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Feynman Physics Lectures V1 Ch33 1962-03-06 Polariztion

Feynman Physics Lectures V1 Ch33 1962-03-06 Polariztion -...

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Unformatted text preview: 33 Polarization 33—1 The electric vector of light In this chapter we shall consider those phenomena which depend on the fact that the electric field that describes the light is a vector. In previous chapters we have not been concerned with the direction of oscillation of the electric field, except to note that the electric vector lies in a plane perpendicular to the direction of propagation. The particular direction in this plane has not concerned us. We now consider those phenomena whose central feature is the particular direction of oscillation of the electric field. In ideally monochromatic light, the electric field must oscillate at a definite frequency, but since the x-component and the y-component can oscillate independ- ently at a definite frequency, we must first consider the resultant effect produced by superposing two independent oscillations at right angles to each other. What kind of electric field is made up of an x-component and a y-component which oscillate at the same frequency? If one adds to an x-vibration a certain amount of y-vibration at the same phase, the result is a vibration in a new direction in the xy-plane. Figure 33~1 illustrates the superposition of different amplitudes for the x-vibration and the y-vibration. But the resultants shown in Fig. 33—1 are not the only possibilities; in all of these cases we have assumed that the x-vibration and the y-vibration are in phase, but it does not have to be that way. It could be that the x-vibration and the y-vibration are out of phase. 1 +2 ly=1 Eyno ax=o tax-i: zx=1 nx=1 33—1 The electric vector of light 33—2 Polarization of scattered light 33—3 Birefringence 33—4 Polarizers 33-5 Optical activity 33—6 The intensity of reflected light 33—7 Anomalous refraction Y Y 1 y y = = E = 1 E = -1 By 1 EV 1 y Y =-1 21:1 Fig. 33—1. Superposition of x-vibrations and y-vibrafions in phase. When the x-vibration and the y-vibration are not in phase, the electric field vector moves around in an ellipse, and we can illustrate this in a familiar way. If we hang a ball from a support by a long string, so that it can swing freely in a horizontal plane, it will execute sinusoidal oscillations. If we imagine horizontal x- and y-coordinates with their origin at the rest position of the ball, the ball can swing in either the x- or y-direction with the same pendulum frequency. By selecting the proper initial displacement and initial velocity, we can set the ball in oscillation along either the x-axis or the y-axis, or along any straight line in the xy-plane. These motions of the ball are analogous to the oscillations of the electric field vector illustrated in Fig. 33—1. In each instance, since the x-vibrations and the y-vibrations reach their maxima and minima at the same time, the x- and 32—05- cillations are in phase. But we know that the most general motion of the ball is motion in an ellipse, which corresponds to oscillations in which the x- and y-directions are not in the same phase. The superposition of x- and y-vibrations which are not in phase is illustrated in Fig. 33—2 for a variety of angles between the phase of the x-vibration and that of the y-vibration. The general result is that the electric vector moves around an ellipse. The motion in a straight line is a particular 33—1 / x = cos cut; 1 By = cos cat; 1 f xx = cos cut; 1 By = -cos(an:+"/¢l; -e 1n/o case corresponding to a phase difference of zero (or an integral multiple of 1r); motion in a circle corresponds to equal amplitudes with a phase difierence of 90° (or any odd integral multiple of 7r/2). In Fig. 33—2 we have labeled the electric field vectors in the x- and y-directions with complex numbers, which are a convenient representation in which to express the phase difierence. Do not confuse the real and imaginary components of the complex electric vector in this notation with the x- and y—coordinates of the field. The x- and y-coordinates plotted in Fig. 33—1 and Fig. 33—2 are actual electric fields that we can measure. The real and imaginary components of a complex electric field vector are only a mathematical convenience and have no physical significance. b c d e cos «it; 1 cos wt; 1 cos cut; 1 cos at; 1 1 cos(ast+"/4); e "A -sin wt; 1 cosmic-0’74); ein/l -cos art; -1 g h i cosart; 1 cosat; 1 cosmt;1 sin wt; -1 -cos(at+3n/4); -e13“/4 cos at; 1 Fig. 33—2. Superposition of x-vibrations and y-vibrations with equal amplitudes but various relative phases. The components Ez and E, are expressed in both real and complex notations. Now for some terminology. Light is linearly polarized (sometimes called plane polarized) when the electric field oscillates on a straight line; Fig. 33—1 illustrates linear polarization. When the end of the electric field vector travels in an ellipse, the light is elliptically polarized. When the end of