S14MeasuringtheSpeedofLight

Measuring the Speed of Light in Air Utilizing Longituninal Modes in an Adjustable-Length, Open-Cavity, HeNe Laser

Matthew McSavaney & Tim Gburek

University of Minnesota

Methods of Experimental Physics Spring 2014

Abstract

This experiment used the properties of an adjustable length, open-cavity, HeNe laser to measure the speed of light in air. The speed of light in air was calculated by measuring the beat frequency between the laser's adjacent longitudinal modes. The calculated value was c(air) = (2.99696 ± 0.00086) ×10^8 m/s. This result is off from the accepted value of 2.997125×10^8 m/s by 0.19 sigma, and is accurate to 1 part in 4,000. The likelihood of getting a result deviating by 0.19 sigma or more from the accepted value is 85%.

Introduction

Lasers do not emit a continuous spectrum of light, but instead generate resonant modes of light as a function of the laser cavity’s length. For a resonant laser, there are an integer number N of half wavelengths between the laser cavity mirrors, corresponding to the allowed wavelengths and frequencies [1]:

Where n is the index of refraction of the medium though which the light travels, and L is the length of the laser cavity. The overall gain profile of the allowed frequency modes is Gaussian due to Doppler broadening within the laser gain medium, as shown in Figure 1.

Fig. 1. Adjacent longitudinal modes exist in the cavity allowing for the determination of the speed of light by measuring the frequency diference between such adjacent modes and plotting these measurements against cavity length [1].

When multiple waves of differing frequencies register on a photodetector, the alternating constructive and destructive interference causes the current generated from the photodetector to oscillate in intensity, in a manner described as beating. The beat frequency, or frequency difference between adjacent longitudinal modes, is defined as [1]:

Notice that beat frequency is proportional to the inverse of the length of the adjustable open cavity. A plot between the two will give a slope equal to one-half the speed of light in a medium of index of refraction n.

Theory

An electrical discharge of 700 to 1500 Volts is applied across the HeNe gas. Electrons from this discharge collide with the He and Ne. He is excited to the 2s0 state, and the Ne is excited to lower (non-laser line) excited states. Collisions between He and Ne raise Ne to the 3s2 state. This generates the desired population inversion [4]. When an electromagnetic field (photon) is applied to the excited electron, the electron will behave as a small electric dipole and will oscillate with the electromagnetic wave, releasing a clone photon as a result. This clone photon will have the same energy, momentum, phase, and spin as the incident photon. In the process of this emission, the excited electron will lose energy and will consequently drop to a lower allowed level to maintain conservation of energy. The electrons are continuously re-excited by the applied voltage, and the process repeats. In particular, the HeNe laser primarily emits at 632.8 nm because Ne has the largest probability of de-exciting from the 3s2 state to the 2p4 state, which are separated by an energy corresponding to 632.8 nm.

Fig. 2. HeNe excitation and lasing process [2].

Fig. 3. Stimulated emission energy level diagram [3].

In order to get the photons traveling in the same direction with the same properties mentioned above (i.e. to get the laser to “lase”), the photons’ path is constrained by bookending the cavity with optical couplers (mirrors). Careful alignment of these output couplers reflects the photons back through the gain medium, amplifying stimulated emission and creating resonant modes within the cavity. Those modes with a gain greater than unity are transmitted out of the laser.

Fig. 4 By continuously reflecting the photons within the medium, stimulated emission of the allowed modes frequencies cascades. Modes of photons with gain greater than unity will be transmitted, creating a coherent beam.

Experimental Setup

We used a Melles Griot 05-LHB-568 HeNe laser tube (632.8 nm) that contains a 0.6 m radius-of-curvature mirror on one side of the gas and a Brewster window on the other side. The Brewster window eliminates modes with polarization orthogonal to the Brewster plane, ensuring that the emitted modes have the same polarization. Without the Brewster window, the adjacent modes would be polarized orthogonal to each other and would produce a beat frequency twice what is expected.

