Measuring the Speed of Light using a Helium Neon Laser

Matthew Klein and Will Chick

University of Minnesota

Methods of Experimental Physics Spring 2013

Abstract

Using Helium Neon, HeNe, laser with an adjustable cavity length, measurements of the speed of light through air will be taken. By lengthening the lasing cavity, the beat frequency of the differing modes under the laser gain curve will decrease. The speed of light can be measured as a function of the slope of the linear fit of the cavity length versus the inverse of this beat frequency. This experiment yielded the speed of light to be 2.99758±0.00072 x 10⁸ m/s compared to the accepted value of 2.99792 x 10⁸ m/s.

Introduction

A Helium-Neon, HeNe, laser functions through the manipulation of the electron energy levels of a mixture of helium and neon atoms by passing an electrical discharge through a mirrored cavity filled with the gas mixture. The electrons will collide with the Helium atoms, exciting them to the 2s₀ state. The electrons mostly excite Neon to lower energy states that correspond to non-laser lines; therefore, the Helium is necessary for an efficient laser.^[3] The excited Helium atoms will collide with Neon atoms exciting the Neon to a higher level (a small difference in energy ~0.2eV must be provided by the environment). Due to quantum mechanical effects, this energy level of Neon is unable to directly emit down to the ground state and must first emit to an intermediate state. Though nearly 200 different emissions are possible, the emission with the greatest probability, a feature further taken advantage of in the design of most HeNe lasers, is the 632.8nm wavelength corresponding to the 3s₂ to 2p₄ emission.^[3] This emission is dominant due to the high population of 3s₂ electrons and the large number of available 2p₄ levels, which readily emit down to ground state.

Figure 1. Diagram of how the entire lasing process works from electron impact to light radiation. [4]

Through a process of stimulated emission, a 632.8 nm photon passes close to another excited atom of Neon and through the interaction of the photon and electron’s electromagnetic fields, causes the Neon atom to itself emit a 632.8 nm photon in phase with the original photon. This process cascades through the excited neon atoms causing a large number of photons with the same wavelength and

phase.^[3] Figure 1 depicts how this occurrence this occurs.

Figure 2. Diagram of standing wave solutions from a string connected to two rigid walls. [5]

In order to maximize the number of photons available to produce the lasing action, the gas excitation chamber is placed between two mirrored walls called the lasing cavity. As the photons are reflected and move from wall to wall, they excite additional photons. However, these two wall reflections result in constructive interference which occurs only at specific frequencies. These standing wave solutions are analogous to a string connected to two rigid walls, seen in figure 2, where only discrete solutions are possible.

Each of these frequencies, known as modes, will have a slightly different wavelength. Compared to a frequency of 474THz for the 632.8nm wavelength, one study^[6] found a spacing of approximately 300MHz between modes, though this spacing will vary with the length of the lasing cavity.

Theory

Because these different modes have different wavelengths, there will be a difference between them called the beat frequency. The beat frequency, Δf, is equal to:

Equation 1

where N is an integer greater than 0, n is the index of refraction of the cavity, and L is the length of the cavity.

The speed of a beam of light, c, is dependent on the medium through which it is travelling. This change in velocity is dependent on the index of refraction, n, of the medium such that the speed in a medium is equal to the speed in a vacuum divided by the index of refraction.

Equation 2

By this equation, it is apparent that a true vacuum must have an index of refraction of 1. Air at standard temperatures and pressures has an index of refraction very close to 1, typically around 1.00026. The function describing the index of refraction of air is well known and is dependent on temperature, pressure, humidity, and the wavelength of the light itself.^[7]

HeNe lasers naturally emit waves in multiple modes given by equation 3^[6]:

Equation 3

where g₁g₂ is the resonator stability of the laser, and m and n are positive integers. g_x is defined as 1- , with R_x being the radius of curvature of a mirror. The fundamental mode of the laser is the TEM₀₀ mode. This mode corresponds to a Gaussian beam profile. Additional mode patterns can be seen in Figure 2. The TEM₀₀ mode will be used for experimentation because it is the easiest to isolate and it also makes minimizing frequency pushing and pulling easier. The laser will be forced into the TEM₀₀ mode by proper alignment of the two reflecting mirrors.

