exactly what we would expect from ray optics: the rays passing through the two slits would remain parallel, and would continue moving in the \(\theta =0\) direction. (In fact there would be images of the two separate slits on the screen, but our analysis was all in terms of angles, so we should not expect it to address the issue of whether there is structure within a set of rays that are all traveling in the \(\theta =0\) direction.)

\[\begin{equation*}
\frac{\lambda}{d} = \frac{\theta}{m} ,
\end{equation*}\]

which can be solved for \(\theta \) to give
\[\begin{equation*}
\theta = \frac{m\lambda}{d} .
\end{equation*}\]

The difference in angle between successive fringes is the
change in \(\theta \) that results from changing \(m\)
by plus or minus one,
\[\begin{equation*}
\Delta\theta = \frac{\lambda}{d} .
\end{equation*}\]

For example, if we write \(\theta_7\) for the angle of the
seventh bright fringe on one side of the central maximum and
\(\theta_8\) for the neighboring one, we have
\[\begin{align*}
\theta_8-\theta_7 &= \frac{8\lambda}{d}-\frac{7\lambda}{d}\\
&= \frac{\lambda}{d} ,
\end{align*}\]

and similarly for any other neighboring pair of fringes.
Although the equation \(\lambda /d=\sin \theta /m\) is only valid for a double slit, it is can still be a guide to our thinking even if we are observing diffraction of light by a virus or a flea's leg: it is always true that

(1) large values of \(\lambda /d\) lead to a broad diffraction pattern, and

(2) diffraction patterns are repetitive.

In many cases the equation looks just like \(\lambda /d =\sin \theta /m\) but with an extra numerical factor thrown in, and with \(d\) interpreted as some other dimension of the object, e.g., the diameter of a piece of wire.

Suppose we replace a double slit with a triple slit, s. We can think of this as a third repetition of the structures that were present in the double slit. Will this device be an improvement over the double slit for any practical reasons?

The answer is yes, as can be shown using figure u. For ease of visualization, I have violated our usual rule of only considering points very far from the diffracting object. The scale of the drawing is such that a wavelengths is one cm. In u/1, all three waves travel an integer number of wavelengths to reach the same point, so there is a bright central spot, as we would expect from our experience with the double slit. In figure u/2, we show the path lengths to a new point. This point is farther from slit A by a quarter of a wavelength, and correspondingly closer to slit C. The distance from slit B has hardly changed at all. Because the paths lengths traveled from slits A and C differ by half a wavelength, there will be perfect destructive interference between these two waves. There is still some uncanceled wave intensity because of slit B, but the amplitude will be three times less than in figure u/1, resulting in a factor of 9 decrease in brightness. Thus, by moving off to the right a little, we have gone from the bright central maximum to a point that is quite dark.

Now let's compare with what would have happened if slit C had been covered, creating a plain old double slit. The waves coming from slits A and B would have been out of phase by 0.23 wavelengths, but this would not have caused very severe interference. The point in figure u/2 would have been quite brightly lit up.

To summarize, we have found that adding a third slit narrows down the central fringe dramatically. The same is true for all the other fringes as well, and since the same amount of energy is concentrated in narrower diffraction fringes, each fringe is brighter and easier to see, t.

This is an example of a more general fact about diffraction: if some feature of the diffracting object is repeated, the locations of the maxima and minima are unchanged, but they become narrower.

Taking this reasoning to its logical conclusion, a diffracting object with thousands of slits would produce extremely narrow fringes. Such an object is called a diffraction grating.

If we use only a single slit, is there diffraction? If the slit is not wide compared to a wavelength of light, then we can approximate its behavior by using only a single set of Huygens ripples. There are no other sets of ripples to add to it, so there are no constructive or destructive interference effects, and no maxima or minima. The result will be a uniform spherical wave of light spreading out in all directions, like what we would expect from a tiny lightbulb. We could call this a diffraction pattern, but it is a completely featureless one, and it could not be used, for instance, to determine the wavelength of the light, as other diffraction patterns could.

All of this, however, assumes that the slit is narrow compared to a wavelength of light. If, on the other hand, the slit is broader, there will indeed be interference among the sets of ripples spreading out from various points along the opening. Figure v shows an example with water waves, and figure w with light.

How does the wavelength of the waves compare with the width of the slit in figure v?

(answer in the back of the PDF version of the book)We will not go into the details of the analysis of single-slit diffraction, but let us see how its properties can be related to the general things we've learned about diffraction. We know based on scaling arguments that the angular sizes of features in the diffraction pattern must be related to the wavelength and the width, \(a\), of the slit by some relationship of the form

\[\begin{equation*}
\frac{\lambda}{a} \leftrightarrow \theta .
\end{equation*}\]

This is indeed true, and for instance the angle between the maximum of the central fringe and the maximum of the next fringe on one side equals \(1.5\lambda/a\). Scaling arguments will never produce factors such as the 1.5, but they tell us that the answer must involve \(\lambda /a\), so all the familiar qualitative facts are true. For instance, shorter-wavelength light will produce a more closely spaced diffraction pattern.

An important scientific example of single-slit diffraction is in telescopes. Images of individual stars, as in figure y, are a good way to examine diffraction effects, because all stars except the sun are so far away that no telescope, even at the highest magnification, can image their disks or surface features. Thus any features of a star's image must be due purely to optical effects such as diffraction. A prominent cross appears around the brightest star, and dimmer ones surround the dimmer stars. Something like this is seen in most telescope photos, and indicates that inside the tube of the telescope there were two perpendicular struts or supports. Light diffracted around these struts. You might think that diffraction could be eliminated entirely by getting rid of all obstructions in the tube, but the circles around the stars are diffraction effects arising from single-slit diffraction at the mouth of the telescope's tube! (Actually we have not even talked about diffraction through a circular opening, but the idea is the same.) Since the angular sizes of the diffracted images depend on \(\lambda \)/a, the only way to improve the resolution of the images is to increase the diameter, \(a\), of the tube. This is one of the main reasons (in addition to light-gathering power) why the best telescopes must be very large in diameter.

What would this imply about radio telescopes as compared with visible-light telescopes?