Fig. 5. Laser cavity, translation stage, FPI, Photodetector.

Fig. 6. Close up of laser cavity.

Fig. 7. Close up of FPI.

The output coupler was mounted on a translation stage controlled by a Vexta, 2-Phase, Model PK266M-E2.0A stepper motor that precisely moved the coupler forward or backward, effectively shortening or lengthening the cavity. The motor was controlled with a program coded in LabVIEW. The iris located between the Brewster window and the output coupler was used to manually attenuate the overall amplitude of the laser gain profile (as monitored on the oscilloscope), which minimized the effects of frequency pushing.

Fig. 8. Experimental schematic of laser caviy [1].

This iris internal to the laser cavity also attenuated gain in the region away from the laser’s optical axis, thus helping to ensure the beam emitted in the fundamental transverse electromagnetic mode (TEM₀₀), seen in Figure 9. Other TEM modes create superfluous beat frequencies. The beam was split by a non-polarizing 50:50 beam splitter into a fast photodetector (used to obtain the beat frequency off the RF spectrum analyzer) and a Fabry-Perot Interferometer (FPI) with a Free Spectral Range (FSR) of 7.5 GHz and a finesse of 200. The FSR must be roughly equivalent to, or greater than, the 1.5 GHz gain bandwidth of the HeNe laser in order to view the mode structure. Finesse is the ratio of the FSR to the resolution (37.5 MHz) of the interferometer. The output of the FPI allowed us to monitor the laser gain profile and mode structure in order to minimize systematic errors introduced by frequency pulling and pushing.

Fig. 9. Transvers electric modes patterns. The TEM00 mode has a Gaussian light intensity profile as seen in the upper left corner [5].

Conditions for generating stable Gaussian beams are a function of the radius of curvature of each mirror and the length of the cavity [4]. By convention in the literature, each mirror has a dimensionless “g-factor”:

where L is the distance between each mirror with radius of curvature R which must satisfy the stability condition [4]:

The shaded regions of the laser stability diagram, Figure 10, are the combinations of g factors that allow lasing for two concave mirrors.

Fig. 10. The two branches of the rectangular hyperbola bound stable two-mirror resonators (shaded region) [4].

Methods

Before taking data you will need to calibrate your measurements:

1. Choose a "zero point" for the length measurementson the translation stage. If there are bad sections of the translation stage, this step will help you keep track of, and avoid them, in later data collection runs.
2. Carefully align the output couplers such that the beam remains in fundemental Gaussian mode over the desired data collection span. This step will require multiple alignment attempts.
3. Choose laser gain profile amplitude (by adjusting the iris internal to the cavity), with your allowed tolerances for frequency pushing variance. This amplitude will naturally vary as you change the length of the laser cavity. Choose an amplitude and allowed tolerances that you can reproduce over the span of your data run.
4. Choose relative mode intensity tolerances for frequency pulling minimization.

The absolute length of the laser cavity cannot be well determined due to uncertainty in the position of the back mirror internal to the laser tube. By dividing the length of the cavity into three components (Figure 8), we relegate the indices of refraction acting in each section to a constant offset.

Taking the derivative with respect to the inverse beat frequency eliminates the unknown terms, and the speed of light in air can be calculated using a weighted least squares regression:

You will need to consider uncertainty in both the length and beat frequency measurements to generate a least squares regression fit of the data. A full analysis will account for each dimension, but if you can demonstrate that one source of uncertainty dominates, a reasonable measure of the speed of light in air can be found by only considering errors in that particlar dimension. Errors in the length measurement are attributable to uncertainties in the following:

1. motor step angle (angle/step)
2. translation stage accuracy (distance/step)
3. lead screw pitch variations

the quality of the coupling of the stepper motor lead screw with the translation stageare ultimately limited by thermal expansion at the micron scale [6].