the electric field vector travels around a circle, we have circular polarization. If the end of the electric vector, when we look at it as the light comes straight toward us, goes around in a counterclockwise direction, we call it right-hand circular polarization. Figure 33—2(g) illustrates right-hand circular polarization, and Fig. 33—2(c) shows left-hand circular polarization. In both cases the light is coming out of the paper. Our convention for labeling left-hand and right-hand circular polarization is consistent with that which is used today for all the other particles in physics which exhibit polarization (e.g., electrons). However, in some books on optics the opposite conventions are used, so one must be careful. We have considered linearly, circularly, and elliptically polarized light, which covers everything except for the case of unpolarized light. Now how can the light be unpolarized when we know that it must vibrate in one or another of these ellipses? If the light is not absolutely monochromatic, or if the x- and y-phases are not kept perfectly together, so that the electric vector first vibrates in one direction, then in another, the polarization is constantly changing. Remember that one atom emits during 10—8 sec, and if one atom emits a certain polarization, and then another atom emits light with a different polarization, the polarizations will change every 10—8 see. If the polarization changes more rapidly than we can detect it, then we call the light unpolarized, because all the effects of the polarization average out. None of the interference effects of polarization would show up with unpolarized light. But as we see from the definition, light is unpolarized only if we are unable to find out whether the light is polarized or not. 33-2 33—2 Polarization of scattered light The first example of the polarization effect that we have already discussed is the scattering of light. Consider a beam of light, for example from the sun, shining on the air. The electric field will produce oscillations of charges in the air, and mo- tion of these charges will radiate light with its maximum intensity in a plane normal to the direction of vibration of the charges. The beam from the sun is unpolarized, so the direction of polarization changes constantly, and the direction of vibration of the charges in the air changes constantly. If we consider light scattered at 90°, the vibration of the charged particles radiates to the observer only when the vibration is perpendicular to the observer’s line of sight, and then light will be polarized along the direction of vibration. So scattering is an example of one means of producing polarization. 33-3 Birefringence Another interesting effect of polarization is the fact that there are substances for which the index of refraction is different for light linearly polarized in one direction and linearly polarized in another. Suppose that we had some material which consisted of long, nonspherical molecules, longer than they are wide, and suppose that these molecules were arranged in the substance with their long axes parallel. Then what happens when the oscillating electric field passes through this substance? Suppose that because of the structure of the molecule, the electrons in the substance respond more easily to oscillations in the direction parallel to the axes of the molecules than they would respond if the electric field tries to push them at right angles to the molecular axis. In this way we expect a different response for polarization in one direction than for polarization at right angles to that direc- tion. Let us call the direction of the axes of the molecules the optic axis. When the polarization is in the direction of the optic axis the index of refraction is different than it would be if the direction of polarization were at right angles to it. Such a substance is called birefringent. It has two refrangibilities, i.e., two indexes of refraction, depending on the direction of the polarization inside the substance. What kind of a substance can be birefringent? In a birefringent substance there must be a certain amount of lining up, for one reason or another, of unsymmetrical molecules. Certainly a cubic crystal, which has the symmetry of a cube, cannot be birefringent. But long needlelike crystals undoubtedly contain molecules that are asymmetric, and one observes this effect very easily. Let us see what effects we would expect if we were to shine polarized light through a plate of a birefringent substance. If the polarization is parallel to the optic axis, the light will go through with one velocity; if the polarization is per- pendicular to the axis, the light is transmitted with a different velocity. An inter- esting situation arises when, say, light is linearly polarized at 45° to the optic axis. Now the 45° polarization, we have already noticed, can be represented as a super- position of the x- and the y—polarizations of equal amplitude and in phase, as shown in Fig. 33—2(a). Since the x- and y-polarizations travel with different