(answer in the back of the PDF version of the book)Double-slit diffraction is easier to understand conceptually than single-slit diffraction, but if you do a double-slit diffraction experiment in real life, you are likely to encounter a complicated pattern like figure aa/1, rather than the simpler one, 2, you were expecting. This is because the slits are fairly big compared to the wavelength of the light being used. We really have two different distances in our pair of slits: \(d\), the distance between the slits, and \(w\), the width of each slit. Remember that smaller distances on the object the light diffracts around correspond to larger features of the diffraction pattern. The pattern 1 thus has two spacings in it: a short spacing corresponding to the large distance \(d\), and a long spacing that relates to the small dimension \(w\).

◊

Why is it optically impossible for bacteria to evolve eyes that use visible light to form images?

\myoptionalsubsection[2]{The principle of least time}

In subsection 12.1.5 and 12.4.5, we saw how in the ray model of light, both refraction and reflection can be described in an elegant and beautiful way by a single principle, the principle of least time. We can now justify the principle of least time based on the wave model of light. Consider an example involving reflection, ab. Starting at point A, Huygens' principle for waves tells us that we can think of the wave as spreading out in all directions. Suppose we imagine all the possible ways that a ray could travel from A to B. We show this by drawing 25 possible paths, of which the central one is the shortest. Since the principle of least time connects the wave model to the ray model, we should expect to get the most accurate results when the wavelength is much shorter than the distances involved --- for the sake of this numerical example, let's say that a wavelength is 1/10 of the shortest reflected path from A to B. The table, 2, shows the distances traveled by the 25 rays.

Note how similar are the distances traveled by the group of 7 rays, indicated with a bracket, that come closest to obeying the principle of least time. If we think of each one as a wave, then all 7 are again nearly in phase at point B. However, the rays that are farther from satisfying the principle of least time show more rapidly changing distances; on reuniting at point B, their phases are a random jumble, and they will very nearly cancel each other out. Thus, almost none of the wave energy delivered to point B goes by these longer paths. Physically we find, for instance, that a wave pulse emitted at A is observed at B after a time interval corresponding very nearly to the shortest possible path, and the pulse is not very “smeared out” when it gets there. The shorter the wavelength compared to the dimensions of the figure, the more accurate these approximate statements become.

Instead of drawing a finite number of rays, such 25, what happens if we think of the angle, \(\theta \), of emission of the ray as a continuously varying variable? Minimizing the distance \(L\) requires

\[\begin{equation*}
\frac{dL}{d\theta} = 0 .
\end{equation*}\]

Because \(L\) is changing slowly in the vicinity of the angle that satisfies the principle of least time, all the rays that come out close to this angle have very nearly the same \(L\), and remain very nearly in phase when they reach B. This is the basic reason why the discrete table, ab/2, turned out to have a group of rays that all traveled nearly the same distance.

As discussed in subsection 12.1.5, the principle of least time is
really a principle of least *or greatest* time. This
makes perfect sense, since \(dL/d \theta =0\) can in general
describe either a minimum or a maximum

The principle of least time is very general. It does not apply just to refraction and reflection --- it can even be used to prove that light rays travel in a straight line through empty space, without taking detours! This general approach to wave motion was used by Richard Feynman, one of the pioneers who in the 1950's reconciled quantum mechanics with relativity. A very readable explanation is given in a book Feynman wrote for laypeople, QED: The Strange Theory of Light and Matter.

\begin{homeworkforcelabel}{diffuse}{1}{}{1}Draw a ray diagram showing why a small light source (a candle, say) produces sharper shadows than a large one (e.g., a long fluorescent bulb). \end{homeworkforcelabel}

\begin{homeworkforcelabel}{gps}{1}{}{2} A Global Positioning System (GPS) receiver is a device that lets you figure out where you are by receiving timed radio signals from satellites. It works by measuring the travel time for the signals, which is related to the distance between you and the satellite. By finding the ranges to several different satellites in this way, it can pin down your location in three dimensions to within a few meters. How accurate does the measurement of the time delay have to be to determine your position to this accuracy? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{lightwithatomwavelength}{1}{}{3}Estimate the frequency of an electromagnetic wave whose wavelength is similar in size to an atom (about a nm). Referring back to figure o on p. 705, in what part of the electromagnetic spectrum would such a wave lie (infrared, gamma-rays, ...)? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{stealth}{1}{}{4}The Stealth bomber is designed with flat, smooth surfaces. Why would this make it difficult to detect via radar? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{alien-pool}{1}{}{5}The natives of planet Wumpus play pool using light rays on an eleven-sided table with mirrors for bumpers, shown in the figure on the next page. Trace this shot accurately with a ruler to reveal the hidden message. To get good enough accuracy, you'll need to photocopy the page (or download the book and print the page) and construct each reflection using a protractor. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{parabolic}{1}{}{6}The figure on the next page shows a curved (parabolic) mirror, with three parallel light rays coming toward it. One ray is approaching along the mirror's center line. (a) Trace the drawing accurately, and continue the light rays until they are about to undergo their second reflection. To get good enough accuracy, you'll need to photocopy the page (or download the book and print the page) and draw in the normal at each place where a ray is reflected. What do you notice? (b) Make up an example of a practical use for this device. (c) How could you use this mirror with a small lightbulb to produce a parallel beam of light rays going off to the right? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{mirrorspeed}{1}{}{7}(answer check available at lightandmatter.com) A man is walking at 1.0 m/s directly towards a flat mirror. At what speed is his separation from his image decreasing? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{belowthebelt}{1}{}{8}If a mirror on a wall is only big enough for you to see yourself from your head down to your waist, can you see your entire body by backing up? Test this experimentally and come up with an explanation for your observations, including a ray diagram.

Note that when you do the experiment, it's easy to confuse
yourself if the mirror is even a tiny bit off of vertical.
One way to check yourself is to artificially lower the top
of the mirror by putting a piece of tape or a post-it note
where it blocks your view of the top of your head. You can
then check whether you are able to see more of yourself both
above *and* below by backing up.
\end{homeworkforcelabel}

\begin{homeworkforcelabel}{convexraytracing}{1}{}{9} In section 12.2 we've only done examples of mirrors with hollowed-out shapes (called concave mirrors). Now draw a ray diagram for a curved mirror that has a bulging outward shape (called a convex mirror). (a) How does the image's distance from the mirror compare with the actual object's distance from the mirror? From this comparison, determine whether the magnification is greater than or less than one. (b) Is the image real, or virtual? Could this mirror ever make the other type of image? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{listmirrorimages}{1}{}{10}As discussed in question 9, there are two types of curved mirrors, concave and convex. Make a list of all the possible combinations of types of images (virtual or real) with types of mirrors (concave and convex). (Not all of the four combinations are physically possible.) Now for each one, use ray diagrams to determine whether increasing the distance of the object from the mirror leads to an increase or a decrease in the distance of the image from the mirror.