*Figure 4:* TEM_mn mode profiles for a linearly polarized beam of light. The m coordinate determines the number of horizontal nodes (m+1) and the n coordinate gives the number of vertical nodes (n+1). Each node is π radians out of phase with its adjacent nodes. The white corresponds to the intensity of the emitted light. [9]

Equation 4

Determining the speed of light can be done very easily by using equation 1. By solving for L and multiplying both sides by n, we get that:

Equation 5

L_HeNe corresponds to the length of the gas chamber, L_fixed the length from the end of the gas chamber to the iris, and ΔL the length from the iris to the output coupler shown in Figure 5. In solving this equation for , the final solution can be found.

Equation 6

The slope of this equation yields a simple expression for calculating c.

Experimental Setup

Figure 6 shows a schematic drawing of the experimental apparatus. A HeNe laser with a main resonant frequency of 632.8nm and a Brewster window, 25 cm from back mirror to output of Brewster window will be used as the light source for the experiment. The mirror in the back of the laser cavity is 99.9% reflective with a radius of curvature of 60 cm. The Brewster window eliminates the circularly polarized portion of the beam without losing any intensity. The laser beam will pass through an iris 1 cm from the Brewster window. The iris is used to attenuate the output power of the laser. An output coupler mirror is placed on a stand 2 cm from the iris that is attached to a stepper motor allowing for the variable distance, ΔL, between the iris and the output coupler to be manipulated. Most of the light will be reflected by the output coupler mirror, 98.5% reflective with a radius of curvature of 45 cm, to excite additional photons in the lasing cavity, but the remaining portion of the beam will be transmitted onto a non-polarizing beam splitter which will divide the beam into two portions.

*Figure 6:* Schematic diagram of the experimental setup. A HeNe laser with a variable length cavity emits a beam which is split by a non-polarizing beam splitter. One half of the beam goes to a RF spectrum analyzer where the beat frequency is measured. The second half goes to a scanning Fabry-Perot which scans the laser gain function to reveal the modes of the laser allowing for corrections to be made for frequency pushing and pulling. (Figure adapted [6])

One portion of the beam will be incident on a scanning Fabry-Perot. The Fabry-Perot works by adjusting the distance between the two mirrors and measures the relative intensities of each of the frequencies. This signal is sent to the oscilloscope where the longitudinal modes can be visualized.

The second portion of the beam will be incident on a photodetector which will be read by an RF spectrum analyzer. The spectrum analyzer gives the difference between the mode frequencies inside of the laser. The maximum value on the spectrum analyzer gives the beat frequency because it is the difference between the two most intense mode frequencies inside of the cavity.

How to Align the Laser

Aligning the laser may seem like a daunting task, but after a few steps it can be done in less than an hour's time.

Step 1. Mount a test laser in a mirror mount. Place each iris one by one right up close the test laser and align their heights to the test laser.

Step 2. Place one iris a few cm away from the test laser and the other on the opposite side of the optical table where the real laser will be placed. Once this is done, adjust the pitch of the test laser until i goes through the center of both iris'. Now he test laser is aligned wih the table.

Step 3. Put the output coupler in the of the test laser, and adjust it until the test laser is centered on the output coupler.

Step 4. Now we can add the test laser into the setup. Put the rear of the laser cavity near the test laser, and use a glass slide to adjust the cavity so it is roughly centered on the test beam. If done correctly, you will see a greenish dot coming out of the brewster window. *hint: This is a good time to turn the lights off

Step 5. We now want to get that dot as close to the center of the output coupler as possible while keeping the back of the cavity centered to the test laser.

