Physics 309 Lab 2 Faraday Effect The Faraday effect is rotation of the plane of light polarization by a magnetic field acting on a material. The rotation angle θ is proportional to the magnetic field and the length of the sample. The proportionality constant is known as the Verdet coefficient V: θ = VLB. The goal of the lab is to measure the Verdet coefficient for two materials: glass and CdMnTe (Cadmium-Manganese-Telluride). A. Apparatus C1 Stationary capacitor plate L Laser C2 Rotor P Polarizer (stationary) B Bearings M2 Mirror D Photodiode housing S Solenoid M Motor Sa Sample Pa Polarization Analyzer (rotating) Physics 309 Lab 2, page 1
Your apparatus consists of a light source, mirrors to direct the light, a stationary polaroid filter, a polaroid filter rotated continuously by a motor, an electronic readout of the rotation angle, a solenoid magnet for the sample, and a photodetector with a gain adjustment. The angle of the rotating polaroid filter is measured by a capacitive technique. There are two pairs of stationary drive electrodes arranged as quadrants of a circle. A bow-tie shaped rotor plate rotates with the polaroid filter. There is a pickup electrode in the middle of the circle of drive electrodes. The capacitance between the rotor and the drive electrode pairs depends on the rotation angle. If we make adjacent drive electrodes have opposite voltages (thus opposite electrodes have the same voltage), the floating rotor voltage will depend on the rotation angle. It s impractical to measure the DC voltage of the floating rotor directly without disturbing it. But if we put AC on the drive electrodes, we induce an angle-dependent AC voltage on the rotor, which induces an AC voltage on the center pickup electrode. The pickup electrode voltage is passed through a phase-sensitive circuit, resulting in a voltage which is positive when the rotor is aligned with one pair of drive electrodes, negative when aligned with the other pair, and zero when halfway between. If we had two half-circle drive electrodes and a half-circle rotor turning at a uniform rate, the voltage would ramp linearly up for a half-turn, then ramp linearly down. But since the photodetector signal goes through two cycles during one turn, it s convenient to use quadrants so we get two cycles of ramp voltage per turn as well. A polaroid filter transmits light with plane polarization parallel to the axis of the filter and strongly attenuates light with plane polarization perpendicular to the axis of the filter. If polarized light is passed through two polaroid filters, the transmitted intensity depends on the angle α between the polarization and the filter. Ideally, Intensity out Intensity in = T( α) = cos 2 ( θ) Physics 309 Lab 2, page 2
If a sample with Faraday rotation φ is placed between the two filters, the transmission becomes T( α) = cos 2 ( θ + φ) Your magnet is many turns of wire on a water-cooled aluminum bobbin. The 5 ridges you see are the cooling tubes. The water must be turned on and flowing if you use more than about 2 amps of current or the magnet will overheat. The winding is in 4 sections, with the left 2 in series with each other, and the right 2 in series with each other. There external connections to the two halves so they can be connected in series or parallel, depending on the power supply available. Note that if the windings are in parallel, the total current is divided between the halves, and you must be careful about making good connections to keep the division approximately equal. Note also that it is possible to connect the halves so their fields cancel rather than add! The magnetic field is proportional to current, but is not exactly uniform with position. You need to know the magnetic field where the sample is (integrated over the sample length). You will map the field with a commercial Hall-effect gaussmeter at one current setting, and extrapolate to other currents. B. Setup 1. Optics Alignment Plug in the AC cord and turn on the motor to check rotation. Turn on the logic power and connect the Capacitive Ramp signal to the oscilloscope. You should see the voltage ramp up and down very roughly +/ 2 volts. Turn the motor off but leave the logic power on. Put a scrap of paper between the laser and the mirror to check the laser. Put the paper between the mirror and the magnet to check that the beam is aimed into the magnet hole. Put the paper between the magnet and the rotating filter to make sure the beam comes out of the magnet hole. Put the paper between the rotor and the diode detector to make sure the beam comes out. Turn the motor on and you should see the light flash as the polaroid filter rotates. The photodiode signal comes from an amplifier with an offset of about -2 volts, and a gain that is controlled remotely. The maximum possible output voltage is very roughly +2 volts. If the gain is too high, the signal will saturate and be very noisy. Connect the Photodetector OutputÓ signal to the oscilloscope. You should see the photodiode voltage vary sinusoidally. Block the light and it should go to -2 volts. The peak voltage should be about +2 volts. If the light doesn t come through, or your photodiode peak voltage is low with no sam- Physics 309 Lab 2, page 3