Draw BIG ray diagrams! Each diagram should use up about half a page of paper.

Some tips: To draw a ray diagram, you need two rays. For one of these, pick the ray that comes straight along the mirror's axis, since its reflection is easy to draw. After you draw the two rays and locate the image for the original object position, pick a new object position that results in the same type of image, and start a new ray diagram, in a different color of pen, right on top of the first one. For the two new rays, pick the ones that just happen to hit the mirror at the same two places; this makes it much easier to get the result right without depending on extreme accuracy in your ability to draw the reflected rays. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{telescopemovehead}{1}{}{11}If the user of an astronomical telescope moves her head closer to or farther away from the image she is looking at, does the magnification change? Does the angular magnification change? Explain. (For simplicity, assume that no eyepiece is being used.) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{invertedforehead}{1}{}{12}In figure g/2 in on page 754, only the image of my forehead was located by drawing rays. Either photocopy the figure or download the book and print out the relevant page. On this copy of the figure, make a new set of rays coming from my chin, and locate its image. To make it easier to judge the angles accurately, draw rays from the chin that happen to hit the mirror at the same points where the two rays from the forehead were shown hitting it. By comparing the locations of the chin's image and the forehead's image, verify that the image is actually upside-down, as shown in the original figure. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{rays-cross-but-no-image}{1}{}{13}The figure shows four points where rays cross. Of these, which are image points? Explain. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{moose}{1}{}{14}Here's a game my kids like to play. I sit next to a sunny window, and the sun reflects from the glass on my watch, making a disk of light on the wall or floor, which they pretend to chase as I move it around. Is the spot a disk because that's the shape of the sun, or because it's the shape of my watch? In other words, would a square watch make a square spot, or do we just have a circular image of the circular sun, which will be circular no matter what? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{flatmirrormagnification}{1}{}{15}Apply the equation \(M=d_i/d_o\) to the case of a flat mirror. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{virtual-image-location}{1}{}{16}(solution in the pdf version of the book) Use the method described in the text to derive the equation relating object distance to image distance for the case of a virtual image produced by a converging mirror. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{focal-length-by-ray-tracing}{1}{}{17}Find the focal length of the mirror in problem 6 .(answer check available at lightandmatter.com) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{rank-focal-lengths}{1}{}{18}Rank the focal lengths of the mirrors in the figure, from shortest to longest. Explain. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{image-combinations}{1}{}{19}(solution in the pdf version of the book) (a) A converging mirror with a focal length of 20 cm
is used to create an image, using an object at a distance of
10 cm. Is the image real, or is it virtual? (b) How about
\(f=20\) cm and \(d_o=30\) cm? (c) What if it was a *diverging*
mirror with \(f=20\) cm and \(d_o=10\) cm? (d) A diverging
mirror with \(f=20\) cm and \(d_o=30\) cm?
\end{homeworkforcelabel}

\begin{homeworkforcelabel}{magnificationdependsondistance}{1}{}{20}(a) Make up a numerical example of a virtual image formed
by a converging mirror with a certain focal length, and
determine the magnification. (You will need the result of
problem 16.) Make sure to choose values of \(d_o\) and \(f\) that
would actually produce a virtual image, not a real one. Now
change the location of the object *a little bit* and
redetermine the magnification, showing that it changes. At
my local department store, the cosmetics department sells
hand mirrors advertised as giving a magnification of 5 times. How
would you interpret this?

(b) Suppose a Newtonian telescope is being used for astronomical observing. Assume for simplicity that no eyepiece is used, and assume a value for the focal length of the mirror that would be reasonable for an amateur instrument that is to fit in a closet. Is the angular magnification different for objects at different distances? For example, you could consider two planets, one of which is twice as far as the other. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{infinitemagnification}{1}{}{21}(a) Find a case where the magnification of a curved
mirror is infinite. Is the *angular* magnification
infinite from any realistic viewing position? (b) Explain
why an arbitrarily large magnification can't be achieved by
having a sufficiently small value of \(d_o\).
\end{homeworkforcelabel}

\begin{homeworkforcelabel}{hear-your-own-whisper}{1}{}{22}A concave surface that reflects sound waves can act just like a converging mirror. Suppose that, standing near such a surface, you are able to find a point where you can place your head so that your own whispers are focused back on your head, so that they sound loud to you. Given your distance to the surface, what is the surface's focal length? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{soldier}{2}{}{23}The figure shows a device for constructing a realistic optical illusion. Two mirrors of equal focal length are put against each other with their silvered surfaces facing inward. A small object placed in the bottom of the cavity will have its image projected in the air above. The way it works is that the top mirror produces a virtual image, and the bottom mirror then creates a real image of the virtual image. (a) Show that if the image is to be positioned as shown, at the mouth of the cavity, then the focal length of the mirrors is related to the dimension \(h\) via the equation

\[\begin{equation*}
\frac{1}{f} = \frac{1}{h}+\frac{1}{h+\left(\frac{1}{h}-\frac{1}{f}\right)^{-1}} .
\end{equation*}\]

(b) Restate the equation in terms of a single variable \(x=h/f\), and show that there are two solutions for \(x\). Which solution is physically consistent with the assumptions of the calculation? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{range-of-magnifications}{1}{}{24}(a) A converging mirror is being used to create a virtual image. What is the range of possible magnifications? (b) Do the same for the other types of images that can be formed by curved mirrors (both converging and diverging). \end{homeworkforcelabel}

\begin{homeworkforcelabel}{image-of-dropping-object}{1}{}{25}A diverging mirror of focal length \(f\) is fixed, and faces down.
An object is dropped from the surface of the mirror, and falls away
from it with acceleration \(g\). The goal of the problem is to find
the maximum velocity of the image.

(a) Describe the motion of the image verbally, and explain why we should
expect there to be a maximum velocity.

(b) Use arguments based on units to determine the form of the solution,
up to an unknown unitless multiplicative constant.

