: Imaging Systems Laboratory II. Laboratory 6: The Polarization of Light April 16 & 18, 2002

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1 : Imaging Systems Laboratory II Laboratory 6: The Polarization of Light April 16 & 18, 22 Abstract. In this lab, we will investigate linear and circular polarization of light. Linearly polarized light can be generated with polarizing materials, where the direction of polarization is determined by the orientation of the material. We will use such materials to study Malus Law. It can also be created via reflection, as is evident from the Fresnel equations. Circularly polarized light is usually generated with the use of waveplates, and its effects will also be studied. Finally, if conditions permit, we will take a look at the polarization of scattered sunlight. Lab write-ups are due one week after you take your data. 1 Theory As is obvious from the name, electromagnetic waves require two traveling-wave components to propagate. These two vector components are (1 the electric field, E, and (2 the magnetic field, B. The two components are orthogonal to each other and mutually orthogonal to the direction of travel, which is in the direction specified by the Poynting vector S. The Poynting vector is the cross product of the two field vectors: 1 S = E B µ If the electric field E is oriented in the x-direction and the magnetic field B in the y-direction, then the two waves propagate in the +z-direction (towards z = +, because: xˆ yˆ = zˆ where the ^ indicates that the vector has unit length (i.e. it is a unit vector, In vacuum (also known as free space, the electric and magnetic fields propagate with identical phases, e.g., E x, = E cos( kz ω t + ϕ ( t 1 B( x, t = B cos( kz ω t + ϕ = E cos( kz ωt + ϕ c where the wave number k = 2π/λ, the angular temporal frequency ω = 2πν, and the velocity of light c = λν. Because the amplitude of the electric field is larger than the magnetic field by the factor c, phenomena associated with the electric field often are much easier to measure than magnetic effects. The direction of the electric-field vector (which may vary with time/position is called the polarization of the radiation. 1

2 Polarized light comes in different flavors; plane (also called linearly, elliptically, and circularly polarized light. Of these the most familiar type is plane polarized, where the electric field vector E points in the same direction for different points in the wave. Plane-polarized waves with E oriented along the x- or y-axis are easy to visualize; just construct a traveling wave oriented along that direction. However, in general the orthogonal electric field components (in the x- and y- directions are independent; they need not have the same amplitude E, frequency ν, or initial phase φ. In this discussion of polarization, the components do have the same frequency, but may have different amplitudes (E x and E y and initial phases (φ x and φ y. Plane-polarized waves at an arbitrary angle θ may be constructed by adding x- and y-components with the same frequency and initial phase but different amplitudes, e.g., E( x, t = xˆ E cos( ˆ, kz ω t + φ + ye, cos( kz ωt + φ The orientation of the linearly polarization electric field vector is: x θ = tan 1 If the amplitudes are identical ( E, x = E, y E, but the initial phases differ by π/2 radians, then the electric field is: E ( E, y E( x, t = xˆ E cos( ˆ kx ωt + ϕ + ye cos( kz ωt + ϕ In this case, the resulting electric vector follows a circular path about the direction of propagation; the electric field is circularly polarized. A similar configuration where the amplitudes in the x- and y-directions are not equal yields elliptically polarized light. Circularly polarized light may be generated by "delaying" the phase of one component of planepolarized light by π/2 radians, or 1/4 of a period. The "device" for introducing the phase delay is called a quarter-wave plate, which is constructed of a material that is structurally anisotropic, meaning that the atomic structure of the material has a definite orientation. In other words, the spacing between atoms in one direction (say, along the x-axis is different than that in another direction (such as along y. These different spacings between atoms result in differences in the atomic forces and thus different indices of refraction along the two directions. The different refractive indices then result in different propagation velocities along x and y, and thus waves with the two polarizations that are in-phase at the input face of a block of material will require different propagation times and thus will arrive out of phase at the output face. Varying the thickness of the block will change the amount of phase delay φ. A phase delay of φ = π/2 is a quarter wave. Such a block is a quarter-wave plate. 3 Procedure This lab consists of three sections, with an optional fourth section if the weather is clear: (1 an investigation of plane-polarized light, including a measurement of the intensity of light after, x y π 2 2

