S16_SpeedofLight

Measurement of the Speed of Light via Beat Frequencies of a HeNe Laser's Longitudinal Modes

Michael Hank & George Sorrell

University of Minnesota

School of Physics and Astronomy

Abstract

Using an open cavity Helium Neon gas laser with an adjustable cavity length, we measured the speed of light in air. By measuring the beat frequencies of the longitudinal modes as a function of cavity length, we measured cair = (2.9968 ± 0.0012)*108 m/s. This is consistent with literature values but not precise enough to distinguish the speed of light in a vacuum, which was a goal for the precision of the experiment.The theoretical limit for accuracy in this experiment is 1 part in 10,000 and our result was accurate to 1 part in 2,500.

Introduction

The speed of light, c, is a fundamental constant in the universe. It serves as the speed at which photons propagate through space and as the maximum speed with which an object can travel. It also has implications in special relativity involving Lorentz contraction and time dilation. These fields motivate an accurate and precise measurement of the speed of light [1].

Gas laser tubes emit Doppler-broadened light by spontaneous emission. By properly aligning two mirrors, the reflections of the spontaneously emitted light can cause stimulated emission, discussed in the Theory section. Due to the boundary conditions of mirrors, only certain wavelengths in the Doppler broadening envelope can constructively interfere and emit laser light. These “modes” of laser light change as a function of the cavity length. Using a photodetector, the photons are converted into a current with a certain beat frequency dependent on the spacings of the modes, from which the speed of light in air can be calculated [2].

Theory

Figure 1. Diagram of stimulated emission. Taken from [3].

Lasers work by the process of stimulated emission (see Figure 1), where incoming photons stimulate electrons to de-excite and release a second photon of equal energy and momentum [4]. This dominates over spontaneous emission when a majority of the electrons are in the higher energy level [4]. For Helium-Neon lasers, this is accomplished by applying a high voltage across the gas tube. As the photons are reflected back and forth in the cavity, this effect causes the number of photons to increase via feedback.

Figure 2. Gain profile for the HeNe laser as taken from [2].

The spectrum of allowed wavelengths is expanded by the Doppler effect due to the random motion of the gas molecules, yielding a Gaussian profile as shown by the blue line in Figure 2 [2]. Boundary conditions of the laser cavity require an integral number of half-wavelengths (standing waves) to fit in the cavity [2]. This corresponds to resonance wavelengths (or longitudinal modes) of ([2])

where n is the index of refraction for the laser cavity’s medium, L is the length of the laser cavity, and N is a positive integer representing the order of the longitudinal mode. These modes are shown by the black lines in Figure 2. In order to maintain a steady state beam, gains due to feedback must compensate for losses from the beam output and scattering. Due to this, a minimum gain threshold is required for lasing to occur at a given frequency [4].

Interfering waves mix in the photodetector, producing beat frequencies with frequencies equal to the difference between adjacent longitudinal modes [2]:

This will cause the current produced by the beam on a detector to vary with frequency 𝛥f. By altering the cavity length and fitting L vs. the reciprocal of the observed beat frequencies, a measurement of cair can be obtained.

Breaking down the length into a laser tube component, a fixed component, and a variable component, the change in cavity length can be related to the speed of light in air:

Apparatus

A gas laser tube was aligned so that it emitted a laser beam through a 99.4% reflective spherical mirror used as an output coupler. The output coupler was placed on a linear translation stage driven by a stepper motor. The translation stage step size was measured to be 2.5µm per step using our stepper motor. This length measurement was made very precisely using a Micro-Epsilon laser triangulation sensor and repeatability tests were performed to ensure that the stage moved the correct number of steps every time. A non-polarizing beam splitter sent the beam to a Fabry-Perot interferometer and a photodetector. The Fabry-Perot output was connected to an oscilloscope, showing the intensities of the longitudinal modes. A reflection from the glass on the Fabry-Perot went to a convex lens, allowing visual confirmation that the laser was in the TEM00 mode. The photodetector signal went to a spectrum analyzer which showed the power spectrum of the signal in frequency space, displaying a peak corresponding to the beat frequency. The setup is shown in Figures 3 and 4.

Figure 3: Diagram of the experiment. NDF stands for Neutral Density Filter and NPBS stands for Non-Polarizing Beam Splitter. The ΔL term shows the variation allowed, for the output coupler position shown ΔL is 0. Original figure.

Before taking data, measurements of the error due to frequency pushing and pulling were taken. Error due to frequency pushing was determined by finding the beat frequency for a range of intensities and setting upper and lower limits on the intensity of the modes for data. Frequency pulling error was determined by finding the standard deviation of several data points taken within the allowed range for frequency pushing. Beat frequency data was taken while pushing on the table to adjust the cavity length by microns to put the relative intensities of the modes in the correct range to minimize frequency pushing and pulling errors and then using the peak finder function on the spectrum analyzer. This measurement was used in conjunction with the known change in length of the laser cavity to calculate the speed of light as shown in the theory section. The scale of the experiment can be seen in Figure 4, where the translation stage can move the output coupler by several millimeters.