F18_Speed_of_Light

Measuring the Speed of Light using Beating Longitudinal Modes in an Open-Cavity HeNe Laser

------- Griffen Rizzo and Jie Thing Lee -------

University of Minnesota

Methods of Experimental Physics II

Fall 2018

Abstract

We propose to determine c, the speed of light by measuring the beating longitudinal modes of a HeNe laser as they relate to the cavity length of the open-cavity laser. We hope to expand and improve upon projects performed by previous MXP students by taking steps to minimize the uncertainties in the measurements, and by investigating phenomena, frequency pushing and pulling. We yield the experimental value of c_vacuum , the speed of light to be 2.9970±0.0003×10^8 m/s while the expected value is 2.9979×10^8 m/s.

Introduction

Figure (1) Laser Gain Profile. Picture taken from citation [2]

In a laser, discrete frequency peaks exist within the gain curve, when two frequency modes exist simultaneously, beats are produced. The frequency of these beats is equal to the difference between the frequencies of the two longitudinal modes. The length of the laser cavity is related to the frequencies that can exist in the laser by the speed of light. As we change the cavity length, the frequencies change, and the speed of light can be calculated from that data.

Theory

Figure (2) Stimulated Emission . Picture taken from citation [6]

Spontaneous emission occurs in a laser, by which the electron randomly releases a photon as it falls back into the ground state. However, stimulated emission is the primary manner by which light is produced in a laser. It occurs when a photon passes by an atom with an excited electron. This causes the electron to fall into a lower state, and causes a photon of the same energy and momentum to be released in phase with the one that passed by.

Nodes of a wave occur every half wavelength, so only a half integer number of wavelengths can exist between the mirrors, which is the cavity length, L. The wavelength is equal to the speed of light, c, divided by the frequency, f. Solving for the frequency gives the equation

(1)

Where N is a whole number, and n is the refractive index of the gas that comprises the laser cavity.

If the mirrors are not aligned in a specific way, the beam reflected off the output coupler will interfere with the beam coming from the back of the laser. This will cause translational modes in the beam and will result in other beat frequencies that will also interfere with the measurement of the spectrum analyzer, so equation (2) is only true for the the Transverse Electromagnetic, or TEM00 state, which is what we will be using for all the measurements in this lab.

In figure 1, The intensity of each of the peaks at various values of n is given by the height of the gain curve at that frequency. However, there is a gain threshold that must be achieved for a peak to sustain itself (Figure 1). If the intensity of a peak is too low, it will not be able to stimulate enough emissions to avoid decaying to zero. So, only the peak, or peaks that are above the gain threshold will actually be emitted from the laser. The difference in the frequency of these peaks, is known as the beat frequency, and is given by the equation

(2)

Frequency Pulling and Pushing

Figure (3) Frequency Pulling. From citation [2]. Figure (4) Frequency Pushing. From citation [2]

If the peaks of the longitudinal modes are not the same height, they will be subject to frequency pulling. This is due to their differences in refraction as they travel to the detector. The signal with the lower intensity will be pulled towards the center of the gain curve, thus artificially changing the beat frequency, and changing the measurement. If the two peaks have the same relative intensity, but the overall intensity of the beam changes, they will be subject to frequency pushing.

An RF spectrum analyzer will be used to measure the beat frequency, , but a Scanning Fabry-Perot Interferometer will also be used to look at the longitudinal modes that compose the signal. With it, we will be able to observe how many peaks there are, and we will be able to compare their heights.

Experimental Setup

Figure (5). A schematic diagram of the Experimental set up. An open cavity HeNe laser emits a beam and it splits into two by the NPBS. One beam is directed into the Scanning Fabry Perot Interferometer to be observe by the oscilloscope, another beam is directed to a silicon amplified photodetector to be analyzed by the rf spectrum analyzer. Reflection from the SFPI is observe by a magnifying lens to ensure TEM_00 mode. Picture taken by citation [5].

The Adjustable Open Cavity Laser design

A Melles Griot 05-LHB-568 HeNe laser emits a light on an external 99.5% reflective output coupler which reflects the light back onto a high reflection mirror in the back of the laser, thus emitting a laser beam. The output coupler is firmly mounted on a translation stage where the stage position with the length of the cavity will be controlled by a stepper motor.

Taking the indices of refraction into account, nL in equations (1) and (2) becomes

(3)

Then, plugging this expression into equation (2), and solving for ∆L, we obtain

(4)

The motor has a definition of 400 steps a revolution which gives a 2.5 µm per step size. Consequently, the stepper motor will be controlled by a program coded in LabVIEW.

