Physics307L F07:People/Ozaksut/Final Report

Measuring the Speed of Light Using Analog Electronics
Anne Ozaksut

''aozaksut@unm.edu, Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico. September 2007.''



Abstract
The speed of light can be determined through interferometry or by using analog electronics. Interferometers are sensitive to vibrations and other environmental factors, so I wish to test the accuracy of the analog electronic method, as it is relatively unaffected by environment or vibration. The speed of light is calculated using measurements of relative changes in distance of light travelled and time between emission and detection of the light pulse. A Time-to-Amplitude (analog) Converter was used because changes in its output voltage (amplitude) can be detected more sensitively than the time between light emission and light detection should the LED and PMT be connected directly to a detector,such as an oscilloscope, to try to resolve the wavefronts. The speed of light was calculated to be $$c=\left(3.02 \pm .24\right)\times10^{8} m/s$$ using data gathered by a single channel analyzer. Compared to the accepted value of 299,792,458 m/s, the relative error is .63%. 

Acknowledgements
I thank Steven J. Koch for his insight in the lab and Matthew Depaula for his partnership in becoming familiar with new equipment and gathering data using the oscilloscope. Additionally, I thank Kyle Martin and Jesse Smith for their work in gathering a second set of data using the single channel analyzer. 

Introduction
Accurately measuring the speed of light propagation is important in substantiating theories about the nature of light, as it is a constant that appears in both astronomical and physical calculations. Attempts to measure the speed of light have been made for centuries by astronomers and physicists. Because the speed of light is very high, it would seem difficult to measure change in distance over change in time given human reaction time and the relatively short distances which could be controlled in an experiment on earth. The famous Michelson-Morley experiment of 1887, initially contrived to prove the existence of ether, was an early lab-confined experiment that used the wave theory of light propagation to produce observable interference effects by splitting a beam of light and offsetting part of it by a measurable amount. The speed of propagation could then be related to the wavelength of the incident beam. Interferometers were used to measure the speed of light into the 1970s, and in 1983, the value 299,792,458 m/s was established as the speed of light by the International Bureau of Weights and Measures. Because sensitive interferometers aren't common, it is important to show that the speed of light can be measured accurately using analog electronics. A Time-to-Amplitude Converter converted tiny changes in signal to a waveform which could be analyzed on a Single Channel Analyzer and converted back into information about the distance and time travelled by a photon originating at a Light Emitting Diode and detected at a Photomultiplier Tube to measure the speed of light to an accuracy of .63%.

Materials and Methods
For the experiment, a light source and a light detector were needed. The source was a UNM machine shop Light Emitting Diode (LED) which emitted green light pulses at a frequency proportional to the 198V of power supplied to it by the Harrison Laboratories # 6207A power supply. A cable was connected from the LED source directly to the input 1 on the EG&G Ortec model 567 Time-to-Amplitude Converter (TAC) to track the emission of each signal. The detector was a model N-134, unknown manufacturer, Photo Multiplier Tube (PMT) which converts volume of incident photons to volume of electrons in an electric current. The PMT was connected to a Canberra Nsec Delay 2058 delay box to ensure the PMT signal reached the TAC after the LED signal. This was a concern because there was a considerable amount of cable between the LED and TAC which would delay the start signal on the TAC. Adding a constant delay to the PMT signal would not affect the result, because only relative changes in data from the first data point are needed in order to calculate the speed of light. The signal was then connected from the delay box to the input 2 of the TAC using cables. The TAC converts the time between input signal 1 (start) and input signal 2 (stop) to a voltage output with a wave amplitude proportional to the time between input 1 and input 2, governed by conversion switches on the front of the TAC (10Vc multiplier over 50ns scaler outputs .2V/ns). Finally, the TAC was connected to the input on the EG&G Ortec model 567 Single Channel Analyzer (SCA) by cable in order to measure the TAC amplitude changes with changes in distance and take data as (x, y)=(voltage amplitude 1=0, distance=0), (voltage amplitude 2-voltage amplitude 1, distance 2-distance 1), (voltage amplitude 3-voltage amplitude 1, distance 3-distance 1), etc. An oscilloscope was used to gather data instead of an SCA for two trials before the process was modified. An advantage of using a single channel analyzer over an oscilloscope is that because the TAC output voltage fluctuates due to either the nature of the TAC mechanism or the fluctuating intensity of light received at the PMT, every output voltage could be recorded independently and precisely and an average of single voltage readings could be taken to get a better average voltage output than if an oscilloscope were used, because the oscilloscope only reads voltage in increments of .02V, and the only method for finding the average voltage output is by using the "average" function on the machine itself. The TAC conversion factor (TAC multiplier) was used to convert measured changes in voltage to changes in time, and then the distance data points were plotted against the corresponding new time data points. Data analysis was done using the Microsoft Excel program. To determine the speed of light, our calculated data points were plotted as (time, distance) and a linear fit curve, calculated by the method of least squares, was superimposed on the data set. The reported speed of light is the slope of that liner fit curve.