(c) Complete the solution by determining the unitless constant.
\end{homeworkforcelabel}

\begin{homeworkforcelabel}{diamond}{1}{}{26}Diamond has an index of refraction of 2.42, and part of the reason diamonds sparkle is that this encourages a light ray to undergo many total internal reflections before it emerges. (a) Calculate the critical angle at which total internal reflection occurs in diamond. (answer check available at lightandmatter.com) (b) Explain the interpretation of your result: Is it measured from the normal, or from the surface? Is it a minimum angle for total internal reflection, or is it a maximum? How would the critical angle have been different for a substance such as glass or plastic, with a lower index of refraction? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{underwaterlens}{1}{}{27}Suppose a converging lens is constructed of a type of plastic whose index of refraction is less than that of water. How will the lens's behavior be different if it is placed underwater? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{refractorvsreflector}{1}{}{28}There are two main types of telescopes, refracting (using a lens) and reflecting (using a mirror, as in figure i on p. 756). (Some telescopes use a mixture of the two types of elements: the light first encounters a large curved mirror, and then goes through an eyepiece that is a lens. To keep things simple, assume no eyepiece is used.) What implications would the color-dependence of focal length have for the relative merits of the two types of telescopes? Describe the case where an image is formed of a white star. You may find it helpful to draw a ray diagram. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{explainlens}{1}{}{29}Based on Snell's law, explain why rays of light passing through the edges of a converging lens are bent more than rays passing through parts closer to the center. It might seem like it should be the other way around, since the rays at the edge pass through less glass --- shouldn't they be affected less? In your answer:

- Include a ray diagram showing a huge, full-page, close-up view of the relevant part of the lens.
- Make use of the fact that the front and back surfaces aren't always parallel; a lens in
which the front and back surfaces
*are*always parallel doesn't focus light at all, so if your explanation doesn't make use of this fact, your argument must be incorrect. - Make sure your argument still works even if the rays don't come in parallel to the axis.

\end{homeworkforcelabel}

\begin{homeworkforcelabel}{camera}{1}{}{30}When you take pictures with a camera, the distance between the lens and the film has to be adjusted, depending on the distance at which you want to focus. This is done by moving the lens. If you want to change your focus so that you can take a picture of something farther away, which way do you have to move the lens? Explain using ray diagrams. [Based on a problem by Eric Mazur.] \end{homeworkforcelabel}

\begin{homeworkforcelabel}{goggles}{1}{}{31}When swimming underwater, why is your vision made much clearer by wearing goggles with flat pieces of glass that trap air behind them? [Hint: You can simplify your reasoning by considering the special case where you are looking at an object far away, and along the optic axis of the eye.] \end{homeworkforcelabel}

\begin{homeworkforcelabel}{imagelocation1}{1}{}{32}(answer check available at lightandmatter.com) An object is more than one focal length from a converging lens. (a) Draw a ray diagram. (b) Using reasoning like that developed in section 12.3, determine the positive and negative signs in the equation \(1/f=\pm1/d_i\pm1/d_o\). (c) The images of the rose in section 4.2 were made using a lens with a focal length of 23 cm. If the lens is placed 80 cm from the rose, locate the image. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{funkylenses}{1}{}{33}The figure shows four lenses. Lens 1 has two spherical surfaces. Lens 2 is the same as lens 1 but turned around. Lens 3 is made by cutting through lens 1 and turning the bottom around. Lens 4 is made by cutting a central circle out of lens 1 and recessing it.

(a) A parallel beam of light enters lens 1 from the left, parallel to its axis. Reasoning based on Snell's law, will the beam emerging from the lens be bent inward, or outward, or will it remain parallel to the axis? Explain your reasoning. As part of your answer, make a huge drawing of one small part of the lens, and apply Snell's law at both interfaces. Recall that rays are bent more if they come to the interface at a larger angle with respect to the normal.

(b) What will happen with lenses 2, 3, and 4? Explain. Drawings are not necessary. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{eyefocallength}{2}{}{34}The drawing shows the anatomy of the human eye, at twice life size. Find the radius of curvature of the outer surface of the cornea by measurements on the figure, and then derive the focal length of the air-cornea interface, where almost all the focusing of light occurs. You will need to use physical reasoning to modify the lensmaker's equation for the case where there is only a single refracting surface. Assume that the index of refraction of the cornea is essentially that of water. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{imagelocation2}{1}{}{35}(answer check available at lightandmatter.com) An object is less than one focal length from a converging lens. (a) Draw a ray diagram. (b) Using reasoning like that developed in section 12.3, determine the positive and negative signs in the equation \(1/f=\pm1/d_i\pm1/d_o\). (c) The images of the rose in section 4.2 were made using a lens with a focal length of 23 cm. If the lens is placed 10 cm from the rose, locate the image. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{imagelocation3}{1}{}{36}(answer check available at lightandmatter.com) Nearsighted people wear glasses whose lenses are diverging. (a) Draw a ray diagram. For simplicity pretend that there is no eye behind the glasses. (b) Using reasoning like that developed in section 12.3, determine the positive and negative signs in the equation \(1/f=\pm1/d_i\pm1/d_o\). (c) If the focal length of the lens is 50.0 cm, and the person is looking at an object at a distance of 80.0 cm, locate the image. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{imageunderwater}{2}{}{37}(a) Light is being reflected diffusely from an object 1.000 m underwater. The light that comes up to the surface is refracted at the water-air interface. If the refracted rays all appear to come from the same point, then there will be a virtual image of the object in the water, above the object's actual position, which will be visible to an observer above the water. Consider three rays, A, B and C, whose angles in the water with respect to the normal are \(\theta_i=0.000°\), \(1.000°\) and \(20.000°\) respectively. Find the depth of the point at which the refracted parts of A and B appear to have intersected, and do the same for A and C. Show that the intersections are at nearly the same depth, but not quite. [Check: The difference in depth should be about 4 cm.]

(b) Since all the refracted rays do not quite appear to have come from the same point, this is technically not a virtual image. In practical terms, what effect would this have on what you see?

