Physics307L:People/Mahony/Formal

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

Light has been a topic of study among scientists since an experiment by Galileo established a lower bound on the speed of light of approximately 60 miles per second during the seventeenth century.[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]. In 1887, Michelson and Morley's famous interferometry experiment refuted the theory of the aether in which light must propagate. This experiment also measured the speed of light to be [3] Equally controversial was Einstein's theory of special relativity, published 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.[4] This theory became widely accepted, and throughout the rest of the 20th century, many experiments were done to more accurately measure this speed.[5] 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.[6] In our experiment, we set out to measure this speed to see if our experimental data matched the accepted value.

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.

We measured the TAC's voltage using a Tektronix TDS 1002 Oscilloscope. The entire experimental setup can be seen in figure 2, with the exception of the polarizers, due to their placement inside the cardboard tube. This voltage corresponded to the time between the LED trigger pulse and the PMT response pulse with the LED at different positions, all 10 cm apart. As the LED got closer to the PMT, the intensity would increase, and this would cause error due to "time walk." The oscilloscope displays a signal by triggering when the signal reaches some threshold. The "time walk" effect 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 causes the oscilloscope and the TAC to trigger at a different time, and the TAC will produce a different voltage. To minimize the error due to time walk (see Figure 1) we used a set of polarizers placed on the PMT and the LED to keep the intensity of the LED pulses constant. We measured the intensity of the LED when it was at its maximum distance from the PMT, and then we rotated the PMT with the polarizer attached so that the intensity of the LED signal remained constant for every measurement with the LED in a closer position.

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

Results and Discussion

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. 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.

I used the chi-square minimization technique to fit the data with a line. The slope of the line and standard error were used in a weighted average to compute the measured speed of light. This value was:

[math]\displaystyle{ 2.941(15)\cdot 10^{8} m/s }[/math]

The exact speed of light is approximately:

[math]\displaystyle{ 2.998\cdot 10^{8} m/s }[/math]

The calculated speed of light was 4 sigma away. Assuming only normally distributed random error, the probability of measuring the same value we did is 0.006%.

Figure 3: Trials 1-6, the accepted speed of light, and the calculated value from the measurements, along with their corresponding uncertainties, are shown.

The supplementary data and analysis can be seen here.

Conclusions

The probability of measuring the same value we did is 0.006%. Assuming only normally distributed random error, the likelihood of this happening 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.

Acknowledgements

Thanks my lab partner Ryan for his help with running the lab, taking data, and finishing up the lab notebook with me. I'd also like to thank Dr. Koch for his helpful explanations of various parts of the setup.

Thanks to A. Barron, who I referred to for help in formatting citations as well as getting a general idea of what I needed to write.[7]

References

1. Boyer, Carl B. "Early Estimates of the Velocity of Light." Isis Vol. 33, No. 1 (Mar., 1941), pp. 24-40 link

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. Retrieved 2009-12-05.

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

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

5. 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

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

7. A. Barron's Final Report

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