S15SpeedofLight

Measuring the Speed of Light via Beat Frequencies of Longitudinal Modes

By Kali Ask and Noah Bittermann

Advised by Kurt Wick

University of Minnesota--Twin Cities

School of Physics and Astronomy

Minneapolis MN 55455

Abstract

The speed of light, c, is significant because it is a fundamental constant in physics. In this experiment the speed of light in air, c_air, from which c may be easily calculated, was measured by analyzing the beat signal between neighboring longitudinal modes of an adjustable open cavity Helium-Neon (HeNe) laser for various cavity lengths. We measured c_air to be (2.997±0.003) x 10⁸ m/s, which is 0.1 sigma away from the accepted value of 2.997 x 10⁸ m/s. Our result is accurate to 1 part in 1,000.

Theory

In our experiment, we took advantage of the particular spectrum of a HeNe laser. This is demonstrated below in Figure 1.

Figure 1: Spectrum of a HeNe Laser

The properties of the spectrum of a HeNe laser relevant to our experiment are due to three effects:

1) Doppler broadening: In their own rest frame molecules of the lasing medium emit light of a single frequency. Due to the Doppler effect, this manifests itself as a Gaussian distribution of frequencies in the laboratory frame, giving the Gaussian envelope demonstrated in Figure 1A [1].

2) Standing wave condition: Only frequencies which are a half-integer multiple of the laser cavity length may exist in the cavity. This gives the discrete set of peaks following the aforementioned envelope, demonstrated in Figure 1B [2].

3) Gain threshold: Only frequencies whose gain is greater than their loss (that is, the gain threshold) may exist in the cavity. Thus only the peaks shown in Figure 1C may exist in the cavity. Typically only one or two exist at a time [2].

Equation (1) describes the frequencies which exist in the cavity. N is a positive integer, c is the speed of light, n is the index of refraction, and L is the laser cavity length. In equation (2), the beat frequency between adjacent modes (that is, modes indexed by N and N+1) is calculated. Rearranging equation (2) gives equation (3). After measuring the inverse beat frequency for various cavity lengths, one may apply linear regression to find the slope and thus c_air.

Experimental Setup

The experimental setup in full is shown below in Figure 2.

Figure 2: Experimental Apparatus

Laser Cavity

The laser cavity itself consist of several different components. The laser tube houses the HeNe lasing medium and one mirror of the laser cavity. A Brewster window is attached to the emitting end of the laser tube, which ensures that the beam is of a single polarization. An iris is located in the cavity and is used to control the power. The output coupler is the other boundary of the laser cavity.

Translation Stage

In order to adjust the cavity length, we placed the output coupler on a translation stage. The stage consists of a platform whose motion was driven by a worm screw, which itself is turned with a stepper motor. We used a LabVIEW program to control the direction of the stage's motion and number of steps taken by the stepper motor. For the details of this program, see the "Diffraction by a Straight Edge" lab in the MXPII Lab Manual. Because we could not located the exact specifications for our translation stage, we did not know how far a single step corresponded to. Thus, in order to measure the laser cavity length, we used calipers.

Fabry-Perot Spectrometer

Once the beam exits the laser cavity, it is split by a non-polarizing beamsplitter (NPBS). Part of the beam is sent to a Fabry-Perot spectrometer whose signal was read with an oscilloscope. This allowed us to view the spectrum of the signal within the cavity, and thus determine the prescence of higher order TEM modes and the effects frequency pushing and pulling, each of which is explained below in the "Methods" section. To diminish back reflections, we placed an iris between the Fabry-Perot spectrometer and the NPBS.

Photodetector and Radio Frequency (RF) Spectrum Analyzer

From the NPBS, a portion of the beam is sent to a photodetector, which we used to measure the beat signal. So that we did not over-saturate the photodetector, we diminished the intensity of the beam with a neutral density filter. In order to read the signal of the photodetector, we used a radio frequency (RF) spectrum analyzer.

Images

Images of our actual experimental apparatus are shown below in Figure 3. While all components shown in Figure 2 were used, our actual configuration differed slightly.

Figure 3: Images of Actual Experimental Apparatus

Methods

Aligning the Laser and Other Pieces of the Apparatus

In order to properly align our primary open cavity laser, that is, the one actually used in our experiment, we used a secondary closed cavity laser. We employed a technique called "walking the beam". Using two irises of the same height, we first ensured that secondary beam was parallel with the table immediately after it left the secondary laser cavity. Next, using mirrors we directed the beam to the vicinity of the laser we were trying to align. In order to ensure that after the beam reflected off the mirrors it was still parallel to the table, we shined it through the same aforementioned irises. After this, we aligned the laser tube of the primary laser with the secondary beam. We aligned the output coupler by adjusting it so that it reflected the secondary beam back in exactly the same direction it came from. We were able to align the Fabry-Perot spectrometer using a similar method. Aligning the NPBS and photodetector was straightforward--we simply placed these two components so that the beam shone on the photodetector's sensor.

