# Wave Optics

## Connection for AP® Courses

If you have ever looked at the reds, blues, and greens in a sunlit soap bubble and wondered how straw-colored soapy water could produce them, you have hit upon one of the many phenomena that can only be explained by the wave character of light. The same is true for the colors seen in an oil slick or in the light reflected from an optical data disk. These and other interesting phenomena, such as the dispersion of white light into a rainbow of colors when passed through a narrow slit, cannot be explained fully by geometric optics. In these cases, light interacts with objects and exhibits a number of wave characteristics. The branch of optics that considers the behavior of light when it exhibits wave characteristics is called “wave optics” (or sometimes “physical optics”).
These soap bubbles exhibit brilliant colors when exposed to sunlight. How are the colors produced if they are not pigments in the soap?
This chapter supports Big Idea 6 in its coverage of wave optics by presenting explanations and examples of many phenomena that can only be explained by the wave aspect of light. You will learn how only waves can exhibit diffraction and interference patterns that we observe in light (Enduring Understanding 6.C). As explained by Huygens’s principle, diffraction is the bending of waves around the edges of a nontransparent object or after passing through an opening (Essential Knowledge 6.C.4). Interference results from the superposition of two or more traveling waves (Enduring Understanding 6.D, Enduring Understanding 6.D.1). Superposition causes variations in the resultant wave amplitude (Essential Knowledge 6.D.2). The interference can be described as constructive interference, which increases amplitude, and destructive interference, which decreases amplitude. Based on an understanding of diffraction and interference of light, this chapter also explains experimental observations that occur when light passes through an opening or set of openings with dimensions comparable to the wavelength of the light – specifically the effects of double-slit, multiple-slit (Essential Knowledge 6.C.3), and single-slit (Essential Knowledge 6.C.2) openings. Another aspect of light waves that you will learn about in this chapter is polarization, a phenomenon in which light waves all vibrate in a single plane. The explanation for this phenomenon is based on the fact that light is a traveling electromagnetic wave (Enduring Understanding 6.A) that propagates via transverse oscillations of both electric and magnetic field vectors (Enduring Understanding 6.A.1). Light waves can be polarized by passing through filters. Many sunglasses contain polarizing filters to reduce glare, and certain types of 3-D glasses use polarization to create an effect of depth on the movie screen.
Big Idea 6 Waves can transfer energy and momentum from one location to another without the permanent transfer of mass and serve as a mathematical model for the description of other phenomena.
Enduring Understanding 6.A A wave is a traveling disturbance that transfers energy and momentum.
Essential Knowledge 6.A.1 Waves can propagate via different oscillation modes such as transverse and longitudinal.
Enduring Understanding 6.C Only waves exhibit interference and diffraction.
Essential Knowledge 6.C.2 When waves pass through an opening whose dimensions are comparable to the wavelength, a diffraction pattern can be observed.
Essential Knowledge 6.C.3 When waves pass through a set of openings whose spacing is comparable to the wavelength, an interference pattern can be observed. Examples should include monochromatic double-slit interference.
Essential Knowledge 6.C.4 When waves pass by an edge, they can diffract into the “shadow region” behind the edge. Examples should include hearing around corners, but not seeing around them, and water waves bending around obstacles.
Enduring Understanding 6.D Interference and superposition lead to standing waves and beats.
Essential Knowledge 6.D.1 Two or more wave pulses can interact in such a way as to produce amplitude variations in the resultant wave. When two pulses cross, they travel through each other; they do not bounce off each other. Where the pulses overlap, the resulting displacement can be determined by adding the displacements of the two pulses. This is called superposition.
Essential Knowledge 6.D.2 Two or more traveling waves can interact in such a way as to produce amplitude variations in the resultant wave.


# Wave Optics

## The Wave Aspect of Light: Interference

### Learning Objectives

By the end of this section, you will be able to:
1. Discuss the wave character of light.
2. Identify the changes when light enters a medium.
We know that visible light is the type of electromagnetic wave to which our eyes respond. Like all other electromagnetic waves, it obeys the equation
where  is the speed of light in vacuum,  is the frequency of the electromagnetic waves, and  is its wavelength. The range of visible wavelengths is approximately 380 to 760 nm. As is true for all waves, light travels in straight lines and acts like a ray when it interacts with objects several times as large as its wavelength. However, when it interacts with smaller objects, it displays its wave characteristics prominently. Interference is the hallmark of a wave, and in  both the ray and wave characteristics of light can be seen. The laser beam emitted by the observatory epitomizes a ray, traveling in a straight line. However, passing a pure-wavelength beam through vertical slits with a size close to the wavelength of the beam reveals the wave character of light, as the beam spreads out horizontally into a pattern of bright and dark regions caused by systematic constructive and destructive interference. Rather than spreading out, a ray would continue traveling straight ahead after passing through slits.
Light has wave characteristics in various media as well as in a vacuum. When light goes from a vacuum to some medium, like water, its speed and wavelength change, but its frequency  remains the same. (We can think of light as a forced oscillation that must have the frequency of the original source.) The speed of light in a medium is , where  is its index of refraction. If we divide both sides of equation

 by
, we get . This implies that
, where  is the wavelength in a medium and that
where  is the wavelength in vacuum and  is the medium’s index of refraction. Therefore, the wavelength of light is smaller in any medium than it is in vacuum. In water, for example, which has , the range of visible wavelengths is  to , or . Although wavelengths change while traveling from one medium to another, colors do not, since colors are associated with frequency.

### Section Summary

1. Wave optics is the branch of optics that must be used when light interacts with small objects or whenever the wave characteristics of light are considered.
2. Wave characteristics are those associated with interference and diffraction.
3. Visible light is the type of electromagnetic wave to which our eyes respond and has a wavelength in the range of 380 to 760 nm.
4. Like all EM waves, the following relationship is valid in vacuum: , where  is the speed of light,  is the frequency of the electromagnetic wave, and  is its wavelength in vacuum.
5. The wavelength  of light in a medium with index of refraction  is . Its frequency is the same as in vacuum.

### Conceptual Questions

### Problems & Exercises


# Wave Optics

## Huygens's Principle: Diffraction

### Learning Objectives

By the end of this section, you will be able to:
1. Discuss the propagation of transverse waves.
2. Discuss Huygens’s principle.
3. Explain the bending of light.
 shows how a transverse wave looks as viewed from above and from the side. A light wave can be imagined to propagate like this, although we do not actually see it wiggling through space. From above, we view the wavefronts (or wave crests) as we would by looking down on the ocean waves. The side view would be a graph of the electric or magnetic field. The view from above is perhaps the most useful in developing concepts about wave optics.
The Dutch scientist Christiaan Huygens (1629–1695) developed a useful technique for determining in detail how and where waves propagate. Starting from some known position, Huygens’s principle states that:
Every point on a wavefront is a source of wavelets that spread out in the forward direction at the same speed as the wave itself. The new wavefront is a line tangent to all of the wavelets.

 shows how Huygens’s principle is applied. A wavefront is the long edge that moves, for example, the crest or the trough. Each point on the wavefront emits a semicircular wave that moves at the propagation speed . These are drawn at a time  later, so that they have moved a distance . The new wavefront is a line tangent to the wavelets and is where we would expect the wave to be a time  later. Huygens’s principle works for all types of waves, including water waves, sound waves, and light waves. We will find it useful not only in describing how light waves propagate, but also in explaining the laws of reflection and refraction. In addition, we will see that Huygens’s principle tells us how and where light rays interfere.
 shows how a mirror reflects an incoming wave at an angle equal to the incident angle, verifying the law of reflection. As the wavefront strikes the mirror, wavelets are first emitted from the left part of the mirror and then the right. The wavelets closer to the left have had time to travel farther, producing a wavefront traveling in the direction shown.
The law of refraction can be explained by applying Huygens’s principle to a wavefront passing from one medium to another (see ). Each wavelet in the figure was emitted when the wavefront crossed the interface between the media. Since the speed of light is smaller in the second medium, the waves do not travel as far in a given time, and the new wavefront changes direction as shown. This explains why a ray changes direction to become closer to the perpendicular when light slows down. Snell’s law can be derived from the geometry in , but this is left as an exercise for ambitious readers.
What happens when a wave passes through an opening, such as light shining through an open door into a dark room? For light, we expect to see a sharp shadow of the doorway on the floor of the room, and we expect no light to bend around corners into other parts of the room. When sound passes through a door, we expect to hear it everywhere in the room and, thus, expect that sound spreads out when passing through such an opening (see ). What is the difference between the behavior of sound waves and light waves in this case? The answer is that light has very short wavelengths and acts like a ray. Sound has wavelengths on the order of the size of the door and bends around corners (for frequency of 1000 Hz, , about three times smaller than the width of the doorway).
If we pass light through smaller openings, often called slits, we can use Huygens’s principle to see that light bends as sound does (see ). The bending of a wave around the edges of an opening or an obstacle is called diffraction. Diffraction is a wave characteristic and occurs for all types of waves. If diffraction is observed for some phenomenon, it is evidence that the phenomenon is a wave. Thus the horizontal diffraction of the laser beam after it passes through slits in  is evidence that light is a wave.

