Beat Frequencies and Hermite-Gaussian Modes of an Open Cavity HeNe Laser
There are four main components of a He-Ne laser: a gas tube, a high voltage source, and two reflective surfaces (a mirror and output coupler). A diagram is shown in The figure below [1]. The mirror at the rear (high reflector) should have a high reflectivity to preserve the power of the laser and the mirror at the front (output coupler) should have a lower reflectivity so the beam is passed out of the laser. An open cavity He-Ne laser has the same components as the He-Ne laser previously described with one important difference. The output coupler of an open cavity laser is not attached to the gain tube, which is comprised of the gas tube, power source, and rear mirror. Instead of an attached output coupler, the light is passed out of a Brewster window which only transmits, and in order to produce lasing, an exterior output coupler must be manually aligned with the gain tube. The external output coupler allows the user to adjust cavity length and manually align the output coupler with the gain tube allowing observations of the various Hermite-Gaussian modes and their corresponding beat frequencies.
Introduction
Understanding the spatial and temporal properties of lasers is particularly important in their application in modern technology. Beat frequencies in particular are crucial in the development of the ring laser gyroscope in which two counter propagating lasers are produced on the same path and the beat frequency produced allows the rotation of the object to be determined. This allows for the development of navigational equipment which does not suffer from mechanical loss like a standard gyroscope would.
Theory
In order to function, the photons in a laser must reflect back and forth a large number of times. This requires precise alignment of the rear mirror and output coupler. If flat mirrors are used, they must be exactly parallel, or the reflections will stray from the edges too rapidly. To this end, curved mirrors are used. Solving the lensmaker’s equation yields two regions determined by the radii of the mirrors, r1 and r2, and distance between them, L, in which the laser will function
For convenience, we transform this into a single stability equation which is more useful in the context of Hermite-Gaussian modes
In reality, gas lasers emit light over a range of frequencies not one identifying frequency. This effect comes from the random thermal motion of gas molecules in the active medium. The emitted light has non-uniform frequency due to the motion of the gas molecules relative to the stimulated emission. The frequencies produced are bound in a Maxwell-Boltzmann distribution and the spreading is called Doppler broadening. The effect is shown in the figure below [2].
Solving Maxwell’s equations for the laser given the boundary conditions set by the mirrors imposes the condition that the output spectrum is discrete rather than continuous. As the relative gain of these discrete wavelengths fall off as dictated by the distribution, a threshold emerges beneath which the gain is not high enough for the frequency to exist continuously in the laser and it will die off. The threshold arises from the reflectance of the output coupler. An output coupler with low reflectance lets more light out, decreasing the average number of reflections of any given photon between the rear mirror and output coupler, decreasing the intensity below a stable level in the Maxwell-Boltzmann distribution. The wavelengths can be directly determined by the length of the laser cavity. Inspecting only the simplest mode of the laser, TEM00, the allowable resonant wavelengths are given by
When a laser with multiple discrete wavelengths is projected onto a detector, then the photocurrent will generate a beat signal. This effect comes from measurement of the signal as an intensity. Intensity goes as the square of the magnetic field and when two signals with similar frequencies are added, the cross product is transformed into two signals which have frequencies equal to the sum and difference of the signal frequencies. Both the signal and sum frequencies are far too high for standard measurement techniques, so the surviving frequencies are the difference frequencies, or the beats. The beat frequency can be predicted as:
Equation 7 yields the fascinating result that the beat frequency only depends on the length of the cavity and not on which frequencies appear in the spectrum. With L ranging from 0.2 – 0.5 m, goes from 300 to 750 MHz. The TEM00 mode represents the solution of the Helmholtz equation as a Gaussian beam; however, this is not the only solution. The Hermite-Gaussian beam represents the full family of solutions to the Helmholtz equation given the boundary conditions imposed by the mirrors. A Hermite-Gaussian Beam has complex amplitude expressed by [3]:
Where Hl(u) is the Hermite polynomial of order l, Al,m is a normalizing coefficient, and W is the beam width. These higher order modes introduce additional beat frequencies to the spectrum. Notably, the spacing of the discrete frequencies within each mode, is the same for all Hermite-Gaussian modes, and as such they will produce the beat frequency from (7). However, the spacing between different modes differs. The beat frequency between two TEMlm modes operating on the same resonance n can be expressed as
Additionally, beats can arise from modes of both different order (l, m) and resonance n. These beat frequencies are given by
By slightly changing the alignment of the output coupler with the rear mirror, the boundary conditions of the system are changed, and one can generate higher Hermite-Gaussian transverse modes TEMlm which have transverse intensity distributions as seen in the figure below [4]
The laboratory set up is shown below:
The challenge then is identifying the mode beating which produced a given beat frequency. Single mode beating cannot be determined as any mode will produce the same frequency, so only by inspecting the transverse image can the mode order be determined. Additionally, from (12) we see that what only matters is the difference in the sum of mode orders leaving an infinite combination of modes which beating against each other would produce the frequency measured. We propose that by looking at the transverse image, a laser which is beating between two distinct modes would present an image that is the superposition of the two images.