Step 6. Turn on the laser, if it is not lasing yet, only fine adjustments need to be done in order to do so. Once lasing, you usually want a gaussian beam which corresponds to a single dot, so i would recommend doing that before fiddling with the other modes.

Data Collection and Analysis

The cavity length is measured via the change in position on the stepper motor. The stepper motor completes one rotation for 800 cycles of a timing frequency. Using a Labview program to provide a specific number of cycles the length of the cavity can be changed in increments of 3.97 microns. The uncertainty of each measurement is given by half a step of the stepper motor. Data was taken every 100 steps to minimize the effect of the uncertainty in each distance measurement, and to not have any overlap in the beat frequencies and their uncertainties. To limit the effects of frequency pushing, all data was taken when the max value of a longitudinal mode was 100 mV. A fluctuation of 10% in power of this measurement will give an uncertainty of 9 kHz, which is easily seen by changing the power by 10% and seeing the change in beat frequency. Using the iris, we were able to maintain 100 mV within 5% limiting the uncertainty in frequency pushing to 4.5 kHz. Frequency pulling; on the other hand, require to make sure the longitudinal modes always have the same ratio of relative intensities between them. This is done by lightly touching the optical table the experiment is set up on. By doing this, the length of the lasing cavity is changed by tenths of a micron causing the relative intensities of the modes to change. This difference in length is taken into account for by the uncertainty due to frequency pulling. Data was kept when the ratio between the mode intensities was within 10% of 1 for a few seconds before and after the point was taken to account for any time delays between the oscilloscope and spectrum analyzer. The uncertainty in the beat frequency from frequency pulling in this experiment was 9 kHz.

A weighted least-squares regression of the cavity length and inverse beat frequency is taken to find the slope ΔL(1/∆f) which will allow the calculation of the speed of light through the air based on Equation 6. The fit is shown in figure 5. This fit yields the speed of light in vacuum to be 2.99758±0.00072 x 10⁸ m/s. This value is .474σ away from the accepted value of 2.99792 x 10⁸ m/s. Figure 6 shows the χ_i for this fit. As seen in this fit there is a sinusoidal phenomenon happening. The period of the sinusoid is the same as our stepper motors, so we know there these values are caused by the stepper motor.

Figure 7. Plot of Least-Squares it for 1/Beat Frequency vs. ΔL. The error bars are too small to be seen on the plot. Figure 8. Plot of χ_i values for he fit of 1/beat frequency vs. ΔL <a name="Discussion"></a>

Discussion

The systematic from the stepper motor limits the analysis that can be done on our data, and what is presented is a best estimate uncertainty based on the experimental uncertainties. The standard lead accuracy of the screw is better than .0006 in/in, the positional bi-directional repeatability is within 50 μin, and the Total Indicator Runout is .003 in/ft[10]. The first two values are much smaller than that being caused by our systematic, which has an uncertainty on the order of tens of microns. The runout, or straightness, of the screw is likely the reason that there is a systematic in our measurement. The screw used in the experiment was not made directly before the experiment and had been used multiple times beforehand which could have increased the runout of the screw more than the quoted manufacturer’s specifications. Further investigation of the runout could be done by adjusting the angle of the motor, or securing it much better to the track, in the case that the weight of the motor was torquing the screw causing the sinusoid.

Parameters could be added to fit the sinusoid to the data set, to eliminate it that way, but with time constraints that analysis could not be done. This would allow for a proper χi2, and would give the proper value of the uncertainty to quote on the final value of c. Fitting the sinusoid should not affect the quoted value of c though, so the value given should be the correct value. Another way would be to take data points on the every half period of the sinusoid, to limit the effects of the systematic. The availability of the equipment was limited at the end of the project, so this was not able to be done. This is most likely the best way of getting rid of the systematic as there is still plenty of lasing range in order to carry it out, and it ignores that there is a sinusoid in the measurements. 10

Additionally, the effects due to frequency pulling could be further limited using a piezoelectric feedback circuit. The circuit would be connected to the output coupler allowing for the table pushing effects to be done much more accurately. The circuit would take the output of the oscilloscope and adjust the length of the cavity until the intensities of each longitudinal mode are even. This would limit the effect of frequency pulling down to approximately 1 kHz.