### Test Prep for AP Courses

### Section Summary

1. An accurate technique for determining how and where waves propagate is given by Huygens’s principle: Every point on a wavefront is a source of wavelets that spread out in the forward direction at the same speed as the wave itself. The new wavefront is a line tangent to all of the wavelets.
2. Diffraction is the bending of a wave around the edges of an opening or other obstacle.

### Conceptual Questions


# Wave Optics

## Young’s Double Slit Experiment

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the phenomena of interference.
2. Define constructive interference for a double slit and destructive interference for a double slit.
Although Christiaan Huygens thought that light was a wave, Isaac Newton did not. Newton felt that there were other explanations for color, and for the interference and diffraction effects that were observable at the time. Owing to Newton’s tremendous stature, his view generally prevailed. The fact that Huygens’s principle worked was not considered evidence that was direct enough to prove that light is a wave. The acceptance of the wave character of light came many years later when, in 1801, the English physicist and physician Thomas Young (1773–1829) did his now-classic double slit experiment (see ).
Why do we not ordinarily observe wave behavior for light, such as observed in Young’s double slit experiment? First, light must interact with something small, such as the closely spaced slits used by Young, to show pronounced wave effects. Furthermore, Young first passed light from a single source (the Sun) through a single slit to make the light somewhat coherent. By coherent, we mean waves are in phase or have a definite phase relationship. Incoherent means the waves have random phase relationships. Why did Young then pass the light through a double slit? The answer to this question is that two slits provide two coherent light sources that then interfere constructively or destructively. Young used sunlight, where each wavelength forms its own pattern, making the effect more difficult to see. We illustrate the double slit experiment with monochromatic (single ) light to clarify the effect.  shows the pure constructive and destructive interference of two waves having the same wavelength and amplitude.
When light passes through narrow slits, it is diffracted into semicircular waves, as shown in (a). Pure constructive interference occurs where the waves are crest to crest or trough to trough. Pure destructive interference occurs where they are crest to trough. The light must fall on a screen and be scattered into our eyes for us to see the pattern. An analogous pattern for water waves is shown in (b). Note that regions of constructive and destructive interference move out from the slits at well-defined angles to the original beam. These angles depend on wavelength and the distance between the slits, as we shall see below.
To understand the double slit interference pattern, we consider how two waves travel from the slits to the screen, as illustrated in . Each slit is a different distance from a given point on the screen. Thus different numbers of wavelengths fit into each path. Waves start out from the slits in phase (crest to crest), but they may end up out of phase (crest to trough) at the screen if the paths differ in length by half a wavelength, interfering destructively as shown in (a). If the paths differ by a whole wavelength, then the waves arrive in phase (crest to crest) at the screen, interfering constructively as shown in (b). More generally, if the paths taken by the two waves differ by any half-integral number of wavelengths [, , , etc.], then destructive interference occurs. Similarly, if the paths taken by the two waves differ by any integral number of wavelengths (, , , etc.), then constructive interference occurs.
 shows how to determine the path length difference for waves traveling from two slits to a common point on a screen. If the screen is a large distance away compared with the distance between the slits, then the angle  between the path and a line from the slits to the screen (see the figure) is nearly the same for each path. The difference between the paths is shown in the figure; simple trigonometry shows it to be , where  is the distance between the slits. To obtain constructive interference for a double slit, the path length difference must be an integral multiple of the wavelength, or
Similarly, to obtain destructive interference for a double slit, the path length difference must be a half-integral multiple of the wavelength, or
where  is the wavelength of the light,  is the distance between slits, and  is the angle from the original direction of the beam as discussed above. We call  the order of the interference. For example,  is fourth-order interference.
The equations for double slit interference imply that a series of bright and dark lines are formed. For vertical slits, the light spreads out horizontally on either side of the incident beam into a pattern called interference fringes, illustrated in . The intensity of the bright fringes falls off on either side, being brightest at the center. The closer the slits are, the more is the spreading of the bright fringes. We can see this by examining the equation
For fixed  and , the smaller  is, the larger  must be, since

.

 This is consistent with our contention that wave effects are most noticeable when the object the wave encounters (here, slits a distance  apart) is small. Small  gives large , hence a large effect.

### Test Prep for AP Courses

### Section Summary

1. Young’s double slit experiment gave definitive proof of the wave character of light.
2. An interference pattern is obtained by the superposition of light from two slits.
3. There is constructive interference when , where  is the distance between the slits,  is the angle relative to the incident direction, and  is the order of the interference.
4. There is destructive interference when .

### Conceptual Questions

### Problems & Exercises


# Wave Optics

## Multiple Slit Diffraction

### Learning Objectives

By the end of this section, you will be able to:
1. Discuss the pattern obtained from diffraction grating.
2. Explain diffraction grating effects.
An interesting thing happens if you pass light through a large number of evenly spaced parallel slits, called a diffraction grating. An interference pattern is created that is very similar to the one formed by a double slit (see ). A diffraction grating can be manufactured by scratching glass with a sharp tool in a number of precisely positioned parallel lines, with the untouched regions acting like slits. These can be photographically mass produced rather cheaply. Diffraction gratings work both for transmission of light, as in , and for reflection of light, as on butterfly wings and the Australian opal in  or the CD pictured in the opening photograph of this chapter. In addition to their use as novelty items, diffraction gratings are commonly used for spectroscopic dispersion and analysis of light. What makes them particularly useful is the fact that they form a sharper pattern than double slits do. That is, their bright regions are narrower and brighter, while their dark regions are darker.  shows idealized graphs demonstrating the sharper pattern. Natural diffraction gratings occur in the feathers of certain birds. Tiny, finger-like structures in regular patterns act as reflection gratings, producing constructive interference that gives the feathers colors not solely due to their pigmentation. This is called iridescence.
The analysis of a diffraction grating is very similar to that for a double slit (see ). As we know from our discussion of double slits in Young's Double Slit Experiment, light is diffracted by each slit and spreads out after passing through. Rays traveling in the same direction (at an angle  relative to the incident direction) are shown in the figure. Each of these rays travels a different distance to a common point on a screen far away. The rays start in phase, and they can be in or out of phase when they reach a screen, depending on the difference in the path lengths traveled. As seen in the figure, each ray travels a distance  different from that of its neighbor, where  is the distance between slits. If this distance equals an integral number of wavelengths, the rays all arrive in phase, and constructive interference (a maximum) is obtained. Thus, the condition necessary to obtain constructive interference for a diffraction grating is
where  is the distance between slits in the grating,  is the wavelength of light, and  is the order of the maximum. Note that this is exactly the same equation as for double slits separated by . However, the slits are usually closer in diffraction gratings than in double slits, producing fewer maxima at larger angles.
Where are diffraction gratings used? Diffraction gratings are key components of monochromators used, for example, in optical imaging of particular wavelengths from biological or medical samples. A diffraction grating can be chosen to specifically analyze a wavelength emitted by molecules in diseased cells in a biopsy sample or to help excite strategic molecules in the sample with a selected frequency of light. Another vital use is in optical fiber technologies where fibers are designed to provide optimum performance at specific wavelengths. A range of diffraction gratings are available for selecting specific wavelengths for such use.

### Test Prep for AP Courses

### Section Summary

1. A diffraction grating is a large collection of evenly spaced parallel slits that produces an interference pattern similar to but sharper than that of a double slit.
2. There is constructive interference for a diffraction grating when , where  is the distance between slits in the grating,   is the wavelength of light, and   is the order of the maximum.

### Conceptual Questions

### Problems & Exercises


# Wave Optics

## Single Slit Diffraction

### Learning Objectives

By the end of this section, you will be able to:
1. Discuss the single slit diffraction pattern.
Light passing through a single slit forms a diffraction pattern somewhat different from those formed by double slits or diffraction gratings.  shows a single slit diffraction pattern. Note that the central maximum is larger than those on either side, and that the intensity decreases rapidly on either side. In contrast, a diffraction grating produces evenly spaced lines that dim slowly on either side of center.
The analysis of single slit diffraction is illustrated in . Here we consider light coming from different parts of the same slit. According to Huygens’s principle, every part of the wavefront in the slit emits wavelets. These are like rays that start out in phase and head in all directions. (Each ray is perpendicular to the wavefront of a wavelet.) Assuming the screen is very far away compared with the size of the slit, rays heading toward a common destination are nearly parallel. When they travel straight ahead, as in (a), they remain in phase, and a central maximum is obtained. However, when rays travel at an angle  relative to the original direction of the beam, each travels a different distance to a common location, and they can arrive in or out of phase. In (b), the ray from the bottom travels a distance of one wavelength  farther than the ray from the top. Thus a ray from the center travels a distance  farther than the one on the left, arrives out of phase, and interferes destructively. A ray from slightly above the center and one from slightly above the bottom will also cancel one another. In fact, each ray from the slit will have another to interfere destructively, and a minimum in intensity will occur at this angle. There will be another minimum at the same angle to the right of the incident direction of the light.
At the larger angle shown in (c), the path lengths differ by  for rays from the top and bottom of the slit. One ray travels a distance  different from the ray from the bottom and arrives in phase, interfering constructively. Two rays, each from slightly above those two, will also add constructively. Most rays from the slit will have another to interfere with constructively, and a maximum in intensity will occur at this angle. However, all rays do not interfere constructively for this situation, and so the maximum is not as intense as the central maximum. Finally, in (d), the angle shown is large enough to produce a second minimum. As seen in the figure, the difference in path length for rays from either side of the slit is , and we see that a destructive minimum is obtained when this distance is an integral multiple of the wavelength.
Thus, to obtain destructive interference for a single slit,
where  is the slit width,  is the light’s wavelength,  is the angle relative to the original direction of the light, and  is the order of the minimum.  shows a graph of intensity for single slit interference, and it is apparent that the maxima on either side of the central maximum are much less intense and not as wide. This is consistent with the illustration in (b).