ple in place, check the following: 1. The laser box may be rotated on its post. It it s loose, ask for help to get it tightened up. 2. The laser box output head may be rotated, aiming the beam up or down. If it s loose, ask for help. 3. The adjustable mirror may not be firmly attached to its mount. If it s loose, ask for help. 4. The mirror mount may be rotated on its post. The mirror post has a set-screw and a pin. Loosen the set screw, make sure the mirror is seated down, and tighten. 5. The two mirror adjustment knobs may be set wrong. One controls the vertical and one controls the horizontal. Aim the beam so it goes into the magnet, out of the magnet, and into the photodiode. When light goes all the way through, use the mirror adjustment knobs to maximize the peak photodetector amplitude. 2. Magnet setup The bench supply has one switch to turn it off and on, and another to select the voltage range (0-8 or 0-16 volts). Do not run more than 2 amps through the magnet without turning on the cooling water! Make sure both water hoses are connected to the apparatus, and turn on the flow at both yellow valves (handles parallel to pipes). You should feel turbulence in the tubes, and the temperature of the magnet. The magnetic field depends on the current, not the voltage directly. The knob on the supply controls the voltage not the current, so changing the load resistance changes the current. The magnet resistance depends on the temperature of the coil, which will depend on how long the magnet has been on or off. The current meter on the supply is not very accurate, so use the black ammeter to measure the current. Set the right switch to DC, and the left switch to 10 amps. Connect the red post of the power supply to the left post of the meter, and the right post of the meter to the red post of the magnet. Connect the black post of the magnet back to the power supply. The magnet has 4 physically separate windings. The left 2 are in series, and the right 2 are in series. You can connect the two magnet halves in series or parallel externally, to match the current-voltage rating of a power supply. If possible, connect the windings in series. The winding resistances are not identical, so if you connect them in parallel, the current will not divide exactly equally. The meter Physics 309 Lab 2, page 4
will show the sum of the two currents. If the windings are in parallel, you should measure the two currents with two meters for precision work. Note that it is possible to mis-connect the windings so the fields subtract instead of add, whether they are in series or parallel. If you have bad connections, their resistance can change the current significantly as well. So wiggle all the wires while watching the current meter, and replace any wires that seem flakey. Do not assume that the magnet current stays constant! The supplies are voltage-regulated, not current-regulated. The magnet resistance changes with temperature (as well as from flakey connections). So make a habit of checking the current regularly. 3. Magnetic field vs current vs position calibration The rotation angle is related to the Verdet coefficient V by θ = VLB, where L is the sample length and B is the magnetic field. So to get the Verdet coefficient, we need the sample length (written on the sample holder) and the magnetic field. If the field is not uniform over the length of the sample, you need to know the the field vs position. How does the total Faraday rotation depend on the magnetic field and sample length if the magnetic field is non-uniform? Sketch what you think the magnetic field lines should look like for your solenoid. How does the direction depend on Z position, and radial position? Sketch what you think the magnetic field Z component will be as a function of position along the solenoid axis, including beyond the end of the solenoid. There is only one Gaussmeter for the class, so you will have to take turns using it. The magnetic field vs current calibration can be done either before or after acquiring the sample data, so take sample data and fit it while waiting for the Gaussmeter. The Gaussmeter works by the Hall effect (the transverse voltage produced by the Lorentz force on a current flowing in a magnetic field) in a small semiconductor sensor in the tip of the probe. The Hall effect is sensitive to a single component of the magnetic field. For this probe it is the component parallel to the long axis of the probe, which is appropriate for measuring the Z component of the magnetic field inside your solenoid. The magnetic field is measured at the tip of the probe. Turn the Gaussmeter on. The flashing number initially on the display is the last probe calibration, which should match the number on the label of the probe. If it doesn t, refer to the instruction book in the case. If it does, press the Enter/Reset button next to the power switch once. The display should flash Zero Probe. Put the probe all the way into the magnetic shield, and press the PB button (one row up from the Enter /Reset button. The display should change to 00.0. When you take the shield off the probe, it should read about 0.5 Gauss if the probe is parallel to the Physics 309 Lab 2, page 5
