Properties of Helium Neon Lasers

Josh Stein and Stephanie Rich

School of Physics and Astronomy University of Minnesota - Twin Cities

Minneapolis, MN 55455

May 15, 2013

Abstract

We construct a Helium Neon (HeNe) laser to determine the relationship between mode beating and the cavity length of the laser. A coherent beam of light is emitted by the laser when standing waves, or longitudinal modes, are produced within the cavity. Usually multiple longitudinal modes oscillate simultaneously at varying frequencies. We intend to relate the difference between adjacent frequencies, the beat frequencies, to the spatial properties of the laser beam and the length of the lasing cavity.

Introduction

Since their development in the 1950s, lasers have been used in numerous fields, including medicine, industry, scientific experiments, and consumer goods. The HeNe laser was the first continuous-wave (CW) laser, meaning the power output of the laser is continuous over time [1]. Since then, HeNe lasers have become the most commonly used lasers for optics experiments as they have a relatively low cost and produce unparalleled beam geometry [2]. This beam geometry allows for visual confirmation of electromagnetic properties, as the shape of the beam is given by the solution to Maxwell's equations. This correlation will be tested by relating the beat frequencies for each beam profile, or transverse mode, to the length of the lasing cavity.

A HeNe laser functions by the stimulated emission of photons produced by the electrical pumping of helium and neon. Electrical pumping produces a population inversion by means of stimulating the gain medium, a HeNe gas, with electrodes on either end of the lasing cavity producing a large DC voltage. When energetic electrons move between the electrodes and collide with the helium, the helium atoms become excited. These atoms collide with the neon atoms with enough energy to push the neon atoms into a higher, stable, state which causes a population inversion. When a photon with proper energy collides with one of these neon atoms, the atom emits an additional photon of this same energy and momentum. This collision and emission process causes a cascade of photons from the neon population that all occur at the same phase angle and frequency. A standing wave is produced by the cascading photons if mirrors at an appropriate distance apart are placed at either end of the gain medium in the lasing cavity [3]. With one 98.5% reflecting mirror that allows a portion of incident photons to pass through it and one highly reflective mirror, a coherent beam of light is emitted – this is diagrammed in Figure (1).

When the 98.5% reflecting mirror, the output coupler, is perfectly aligned with the lasing cavity, the laser produces a Gaussian profile beam. This beam is called the fundamental transverse mode, or , as these modes oscillate normal, or transverse, to the propagation of the beam . These modes can be described by solving the wave equation for electromagnetic waves for various forms of symmetry. When the laser cavity is not radially symmetric, the profile of the beam becomes patterns of lobes arranged in rows and columns [4]. The beam itself is formed by the longitudinal modes oscillating in the direction of beam propagation. Because the laser cavity is significantly larger than the wavelengths of these modes, their frequencies are too fast to be measured directly. However, the difference between the frequencies can be measured, with this frequency known as the beat frequency. Beat frequencies are also observed between higher order transverse modes and

and the standing wave requirement of it is trivial to show that the standing wave frequency for a laser is as follows [3].

Where f is the standing wave frequency, c is the speed of light in vacuum, d is the length of the resonating chamber, and q is an integer. Since the standing wave frequencies are too large to be measured directly, it is the beat frequencies that we are interested in. The beat frequency is easily calculable by taking the difference between two adjacent modes, i.e. q and q+1 [3].

denotes the beat frequency for longitudinal modes and . This is also known as the free spectral range or . Transverse modes above the fundamental mode are found by solving the wave equation for electromagnetic waves, a result of Maxwell's equations.

The solution for U(r,t) gives the spatial and temporal properties of the wave and is the Laplacian [4]. Again, c is the speed of light in vacuum. The spatial properties of a laser beam are determined by the boundary conditions imposed on it by the shape of the laser cavity. In a cavity which is not radially symmetric, equation (3) is solved in terms of rectangular coordinates x, y and z. This solution is comprised of Hermite polynomials of indices m and n. For the purposes of our experiment, the solution and the Hermite polynomials are unimportant. We are only concerned with the polynomial indices. The spacing between an arbitrary transverse mode,

, and is given by J. Henningsen as

In equation (4) m and n are the indices of the transverse mode, denoted as , and represent the radius of curvature of each mirror surrounding the resonating cavity [3]. The cavity length is shown again as d . The arctangent term is known as the Gouy phase and indicates that the beam picks up a phase shift of

as it passes through a focus, the spherical output coupler [4]. This shows that the free spectral range is scaled by the sum of a mode's indices and a phase shift depending on the cavity length. For high order modes, the free spectral range is greatly increased by the m+n multiple. Since longitudinal modes oscillate along with transverse modes,

must be modified to account for this translation of beat frequency as follows

Where k = -3, -2, …, 0, 1, 2 which account for the offset of transverse modes introduced by additional and longitudinal modes. The modes are described by lobes of light arranged in m+1 columns by n+1 rows. This is diagrammed below in figure (2). The particular transverse mode that the beam is in at any given time is independent of its frequency and determined only by boundary conditions. This means that a given transverse mode may beat against multiple adjacent frequencies of the same mode.