(c) In the case where the angles are all small, use algebra and trig to show that the refracted rays do appear to come from the same point, and find an equation for the depth of the virtual image. Do not put in any numerical values for the angles or for the indices of refraction --- just keep them as symbols. You will need the approximation \(\sin\theta\approx \tan\theta\approx \theta\), which is valid for small angles measured in radians. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{least-time-refraction}{2}{}{38}Prove that the principle of least time leads to Snell's law. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{field-of-view}{1}{}{39}(solution in the pdf version of the book) Two standard focal lengths for camera lenses are 50 mm (standard) and 28 mm (wide-angle). To see how the focal lengths relate to the angular size of the field of view, it is helpful to visualize things as represented in the figure. Instead of showing many rays coming from the same point on the same object, as we normally do, the figure shows two rays from two different objects. Although the lens will intercept infinitely many rays from each of these points, we have shown only the ones that pass through the center of the lens, so that they suffer no angular deflection. (Any angular deflection at the front surface of the lens is canceled by an opposite deflection at the back, since the front and back surfaces are parallel at the lens's center.) What is special about these two rays is that they are aimed at the edges of one 35-mm-wide frame of film; that is, they show the limits of the field of view. Throughout this problem, we assume that \(d_o\) is much greater than \(d_i\). (a) Compute the angular width of the camera's field of view when these two lenses are used. (b) Use small-angle approximations to find a simplified equation for the angular width of the field of view, \(\theta \), in terms of the focal length, \(f\), and the width of the film, \(w\). Your equation should not have any trig functions in it. Compare the results of this approximation with your answers from part a. (c) Suppose that we are holding constant the aperture (amount of surface area of the lens being used to collect light). When switching from a 50-mm lens to a 28-mm lens, how many times longer or shorter must the exposure be in order to make a properly developed picture, i.e., one that is not under- or overexposed? [Based on a problem by Arnold Arons.] \end{homeworkforcelabel}

\begin{homeworkforcelabel}{nearsighted}{1}{}{40}A nearsighted person is one whose eyes focus light too strongly, and who is therefore unable to relax the lens inside her eye sufficiently to form an image on her retina of an object that is too far away.

(a) Draw a ray diagram showing what happens when the person tries, with uncorrected vision, to focus at infinity.

(b) What type of lenses do her glasses have? Explain.

(c) Draw a ray diagram showing what happens when she wears glasses. Locate both the image formed by the glasses and the final image.

(d) Suppose she sometimes uses contact lenses instead of her glasses. Does the focal length of her contacts have to be less than, equal to, or greater than that of her glasses? Explain. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{flatworm}{1}{}{41}Fred's eyes are able to focus on things as close as 5.0 cm. Fred holds a magnifying glass with a focal length of 3.0 cm at a height of 2.0 cm above a flatworm. (a) Locate the image, and find the magnification. (b) Without the magnifying glass, from what distance would Fred want to view the flatworm to see its details as well as possible? With the magnifying glass? (c) Compute the angular magnification. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{binoculars}{2}{}{42}Panel 1 of the figure shows the optics inside a pair of binoculars.
They are essentially a pair of telescopes, one for each eye. But to make them more compact,
and allow the eyepieces to be the right distance apart for a human face, they incorporate
a set of eight prisms, which fold the light path. In addition, the prisms make the image upright.
Panel 2 shows one of these prisms, known
as a Porro prism. The light enters along a normal, undergoes
two total internal reflections at angles of 45 degrees with respect to the back surfaces,
and exits along a normal. The image of the letter R has been flipped across the horizontal.
Panel 3 shows a pair of these prisms glued together. The image will be flipped across
both the horizontal and the vertical, which makes it oriented the right way for the user
of the binoculars.

(a) Find the minimum possible index of refraction for the glass used in the prisms.

(b) For a material of this minimal index of refraction, find the fraction of the incoming
light that will be lost to reflection in the four Porro prisms on a each side of a pair
of binoculars. (See
section 6.2.)
In real, high-quality binoculars, the optical surfaces of the prisms
have antireflective coatings, but carry out your calculation for the case where there is
no such coating.

(c) Discuss the reasons why a designer of binoculars might or might not want to use a material
with exactly the index of refraction found in part a.
\end{homeworkforcelabel}

\begin{homeworkforcelabel}{eyeglasses-no-mag}{1}{}{43}It would be annoying if your eyeglasses produced a magnified or reduced image. Prove that when the eye is very close to a lens, and the lens produces a virtual image, the angular magnification is always approximately equal to 1 (regardless of whether the lens is diverging or converging). \end{homeworkforcelabel}

\begin{homeworkforcelabel}{label-fringes}{1}{}{44}The figure shows a diffraction pattern made by a double slit, along with an image of a meter stick to show the scale. Sketch the diffraction pattern from the figure on your paper. Now consider the four variables in the equation \(\lambda /d=\sin \theta /m\). Which of these are the same for all five fringes, and which are different for each fringe? Which variable would you naturally use in order to label which fringe was which? Label the fringes on your sketch using the values of that variable. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{matchpatterns}{1}{}{45}Match gratings A-C with the diffraction patterns 1-3 that they produce. Explain.

\end{homeworkforcelabel}

\begin{homeworkforcelabel}{yellow-red}{1}{}{46}The figure below shows two diffraction patterns. The top one was made with yellow light, and the bottom one with red. Could the slits used to make the two patterns have been the same?

\end{homeworkforcelabel}

\begin{homeworkforcelabel}{find-actual-wavelength}{1}{}{47}The figure on p. 807 shows a diffraction pattern made by a double slit, along with an image of a meter stick to show the scale. The slits were 146 cm away from the screen on which the diffraction pattern was projected. The spacing of the slits was 0.050 mm. What was the wavelength of the light?(answer check available at lightandmatter.com) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{microscopeblue}{1}{}{48}Why would blue or violet light be the best for microscopy? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{compare-diffraction-same-light}{1}{}{49}The figure below shows two diffraction patterns, both made with the same wavelength of red light. (a) What type of slits made the patterns? Is it a single slit, double slits, or something else? Explain. (b) Compare the dimensions of the slits used to make the top and bottom pattern. Give a numerical ratio, and state which way the ratio is, i.e., which slit pattern was the larger one. Explain.

