Lab #13: Polarization Introduction In this experiment we will investigate various properties associated with polarized light. We will study both its generation and application. Real world applications include LCD displays in your computer, efficient filters, glare-reducing sunglasses, measurements of properties of remote gases, measurements of properties of particulates and objects, the phase-contrast microscope, and stress test procedures, to name but a few. Theory In any transverse wave, such as light, the vibration is perpendicular to the direction of the propagation of the wave. For light traveling in the z-direction, the electric field is perpendicular to the z-axis, oscillating in the x-y plane. (The magnetic field is also perpendicular to the z-axis). If the vibration of any transverse wave is fixed along a certain direction in space (say for this example, with the electric field along the x-axis and the magnetic field along the y-axis), the wave is said to be linearly polarized. If the direction of oscillation is randomly oriented, then the wave is said to be unpolarized. Typically, waves produced by a single source are polarized, such as waves on a string produced by the regular vibration of one end, or electromagnetic waves produced by a single atom. On the other hand, waves produced by a collection of different sources are usually unpolarized. This is the case for ordinary sources of light such as incandescent bulbs, where millions of atoms act independently, emitting light with uncorrelated polarizations. The electric field for such light can be resolved into x- and y-components with, on the average, the same amplitudes. There are four phenomena which polarize light: 1) absorption 2) scattering 3) reflection 4) birefringence (also called double refraction). In this experiment you will investigate these ways of creating polarized light, and study its properties. Some more detailed theory. As already noted, electromagnetic waves are transverse waves in which the vibrations of the electric and magnetic fields are perpendicular to each other and to the direction of propagation. At most frequencies it is the electric field that interacts more strongly with
matter, so its orientation in the plane perpendicular to the direction of propagation is important in understanding the way a wave will interact with a medium. Ordinary, or natural, light is a superposition of a multitude of independent waves emitted from different atoms at the source. In each of these waves (or wave packets) the electric field oscillates in a unique direction in the plane perpendicular to the propagation axis (the transverse plane), but since the wave packets are randomly oriented there is no preferred direction of the resulting electric field. On the average, the electric field is of equal intensity in all directions in the transverse plane and the wave is said to be unpolarized. Conversely, polarized light waves are those in which the intensity of the electric field varies with the direction in the transverse plane. Linearly- or plane-polarized light is light for which the electric field vibrations are restricted to one direction. In most parts of this experiment you will use Polaroid polarizing filters as analyzers for determining the state of polarization of light; they may also be used to create polarized light. As explained below, these filters polarize light by the process of selective absorption. Dichroism Certain solids such as tourmaline have a natural anisotropic crystal structure which renders them electrically conducting (at optical frequencies) in one direction but not in others. Figure 1 shows unpolarized light falling on a thin slab of tourmaline cut along the appropriate planes. We can regard the light as split into two orthogonal linearly-polarized components. One of these is strongly absorbed (in Figure 1 it is the horizontal component), while the other is transmitted with much less loss. The light emerging from the crystal will then be linearly polarized. This phenomenon is called dichroism. Figure 1. Illustration of dichroism. In 1852 it was discovered that crystals of the material now called herapathite are dichroic and that in sufficient thickness they will completely absorb light polarized in one direction. Later a way was found to stabilize and align the tiny, needle shaped crystals of herapathite in 2
a matrix of cellulose acetate or polyvinyl, giving a material now known as Polaroid. Today, Polaroid is used in a multitude of applications and is known as an efficient and economical polarizer, its efficiency being highest for green and yellow light. The polarizers in this experiment are graduated around the circumference in degrees. Only the light with an E-component in the 0º - 180º plane will be transmitted. When the polarizers are put in their holders correctly, the graduations on the ring can be seen through a hole in the holder, opposite a white line. This can be used to orient the polarizers accurately. Malus s Law Suppose linearly polarized light falls on an ideal polarizer whose transmission axis is at an angle θ to the direction of polarization of the incident light. What is the intensity of the light transmitted by the polarizer? The polarizer will allow through only that component of the field which is parallel to its transmission axis. If the incident electric field is E o, then the transmitted electric field is E 0 cos!. The intensity I of the light reaching the detector is proportional to the square of the electric field, so ( ) 2 (1) I! E 0 cos" The maximum intensity I 0 occurs when the angle between the transmission axis of