### Test Prep for AP Courses

### Section Summary

1. A single slit produces an interference pattern characterized by a broad central maximum with narrower and dimmer maxima to the sides.
2. There is destructive interference for a single slit when
, where  is the slit width,
  is the light’s wavelength,
  is the angle relative to the original direction of the light, and
  is the order of the minimum. Note that there is no
 minimum.

### Conceptual Questions

### Problems & Exercises


# Wave Optics

## Limits of Resolution: The Rayleigh Criterion

### Learning Objectives

By the end of this section, you will be able to:
1. Discuss the Rayleigh criterion.
Light diffracts as it moves through space, bending around obstacles, interfering constructively and destructively. While this can be used as a spectroscopic tool—a diffraction grating disperses light according to wavelength, for example, and is used to produce spectra—diffraction also limits the detail we can obtain in images. (a) shows the effect of passing light through a small circular aperture. Instead of a bright spot with sharp edges, a spot with a fuzzy edge surrounded by circles of light is obtained. This pattern is caused by diffraction similar to that produced by a single slit. Light from different parts of the circular aperture interferes constructively and destructively. The effect is most noticeable when the aperture is small, but the effect is there for large apertures, too.
How does diffraction affect the detail that can be observed when light passes through an aperture? (b) shows the diffraction pattern produced by two point light sources that are close to one another. The pattern is similar to that for a single point source, and it is just barely possible to tell that there are two light sources rather than one. If they were closer together, as in (c), we could not distinguish them, thus limiting the detail or resolution we can obtain. This limit is an inescapable consequence of the wave nature of light.
There are many situations in which diffraction limits the resolution. The acuity of our vision is limited because light passes through the pupil, the circular aperture of our eye. Be aware that the diffraction-like spreading of light is due to the limited diameter of a light beam, not the interaction with an aperture. Thus light passing through a lens with a diameter  shows this effect and spreads, blurring the image, just as light passing through an aperture of diameter  does. So diffraction limits the resolution of any system having a lens or mirror. Telescopes are also limited by diffraction, because of the finite diameter  of their primary mirror.
Just what is the limit? To answer that question, consider the diffraction pattern for a circular aperture, which has a central maximum that is wider and brighter than the maxima surrounding it (similar to a slit) [see (a)]. It can be shown that, for a circular aperture of diameter , the first minimum in the diffraction pattern occurs at  (providing the aperture is large compared with the wavelength of light, which is the case for most optical instruments). The accepted criterion for determining the diffraction limit to resolution based on this angle was developed by Lord Rayleigh in the 19th century. The Rayleigh criterion for the diffraction limit to resolution states that two images are just resolvable when the center of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other. See (b). The first minimum is at an angle of , so that two point objects are just resolvable if they are separated by the angle
where  is the wavelength of light (or other electromagnetic radiation) and  is the diameter of the aperture, lens, mirror, etc., with which the two objects are observed. In this expression,  has units of radians.
Diffraction is not only a problem for optical instruments but also for the electromagnetic radiation itself. Any beam of light having a finite diameter  and a wavelength  exhibits diffraction spreading. The beam spreads out with an angle  given by the equation . Take, for example, a laser beam made of rays as parallel as possible (angles between rays as close to  as possible) instead spreads out at an angle , where  is the diameter of the beam and  is its wavelength. This spreading is impossible to observe for a flashlight, because its beam is not very parallel to start with. However, for long-distance transmission of laser beams or microwave signals, diffraction spreading can be significant (see ). To avoid this, we can increase . This is done for laser light sent to the Moon to measure its distance from the Earth. The laser beam is expanded through a telescope to make  much larger and  smaller.
In most biology laboratories, resolution is presented when the use of the microscope is introduced. The ability of a lens to produce sharp images of two closely spaced point objects is called resolution. The smaller the distance  by which two objects can be separated and still be seen as distinct, the greater the resolution. The resolving power of a lens is defined as that distance . An expression for resolving power is obtained from the Rayleigh criterion. In (a) we have two point objects separated by a distance . According to the Rayleigh criterion, resolution is possible when the minimum angular separation is
where  is the distance between the specimen and the objective lens, and we have used the small angle approximation (i.e., we have assumed that  is much smaller than ), so that .
Therefore, the resolving power is
Another way to look at this is by re-examining the concept of Numerical Aperture () discussed in Microscopes. There,  is a measure of the maximum acceptance angle at which the fiber will take light and still contain it within the fiber. (b) shows a lens and an object at point P. The  here is a measure of the ability of the lens to gather light and resolve fine detail. The angle subtended by the lens at its focus is defined to be . From the figure and again using the small angle approximation, we can write
The  for a lens is , where  is the index of refraction of the medium between the objective lens and the object at point P.
From this definition for , we can see that
In a microscope,  is important because it relates to the resolving power of a lens. A lens with a large  will be able to resolve finer details. Lenses with larger  will also be able to collect more light and so give a brighter image. Another way to describe this situation is that the larger the , the larger the cone of light that can be brought into the lens, and so more of the diffraction modes will be collected. Thus the microscope has more information to form a clear image, and so its resolving power will be higher.
One of the consequences of diffraction is that the focal point of a beam has a finite width and intensity distribution. Consider focusing when only considering geometric optics, shown in (a). The focal point is infinitely small with a huge intensity and the capacity to incinerate most samples irrespective of the  of the objective lens. For wave optics, due to diffraction, the focal point spreads to become a focal spot (see (b)) with the size of the spot decreasing with increasing . Consequently, the intensity in the focal spot increases with increasing . The higher the , the greater the chances of photodegrading the specimen. However, the spot never becomes a true point.

### Test Prep for AP Courses

### Section Summary

1. Diffraction limits resolution.
2. For a circular aperture, lens, or mirror, the Rayleigh criterion states that two images are just resolvable when the center of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other.
3. This occurs for two point objects separated by the angle , where  is the wavelength of light (or other electromagnetic radiation) and  is the diameter of the aperture, lens, mirror, etc. This equation also gives the angular spreading of a source of light having a diameter .

### Conceptual Questions

### Problems & Exercises


# Wave Optics

## Thin Film Interference

### Learning Objectives

By the end of this section, you will be able to:
1. Discuss the rainbow formation by thin films.
The bright colors seen in an oil slick floating on water or in a sunlit soap bubble are caused by interference. The brightest colors are those that interfere constructively. This interference is between light reflected from different surfaces of a thin film; thus, the effect is known as thin film interference. As noticed before, interference effects are most prominent when light interacts with something having a size similar to its wavelength. A thin film is one having a thickness  smaller than a few times the wavelength of light, . Since color is associated indirectly with  and since all interference depends in some way on the ratio of  to the size of the object involved, we should expect to see different colors for different thicknesses of a film, as in . Some of the earliest measurements of such films and their effects were conducted by Agnes Pockels, a self-taught German chemist who investigated the characteristics of soapy and greasy films in water. Using homemade materials, Pockels developed a trough for measuring surface films and began conducting experiments. While scientific and societal barriers for women prevented her from publishing on her own, renowned scientist Lord Rayleigh supported her efforts and pushed for her work to be shared in the journal Nature. The trough Pockels invented became the basis for the contemporary version, as described below.
What causes thin film interference?  shows how light reflected from the top and bottom surfaces of a film can interfere. Incident light is only partially reflected from the top surface of the film (ray 1). The remainder enters the film and is itself partially reflected from the bottom surface. Part of the light reflected from the bottom surface can emerge from the top of the film (ray 2) and interfere with light reflected from the top (ray 1). Since the ray that enters the film travels a greater distance, it may be in or out of phase with the ray reflected from the top. However, consider for a moment, again, the bubbles in . The bubbles are darkest where they are thinnest. Furthermore, if you observe a soap bubble carefully, you will note it gets dark at the point where it breaks. For very thin films, the difference in path lengths of ray 1 and ray 2 in  is negligible; so why should they interfere destructively and not constructively? The answer is that a phase change can occur upon reflection. The rule is as follows:
When light reflects from a medium having an index of refraction greater than that of the medium in which it is traveling, a

