Measuring the Speed of Light -Caitlyn Kloeckl

Measuring the Speed of Light Using a Helium Neon Laser

Caitlyn Kloeckl

University of Minnesota

Methods of Experimental Physics II, Fall 2019

Abstract

The purpose of this experiment was to measure the speed of light using a helium neon laser. By varying the laser cavity length, various frequencies were produced in the laser cavity. The various frequencies interfered and produced a beat frequency which was measured. Plotting this beat frequency against the cavity length and implementing a linear fit, the speed of light was determined from the slope of the line. The value for the speed of light in air was measured to be (3.0019 ± 0.0034)*10⁸ m/s.

Introduction

The speed of light has been known to within 5% since the late 17th century,¹ and the measurement has steadily improved until the 1983 redefinition of the meter in terms of the speed of

light.² At the time of the first measurement, the speed of light was a mere curiosity. Since then, new physics and technology have been developed which depend on the precise value of the

speed of light. The following method for measuring the speed of light has been successfully deployed⁴ in recent years and measured the speed of light in air as 2.9972±0.0002

10⁸ m/s. The

following experiment has replicated similar results using an open-cavity helium-neon laser.

Theory

A laser creates a standing electromagnetic wave in a cavity containing a gain medium-in this case, the helium-neon mixture. When voltage is applied across the tube, a discharge arc excites the atoms within the tube. Since the direction of motion is random, the component of the velocity directed along the cavity is also random. This means an emitted photon has a random initial velocity. So, the photons emitted are Doppler-shifted by a random amount. The doppler shift causes changes in wavelength, as depicted below. Because of the doppler shift, the light in the cavity varies in a discrete set of wavelengths.

After passing through the cavity, the discrete set of permitted frequencies appear with a Gaussian gain profile centered at the optimal wavelength for the laser:

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:

Experimental Setup

We constructed an open-cavity helium-neon laser operating at 632 nm. Everything was mounted to an optics table. An output coupler was placed on the programmable translation stage (stepper motor). This gave us fine control of the cavity length. We used mirrors to direct the beam onto a photodetector, which measured the intensity and frequency of the beam. We then connected the output of the photodetector to the spectrum analyzer, which isolated the beat frequency. Everything is shown below:

This set of data has 139 unique stepper motor values along with the respective inverse beat frequency. As is noted by the red line, a linear fit was applied to this plot. This fit was calculated by using the MATLAB fit function along with code provided by the advisors of this class. The fitting algorithm uses a least squares algorithm to fit the weighted data to the linear model. This produced a value of (6.6625 ± 0.0074)*10^-9 for the slope of the line. Taking 2 over this value and propagating the error, the value for c, the speed of light in air, was measured to be:

(3.0019 ± 0.0034)*10⁸ m/s

The following is the weighted residual plot for each motor step. This results in a P-value of 0.99, reduced Chi squared value of 0.847

References

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- Romer, M. (1676). A demonstration concerning the Motion of Light, communicated from Paris, in the Journal des Scavans, and here made English. Philosophical Transactions. Retrieved from https://royalsocietypublishing.org/doi/10.1098/rstl.1677.0024, accessed 9-29-2020.

- Brickner, R. G.; Kappers, L. A.; Lipschultz, F. P.; (1979). Determination of the speed of light by measurement of the beat frequency of internal laser modes, American Journal of Physics 47, 1086-7. doi/:10.1119/1.11980.

- D’Orazio, D. J.; Pearson, M. J.; Schultz, J. T.; Sidor, D.; Best, M. W.; Goodfellow, K. M,; Scholten, R. E.; and White, J. D. (2010) Measuring the speed of light using beating longitudinal modes in an open-cavity HeNe laser, American Journal of Physics 78(5), 524-8. doi:10.1119/1.3299281.

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- Wang, C. P.; Varwig, R. L. (1979) Competition of Longitudinal and Transverse Modes in a CW HF Chemical Laser, The Aerospace Corporation/DARPA. Retrieved from https://apps.dtic.mil/dtic/tr/fulltext/u2/a080382.pdf, accessed 10/12/2020.

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