Faraday Rotation Objectives o To verify Malus law for two polarizers o To study the effect known as Faraday Rotation Introduction Light consists of oscillating electric and magnetic fields. Light is polarized when the electric field oscillatess in a unique direction. The plane of polarization is determined by the axis of the polarizer. Only the component of the incident light with an electric field oscillating in that plane can pass through the polarizer. All other light is unable too pass through. Once the light has passed through the filter, it is linearly polarized. 1 When two polarizers are placed in line, the second polarizer is called the analyzer. If the polarizers are at 9 with respect to each other, then no light will pass through the analyzer; this is called the extinction point. If the polarizers are at with respect to each other, then all the light will pass through the analyzer. In the range between and 9 a fraction of light passes through the analyzer. This relationship between two polarizers is illustrated in Figure 1. Figure 1. Two sheets of polarizing material, called the polarizer and the analyzer, may be used to adjust the polarization direction and intensity of the lightt reaching the photocell. This can be done by changing the angle θ between the transmission axes of the polarizer and analyzer. When unpolarized light passes through the first polarizer, the intensity of light is reduced to 5 % of its initial value. If the light entering the analyzer has an intensity S, the intensity of light that exits the analyzer is given by Malus law, 2 S = S cos θ (1)
Faraday Rotation In 1845, Michael Faraday discovered that linearly polarized light passing through a material in the presence of a magnetic field experiences a rotation, which is now referred to as Faraday rotation.. 2 This was the first experimental evidencee of a relationship between light and electromagnetism. Figure 2 shows the rotation of the plane of polarization after passing through the material. The light is propagating in the x-direction, but the light is polarized along the z-axis. The horizontal arrows throughh the sample represent the magnetic field. The magnetic field must be parallel to the propagation of the light in order for the rotation to occur. 3 Figure 2. Incoming light is polarized along the z-axis and is propagating along the x-axis. The rotation is evident when comparing the incoming and outgoing red arrows representing the planes of polarization. Some materials, like calcite, are always birefringent, materials that are anisotropic in that the speed of light is not the same in all directions. When a light ray is incident on this type of material, it is separated into two separate rays that have orthogonal polarization directions. Consequently, the two beams travel at different speeds through the material. 4 Faraday rotation is a result of a magnetic resonance that causes waves to be decomposed into two circularly polarized rays that propagate at different speeds, a property known as circular birefringence. The rays recombine upon emergence from the material. However, because there is difference in propagation speed, they do so with a net phase offset,, resulting in a rotation of the angle of linear polarization. The amount of rotation is given by, π ΔnL φ = Λ =Λ λ L B dl BL (2) where Δn is the magnitude of circular birefringence, L is the length of the sample material, λ is the wavelength of the light, and B is the magnetic field. The Verdet constant Λ indicates the relative ability of light to rotate within the material. 3 The larger the constant for the material, the larger the effect. Some materials have extremely high Verdet constants. Terbium gallium garnet is one (Λ 4 rad/t-m)of over 45 can be achieved. In this experiment, two methods for determining the Verdet constant will be used. The first By placing a rod of this material in a strong magnetic field, Faraday rotation angles method of determining the Verdet constant will use Equation 2 and involves adjusting the analyzer angle for maximum intensity with and withoutt the magnetic field, so that the rotation angle φ can be determined. The second method of determining the Verdet constant combines Malus Law and Equation 2. Malus Law can be rewritten in terms of the measured voltages from the photodetector used in this experiment to give
1 V B θ = = cos = B V (3) θ > = cos c B 1 V B V (4) Using these relationships, the angle can be found from the voltages for the casess with no magnetic field and with a magnetic field present. V refers to the maximum voltage when the magnetic field is present. Next, the angle φ can be found from the difference of Equation (3) and Equation (4). φ = θ θ B> B= (5) II. Experimental Set-up Equipment The TeachSpin Faraday Rotation Apparatus (FR1-A) has four basic components. First, there is a laser that produces 65 nm light with a power output of 3 mw. Within the light source container is the first of two polarizing filters that polarizes 95 % of the light. The laser has adjustmentt knobs so that the light can be directed through the center of the solenoid. 4 Secondly, theree is the solenoid that has a length of.15 m. The coil has 14 turns per layer with 1 layers. The third component is the analyzer, the second polarizer. The analyzer is able to rotate through 36. The fiducial marks on the analyzer occur in 1 increments. Lastly, the light is 4 directed into a photodiode whichh is a linear photon detector connected to three resistors. The detector is connected to a voltmeter. The voltage corresponds to the relative intensity of the light passing to the photodiode. This device is pictured in Figure 3. Figure 3. This is a photograph of the TeachSpin Faraday Rotation Apparatus. The apparatus is made up of four main components including the light source, solenoid, analyzer, and photodiode. The light source includes a polarizer at the exit.
