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Experiment 12 Polarized Light 1 Introduction 2 Brewster Angle

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Experiment 12 Polarized Light 1 Introduction 2 Brewster Angle Experiment 12 Polarized Light 1 Introduction Polarization is a wave properly where waves can oscillate in different direc- tions while having the same propagation direction. This is seen commonly in transverse waves (eg, light, gravitational, etc). Much of this week's lab is show-and-tell, where you will see some inter- esting phenomena using polarization and we want you to describe what you saw and provide an explanation for it. We will do some calculations, but the majority of the work is observation. 2 Brewster Angle Recall the problem of an electromagnetic wave reflecting off a dielectric. We will consider the two linear polarizations of light: in P polarization, the E- field is in the plane formed by the incident propagation vector and the surface normal (think P for Plane), and the S polarization which is perpendicular to it (the S is from the German \senkrecht" meaning perpendicular, or note that the S polarization E-field is tangent to the Surface off which the wave is reflecting). If we apply the required continuity expressions to Maxwell's equations, we can find the following relationships regarding the intensity of the reflected wave: 2 n1 cos θi − n2 cos θt RS = n cos θ + n cos θ 1 i 2 t (1) 2 n1 cos θt − n2 cos θi RP = n1 cos θt + n2 cos θi n1 A substitution can be made to replace sin θt by sin θi, but the expressions n2 get messier. If we make a plot of these coefficients as functions of the incident angle (such as what is shown in figure 1), we find some interesting properties. The reflectivities both approach unity as the incident light approaches the surface tangent. This property is exploited to focus x- and γ- rays for telescopes, as 1 even very high energy photons will reflect at near-tangent incidence angles, rather than penetrate the substance or scatter. However, the most important property we see is that at some angle, the reflection coefficient of the P polarization vanishes. This happens when the reflected and transmitted wavevectors are perpendicular to each other. As the polarization and propagation vectors must be perpendicular, the reflected wave cannot propagate and so vanishes. You can show from Snell's Law that nt this happens at tan θB = , where θB is Brewster's angle (named after Sir ni David Brewster, a Scottish physicist of considerable note in the 19th century), and nt and ni are the indicies of refraction in the transmitted and incident media. 1 P-polarization S-polarization 0.8 0.6 Reflection coefficient 0.4 0.2 0 0 10 20 30 40 50 60 70 80 90 Incident angle (degrees) Figure 1: Reflection coefficients for light on an air-water interface. 2.1 Procedure We'll measure the index of refraction of a prism by finding Brewster's angle. We will use spectrometers with polarizer mounts to make this measurement. • Place the sodium lamp behind the slits. • Set the telescope cross hairs on the image of the slits. 2 • Record the reading from the base of the telescope. Use this to correct all other readings from the spectrometer. • Place a polarizer on the telescope mount. • Place the prism on the table. • Rotate the telescope by about 60 degrees. • Place a prism on the prism table such that the reflection off the incident surface goes into the telescope. You may need to rotate the telescope to achieve this. • We know that the index of refraction of the prisms is around 1.7, so we know that Brewster's angle will be around 60 degrees. Further, we know that near Brewster's angle, the P-polarization reflection coefficient is very small, while that of the S-polarization is rather a bit larger. We can use this information to help narrow the search window. • Find Brewster's angle by finding the telescope angle and polarization angle such that the reflection vanishes. As you rotate the telescope you'll also need to rotate the table the prism sits on so the reflection enters the telescope. This will probably be easiest by first finding the polarization state of minimum intensity and tweaking things slowly from there. 2.2 Analysis nt As given above, tan θB = . We know ni ≈ 1, so nt = tan θB. How well ni does this compare to the value we expect, given the previous measurements we've made with these prisms? 3 Circular Polarization An alternate set of basis vectors for expressing the polarization of light is that of left and right circular polarization (LCP and RCP for short). LCP is defined as a rotation in the positive sense (counter-clockwise) when the propagation vector is pointing at you, and RCP is the other direction. 3 What do we mean by \rotating" polarization? Consider E-fields oscillat- ing in the x- and y- directions with a phase difference of π=2: E = E cos(!t) x 0 (2) Ey = E0 cos(!t − π=2) At t = 0, Ex will be a maximum and Ey will be zero, so the sum of them will point in +^x.As t increases, Ex begins decreasing and Ey begins increasing. At !t = π=4 they will be equal, and the vector sum will point into the first quadrant. As !t approaches π=2, Ex approaches zero and Ey approaches its maximum, with the vector sum now pointing in +^y. As t continues to increase the vector sum continues to rotate around. This example illustrates LCP, and the opposite (the π=2 being added rather than subtracted) is RCP, assuming the propagation vector points up out of the page. So if you have two E-fields with the same amplitude and the correct phase, you can create circularly polarized light. If the amplitudes are not the same or the phase is not π=2 you create elliptically polarized light. To create circularly polarized light you must have a quarter-wave plate, which is an optical element made of a birefringent material cut to a thickness such that that waves arriving in phase leave the plate a quarter of a phase apart. Light incident normally on the plate with the linear polarization at an angle of π=4 to the crystal axis will leave the crystal with circular polarization. Quarter-wave plates only work for a very well-defined wavelength, and the ones we have are for the sodium doublet wavelength. Due to their poor condition, we can do little with them other than demonstrate the addition of alternate polarization modes. 3.1 Prodecure • Remove the prism from the spectrometer table. • Point the telescope at the slits. • Place a polarizer between the sodium lamp and the slits on the colli- mator. • Rotate one of the polarizers until the polarization axes are perpendic- ular to each other, ie there is no light passing through them. • Place the circular polarizer on the collimator mount. 4 • You should now be able to see light in the telescope. If there isn't, rotate the circular polarizer by 45 degrees. Rotate the polarizer on the telescope mount through 180 degrees while looking through the telescope. Does the intensity vanish anywhere? 4 Double Refraction Now, we will perform several observations on the light table. 4.1 Calcite The Vikings knew that a naturally birefringent material they called a \sun- stone" (now known as calcite) could be used to assist in navigation at sea. While we don't have longboats or nearby settlements to raid, we do have a few pieces of calcite that we can use to make some observations. • Place a calcite crystal over a small hole in a piece of black paper. Observe the motion of the two dots seen through the crystal as the crystal is rotated. Use an analyzer to study the polarization of the two dots. Note which dot appears to be closer to the surface. Make sketches showing the relative orientation the the polarizer axis to the crystal axes for the various positions. • Place a second calcite crystal on top of the first. Repeat these obser- vations. Notice which dot appears to stand still as the crystal is rotated. This is the O-ray (or Ordinary), the other is the E-ray (or Extraordinary). Which dot appears nearer the top surface? Why? Is the index of refraction greater for the O-ray or the E-ray? Does the calcite crystal have positive or negative birefringence? Explain what you see with two crystals. 4.2 Stress lines in acrylic Also on the light table is a small piece of acrylic in a holder with a screw. Apply stress to the crystal by means of the screw. Observe the plastic with polarizers. What do you see as you change the amount of stress? 5 4.3 Strange colors in syrup The final item on the light table is a bottle of syrup. Place the bottle between two polarizers and observe the psychedelic colors as one of the polarizers is rotated. What's happening here? 5 Rotation of Plane of Polarization and Scat- tering in Sugar Solution Send a beam of polarized light through a tube of sugar solution. Observe the scattered light note its polarization. How does it look when viewed from different azimuthal angles? 6.
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