Earth s field Take the mirror off the apparatus to make room for the Gaussmeter. Connect your bench supply and ammeter to BOTH of the windings of the solenoid and set it to a large current (why?). Note the current, and whether the connection is series or parallel. Measure the magnetic field every 5mm, from 0mm defined as the end of the solenoid, to as far as you can get the probe. Check the current again at the end of the measurement. Repeat the process a few times. Plot your data. Is the reproducibility of your repeated measurements consistent with the 0.1 Gauss resolution of the meter, or is there some other contribution to the error? How big? What else might be contributing? C. Manual Observation of Faraday Rotation Connect the Capacitive Ramp and Photodetector output both to the oscilloscope. Display both signals in chop mode. They should be roughly in phase with each other. Set the oscilloscope to xy mode (set the time knob to below the slowest sweep speed). If the ramp is in channel 1 and the photodiode signal is in channel 2, you should get a curve on the scope that looks like an integral sign. If the ramp and photodiode signals are exactly in phase, you will get a single curve. If they are a bit out of phase, the curve with be a loop. There are actually two full cycles of the voltage for one turn of the apparatus, and they are typically slightly different, so you may be able to recognize 2 different loops. The Faraday effect shows up as a change in the rotation angle for maximum transmission. It s most convenient for measurements if the curve is a loop, because then the Faraday effect moves the curves apart when the magnetic field is one sign, and moves them closer together when the field is the other sign. The relative phase of the two signals can be changed by rotating the polaroid filter on the laser head (you have to loosen the set screws to do it). Loosen the post setscrew on the mirror and lift it off. Put the clear glass sample inside the magnet, and put the mirror back. With the magnet off but the motor running, set the oscilloscope to XY mode with the ramp voltage in channel 1 and the photodiode signal in channel 2. Watch the upward and downward zero-crossing points, and turn the magnet on and off. What do the zero crossing points do? If you don t see a definite effect, increase the vertical gain on channel 1 (which controls the horizontal gain in XY mode). Also use a higher magnetic field. How many degrees of rotation does the change of the zero crossing points correspond to? What do you estimate is the error on your measurement of the angle? Physics 309 Lab 2, page 6
What happens when you reverse the magnetic field? The other sample is CdMnTe, cadmium-manganese-telluride, with a much higher Verdet coefficient. It is much less transparent (you can look through the sample holder to check), so the intensity signal will be smaller. You will probably want to change the amplifier gain by toggling the dip switches on the plug-in circuit. Set the photodiode amplifier gain to the highest value that does not saturate the output. The integer you type in the box is converted non-linearly to the gain when you press the button. Repeat the manual measurement of the change in the zero-crossing points between magnet-on and magnet-off, for the CdMnTe sample. How many degrees of rotation does it correspond to? What is your estimated error? Calculate the Verdet coefficients for CdMnTe and glass from your manual measurements of the rotation angle, and the magnetic field and sample length. List the contributions to the estimated errors on your values in a table. Compute the sum in quadrature (square root of sum of squares) of the error contributions. You estimate the magnetic field in the sample from your field vs current measurements. The resolution of the Gaussmeter is 0.1 Gauss, and the calibration accuracy is 2%. How do these enter the error in the field? How does the reproducibility of the field vs position data enter? How does the accuracy and resolution of the ammeter enter? What if you use the same ammeter for calibration of the magnetic field and for taking rotation data? How well do you know where the sample was in the magnetic field? How much error is contributed by this? How do you average over the non-zero length of the sample? You can significantly reduce the contribution of the rotation angle measurement error in two ways. You can measure the angle more accurately by computer measurement and fitting, and you can increase the range of rotation angle by going to higher magnetic field. D. Computer data acquisition Remove the sample from the apparatus. Connect the photodetector and ramp signals to the interface board inputs with the appropriate range. Set the number of samples and µsec/sample to values that will record about 3 full periods. Acquire the data. Save the data to a file. Print the data in A,B vs time mode, and B vs A mode and tape in your notebook. Copy the postscript file if you want it for your formal writeup and/or oral presentation. Physics 309 Lab 2, page 7
