WJP X, XXXX.XX Wabash (20XX) Journal of Physics 1 A Method of Directly Measuring the Speed of Light Using Different Optical Path Lengths Thomas F. Pizarek, Adam L. Fritsch, and Samuel R. Krutz Department of Physics, Wabash College, Crawfordsville, IN 47933 (Dated: October 3, 2008) In this experiment, the speed of light was measured directly by observing the difference in time that it took for laser light pulses to travel optical paths of different lengths. An optical chopper was used to generate light pulses; different paths were created with a glass slide acting as a beam splitter and a square loop of mirrors. This method yielded a result of c = (1.008 ± 0.017) 10 8 m/s. Physicists have been making measurements of the speed of light since Galileo first proposed an experiment in the early 1600s. In 1879, Albert Abraham Michelson measured the speed of light to an unprecedented degree of accuracy using a method somewhat similar to the one used in this experiment.[1] The currently accepted value of the speed of light is c = 299, 792, 458 m/s. Accurate knowledge of the speed of light is still important today as telecommunications technology rapidly advances. The use of fiber optic cables relies closely upon knowledge of the speed of light in order to determine the rate at which information can be sent. This is also important in global positioning. Satellite receivers used for this purpose triangulate the coordinates of radio signals; this could not be very accurate, and therefore not useful, without knowing well the speed at which these signals travel, c. The mathematical basis of this experiment is simple: the definition of velocity. By creating a pulse of light, splitting it in two, sending its parts down paths of different known lengths, and measuring the relative time delay in the travel of the two parts, the speed of light can be easily deduced. In order to do such an experiment, we used a helium-neon (HeNe) laser, an optical chopper, a glass slide, a number of lenses to focus and collimate the beam, a variety of mirrors, a pair of photodiodes, and an oscilloscope attached to a laptop computer. (Fig. 1) After being emitted from the laser apparatus, the beam is focused with a convex lens of short focal length. The optical chopper - a slotted disc that spins at a variable frequency - is placed at the focus of the beam, where the waist is narrowest, so that it can cut the beam as quickly as possible and thereby give us a fine time resolution. Another convex lens is placed a distance of its focal length from the chopper in order to collimate the beam. The placement of this lens is particularly important, since it greatly affects how much the beam spreads as it travels. In order to make fine adjustments to its position, it was mounted on a translation stage. To split the beam into two, we pass it through a glass slide such that a low-intensity beam is reflected off of the slide and onto a photodiode, while a more intense beam passes through the slide. The transmitted beam is sent around a loop of four mirrors before being focused onto the other
WJP X, XXXX.XX Wabash (20XX) Journal of Physics 2 photodiode. Diode 1 Laser Chopper Ch 1 Ch 2 Scope Laptop Diode 2 FIG. 1: This diagram illustrates the concept of our experimental setup. We send a laser beam through a focusing lens so that the beam can be cut with an optical chopper more precisely. Once past the chopper, the beam is collimated with a second lens. Then, the laser is split into two beams using a glass slide. One beam is directed straight to a photodiode (Diode 1), while the other is sent around a set of mirrors and eventually reaches another photodiode (Diode 2). These photodiodes are both connected to an oscilloscope where their signals can be visualized. By triggering off of one of the diodes, a timing difference can be seen. A software program on a laptop connected to the oscilloscope can then take a snapshot of the two relative signals. By analyzing these data with an error function fit (erf fit), a precise difference in time of the two paths can be determined. This relative time difference, along with the measured difference in path length of the two beams, is used to calculate the value of c. To take data, we varied the path length difference by making the beam travel around the mirror loop different numbers of times. We configured the mirrors in two different ways and were able to obtain path length differences of 16.4 m, 18.6 m, 37.2 m, 55.8 m. For each path length difference, we took four relative time delay measurements by switching the channels on the oscilloscope corresponding to the short and long