Lasers are often described as focusing light of a single wavelength to a single point. The light emitted is therefore spatially and temporally coherent, because the electromagnetic waves are all in phase and all transverse positions the emissions have non-varying phase relationships.
In free space, light can exist at any wavelength and frequency. In a laser, however, light is bounced off two opposite mirrors and creates a standing wave in the longitudinal direction. This largely restricts the wavelengths of the light because an integer number of half-wavelengths of the light must fit in the space between the mirrors. All lasers can be described by the number of nodes, n, m, and q, in the x, y, and z directions, respectively.
Figure 1. Two different modes of a laser. The transverse mode is described by the integers n and m while the longitudinal mode is described by q. n, m, and q all describe the number of wavelengths of light contained within the laser cavity.
It is thus possible to describe the length of the laser cavity by L=qλ/2, where q is an integer and λ is the wavelength of emitted light. Since the wavelength is restricted by the boundary conditions, the frequency of emitted light is as well. The wavelength and frequency of electromagnetic waves are related by c=λf, so f=qc/2L, where q is an integer, c is the speed of light, and L is the length of the cavity.
n and m describe the pattern of waving in the transverse direction. There are two classes of transverse modes: Gauss-Hermite modes and Gauss-Laguerre modes. Gauss-Hermite modes have a very grid-like structure, and are described by the equation
where Hn and Hm are Hermite polynomials. The simplest form is when the Hermite polynomials are both equal to one; this yields
This pattern is a simple dot! The transverse intensity is a gaussian, and for this reason the pattern is called the Gauss-Hermite mode. Gauss-Laguerre modes use Laguerre polynomials in the place of Hermite polynomials. Laguerre polynomials are described in polar coordinates, using r and θ to describe position. The patterns therefore have circular symmetry, and look overall circular.
After aligning the commercial laser, I observed the patterns reflected on the wall of light emitted from the laser, passing through a diverging lens and then through a Fabry-Perot interferometer. The interferometer was composed of two lenses whose angle and separation distance I could change. I noticed a pattern of circular rings that seemed to converge or diverge when I changed the separation distance. As I kept increasing the separation distance, the outer rings got closer and closer together: this is because the number of half-wavelengths of light that fit in the cavity kept increasing.
I then measured this pattern using a Spectrum Analyzer connected to an oscilloscope. One of the inputs of the oscilloscope measured the voltage applied to the analyzer, and the other input displayed the detection of light. The voltage applied to the analyzer had a deep well pattern. The other input mimicked the wells, but also had clear areas of peaks in between the walls of each well.
Each of these areas was made up of three little peaks. The middle one was the highest, and the adjacent ones had the same amplitude, although it was slightly lower than that of the middle. The number of peaks corresponded with the number of nodes in the transverse plane. Since I saw three peaks, there were two nodes in the cavity.
The distance between large peaks depended on the size of the crystal within the analyzer, which was a known value, and the distance between the small peaks depended on the transverse size of the laser cavity. They were thus polarized in opposite directions, and I attempted to use a polarizer to see this phenomenon. I used the oscilloscope to measure the distances between the peaks, and was then able to use the wavelength of the emitted light to calculate the size of the laser cavity. I did this for the commercial laser as well as the open-cavity laser, and found widths of 2.4 cm and 0.35 cm, respectively. When I measured the width of each laser, these numbers were very close to the actual dimensions.
If we didn’t know the length of the analyzer crystal, we could calculate it using measurable quantities in the lab. I initially measured the width of the laser chamber using Δf1=Δf2Δt1/Δt2, where Δf1 was the frequency along the width of the chamber, Δt1 was the time between the smaller peaks as measured by the oscilloscope, Δt2 is the time between the larger peaks on the scope, and Δf2 is a property of the analyzer. Since we also know that L1=c/2Δf1, where L1 is the length of the cavity, we find that Δf2=cΔt2/2L1Δt1. Inserting my data into this equation, I found the analyzer to have a free spectral range of 7.56 GHz with an uncertainty of .52 GHz. This is easily within error of the actual value of 7.4 GHz.
I then placed a camera so that it could record the patterns created by the open-cavity laser. By making small adjustments on the variable mirror, I could see a wide variety of patterns being recorded by the camera. Some of these patterns had a very linear geometry while others had a circular geometry more likely associated with the Gauss-Laguerre models discussed previously. Some of these images are shown below.
Figure 2. These are just some of the patterns I was able to see from the open-cavity laser. A camera connected to a computer recorded these images.
I then set up an image of two adjacent spots, and measured how their separation changed depending on the distance from the output of the laser to the camera. The closest and farthest images are compared in Figure 3.
Figure 3. The left image is at a distance of 21 cm from the edge of the laser and the image to the right is at a distance of 127 cm from the laser. It is clear that the separation between the two spots increases with distance.
In an ideal laser, the separation between these two spots should remain constant, because the laser beam should not vary in diameter. However, most lasers have a slight divergence. By graphing separation of spots vs distance between laser and sensor, I calculated the divergence of my open-cavity laser. The relation between these two variables was clearly linear, and I fit a line to the graph. The graph allowed me to extrapolate the separation of the two spots at any distance from the mirror. I then measured the reflection of the spots against two walls in order to convert my equation to units of distance rather than of distances in the computer program. I could set up a triangle with one leg as the distance between two points and the other leg their differences in separation. As shown in Figure 4, the angle facing the differences in separation is related to the divergence of the laser. I calculated this divergence to be 0.28 degrees.
Figure 4. By measuring the radius of the laser beam at two distances, we can calculate the divergence angle. If we use two points 2 meters apart, we find that tana=(y2-y1)/1. In this way we can calculate the angle of divergence θ=2a.
Inside the laser cavity, there is a location z0 when the transverse waves are all aligned in a plane. This location is the beam waist, and depends on the curvatures of the two mirrors within the laser. Theoretically, a mirror focuses light onto a specific point, called the focus. In reality, this focus is not precisely a point, but has a clearly defined width: this is the beam waist. The diverging rays from the laser also converge to this point, so I extrapolated my equation to find where it would theoretically hit the x-axis: this occurred at x=-1.77186 cm, so the beam waist of my laser was 1.77186 cm from the second mirror inside the laser.
My laser was made up of one concave mirror and one planar mirror, so light was focused to the beam waist only by the concave mirror. Thus the curvature of the mirror was equal to the distance between it and the focus. I found this curvature to have a radius of 8.22814 cm.
I never really knew how lasers work, so this experiment was very interesting for me. I learned a lot about how theoretical lasers differ from experimental lasers, and how some experimental phenomena are dependent on the specific properties of each laser. The theory was sometimes difficult for me to grasp, but by the end of the experiment I had a much better understanding of the physics of lasers.