Scattering and Polarization

C HAP T E R 13

13.1 these two molecules interferes de­ 1 3.2 INTRODUCTION structively. as in Figure 12.4 (along RAYLEIGH SCATTERING the x-axis). and this Is so for any - sideways direction from which you - So far we have considered what look at the beam. Air. on the other When white light scatters from hap pens when light encounters ma­ hand, is a gas. so there is no guar­ some molecules. it scatters selec­ terial obstacles of a size much antee that there will be another tively because part of the light is ab ­ greater than the wavelength (geo­ molecule half a wavelength beyond sorbed at the resonant frequ encies metrical optics) or of a size so small the first-sometimes there may be a of the molecules-the scattered as to be comparable with the wave­ few extra molecules around one light Is then colored. For man y length of light (wave optics). But point, sometimes a few less. You see other molecules. however, the im­ you can also observe effects due the scattering from these pOints be­ portant resonant frequenCies are to even smaller obstacles, much cause of these fluctuations. (The significantly higher than visible fre­ smaller than the wavelength of visi­ TRY IT suggests ways to enhance quencies. White light nevertheless ble light. When light interacts with the scattering in air.) becomes colored when it scatters an isolated object that small, it Scattering is selective in several from these molecules-the higher shakes all the charges in the object, ways: light of certain wavelengths the frequency of the incident light. which th en radiate in all directions. is scattered more than light of other the more light will be scattered. This phenomenon is called scatter­ wavelengths. and light of one polar­ ThiS type of scattering is called ing. ization (Sec. 1.3B) is scattered Rayleigh scattering and occurs To scatter with appreciable inten­ more than light of another polari­ whenever the scattering particles sity, light must encounter many zation. Because our eyes are not are much smaller than the incident isolated small objects: for example, very sensitive to the polarization of wavelength and have resonances at the m olecules that constitute the light. we have not yet discussed the frequenCies higher than those of air. Indeed, it is onJy because light phenomena associated with it. visible light. Eq.uivalently, we may does scatter in air that you can Scattering provides us with the op­ write the rule for Rayleigh scatter­ see the beams of Figures 1.3, 1.4, portunity to do so, even as it ac­ ing: and 8.19b. Without scattering, the tually prOvides most of the polarized light's path through the air would light around us. Much of this chap­ The shorter the wavelength be invisible and the camera could ter will therefore deal with polarized of the incident light, the not record it. light, produced by scattering as well more light is scattered This simple explanation, offered as by other means. when these figu res were intro­ duced, now bears closer scrutiny. It This result was worked out in detail would seem that when propagating TR Y IT by the same Lord Rayleigh we met in any dense medium, such as in Sections 10.5B and 12.5C. It glass, light should be scattered by FOR SECTION 13.1 says that blue light will be scattered the many molecules that are pres­ Light beams more than red light. In fact, for in­ ent. Instead. as we know. it contin­ cident broad-band white light, the ues to propagate in a sharp beam, To see the path of light rays, you need intenSity of scattered 400-nm light as in vacuum, only with a different tiny objects in the path that will scatter Is almost ten times as great as that part of the light to your eye. To see the speed. This is because there are of 700-nm light. beam of your flashlight, put larger One consequence of Rayleigh scat­ many molecules present in glass, particles in the air, such as dust motes, and wh enever there is one molecule smoke, or chalk dust. In their presence tering is the sky's blue color. Light to scatter light, there will be an­ yo u should be able to trace the light's reaching your eyes from the sky is other one for which the light path path, including reflections from mirrors sunlight that has been scattered by to .your eye is half a wavelength and water surfaces, to test the law ofreflec­ the air molecules (Fig. 13.1 ) and is longer. The scattered light from tion (Sec. 2.4) and Snell's law (Sec. 2.5). therefore predominately blue. Since 347 CHAPTER I3, SCATTERING AND POLARIZATION