velocities, their phases change at a different rate as the light passes through the substance. So, although at the start the x- and y-vibrations are in phase, inside the material the phase difference between x- and y-vibrations is proportional to the depth in the substance. As the light proceeds through the material the polarization changes as shown in the series of diagrams in Fig. 33—2. If the thickness of the plate is just right to introduce a 90° phase shift between the x- and y-polarizations, as in Fig. 33—2(c), the light will come out circularly polarized. Such a thickness is called a quarter-wave plate, because it introduces a quarter-cycle phase difference between the x- and the y-polarizations. If linearly polarized light is sent through two quarter-wave plates, it will come out plane-polarized again, but at right angles to the original direction, as we can see from Fig. 33—2(e). One can easily illustrate this phenomenon with a piece of cellophane. Cello- phane is made of long, fibrous molecules, and is not isotropic, since the fibers lie preferentially in a certain direction. To demonstrate birefringence we need a 33—3 CELLOPHANE \POLAROID/ Fig. 33-3. An experimental demon- stration of the birefringence of cellophane. The electric vectors in the light are indi- cated by the dotted lines. The pass axes of the polaroid sheets and optic axes of the cellophane are indicated by arrows. The incident beam is unpolarized. beam of linearly polarized light, and we can obtain this conveniently by passing unpolarized light through a sheet of polaroid. Polaroid, which we will discuss later in more detail, has the useful property that it transmits light that is linearly polarized parallel to the axis of the polaroid with very little absorption, but light polarized in a direction perpendicular to the axis of the polaroid is strongly absorbed. When we pass unpolarized light through a sheet of polaroid, only that part of the unpolarized beam which is vibrating parallel to the axis of the polaroid gets through, so that the transmitted beam is linearly polarized. This same property of polaroid is also useful in detecting the direction of polarization of a linearly polarized beam, or in determining whether a beam is linearly polarized or not. One simply passes the beam of light through the polaroid sheet and rotates the polaroid in the plane normal to the beam. If the beam is linearly polarized, it will not be transmitted through the sheet when the axis of the polaroid is normal to the direction of polarization. The transmitted beam is only slightly attenuated when the axis of the polaroid sheet is rotated through 90°. If the transmitted in- tensity is independent of the orientation of the polaroid, the beam is not linearly polarized. To demonstrate the birefringence of cellophane, we use two sheets of polaroid, as shown in Fig. 33—3. The first gives us a linearly polarized beam which we pass through the cellophane and then through the second polaroid sheet, which serves to detect any effect the cellophane may have had on the polarized light passing through it. If we first set the axes of the two polaroid sheets perpendicular to each other and remove the cellophane, no light will be transmitted through the second polaroid. If we now introduce the cellophane between the two polaroid sheets, and rotate the sheet about the beam axis, we observe that in general the cellophane makes it possible for some light to pass through the second polaroid. However, there are two orientations of the cellophane sheet, at right angles to each other, which permit no light to pass through the second polaroid. These orientations in which linearly polarized light is transmitted through the cellophane with no effect on the direction of polarization must be the directions parallel and per- pendicular to the optic axis of the cellophane sheet. We suppose that the light passes through the cellophane with two different velocities in these two different orientations, but it is transmitted without changing the direction of polarization. When the cellophane is turned halfway between these two orientations, as shown in Fig. 33—3, we see that the light transmitted through the second polaroid is bright. It just happens that ordinary cellophane used in commercial packaging is very close to a half-wave thickness for most of the colors in white light. Such a sheet will turn the axis of linearly polarized light through 90° if the incident linearly polarized beam makes an angle of 45° with the optic axis, so that the beam emerging from the cellophane is then vibrating in the right direction to pass through the second polaroid sheet. If we use white light in our demonstration, the cellophane sheet will be of the proper half-wave thickness only for a particular component of the white light, and the transmitted beam will have the color of this component. The color trans- mitted depends on the thickness of the cellophane sheet, and we can vary the effective thickness of the cellophane by tilting it so that the light passes through the cellophane