3 passing through two polarizers oriented at a relative angle θ, (2 an investigation of the mechanism for generating plane-polarized light by reflection from a plane surface, (3 an investigation of circularly polarized light, and (4 polarization by scattering (optional, if outside sky is clear and blue. Linear and circular polarizers are available for your use in this lab, as well as quarter- and half-wave plates. Light Source Stationary Polarizer Rotatable Polarizer CCD Camera Figure 1. Experimental set-up for the study of plane polarization. 3.1 Plane-Polarized Light (also called Linearly Polarized The most common mechanism for generating polarized light from unpolarized light is to use a filter which passes only those electric fields whose vectors point in the desired direction. This is the device used in common sunglasses. For our measurements, set up the components as shown in Figure 1. Then, 1. Measure the baseline intensity of the source using the CCD camera (with lens. You may have to attenuate the light source with a piece of paper and/or stop down the aperture of the lens. 2. Insert one polarizer in front of the detector and measure the intensity relative to the original unfiltered light. How much light does one polarizer let through? Next, increase the intensity of the light until you nearly get a saturated image on the CCD. 3. Add a second polarizer to the path and orient it first to maximize and then to minimize the intensity of the transmitted light. Measure both intensities. 3

4 4. With the first two polarizers oriented at 9 relative to one another, add a third polarizer BETWEEN the first two oriented at an angle θ π/4 radians (45. Measure the intensity on the detector. 5. Next we will try to confirm Malus Law. Two polarizers must again be used: one to create linearly polarized light, and one to test for the polarization. The second polarizer often is called an analyzer, and if possible, use one of the rotatable polarizers on the optical mounts. Set up the polarizers and adjust the light intensity so that when they are aligned, the image on the camera is again nearly saturated. 6. Measure the transmitted light for different relative angles θ (say, every 1 or so by rotating the second polarizer. Grab an image of the source through the two polarizers and determine the average pixel value for your light source at each angle θ. The data should follow Malus' law: I = I cos 2 θ. Screen Laser Polarizer θi θr 3.2 Polarization by reflection Figure 2. The basic arrangement for the measurement of Brewster s Angle. 1 View light reflected from a glossy surface (such as a waxed floor or tabletop through the polarizer. Rotate the polarizer to find the direction of greatest transmission. Infer the direction of dominant polarization of the reflected light. 2 Use the setup shown in Figure 2 above so that the laser beam reflects from a piece of glass. As always, NEVER LOOK DIRECTLY AT A LASER SOURCE. You could cause damage to your eye. The prism shown is really just a piece of glass with a flat surface for our purposes today; we won t be using it to disperse or internally reflect light as we have in the past. Place one of the linear polarizers between the laser and the prism. Focus your attention on the 4

5 intensity of the beam reflected from the front surface of the prism you may need to darken the room. Rotate the prism on the paper sheet with angular markings to vary the incident angle θ i. At each θ i, rotate the polarizer to minimize the transmitted intensity. At a particular θ i, the intensity of the reflected beam should be essentially zero; θ i should be approximately 6. This angle θ i is Brewster's angle, where the electric-field vector parallel to the plane of incidence is not reflected. Measure Brewster s angle at least 4 times and average to get your final result. Remove the polarizer from the incident beam and use it to examine the state of polarization of the reflected light. 3.3 Circular Polarization As stated above, circularly polarized light is generated by delaying (or advancing the relative phase of one orthogonal component of linearly polarized light by 9 = π/2 radians. A circular polarizer is constructed as an open-faced sandwich of a linear polarizer followed by a quarter-wave plate oriented at the proper angle; the quarter-wave plate delays one polarization by λ/4 π/2 radians. Because it is an asymmetric sandwich, the circular polarizer works only for light passing through in the proper direction (linear polarizer first. 1 Construct a circular polarizer with the components available and place it over a shiny surface (a coin works well. Shine a light source on the arrangement from above, and test the behavior as you rotate the linear polarizer. Make sure you can see a variation in intensity of the light reflected by the coin. You should be able to see one clear minimum in intensity between and 9 degrees. 2 Orient your CCD camera so you can grab images of the coin at different orientation angles. Measure the intensity at several (~7 different angles from to 9 degrees, including one near the minimum intensity. 3.4 Polarization by Scattering (optional, if outside sky is clear and blue Examine the scattered blue skylight for linear polarization. Look at several angles measured relative to the sun. Try to determine the direction measured relative to the sun for which the light is most completely linearly polarized. This knowledge is useful to determine the direction of polarization of any linear polarizer. You also should check the skylight for circular polarization. 4 Analysis In your writeups, be sure to include the following items. 1 Plot the expected curve for Malus Law together with your experimental data. 2 Note what happened when the third polarizer was inserted between the two crossed polarizers and explain. 5

6 3 State your final result for Brewster s angle with σ / N uncertainty. 4 Graph your results for the brightness of the coin as a function of the orientation angle of the linear polarizer. Explain why the minimum occurs where it does. 6