\end{homeworkforcelabel}

\begin{homeworkforcelabel}{overlappingrainbows}{1}{}{50}When white light passes through a diffraction grating, what is the smallest value of \(m\) for which the visible spectrum of order \(m\) overlaps the next one, of order \(m+1?\) (The visible spectrum runs from about 400 nm to about 700 nm.) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{star}{1}{}{51}For star images such as the ones in figure y, estimate the angular width of the diffraction spot due to diffraction at the mouth of the telescope. Assume a telescope with a diameter of 10 meters (the largest currently in existence), and light with a wavelength in the middle of the visible range. Compare with the actual angular size of a star of diameter \(10^9\) m seen from a distance of \(10^{17}\) m. What does this tell you? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{compare-double-slits}{1}{}{52}The figure below shows three diffraction patterns. All were made under identical conditions, except that a different set of double slits was used for each one. The slits used to make the top pattern had a center-to-center separation \(d=0.50\) mm, and each slit was \(w=0.04\) mm wide. (a) Determine \(d\) and \(w\) for the slits used to make the pattern in the middle. (b) Do the same for the slits used to make the bottom pattern.

\end{homeworkforcelabel}

\begin{homeworkforcelabel}{grating}{1}{}{53}(answer check available at lightandmatter.com) The beam of a laser passes through a diffraction grating, fans out, and illuminates a wall that is perpendicular to the original beam, lying at a distance of 2.0 m from the grating. The beam is produced by a helium-neon laser, and has a wavelength of 694.3 nm. The grating has 2000 lines per centimeter. (a) What is the distance on the wall between the central maximum and the maxima immediately to its right and left? (b) How much does your answer change when you use the small-angle approximations \(\theta\approx\sin\theta\approx\tan\theta\)? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{kidney}{1}{}{54}Ultrasound, i.e., sound waves with frequencies too high to be audible, can be used for imaging fetuses in the womb or for breaking up kidney stones so that they can be eliminated by the body. Consider the latter application. Lenses can be built to focus sound waves, but because the wavelength of the sound is not all that small compared to the diameter of the lens, the sound will not be concentrated exactly at the geometrical focal point. Instead, a diffraction pattern will be created with an intense central spot surrounded by fainter rings. About 85% of the power is concentrated within the central spot. The angle of the first minimum (surrounding the central spot) is given by \(\sin \theta =\lambda/b\), where \(b\) is the diameter of the lens. This is similar to the corresponding equation for a single slit, but with a factor of 1.22 in front which arises from the circular shape of the aperture. Let the distance from the lens to the patient's kidney stone be \(L=20\) cm. You will want \(f>20\) kHz, so that the sound is inaudible. Find values of \(b\) and \(f\) that would result in a usable design, where the central spot is small enough to lie within a kidney stone 1 cm in diameter. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{unphysical-diffraction}{1}{}{55}Under what circumstances could one get a mathematically undefined result by solving the double-slit diffraction equation for \(\theta \)? Give a physical interpretation of what would actually be observed. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{ultrasound-imaging-versus-heating}{1}{}{56}When ultrasound is used for medical imaging, the frequency may be as high as 5-20 MHz. Another medical application of ultrasound is for therapeutic heating of tissues inside the body; here, the frequency is typically 1-3 MHz. What fundamental physical reasons could you suggest for the use of higher frequencies for imaging? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{illuminable}{2}{}{57}Suppose we have a polygonal room whose walls are mirrors, and there a pointlike
light source in the room. In most such examples, every point in the room ends up
being illuminated by the light source after some finite number of reflections.
A difficult mathematical question, first posed in the middle of the last century,
is whether it is ever possible to have an example in which the whole room is
*not* illuminated. (Rays are assumed to be absorbed if they strike exactly at a vertex of the polygon,
or if they pass exactly through the plane of a mirror.)

The problem was finally solved in 1995 by G.W. Tokarsky, who found an example of a room that was not illuminable from a certain point. Figure 57 shows a slightly simpler example found two years later by D. Castro. If a light source is placed at either of the locations shown with dots, the other dot remains unilluminated, although every other point is lit up. It is not straightforward to prove rigorously that Castro's solution has this property. However, the plausibility of the solution can be demonstrated as follows.

Suppose the light source is placed at the right-hand dot. Locate all the images formed by single reflections. Note that they form a regular pattern. Convince yourself that none of these images illuminates the left-hand dot. Because of the regular pattern, it becomes plausible that even if we form images of images, images of images of images, etc., none of them will ever illuminate the other dot.

There are various other versions of the problem, some of which remain unsolved. The book by Klee and Wagon gives a good introduction to the topic, although it predates Tokarsky and Castro's work.

References:

G.W. Tokarsky, “Polygonal Rooms Not Illuminable from Every Point.” Amer. Math. Monthly 102, 867-879, 1995.

D. Castro, “Corrections.” Quantum 7, 42, Jan. 1997.

V. Klee and S. Wagon, *Old and New Unsolved Problems in Plane Geometry and Number Theory*. Mathematical
Association of America, 1991.
\end{homeworkforcelabel}

\begin{homeworkforcelabel}{peaucellier}{2}{}{58}A mechanical linkage is a device that changes one type of motion into another. The most familiar example occurs in a gasoline car's engine, where a connecting rod changes the linear motion of the piston into circular motion of the crankshaft. The top panel of the figure shows a mechanical linkage invented by Peaucellier in 1864, and independently by Lipkin around the same time. It consists of six rods joined by hinges, the four short ones forming a rhombus. Point O is fixed in space, but the apparatus is free to rotate about O. Motion at P is transformed into a different motion at \(\text{P}'\) (or vice versa).

Geometrically, the linkage is a mechanical implementation of
the ancient problem of inversion in a circle.
Considering the case in which the rhombus is folded flat, let
the \(k\) be the distance from O to the point where P and \(\text{P}'\)
coincide. Form the circle of radius \(k\) with its center at O.
As P and \(\text{P}'\) move in and out, points on the inside of the
circle are always mapped to points on its outside, such that
\(rr'=k^2\). That is, the linkage is a type of analog computer
that exactly solves the problem of finding the inverse of a
number \(r\). Inversion in a circle has many remarkable geometrical properties, discussed in
H.S.M. Coxeter, *Introduction to Geometry*, Wiley, 1961.
If a pen is inserted through a hole at P, and \(\text{P}'\) is traced over
a geometrical figure, the Peaucellier linkage can be used to draw a kind
of image of the figure.

A related problem is the construction of pictures, like the one in the bottom panel of the figure, called anamorphs. The drawing of the column on the paper is highly distorted, but when the reflecting cylinder is placed in the correct spot on top of the page, an undistorted image is produced inside the cylinder. (Wide-format movie technologies such as Cinemascope are based on similar principles.)