Transverse ElectroMagnetic (TEM) Modes

TEM modes are different patterns of intensity in the cross-section of the laser beam, as shown in Figure 4A. TEM₀₀ is a Gaussian distribution of intensity. All other modes, which are called higher-order TEM modes, have different patterns of intensity [4]. The principles developed in the theory section, namely equation (1), only hold when the beam is in TEM₀₀. This is because higher order TEM modes introduce additional beat frequencies which complicate the reading of the RF spectrum analyzer. Our strategy for determining and mitigating the presence of higher order TEM modes was two-fold. First, to put the beam in a TEM mode close to TEM₀₀, we used a divergent lens to expand the beam cross-section so that it was more visible and made small adjustments to the alignment of the output coupler until the intensity distribution was approximately Gaussian. To fine tune the intensity distribution of the cross-section, we used the Fabry-Perot spectrometer. A Fabry-Perot spectrometer signal in which higher order TEM modes are present is shown in Figure 4B. To ensure that the beam was in TEM₀₀ we adjusted the output coupler until small extraneous peaks such as the middle peak in Figure 4B disappeared.

Figure 4: Images of TEM Modes (B, taken from [5] Paschotta, Rudiger. "Resonator Modes.") and Their Effect on the Fabry-Perot Spectrometer Signal (A):

Frequency Pushing and Frequency Pulling

We also used the Fabry-Perot Spectrometer to determine and mitigate the effects of frequency pulling and pushing. Frequency pulling is the result of different frequencies having different indices of refraction. It has the net effect of pulling the frequencies of the longitudinal modes closer together and towards the central frequency, thus decreasing the beat frequency. The decrease in the beat frequency of two adjacent modes is related to their relative intensities. In particular, modes of the same relative intensities do not experience this effect [2]. Thus, to reduce the effects of frequency pulling, measurements were only taken when the peaks displayed on the oscilloscope were of the same relative height. Figure 5A shows the Fabry-Perot spectrometer reading of two sets of modes which experience different pulling effects. By gently pushing on the air cushioned optics table on which this experiment was performed, changing the cavity length on the order of microns, the relative power of the two modes were kept within approximately ± 5 mV. Frequency pushing is caused by the sensitivity of a longitudinal mode's frequency to the power in the cavity. It has the net effect of increasing the difference between the frequencies of the longitudinal modes, and thus of increasing the beat frequency [2]. Figure 5B shows the Fabry-Perot spectrometer reading of two sets of modes which experience different pushing effects. This source of error was minimized by taking measurements of the beat frequency only when the total power in the cavity was at a predetermined quantity (140 mV for each mode), which was maintained within approximately ±5 mV.

Figure 5: Demonstration of Frequency Pushing and Pulling Effects on Fabry-Perot Spectrometer Signal

Measuring Length

Because one end of the laser cavity is inside of the laser tube, we cannot accurately measure L. Furthermore, since the laser beam travels through both air and the HeNe lasing medium, the index of refraction n is comprised of two parts: that of air, and that of the HeNe medium, which is unknown. Both of these issues were solved by manipulating (3). By dividing the cavity length into three parts as shown in Figure 2 and combining indices of refraction we obtain equation (6), which may be substituted in equation (3) to obtain equation (7). Since all unknowns of equation (7) are in the intercept, it is straightforward to apply linear regression and determine c_air.

(6) [2]

(7)

Results and Analysis

To determine the effect of frequency pushing on our measurement of the beat frequency, for a fixed cavity length we measured how changing the absolute power of the two modes affected the beat frequency when their relative intensities were the same. To determine the effect of frequency pulling, for a fixed cavity length we measured how changing the relative intensity of the two modes affected the beat frequency while the total power in the cavity was constant. These results are plotted in Figure 6. Using the regression equations for these lines, we were able to calculate how our bounds for the relative intensity of the modes and their absolute power affected our errors in measuring the beat frequency. We found that maintaining the absolute power of the modes within ±5 mV corresponded to an error of 20 kHz, while maintaining the relative intensities of the modes within ±5 mV corresponded to an error of 8 kHz.

Figure 6: Frequency Pulling and Pushing Regression Plots

Figure 7 shows the linear regression plot and and chi plot of our inverse beat frequency measurements for various cavity lengths. We measured c_air to be (2.997±0.003) x 10⁸ m/s, which is 0.1 sigma away from the accepted value of 2.997 x 10⁸ m/s and accurate to 1 part in 1,000. Looking to the chi plot, there is a clear periodic trend, namely a sawtooth pattern. This systematic error may have been caused by temperature fluctuations within the spectrum analyzer, which would have distorted its measurements. To investigate this, one could turn off the spectrum analyzer between measurements and allow it to cool down.

Figure 7: Linear Regression and Chi Plot of Inverse Beat Freuqency Measurements for Various Cavity Lengths

Acknowledgements

We would like to thank our advisor Kurt Wick as well as Elias Puchner and Clement Pryke for their help with this project.

References

[1] Saleh, Bahaa E. A., and Malvin Carl. Teich. Fundamentals of Photonics. New York: Wiley, 1991. Print.

[2] D’Orazio, Daniel J., Mark J. Pearson, Justin T. Schultz, Daniel Sidor, Michael W. Best, Kenneth M. Goodfellow, Robert E. Scholten, and James D. White. "Measuring the Speed of Light Using Beating Longitudinal Modes in an Open-cavity HeNe Laser." American Journal of Physics 78.5 (2010): 524. Web.

[3] Gregoric, Vince. "A Precise Measurement of the Speed of Light in Air from the Frequency Separation of Longitudinal Modes in an Open-cavity HeNe Laser." VinceGregoric / Laser Teaching Center Biography. Stony Brook University, June 2012. Web. 09 Mar. 2015.

[4] Paschotta, Rudiger. "Resonator Modes." Encyclopedia of Laser Physics and Technology. Web. 11 Mar. 2015.