3 4 8

the direct rays from the sun have There are a number of other ex­ or sugar, talcum powder, ch alk , the some of the blue part of the spec­ amples where blue coloring is due white spots of moths and butter­ trtim scattered out of them they to Rayleigh scattering. We've al­ flies, white paper, fog, snow, beaten should look slightly yellowish. When ready noted (Sec. 9.9E) that fine egg white, and beer foam all look the sun is overhead, and if the sky black pigment mixed into white white for the same reason. The is very clear, this is a small effect. paint gives it a bluish cast for just white pattern in star rubies and However , if there are lots of tiny this reason. In fact, da Vinci noted sapphires, and in tiger's-eye quartz du st and smoke particles in the air, this and attributed it to the same are Similarly due to scattering from the effect is larger. It becomes even cause that makes the sky blue. The large inclusions. Likewise, the clear larger as the sun sets; the direct writer George MacDonald correctly albumen becomes white as an egg rays from it to your eyes must pass identifies another blue with the is cooked because the protein mol­ through more and more atmo­ blue of the sky when he writes, ecules are freed of their surface wa­ sphere, so these rays are depleted of "Where did you get those eyes so ter and are then able to coagulate more and more of the shorter wave­ blue?IOut of the sky as I came into large clumps, which scatter lengths, and the sun looks redder through." Your beautiful blue eyes nonselectively. When the watery and redder. (See the TRY IT.) are due to scattering from small, whey from milk is made into cheese widely separated particles in your (such as ricottal, the cheese is IrIses. Similarly, the moonstone white because of the same coagula­ owes its blue sheen to Rayleigh tion process. scattering. Yellow lights are often u sed as fog You can see the same effect in lights or headlights on cars because fine smoke, say from a wood fire. yellow is as easily detected by your The smoke looks bluish when illu­ eyes as white but is scattered less in minated from the side and viewed a fine mist, when the droplets are against a dark background, so only very small compared to the wave­ scattered light reaches your eyes. If length of visible light. Un fortu­ instead the smoke is seen against a nately, more often the droplets are bright background, it looks red or larger, and the yellow is then scat­ brown (due to the removal of blue tered as well. by scattering). If the smoke gets too Particles may, of course, also pro­ thick, the particles become dense duce colors by selective absorption enough that all light is repeat­ and r€iflection. The smog we all edly scattered, and the light that know and love takes its brown color emerges sideways is white or gra,y. because of absorption by particles A similar effect can occur when of nitrous oxide, which has reso­ the particles become larger. For ex­ nances at visible frequencies. Not ample, the smoke rising from the all particles in the sky produce ugly end of a Cigarette is bluish, but af­ colors, however. For example, in ter you inhale and exhale it, the 1883 the volcano Krakatoa erupted smoke looks gray or white. Here you spectacularly, spewing micrometer­ have covered the smoke particles size particles into the atmosphere with mOisture, making them much in such abundance that all over the larger. They are then large enough world there were unusually colorful to scatter light of all wavelengths sunrises and sunsets for three equally-as in the reflections of geo­ years! metrical optics-and thus they look Scattering by the air or the parti­ fiGURE 13.1 white. This is also why clouds are cles in it is thus responsible for aer­ white: The water droplets in clouds ial perspective (Sec. 8.6El, wh ich When A looks at point C in the sky, only scattered light from the sun reaches her may be fifty times as large as the makes distant dark hills look blue eyes. As short-wavelength light is wavelength of visible light. With so and distant snow-clad peaks look scattered most, the sky looks blue to many droplets, and thus so many yellow (see Plate 8.4). The purer and her. Direct rays from the sun, from surfaces to reflect the light, the more transparent the air, the bluer which the blue end of the spectrum has been removed by scattering, vary from clouds scatter almost all the light those dark hills. white to yellow to red, depending on and look white even though the in­ how much atmosphere they have dividual drops are nearly transpar­ traversed and how much dust is in the ent. (Of course, very dense clouds air-that is, on how much scattering the light has suffered. Thus to B, who sees will not transmit light-they either the sun setting, the sun appears much absorb it or reflect it upward-so redder than to A. they look black.) Small grains of salt

• 13.3 POLARIZATION DUE TO SCATTERING

349 TRY IT need by sunlight. For example, if a good One dial causes a horizontal line to exposure by sunlight was _ sec, then, be drawn on a screen, the other with the moon in the same position as FO R SECTION 13.2 makes a vertical line. If you simul­ the sun was, you might open your lens Blue skies taneously turn both dials in the an additional three f-stops and make a same direction ("in phase"), you 20-minute exposure. Most color film 0 A good source of small scattering draw a 45 line. Another toy is a won't give accurate color with such a particles is milk, whose solid particles are long exposure-Kodak recommends maze with two knobs that control much smaller than the wavelength of Kodacolor 400 if you want prints, or its tilt in two perpendicular direc­ visible light. You can use these to make a Kodachrome 25 (daylight) with a CC10M tions. Simultaneous control of both blue "sky" and a red "sunset." In a dark filter if you want slides. knobs allows you to roll a ball in any room, shine a light beam, say from a direction through the maze. Analo­ flashlight, through a clear glass of water. Look at the beam from the side so you gously, any direction of the electric can see the scattered light. (It helps if field in the x-y plane can be thought there is a black background.) At the same 13.3 of as consisting of two components time look at the direct, transmitted light, POLARIZATION DUE that lie in two mutually perpendic­ for example, by reflecting it from a piece TO SCATTERING ular directions. Further, these di­ of white paper. Now add a little milk, rections can be any pair that we one or two drops at a time, and stir the choose, not necessarily the x- and water. The scattered light will become The same Rayleigh scattering that y-directions (Fig. 13.2e). While this bluish as the transmitted light becomes gives us the blue sky also polarizes seems like only a way to think yellowish and then reddish. As you add the scattered light. In order to see about things now, we11 see that Na­ even more milk, the scattered light becomes white because of repeated how this comes about, let's return ture thinks about them in just this scattering, and you have made a white to some of the basic ideas of polari­ way (Sec. 13.6). "cloud, " zation, which were first introduced What about unpolarized light? You can see the blue of the air directly in Section 1.38. Such light is polarized in all differ­ if you have enough air with a dark ent directions (perpendicular to the background, and something black with direction of propagation); the direc­ which to compare the color of the air. An A. Polarized light tion of the electric field varies rap­ otherwise dark room viewed from the idly and randomly-it has no pre­ outside through an open window makes Light, recall, is a transverse wave­ ferred direction. We can think of a good dark background. To block all the electric field is always perpen­ this as a wave with two components extraneous light, view the window through a long mailing tube. Since you'll dicular to the direction of propaga­ that have a rapid and random want to be as far as possible from the tion of the wave (the direction of the variation of their relative phase­ window (30 or 40 m ), cover the far end ray). For example, if the wave is the two components are incoher­ of the tube with some aluminum foil in traveling in the z-direction (Fig. ent. It is like the result of turning which you've made a small (several 13.2al, the electriC field may be in the two dials of the Etch-A-Sketch® millimeters) hole. To avoid light entering the x-direction, in the y-direction, back and forth with no particular the hole at an angle, wrap a piece of or in any other direction within the relation between them-you then black paper around the far end of the x-y plane. If the light wave's electriC get a line that wiggles around in all tube, so it extends 15 or 20 cm beyond field is always parallel to the x-axis, directions, going no place special. the foil. Look toward the window when we say the light is linearly polar­ Figure 13.2f shows how we']] indi­ the sun is to one side or overhead. Then the only light entering the tube will be ized in the x-direction (Figs. 13.2b cate such unpolarized light. the sunlight that is scattered by the air and c). Similarly, a wave whose With this information, let's see between you and the window. This electric field is always parallel to the why Rayleigh scattering produces should look distinctly bluish compared to y-axis is linearly polarized in the y­ polarized light. the dark surrounding of the aperture in direction. the foil. What happens to the blue as you Suppose the wave is linearly po­ change your distance from the window? larized in some other direction, say B. Polarization due Since Rayleigh scattering also occurs at at an angle of 450 between the x­to Rayleigh scattering night (except that there is less light to and y-axes. We may nevertheless scatter), a moonlight photograph taken think of this wave as consisting of A simple model can illustrate how with a long enough exposure time should show the same colors as one two in-phase waves (Fig. 13.2dl, scattering produces polarization. taken by daylight. Try it on a clear night one linearly polarized in the x-direc­ Figure 13.3 shows two long jump during a full moon. The moonlight is tion (the x-component) and one in ropes that are joined togeth er at then almost a factor of 1rf' weaker than the y-direction (the y-component). their centers by a ring and then sunlight. Because of the failure of There are a number of gadgets that stretched out tightly at right angles, reciprocity of the film at long exposures illustrate this concept. One, called so as to form a (horizontal) cross. (Sec. 4.6), you should make an exposure "Etch-A-Sketch®," allows you to The ring is like a scatterer in this about 1(f times as great as you would draw a picture by turning two dials. sense: If one end of one rope is wig­ CHAPTER 13, SCATTERING AND POLARlZATlON

