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Microwave Interferometry: Measuring index of refraction By: Carissa Erickson Ryan Forde Objectives The objectives of this lab were to become familiar with microwave circuits and to determine the refractive index of several materials. The microwave circuit used was an interferometer. Theory Microwaves are electromagnetic radiation. The range usually considered to be microwaves has frequencies between 1.5 GHz and 25.0 GHz. They were generated in our setup by a reflex Klystron, then sent through an interferometer. The interferometer allowed the phase shift caused by a sample to be measured. Reflex Klystron In a klystron, microwaves are produced by accelerating a beam of electrons. The reflex klystron is different from other types of klystron in that it uses a reflector plate instead of an output cavity. The electron beam is modulated by passing it through an oscillating resonant cavity. The feedback required to maintain oscillations within the cavity is obtained by reversing the beam and sending it back through the cavity. The electrons in the beam are velocity-modulated before the beam passes through the cavity the second time and will give up the energy required to maintain oscillations. The beam is turned around by a negatively charged electrode called the reflector plate. Interferometer In an interferometer, a beam of radiation is split into two paths. The paths are adjusted so that they are 180 out of phase and no signal is received. A sample introduced into one of the arms induces a phase shift. The induced phase shift can be measured by adjusting the path in one of the interferometer arms. Measuring this adjustment provides a measure of how much phase was induced by the sample. Induced Phase Shift The induced phase shift depends on equation (1). From this equation, we can derive an expression that will allow the index of refraction of the material to be computed. Equation (4) depends only on the measured phase shift and the thickness, d, of the material. d (1) k k 0 dx k k 0 d 0 (2) (3) (4) k k 0 1d k 0 c 2 f n 1d 2 f 1d c v c c r c 1 2f d 2 r 1 d Index of Refraction (Permittivity) The materials used as samples in this type experiment must all be dielectrics. This means that their electrons are not free as in a conductor, but bound to the atoms in the material. They are affected by an electric field, however. The electric field produces a dipole moment by shifting the electron orbits in the opposite direction to the electric field. The shifted electron orbits cause an electric field to form in the dielectric and this field interacts with the electromagnetic wave passing through. This interaction causes the difference in permittivity between a dielectric and free space. (5) 2 ne r 0 1 2 m 0 Where r is the permittivity of the dielectric, 0 is the permittivity of free space, n is the number of molecules per unit volume, e is the electronic charge, 0 is the frequency of bound motion and m is the mass of the nucleus of the molecules of the dielectric in kg. Introduction to Wave Phenomena by A. Hirose and K. E. Lonngren Apparatus Waveguide Tuner E-H Tuner Coupler XXXXXX Klystron Gain Horns Sample Hybrid matched tee Phase Shifter Detector Attenuator Isolator Equipment Klystron: This was the microwave generator. Its operation is described above. E-H Tuner: This was used to match the radiation generated by the klystron to the waveguide. It probably consists of some kind of resonant cavity whose properties can be adjusted. A matched signal has a much higher amplitude and should suffer lower loss in the waveguide. Waveguide Tuner: Gain Horns: Some of the samples attenuate the microwave beam. These amplified the signal by 5dB so that it was easier to detect. Sample: This is the piece of material whose index of refraction is to be measured. Phase Shifter: This was used to adjust the phase of the microwave beam in one arm of the interferometer. The exact mechanism for this device is not known, but it probably operates by applying a magnetic field across the waveguide. Attenuator: Attenuation was used to match the amplitudes of the two signals so that they cancel as completely as possible when they are out of phase. Isolator: This was used to prevent radiation from traveling back to the source. Procedure 1. The corresponding voltage was measured in steps of 0.01 units on the phase shifter between 0 and 0.11. This included the first maximum as well as the first minimum on the voltage curve. The voltage curve was plotted and is shown in Figure 1. voltage Voltage vs Position 0.6 0.5 0.4 0.3 0.2 0.1 0 0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0.1200 position Figure 1: Voltage vs Position 2. The formula for the voltage sine wave was determined from Figure 1: V .254 .254 sin( ) (1) 3. Since the voltage, V, was measured experimentally, the phase shift for each voltage, , was determined from (1). V .254 arcsin .254 (2) The error in the phase shift was: 1 V .254 . 