Measuring the Speed of Light Using a HeNe Laser

Liam Thompson

University of Minnesota

Methods of Experimental Physics II

Fall 2019

Abstract:

The speed of light in air was measured using the beat frequency of two longitudinal modes of a HeNe laser. The beat frequencies were measured over a range of different laser cavity lengths. This experiment produced a value of the speed of light of c = (2.99777±0.00006(stat)±0.00013(syst))×10^8 m/s. The combined uncertainty of 14 km/s is 1.3 standard deviations less than the actual value of c = 299,792,458 m/s. The statistical uncertainty of 6 km/s achieved in this experiment is roughly 5 times more precise than previous MXPII experiments.

Introduction:

The speed of light is arguably the most important physical constant. In 1972, it was measured with a precision of 1 m/s using a laser interferometer [1]. This was used to define the speed of light in a vacuum to be a constant of c = 299,792,458 m/s. This experiment will try to measure the speed of light using a HeNe laser to as high a precision as possible.

Theory:

A HeNe laser contains a hot plasma of Helium and Neon atoms. These atoms are moving rapidly, which means that there will be a Doppler shift in the light that they emit. This produces a Gaussian distribution of emitted energies, which is the blue line in the figure below. A laser is a resonant cavity, which means that only wavelengths that are half integer multiples of the cavity length will experience constructive interference. Frequencies that do not experience constructive interference will not have a high enough gain to lase. This imposes the condition that the frequencies will by c/2nL, where n is the refractive index, and L is the cavity length. Two frequencies that are separated will produce a beat frequency in the intensity of the laser. This is roughly 370 MHz, which is slow enough that it can be measured with a photodetector.

Fig.1 Shows Doppler broadening and the longitudinal modes of a HeNe laser [2]

Experimental Setup:

Fig.2 Experimental Apparatus

Fig.2 is the experimental apparatus used for this measurement. The output coupler is a partially transparent mirror, and is the end of the laser cavity. The beam is split and one component is sent into a photodetector, while the other component is sent into the scanning Fabry-Perot interferometer. The light that is sent into the interferometer passes through a quarter wave-plate, and then a magnifying lens. The quarter wave-plate significantly reduces retro-reflection, while the magnifying lens is essential to align the Fabry-Perot.

The speed of light is calculated by measuring the beat frequency over a range of cavity lengths. The speed of light can then be calculated from the slope of the fit between 1/f and ΔL using the following equation [3].

The interferometer is connected to a digital oscilloscope in Labview. The scanning Fabry-Perot interferometer allows you to visualize the mode structure of the laser light. This is used to minimize the effects of frequency pulling, which is a shift in the frequency from the location of longitudinal modes on the gain curve in Fig.1. The output of the interferometer can be seen in Fig.3. A program constantly checks the percent difference in amplitudes between the two peaks and only measures the beat frequency when the peaks have been within 15% for 0.5 seconds. This significantly reduces the uncertainty in the beat frequency from a non-linear effect called frequency pulling.

Fig.3 Output of the scanning Fabry-Perot interferometer

The photodetector is connected to an RF spectrum analyzer which transforms the data into frequency-space. This is then accessed by the same Labview program. The location and intensity of the beat frequency is recorded.

The measurement was performed in step size of 1 mm over a total distance of 14 mm, with about 50 data points at each position.

Frequency Pushing and Pulling:

A fit was found for the equation relating 1/f and ΔL. Fig.4 is a plot of the residuals of that fit versus the intensity measured at the photodetector. This does not appear linear, but it was linear when the measurement was performed over a much smaller distance.

Fig.4 Residuals of fit between 1/f and ΔL

The reason that Fig.4 is not linear can be clearly seen in Fig.5. Fig.5 shows that there is a linear relationship between the slope of frequency pushing versus the length of the laser cavity. Using the slopes and intercepts of each fit, the effects of frequency pushing were corrected. This correction allowed for a significant increase in the precision of the frequency measurement.

Frequency pulling was measured as well, by taking measurements at the same ΔL, but with varying percent peak differences. This can be seen in Fig.6. The slope of this fit was 57 Hz/dB. From this slope, the allowed percent peak difference to take a measurement was chosen to be 15%. A smaller percent peak difference reduces the uncertainty in frequency, but is more difficult to achieve a sufficiently stable laser. It is also important to choose settings on the RF spectrum analyzer so that the resolution of the measurement of the beat frequency is small compared to the standard deviation. This requires increasing the time constant of the measurement and limiting the range of frequencies considered. A range of 500 KHz and a time constant of 0.5 seconds were used which resulted in a resolution of roughly 800 Hz.

Results

Fig.6 Frequency pulling in a HeNe laser measured at ΔL = 0.

Using the corrections for frequency pushing, and minimizing the uncertainties due to frequency pulling, the average uncertainty in the measurement of the frequency was reduced to 1400 Hz. This was calculated by taking the standard deviation of the data at each measurement location.

Fig.7 A linear fit between 1/f and ΔL. The speed of light was calculated from the slope of this graph.

Fig.7 is a graph of the data used to find the speed of light. This resulted in a value of c = (2.99777±0.00006(stat)±0.00013(syst))×10^8 m/s. Using the combined uncertainty of 14 km/s, the speed of light is 1.3 standard deviations smaller than the actual value. The statistical uncertainty is precise to 1 part in 50,000, which is a significant improvement over previous MXPII projects [4]. This is sufficiently precise that it can easily distinguish between the speed of light in a vacuum, and the speed of light in air, as this result was 8 standard deviations less than the calculation where the refractive index was set to 1. This was accomplished by measuring over a significantly longer distance, and through the methods in which the non-linear effects were accounted for. The reduced chi-squared value for this data is 0.96, which is close to the desired value of 1.

The systematic uncertainties in this experiment are larger than the statistical by roughly a factor of two. The two dominant uncertainties are both due to the thermal expansion of steel of 1.1×10^5 /K. The length of the translation stage will change with the temperature. This will directly affect the measurement of ΔL. The length before ΔL = 0 was assumed to be constant. This is not the case, if the temperature of the room changes during the experiment. This systematic uncertainty in dependent on the total length of the laser divided by the total distance measured over. Because of this, it is extremely important to measure over as long a distance as possible to reduce this systematic uncertainty.

This experiment can be further improved in 2 ways. The first is to measure over a longer distance, which will reduce both the statistical and some of the systematic uncertainties. This can at most increase the total precision by roughly a factor of 2 as then the precision becomes comparable to the uncertainty in the calibration of the translation stage. To improve the results beyond this point would require building a quadrature interferometer, or some other independent device to measure the location of the output coupler. Combining these two methods could potentially allow for a total precision of roughly 1 in 10^5, or 3 km/s.

References:

[1] K. M. Evenson et al., Phys. Rev. Lett.,29, 1346,(1972)

[2] Daniel J. D’Orazio, Mark J. Pearson, Justin T.Schultz, Daniel Sidor, Michael W. Best et al., :Am. J. Phys.78, 524 (2010)

[3] B.E.A Saleh, M.C Teich, Fundamentals of Pho-tonics, 2nd ed.(Wiley, Hoboken, NJ, 2007),

[4] Griffen Rizzo and Jie Thing Lee, Measuring theSpeed of Light using Beating Longitudinal Modesin an Open-Cavity HeNe Laser, MXP Database,Student Project, PHYS4052WFall2018