350

x x x -y plane

+--+----o~ z ttt • z ..r----~------__ z

y

(a) (b) (e)

y y Electri c field y

Electric fi eld One component of electric ------~~~--~------x fi eld ~L-~----~~~------x ------r--~------x

x-component Anoth er co mpo nent of electric field of electric field (d) (e) (f)

Since the electric field of the u npo­ FIGU RE 13.2 larized wave pOints in all directions in the x-y plane, the charges in the (a) A light wave traveling in the z­ direction can have its electric field only scatterer will oscillate in all those in the x-V plane (or in any parallel plane). directions, but not in the z-dlrec­ (b) The electric field of a light wave that tion. As usual, we may think of my is linearly polarized in the x-direction such oscillations in the x-y plane as (al so commonly, but less appropriately, x - y­ called plane polarized). (e) A shorthand consisting of just an and a notation for the same wave as in (b). component. Only the x-componen t (d) Any electric field in the x-V plane can ! of oscillation radiates in the y-dl­ be thought of as a combination of a field (a) rection-the y-component cannot in the x-direction with one in the y­ (because the scattered wave is direction. (e) An electric field in one direction within a plane can be thought transverse) and there is no z-com­ of as a combination of fields in any two ponent (because the incident wave mutually perpendicular directions within is transverse). Hence, any light ra­ that plane. (f) Unpolarized light (traveling diated in the y-direction (to E or in the z-direction). J E 2 ) is linearly polarized in the x -di­ rection. Similarly, the light reach­ gled u p and down [linearly polarized ing observers at E J and E4 is lin­ in the x-direction). a wave propa­ .. early polarized in the y-direction. gates down that string, wiggling the (b) An observer at E s , however, sees the ring up and down (Fig. I3.3a). This incident wave as well as light scat ­ causes a (scattered) wave oscillating tered in the forward direction, both up and down to develop on the FIGURE 13.3 of which are unpolarized. cross rope. However, if the first rope Two crossed ropes are joined at their For the blue, Rayleigh scattered is wiggled sideways (linearly polar­ centers by a very light ring that can slide sky light this means that the b lue ized in the y-direction), it cannot freely along the ropes. (a) Wiggling one light is linearly polarized for light set up a wave in the cross rope be­ rope up and down (at the arrow) coming from pOints in the sky 90° produces a wave. This wave, in turn, cause the ring slides along tha t away from the sun (observers E J to creates an up-and-down wave in the rope and doesn't move it-there is cross rope. (b) The first rope is wiggled E 4)' However, the light coming from no motion transverse to the cross sideways and simply slides over the cross... directions near the sun (observer rope (Fig. 13.3b). Thus. if we wiggle rope, producing no wave in it. Es) or opposite the sun is unpo]ar ­ the first rope in both x- and y-direc­ ized. For regions in between, the tions, making an unpolarized light is partially polarized- a mix­ wave, the only wave resulting in the ing of electromagnetic waves. Imag­ ture of polarized and unpo!arized cross rope would be linearly polar­ ine that an unpolarized wave trav­ light. If there are large particles in ized in the vertical, x-direction. eling in the z-direction strikes a the air (as in s mog). the forces The same idea applies to scatter­ small scatterer a t 0 (Fig. 13.4). within them may cause the charges 13.3 POLARIZATION DUE TO SCATTERING

351 ans. Octopuses, for example, have even been trained to respond to 90° x changes In the direction of polari­ zation-so if you can't get a polar­ Izing filter for the TRY IT. you can / Incident unpolarized 7 wave use a trained octopus.

