S21_SpeedofLight

Determination of the speed of light from the longitudinal modes beat frequencies of an open-cavity helium-neon laser

Mohamad Taim and Nicholas Kuder

University of Minnesota, Minneapolis, MN, USA

We empirically determined the speed of light by measuring the variation in longitudinal mode frequencies, or the beat frequencies, of an adjustable-length, open-cavity helium-neon laser as a function of its cavity length. The TEM₀₀ mode lasing output of the laser was analyzed using a fast frequency photodiode detector and a radio frequency spectrum analyzer. A Fabry-Perot interferometer was used to monitor the intensity of the longitudinal modes and we found that the phenomena of frequency pushing and pulling had little effects on the beat frequency measurements. Plotting the reciprocal of the beat frequency as a function of the change in cavity length, the speed of light was found, by using linear weighted least squares regression, to be (2.997 ± 0.003) × 10⁸ ms⁻¹. This value is 0.3σ away from the defined value of speed of light and is accurate to 1 part in 3200.

INTRODUCTION

A He-Ne laser tube is mounted on an optical post and a 99.9% reflective output coupler (ThorLabs) with a radius of 45 cm is secured on a translation stage (Micro-Controle), which is controlled by a stepper motor (PhobOTronix), completing the open cavity of the laser system. Between the output coupler and the laser tube is an adjustable iris to block unwanted transverse modes.

The speed of light, 𝑐, is an important fundamental constant in many scientific fields, from electrodynamics to general relativity. 𝑐 is not just about describing the propagation property of electromagnetic waves, but it also happens to be the upper bound to the propagation speed of signals and to the speeds of all material particles.

Large numbers of determinations have been made since Galilei Galileo first attempted to prove the quantifiability of speed of light by using two lanterns placed on top of two separate hills and measuring the time taken for the light signals to cross the valley [1]. Most contemporary attempts on measuring the speed of light employed variations of the two common experimental methods: time of flight, and measurement of the frequency and wavelength of a radiation [2]. Earlier papers described techniques like Foucalt’s rotating mirror method [3], Kerr electro-optical shutter [4, 5], and microwaves [6].

Our experiment utilized the resonance condition of an open cavity, first done by Essen and Gordon-Smith in 1947 with a cylindrical copper cavity resonator as described in Ref. [7]. Their value of 𝑐 was accurate to 3 parts in 100,000. In 2010, the determination in Ref. [8] utilized a Helium-Neon (He-Ne) laser to make a determination that was accurate to within 1 in 4100. We also used a He-Ne laser system to determine 𝑐 by studying the longitudinal mode frequencies of the lasing beam and its relationship with the length of the laser cavity.

THEORY

A He-Ne laser system consists of a cavity in between two highly reflective mirrors, or a Fabry-Perot cavity. A 99.9% reflective mirror is on one end and a 99% reflective output coupler is on the other with the output coupler letting through a little amount of light that becomes the laser beam. The cavity is filled with a mixture of helium and neon and together, the gas-filled cavity is called the medium.

When a DC current is pumped into the medium, the electrons in it excite into higher levels of energy and the medium is said to undergo population inversion, a process in which the number of excited atoms is more than the number of ground atoms. These excited electrons then decay to lower levels of energy and release photons. This is the spontaneous emission and the photons are emitted in random directions and phases. Those that are parallel to the horizontal axis of the laser tube get reflected by the two mirrors many, many times. This leads to stimulated emission when the photons interact with other excited electrons in the medium. The result is an avalanche of photons with similar orientation, wavelength, and phase that comes out of the output coupler as a bright, nearly monochromatic, single beam of laser.

The Fabry-perot cavity also imposes a resonance condition on the system in that the laser will not operate with just any wavelength. The lasing output will only have wavelengths, λ, that satisfy the equation

λ = 2𝑛𝐿 ∕ 𝑘

Laser stands for light amplification by the stimulated emission of radiation. The process of lasing consists of particles in a gain medium, He-Ne in our case, being excited into a higher energy state and emitting photons as they decay back into their lower energy state. Two types of emission may occur, spontaneous and stimulated. For both cases the energy of the photon is the difference in the energy states. For the case of spontaneous emission the photon will have random a phase and momentum. In the case of stimulated emission a photon interacts with an excited particle in the gain medium and causes it to decay and emit a second photon. The second photon in stimulated emission will have identical phase and momentum as that of the first. A laser consists of a gain medium with a voltage applied, to excite the particles in it, and two mirrors on opposite ends, parallel to each other with one being slightly transparent, 0.1% transparent in our case. The mirrors cause the photons to bounce back and forth within the cavity amplifying the stimulated emission until they exit through the slightly transparent mirror.