Since the PMT is very sensitive to changes in light, the emitter-detector system was completely enclosed in a long cardboard tube to try to eliminate ambient light in the lab. Because light is an EM wave, and we are analyzing the speed of light using electronics, the delay box or delay cables on the detector side were necessary in order to maintain a positive difference in time between emission and detection for all distances between LED and PMT. Because exposing the PMT to bright light can damage the instrument, we had some uncertainty in where exactly the detector was positioned in the length of the tube. Additionally, we had uncertainty in where exactly the light source was located within the LED apparatus. However, these factors did not affect the speed we measured because changes in time (as a voltage output on the oscilloscope or SCA) and changes in distance (on the meter stick) can be accurately measured from our first data point. Because the TAC triggers at a constant voltage, we attempted to reduce the effect of timewalk on the experiment by adjusting the polarization of the LED-PMT system to maintain a constant intensity of light at the PMT, independent of the distance between TAC and PMT. A cable was connected from the PMT to the Tektronix TDS 1002 two channel digital storage oscilloscope, 60MHz 1GS/s, the orientation of the PMT relative to the LED was adjusted until the PMT current output read the same value each time before taking each TAC reading.

Results


First data sets using the oscilloscope as the voltmeter for the TAC output. The speed of light was later more accurately determined when using the single channel analyzer as the voltmeter for the TAC output. This data is displayed for comparison.



A plot of distance over time using the first set of data gathered using the oscilloscope.



A plot of distance over time using the second set of data gathered using the oscilloscope.



Single channel analyzer raw data



A plot of distance over time using the data gathered using the single channel analyzer.

Discussion
Trial 1 $$c=\left(3.54 \pm 1.82\right)\times10^{8} m/s$$

Trial 2 $$c=\left(2.64 \pm 0.78\right)\times10^{8} m/s$$

Trial 3 using the single channel analyzer $$c=\left(3.02 \pm .24\right)\times10^{8} m/s$$

Using the method described above, the speed of light was measured to be $$c=\left(3.54 \pm 1.82\right)\times10^{8} m/s$$ for trial 1, $$c=\left(2.64  \pm 0.78\right)\times10^{8} m/s$$ for trial 2, and $$c=\left(3.02  \pm .24\right)\times10^{8} m/s$$ for trial 3. The weighted average of the two oscilloscope trials is $$c=\left(2.78 \pm .717\right)\times10^{8} m/s$$. The advantage of using the analog electronics method to measure the speed of light is that the measurement doesn't rely on the speed of signal detection in an oscilloscope or other device for light travel of 5 meters at most. Instead, the TAC follows each signal from the LED to the PMT and outputs a voltage proportionate to the time between signals at the LED and PMT which the oscilloscope or SCA is more sensitive to. A disadvantage encountered during the first two trials using the oscilloscope is that the oscilloscope's voltage detection is limited to increments of .02V. For the TAC multiplier settings chosen for the first two trials, a change in distance of 10 centimeters corresponded to a delay of 3E-10 seconds in the TAC, which corresponded to a change in output voltage of only .0067 V. In future trials using the oscilloscope, more accurate measurements of voltage could be made by using a TAC multiplier of greater than 1V/50ns. A single channel analyzer was used to gather data for a third trial because its precise voltage detection is limited to increments of about 5E-6V [[Media:sca spreadsheet.xlsx]]. Comparing the result of trial three to the accepted value for the speed of light, our relative error is only .63%. 

Conclusion
Using analog electronics, the speed of light was measured to be $$c=\left(3.02 \pm .24\right)\times10^{8} m/s$$. Compared to the accepted value of the speed of light, the measured value was accurate to 99.36%, and precise to 92.1%. The most significant error probably comes from the variance in voltage output in the TAC, because our measurement was more accurate than precise. We attempted to reduce the variance in time of our data points by collecting voltage for long periods of time (500,000+ readings), but the variance could be further reduced by collecting data for longer periods of time at each distance point (1,000,000+ readings, perhaps).

Literature Cited
1. http://www.patentstorm.us/patents/5589938-description.html

2. http://www.is.wayne.edu/mnissani/a&s/light.htm

3. http://scienceworld.wolfram.com/physics/Michelson-MorleyExperiment.html

4. http://www.nature.com.libproxy.unm.edu/nature/journal/v239/n5367/pdf/239065a0.pdf

5. http://www.bipm.org/en/home/

6. http://en.wikipedia.org/wiki/Least_squares

7. http://www-hep.phys.unm.edu/~gold/phys307L/manual.pdf