24.6

Polarization

| Polarized Electromagnetic Waves

One of the essential features of electromagnetic waves is that they are transverse waves, and because of this feature they can be polarized. Figure 24-18 illustrates the idea of polarization by showing a transverse wave as it travels along a rope toward a slit. The wave is said to be linearly polarized, which means that its vibrations always occur along one direction. This direction is called the direction of polarization. In part a of the picture, the direction of polarization is vertical, parallel to the slit. Consequently, the wave passes through easily. However, when the slit is turned perpendicular to the direction of polarization, as in part b, the wave cannot pass, because the slit prevents the rope from oscillating. For longitudinal waves, such as sound waves, the notion of polarization has no meaning. In a longitudinal wave the direction of vibration is along the direction of travel, and the orientation of the slit would have no effect on the wave.

Figure 24-18 A transverse wave is linearly polarized when its vibrations always occur along one direction. (a) A linearly polarized wave on a rope can pass through a slit that is parallel to the direction of the rope vibrations, but (b) cannot pass through a slit that is perpendicular to the vibrations.

In an electromagnetic wave such as that in Figure 24-4, the electric field oscillates along the y axis. Similarly, the magnetic field oscillates along the z axis. Therefore, the wave is linearly polarized, with the direction of polarization taken arbitrarily to be that along which the electric field oscillates. If the wave is a radio wave generated by a straight-wire antenna, the direction of polarization is determined by the orientation of the antenna. In comparison, the visible light given off by an incandescent light bulb consists of electromagnetic waves that are completely unpolarized. In this case the waves are emitted by a large number of atoms in the hot filament of the bulb. When an electron in an atom oscillates, the atom behaves as a miniature antenna that broadcasts light for brief periods of time, about 10−8 seconds. However, the directions of these atomic antennas change randomly as a result of collisions. Unpolarized light, then, consists of many individual waves, emitted in short bursts by many “atomic antennas,” each with its own direction of polarization. Figure 24-19 compares polarized and unpolarized light. In the unpolarized case, the arrows shown around the direction of wave travel symbolize the random directions of polarization of the individual waves that comprise the light.

Figure 24-19 In polarized light, the electric field of the electromagnetic wave fluctuates along a single direction. Unpolarized light consists of short bursts of electromagnetic waves emitted by many different atoms. The electric field directions of these bursts are perpendicular to the direction of wave travel but are distributed randomly about it.

Linearly polarized light can be produced from unpolarized light with the aid of certain materials. One commercially available material goes under the name of Polaroid. Such materials allow only the component of the electric field along one direction to pass through, while absorbing the field component perpendicular to this direction. As Figure 24-20 indicates, the direction of polarization that a polarizing material allows through is called the transmission axis. No matter how this axis is oriented, the intensity of the transmitted polarized light is one-half that of the incident unpolarized light. The reason for this is that the unpolarized light contains all polarization directions to an equal extent. Moreover, the electric field for each direction can be resolved into components perpendicular and parallel to the transmission axis, with the result that the average components perpendicular and parallel to the axis are equal. As a result, the polarizing material absorbs as much of the electric (and magnetic) field strength as it transmits.

Figure 24-20 With the aid of a piece of polarizing material, polarized light may be produced from unpolarized light. The transmission axis of the material is the direction of polarization of the light that passes through the material.

| Malus’ Law

Once polarized light has been produced with a piece of polarizing material, it is possible to use a second piece to change the polarization direction and simultaneously adjust the intensity of the light. Figure 24-21 shows how. As in this picture the first piece of polarizing material is called the polarizer, and the second piece is referred to as the analyzer. The transmission axis of the analyzer is oriented at an angle relative to the transmission axis of the polarizer. If the electric field strength of the polarized light incident on the analyzer is E, the field strength passing through is the component parallel to the transmission axis, or . According to Equation 24.5b, the intensity is proportional to the square of the electric field strength. Consequently, the average intensity of polarized light passing through the analyzer is proportional to . Thus, both the polarization direction and the intensity of the light can be adjusted by rotating the transmission axis of the analyzer relative to that of the polarizer. The average intensity of the light leaving the analyzer, then, is

| (24.7)

where is the average intensity of the light entering the analyzer. Equation 24.7 is sometimes called Malus’ law, for it was discovered by the French engineer Ètienne Louis Malus (1775–1812). Example 7 illustrates the use of Malus’ law.

Figure 24-21 Two sheets of polarizing material, called the polarizer and the analyzer, may be used to adjust the polarization direction and intensity of the light reaching the photocell. This can be done by changing the angle between the transmission axes of the polarizer and analyzer.

| Example 7 | | | Using Polarizers and Analyzers

What value of should be used in Figure 24-21 so that the average intensity of the polarized light reaching the photocell will be one-tenth the average intensity of the unpolarized light?

Problem solving insight Remember that when unpolarized light strikes a polarizer, only one half of the incident light is transmitted, the other half being absorbed by the polarizer.