FIG. 1. A basic diagram of a laser

The mirrors allow only for standing waves from the photons, like two tied ends of a rope, thus the allowed wavelengths of light from the laser are

λ = 2𝑛𝐿 ∕ 𝑘, (1)

where 𝑘 is any integer, 𝑛 is the index of refraction, and 𝐿 is the cavity length. The frequency is thus given by

𝑓 = 𝑘𝑐 ∕ 2𝑛𝐿, (2)

where 𝑐 is the speed of light. Beats occur when two close, but different, frequencies interfere with each other, thus beats are created from adjacent frequencies within the laser. The beat frequency is simply the difference in the frequencies interfering with each other. The beat frequency for the laser can thus be given as

𝛥𝑓 = 𝑓 + 1 - 𝑓 = 𝑐 ∕ 2𝑛𝐿, (3)

where 𝛥𝑓 is the beat frequency. For our experiment this result has a few flaws in it, it assumes that there is only one index of refraction and that we can measure the cavity length. Some of the other indices of refraction include that of the He-Ne, and that of glass. Furthermore it is not possible for us to measure the length inside of the laser tube, the only measurable distance is outside of the tube. When considering multiple indices of refraction

𝑛𝐿 = 𝑛ₕₑₙₑ𝐿ₕₑₙₑ + 𝑛ₐᵢᵣ𝐿ₐᵢᵣ. (4)

Provided the distance in air is increased by a distance of 𝛥𝐿 this becomes

𝑛𝐿 = 𝑛ₕₑₙₑ𝐿ₕₑₙₑ + 𝑛ₐᵢᵣ(𝐿ₖ + 𝛥𝐿). (5)

Substituting this into Eq. 3 and solving for 𝛥𝐿 yields

𝛥𝐿 = 𝑐 ∕ 2𝑛ₐᵢᵣ𝛥𝑓 - 𝑛ₕₑₙₑ𝐿ₕₑₙₑ ∕ 𝑛ₐᵢᵣ - 𝐿ₖ. (6)

Thus the slope of the change in cavity length as a function of inverse beat frequency is proportional to the speed of light. It should also be noted that any other lengths and indices of refraction would simply be absorbed into the y-intercept.

EXPERIMENTAL SETUP & METHODOLOGY

Laboratory Setup

The laser beam travels through a non-polarizing beam splitter, which directs half of the beam to a fast frequency photodiode detector (Thorlabs model PDA8A) and the other half into a scanning Fabry-Perot interferometer (SFPI) (Tropel model 240). The signal from the photodetector is sent to a radio frequency (RF) spectrum analyzer (Rigol model DSA815), which displays the beat frequency. The beam from the SFPI is directed to another photodetector, which is connected to an oscilloscope that will display the relative intensities of the modes. Both the spectrum analyzer and the SFPI are assumed to be calibrated correctly.

The measurements on beat frequency will be affected by two optical phenomena, namely frequency pulling and pushing. Frequency pulling occurs when there are varying index of refraction near the resonance transition. This causes the difference in mode frequencies, depending on its position under the gain profile curve, to have a smaller value [9]. In other words, the two mode frequencies are being pulled toward the center. This effect is more pronounced when we have mode frequencies that are not at similar intensity.

Meanwhile, frequency pushing is when we have an increase in gain, or increase in refractive index, due to reasons that are myriad and complex, such as the ambient temperature and pressure. This phenomenon increases the beat frequency and also the total intensity [9]. We can prevent this by taking data only when the total intensity is not changing.

The SFPI serves to monitor the relative intensities of the longitudinal modes. The iris in between the Brewster window and the output coupler is used to adjust the total intensity of the laser beam. These measures are taken to minimize the uncertainty in beat frequency measurements.

The schematic diagram of the experiment is shown below in Fig. 2 [8].

FIG. 2. The schematic diagram of the experimental setup.

Calibration

The stepper motor, controlled via a LabVIEW (version 2018) program, is used to move the translation stage by a known number of steps. After the translation stage translated the distance, a digital caliper, fixed on the translation stage using tapes to minimize human error, is used to measure the total distance moved by the translation stage. This is the change in laser cavity length, 𝛥𝐿. This process is repeated until 21 data points have been collected. The change in length per step is then determined by the slope of the data using linear regression.