The apparatus came with glass rods of diameter.5 m and.1 m long. These rods are designed to be placed inside the solenoid so that Verdet constant measurements can be carried out on the rods. 4 The rods are made of an unknown material and the Verdet constant for these rods is unknown. A DC power supply is connected to the solenoidd so that a current can be deliveredd to the coils. There is a power supply for the laser built into the amplifier box, which is used to amplify the signal from the photodetector. A voltmeter is used to read the amplified signal from the photodetector. A PASCO magnetic field sensor was used to determine the magnetic field within the solenoid. The sensor has a Hall Effect probe that was inserted into the solenoid for the measurements. The sensor has 8 notches on the long probe which were separated by.1 m. Measurements were taken by lining up each notch at the edge of the wooden case for the solenoid. The field was calculated for a wide range of currents. 5 The magnetic field in the solenoid is depicted in Figure 4. If the sample rod is placed at the center of the solenoid, its end will lie near the 5.5 mark. Thus, the rod will be in a relatively constant magnetic field. The end of the solenoid lies at the 3 mark. Notice how abruptly the magnetic field decreasess near the end of the solenoid. Thus, it is clear that the sample must be placed at the center of the solenoid. Figure 4. The magnetic field inside the solenoid, startingg at one end of the solenoid, is plotted as a function of the current in the solenoid. 1. 8 A 1. 6 A 1. 4 A 1. 2 A 1. A. 8 A. 6 A. 4 A. 2 A
III. Experimental Procedures Malus Law There is an entrance polarizer in the laser light source compartment. The second polarizer is the analyzer that will be rotated during this exercise to validate Malus law. Make sure that the solenoid is empty. Before taking measurements, align the laser so that it is directed through the center of the solenoid and into the photodiode. It may be helpful to use a note card to verify that light is passing through the solenoid. The light source direction can be modified with the white knobs on the outside of the light source compartment. The height of the photodiode can also be adjusted using the white knob at the base of the detector. The solenoid should be empty and no current should be passing through it. The maximum voltage across the detector must not exceed.3 V because the detector will become saturated and become nonlinear past this point. The resistance on the photodiode should be chosen so that the voltage is within the linear range, less than.3 V. To begin taking data, choose an initial point on the analyzer to start measurements ( ). Rotate the analyzer in five degree increments for 36. At each 5 increment, record the angle and the voltage reading. When the laser is turned off, find the background voltage produced from the electronics and the lights in the room. Make a graph of voltage versus angle, and confirm that the voltage versus angle plot follows the cosine squared function given by the voltage form of Malus Law in Equation 6 below. Take the background light into account by subtracting any background voltage from measured voltages. V = V (6) cos 2 θ Determination of Verdet Constant, Λ Method I To calculate the Verdet constant, the glass rod must be placed inside the solenoid. In order to insert the sample rod, the analyzer, photodiode, and light source need to be removed from the platform so that you have full access to the center of the solenoid. It is important not to scratch the ends of the rods. A soft material like a cotton swab should be used to insert the rod. Again, the light beam must be aligned so that it is directed along the central axis of the solenoid and into the photodiode. Method I is comprised of three steps that should be repeated for eight values of the magnetic field. 1. Set the analyzer to 9 with respect to the entrance polarizer. This may not be at the 9 marker on the analyzer, but this point should correspond to the extinction point where the light intensity passing through the analyzer (or the voltage reading) is the smallest. The method requires one to accurately determine the extinction point. Note: Be careful when you are determining the extinction point. Since the minimum of the cos 2 θ curve is sort of broad, it is difficult to know if one is truly at the mid-point of the minimum. However, as one gains experience and understanding of the method, one can