paths without changing the triggering settings on the oscilloscope and also by physically switching the diodes from short to long paths and vice versa. For all of these measurements, the chopper frequency was set to 3 khz. In order to obtain time values from the data obtained by the oscilloscope, we fit an error function (erf) to the voltage curve. This approach is valid because the laser beam has a gaussian profile and the error function corresponds to the integral of a gaussian curve. After our first few runs, we saw that the erf fits did not agree well with the data because we had not captured enough of the information to the right of the rise in voltage to be sure that the curve had leveled off completely. We repeated these runs more carefully and were able to get much better fits. (Fig. 2) It may be possible to improve this even further by adjusting the time scale on the oscilloscope to cover a larger data range. After taking data for all of our path lengths, diode configurations, and wiring configurations, we used erf fits to find time values for the short and long paths of each run and subtracted these values to obtain the relative time
WJP X, XXXX.XX Wabash (20XX) Journal of Physics 3 Before Voltage Time (s) After Voltage Time (s) FIG. 2: Here is a comparison of the error function (erf) fits of two runs along the same path length. In the first run (labeled Before ), insufficient data was collected to definitively see where the top of the curve became level. This resulted in an erf fit that did not match the data well. In the second run (labeled After ), better attention was paid to the image on the oscilloscope to be sure that the curve had a slope near zero after rising. The erf fit matches this data much better.
WJP X, XXXX.XX Wabash (20XX) Journal of Physics 4 delays. We averaged the values from the different diode configurations from each path length and then plotted these points. (Fig. 3) Using the uncertainties in time given by the erf fits, we applied error bars to the time values in our chart. Because we have time delay as a function of distance, we were able to arrive at our value of c by fitting a line to the data and then inverting the slope of the fit line. To calculate the uncertainty in c, we used the following procedure c = A 1 (1) ( δc = c 1 δa ) 2 (2) A = c 2 (δa) (3) where A is the slope of the fit line. This procedure yielded a value of c = (1.008 ± 0.017) 10 8 m/s. FIG. 3: This is a plot of our relative time delay data as a function of path length difference. We used a linear fit and then inverted the slope of the line to obtain our value of c. Our value obviously does not agree with the accepted value of c = 299, 792, 458 m/s. Confronted with this, we decided to take further data varying the chopper frequency between 1 khz and 4 khz. (Fig. 4) We expected the data to generally improve as we increased the frequency, because the higher velocity of the chopper wheel will yield a more clearly defined pulse signal, which would give better time resolution. Accordingly, the calculated value of c improved as we increased the frequency from 1 khz to 3 khz. However, when we increased the frequency further, from 3 khz to 4 khz, the results got worse. This may be due to instability in the chopper at frequencies near the maximum of its range.
WJP X, XXXX.XX Wabash (20XX) Journal of Physics 5 1.2 10 8 C Value (m/s) 1 10 8 T* C Value (m/s) 8 10 7 6 10 7 4 10 7 T** 2 10 7 0 0.5 1 1.5 2 2.5 3 3.5 4 Frequency (khz) FIG. 4: This plot shows how our c value changed as we varied the optical chopper s frequency for two different path length differences. The upper data set (with trend line T *) corresponds to a 55.8 m path length difference and the lower data set (with trend line T **) corresponds to a path length difference of 18.6 m. The c value was best at a chopper frequency of 3 khz. The value c = (1.008 ± 0.017) 10 8 m/s obtained by this experiment affirms that the speed of light is finite and can be directly measured in an undergraduate laboratory setting. The accuracy of the measurement could possibly be improved by using an optical chopper with a higher frequency capability. It may be even better to use a different method to create the light pulses; an acoustic modulator, for example. Also, modifying the time scale on the oscilloscope to gather more data could help to create better erf fits. The method used in this experiment has potential to be improved in the future and the scope of the experiment has important historical and practical aspects. [1] Michael Fowler. The Speed of Light. http://galileoandeinstein.physics.virginia.edu/lectures/spedlite.html