If the film in  is a soap bubble (essentially water with air on both sides), then there is a  shift for ray 1 and none for ray 2. Thus, when the film is very thin, the path length difference between the two rays is negligible, they are exactly out of phase, and destructive interference will occur at all wavelengths and so the soap bubble will be dark here.
The thickness of the film relative to the wavelength of light is the other crucial factor in thin film interference. Ray 2 in  travels a greater distance than ray 1. For light incident perpendicular to the surface, ray 2 travels a distance approximately  farther than ray 1. When this distance is an integral or half-integral multiple of the wavelength in the medium (, where  is the wavelength in vacuum and  is the index of refraction), constructive or destructive interference occurs, depending also on whether there is a phase change in either ray.
Thin-film interference has created an entire field of research and industrial applications. Its foundations were laid by Irving Langmuir and Katharine Burr Blodgett, working at General Electric in the 1920s and 1930s. Langmuir had pioneered a method for producing ultra-thin layers on materials. Blodgett built on these practices by creating a method to precisely stack and compress these layers in order to produce a film of a desired thickness and quality. The device they developed became known as the Langmuir-Blodgett trough, built from principles developed by Agnes Pockels and still used in laboratories today. The earliest widely applied use of these principles was non-reflective glass, which Blodgett patented in 1938 and which was used almost immediately in the making of the film Gone With the Wind. The film is viewed as a tremendous leap in cinematography; cameras, microscopes, telescopes, and many other instruments rely on Blodgett's invention as well.
Thin film interference is most constructive or most destructive when the path length difference for the two rays is an integral or half-integral wavelength, respectively. That is, for rays incident perpendicularly,  or . To know whether interference is constructive or destructive, you must also determine if there is a phase change upon reflection. Thin film interference thus depends on film thickness, the wavelength of light, and the refractive indices. For white light incident on a film that varies in thickness, you will observe rainbow colors of constructive interference for various wavelengths as the thickness varies.
Another example of thin film interference can be seen when microscope slides are separated (see ). The slides are very flat, so that the wedge of air between them increases in thickness very uniformly. A phase change occurs at the second surface but not the first, and so there is a dark band where the slides touch. The rainbow colors of constructive interference repeat, going from violet to red again and again as the distance between the slides increases. As the layer of air increases, the bands become more difficult to see, because slight changes in incident angle have greater effects on path length differences. If pure-wavelength light instead of white light is used, then bright and dark bands are obtained rather than repeating rainbow colors.
An important application of thin film interference is found in the manufacturing of optical instruments. A lens or mirror can be compared with a master as it is being ground, allowing it to be shaped to an accuracy of less than a wavelength over its entire surface.  illustrates the phenomenon called Newton’s rings, which occurs when the plane surfaces of two lenses are placed together. (The circular bands are called Newton’s rings because Isaac Newton described them and their use in detail. Newton did not discover them; Robert Hooke did, and Newton did not believe they were due to the wave character of light.) Each successive ring of a given color indicates an increase of only one wavelength in the distance between the lens and the blank, so that great precision can be obtained. Once the lens is perfect, there will be no rings.
The wings of certain moths and butterflies have nearly iridescent colors due to thin film interference. In addition to pigmentation, the wing’s color is affected greatly by constructive interference of certain wavelengths reflected from its film-coated surface. Car manufacturers are offering special paint jobs that use thin film interference to produce colors that change with angle. This expensive option is based on variation of thin film path length differences with angle. Security features on credit cards, banknotes, driving licenses and similar items prone to forgery use thin film interference, diffraction gratings, or holograms. Australia led the way with dollar bills printed on polymer with a diffraction grating security feature making the currency difficult to forge. Other countries such as New Zealand and Taiwan are using similar technologies, while the United States currency includes a thin film interference effect.

### Problem-Solving Strategies for Wave Optics

Step 1. Examine the situation to determine that interference is involved. Identify whether slits or thin film interference are considered in the problem.
Step 2. If slits are involved, note that diffraction gratings and double slits produce very similar interference patterns, but that gratings have narrower (sharper) maxima. Single slit patterns are characterized by a large central maximum and smaller maxima to the sides.
Step 3. If thin film interference is involved, take note of the path length difference between the two rays that interfere. Be certain to use the wavelength in the medium involved, since it differs from the wavelength in vacuum. Note also that there is an additional  phase shift when light reflects from a medium with a greater index of refraction.
Step 4. Identify exactly what needs to be determined in the problem (identify the unknowns). A written list is useful. Draw a diagram of the situation. Labeling the diagram is useful.
Step 5. Make a list of what is given or can be inferred from the problem as stated (identify the knowns).
Step 6. Solve the appropriate equation for the quantity to be determined (the unknown), and enter the knowns. Slits, gratings, and the Rayleigh limit involve equations.
Step 7. For thin film interference, you will have constructive interference for a total shift that is an integral number of wavelengths. You will have destructive interference for a total shift of a half-integral number of wavelengths. Always keep in mind that crest to crest is constructive whereas crest to trough is destructive.
Step 8. Check to see if the answer is reasonable: Does it make sense? Angles in interference patterns cannot be greater than , for example.

### Test Prep for AP Courses

### Section Summary

1. Thin film interference occurs between the light reflected from the top and bottom surfaces of a film. In addition to the path length difference, there can be a phase change.
2. When light reflects from a medium having an index of refraction greater than that of the medium in which it is traveling, a  phase change (or a  shift) occurs.

### Conceptual Questions

### Problems & Exercises


# Wave Optics

## Polarization

### Learning Objectives

By the end of this section, you will be able to:
1. Discuss the meaning of polarization.
2. Discuss the property of optical activity of certain materials.
Polaroid sunglasses are familiar to most of us. They have a special ability to cut the glare of light reflected from water or glass (see ). Polaroids have this ability because of a wave characteristic of light called polarization. What is polarization? How is it produced? What are some of its uses? The answers to these questions are related to the wave character of light.
Light is one type of electromagnetic (EM) wave. As noted earlier, EM waves are transverse waves consisting of varying electric and magnetic fields that oscillate perpendicular to the direction of propagation (see ). There are specific directions for the oscillations of the electric and magnetic fields. Polarization is the attribute that a wave’s oscillations have a definite direction relative to the direction of propagation of the wave. (This is not the same type of polarization as that discussed for the separation of charges.) Waves having such a direction are said to be polarized. For an EM wave, we define the direction of polarization to be the direction parallel to the electric field. Thus we can think of the electric field arrows as showing the direction of polarization, as in .
To examine this further, consider the transverse waves in the ropes shown in . The oscillations in one rope are in a vertical plane and are said to be vertically polarized. Those in the other rope are in a horizontal plane and are horizontally polarized. If a vertical slit is placed on the first rope, the waves pass through. However, a vertical slit blocks the horizontally polarized waves. For EM waves, the direction of the electric field is analogous to the disturbances on the ropes.
The Sun and many other light sources produce waves that are randomly polarized (see ). Such light is said to be unpolarized because it is composed of many waves with all possible directions of polarization. Polaroid materials, invented by the founder of Polaroid Corporation, Edwin Land, act as a polarizing slit for light, allowing only polarization in one direction to pass through. Polarizing filters are composed of long molecules aligned in one direction. Thinking of the molecules as many slits, analogous to those for the oscillating ropes, we can understand why only light with a specific polarization can get through. The axis of a polarizing filter is the direction along which the filter passes the electric field of an EM wave (see ).
 shows the effect of two polarizing filters on originally unpolarized light. The first filter polarizes the light along its axis. When the axes of the first and second filters are aligned (parallel), then all of the polarized light passed by the first filter is also passed by the second. If the second polarizing filter is rotated, only the component of the light parallel to the second filter’s axis is passed. When the axes are perpendicular, no light is passed by the second.
Only the component of the EM wave parallel to the axis of a filter is passed. Let us call the angle between the direction of polarization and the axis of a filter . If the electric field has an amplitude , then the transmitted part of the wave has an amplitude  (see ). Since the intensity of a wave is proportional to its amplitude squared, the intensity  of the transmitted wave is related to the incident wave by
where  is the intensity of the polarized wave before passing through the filter. (The above equation is known as Malus’s law.)

### Polarization by Reflection

By now you can probably guess that Polaroid sunglasses cut the glare in reflected light because that light is polarized. You can check this for yourself by holding Polaroid sunglasses in front of you and rotating them while looking at light reflected from water or glass. As you rotate the sunglasses, you will notice the light gets bright and dim, but not completely black. This implies the reflected light is partially polarized and cannot be completely blocked by a polarizing filter.
 illustrates what happens when unpolarized light is reflected from a surface. Vertically polarized light is preferentially refracted at the surface, so that the reflected light is left more horizontally polarized. The reasons for this phenomenon are beyond the scope of this text, but a convenient mnemonic for remembering this is to imagine the polarization direction to be like an arrow. Vertical polarization would be like an arrow perpendicular to the surface and would be more likely to stick and not be reflected. Horizontal polarization is like an arrow bouncing on its side and would be more likely to be reflected. Sunglasses with vertical axes would then block more reflected light than unpolarized light from other sources.
Since the part of the light that is not reflected is refracted, the amount of polarization depends on the indices of refraction of the media involved. It can be shown that reflected light is completely polarized at a angle of reflection , given by
where  is the medium in which the incident and reflected light travel and  is the index of refraction of the medium that forms the interface that reflects the light. This equation is known as Brewster’s law, and  is known as Brewster’s angle, named after the 19th-century Scottish physicist who discovered them.