the polarizer and the direction of polarization of the incident light is zero. We can then express Eq. 1 as I = I 0 cos 2! (2) This is known as Malus Law, having first been published in 1809 by Etienne Malus, an engineer in the army of Napoleon. DANGER Laser radiation can cause retinal damage and blindness if allowed to be focused into the eye. DO NOT LOOK DIRECTLY INTO THE LASER!! Viewing the beam from the side, or viewing a pattern on the wall or screen, is not harmful. Experiment: As soon as you arrive at your apparatus, switch on the laser. This will allow it to warm up and stabilize while you are getting ready to make measurements. Apparatus: 1 Helium-Neon laser (λ = 632.8 nm) 1 optical bench 1 screen 1 photodiode light detector (mounted) 1 digital voltmeter (DVM) 1 quarter-wave (λ/4) calibrated retarder 2 Polaroid polarizing filters with optical bench mounts 1 glass plate on a rotating circular mount 3
1 colloidal suspension in a mounted transparent container Procedure 1. Polarization of Light and Malus s Law Mount a Polaroid filter on the optical bench so that the laser beam passes through its centre. Since the absorption across the surface of the Polaroid can vary for a fixed angle relative to the electric vector of the light, check that, as the filter is rotated, the laser beam passes through the same spot on the filter. Place the screen downstream. Now rotate the Polaroid while observing qualitatively the behaviour of the transmitted intensity on the screen. Is the laser beam unpolarized, partially polarized, or linearly polarized? (The question of why the laser has this state of polarization will be addressed in Exercise 4). Remove the screen so that the laser beam strikes the photodetector. Adjust the position of the laser and the detector until the beam exactly enters the opening of the detector. Once adjusted, be careful not to touch the laser, or you may upset the adjustment (and your measurements!). Connect the digital voltmeter (DVM) to the detector and select an appropriate scale. Measure the transmitted light intensity I as a function of θ, the angle between the polarization direction of the laser and the transmission axis of the polarizer. (Remember, this linearly polarized laser is effectively a combination of a randomly polarized laser and a Polaroid). Vary θ from 0º to 180º in steps of 10º and make a graph of I versus cos 2 θ. Comment on your observations. 2. Two Polarizer Experiment Orient the polarizer set-up for the last exercise so that the transmitted intensity is zero. Now insert a second polarizer between the first polarizer and the laser, again making sure that it is centred. Note that the beam is now partially transmitted for almost all orientations of this second polarizer. For what orientation is the transmitted light most intense? Using a similar approach to that used to derive Malus s Law, deduce that the transmitted intensity varies with the orientation angle of this second polarizer as I = I 0 sin 2 2! (3) Put the derivation in your lab notebook. Test this expression by measuring I for various angles, and plotting a graph of I versus sin 2 2!. 3. Production of Circularly Polarized Light One of the more unusual states of polarization is circular, where the electric field vector E of the light has a constant magnitude and rotates in the transverse plane at the frequency of the light wave. Such a motion can be produced by superposing two perpendicular oscillating 4
electric fields (say E x and E y ) which have the same amplitude but have a phase difference of π/2. To produce such light, we make use of the fact that some materials are anisotropic with respect to the propagation of light. In such cases, light polarized in different directions travels at different speeds through the material. In other words, the index of refraction of these materials depends on direction. We define the fast axis of such a material as the direction of E for which the wave would travel fastest. If linearly polarized light is incident on a slab of this material so that its electric vector is at an angle to the fast axis (chosen as the x-direction in Fig. 2), then the E x and E y components which started out in phase will gradually develop a phase difference!" because they travel at different speeds. If the thickness of the material is d and the indices of refraction for E x and E y are n x, and n y, then ( ) n x % n y d (4)!" = 2# $ where λ is the wavelength of the light. If n x! n y d is chosen to be equal to! 4, then!" = # 2 as required. Such a device is known as a quarter-wave retarding plate, or just quarter-wave plate, and for arbitrary values of θ it turns linearly polarized light into elliptically polarized light. However, if θ is chosen to make E x = E y, then circularly polarized light results. Figure 2. Light propagation in an anisotropic medium. Method Replace the Polaroid that is nearest the laser by the quarter-wave plate. Rotate the remaining Polaroid through 360º while observing the transmitted intensity. Now change the orientation of the fast axis of the quarter-wave plate by rotating the plate in its holder, and again examine the transmitted intensity as you rotate the Polaroid. By repeating this a few times, you should find an orientation of the quarter-wave plate for which the transmitted intensity is 5