TRY IT

FOR SECTION 13.38 Polarization of the sky

You will need a polarizing filter such as used in polarizing sun glasses, viewers for 3-D movies, or polarizing camera filters. Alternatively, Figure 13.6 shows how you can make one. You can detect FIGURE 13.4 While our eyes are not sufficiently the presence of polarized light by noting An unpolarized wave traveling in the z­ sensitive to the polarization of light if the light becomes alternately dark and direction strikes a scatterer at O. The for this navigational trick, we can bright as you rotate the filter around your line of sight. wave that scatters along the y-axis is still use it if we have the help of po­ linearly polarized in the x-direction, larizing devices, which weU de­ First use the filter to convince yourself while that which scatters along the x-axis that sunlight is unpolarized. In order to scribe below. With one such device is linearly polarized in the y-direction. avoid looking directly at the sun, use a That which scatters in the z-direction is (cordierlte, a dichroic crystal-Sec. pinhole to project its image. Cover the unpolarized. 13.5), which they called a "sun pinhole with the filter and rotate the filter stone," Vikings are said to have while you look at the image. Does the used the polarization of the sky's image of the sun become darker and to oscillate in other directions than light to naVigate. Even today, the lighter? Next look through the filter at that of the electric field of the inci­ polarization of the sky In the twi­ some blue sky, 9(f' away from the sun. dent wave, so the scattered light is light is of use to airplane naVigators Again, rotate the filter. Is the light from polarized less, If at all. Repeated who fly over the poles. that part of the sky polarized, as it's scattering, as In clouds, causes the A polarizing device that transmits supposed to be? Try other parts of the sky. The more the polarization, the light to come out polarized in all di­ light of one polarization, but not greater the difference between dark and light of a perpendicular polariza­ rections-that is, unpolarized-so light as you rotate the filter. Where is the light from clouds is not polarized. tion, is called a polarizing .filter. polarization greatest and where least? . (The TRY IT tells you how to verify Because of the polarization of the How would you locate the sun by using some of these statements.) light from the blue sky, a properly the polarization of sky light? Also look at It is possible to determine the lo­ oriented polarizing filter in front of clouds and smog. cation of the sun by measuring the your camera can block out that Check these ideas with the blue " sky" polarization of the light from the light, increasing the contrast be­ you made for the TRY IT for Section 13.2. blue sky. Many insects and arach­ tween the sky and the white clouds. From the side and from above, look nids seem to use this as a naviga­ Light can also undergo Rayleigh through your polarizing filter at the blue scattered light. Also look end on into the tional device. For example, the wig­ scattering from small scatterers un­ transmitted beam. Additionally, place the gle dance of bees, by which they der water. The resulting polarized filter between the flashlight and the communicate the direction of food, light may be, as for airborne in­ milky water. Look from the side and depends on the direction of the sects, a useful clue to underwater notice what happens to the scattered light's polarization. The sensitivity denizens for navigation. It may also light as you rotate the polarizer. Also of the bee's eye to polarization is provide a stable reference to help notice what happens to the transmitted different at different wavelengths, the animal stay In one place (sta­ red " sunset." Explain what you see, reaching a maximum at 355 nm. tion keeping). It may help to im­ using Figure 13.4. Repeat these This suggests that a pattern of po­ prove the animal's visual contrast experiments as you add more milk. In larized light appears as a colored (as it helped the camera "see" the particular, when you've added more milk and you look through the filter at the pattern to the bee. (Why?) These clouds). For these and possibly beam from the side, you should notice and other insects can locate the sun other reasons, numerous under­ that there is polarized light coming from even when it is behind a cloud by water animals show a sensitivity the water close to the flashlight, but not detecting the polarization of the to polarized light: crustaceans, from the water farther from the flashlight. I light from a patch of blue sky. cephalopods, fishes, and amphibi­ Why does this happen? I CHAPTER 13: SCAITERlNG AND POLARIZATION

352 13.4 In the plane of the figure (the x-y are perfectly free to radiate in the POLARIZATION DUE plane), as drawn, and one polarized direction of the reflected beam. TO REFLECTION perpendicular to the plane of the There is nothing unusual in this figure (the z-dlrection). case, so there is a reflected beam of - Let's consider the first compo­ this polarization. Polarized light may be made in nent. The direction of polarization Thus, if an unpolarized beam ar­ other ways beside scattering. Prob­ of the incident beam must be, as rives at Brewster's angIe of inci­ ably the second most common shown, perpendicular to the inci­ dence, only one component of polar­ source of polarized Ugh t also relies dent ray. Similarly, the direction ization is reflected. The reflected on the transverse nature of light­ of polarization of the transmitted beam is then linearly polarized in polarization by rl4lection. beam must be perpendicular to the the z-direction. At nearby angles When light in air strikes a transmitted ray. This means that this is almost true-that is, of the smooth glass surface at an angle of the electric field in the glass, and light polarized in the x-y plane, very incidence at (Fig. 13.5), it Wiggles thus the direction in which the little is reflected. Hence, th e re­ the charges at the surface of the charges oscillate there, is perpen­ flected light is partially polarized,

glass. There Is a direction ar in dicular to the transmitted ray. But consisting of a large component of which th e radiation emitted from it is the radiation from these oscil­ one polarization and a small com­ all these ch arges is in phase. This lating charges that produces the re­ ponent of the other. Since you often