### Polarization by Scattering

If you hold your Polaroid sunglasses in front of you and rotate them while looking at blue sky, you will see the sky get bright and dim. This is a clear indication that light scattered by air is partially polarized.  helps illustrate how this happens. Since light is a transverse EM wave, it vibrates the electrons of air molecules perpendicular to the direction it is traveling. The electrons then radiate like small antennae. Since they are oscillating perpendicular to the direction of the light ray, they produce EM radiation that is polarized perpendicular to the direction of the ray. When viewing the light along a line perpendicular to the original ray, as in , there can be no polarization in the scattered light parallel to the original ray, because that would require the original ray to be a longitudinal wave. Along other directions, a component of the other polarization can be projected along the line of sight, and the scattered light will only be partially polarized. Furthermore, multiple scattering can bring light to your eyes from other directions and can contain different polarizations.
Photographs of the sky can be darkened by polarizing filters, a trick used by many photographers to make clouds brighter by contrast. Scattering from other particles, such as smoke or dust, can also polarize light. Detecting polarization in scattered EM waves can be a useful analytical tool in determining the scattering source.
There is a range of optical effects used in sunglasses. Besides being Polaroid, other sunglasses have colored pigments embedded in them, while others use non-reflective or even reflective coatings. A recent development is photochromic lenses, which darken in the sunlight and become clear indoors. Photochromic lenses are embedded with organic microcrystalline molecules that change their properties when exposed to UV in sunlight, but become clear in artificial lighting with no UV.

### Liquid Crystals and Other Polarization Effects in Materials

While you are undoubtedly aware of liquid crystal displays (LCDs) found in watches, calculators, computer screens, cellphones, flat screen televisions, and other myriad places, you may not be aware that they are based on polarization. Liquid crystals are so named because their molecules can be aligned even though they are in a liquid. Liquid crystals have the property that they can rotate the polarization of light passing through them by . Furthermore, this property can be turned off by the application of a voltage, as illustrated in . It is possible to manipulate this characteristic quickly and in small well-defined regions to create the contrast patterns we see in so many LCD devices.
In flat screen LCD televisions, there is a large light at the back of the TV. The light travels to the front screen through millions of tiny units called pixels (picture elements). One of these is shown in  (a) and (b). Each unit has three cells, with red, blue, or green filters, each controlled independently. When the voltage across a liquid crystal is switched off, the liquid crystal passes the light through the particular filter. One can vary the picture contrast by varying the strength of the voltage applied to the liquid crystal.
Many crystals and solutions rotate the plane of polarization of light passing through them. Such substances are said to be optically active. Examples include sugar water, insulin, and collagen (see ). In addition to depending on the type of substance, the amount and direction of rotation depends on a number of factors. Among these is the concentration of the substance, the distance the light travels through it, and the wavelength of light. Optical activity is due to the asymmetric shape of molecules in the substance, such as being helical. Measurements of the rotation of polarized light passing through substances can thus be used to measure concentrations, a standard technique for sugars. It can also give information on the shapes of molecules, such as proteins, and factors that affect their shapes, such as temperature and pH.
Glass and plastic become optically active when stressed; the greater the stress, the greater the effect. Optical stress analysis on complicated shapes can be performed by making plastic models of them and observing them through crossed filters, as seen in . It is apparent that the effect depends on wavelength as well as stress. The wavelength dependence is sometimes also used for artistic purposes.
Another interesting phenomenon associated with polarized light is the ability of some crystals to split an unpolarized beam of light into two. Such crystals are said to be birefringent (see ). Each of the separated rays has a specific polarization. One behaves normally and is called the ordinary ray, whereas the other does not obey Snell’s law and is called the extraordinary ray. Birefringent crystals can be used to produce polarized beams from unpolarized light. Some birefringent materials preferentially absorb one of the polarizations. These materials are called dichroic and can produce polarization by this preferential absorption. This is fundamentally how polarizing filters and other polarizers work. The interested reader is invited to further pursue the numerous properties of materials related to polarization.

### Test Prep for AP Courses

### Section Summary

1. Polarization is the attribute that wave oscillations have a definite direction relative to the direction of propagation of the wave.
2. EM waves are transverse waves that may be polarized.
3. The direction of polarization is defined to be the direction parallel to the electric field of the EM wave.
4. Unpolarized light is composed of many rays having random polarization directions.
5. Light can be polarized by passing it through a polarizing filter or other polarizing material. The intensity  of polarized light after passing through a polarizing filter is  where  is the original intensity and  is the angle between the direction of polarization and the axis of the filter.
6. Polarization is also produced by reflection.
7. Brewster’s law states that reflected light will be completely polarized at the angle of reflection , known as Brewster’s angle, given by a statement known as Brewster’s law: , where  is the medium in which the incident and reflected light travel and  is the index of refraction of the medium that forms the interface that reflects the light.
8. Polarization can also be produced by scattering.
9. There are a number of types of optically active substances that rotate the direction of polarization of light passing through them.

### Conceptual Questions

### Problems & Exercises


# Wave Optics

## *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light

### Learning Objectives

By the end of this section, you will be able to:
1. Discuss the different types of microscopes.
Physics research underpins the advancement of developments in microscopy. As we gain knowledge of the wave nature of electromagnetic waves and methods to analyze and interpret signals, new microscopes that enable us to “see” more are being developed. It is the evolution and newer generation of microscopes that are described in this section.
The use of microscopes (microscopy) to observe small details is limited by the wave nature of light. Owing to the fact that light diffracts significantly around small objects, it becomes impossible to observe details significantly smaller than the wavelength of light. One rule of thumb has it that all details smaller than about  are difficult to observe. Radar, for example, can detect the size of an aircraft, but not its individual rivets, since the wavelength of most radar is several centimeters or greater. Similarly, visible light cannot detect individual atoms, since atoms are about 0.1 nm in size and visible wavelengths range from 380 to 760 nm. Ironically, special techniques used to obtain the best possible resolution with microscopes take advantage of the same wave characteristics of light that ultimately limit the detail.
The most obvious method of obtaining better detail is to utilize shorter wavelengths. Ultraviolet (UV) microscopes have been constructed with special lenses that transmit UV rays and utilize photographic or electronic techniques to record images. The shorter UV wavelengths allow somewhat greater detail to be observed, but drawbacks, such as the hazard of UV to living tissue and the need for special detection devices and lenses (which tend to be dispersive in the UV), severely limit the use of UV microscopes. Elsewhere, we will explore practical uses of very short wavelength EM waves, such as x rays, and other short-wavelength probes, such as electrons in electron microscopes, to detect small details.
Another difficulty in microscopy is the fact that many microscopic objects do not absorb much of the light passing through them. The lack of contrast makes image interpretation very difficult. Contrast is the difference in intensity between objects and the background on which they are observed. Stains (such as dyes, fluorophores, etc.) are commonly employed to enhance contrast, but these tend to be application specific. More general wave interference techniques can be used to produce contrast.  shows the passage of light through a sample. Since the indices of refraction differ, the number of wavelengths in the paths differs. Light emerging from the object is thus out of phase with light from the background and will interfere differently, producing enhanced contrast, especially if the light is coherent and monochromatic—as in laser light.
Interference microscopes enhance contrast between objects and background by superimposing a reference beam of light upon the light emerging from the sample. Since light from the background and objects differ in phase, there will be different amounts of constructive and destructive interference, producing the desired contrast in final intensity.  shows schematically how this is done. Parallel rays of light from a source are split into two beams by a half-silvered mirror. These beams are called the object and reference beams. Each beam passes through identical optical elements, except that the object beam passes through the object we wish to observe microscopically. The light beams are recombined by another half-silvered mirror and interfere. Since the light rays passing through different parts of the object have different phases, interference will be significantly different and, hence, have greater contrast between them.
Another type of microscope utilizing wave interference and differences in phases to enhance contrast is called the phase-contrast microscope. While its principle is the same as the interference microscope, the phase-contrast microscope is simpler to use and construct. Its impact (and the principle upon which it is based) was so important that its developer, the Dutch physicist Frits Zernike (1888–1966), was awarded the Nobel Prize in 1953.  shows the basic construction of a phase-contrast microscope. Phase differences between light passing through the object and background are produced by passing the rays through different parts of a phase plate (so called because it shifts the phase of the light passing through it). These two light rays are superimposed in the image plane, producing contrast due to their interference.
A polarization microscope also enhances contrast by utilizing a wave characteristic of light. Polarization microscopes are useful for objects that are optically active or birefringent, particularly if those characteristics vary from place to place in the object. Polarized light is sent through the object and then observed through a polarizing filter that is perpendicular to the original polarization direction. Nearly transparent objects can then appear with strong color and in high contrast. Many polarization effects are wavelength dependent, producing color in the processed image. Contrast results from the action of the polarizing filter in passing only components parallel to its axis.
Apart from the UV microscope, the variations of microscopy discussed so far in this section are available as attachments to fairly standard microscopes or as slight variations. The next level of sophistication is provided by commercial confocal microscopes, which use the extended focal region shown in (b) to obtain three-dimensional images rather than two-dimensional images. Here, only a single plane or region of focus is identified; out-of-focus regions above and below this plane are subtracted out by a computer so the image quality is much better. This type of microscope makes use of fluorescence, where a laser provides the excitation light. Laser light passing through a tiny aperture called a pinhole forms an extended focal region within the specimen. The reflected light passes through the objective lens to a second pinhole and the photomultiplier detector, see . The second pinhole is the key here and serves to block much of the light from points that are not at the focal point of the objective lens. The pinhole is conjugate (coupled) to the focal point of the lens. The second pinhole and detector are scanned, allowing reflected light from a small region or section of the extended focal region to be imaged at any one time. The out-of-focus light is excluded. Each image is stored in a computer, and a full scanned image is generated in a short time. Live cell processes can also be imaged at adequate scanning speeds allowing the imaging of three-dimensional microscopic movement. Confocal microscopy enhances images over conventional optical microscopy, especially for thicker specimens, and so has become quite popular.
The next level of sophistication is provided by microscopes attached to instruments that isolate and detect only a small wavelength band of light—monochromators and spectral analyzers. Here, the monochromatic light from a laser is scattered from the specimen. This scattered light shifts up or down as it excites particular energy levels in the sample. The uniqueness of the observed scattered light can give detailed information about the chemical composition of a given spot on the sample with high contrast—like molecular fingerprints. Applications are in materials science, nanotechnology, and the biomedical field. Fine details in biochemical processes over time can even be detected. The ultimate in microscopy is the electron microscope—to be discussed later. Research is being conducted into the development of new prototype microscopes that can become commercially available, providing better diagnostic and research capacities.