Plot the data with Gnuplot. The columns are Channel A voltage, Channel B voltage, Channel A time, Channel B time. Plot A and B vs time, and B vs A. Print your plots. Since the upward and downward ramps are different, we need two different functions to fit the data. We also may have a difference between the two up-ramps in a single rotation of the apparatus, and between the two down-ramps in a single rotation. We need to tell Gnuplot somehow what range of data to fit. The simplest way is to put a pair of blank lines in the file between different parts of the data set. Open the file with a text editor. The data in the column with the ramp voltage will increase uniformly to a maximum (or decrease uniformly to a minimum), then the slope will reverse. Scroll until you find the place where the slope reverses, and insert two blank lines. Continue scrolling to the next slope reversal, and insert two blank lines there. Repeat this until the end of the file. Gnuplot considers double blank lines to mark an index boundary in the file. To plot the first part of a file by itself, type plot datafile index 0 using... To fit the second part of a file without the other parts, type fit f(x) datafile index 1 using... via... Fit the Channel A and Channel B vs time in the different sections of your file, using appropriate functions (linear for the ramp, offset sinusoid for the photodiode). Plot the residuals. What do you conclude? Manual editing of the data file to separate the sections is pretty tedious, and we will be dealing with many data files, so a program to do it automatically has been written for you. Make now use of the file mp.plt, which separates the up and down slopes of your data file (the old file is not touched, hopefully!) Make a gnuplot call-file that takes a file-name as a parameter, which defines your functions and initial parameters, and fits the datafile. You can then make a load file that calls this call file multiple times with the datafile.0, datafile.1, datafile.2, datafile.3, etc. files. Repeat the fits you did the previous section using this method. E. Fitting for Faraday Rotation Angle The sawtooth voltage is related to angle by V 1 ( θ) = g 1 θ + h 1 where h 1 is an offset and g 1 is a gain or slope. We can solve this for θ : Physics 309 Lab 2, page 8
θ( V 1 ) = V h 1 1 g 1 The photodiode voltage is related to the light intensity by V 2 ( L) = g 2 L + h 2 where where h 1 is an offset and g 1 is a gain or slope. The light intensity vs rotation angle when the polaroid filter is rotated is L( θ) = L 0 cos 2 ( θ + φ) where φ is the angle where the light intensity is maximum, and θ is the rotation angle relative to that angle. If we change the magnetic field, the φ parameter will change due to the Faraday rotation. So to measure the Faraday rotation, we need to look at how φ changes. The photodiode voltage is related to the rotation angle by V 2 ( ) + h 2 ( θ) = g 2 L 0 cos 2 θ + φ It turns out to be useful to use the identity cos 2 cos 2α ( α) = ( ) +1 = 2 to write an equivalent function V 2 sin 2θ + 2φ + π 2 ( θ) = g 2 L 0 2 sin ( 2α + π 2 ) + 1 2 ( ) +1 We can insert the solution for θ( V 1 ) to get sin 2 ( ) = g 2 L 0 V 2 V 1 We can then re-write this as + h 2 V 1 h 1 + 2φ + π 2 +1 g 1 + h 2 2 V 2 ( V 1 ) = g L 2 0 2 sin 2 V 1 + 2h 1 g 1 g 1 + π 2 + 2φ g + 2 L 0 2 + h 2 Physics 309 Lab 2, page 9
which is equivalent to V 2 ( V 1 ) = asin( bv 1 + c) + d This is a fairly simple 4-parameter function to fit, although the fit parameters have complicated relations to the physical parameters: a = g 2 L 0 2 b = 2 g 1 c = 2h 1 g 1 + π 2 + 2φ d = g 2L 0 2 + h 2 But we don t really care much about all the gain and offset parameters, or the L 0 parameter, only about the Faraday rotation. It shows up only in the c parameter. So if we plot the c parameter vs magnetic field, we should get a line whose slope is twice the Faraday rotation angle vs magnetic field. We actually have 4 different data segments: two up and two down. Each will have a different offset, but should have the same slope vs magnetic field. There is an intrinsic phase-ambiguity in the sine function: sin( α) = sin( α + 2πn) where n is an integer. So it is possible for your fit to give a c parameter that is off by exactly 2πn. The solution your fit picks will probably depend on the initial parameter values you give it. The slope of voltage vs angle, g 1, is negative for the down ramps. So you might expect the b parameter to be negative for the down ramps. But there is another phase ambiguity: sin( α) = sin( α) = sin( α + π). You can get an identically good fit with b and c, or with b and c + π. ( ) = asin( bv 1 + c) + d is The reason I recommend that you use the functional form V 2 V 1 that you can get fits to all 4 segments of the sawtooth with similar a, b, and d parameters (by using the above phase ambiguity). The two up-ramps will have similar c parameters, and the two down ramps will have similar d parameters. The slope of c vs magnetic field will be opposite for the up and down ramps. Fit each of your data file segments to the a,b,c,d parameterization. Plot the function overlays on the data for each data file segment, and record the parameters. Does the parameterization fit the data well? How consistent are the parameters between file segments? Physics 309 Lab 2, page 10