is the reflected beam, at ar = at. In flected ray. Because light is trans­ look at Objects at angles n ear any oth er direction (in the air) the verse, these charges cannot radiate Brewster's angle, much of th e re­ radiation from the different charges along their direction of oscillation. flected light that you see is polar­ interferes des tructively. Similarly, Hence there cannot be a reflected ized. (The TRY IT invites you to th ere Is a d irection at of construc­ ray perpendicular to the transmit­ check this.) tive in terference between the inci­ ted ray. Thus, the intensity of the The Situation is different for light dent radia tion and that from the reflected ray is zero for this special reflected from metals. Visible ligh t glass atoms. This is th e transmitted angle of incidence, called Brew­ is not transmitted into metals be­ beam, given by Snell's law. Again ster's angle, for which the reflected cause there are so many electrons destructive interference eliminates beam is perpendicular to the trans­ free to move parallel to the surface, rays in any other dir ection in the mitted beam. (This is named after which cancel any internal electric glass. We've drawn th e figure for a Sir David Brewster, who also in­ fields (Sec. 2.3B). Electrons moving special case we want to examine, vented the kaleidoscope.) Since an­ freely parallel to the surface can ra­ where the transmitted and re­ gles of refraction depend on the two diate in all directions away from the flected rays just happen to be at media involved, so does Brewster's surface and can hence create re­ right angles to each other. If the in­ angle. (Appendlx L gives a mathe­ flected beams of both polariza­ cident light Is unpolarized, we can matical expression.) Brewster's an­ tions-the reflected light is not lin­ consider it as consisting of two gle is typically near 56°, the value early polarized. components: one linearly polarized for light in air incident on glass. Because much of the specular­ Now conSider the other compo­ ly reflected light from nonmetallic nent of the incident light-linearly surfaces is at least partially polar­ FIGURE 13.5 polarized in the z-direction (i.e., ized, polarizing filters are often perpendicular to the plane of the Incident, transmitted, and reflected ray used as sunglasses. If they are ori­ directions for the case when the angle of figure). Here the charges in the ented so as to remove the compo­ incidence is equal to Brewster's angle. glass oscillate in the z-direction and nent of light that is polarized hori­ zontally, such sunglasses elim inate specular reflections (glare) from roadways, lakes, and other hor izon ­ tal surfaces. On the other h and, light reflected diffusely will have been reflected or scattered several times and will thus be unpolarized. Only half of such light is blocked by your sunglasses (the horizontal component), so you still see th e road itself but with the glare re­ duced. For the same reason, airport control towers and the bridges of ocean liners often have sheets of po­ larizing filters over their windows. The surface reflections shown in 13.4 POLARIZATION DUE TO REFLECTION

353

T R Y IT

f OR SECTION 13.4 Polarization of reflected light

You will need a polarizing filter, as in the TRY IT for Section 13.38. Look through Incident this filter at light reflected from shiny unpolarized a light Tube surface (not metal) such as a polished desk or floor, a shiny table top, a piece of smooth white paper, or a piece of glass with some black paper behind it. Position yourself arid a light so you see a FIGURE 13.6 Plate 9.7a were eliminated in Plate good specular reflection of the light. Rotate the filter to see if the light is 9.7b by means of a polartzing filter A polarizing filter made of Brewster polarized, as you did before. Compare in front of the camera lens, oriented windows. Each piece of glass is at Brewster's angle, e8 , to the incident light the amount of polarization that you see so as to block them. direction. The light that reaches the when the light strikes the surface at near observer consists predominantly of the normal incidence, at intermediate POND E R component of polarization in the plane angles, and at glancing incidence. At of the figure. The inside of the tube which angle do you expect to see the should be black, so the reflected light is What was the orientation of the greatest polarization? Do you? absorbed. The angle 4> is equal to polarizing filter? Repeat the experiment with a diffusely 90° - e8 . For a glass in air, eB = 56.3°, so 4> = 33.7". reflecting surface such as a piece of cloth or a piece of white paper with a matte Several other devices take advan­ finish. Why isn 't the light polarized here? tage of the polarization at Brew­ Also try a shiny metal surface, such as a ster's angle. At n ormal incidence, losses by reflection, these windows polished cooking pan, a stainless steel each surface of a piece of glass re­ are positioned so the beam strikes knife blade, or the chrome surfaces of a flects 4% of the incident intensity, them at Brewster's angle. They are car. so 92% is transmitted. But suppose then called Brewster windows. Af­ Light reflected from the road has both you want to transmit light through ter 100 passages back and forth a diffuse part (by which you see the road) and a specular part (the glare). On a 100 pieces of glass. Then 0.92 of through these windows, light of one sunny day, look at a road through your the incident beam is transmitted by polarization is completely lost, but polarizing filter, and notiEe what the first piece, 0.92 of that by the the other polarization is not dimin­ happens to the glare as you rotate the second, etc. After 100 pieces, only ished by reflections-the laser light filter. produced this way is therefore po­ Light in the primary rainbow has been (0.92)100 = 2.4 X 10-4 larized. reflected once (Sec. 2.68), so you can of the original intensity has been The idea of the Brewster window expect it to be polarized also. If you trans mitted-that is, hardly any­ can be used to make a polarizing fil­ think of how a mirror located at some thing. You can do much better if ter of the type needed in many of point on the rainbow would have to be you slant the pieces of glass so that the TRY IT's. About five pieces of oriented if it were to reflect sunlight to your eyes (as the droplets do), you the light is incident on them at glass (microscope slides work well) should be able to convince yourself that Brewster's angle. At this angle, are positioned one behind the oth­ the light from the rainbow must be about 15% of one polarization com­ er, each at Brewster's angle to the polarized along the direction of the arc ponent is reflected at each of the incident beam (Fig. 13.6). While not (rather than radially). That is, light from 200 surfaces. However, none of the as effective as 100 pieces of glass, the (horizontal) top of the bow should be other polartzation component is re­ this device considerably diminishes polarized horizontally, whereas light flected. Since all of that polariza­ the component of light polarized from the vertical arms of the bow (near tion is transmitted, it can comfort­ perpendicular to the plane of the the pot of gold) should be polarized ably pass through all 100 pieces of figure, but still passes the compo­ vertically. Check this with a polarizing glass. Hence, if unpolarized light is nent polarized in that plane-it filter the next time you see a rainbow (or make a rainbow with your garden hose). incident on this arrangement, half therefore acts as a polarizing filter. What do you expect happens in the of the intensity is transmitted. Like all other polarizing filters, you secondary rainbow? Does it? This idea is often used in gas la­ can use this to detect polarized sers (Sec. '15.4). In such a laser, the light. Unlike other filters, however, light is reflected back and forth be­ you can use this to tell the direction tween two mirrors about 100 Urnes. oj polarization of the light, since For precise adjustment, the mirrors the easily visible slant of the glass are situated outside glass windows tells you which component the that contain the gas. To avoid Brewster window passes. CHAPTER 13: SCATTERING AJlD POLARIZATION

