# Measurement of the Speed of Light using a Time of Flight Approach

## Contact Information

Author:Tom Mahony

Experiment conducted by Tom Mahony and Ryan Long

University of New Mexico Physics Department

800 Yale Blvd NE Albuquerque, New Mexico 87131-0001

## Abstract

The speed of light is an important fundamental constant in physics. The invariance of the speed of light is the foundation of special relativity, but the speed of light has importance in many other fields, including electromagnetism and cosmology. Several methods have been implemented to precisely measure the speed of light including time of flight measurements, interferometry, and cavity resonance techniques. In this experiment we measured the speed of light using a time of flight approach. We used a Time-Amplitude Converter (TAC) to measure the delay between a pulsed light emitting diode (LED) and a photomultiplier tube (PMT). We then positioned the LED at different distances from the PMT to measure different delays. We then converted the voltages that corresponded to the different delays to units of time. Fitting this data to a line using linear regression, we calculated the the slope of the line, which gave us our measured speed of light of 2.941(15)*10^8 m/s.

## Introduction

Though light had long been thought to travel instantaneously, an experiment by Galileo during the seventeenth century that established a lower bound on the speed of light of approximately 60 miles per second set off a series of experiments investigating the nature of light.[1]. One of the earliest published results of the measurement of the speed of light was done by Albert Michelson in 1879, during his time spent at the Naval Academy. Michelson reported a speed of light of 299944±51 km/s.[2]. During the 1800's scientists debated whether light must propagate through an aether medium, in much the same way sound waves require a medium to travel. In 1887, Michelson and Morley's famous interferometry experiment refuted this aether theory, but showing that light traveled at the same speed regardless of its alignment on Earth.[3] Though this result was initially met with skepticism, the experiment was repeated many more times throughout the early 20th century (In fact modern adaptations of the Michelson-Morley experiment have been carried out using much more advance techniques[4]). Not long after, Einstein published his theory of special relativity in 1905, which proposed that the speed at which light propagated was a fundamental constant, invariant of the speed of the reference frame in which it was observed.[5] This theory became widely accepted, and throughout the rest of the 20th century, many experiments were done to more accurately measure this speed. Different techniques were used, including cavity resonators[6] and heterodyne laser measurements[7] which reported values for the speed of light of 299,792.5±3 km/sec and 299,792,459.0±0.8 m/s respectively. In 1983 the meter was redefined by the CGPM as the distance traveled by light in 1/299,792,458 seconds, giving the speed of light the exact value of 299,792,458 meters/second.[8] In our experiment, we set out to measure this speed using a time of flight measurement. We compared it against the exact value to determine if this method was satisfactory.

## Methods

Figure 1: Time Walk Effect- As the amplitude of a signal varies, the shape of the pulse changes. This change in the pulse shape causes the threshold for a trigger to be shifted in time. This shift, or delay, causes the instrument to trigger at a different time and introduces systematic error in the experiment. We attempted to minimize the time walk effect by using polarizers to maintain a constant intensity from the LED.

We positioned a photomultiplier tube (PMT) powered by a Bertran 313B Power Supply on one end of a carboard tube. We placed a LED in the other end, powered by a Harrison Laboratories 6207A PSU. We measured the time difference between the LED's pulse and the photomultiplier's response with a Ortec 567 TAC/SCA Module plugged into a Harshaw NQ-75 NIM Bin. We placed a Canberra 2058 Delay Module between the PMT and the TAC to guarantee the response pulse would be received by the TAC after the triggering pulse from the LED. The entire experimental setup can be seen in figure 2, with the exception of the polarizers, due to their placement inside the cardboard tube.

We measured the TAC's voltage using a Tektronix TDS 1002 Oscilloscope. This voltage corresponded to the time difference between the LED trigger pulse and the PMT response pulse. The LED was initially placed at some fixed distance from the PMT, and then was positioned closer in 10cm steps. As the LED moved closer to the PMT, the intensity of the pulse increased, and this would have caused error due to "time walk." We minimized this effect by taking making note of the intensity of the LED at its farthest position and using the polarizers mounted inside the carboard tube to reduce the intensity of the latter measurements so that they would match the first. The time walk effect can cause significant systematic error if it is not addressed. It arises from the way an instrument such as an oscilloscope displays a signal. The oscilloscope displays a signal by triggering at some given threshold of the falling or rising edge of the input signal. The "time walk" effect (see figure 1) is the change in time of this trigger signal due to a change in amplitude of the input signal. In this experiment, a change in intensity of the LED signal, seen by the amplitude of the PMT signal, causes the oscilloscope and the TAC to trigger at a different time, and the TAC will produce a different voltage.

The experiment consisted of six trials, with several differences. The first trial was done using 10 different positions for the LED. The last 5 trials used 11 positions for the LED, and the TAC signal was averaged on the oscilloscope using its averaging function in an effort to reduce the noise of the signal. Trials 1-5 were conducted with Tom moving the LED circuit, and Ryan reading the oscilloscope to take the data. Trial 6 had their roles reversed to see if it would affect the data in a substantial way.