Errors in the beat frequency measurements are attributable to the effects of frequency variability caused by frequency pulling and pushing. Frequency pulling occurs when the adjacent longitudinal modes are emitted from the laser over slightly different cavity lengths (on the order of microns) and therefore each mode passes through slightly different amounts of each medium (and related index of refraction). This generates a quantifiable difference in their relative intensities such that the decomposition of the length equation (mentioned above) does not hold identically for both modes. Applying slight pressure to the optical table changes the cavity length on the order of microns, and a measurement of the beat frequency is taken when the relative intensities are within a chosen tolerance. Frequency pushing arises from variability in the gain curve amplitude as the field intensity in the laser cavity increases. As the gain curve amplitude increases, so does the beat frequency [1]. By keeping the length of the cavity constant, and varying amplitude of the laser (by adjusting the iris inside the cavity), we can correlate the gain curve amplitude tolerance, to a kHz uncertainty.

Fig. 11. Frequency pulling.

Fig. 12. Frequency pushing.

A Note on Translation Stages

The translation stage used was built by Micro-Controle (no model number listed), which was bought out by Newport in 1991 [7]. The specifications of the translation stage are unknown to our advisor and TA, and no documentation exists. The distance traveled per step was estimated by averaging the total distance traveled by the total number of steps. But averaging washes out the details of individual steps, some steps may move too far forward, and some steps may not move far enough. A periodic pattern in you chi plot may indicate that data was taken over a bad section of the translation stage.

Discussion

The calculated speed of light in air using a least squares regression of delta L vs. 1/(delta f) was c(air) = (2.99696 ± 0.00086) ×10^8 m/s. This result is off from the accepted value of 2.997125×10^8 m/s by 0.19 sigma, and is accurate to 1 part in 4,000. The likelihood of getting a result deviating by 0.19 sigma or more from the accepted value is 85%. In accounting for frequency pushing and pulling, it was found that a deviation of 3.85% from a set gain curve amplitude corresponded to an uncertainty in the beat frequency of 3.51 kHz due to frequency pushing. When considering the ratio of relative intensities of the adjacent longitudinal modes, it was found that a deviation of 8.75% from a ratio of 1 corresponded to an uncertainty in beat frequency of 1.19 kHz due to frequency pulling. This experiment provided an accurate means of determining of the speed of light in air and was an excellent introduction to the study of lasers, optics, and statistical analysis that expanded upon analytical and experimental techniques used in previous coursework.

Fig. 13. Fit of change in length versus inverse beat frequency, and associated chi plot. Anomalies in chi plot are due to taking data over an imperfect span of the translation stage.

Acknowledgments

Thanks to Kurt Wick, Greg Pawloski, Clem Pryke, and Matt Klein for their invaluable help, patience, and guidance.

References

[1] D’Orazio, D. et al. 2010. Measuring the speed of light using beating longitudinal modes in an open-cavity HeNe laser. Am. J. Phys. 78 (5)

[2] Goldwasser, Samuel M. “Helium-Neon Lasers.” Sam’s Laser FAQ. Web. 14 March 2014. <http://www.repairfaq.org/sam/laserhen.htm#hentoo0>.

[3] "Stimulated Emission." Wikipedia. Wikimedia Foundation, 16 Feb. 2014. Web. 10 Mar. 2014.

[4] Saleh, B.E.A. and M.C. Teich. Fundamentals of Photonics. John Wiley & Sons, Inc., 1991.

[5] "Transverse Mode." Wikipedia. Wikimedia Foundation, 22 Feb. 2014. Web. 9 Mar. 2014 <http://en.wikipedia.org/wiki/Transverse_mode>.

[6] "Motion Basics and Standards." Newport Corp., Web. 9 May 2014. <https://www.newport.com/Motion-Basics-and-Standards/140230/1033/content.aspx>

[7] "Company Overview." Company Overview. Web. 9 May 2014. <http://www.newport.com/cms/company/company-overview>

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