354

13.5 laroid's-not all wavelengths are ab­ 12 POLARIZATION D UE sorbed equally, so the transmitted TO ABSORPTION light is colored. Such crystals are called dichroic because even unpo­ 10~~~2 - larized light passing through them One of the most commonly used in one direction becomes a different 9 ( ) ( ) 3 visible-ligh t linear polarizers ab­ color than light that passes through sorbs one compon ent of polariza­ them in another direction. * Be­ tion, while transm itting the per­ cause of the color produced, these pendicular components. Polaroid dichroic crystals are not usually 8~~~' consists of a parallel array of long­ used as polarizing filters. The word 6 chain molecules whose electrons dichroic has come to mean any ma­ (al cannot freely move across the nar­ terial that produces polarized light row molecules. When incident light by absorption, so Polaroid is con­ is polarized so its electric field sidered dichroic. pulls across the molecules, it can't The back of your eye contains a make the electrons move. Hence dichroic material: the yellow pig­ they don't radiate, and the inci­ ment (macula luteal that absorbs ! dent wave continues unaffected­ light between 430 and 490 nm (b) Polarization that component of polarization is (blue) and covers your fovea (the transmitted. However, when the place corresponding to the center of electric field of the light drives elec­ your field of view-see Sec. 5.2B). FIGURE 13.7 trons along the long molecules, the The dichroism in this case consists electJ"ons do move and absorb the of stronger absorption when the di­ (a) The dichroic pigment in the macula is arranged radially. (b) Haidinger's light's energy. Thus only light of rection of polarization of light is lutea brush, when the light is polarized one p olariza tion (across the mole­ perpendicular to the pigment fibers vertically (arrow). cules) is transmitted. than when it is parallel. The pig­ This type of polarizing filter was ment in the macula is arranged ra­ invented by Edwin H. Land in 1928 dially, like the spokes of a wheel. as when he was a 19-year-old under­ shown schematically in Figure TRY IT graduate. (His interest had been 13.7a. Suppose that white light in­ stimulated by reading about some cident on your eye is polarized ver­ fOR SECnON 13.5 polarizing crystals that had been tically (I.e., parallel to fibers 6 and Haidinger's brush discovered in the 1850s when a 12 in Fig. 13.7a). As a consequence physician's pupil, for some peculiar of the dichroism, the vertical fibers The best way to look for Haidinger's brush is to use a linear polarizing filter reason, put drops of iodine in the (6 and 12) absorb the blue part of and a light blue, nonshiny background, urine of a dog who had previously this light least, and fibers perpen­ such as a large piece of blue construction been fed quinine.) Sheets of Polar­ dicular to this direction (fibers 3 paper. Brightly illuminate the paper, and oid material, usually mounted be­ and 9) absorb it most strongly. look through the filter at a point near the tween thin sheets of glass or plas­ Therefore, you see a horizontal dark center of the paper. Because the image tic, form the polarizing filters you yellow line (Fig. 13.7b) known as will fade as your eye adapts, it helps to often find in sunglasses. Whereas Haidinger's brush. As shown in rotate the filter occasionally about your they absorb most of the visible light the figure, the yellow brush is often line of sight. If you don't see the yellow· of the appropriate polarization, they accompanied by adjacent blue re­ brown brush at first, don't be dismayed; are not completely effective at the gions, presumably due to simulta­ it took Helmholtz twelve years to see it after he first learned of the phenomenon shorter wavelengths. For this rea­ neous color contrast (Sec. 10.6AJ. from Haidinger. You should be able to son very bright horizontally polar­ If. instead, the direction of polari­ do much better-perhaps a few minutes. ized ligh t (e.g., bright sunlight zation were horizontal, the yellow Once you see the brush, notice how it polarized by reflection from wa­ brush would be vertical-it always rotates as you slowly rotate the filter. This ter) may look deep blue or violet lies perpendicular to the direction provides another technique for through your polarizing sunglasses, of polarization. The TRY IT tells you determining the direction of polarization instead of black as it would for an how to look for Haidinger's brush. of the filter, because the brush is always ideal polariZing filter. perpendicular to it. Various naturally occurring crys­ Another good background is the blue sky. Use your polarizing filter to see the tals absorb one component of polar­ "Greek dis, twice, plus chros, color. brush there. Because the light from the ization more than they do the per­ Crystals exhibiting three colors, for blue sky is already polarized (Sec. 13.3B), pen dicular component. The semi­ light passing through them in three different directions, are called trichroiC some people are able to see Haidinger's precious stone tourmaline is one (Greek treis, three). Generally, all such brush in the sky without the aid of a example. However, tourmaline's ab­ crystals are called pleochroic (Greek filter. In what direction should you look? sorption is more selective than Po- pleion, more). Try it! 13.6 POLARIZING AND ANALYZING