RESULTS & DATA ANALYSIS

In order to calibrate the translation stage and obtain a relationship between the number of steps, n, and the translated distance, 𝛥𝐿, which corresponds to the difference in length of the laser cavity, we measured 𝛥𝐿 for a known value of 𝑛. We found from the plot in Fig. 3, for 𝑛 = 10,

𝛥𝐿 = 2.5 × 10⁻⁵ m

The uncertainty in 𝛥𝐿 is σ = ± 1.0 × 10⁻⁵ m, which is dictated by the tolerance of the digital caliper.

FIG. 3. The plot for the calibration of the translation stage.

Using the iris to ensure that the mode is in TEM₀₀ mode (single circular beam), we then measured the beat frequencies as a result of varying the length of the laser cavity. The values for the reciprocal of beat frequency, 1 ∕ 𝛥𝑓, as a function of 𝛥𝐿 is plotted in Fig. 4.

FIG. 4. Reciprocal beat frequency as a function of the change in the open-cavity length.

For each 𝛥𝐿, the value for 1 ∕ 𝛥𝑓 was averaged over 100 sweep cycles to minimize the uncertainty on frequency reading by an order of magnitude.

Linear regression using the weighted least squares method was performed and the data plotted in Fig. 4 yielded the final result for the speed of light in air to be

𝑐 = (2.997 ± 0.003) × 10⁸ ms⁻¹

Our measured value is accurate to within 0.3σ of the defined value 𝑐 = 299,792,458 ms⁻¹ [10]. As can be seen in Fig. 5}, the linear fit residuals show no structure, which indicates good data. Specifically, the reduced chi squared is 𝜒² = 0.922 and the P-value is 𝑃 = 0.58.

FIG. 5. Residuals from the linear fit.

The uncertainty in the final result can be explained by the phenomena frequency pushing and pulling. Although we found no correlation between the two phenomena and the beat frequency. Specifically, we found that the relative intensities of the two modes observed on the SFPI oscilloscope had little effect on the value of beat frequency on the spectrum analyzer. Further investigation also found little effect on beat frequency caused by the total intensity, which we controlled by the iris. The calculated speed of light did not account for frequency pushing, as it was shown to have no effect in our experiment. Furthermore, frequency pulling was not considered in the calculations for multiple reasons. The data taken for the mode frequencies and amplitudes only occurred once per cavity length. Provided the experiment was repeated this would be done with every measurement of beat frequency as the mode frequencies and amplitudes may change even with constant cavity length. Due to this fact it is likely that this data is not a good indicator to keep, or discard beat frequency data points. Despite that some data analysis was done with it. In a fit not shown data points were taken where the ratio of modal frequencies was between 0.3 and 1.7 and all others discarded, resulting in only six data points out of one hundred. The fit resulted in a value of 3.013·10⁸ ± 5.62·10⁵ms^-1, inconsistent with the accepted value of the speed of light. This is likely due to the small sample size as well as limited measurements of the modal frequencies and amplitudes.

In future experiments, one can reduce the uncertainty caused by frequency pulling by implementing an electronic feedback system that can make slight adjustments to the cavity length in order to maintain similar relative intensities at all time. Other sources of error like the misalignment of the optical equipment, and the thermal expansion of the optical table can also be looked into. Additionally it is advised for future experiments measurements be taken of the modal amplitudes for each beat frequency measurement, then discard individual beat frequency measurements before averaging the beat frequency measurements for a given cavity length.

CONCLUSION

By varying the length of the open cavity of a He-Ne laser, we determined the speed of light to be 𝑐 = (2.997 ± 0.003) × 10⁸ ms⁻¹. This value is 0.3σ away from the defined value 𝑐 = 299,792,458 ms⁻¹. We found that the phenomena frequency pulling and pushing had little to no effects on our measurements and thus, negligible uncertainties on our final value of 𝑐.

ACKNOWLEDGEMENTS

Our thanks to Professor Kurt Wick for his guidance and enthusiasm throughout the research; to Professors Dan Dahlberg and Daniel Cronin-Hennesy for their lectures; to Kevin Booth for the constant help with various problems. We express gratitute to the Department of Physics for the equipments and the permission to run the experiment.

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