Make a call-file that initializes the parameters to typical values, and fits the file whose name is passed in as a parameter to the a,b,c,d parameterization. Make a load-file that defines the a,b,c,d parameterization, then calls the call-file once for each data file segment. Have the load-file re-name the file that contains the results of the 4 fits, by including a line like! mv fit.log newname.log This will cause Linux to rename fit.log to newname.log F. Faraday Rotation Data Get a sample of CdMnTe or glass, and note the number of the sample holder (so you can get the same one later). Put your sample in the solenoid and note where the sample is relative to the end of the solenoid (so you know what part of your field map to use). Adjust the gain of the photodiode amplifier to give the largest signal that does not saturate. Set the samples and time per sample to get somewhat more than 2 cycles of the ramp. Making sure that you power both windings, acquire data at the maximum current your bench supply can reach. Write down the current reading, both before and after acquiring data. If the current is significantly different, check for loose connections. Reverse the current and acquire data there, also noting the current before and after. Note you will have to reverse the ammeter connections when you reverse the current. Also acquire data at zero current. Split your data files into sections, and fit the photodiode vs ramp for each section of each file. Record all the fit parameters. The parameters should be essentially the same for all sections of all files, except the c parameter that differs due to Faraday rotation (and due to up and down ramps). If the other parameters differ, check whether the fit has chosen a mathematically equivalent solution (like adding 2Nπ to the phase, changing the sign of the amplitude and also changing the phase by π, etc.) Try repeating the questionable fit with starting values for the parameters that fit the expected pattern. Create a file in a text editor, with one column for the magnet current, 4 columns for the Faraday rotation sensitive parameter from your photodiode vs ramp fits for the 2 upramps and 2 down-ramps, and more columns for the errors from the fits. You should be able to manually recognize the two different up and two different down ramps by their slightly different parameter values. Make one plot with all 4 columns plotted vs magnet current. You should have 2 parallel lines with positive slope and two with negative slope. You may be able to recognize some swaps of data between columns, if so, fix them. Replot the data with appropriate sign reversals so you get 4 parallel lines. Physics 309 Lab 2, page 11
Fit the slopes of the lines, and average them together. What is the error on the average? What is the effect of current measurement errors on the slope error? G. High Current Data The Faraday rotation angle is small, especially for the glass sample, and it is useful to get to higher current. There is one large power supply on a cart in the lab that can get to higher magnetic field values than is possible with the small bench supplies. It will also put out enough current to blow its own fuse (15 amps). The left-hand knob sets the voltage (the inner part is a fine adjustment). Note that zero on the knob still produces a substantial output voltage when the supply is turned on. The right hand knob sets a current limit increasing the voltage knob setting will not increase the current beyond the current limit setting. The on-off switch is next to the adjustment knobs and the pilot light. The switch in the lower right corner is a polarity reversing switch. The meters on the power supply are not very accurate; you still need to use the black ammeters. Connect the power supply to your ammeter and solenoid. Set the supply to give as high a current as possible without blowing the fuse (which is possible if the windings are connected in parallel). Take data files with the same conditions you used for the others at this current, then reverse current, then zero current, for both samples. Split the files, and fit them. You should be able to add c parameters from this data to the data you took at lower current with the bench supply. H. Calculating Verdet Coefficients and Errors Calculate the Verdet coefficient for the glass and CdMnTe samples. Calculate the errors on your Verdet coefficients from all the sources you can think of. List the contributions in a table. Compute the sum in quadrature of the errors. Physics 309 Lab 2, page 12