at an angle, consequently through a longer path in the cellophane. As the sheet is tilted the transmitted color changes. With cellophane of different thicknesses one can construct filters that will transmit different colors. These filters have the interesting property that they transmit one color when the two polaroid sheets have their axes perpendicular, and the complementary color when the axes of the two polaroid sheets are parallel. Another interesting application of aligned molecules is quite practical. Certain plastics are composed of very long and complicated molecules all twisted together. When the plastic is solidified very carefully, the molecules are all twisted in a mass, so that there are as many aligned in one direction as another, and so the plastic is not particularly birefringent. Usually there are strains and stresses introduced when the material is solidified, so the material is not perfectly homo- 33—4 geneous. However, if we apply tension to a piece of this plastic material, it is as if we were pulling a whole tangle of strings, and there will be more strings preferen- tially aligned parallel to the tension than in any other direction. So when a stress is applied to certain plastics, they become birefringent, and one can see the effects of the birefringence by passing polarized light through the plastic. If we examine the transmitted light through a polaroid sheet, patterns of light and dark fringes will be observed (in color, if white light is used). The patterns move as stress is applied to the sample, and by counting the fringes and seeing where most of them are, one can determine what the stress is. Engineers use this phenomenon as a means of finding the stresses in odd-shaped pieces that are difficult to calculate. Another interesting example of a way of obtaining birefringence is by means of a liquid substance. Consider a liquid composed of long asymmetric molecules which carry a plus or minus average charge near the ends of the molecule, so that the molecule is an electric dipole. In the collisions in the liquid the molecules will ordinarily be randomly oriented, with as many molecules pointed in one direc- tion as in another. If we apply an electric field the molecules will tend to line up, and the moment they line up the liquid becomes birefringent. With two polaroid sheets and a transparent cell containing such a polar liquid, we can devise an arrangement with the property that light is transmitted only when the electric field is applied. So we have an electrical switch for light, which is called a Kerr cell. This effect, that an electric field can produce birefringence in certain liquids, is called the Kerr effect. 33—4 Polarizers So far we have considered substances in which the refractive index is different for light polarized in different directions. Of very practical value are those crystals and other substances in which not only the index, but also the coefficient of ab- sorption, is different for light polarized in different directions. By the same argu- ments which supported the idea of birefringence, it is understandable that absorp- tion can vary with the direction in which the charges are forced to vibrate in an anisotropic substance. Tourmaline is an old, famous example and polaroid is another. Polaroid consists of a thin layer of small crystals of herapathite (a salt of iodine and quinine), all aligned with their axes parallel. These crystals absorb light when the oscillations are in one direction, and they do not absorb appreciably when the oscillations are in the other direction. Suppose that we send light into a polaroid sheet polarized linearly at an angle 0 to the passing direction. What intensity will come through? This incident light can be resolved into a component perpendicular to the pass direction which is proportional to sin 0, and a component along the pass direction which is pro- portional to cos 0. The amplitude which comes out of the polaroid is only the cosine 0 part; the sin 0 component is absorbed. The amplitude which passes through the polaroid is smaller than the amplitude which entered, by a factor cos 0. The energy which passes through the polaroid, i.e., the intensity of the light, is proportional to the square of cos 0. Cos 2 0, then, is the intensity transmitted when the light enters polarized at an angle 0 to the pass direction. The absorbed intensity, of course, is sin2 0. An interesting paradox is presented by the following situation. We know that is is not possible to send a beam of light through two polaroid sheets with their axes crossed at right angles. But if we place a third polaroid sheet between the first two, with its pass axis at 45° to the crossed axes, some light is transmitted. We know that polaroid absorbs light, it does not create anything. Nevertheless, the addition of a third polaroid at 45° allows more light to get through. The analysis of this phenomenon is left as an exercise for the student. One of the most interesting examples of polarizatio...
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