### Section Summary

1. To improve microscope images, various techniques utilizing the wave characteristics of light have been developed. Many of these enhance contrast with interference effects.

### Conceptual Questions


# Special Relativity

## Connection for AP® Courses

In this chapter you will be introduced to the theory of special relativity, which was first described by Albert Einstein in the year 1905. The chapter opens with a discussion of Einstein’s postulates that form the basis of special relativity. You will learn about an essential physics framework that is used to describe the observations and measurements made by an observer in what is called the “inertial frame of reference” (Enduring Understanding 3.A). Special relativity is a universally accepted theory that defines a relationship between space and time (Essential Knowledge 1.D.3). When the speed of an object approaches the speed of light, Newton’s laws no longer hold, which means that classical (Newtonian) mechanics (Enduring Understanding 1.D) is not sufficient to define the physical properties of such a system. This is where special relativity comes into play. Many interesting and counterintuitive physical results follow from the theory of special relativity. In this chapter we will explore the concepts of simultaneity, time dilation, and length contraction.
Further into the chapter you will find information that supports the concepts of relativistic velocity addition, relativistic momentum, and energy (Enduring Understanding 4.C). Learning these concepts will help you understand how the mass (Enduring Understanding 1.C and Essential Knowledge 4.C.4) of an object can appear to be different for different observers and how matter can be converted into energy and then back to matter so that the energy of the system remains conserved. (Essential Knowledge 1.C.4 and Enduring Understanding 5.B). The information and examples presented in the chapter support Big Ideas 1, 3, 4, and 5 of the AP® Physics Curriculum Framework.
The content of this chapter supports:
Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure.
Enduring Understanding 1.C Objects and systems have properties of inertial mass and gravitational mass that are experimentally veriﬁed to be the same and that satisfy conservation principles.
Essential Knowledge 1.C.4 In certain processes, mass can be converted to energy and energy can be converted to mass according to

, the equation derived from the theory of special relativity.
Enduring Understanding 1.D Classical mechanics cannot describe all properties of objects.
Essential Knowledge 1.D.3 Properties of space and time cannot always be treated as absolute.
Big Idea 3 The interactions of an object with other objects can be described by forces.
Enduring Understanding 3.A All forces share certain common characteristics when considered by observers in inertial reference frames.
Essential Knowledge 3.A.1 An observer in a particular reference frame can describe the motion of an object using such quantities as position, displacement, distance, velocity, speed, and acceleration.
Big Idea 4 Interactions between systems can result in changes in those systems.
Enduring Understanding 4.C Interactions with other objects or systems can change the total energy of a system.
Essential Knowledge 4.C.4 Mass can be converted into energy and energy can be converted into mass.
Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws.
Enduring Understanding 5.B The energy of a system is conserved.
Essential Knowledge 5.B.11 Beyond the classical approximation, mass is actually part of the internal energy of an object or system with

.
It is important to note that although classical mechanic, in general, and classical relativity, in particular, are limited, they are extremely good approximations for large, slow-moving objects. Otherwise, we could not use classical physics to launch satellites or build bridges. In the classical limit (objects larger than submicroscopic and moving slower than about 1% of the speed of light), relativistic mechanics becomes the same as classical mechanics. This fact will be noted at appropriate places throughout this chapter.


# Special Relativity

## Einstein’s Postulates

### Learning Objectives

By the end of this section, you will be able to:
1. State and explain both of Einstein’s postulates.
2. Explain what an inertial frame of reference is.
3. Describe one way the speed of light can be changed.
Have you ever used the Pythagorean Theorem and gotten a wrong answer? Probably not, unless you made a mistake in either your algebra or your arithmetic. Each time you perform the same calculation, you know that the answer will be the same. Trigonometry is reliable because of the certainty that one part always flows from another in a logical way. Each part is based on a set of postulates, and you can always connect the parts by applying those postulates. Physics is the same way with the exception that all parts must describe nature. If we are careful to choose the correct postulates, then our theory will follow and will be verified by experiment.
Einstein essentially did the theoretical aspect of this method for relativity. With two deceptively simple postulates and a careful consideration of how measurements are made, he produced the theory of special relativity.

### Einstein’s First Postulate

The first postulate upon which Einstein based the theory of special relativity relates to reference frames. All velocities are measured relative to some frame of reference. For example, a car’s motion is measured relative to its starting point or the road it is moving over, a projectile’s motion is measured relative to the surface it was launched from, and a planet’s orbit is measured relative to the star it is orbiting around. The simplest frames of reference are those that are not accelerated and are not rotating. Newton’s first law, the law of inertia, holds exactly in such a frame.
The laws of physics seem to be simplest in inertial frames. For example, when you are in a plane flying at a constant altitude and speed, physics seems to work exactly the same as if you were standing on the surface of the Earth. However, in a plane that is taking off, matters are somewhat more complicated. In these cases, the net force on an object, , is not equal to the product of mass and acceleration, . Instead,  is equal to  plus a fictitious force. This situation is not as simple as in an inertial frame. Not only are laws of physics simplest in inertial frames, but they should be the same in all inertial frames, since there is no preferred frame and no absolute motion. Einstein incorporated these ideas into his first postulate of special relativity.
As with many fundamental statements, there is more to this postulate than meets the eye. The laws of physics include only those that satisfy this postulate. We shall find that the definitions of relativistic momentum and energy must be altered to fit. Another outcome of this postulate is the famous equation .

### Einstein’s Second Postulate

The second postulate upon which Einstein based his theory of special relativity deals with the speed of light. Late in the 19th century, the major tenets of classical physics were well established. Two of the most important were the laws of electricity and magnetism and Newton’s laws. In particular, the laws of electricity and magnetism predict that light travels at  in a vacuum, but they do not specify the frame of reference in which light has this speed.
There was a contradiction between this prediction and Newton’s laws, in which velocities add like simple vectors. If the latter were true, then two observers moving at different speeds would see light traveling at different speeds. Imagine what a light wave would look like to a person traveling along with it at a speed . If such a motion were possible then the wave would be stationary relative to the observer. It would have electric and magnetic fields that varied in strength at various distances from the observer but were constant in time. This is not allowed by Maxwell’s equations. So either Maxwell’s equations are wrong, or an object with mass cannot travel at speed . Einstein concluded that the latter is true. An object with mass cannot travel at speed . This conclusion implies that light in a vacuum must always travel at speed  relative to any observer. Maxwell’s equations are correct, and Newton’s addition of velocities is not correct for light.
Investigations such as Young’s double slit experiment in the early-1800s had convincingly demonstrated that light is a wave. Many types of waves were known, and all travelled in some medium. Scientists therefore assumed that a medium carried light, even in a vacuum, and light travelled at a speed  relative to that medium. Starting in the mid-1880s, the American physicist A. A. Michelson, later aided by E. W. Morley, made a series of direct measurements of the speed of light. The results of their measurements were startling.
The eventual conclusion derived from this result is that light, unlike mechanical waves such as sound, does not need a medium to carry it. Furthermore, the Michelson-Morley results implied that the speed of light  is independent of the motion of the source relative to the observer. That is, everyone observes light to move at speed  regardless of how they move relative to the source or one another. For a number of years, many scientists tried unsuccessfully to explain these results and still retain the general applicability of Newton’s laws.
It was not until 1905, when Einstein published his first paper on special relativity, that the currently accepted conclusion was reached. Based mostly on his analysis that the laws of electricity and magnetism would not allow another speed for light, and only slightly aware of the Michelson-Morley experiment, Einstein detailed his second postulate of special relativity.
Deceptively simple and counterintuitive, this and the first postulate leave all else open for change. Some fundamental concepts do change. Among the changes are the loss of agreement on the elapsed time for an event, the variation of distance with speed, and the realization that matter and energy can be converted into one another. You will read about these concepts in the following sections.