Experimental Methods and Set-up
A spherical mirror with known transmission and curvature was used as an output coupler. The output coupler was fixed on a translation stage to allow adjustments of cavity length and measurement of the relative distance between cavity lengths. The translation stage was mounted on the table at various points within both stability regions described in (5) and (6) for data collection at various lengths. Fine adjustment of the output coupler’s orientation in the stand was used to vary the Hermite-Gaussian mode of the laser. The light was divided using a beam splitter and sent to a Scanning Fabre-Perrot Interferometer (SFPI) and a photodetector. The photodetector measures the laser intensity producing the beat signal described and a spectrum analyzer was used to measure the beat frequencies at each length and orientation. The SFPI functions by varying a small resonator cavity length and measuring resonance from a piezoelectric device. The known properties of a Fabre-Perrot resonator are used to produce a true frequency spectrum of the laser. However, the method does not allow for absolute frequency measurement, instead the free spectral range of the SFPI is known and the distance at which a signal repeats itself allows for a measurement of the spacing of the spectrum which gives rise to beat frequencies. The SFPI data was displayed on an oscilloscope and was used to verify the beat signals on the spectrum analyzer. A mirror was also placed and removed in the beam path to reflect the light through a lens and onto a surface to view the transverse image of the laser. The transverse image was also used to confirm the Hermite-Gaussian modes present by comparison to the known transverse images of pure modes to confirm the beat frequencies measured. The experimental setup is shown in the diagram below
Data
Data was collected in both regions of stability for the laser at a total of 15 relative cavity lengths due to the inability to know exactly where the rear mirror is within the gain tube. Data on one of the predicted beat frequency lines was used to determine the true cavity length by minimization the reduced chi-squared The plot below shows strong agreement of the beat frequencies measured with the prediction from Hermite-Gaussian modes. The reduced chi-squared of the full data set was found to be 0.797.
The SFPI was used to confirm beat frequencies appearing with limited success. Only in the simplest cases could frequencies be easily extracted as shown in the two figures below. Even when such frequencies can be measured, it is impossible to tell what modes they are produced by. All beating between single modes are identical and those in which the difference in the sum of l and m mode numbers is the same will produce the same beats.
Instead, we theorize that the way to determine what modes are involved is by viewing the transverse image of the laser. Single mode beating is then easily identifiable as we can immediately identify modes by their transverse images as shown in a previous figure. Beating between modes we predict will show a transverse image which is the superposition of the modes involved. Using ImageJ software we analyzed corresponding intensity profiles for single modes and images which showed beating between modes. Below, we shown a (0,0) mode and a (1,1) mode as well as a superposition of the two which had a beat frequency corresponding to beating between the two modes.
Conclusion
Cavity length and orientation of the output coupler with respect to the gain tube were varied to produce beat frequencies measured using a photodiode and spectrum analyzer. Data was collected for cavity lengths in both regions of stability and compared to those predicted by the Hermite-Gaussian modes. The beat frequencies measured fit the predicted model with a reduced chi-squared of 0.797. Transverse images of the laser were analyzed using ImageJ and data from a SFPI was used to confirm the beat frequencies appearing with limited success. It is our recommendation that future study is conducted to verify the appearance of the transverse image of a laser as a method to predict the beat frequencies occurring.
References
[1] “Helium-Neon Laser.” Wikipedia, 25 Nov. 2003, en.wikipedia.org/wiki/Helium–neon_laser. Accessed 26 Feb. 2019.
[2] 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.
[3] Henningsen, Jes. “Teaching Laser Physics by Experiments.” American Journal of Physics, vol. 79, 2011, pp. 85-93., doi:10.1119/1.3488984.
[4] B.E.A. Saleh, M.C. Teich, “Fundamentals of Photonics, Second”, Wiley, New York, 2012.
Note: MATLAB code used is attached below as well as raw data files