Figure 2: Panorama of the experimental setup. Every instrument is labeled. The polarizers cannot be seen, due to their placement inside the cardboard tube.

After taking measurements, the voltages were converted to time values using the appropriate conversion ration of 1V to 10 ns. Then, for each trial, these values were combined with their corresponding positions in a linear regression to find a line of best fit. The slope of this line was the speed of light for the trial. These values were then combined in a weighted average using their uncertainties as the weighting factors to yield the final measured speed of light for the lab. The calculations were done using Microsoft Excel 2007 with the exception of the linest function, which was done using google docs spreadsheet built in functions (since Excel returned errors when working with that many decimal places).

## Results

In the first trial, we measured the voltage of the TAC with the LED in 10 different positions. For every subsequent trial, we measured the voltage with the LED at 11 positions. After the first trial, we used the averaging function on the oscilloscope. This function took a time average of a signal, which reduced the noise, so we could better measure its voltage. Since the TAC was set to produce a 10V signal for a 100 ns delay, we used this ratio of 1V/10ns to convert our measured voltages into times. See figures 3-8 for the linear regression of each trial.

I used the chi-square minimization technique to fit the data for each trial with a line of best fit. The slope of these lines and standard errors were used in a weighted average to compute the final measured speed of light (see figure 9). This value was:

$2.941(15)\cdot 10^{8} m/s$

The exact speed of light is approximately:

$2.998\cdot 10^{8} m/s$

The calculated speed of light was 4 sigma away. Assuming only normally distributed random error, the probability of measuring a value 4 sigma away from the mean is 0.006%. Therefore, it is highly likely that some systematic error was present in this experiment.

Figure 9: The blue diamonds represent the values of the speed of light for each trial along with one standard error of the mean vertical error bars. The final weighted average speed of light is shown by the green triangle. The exact value of the speed of light is represented by the red square.

The supplementary data and analysis is readily available.[9]

## Conclusions

The probability of measuring a value 4 sigma from the means is 0.006%. Because our value was 4 sigma from the accepted value, assuming only normally distributed random error, the likelihood of measuring this value again is quite low. I conclude that the experimental data deviated from the accepted value due to systematic error. I believe the cause of this error was inadequate minimization of the time walk effect caused by the reliance on human judgment in determining when the intensity of the LED pulse signal matched the original signal. This error might be reduced by the use of a computer to measure the LED signal, rather than using the screen of an oscilloscope. This method is far more quantitative, and I believe it would yield more accurate results, however, it would require a data acquisition card with a much faster sample rate than the card already present in Junior lab.

## Acknowledgements

• I thank Ryan for his help with the electronic lab notebook, data acquisition, and data analysis. I'd also like to thank Dr. Koch for his helpful explanations of various parts of the setup.
• I thank A. Barron for his open access lab notebook, which provided me with insight on reference formatting using biblio.[10]
• I thank Dr. Koch for his guidance in the initial setup of the experiment.

## References

1. Boyer, Carl B. "Early Estimates of the Velocity of Light." Isis Vol. 33, No. 1 (Mar., 1941), pp. 24-40 http://www.jstor.org/stable/330649 [Gal]
2. Michelson, Albert A. "Experimental Determination of the Velocity of Light: Made at the U.S. Naval Academy, Annapolis." 20 February 1880. Washington: Nautical Almanac Office. http://www.gutenberg.org/files/11753/11753-h/11753-h.htm.

[Mich]

3. Michelson, Albert Abraham and Morley, Edward Williams (1887), "On the Relative Motion of the Earth and the Luminiferous Ether", American Journal of Science 34: 333–345

[Michelson]

4. Muller, Holger and Herrmann, Sven and Braxmaier, Claus and Schiller, Stephan and Peters, Achim. "Modern Michelson-Morley Experiment using Cryogenic Optical Resonators." Physical Review Letters Vol 21 Num 2 Pg. 020401-1 - 020401-4. http://prola.aps.org.libproxy.unm.edu/pdf/PRL/v91/i2/e020401.

[Modmich]

5. Walker, John and Einstein, Albert (1905) "On the Electrodynamics of Moving Bodies", Annalen der Physik 17: 89. (English translation of original article.)

[SR]

6. Essen, L. "The Velocity of Propagation of Electromagnetic Waves Derived from the Resonant Frequencies of a Cylindrical Cavity Resonator." Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. Vol. 204. No. 1077 (Dec. 7, 1950). pp. 260-277. http://www.jstor.org/stable/98433.

[Essen]

7. Measurement of the speed of light. T. G. Blaney C. C. Bradley G. J. Edwards B. W. Jolliffe D. J. E. Knight W. R. C. Rowley K. C. Shotton & P. T. Woods Nature 251, 46 (1974) | doi:10.1038/251046a0. http://www.nature.com/nature/journal/v251/n5470/pdf/251046a0.pdf

[Blaney]

8. Base unit definitions: Meter. Nov 15 2009. http://physics.nist.gov/cuu/Units/meter.html

[NIST]

9. Supplementary data

[Data]

10. A. Barron's Final Report

[Barron]