Show that the Peaucellier linkage does *not* convert correctly between
an image and its anamorph, and design a modified version of the linkage that does.
Some knowledge of analytic geometry will be helpful.
\end{homeworkforcelabel}

\begin{homeworkforcelabel}{constant-thickness}{1}{}{59}The figure shows a lens with surfaces that are curved, but whose thickness is constant along any horizontal line. Use the lensmaker's equation to prove that this “lens” is not really a lens at all.(solution in the pdf version of the book) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{very-weak-refraction}{1}{}{60}Under ordinary conditions, gases have indices of refraction only a little greater than that of vacuum, i.e., \(n=1+\epsilon\), where \(\epsilon\) is some small number. Suppose that a ray crosses a boundary between a region of vacuum and a region in which the index of refraction is \(1+\epsilon\). Find the maximum angle by which such a ray can ever be deflected, in the limit of small \(\epsilon\). \hwhint{hwhint:very-weak-refraction} \end{homeworkforcelabel}

\begin{homeworkforcelabel}{close-to-focal-length}{1}{}{61}A converging mirror has focal length \(f\). An object is located at a distance \((1+\epsilon)f\) from the mirror, where \(\epsilon\) is small. Find the distance of the image from the mirror, simplifying your result as much as possible by using the assumption that \(\epsilon\) is small. \hwans{hwans:close-to-focal-length} \end{homeworkforcelabel}

Equipment:

concave mirrors with deep curvature

concave mirrors with gentle curvature

convex mirrors

1. Obtain a curved mirror from your instructor. If it is silvered on both sides, make sure you're working with the concave side, which bends light rays inward. Look at your own face in the mirror. Now change the distance between your face and the mirror, and see what happens. Explore the full range of possible distances between your face and the mirror.

In these observations you've been changing two variables at once: the distance between the object (your face) and the mirror, and the distance from the mirror to your eye. In general, scientific experiments become easier to interpret if we practice isolation of variables, i.e., only change one variable while keeping all the others constant. In parts 2 and 3 you'll form an image of an object that's not your face, so that you can have independent control of the object distance and the point of view.

2. With the mirror held far away from you, observe the image of something behind you, over your shoulder. Now bring your eye closer and closer to the mirror. Can you see the image with your eye very close to the mirror? See if you can explain your observation by drawing a ray diagram.

--------------------\(>\) *turn page*

3. Now imagine the following new situation, but *don't
actually do it yet*. Suppose you lay the mirror face-up on a
piece of tissue paper, put your finger a few cm above the
mirror, and look at the image of your finger. As in part 2,
you can bring your eye closer and closer to the mirror.

Will you be able to see the image with your eye very close to the mirror? Draw a ray diagram to help you predict what you will observe.

Prediction:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

Now test your prediction. If your prediction was incorrect, see if you can figure out what went wrong, or ask your instructor for help.

4. For parts 4 and 5, it's more convenient to use concave mirrors that are more gently curved; obtain one from your instructor. Lay the mirror on the tissue paper, and use it to create an image of the overhead lights on a piece of paper above it and a little off to the side. What do you have to do in order to make the image clear? Can you explain this observation using a ray diagram?

--------------------\(>\) *turn page*

5. Now imagine the following experiment, but *don't do
it yet*. What will happen to the image on the paper if you
cover half of the mirror with your hand?

Prediction:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

Test your prediction. If your prediction was incorrect, can you explain what happened?

6. Now imagine forming an image with a convex mirror (one that bulges outward), and that therefore bends light rays away from the central axis (i.e., is diverging). Draw a typical ray diagram.

Is the image real, or virtual? Will there be more than one type of image?

Prediction:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

Test your prediction.

Equipment:

optical benches

converging mirrors

illuminated objects

1. Set up the optical bench with the mirror at zero on the centimeter scale. Set up the illuminated object on the bench as well.

2. Each group will locate the image for their own value of the object distance, by finding where a piece of paper has to be placed in order to see the image on it. (The instructor will do one point as well.) Note that you will have to tilt the mirror a little so that the paper on which you project the image doesn't block the light from the illuminated object.

Is the image real or virtual? How do you know? Is it inverted, or uninverted?

Draw a ray diagram.

3. Measure the image distance and write your result in the table on the board. Do the same for the magnification.

4. What do you notice about the trend of the data on the board? Draw a second ray diagram with a different object distance, and show why this makes sense. Some tips for doing this correctly: (1) For simplicity, use the point on the object that is on the mirror's axis. (2) You need to trace two rays to locate the image. To save work, don't just do two rays at random angles. You can either use the on-axis ray as one ray, or do two rays that come off at the same angle, one above and one below the axis. (3) Where each ray hits the mirror, draw the normal line, and make sure the ray is at equal angles on both sides of the normal.

5. We will find the mirror's focal length from the instructor's data-point. Then, using this focal length, calculate a theoretical prediction of the image distance, and write it on the board next to the experimentally determined image distance.

This exercise was created by Dan MacIsaac.

Equipment:

eyeglasses

diverging lenses for students who don't wear glasses, or who use glasses with converging lenses

rulers and metersticks

scratch paper

marking pens

Most people who wear glasses have glasses whose lenses are outbending, which allows them to focus on objects far away. Such a lens cannot form a real image, so its focal length cannot be measured as easily as that of a converging lens. In this exercise you will determine the focal length of your own glasses by taking them off, holding them at a distance from your face, and looking through them at a set of parallel lines on a piece of paper. The lines will be reduced (the lens's magnification is less than one), and by adjusting the distance between the lens and the paper, you can make the magnification equal 1/2 exactly, so that two spaces between lines as seen through the lens fit into one space as seen simultaneously to the side of the lens. This object distance can be used in order to find the focal length of the lens.

1. Use a marker to draw three evenly spaced parallel lines on the paper. (A spacing of a few cm works well.)

2. Does this technique really measure magnification or does it measure angular magnification? What can you do in your experiment in order to make these two quantities nearly the same, so the math is simpler?

3. Before taking any numerical data, use algebra to find the focal length of the lens in terms of \(d_o\), the object distance that results in a magnification of 1/2.