become more adept at finding it. Also, a shadow over the photodiode may be present when you rotate the analyzer that in turn changes the voltage readings. Therefore, when rotating the analyzer, make a small rotation, and then completely move your arm/hand from above the apparatus before taking a voltage reading. Maintain consistent measuring techniques through the experiment. 2. Turn on the current to the solenoid. The presence of the magnetic field will cause the voltage to change so that the analyzer is no longer at an extinction point. 3. Rotate the analyzer so that it returns to the extinction point. The amount that the analyzer is rotated is the angle φ needed to find the Verdet constant. The magnetic field B is determined by the amount of current through the coils given by Equation (7) where I is the current in amps. mt ( 11.1 A ) B = I (7) Now, the Verdet constant can be determined using Equation (2) where φ is the angle the analyzer was rotated to return to the extinction point, and L, the length of the sample, which is.1 m (notice this is not the length of the solenoid, but the length of the sample). Calculate the Verdet constants from all trials. Note: The units for Λ should be in rad/(mtm). Report the average and standard deviation for the measured Verdet constants. Plot φ versus BL. Note that φ is the angle required to rotate the analyzer that should be between and 1. What does the slope of φ versus BL represent? Method II Method II does not rely on searching for the extinction point. In the first part of this exercise, the cos 2 θ curve was measured as a function of θ when there was no applied magnetic field and no rod inside the solenoid. When the rod is inside the solenoid and a magnetic field is applied to it, the cos 2 θ curve shifts to the left or right, depending on the direction of the magnetic field. 1. With the rod inside the solenoid and I = A and the laser turned on, rotate the analyzer to find the center of a maximum on the cos 2 θ curve. Record the photovoltage V and angle θ at which this occurs. Then, in 5 steps up to θ + 9, record the photovoltage and angle in a data table. 2. Repeat step 1 for the following currents: 1.8, 2.1, and 2.5 A. 3. For each of the four sets of data collected in steps 1 and 2, plot each set of data, determine the range of θ on the plot that is linear. Replot the data for the range of θ that is linear, then do a linear fit for that data. Record the equations for the linear fits in your lab notebook in the form: V = V + aθ, where V is the y intercept and a is the slope of the line. 4. Pick a value of the photovoltage in the middle of your measured range, which should correspond to the middle of your linear fits. Use this value to calculate a corresponding value of the angle θ for each of the four linear fits. 5. For each of the non-zero currents (and their corresponding magnetic fields), you can now calculate φ I = Δθ I and the Verdet constant from Equation 2, Λ = φ I / BL. 6. Determine the average and standard deviation for the measured Verdet constants.
Questions 1. Determine the uncertainties for the Verdet constants determined by the two methods. Explain or demonstrate mathematically how you arrived at these uncertainties. 2. Do the Verdet constants determined by the two methods agree with each other? How can you tell? If they are not consistent, how can you tell which result is better? 3. Which of the two methods is better? Explain your thinking. 4. What effect on your results, if any, would result if the direction of the magnetic field in the solenoid was reversed? 5. If the light source was a different wavelength, would that make a difference in the Verdet calculations? 6. Does the light undergo a rotation in its plane of polarization if there is no sample material placed in the solenoid? References 1. P.A. Tipler and G. Mosca, Physics for Scientists and Engineers, 5th ed. (W.H. Freeman and Company, New York, 24) pp. 121-127. 2. F.J. Loeffler, A Faraday rotation experiment for the undergraduate physics laboratory, Am. J. Phys. 51, 661-662 (July 1983). 3. A. Jain, J. Kumar, F. Zhou, L. Li and S. Tripathy, A simple experiment for determining Verdet constants using alternating current magnetic fields, Am. J. Phys. 67, 714-717 (August 1999). 4. Faraday Rotation: Guide to the Experiment, TeachSpin, Inc. Buffalo, NY. (23). 5. Instruction Sheet for the PASCO Model C1-652A, PASCO scientific, Roseville, CA. (1999)