### Test Prep for AP Courses

### Section Summary

1. Relativity is the study of how different observers measure the same event.
2. Modern relativity is divided into two parts. Special relativity deals with observers who are in uniform (unaccelerated) motion, whereas general relativity includes accelerated relative motion and gravity. Modern relativity is correct in all circumstances and, in the limit of low velocity and weak gravitation, gives the same predictions as classical relativity.
3. An inertial frame of reference is a reference frame in which a body at rest remains at rest and a body in motion moves at a constant speed in a straight line unless acted on by an outside force.
4. Modern relativity is based on Einstein’s two postulates. The first postulate of special relativity is the idea that the laws of physics are the same and can be stated in their simplest form in all inertial frames of reference. The second postulate of special relativity is the idea that the speed of light  is a constant, independent of the relative motion of the source.
5. The Michelson-Morley experiment demonstrated that the speed of light in a vacuum is independent of the motion of the Earth about the Sun.

### Conceptual Questions


# Special Relativity

## Simultaneity And Time Dilation

### Learning Objectives

By the end of this section, you will be able to:
1. Describe simultaneity.
2. Describe time dilation.
3. Calculate γ.
4. Compare proper time and the observer’s measured time.
5. Explain why the twin paradox is a false paradox.
Do time intervals depend on who observes them? Intuitively, we expect the time for a process, such as the elapsed time for a foot race, to be the same for all observers. Our experience has been that disagreements over elapsed time have to do with the accuracy of measuring time. When we carefully consider just how time is measured, however, we will find that elapsed time depends on the relative motion of an observer with respect to the process being measured.

### Simultaneity

Consider how we measure elapsed time. If we use a stopwatch, for example, how do we know when to start and stop the watch? One method is to use the arrival of light from the event, such as observing a light turning green to start a drag race. The timing will be more accurate if some sort of electronic detection is used, avoiding human reaction times and other complications.
Now suppose we use this method to measure the time interval between two flashes of light produced by flash lamps. (See .) Two flash lamps with observer A midway between them are on a rail car that moves to the right relative to observer B. Observer B arranges for the light flashes to be emitted just as A passes B, so that both A and B are equidistant from the lamps when the light is emitted. Observer B measures the time interval between the arrival of the light flashes. According to postulate 2, the speed of light is not affected by the motion of the lamps relative to B. Therefore, light travels equal distances to him at equal speeds. Thus observer B measures the flashes to be simultaneous.
Now consider what observer A sees happening. Since both lamps are the same distance from her in her reference frame and the train is moving to the right, she perceives the flash from the right-hand bulb occurring before the left-hand bulb. Here a relative velocity between observers affects whether two events are observed to be simultaneous. Simultaneity is not absolute..
This illustrates the power of clear thinking. We might have guessed incorrectly that if light is emitted simultaneously, then two observers halfway between the sources would see the flashes simultaneously. But careful analysis shows this not to be the case. Einstein was brilliant at this type of thought experiment (in German, “Gedankenexperiment”). He very carefully considered how an observation is made and disregarded what might seem obvious. The validity of thought experiments, of course, is determined by actual observation. The genius of Einstein is evidenced by the fact that experiments have repeatedly confirmed his theory of relativity.
In summary: Two events are defined to be simultaneous if an observer measures them as occurring at the same time (such as by receiving light from the events). Two events are not necessarily simultaneous to all observers.

### Time Dilation

The consideration of the measurement of elapsed time and simultaneity leads to an important relativistic effect.
Suppose, for example, an astronaut measures the time it takes for light to cross her ship, bounce off a mirror, and return. (See .) How does the elapsed time the astronaut measures compare with the elapsed time measured for the same event by a person on the Earth? Asking this question (another thought experiment) produces a profound result. We find that the elapsed time for a process depends on who is measuring it. In this case, the time measured by the astronaut is smaller than the time measured by the Earth-bound observer. The passage of time is different for the observers because the distance the light travels in the astronaut’s frame is smaller than in the Earth-bound frame. Light travels at the same speed in each frame, and so it will take longer to travel the greater distance in the Earth-bound frame.
To quantitatively verify that time depends on the observer, consider the paths followed by light as seen by each observer. (See (c).) The astronaut sees the light travel straight across and back for a total distance of , twice the width of her ship. The Earth-bound observer sees the light travel a total distance . Since the ship is moving at speed  to the right relative to the Earth, light moving to the right hits the mirror in this frame. Light travels at a speed  in both frames, and because time is the distance divided by speed, the time measured by the astronaut is
This time has a separate name to distinguish it from the time measured by the Earth-bound observer.
In the case of the astronaut observe the reflecting light, the astronaut measures proper time. The time measured by the Earth-bound observer is
To find the relationship between  and , consider the triangles formed by  and . (See (c).) The third side of these similar triangles is , the distance the astronaut moves as the light goes across her ship. In the frame of the Earth-bound observer,
Using the Pythagorean Theorem, the distance  is found to be
Substituting  into the expression for the time interval  gives
We square this equation, which yields
Note that if we square the first expression we had for , we get . This term appears in the preceding equation, giving us a means to relate the two time intervals. Thus,
Gathering terms, we solve for :
Thus,
Taking the square root yields an important relationship between elapsed times:
where
This equation for  is truly remarkable. First, as contended, elapsed time is not the same for different observers moving relative to one another, even though both are in inertial frames. Proper time  measured by an observer, like the astronaut moving with the apparatus, is smaller than time measured by other observers. Since those other observers measure a longer time , the effect is called time dilation. The Earth-bound observer sees time dilate (get longer) for a system moving relative to the Earth. Alternatively, according to the Earth-bound observer, time slows in the moving frame, since less time passes there. All clocks moving relative to an observer, including biological clocks such as aging, are observed to run slow compared with a clock stationary relative to the observer.
Note that if the relative velocity is much less than the speed of light (), then  is extremely small, and the elapsed times  and  are nearly equal. At low velocities, modern relativity approaches classical physics—our everyday experiences have very small relativistic effects.
The equation  also implies that relative velocity cannot exceed the speed of light. As

 approaches ,

 approaches infinity. This would imply that time in the astronaut’s frame stops at the speed of light. If  exceeded , then we would be taking the square root of a negative number, producing an imaginary value for .
There is considerable experimental evidence that the equation  is correct. One example is found in cosmic ray particles that continuously rain down on the Earth from deep space. Some collisions of these particles with nuclei in the upper atmosphere result in short-lived particles called muons. The half-life (amount of time for half of a material to decay) of a muon is
 when it is at rest relative to the observer who measures the half-life. This is the proper time
. Muons produced by cosmic ray particles have a range of velocities, with some moving near the speed of light. It has been found that the muon’s half-life as measured by an Earth-bound observer () varies with velocity exactly as predicted by the equation . The faster the muon moves, the longer it lives. We on the Earth see the muon’s half-life time dilated—as viewed from our frame, the muon decays more slowly than it does when at rest relative to us.
Another implication of the preceding example is that everything an astronaut does when moving at  of the speed of light relative to the Earth takes 3.20 times longer when observed from the Earth. Does the astronaut sense this? Only if she looks outside her spaceship. All methods of measuring time in her frame will be affected by the same factor of 3.20. This includes her wristwatch, heart rate, cell metabolism rate, nerve impulse rate, and so on. She will have no way of telling, since all of her clocks will agree with one another because their relative velocities are zero. Motion is relative, not absolute. But what if she does look out the window?

### The Twin Paradox

An intriguing consequence of time dilation is that a space traveler moving at a high velocity relative to the Earth would age less than her Earth-bound twin. Imagine the astronaut moving at such a velocity that , as in . A trip that takes 2.00 years in her frame would take 60.0 years in her Earth-bound twin’s frame. Suppose the astronaut traveled 1.00 year to another star system. She briefly explored the area, and then traveled 1.00 year back. If the astronaut was 40 years old when she left, she would be 42 upon her return. Everything on the Earth, however, would have aged 60.0 years. Her twin, if still alive, would be 100 years old.
The situation would seem different to the astronaut. Because motion is relative, the spaceship would seem to be stationary and the Earth would appear to move. (This is the sensation you have when flying in a jet.) If the astronaut looks out the window of the spaceship, she will see time slow down on the Earth by a factor of . To her, the Earth-bound sister will have aged only 2/30 (1/15) of a year, while she aged 2.00 years. The two sisters cannot both be correct.
As with all paradoxes, the premise is faulty and leads to contradictory conclusions. In fact, the astronaut’s motion is significantly different from that of the Earth-bound twin. The astronaut accelerates to a high velocity and then decelerates to view the star system. To return to the Earth, she again accelerates and decelerates. The Earth-bound twin does not experience these accelerations. So the situation is not symmetric, and it is not correct to claim that the astronaut will observe the same effects as her Earth-bound twin. If you use special relativity to examine the twin paradox, you must keep in mind that the theory is expressly based on inertial frames, which by definition are not accelerated or rotating. Einstein developed general relativity to deal with accelerated frames and with gravity, a prime source of acceleration. You can also use general relativity to address the twin paradox and, according to general relativity, the astronaut will age less. Some important conceptual aspects of general relativity are discussed in General Relativity and Quantum Gravity of this course.
In 1971, American physicists Joseph Hafele and Richard Keating verified time dilation at low relative velocities by flying extremely accurate atomic clocks around the Earth on commercial aircraft. They measured elapsed time to an accuracy of a few nanoseconds and compared it with the time measured by clocks left behind. Hafele and Keating’s results were within experimental uncertainties of the predictions of relativity. Both special and general relativity had to be taken into account, since gravity and accelerations were involved as well as relative motion.

