Measuring the Speed of Light via Time of Flight
Author: Ryan Long
Experimentalists: Ryan Long & Tom Mahony
The University of New Mexico
Department of Physics & Astronomy
The speed of light is a very large value, nevertheless, the speed of light can be measured using relatively simple time of flight methods. Experiments of this sort have been carried out since Isaac Beeckman and Galileo Galilei first tried in the early 1600s.  In Junior Lab at University of New Mexico, we measure the speed of light by measuring flight time of LED pulses over the course of a short distance. A major obstacle to overcome in this experiment is the occurence of "time walk", this can cause major systematic error, if not addressed properly. We obtain a value of 29.448 ± .1424 cm/ns, which is inconsistent with the accepted value of 29.979 cm/ns , indicating some source of systematic error. We discuss possibilities for removing this systematic error in future work.
Perhaps one of the most well known and frequently used constants of physics both classical and modern, is the speed of light, denoted by a lower case “c”. The speed of light constant is used for many purposes from calculating the distance to astronomical events, to understanding quantum mechanics. Today the value for the speed of light is defined as 29.9792 cm/ns, this exact value comes from the National Institute of Standards and Technology. Measuring the speed of light can be achieved numerous ways with modern technology, some include radio interferometry, or methane stabilized lasers . We measured the speed of light using a relatively simple method which involves measuring time delay of an LED pulse using a photomultiplier tube and a Time amplitude converter or simply (TAC). The photomultiplier is a device sensitive enough to measure individual photons. When the cathode of the PMT receives incident photons, photoelectrons are ejected from an anode inside the PMT,  the resultant charge pulse intervals from the photoelectrons are then converted into amplitudes by the TAC and displayed on an oscilloscope. The voltage amplitudes can then be converted to time and divided by the distance to obtain the speed of light. This simple, yet effective experiment yielded decent results for my partner, Tom and I.
In order to measure the speed of light, we set up a long, opaque tube made of cardboard with a pulsating LED light source in one end and the photomultiplier tube on the other end of the tube. We connected the photomultiplier tube with BNC cables first to a delay module (Canberra Industries, Inc. Meriden, CT). We then connected the delay module to a Tektronix TDS 1002 digital oscilloscope in parallel with the “Stop” input on the Ortec 567 TAC/SCA Module. The Light source is also connected to the TAC in the “start” input. The delay module was set to 9 ns, to make certain that the stop signal from the PMT arrived after the start signal from the LED. This procedure is necessary due to a substantially longer BNC cable connecting the LED and the TAC. See Figure 1 for an illustration of our setup. Note: the LED was not labeled, we were unsure of the origin of the LED, it is pictured in figure 3.
The light source is mounted on a meter stick so that we can measure various distances of light travel time, which ideally would lower our uncertainty. However this introduces a possible source of systematic error. As the LED is moved closer to the PMT, the PMT amplitude rises due to heightened intensity of photon bombardment. This issue is known as “time walk”. In order to reduce this time walk or fluctuation in voltage, a polarizer is mounted to the front of the PMT. As the LED is pushed down the tube toward the PMT, we rotate the PMT to keep the intensity as continuous as possible. Figure 2 shows the differences in amplitude as a result of "time walk".
We used six trials of recording data for our experiment. Each trial, we moved the LED closer to the PMT in 10 cm increments and recorded the amplitude for each increment. The first trial was only measured with nine increments of 10 cm because we started at a different mark on the meter stick. For trials 1-5, one experimenter (TM) positioned the PMT while the other experimenter (RL) rotated the PMT in order to maintain maximum continuous voltage, and also recorded values from the oscilloscope. For the last trial we switched roles to test for any systematic error in our procedural roles, however there was no significant change in data collection. Our raw data can be found in the google docs spreadsheet below.
Results and Analysis
For my analysis, I performed a linear regression analysis of my data using the "LINEST" function of Excel (Microsoft Office 2008). To compute my final value for the speed of light, I computed a weighted average and corresponding weighted uncertainty. My excel sheet can be downloaded here.
My calculated value is:
The accepted value from NIST is:
Figure 4 shows each measurement from trials one through six, my calculated weighted mean, and the accepted value. The error bars on the figure also demonstrate that my calculated value is inconsistent with the exact value from NIST. I speculate that this inconsistency is attributed to some systematic error in our experiment. Figure 5 shows raw data for each trial, our measurements for trials 2-6 were very consistent.
Our data collection was very consistent, in the sense that we did not have wild fluctuations in measurement but there was obvious systematic error in our experiment. Time walk was most likely responsible for a large portion of it, perhaps the photomultiplier could be modified so that it is stationary with only a rotating polarizer, instead of rotating the entire PMT. Another possible source of error is from the positioning of the LED with respect to the PMT, it is nearly impossible to make perfect increments of ten centimeters. This error would probably be reduced by simply taking more measurements, or also having a more stationary PMT as mentioned previously. Lastly measuring the voltages on the oscilloscope is not a very precise process, so in our final day of collecting data, we attempted to incorporate DAQ hardware to measure our voltages, but due to time constraints and a lack of a proper DAQ card, we did not have any success with this venture.
My special thanks to my partner Tom Mahony for his essential contributions to data acquisition, analysis, and overall exceptional work ethic. I also would like to thank Dr. Koch, and Pranav Rathi, for their assistance in setting up the experiment.
 Boyer, CB (1941). "Early Estimates of the Velocity of Light". Isis 33 (1): 24. doi:10.1086/358523. (from wikipedia)
 K.M. Evenson et al “Speed of Light from Direct Frequency and Wavelength Measurements of the Methane-Stabilized Laser” Physical Review Letters, Vol. 29, no. 19, (1972)
 R. Foord, R. Jones, C.J. Oliver, E.R. Pike “The Use of Photomultiplier Tubes for Photon Counting” Applied Optics, Vol. 8, No. 10, pg. 1 (1969)
 National Institute of Standards and Technology. (Constants, Units and Uncertainty) http://www.nist.gov/index.html