independent of the orientation of the Polaroid. For this condition, what is the angle between the laser (linear) polarization and the special axis of the quarter-wave plate? Explain why rotating the Polaroid has no effect on the transmission. 4. Polarization of Light by Refection: Brewster s Angle When a beam of light strikes a block of glass, part of it is reflected and part is transmitted. If the incident light is unpolarized, then for all angles of incidence other than zero, the reflected and transmitted rays are partially or totally polarized. In 1812 Brewster discovered that for one particular angle of incidence, called the polarizing or Brewster s angle, the reflected wave is plane polarized in a direction perpendicular to the plane of incidence. Using the theory of transmission and reflection of electromagnetic waves, it can be shown that this occurs when the reflected and transmitted rays are mutually perpendicular, as illustrated in Figure 3. Figure 3. Incident and reflected rays at the Brewster angle i b to the normal. In your lab notebook you should use Snell s Law, and the fact that the angle of incidence is equal to the angle of reflection, to show that this condition gives tani b = n (5) where i b is Brewster s angle and n is the refractive index of the glass. The experiment you are about to do really should be done with unpolarized light, since its objective is to show how reflection of unpolarized light can produce polarized light. However, as you have determined in Part 1, the output of the laser you are using is already polarized, and for a reason which is very closely related to the subject of this exercise. Thus what you will observe is that light polarized parallel to the plane of incidence is not reflected at all when it is incident at Brewster s angle. This implies that if unpolarized light were incident at Brewster s angle, only the component polarized perpendicular to the plane of incidence would be reflected, thus producing linearly polarized light. 6
Method Mount a Polaroid filter in front of the laser with its axis horizontal and, with the mounting screws loosened, rotate the laser in its mount until the transmission is maximized. Put the glass plate with its mount on the optical bench. If necessary, adjust the height of the mount so that the beam strikes the glass. Adjust the scale on the mount so that it reads zero when the beam is directed back on itself (i.e. an angle of incidence of zero). The angle of incidence can then be read directly off this scale. Put a polarizer in the holder attached to the mount, and use it to investigate the polarization of the reflected beam. Remove the polarizer and turn out the lights around you. Vary the angle of incidence until the intensity of the reflected beam is virtually zero. (A slight rotation of the Polaroid in front of the laser may improve the degree of extinction). Measure Brewster s angle and compute the refractive index of the glass. Is this a good way to measure the refractive index? Is there a better way? What do you conclude about the plane of polarization of initially unpolarized light reflected at Brewster s angle? The reason that the output of your laser is polarized is that the tube containing the helium and neon gas is sealed off at both ends with windows that are set at Brewster s angle, as shown in Fig. 4. Considering that the laser light must pass through these windows as it bounces back and forth between the end mirrors, can you explain why the laser output is polarized? Figure 4. Laser tube with end windows tilted at the Brewster angle i b. 5. Polarization by Scattering When a beam of unpolarized light passes through a volume containing small particles in suspension, the light scattered sideways is found to be partially linearly polarized. For scattering at right angles to the incident light, the electric field of the scattered light is found to be perpendicular to the plane determined by the direction of the incident and the scattered light. 7
Figure 5. Polarization induced by sideways scattering of light. This behaviour can be understood by considering Figure 5. The unpolarized incident light induces oscillations of the atoms and molecules in all directions normal to its own direction. But since light is a transverse wave, the electric field of the scattered light oscillates in those directions obtained by projecting the induced oscillations of the atoms and molecules into a plane perpendicular to the scattering direction. It follows that the light will be linearly polarized, with the maximum polarization occurring at right angles to the incident light. (Particles that are small compared with the wavelength of light show this effect quite well; for larger particles the induced oscillations are more complicated, and the degree of polarization is less). A familiar example of this effect is the polarization of the light from the sky, which is greatest in a direction of 90º from the direction of the sun. Method Use a Polaroid filter to set the laser polarization to vertical. (There is nothing special about vertical instead of horizontal; it is just slightly more convenient for observation). Set up the container containing the colloidal suspension so that the laser beam passes through it lengthwise. Only a small fraction of the total light in the laser beam is scattered, and the intensity in any one direction is not great enough to be recorded with the photodetector, so in this exercise the eye must be used as the detector. Using a piece of Polaroid as an analyzer, observe the polarization of the light scattered from the suspension in different directions. Be sure to include views at 90º to the incident beam, both from the side and from the top. DO NOT LOOK DIRECTLY INTO THE LASER BEAM!! Describe and explain your observations with diagrams. If the laser were unpolarized and you wanted to produce vertically polarized light by scattering, where would you look? 8