### Section Summary

1. Two events are defined to be simultaneous if an observer measures them as occurring at the same time. They are not necessarily simultaneous to all observers—simultaneity is not absolute.
2. Time dilation is the phenomenon of time passing slower for an observer who is moving relative to another observer.
3. Observers moving at a relative velocity  do not measure the same elapsed time for an event. Proper time  is the time measured by an observer at rest relative to the event being observed. Proper time is related to the time  measured by an Earth-bound observer by the equation
   where

4. The equation relating proper time and time measured by an Earth-bound observer implies that relative velocity cannot exceed the speed of light.
5. The twin paradox asks why a twin traveling at a relativistic speed away and then back towards the Earth ages less than the Earth-bound twin. The premise to the paradox is faulty because the traveling twin is accelerating. Special relativity does not apply to accelerating frames of reference.
6. Time dilation is usually negligible at low relative velocities, but it does occur, and it has been verified by experiment.

### Conceptual Questions

### Problems & Exercises


# Special Relativity

## Length Contraction

### Learning Objectives

By the end of this section, you will be able to:
1. Describe proper length.
2. Calculate length contraction.
3. Explain why we don’t notice these effects at everyday scales.
Have you ever driven on a road that seems like it goes on forever? If you look ahead, you might say you have about 10 km left to go. Another traveler might say the road ahead looks like it’s about 15 km long. If you both measured the road, however, you would agree. Traveling at everyday speeds, the distance you both measure would be the same. You will read in this section, however, that this is not true at relativistic speeds. Close to the speed of light, distances measured are not the same when measured by different observers.

### Proper Length

One thing all observers agree upon is relative speed. Even though clocks measure different elapsed times for the same process, they still agree that relative speed, which is distance divided by elapsed time, is the same. This implies that distance, too, depends on the observer’s relative motion. If two observers see different times, then they must also see different distances for relative speed to be the same to each of them.
The muon discussed in  illustrates this concept. To an observer on the Earth, the muon travels at  for  from the time it is produced until it decays. Thus it travels a distance
relative to the Earth. In the muon’s frame of reference, its lifetime is only . It has enough time to travel only
The distance between the same two events (production and decay of a muon) depends on who measures it and how they are moving relative to it.
The Earth-bound observer measures the proper length , because the points at which the muon is produced and decays are stationary relative to the Earth. To the muon, the Earth, air, and clouds are moving, and so the distance  it sees is not the proper length.

### Length Contraction

To develop an equation relating distances measured by different observers, we note that the velocity relative to the Earth-bound observer in our muon example is given by
The time relative to the Earth-bound observer is , since the object being timed is moving relative to this observer. The velocity relative to the moving observer is given by
The moving observer travels with the muon and therefore observes the proper time . The two velocities are identical; thus,
We know that . Substituting this equation into the relationship above gives
Substituting for  gives an equation relating the distances measured by different observers.
If we measure the length of anything moving relative to our frame, we find its length  to be smaller than the proper length  that would be measured if the object were stationary. For example, in the muon’s reference frame, the distance between the points where it was produced and where it decayed is shorter. Those points are fixed relative to the Earth but moving relative to the muon. Clouds and other objects are also contracted along the direction of motion in the muon’s reference frame.
People could be sent very large distances (thousands or even millions of light years) and age only a few years on the way if they traveled at extremely high velocities. But, like emigrants of centuries past, they would leave the Earth they know forever. Even if they returned, thousands to millions of years would have passed on the Earth, obliterating most of what now exists. There is also a more serious practical obstacle to traveling at such velocities; immensely greater energies than classical physics predicts would be needed to achieve such high velocities. This will be discussed in Relatavistic Energy.
Why don’t we notice length contraction in everyday life? The distance to the grocery shop does not seem to depend on whether we are moving or not. Examining the equation , we see that at low velocities () the lengths are nearly equal, the classical expectation. But length contraction is real, if not commonly experienced. For example, a charged particle, like an electron, traveling at relativistic velocity has electric field lines that are compressed along the direction of motion as seen by a stationary observer. (See .) As the electron passes a detector, such as a coil of wire, its field interacts much more briefly, an effect observed at particle accelerators such as the 3 km long Stanford Linear Accelerator (SLAC). In fact, to an electron traveling down the beam pipe at SLAC, the accelerator and the Earth are all moving by and are length contracted. The relativistic effect is so great than the accelerator is only 0.5 m long to the electron. It is actually easier to get the electron beam down the pipe, since the beam does not have to be as precisely aimed to get down a short pipe as it would down one 3 km long. This, again, is an experimental verification of the Special Theory of Relativity.

### Summary

1. All observers agree upon relative speed.
2. Distance depends on an observer’s motion. Proper length   is the distance between two points measured by an observer who is at rest relative to both of the points. Earth-bound observers measure proper length when measuring the distance between two points that are stationary relative to the Earth.

3. Length contraction  is the shortening of the measured length of an object moving relative to the observer’s frame:

### Conceptual Questions

### Problems & Exercises


# Special Relativity

## Relativistic Addition of Velocities

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate relativistic velocity addition.
2. Explain when relativistic velocity addition should be used instead of classical addition of velocities.
3. Calculate relativistic Doppler shift.
If you’ve ever seen a kayak move down a fast-moving river, you know that remaining in the same place would be hard. The river current pulls the kayak along. Pushing the oars back against the water can move the kayak forward in the water, but that only accounts for part of the velocity. The kayak’s motion is an example of classical addition of velocities. In classical physics, velocities add as vectors. The kayak’s velocity is the vector sum of its velocity relative to the water and the water’s velocity relative to the riverbank.

### Classical Velocity Addition

For simplicity, we restrict our consideration of velocity addition to one-dimensional motion. Classically, velocities add like regular numbers in one-dimensional motion. (See .) Suppose, for example, a girl is riding in a sled at a speed 1.0 m/s relative to an observer. She throws a snowball first forward, then backward at a speed of 1.5 m/s relative to the sled. We denote direction with plus and minus signs in one dimension; in this example, forward is positive. Let  be the velocity of the sled relative to the Earth,  the velocity of the snowball relative to the Earth-bound observer, and  the velocity of the snowball relative to the sled.
Thus, when the girl throws the snowball forward, . It makes good intuitive sense that the snowball will head towards the Earth-bound observer faster, because it is thrown forward from a moving vehicle. When the girl throws the snowball backward, . The minus sign means the snowball moves away from the Earth-bound observer.

### Relativistic Velocity Addition

The second postulate of relativity (verified by extensive experimental observation) says that classical velocity addition does not apply to light. Imagine a car traveling at night along a straight road, as in . If classical velocity addition applied to light, then the light from the car’s headlights would approach the observer on the sidewalk at a speed . But we know that light will move away from the car at speed  relative to the driver of the car, and light will move towards the observer on the sidewalk at speed , too.
Velocities cannot add to greater than the speed of light, provided that  is less than  and  does not exceed . The following example illustrates that relativistic velocity addition is not as symmetric as classical velocity addition.

### Doppler Shift

Although the speed of light does not change with relative velocity, the frequencies and wavelengths of light do. First discussed for sound waves, a Doppler shift occurs in any wave when there is relative motion between source and observer.
In the Doppler equation,  is the observed wavelength,  is the source wavelength, and  is the relative velocity of the source to the observer. The velocity  is positive for motion away from an observer and negative for motion toward an observer. In terms of source frequency and observed frequency, this equation can be written
Notice that the – and + signs are different than in the wavelength equation.
The relativistic Doppler shift is easy to observe. This equation has everyday applications ranging from Doppler-shifted radar velocity measurements of transportation to Doppler-radar storm monitoring. In astronomical observations, the relativistic Doppler shift provides velocity information such as the motion and distance of stars.

### Test Prep for AP Courses

### Section Summary

1. With classical velocity addition, velocities add like regular numbers in one-dimensional motion: , where  is the velocity between two observers,

 is the velocity of an object relative to one observer, and  is the velocity relative to the other observer.
2. Velocities cannot add to be greater than the speed of light. Relativistic velocity addition describes the velocities of an object moving at a relativistic speed:

3. An observer of electromagnetic radiation sees relativistic Doppler effects if the source of the radiation is moving relative to the observer. The wavelength of the radiation is longer (called a red shift) than that emitted by the source when the source moves away from the observer and shorter (called a blue shift) when the source moves toward the observer. The shifted wavelength is described by the equation
     is the observed wavelength,  is the source wavelength, and  is the relative velocity of the source to the observer.

### Conceptual Questions

### Problems & Exercises