Apparatus

In order to measure the beat frequency of the laser output against the cavity length, we utilize a ThorLabs DET10A photodetector wired to Rigol DSA815 Spectrum Analyzer to measure the beats, and we have the output coupler mounted on a track controlled by a stepper motor so the cavity length can be precisely and minutely adjusted. The particular stepper motor we use travels 1/8th of an inch for every 800 steps giving us high accuracy in our cavity length measurements. The laser head for this experiment is a Melles Griot LHB-568 one Brewster window laser head. This head has a fixed highly reflective mirror with radius of curvature

= 60 cm and a reflection greater than 99%. The output coupler we used has a radius of curvature

= 45 cm and 98.5% reflection. A mockup of our apparatus is shown in figure (3). As can be seen in equation (4), as the cavity length, d, approaches the radius of curvature of the output coupler, , the Gouy phase approaches

resulting in the equation

So that all modes between and modes for which the difference between their m+n value is a multiple of two will meet at this point [3]. Additionally, when d equals , fbeat is no longer defined and the laser cannot produce a beam. Because of this, we measured ten frequencies for seven different transverse modes between the terminal of the laser head's Brewster window and 45 centimeters, the radius of curvature of the output coupler where lasing ceases to be possible.

To record data, we designed a LabView program to power the stepper motor, programming it so that the total path, from Brewster window to 45 centimeters, was equally divided into nine sections. By tilting the output coupler, and therefore altering the boundary conditions imposed on the beam, different TEM modes could be selected. To determine which mode we had the laser in, we utilized a beam splitter so that we got two identical beams, one which was measured by the photodetector, and the other reflected off onto a screen so that it could be compared with figure (2) to determine the mode indices. In a cavity of length 30-75 cm, ours being 31.7 cm from mirror 1 to the end of the Brewster window, three or four modes usually oscillate simultaneously. This was seen in our measurements as we would select some mode and observe three or four oscillating frequencies on the spectrum analyzer. As mentioned previously, the individual mode frequencies are two fast to be measured directly so what we observed on the analyzer showed beats between the adjacent modes. These frequencies were recorded for each of the ten positions set by the stepper motor for each of the seven modes we measured. Since the stepper motor traveled 1/8th inch for every 800 steps, we were able to precisely measure the length of the cavity by adding the number of steps per movement multiplied by 0.3175/800 centimeters per step, the 31.7 cm cavity length and a measured 1.6 millimeter offset from the edge of the Brewster window. The value k in equation (4) was taken as a fit parameter because it was not apparent from the data displayed on the spectrum analyzer. The additional

term represented adjacent modes or longitudinal modes, both of which were present in nearly every transverse mode, but could not be directly detected by eye or photodetector. By adjusting k we were able to observe the offset produced by unseen longitudinal modes.

Analysis

In order to compare our measurements with theory, we were able to plug in relevant parameters to equation (4), , and then match the theoretical curves to our data by varying the value of k in equation (5). Additionally, the length of the cavity, 31.7 centimeters, wasn't known to high accuracy and was added to the stepper motor distance values as another fit parameter. The laser head is housed in an opaque enclosure so its length could only be approximately measured from the terminus of the Brewster window to the edge of the enclosure. However, mirror 1 is recessed within the enclosure so the approximate measurement had to be adjusted based on our beat values.

The values for and were well known as they were provided by the manufacturer. This made it easy to create a matlab function of equation (5) with independent variables of cavity length d, mode indices m and n, and constant k. The mode indices were well known parameters as they could be visually seen and must be integers. This meant only d and k had to be varied to obtain a good fit. It was easily determined where the 45 cm cavity length was located as the laser could no longer produce a beam at this point, meaning we only had to determine the initial position cavity length. This was accomplished by setting a value for d in matlab that swept small steps from 30 cm to 45 cm and matching it with the

data as that is simply the free spectral range. With this method we determined the zero step position to have a cavity length of 31.7 cm. Each position of the output coupler was an integer number of steps so the cavity length at any given position was determined by adding the initial cavity length to the number of steps multiplied by .3175/800 centimeters per step. With the initial length of the cavity determined, the rest of the data sets were fit by using different values of k to match theory with our results. We found k values to be between negative 1 and positive one. However, for other modes which we were unable to measure, k ranges from negative 3 to positive 2.