254 2 V .254 .254 (3) 1- V .254 V V (.254) (.254) .254 .254 2 .254 .254 2 .254 .254 .254 (4) 4. The phase shifts were linearized and plotted with respect to phase shifter position. This is shown in Figure 2. Phase Shift vs Position Phase Shift 500 400 y = 144876x3 - 32832x2 + 5553.9x - 55.173 300 200 100 0 0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0.1200 Position Figure 2: Phase Shift vs Position 5. The best fit line to the Phase Shift vs Position curve was described by the equation shown on Figure 2, or: = 144876x3 - 32832x2 + 5553.9x - 55.173 (5) Where: = phase shift x = phase shifter dial position The error in the best fit value was: 434628x 2 65664 x 5553.9 (6) 6. (5) was used to determine the phase shift due to the change in phase shifter dial position. This value corresponded to the distance between the phase shift due to the material and the next minimum in the voltage curve. In order to isolate the phase shift due to the material the value was subtracted from 3600, which corresponded to the next minimum in the voltage curve. Observations Table 1: Thickness and Phase Shifter Position Data Sample Thickness (m +/- .000005) glass1 0.00228 glass2 0.00217 glass3 0.00444 glass4 0.00582 plates1 0.00571 plates2 0.01266 plates2/water 0.00011 bee's wax 0.00092 ice 0.01600 Trial 1 Trial 2 Trial 3 Trial 4 Avg (+/- .0005) 0.0777 0.0805 0.0562 0.0450 0.0730 0.0540 0.0162 0.0965 0.0578 (+/- .0005) 0.0460 0.0790 0.0571 0.0460 0.0690 0.0555 0.0155 0.0954 0.0577 (+/- .0005) 0.0795 0.0545 0.0540 0.0431 0.0727 0.0525 0.0128 0.0979 0.0580 (+/- .0005) (+/- .0005) 0.0786 0.0799 0.0558 0.0447 0.0716 0.0540 0.0146 0.0966 0.0578 0.0803 0.0555 0.0140 Analysis 1. The Dielectric constant was determined for each material using the following formula: c 1 2f d 2 (6) The error was calculated using the following formula: 2 2 c 2 2 d 2 f 2 2 f 2d 3 f 3d 2 2 f d 2 The results are shown in Table 2: c d f (7) fd 2 df 2 fd Table 2: Analysis Results Sample Thickness glass1 glass2 glass3 glass4 plates1 plates2 bee's wax ice (mm +/- .0005) 0.00228 0.00217 0.00444 0.00582 0.00571 0.01266 0.00920 0.01600 Phase Shift Error (degrees) 111 107 182 220 133 188 54 176 (degrees) 15 15 16 17 15 16 16 16 Dielec Const Error (relative) 11.5 11.7 9.09 8.11 4.57 2.99 1.66 2.37 Accep Value (relative) 1.3 1.4 0.30 0.075 0.14 0.12 0.18 0.10 (relative) 5-10 5-10 5-10 5-10 ? ? 2-3 75 Discussion The experimental values of the dielectric constants for the samples were compared to the expected values in Table 3. Most of the values were close to the accepted values. There was a large discrepancy for the measurement of ice, which can be explained by the difficulty of correctly measuring its thickness and the difficulty of getting a sheet of uniform thickness. The beeswax thickness was also difficult to measure because of its compressibility and this may have contributed to the error in its dielectric constant. The glass was assumed to all be of the same type and so all the samples should have had the same dielectric constant. They did not, probably because of adjustments to the attenuators and other parts of the equipment between measurements. There were several random errors associated with this experiment. The microwaves produced by the klystron depended on the beam voltage, which fluctuated. We minimized the effect of this error by making sure the voltage was zero before and after each reading. The minimum phase was not sharply defined, so we estimated the middle. The only systematic error that may have occurred was the possible frequency dependence of the attenuators. The most significant source of error in this experiment was the voltage dependence of the klystron beam. Conclusion The experimental values for the dielectric constants of the sample materials did not all agree within the calculated error range with the accepted values. Further research should be conducted on the possible sources of error with the equipment so that more reliable data could be collected.