4. Measure the object distance that results in a magnification of 1/2, and determine the focal length of your lens.

1. Two sources separated by a distance \(d=2\) cm make circular ripples with a wavelength of \(\lambda=1\) cm. On a piece of paper, make a life-size drawing of the two sources in the default setup, and locate the following points:

A. The point that is 10 wavelengths from source #1 and 10 wavelengths from source #2.

B. The point that is 10.5 wavelengths from #1 and 10.5 from #2.

C. The point that is 11 wavelengths from #1 and 11 from #2.

D. The point that is 10 wavelengths from #1 and 10.5 from #2.

E. The point that is 11 wavelengths from #1 and 11.5 from #2.

F. The point that is 10 wavelengths from #1 and 11 from #2.

G. The point that is 11 wavelengths from #1 and 12 from #2.

You can do this either using a compass or by putting the next page under your paper and tracing. It is not necessary to trace all the arcs completely, and doing so is unnecessarily time-consuming; you can fairly easily estimate where these points would lie, and just trace arcs long enough to find the relevant intersections.

What do these points correspond to in the real wave pattern?

2. Make a fresh copy of your drawing, showing only point F and the two sources, which form a long, skinny triangle. Now suppose you were to change the setup by doubling \(d\), while leaving \(\lambda \) the same. It's easiest to understand what's happening on the drawing if you move both sources outward, keeping the center fixed. Based on your drawing, what will happen to the position of point F when you double \(d\)? How has the angle of point F changed?

3. What would happen if you doubled *both* \(\lambda\) and \(d\)
compared to the standard setup?\_\_\_\_\_\_\_\_\_

4. Combining the ideas from parts 2 and 3, what do you think would happen to your angles if, starting from the standard setup, you doubled \(\lambda \) while leaving \(d\) the same?\_\_\_\_\_\_\_\_\_

5. Suppose \(\lambda\) was a millionth of a centimeter, while \(d\) was still as in the standard setup. What would happen to the angles? What does this tell you about observing diffraction of light?

Equipment:

rulers

computer with web browser

The following page is a diagram of a single slit and a screen onto which its diffraction pattern is projected. The class will make a numerical prediction of the intensity of the pattern at the different points on the screen. Each group will be responsible for calculating the intensity at one of the points. (Either 11 groups or six will work nicely -- in the latter case, only points a, c, e, g, i, and k are used.) The idea is to break up the wavefront in the mouth of the slit into nine parts, each of which is assumed to radiate semicircular ripples as in Huygens' principle. The wavelength of the wave is 1 cm, and we assume for simplicity that each set of ripples has an amplitude of 1 unit when it reaches the screen.

1. For simplicity, let's imagine that we were only to use two sets of ripples rather than nine. You could measure the distance from each of the two points inside the slit to your point on the screen. Suppose the distances were both 25.0 cm. What would be the amplitude of the superimposed waves at this point on the screen?

Suppose one distance was 24.0 cm and the other was 25.0 cm. What would happen?

What if one was 24.0 cm and the other was 26.0 cm?

What if one was 24.5 cm and the other was 25.0 cm?

In general, what combinations of distances will lead to completely destructive and completely constructive interference?

Can you estimate the answer in the case where the distances are 24.7 and 25.0 cm?

2. Although it is possible to calculate mathematically the amplitude of the sine wave that results from superimposing two sine waves with an arbitrary phase difference between them, the algebra is rather laborious, and it become even more tedious when we have more than two waves to superimpose. Instead, one can simply use a computer spreadsheet or some other computer program to add up the sine waves numerically at a series of points covering one complete cycle. This is what we will actually do. You just need to enter the relevant data into the computer, then examine the results and pick off the amplitude from the resulting list of numbers. You can run the software through a web interface at http://lightandmatter.com/cgi-bin/diffraction1.cgi.

3. Measure all nine distances to your group's point on the screen, and write them on the board - that way everyone can see everyone else's data, and the class can try to make sense of why the results came out the way they did. Determine the amplitude of the combined wave, and write it on the board as well.

The class will discuss why the results came out the way they did.

Equipment:

slit patterns, lasers, straight-filament bulbs

You have a mask with a bunch of different double slits cut out of it. The values of w and d are as follows:

pattern A | w=0.04 mm | d=.250 mm |

pattern B | w=0.04 mm | d=.500 mm |

pattern C | w=0.08 mm | d=.250 mm |

pattern D | w=0.08 mm | d=.500 mm |

Predict how the patterns will look different, and test your prediction. The easiest way to get the laser to point at different sets of slits is to stick folded up pieces of paper in one side or the other of the holders.

This is just like station 1, but with single slits:

pattern A | w=0.02 mm |

pattern B | w=0.04 mm |

pattern C | w=0.08 mm |

pattern D | w=0.16 mm |

Predict what will happen, and test your predictions. If you have time, check the actual numerical ratios of the w values against the ratios of the sizes of the diffraction patterns

This is like station 1, but the only difference among the sets of slits is how many slits there are:

pattern A | double slit |

pattern B | 3 slits |

pattern C | 4 slits |

pattern D | 5 slits |

Hold the diffraction grating up to your eye, and look through it at the straight-filament light bulb. If you orient the grating correctly, you should be able to see the \(m=1\) and \(m=-1\) diffraction patterns off the left and right. If you have it oriented the wrong way, they'll be above and below the bulb instead, which is inconvenient because the bulb's filament is vertical. Where is the \(m=0\) fringe? Can you see \(m=2\), etc.?

*Station 5 has the same equipment as station 4. If you're assigned to station 5 first, you should actually do activity 4 first, because it's easier.*

Use the transformer to increase and decrease the voltage across the bulb. This allows you to control the filament's temperature. Sketch graphs of intensity as a function of wavelength for various temperatures. The inability of the wave model of light to explain the mathematical shapes of these curves was historically one of the reasons for creating a new model, in which light is both a particle and a wave.

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.

[1] There is a standard piece of terminology
which is that the “focal point” is the point lying on the optical
axis at a distance from the mirror equal to the focal length.
This term isn't particularly helpful, because it names a location where nothing
normally happens. In particular, it is *not* normally the place where
the rays come to a focus! --- that would be the *image* point.
In other words, we don't normally have \(d_i=f\), unless perhaps
\(d_o=\infty\). A recent online discussion among some physics teachers
(https://carnot.physics.buffalo.edu/archives, Feb. 2006) showed that
many disliked the terminology, felt it was misleading, or didn't know it and would have
misinterpreted it if they had come across it. That is, it appears to be what grammarians
call a “skunked term” --- a word that bothers half the population when it's used
incorrectly, and the other half when it's used correctly.