355 13.6 unpolarized, and the polarizer ori­ polarized light (the polarizer and POLARIZING ented to pass only vertically polar­ analyzer are crossed-Fig. 13 .8b). AND ANALYZING ized light. Suppose the analyzer is Since only vertically polarized light oriented to pass only horizontally strikes the analyzer, no light passes - through it-there is no transmitted A polarizing filter can be used in beam reaching the observer. If in­ two different ways. It can be used stead the analyzer is oriented verti­ as a polarizer-because it trans­ FIGURE 13.8 cally (the polarizer and analyzer are mits only one component of polarI­ (a) Incident light strikes the first parallel-Fig. 13.8c), it passes all zation, incident unpolarized light polarizing filter (the polarizer) . The light the light striking it-the observer striking a polarizing filter results in that passes th rough the polarizer (the then sees the full intermediate in­ polarized transmitted light. Alter­ intermediate light) strikes the second tensity (assuming ideal polarizing polarizing filter (the analyzer). The natively, it can be used as an ana­ amount of light that passes through the filters). lyzer-you can detect the presence analyzer (the transmitled light) depends Thus you can use an analyzer of polarized light with it, as in the on the relative angle of the polarizer and to tell If the polarizer is doing its TRY IT for Section 13.3B, and even analyzer. (b) Crossed polarizer and j\:lb. Makers of polarizing sunglasses analyzer: incident unpolarized light direction the of polarization once striking a vertically oriented polarizer often give you a little analyzer so the filter is calibrated. (You can cal­ produces vertically polarized light, which you can convince yourself that the ibrate a polarizing filter-I.e., find is not passed by a horizontally oriented glasses do polarize (without break­ the direction of polarization that It analyzer. (e) Parallel polarizer and ing the pair in half). p asses-by looking at the light re­ analyzer: the same as in (b), but now the vertically polarized light is passed by the P ONDER flected n ear Brewster's angle from a vertically oriented analyzer. (d) Now the shiny floor , since that light must be analyzer, oriented at an intermediate horizontally polarized.) angle, passes only that component of What transmitted light results if the polarizer is oriented horizontally in Figure 13.8a shows a set-up us­ the intermediate light polarized at that same angle. A weaker transmitled light Figures 13.8b and c? ing two polarizing filters, one as a results. Note that (b) to (d) represent polarizer and one as an analyzer. head-on views of the light beam and of Le t's start with the incident light the filters. Suppose now that the analyzer is oriented at some other angle to th e polarizer (Fig. 13.8d). Now we m ust think of the vertically polarized in­ . . ~ ~ A:l ~~ termediate light as consisting of two components (as In Fig. 13.2el. 0 f 0 one along the angle of the analyzer, f 7 7 \ and one oriented perpendicular to In ci dent light Polarizer Intermediate light Analyzer Transmitted light (a) it. Only the former component (th at parallel to the analyzer's orienta­ tion) passes through the analyzer. There is a transmitted beam in this No case, but it has less intenSity than transmitted in the case where the analyzer and light polarizer are parallel. (Malus's law. the mathematical relation b etween the intenSity of the transmitted light and the angle between the an­ alyzer and polarizer, is given in Ap­ pendix M.) Thus, by adjusting the angle of orientation between the polarizer and analyzer, you can control the 1 ! intensity of the transmitted light. (e) This is often a convenient way of * contrOlling light intensity b ecause it doesn't modify the size or shape of the beam (as a mechanical dia­ phragm wouldl. and, at least for ideal polarizing filters, it doesn't af­ /' fect the color temperature (as dim­ ming an incandescent bulb would). (d)* Since the analyzer tests the polar­ CHAPTER 13: SCATTERING AND POLARIZATION

356 ization of the intermediate light, we the analyzer-some of the light is *13.7 can use this set-up to examine the therefore transmitted through the CONTROLLING effect of various types of obstacles entire system. (If the filters are POLARIZATION WITH on polarized ligh t. For example, in ideal, the transmitted intensity is ELECTRIC FIELDS Section 13.3B we mentioned that one-fourth of the original intensity.) polarized light that suffers repeated This Is a dramatic demonstration - scatterings, as in clouds, becomes that not only is It useful to think of We can carry the "magic" of the last unpolarized-it is depolarized. The any electric field In terms of com­ section further and arrange that all TRY IT shows how you canverify that ponents, it is necessary-it is the of the intensity be transmitted. statement using this set-up and way Nature works. Suppose the polarizer (a) Is ori­ depolarizers such as waxed paper or ented vertically (Fig. 13.9). Light ground glass. passing through It will then be po­ A remarkable phenomenon oc­ laIized vertically. If the n ext polar­ curs when you introduce a third po­ izing filter (b) is oriented at a slight laIizing filter, in serted between the TRY IT angle from the vertical, the light polarizer and the analyzer. Initially, that passes through It will be polar­ with only two filters present, no FOR SECTION 13.6 Ized in this tipped direction, but light passed through the crossed Depolarization by multiple scattering the reduction of its Intensity is n eg­ polarizer and analyzer; when you ligible. If now we add another polar­ looked through them they appeared You will need two polarizing filters; a izing filter (c) that is tipped still fur­ dark. But now you insert the third polarizer and an analyzer, as in Figure ther from the vertical, at a small polarizing filter between them, ori­ 13.B. Place one behind the other with angle from the previous one, it Will their polarization directions crossed, so ented at an angle of 45°, an orien­ pass light polarized in this new dl­ no light passes through them. A good b~tween rection. If we continue this process, tation midway those of the source of multiple scattering is a piece of polarizer and the analyzer. When waxed paper or ground glass (which are with enough filters we can r otate you look through this three-filter translucent rather than transparent the angle of polarization lo any dl­ combination, you see light. This is because they repeatedly scatter the light). rectlon we choose; and if the angle qUite remarkable when you think If the light is depolarized by this multiple between successive polarizing filters abou t it; the filter you insert does scattering, the analyzer will pass some of is small enough, the Intensity is not nothing but absorb some of the it because one component of the then diminished. light, yet some light is now trans­ unpolarized light will be parallel to the Thus, With such a twisted stack mitted after you inserted an ab­ analyzer's orientation. Hold a piece of of ideal polarizing filters, we could waxed paper between the crossed sorber! rotate the angle of polarization by polarizer and analyzer and look through We can understand this magiC this arrangement at a light source. The 90° Without loss of intensity. S uch by Simply repea ting our previous waxed paper should intercept only a part a device could then be used to p ass analyses. At each polarizing filter of the intermediate beam, so you can light through a crossed polarizer we must think of the light as con­ still see the polarizer through the and analyzer when inserted be­ sisting of two components: one analyzer. Can you see any light passing tween them. along the polarizing filter's orienta­ through the waxed paper when the If we could make such a stack tion and one perpendicular to it. polarizer and analyzer are crossed? What whose total angle of twist could be Thu s you start with unpolarized happens as you rotate the analyzer? Does changed fast enough, we could the polarized fight that strikes the waxed light incident on the vertically ori­ have a shutter-a zero angle of paper remain polarized after passing ented polarizer. As in Figure 13.8, twist would pass no light through through it? only the vertical component passes. the crossed polarizer and analyzer, When only the crossed analyzer is while a 90° angle would pass all the present, the vertical component in­ light emerging from the polarizer cident on it is blocked, so no light (that is, half of the inten sity of the FIGURE 13.9 is transmitted (Fig. 13.8b). When. original unpolarized light). It is pos­ the third polarizing filter is inserted Successive polarizing filters oriented at sible to create the effect of uch at a 45° angle, h owever, some of the slightly different angles rotate the stacks out of layers of suitable mol­ direction of polarization. The intensity of vertically polarized light incident on light is unchanged after filter a if ecules and to change the amoun t of It is transmitted by it as light po­ successive orientations change only twist with electricjields. There are larized at 45° (as in Fig. 13.8d). slightly. several ways of dOing this. This light, now Incident on the hor­ izontally oriented analyzer, can be thought of as consisting of both a vertical and a horizontal compo­ nent, as in Figure 13.2d. Since / there is now a horizontal compo­ o I a o nent present, it can pass through b c '13.7 CONTROLLING POLARIZATION WITH ELECTIC FIELDS

