F20SpeedofLight

Measuring the Speed of Light Using an Open-Cavity HeNe Laser

Caitlyn Kloeckl & Matthew Granstrom

Abstract

We used a helium-neon laser to measure the speed of light in air. Interactions between the boundary conditions and the gain profile of the laser create longitudinal modes in the cavity. The spacing of these modes depends on the optical path length of the cavity. Observing the changing intensity of the laser beam allows measurement of the beat frequency, which gives the mode spacing. We measured the beat frequency at a variety of cavity lengths. By performing linear regression, we determined the speed of light in air to be 3.0038±0.0039×108 meters per second.

Introduction

The speed of light has been known to within 5% since the late 18th century. Initial measurements were made by astronomers observing the motion of Jupiter’s moons at different times of the year [1]. Further enhancements were made using rotating mirrors and measurements of electromagnetic constants. This allowed steady improvement 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 there were no instruments or techniques which could put this to use. However, since then, we have developed new physics and technology which depends on this precise value. Many laboratory techniques, such as interferometry, measurements of particle energy, and distance measurements, depend on knowing precisely 1 how fast light travels, both in vacuum and in air. Beyond the lab, GPS location, fiber-optic communications, spacecraft communication, and many other technologies critical to modern life depend on precise knowledge of this fundamental constant.

The discovery of the laser enabled many useful lab techniques, including several methods to measure the speed of light. One such method uses interference between longitudinal modes in an open-cavity helium neon laser. We attempted this measurement. We varied the cavity length, measured the intensity of the beam using a photodetector, and used a spectrum analyzer to determine the beat frequency of the intensity. A linear fit allowed us to determine the speed of light.

Theory

A laser is a gas-filled tube with mirrors at both ends. Applying a voltage creates a discharge arc in the gas, exciting the atoms of helium and neon inside. A few meta-stable states in the energy spectrum of helium atoms allow collisions between atoms to excite neon atoms. This creates a population inversion, where a majority of the neon atoms are in an excited state. The poulation inversion allows certain wavelengths of light to be amplified through stimulated emission, where a photon causes an excited atom to emit another identical photon. This causes a large number of photons to cascade through the laser cavity, all with the same wavelength and phase. This creates a standing electromagnetic wave with wavelength corresponding to the distance between the energy levels of the neon atoms.

The random motion of the helium atoms lead to a random shift in frequency, caused by the Doppler effect. This replaces the perfectly monochromatic light a naïve analysis would predict with a Gaussian amplitude response across a range of frequencies about 1 GHz across.

When we apply the boundary conditions on the laser cavity, we see that the mirrors at the ends require there to be no electric field. Since there are nodes at both ends of the cavity, only waves with a half-wavelength which cleanly divides the cavity length is permitted to lase in the cavity. This combines with the Gaussian response profile to give a series of very narrow frequency peaks with spacing related to the cavity length:

Here, Λ is the optical path length of the cavity--essentially the length divided by the index of refraction--and k is some integer. If we generalize to the case where the cavity isn't homogeneous, in particular if some of the cavity is air, some of it is helium-neon gain medium, and some is glass, we get

where I've written the length in air as some constant length L₀ plus a varying term 𝛿L. If we consider this in terms of frequencies rather than wavelengths, and subtract the frequency with k nodes from the frequency with k+1 modes, we get

Since everything in that first term is constant, if we measure the beat frequency as we change the length, we can perform a linear regression to determine the speed of light. So long as there are only two wave modes lasing in the cavity, observing their interference as the cavity length changes is enough to measure the speed of light.

There are a few complicating factors that affect this analysis. First, there are other ways to set up a standing wave in the laser cavity that do not obey this model. However, they all have significantly wider beams than the mode we've described here. Therefore, placing an iris in the laser cavity can selectively absorb these unnecessary laser modes. Second, the frequencies produced by the laser beam are not exactly the frequencies predicted by the model. Two dominant nonlinear effects, frequency pushing and freqency pulling, cause the intensity of the laser modes in the cavity to change the frequencies at which the modes lase. Essentially, the further apart the intensities of the two interfering modes are, the worse our model will predict the beat frequency, since frequency pulling means the waves are spread out asymmetrically from the peak gain frequency of the helium-neon. Additionally, if each of our measurements are taken at a different intensity, the changing optical properties of the gain medium cause the wave frequencies to be slightly different to the geometrically permitted frequencies. To mitigate these effects, it is important to take data when the intensity of both modes in the laser cavity are the same, and when the intensity matches some reference value.

Apparatus

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:

Results

The following figure depicts the plot of the inverse beat frequency against the cavity length:

This set of data has 139 unique Δ L values along with the respective inverse beat frequency after filtering. 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

Conclusions

We have shown that it is feasible to measure the speed of light in air by measuring interference between the longitudinal modes of a helium-neon laser. The effects of the boundary conditions on the standing wave and the gain curve of the helium-neon gas combined to select a small number of longitudinal modes, and the spacing between the modes depends only on the speed of light and the length of the laser cavity. Using a spectrum analyzer, we have measured the beat frequency at a variety of lengths and used least-squares linear regression to determine the speed of light. We were successfully able to control for the effects of frequency pulling on the beat frequency measurements. Our measurement of (3.0019 ± 0.0034)*10⁸ m/s differs by 1.5σ from the literature value of the speed of light in air. Much of this deviation can be attributed to frequency pushing, which we were unable to sufficiently mitigate using data filtering.

References

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