Scanning Fabry-Perot Interferometer (FPI)

The laser beam splits in two by a non-polarizing beam splitter. One of them is directed into a Tropel Model 7600 Fabry-Perot interferometer and the other into a silicon amplified photodetector.

Even though we can obtain the beat frequency from the RF Spectrum Analyzer which we can calculate the speed of light, the scanning Fabry-Perot Interferometer is required in this experiment to analyze the beat pattern and a rough estimate of the beat frequency to make sure the RF Spectrum Analyzer is providing the correct information. More importantly, the beat pattern which shows different peak heights which needs to be observed during data collection for the pulling and pushing effect.The Fabry-Perot interferometer (SFPI) is a detector that is connected to a digital oscilloscope. The digital oscilloscope is configured until it displays peaks within a free spectral range (FSR) that will be equivalent to or greater than the 7.5GHz gain bandwidth of the SFPI. This configuration will allow better view on the laser gain profile. The laser gain profile will provide information about the intensity and different mode structures which is important to determine the condition of the laser beam. Also, an iris is located between the Brewster window and the output coupler to manually clean up the overall amplitude of the laser gain profile. A reflection from the FPI reflects back onto a magnifying lens as shown in Figure 1. Thus, we can analyze the laser beam pattern while adjusting the output coupler and identify the TEM00 mode which is required for this experiment.

RF Spectrum Analyzer

The second beam from the non-polarizing beam splitter is directed to a silicon amplified photodetector and analyzed by the RIGOL DSA 815 RF Spectrum Analyzer. The RF Spectrum Analyzer will show the beat frequency which is the difference between modes in the laser. Also, the analyzer offers to set a start, stop and centralize the beat frequency which will display accurate data. In principle, a frequency counter could replace the analyzer but a basic counter will not provide any insight of other forms of transverse mode frequency.

Figure (6). Actual experimental apparatus set-up in lab. Original Picture.

Results and Analysis

The value of c_vacuum, the speed of light was measured to be 2.9970±0.0003×10^8 m/s whereas the expected value is 2.9979×10^8 m/s which gave us a precision of 0.026% error. The accepted value falls out of range of the uncertainty of the experimental value which was 2.6σ away. The accepted value of c_air is 2.9971×10^8 m/s which show that the uncertainty of ±0.0003 is small enough to differentiate between the speed of light in air and in vacuum.

Figure 7. The Pulling Effect graph with beat frequency vs the Log(Ratio) of 2 heights of longitudinal modes. Graph shows no trend which indicated that pulling effect have no effect.

Figure 8. The pushing effect shows the linear relationship between the intensity of the laser and the beat frequency. The slope was used as a correction factor to obtain the adjusted beat frequency.

Figure 9. The measurement of speed of light. Plotting distance vs the inverse beat frequency provides the slope which is c/2n. Horizontal error bars are shown but to small to be analyzed.

Figure 10. Chi_i results for the pushing effect shows no trend and good data.

From this experiment, the uncertainty of c was dominated by the effects of the frequency resolution compared to frequency pushing or other factors.

Acknowledgement

This experiment was based on the paper by Daniel J. D’Orazio [2]. I would like to thank our advisor Kurt Wick as well for his assist on this project.

References

[1] D’Orazio, Daniel J., et al. “Measuring the Speed of Light Using Beating Longitudinal Modes in an Open-Cavity HeNe Laser.” American Journal of Physics, vol. 78, no. 5, 2010, pp. 524–528., doi:10.1119/1.3299281.

[2] 21A. M. Lindberg, “Mode frequency pulling in He–Ne Lasers,” Am. J. Phys. 67, 350–353 1999.

[3] Saleh, B. E. A., and M. C. Teich. Fundamentals of Photonics. 2nd ed., John Wiley & Sons, Inc., 2007.

[4] R. Paschotta, article on 'resonator modes' in the Encyclopedia of Laser Physics and Technology, 1. edition Octobe r 2008, Wiley-VCH .

[5] Hank Michael, and Sorrell George, Measurement of the Speed of Light via Beat Frequencies of a HeNe Laser’s Longitudinal Modes, MXP Database, Student Project, PHYS4052WSpring2016.

[6] Shaik, Asif. “Absorption of Radiation, Spontaneous Emission and Stimulated Emission.” Physics and Radio-Electronics, 2018, www.physics-and-radio-electronics.com/blog/absorption-of-radiation-spontaneous-emission-and-stimulated-emission/.