357

A. Liquid crystal displays

Liquid crystals are liquid because their molecules move about within the liquid ; they are crystals because their molecules orient themselves in an array. In one type of liquid crys­ tal (called nematic). all the mole­ cules in a given layer are oriented parallel to each other and to the layer (Fig. 13.10a). The direction of (a) orientation depends on the environ­ ment at the surface. For example. a piece of glass whose surface has \ been rubbed with a cloth or paper l'WJight in a given direction will have micro­ transmitted scopic grooves (a few atoms deep) in that direction. Nematic molecules adjacent to that surface of the glass will then orient themselves along these grooves. If the liquid crystal Battery lies between two glass plates whose (b) surfaces have been rubbed in a given direction, then all the mole­ cules will lie oriented in that direc­ to the plane of the figure. However, FIGURE 13.11 if a battery of a few volts is con­ tion, as in the figure. If now one of (a) A twisted nematic cell allows light to the glass plates is rotated, the mol­ nected to transparent conductors pass th rough a crossed polarizer and ecules closest to that plate will be on the two plates, the molecules ana lyzer. (b) With an electric field rotated along with it, producing a tend to line up parallel to the elec­ applied between the plates of the cell , twist in the layers of the molecules tric field-that is, perpendicular to no light emerges from the analyzer. (Fig. 13.10b). If the thickness of the plates (Fig. 13.lOc). Light inci­ this layer of liquid crystal is large dent on the top of the cell now will electric field, incident light is polar­ compared with the wavelength of not have its polarization altered. If ized vertically, say, by the polarizer, visible light, this twisted nematic the battery is then disconnected, is rotated 900 by the cell, passes ceU is similar to a twisted stack of the molecules return to the align­ through the analyzer, is reflected by polarizing filters in the following ment of Figure 13. lOb. the mirror back through the ana­ sense-if light is polarized in the This cell thus provides a conve­ lyzer (still polarized horiw ntallyl. is plane of the figure and incident on nient switch for light. The cell is rotated 90° by the cell to become ver­ the top of the cell. it emerges from placed, suitably oriented, between tically polarized again, and passes the bottom polarized perpendicular a crossed polarizer and analyzer. back out through the polarizer. When there is no electric field, light If you look at this display from the passes through the system (Fig. front, you see this reflected light, FIGURE 13.10 13. 11a). When the electric field is and the display looks bright. When turned on, however, no light passes the electric field is turned on, how­ (a) The elongated molecules of a nematic (Fig. 13.11bJ. ever, no light reaches the m irror liquid crystal between two suitably prepared glass plates. (b) A twisted This arrangement is commonly and hence none is reflected-the nematic cell: the same as (a), but now used in display devices, such as the display looks dark. If the electr ic the bottom plate has been rotated 90°, display in digital watches (liqu id field is applied only in certain re­ so the molecules of the liquid crystal crystal displays, LCD's). In that gions that form the segments of a closest to the bottom are now oriented case, the same set-up shown in the number, you see a dark ' number perpendicular to the plane of the figure. (c) The same cell as in (b), but with an figure is used with a mirror after against a bright display. Figu re electric field between the plates. the analyzer. Thus, with no applied 13.12 shows one way of controlling the location of the electric field to (a) (b) (e) form numerals.

~~~~~~Ctf1f B. Pockets and Kerr celts ,~=~~~~jfo ~1r:@j~ / j riiiiiiiJ.7iiiiiii~- "-----'...-Y Direction of electric field Certain solids, whose crystal struc­ due to battery tures are suffiCiently asymmetric,