Reasoning Both the polarizer and the analyzer reduce the intensity of the light. The polarizer reduces the intensity by a factor of one-half, as discussed earlier. Therefore, if the average intensity of the unpolarized light is , the average intensity of the polarized light leaving the polarizer and striking the analyzer is . The angle must now be selected so that the average intensity of the light leaving the analyzer will be . Malus’ law provides the solution.
Solution Using and in Malus’ law, we find

When in Figure 24-21, the polarizer and analyzer are said to be crossed, and no light is transmitted by the polarizer/analyzer combination. As an illustration of this effect, Figure 24-22 shows two pairs of Polaroid sunglasses in uncrossed and crossed configurations.

Figure 24-22 When Polaroid sunglasses are uncrossed (top photograph), the transmitted light is dimmed due to the extra thickness of tinted plastic. However, when they are crossed (bottom photograph), the intensity of the transmitted light is reduced to zero because of the effects of polarization. (Diane Schiumo/Fundamental Photographs)

The physics of IMAX 3-D films. An exciting application of crossed polarizers is used in viewing IMAX 3-D movies. These movies are recorded on two separate rolls of film, using a camera that provides images from the two different perspectives that correspond to what is observed by human eyes and allow us to see in three dimensions. The camera has two apertures or openings located at roughly the spacing between our eyes. The films are projected using a projector with two lenses, as Figure 24-23 indicates. Each lens has its own polarizer, and the two polarizers are crossed (see the drawing). In one type of theater, viewers watch the action on-screen using glasses with corresponding polarizers for the left and right eyes, as the drawing shows. Because of the crossed polarizers the left eye sees only the image from the left lens of the projector, and the right eye sees only the image from the right lens. Since the two images have the approximate perspectives that the left and right eyes would see in reality, the brain combines the images to produce a realistic 3-D effect.

Figure 24-23 In an IMAX 3-D film, two separate rolls of film are projected using a projector with two lenses, each with its own polarizer. The two polarizers are crossed. Viewers watch the action on-screen through glasses that have corresponding crossed polarizers for each eye. The result is a 3-D moving picture, as the text discusses.

Conceptual Example 8 illustrates an interesting result that occurs when a piece of polarizing material is inserted between a crossed polarizer and analyzer.

| Conceptual Example 8 | | | How Can a Crossed Polarizer and Analyzer Transmit Light?

As explained earlier, no light reaches the photocell in Figure 24-21 when the polarizer and analyzer are crossed. Suppose that a third piece of polarizing material is inserted between the polarizer and analyzer, as in Figure 24-24a. Does light now reach the photocell?

Figure 24-24 (a) Light reaches the photocell when a piece of polarizing material is inserted between a crossed polarizer and analyzer. (b) The component of the electric field parallel to the transmission axis of the insert is . (c) Light incident on the analyzer has a component parallel to its transmission axis.

Reasoning and Solution Surprisingly, the answer is yes, even though the polarizer and analyzer remain crossed. To show why, we note that if any light is to pass through the analyzer, it must have an electric field component parallel to the transmission axis of the analyzer. Without the insert, there is no such component. But with the insert, there is, as parts b and c of the drawing illustrate. Part b shows that the electric field E of the light leaving the polarizer makes an angle with respect to the transmission axis of the insert. The component of the field that is parallel to the insert’s transmission axis is , and this component passes through the insert. Part c of the drawing illustrates that the field () incident on the analyzer has a component parallel to the transmission axis of the analyzer—namely, . This component passes through the analyzer, so that light now reaches the photocell.
Light will reach the photocell as long as the angle is between 0 and 90°. If the angle is 0° or 90°, however, no light reaches the photocell. Can you explain, without using any equations, why no light reaches the photocell under either of these conditions?
Related Homework: Problems 37 and 40

Need more practice?

Interactive LearningWare24.1
The drawing shows three sheets of polarizing material. The orientation of the transmission axis for each is labeled relative to the vertical direction. A beam of unpolarized light, whose intensity is 1550 W/m2, is incident on the first sheet.What is the intensity of the beam transmitted through the three sheets when , , and ?

Related Homework: Problem 36

An application of a crossed polarizer/analyzer combination occurs in one kind of liquid crystal display (LCD). LCDs are widely used in pocket calculators and cell phones. The display usually consists of blackened numbers and letters set against a light gray background. As Figure 24-25 indicates, each number or letter is formed from a combination of liquid crystal segments that have been turned on and appear black. The liquid crystal part of an LCD segment consists of the liquid crystal material sandwiched between two transparent electrodes, as in Figure 24-26. When a voltage is applied between the electrodes, the liquid crystal is said to be “on.” Part a of the picture shows that linearly polarized incident light passes through the “on” material without having its direction of polarization affected. When the voltage is removed, as in part b, the liquid crystal is said to be “off” and now rotates the direction of polarization by 90°. A complete LCD segment also includes a crossed polarizer/analyzer combination, as Figure 24-27 illustrates. The polarizer, analyzer, electrodes, and liquid crystal material are packaged as a single unit. The polarizer produces polarized light from incident unpolarized light. With the display segment turned on, as in Figure 24-27, the polarized light emerges from the liquid crystal only to be absorbed by the analyzer, since the light is polarized perpendicular to the transmission axis of the analyzer. Since no light emerges from the analyzer, an observer sees a black segment against a light gray background, as in Figure 24-25. On the other hand, the segment is turned off when the voltage is removed, in which case the liquid crystal rotates the direction of polarization by 90° to coincide with the axis of the analyzer. The light now passes through the analyzer and enters the eye of the observer. However, the light coming from the segment has been designed to have the same color and shade (light gray) as the background of the display, so the segment becomes indistinguishable from the background.