With the best estimate of the uncertainty based on the experimental conditions, this experiment was just within precision to determine the difference between the speed of light through a vacuum and through air. The uncertainty required to do so is .0008 x108 m/s, and the uncertainty given experimentally is .00072 x 108 m/s.

References

3. Saleh, B. E. A. and Teich, M. C. (2001) Electromagnetic Optics, in Fundamentals of Photonics, John Wiley & Sons, Inc., New York

4. Dr. W. Luhs. MEOS GmbH D-79427 Eschbach - November 1999 / July 2003

5. Wil Offermans, Studio E Music Creation. http://www.forthecontemporaryflutist.com/etude/etude-02.html

6. Daniel J. D’Orazio, Mark J. Pearson, Justin T. Schultz, Daniel Sidor, Michael W. Best et al., “Measuring the speed of light using beating longitudinal modes in an open-cavity HeNe laser American Journal of Physics. Vol. 78, pg 524 (2010)

7. J. A. Stone and J. H. Zimmerman, NIST Metrology Engineering Toolbox, Index of Refraction of Air Vacuum Wavelength and Ambient Conditions Based on Modified Edlén Equation, emtoolbox.nist.gov/Wavelength/Edlen.asp.

8. Åsa M. Lindberg, “Mode frequency pulling in He–Ne lasers”. American Journal of Physics. Vol. 67, No. 4, pg 350 (April 1999)

9. Mellish, Bob. "Transverse Mode." Wikipedia. Wikimedia Foundation, 04 Feb. 2013. Web. 13 May 2013. .

Automation

The point of this section is to answer the question "Can this experiment be automated?", as in can we read the beat frequency values from the spectrum analyzer onto the computer consistantly with minimizing the uncertainties due to frequency pushing and pulling. The answer is yes, but I won't delve into the exact values and diagrams of the programs because that could be the subject of a future project.

Getting the beat frequency from the spectrum analyzer to the computer is pretty trivial, Labview has a program that does just that, but the trick is to zoom in enough on the SA for the beat measurement to be sufficiently accurate. All you have to do is create a loop that decreases the span of the screen at each incriment, this will ensure that the beat frequency stays in range the entire time. I would recommend making this a seperate VI, and then add it to the stepper motor vi so that it starts after the motor finishes moving the stage.

Now onto the difficult part of automation. As we know, there is only a beat frequency on the SA when there are two or more longitudinal modes on the scope. So we need to figre out a way to maintain the modes while attempting to keep them as even as possible. From the paper above, it is noted that we pushed down on the table to move the maintain both modes, and I used a similar process to accomplish this with a computer program. I used an electromagnet to pull down on the table, like a finger would, in order to maintain both modes. You'll want to send the signal from the Fabry-Perot into the computer using a DAQ board. Then you will want to find the peaks amplitude and position. This would arguably be the hardest part of the programming if LabView did'nt already have a peak finder. The FP signal is sent into the peak finding VI, and it gives the amplitudes and locations of the peaks, in all honesty the positions don't actually matter as long as you know which one is first. Then we take the amplitudes and find difference to sum ratio which gives an absolute ratio between -1 and 1 which is very important for this next part. The ratio will be sent into a PID controller to control the electromagnet.

All you do is start up the stabilizing program, and then the data taking program and sit back and relax, and maybe you could have data that ends up similar to this.

Value for speed of light: 2.9963 ± 0.0018 x 10⁸ m/s

• • This is 5 parts in 10,000 away from accepted c.

  • the chi graph does give the proper period sinosoid, and is uniform as expected.

  • The value of c given was when the reduced chi² was 1, it is .9 sigma away from the accepted value.