The uncertainty in our beat measurements was due to small fluctuations on the spectrum analyzer and noise introduced by the larger intensities of the higher order modes. As the output coupler moved farther from the laser head, goodness of fit decreased. This is primarily due to the light emitted from the head spreading more before reaching the output coupler, causing incoherent light to reach the the photodetector, producing noise. For higher order modes, the fluctuations seen on the spectrum analyzer were several times larger than the fluctuations on low order modes. This caused the fits for high order modes to deviate farther from the theoretical curve.

Results

Despite excess noise from incoherent light and high intensity modes, we found high agreement between data and theory. Reduce chi squared values for each data set ranged from .6 to 3.6 and in our highest order mode with the greatest fluctuations a reduce chi squared value of 9.7 is found. However this high order mode matches the shape of the curve for most of the data suggesting a systematic error. This error was most likely due to the complexity of the

mode. It was a difficult mode to obtain and maintain. This difficulty and complexity led to a larger amount of noise, as well as more adjacent modes, meaning that for some data points, the true frequency may have been displaced or washed out by more dominant beats. The plot of our data with the theoretical curves for each mode are displayed in figure (4).

The agreement between measurement and theory is evident not only in the proximity of each datum to its respective curve but also in the behavior of each set as the cavity length approaches 45 cm, the radius of curvature of the output coupler. As stated in the apparatus section, modes for which the sum of their indices differ by multiples of two, do in fact meet.

, , and all meet at approximately 350 MHz while , , and

meet at roughly 175 MHz just as theory suggests. Additionally, ceases to lase at 45 cm just as all other modes do since theory tells us that beat frequency is undefined at this length. These results provide a visual confirmation of Maxwell's equations as equations (4) and (5) are derived directly from the wave equation for electromagnetic waves.

The chi squared values for each set of data varied as higher order modes produced more noise. The particular mode that we measured frequencies for, despite knowing the visible mode that the beam was set in, were not always known because multiple modes would oscillate at once. This property of mode beating is probably cause for several of our larger errors, especially because the largest errors occur in high order modes. A table of the reduced chi squared values is shown in figure 5. As previously mentioned, the best agreement was for lower order modes, demonstrated by the reduced chi squared values. The best agreement was found for

whereas the mode with the lowest agreement is .

Conclusion

The greatest challenge with this experiment was the time constraints. Despite being a straightforward apparatus and measurement scheme, we were unable to take measurements until the last three weeks of the semester due to complications with a previously attempted experimental setup. In the future, we would choose a more feasible experiment in order to take more data and improve the process of data taking. Had we started this experiment earlier, we would have been able modify and optimize the apparatus, as well as simply taking more data and measuring the beats for each mode several times. Additionally, more information can be taken from our data. Namely the speed of light in vacuum, c, can be found because the free spectral range, which has c/2 as a constant, is very prominent in the beat frequency equation. Data for higher order modes could be found as well as Laguerre-Gaussian modes, which are the solution to the wave equation of a radially symmetric resonating chamber. Having shown the agreement of theory with measured data for the relationship between beat frequencies and cavity length for a 45 cm output coupler, flat mirrors or mirrors with smaller or larger radii of curvature could be used in the future to further test the theory.

References

[1] Dr. Rudiger Paschotta, “Continuous-Wave Operation,” RP Photonics Encyclopedia,

[2] Dr. W. Luhs, Experiment 06: Helium Neon Laser, MEOS GmbH D-79427 Eschbach (2003)

[3] Jes Henningsen, “Teaching Laser Physics By Experiments,” Am. J. Phys. 79, 85 (2011)

[4] H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966)

[5] DrBob at en.wikipedia

[6] Original figures by Josh Stein and Stephanie Rich

[7] Original Figures by Josh Stein

Acknowledgments

Special thanks to our advisor, Mr. Kurt Wick for his help and guidance, Professor Paul Crowell who provided invaluable optics information, Dr. Sam Goldwasser who provided the laser heads and answered many questions, and Will Chick and Matt Klein who allowed us use of their equipment.