Figure 24-25 Liquid crystal displays (LCDs) use liquid crystal segments to form the numbers.

Figure 24-26 A liquid crystal in its (a) “on” state and (b) “off” state.

Figure 24-27 An LCD incorporates a crossed polarizer/analyzer combination. When the LCD segment is turned on (voltage applied), no light is transmitted through the analyzer, and the observer sees a black segment.

The physics of a liquid crystal display.

Color LCD display screens and computer monitors are popular because they occupy less space and weigh less than traditional units do. An LCD display screen, such as that in Figure 24-28, uses thousands of LCD segments arranged like the squares on graph paper. To produce color, three segments are grouped together to form a tiny picture element (or “pixel”). Color filters are used to enable one segment in the pixel to produce red light, one to produce green, and one to produce blue. The eye blends the colors from each pixel into a composite color. By varying the intensity of the red, green, and blue colors, the pixel can generate an entire spectrum of colors.

Figure 24-28 Want to see a photograph of yourself? Just look at the LCD display on a properly equipped cell phone. (Rubberball/Getty Images, Inc.)

| The Occurrence of Polarized Light in Nature

Polaroid is a familiar material because of its widespread use in sunglasses. Such sunglasses are designed so that the transmission axis of the Polaroid is oriented vertically when the glasses are worn in the usual fashion. Thus, the glasses prevent any light that is polarized horizontally from reaching the eye. Light from the sun is unpolarized, but a considerable amount of horizontally polarized sunlight originates by reflection from horizontal surfaces such as that of a lake. Section 26.4 discusses this effect. Polaroid sunglasses reduce glare by preventing the horizontally polarized reflected light from reaching the eyes.

The physics of Polaroid sunglasses.

Polarized sunlight also originates from the scattering of light by molecules in the atmosphere. Figure 24-29 shows light being scattered by a single atmospheric molecule. The electric fields in the unpolarized sunlight cause the electrons in the molecule to vibrate perpendicular to the direction in which the light is traveling. The electrons, in turn, reradiate the electromagnetic waves in different directions, as the drawing illustrates. The light radiated straight ahead in direction A is unpolarized, just like the incident light. But light radiated perpendicular to the incident light in direction C is polarized. Light radiated in the intermediate direction B is partially polarized.

Figure 24-29 In the process of being scattered from atmospheric molecules, unpolarized light from the sun becomes partially polarized.

The physics of butterflies and polarized light.

Researchers have now discovered that at least one butterfly species uses polarized light to attract members of the opposite sex. The butterfly species Heliconius has patterns on its wings that cause light reflected from them to be polarized. This polarized light, invisible to the human eye but visible to other butterflies, is attractive to potential mates. When males were shown the female wings with their polarized light patterns, they swarmed toward the wings. When the males were shown the wings through a filter that blocked out the polarization effects, they largely ignored the wings. Figure 24-30 shows the polarized light reflected from the wings of the Heliconius cydno butterfly (top) and the H. melpomene malleti butterfly (bottom), which does not produce polarized light. Note the right halves of the wings, seen through a polarizing filter; the Heliconius cydno wing clearly exhibits the polarized pattern, while that of the H. melpomene malleti does not. There is also experimental evidence that some bird species use polarized light as a navigational aid.

Figure 24-30 The right halves of these butterfly wings are seen through a polarizing filter. The wing on the Heliconius cydno (top) exhibits polarization patterns, while that on the H. melpomene malleti (bottom) does not. (Courtesy Alison Sweeney, Duke University. Image from Nature 423, 31–32, May 1, 2003. Reproduced with permission.)

Check Your Understanding 4

The drawing shows two sheets of polarizing material. The transmission axis of one is vertical, and that of the other makes an angle of 45° with the vertical. Unpolarized light shines on this arrangement from the left and from the right. From which direction does at least some of the light pass through both sheets: (a) from the left, (b) from the right, (c) from either direction, or (d) from neither direction? What is the answer when the light is horizontally polarized? (The answers are given at the end of the book.)

Background: Light passes through a sheet of polarizing material only when the electric field of the light wave has a component parallel to the transmission axis of the material.
For similar questions (including calculational counterparts), consult Self-Assessment Test 24.1, which is described next.

Test your understanding of the material in Sections 24.1, 24.2, 24.3, 24.4, 24.5 and 24.6 :

· | The Nature of Electromagnetic Waves
· | The Electromagnetic Spectrum
· | The Speed of Light
· | The Energy Carried by Electromagnetic Waves
· | The Doppler Effect and Electromagnetic Waves
· | Polarization

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.