# Laguerre-Gaussian Modes: Post #1

Now that I’ve been on campus a week, I have a clearer idea of my project. My research advisor, Professor Novikova, and I modified the research proposal to reflect an exciting development in the lab. It turns out that the Quantum Optics Lab at William and Mary has embarked on a collaboration with its counterpart at Louisiana State University. LSU is currently applying machine learning techniques to optics. Machine learning is a branch of artificial intelligence. It involves algorithms that make decisions by learning patterns from data rather than by relying on specific instructions. Our collaborators at LSU are using machine learning to teach their computers to recognize and analyze laser modes. Right now, they need experimental data to test their algorithm’s accuracy. I get to collect this data. If successful, this algorithm will have applications in quantum computing and quantum information, especially with regards to data storage.

Our collaborators need images of six basic Laguerre-Gaussian laser modes. These modes correspond to p=0,1 and l=-1,0,1. As mentioned in the abstract, the “p” refers to the radial index, and the “l” to the azimuthal index of the beam. Therefore, p indicates the overall beam size, while l is the phase change the beam experiences. (It is known that Laguerre-Guassian modes carry orbital angular momentum, and the l index also represents this characteristic). The six basic LG modes we need are |-1,0>, |0,0>, |1,0>, |-1,1>, |0,1>, and |1,1>, where the first number in the ordered pair corresponds to l and the second to p. The modes look like this (what you see is the cross section of the incident beam – I will describe more about the setup in a few paragraphs):

Basic 6 LG Modes

Once we have these six basic modes, we can also superimpose them to create new modes. Our collaborators need all the possible combinations ofÂ two out of the six basic modes. The beams can be added equally, or their amplitudes can be altered so that one mode dominates according to a certain percentage. I will therefore collect combinations weighted as 50%-50%, 60%-40%, 70%-30%, 80%-20%, and 90%-10%. (The reversed percentages are also possible, but because of symmetry, they are unnecessary – See image below). There are fifteen possible unique superpositions, so in the end I will have 15 x 5 = 75 superposition images, plus the 6 original modes, for 81 images total. The superpositions will look like this, where the exterior row and column show the basic six modes:

Now that I’ve given some background information, I’d like to report on what I’ve actually been doing for the past five days. It’s been a lot of learning in such a short time! I spent the first two and a half days familiarizing myself with all the optical elements in the lab. My first task, a rite of passage for optics students, was to align a laser into an optical fiber. The process was very difficult since the instruments are so sensitive. Furthermore, our lasers are infrared, so we can’t actually see them! Instead, we have to use an IR viewer or intercept the beam with an IR card.

To align the laser, we have to follow many steps to capture the light and slowly shrink it to the size of the narrowest fiber. First, I set up mirrors to direct the beam through a tiny lens. Once I ensured the beam emerged from the lens completely, I attached it to a wide optical fiber. I used the IR viewer to see if any of the light emerged through the fiber. I only observed a tiny pinprick, so I adjusted the mirror angles to increase the brightness. Then, I had to confirm numerically that all the available light made it through. I measured the power of the incoming light and compared it to the power of the exiting light using a voltmeter (since voltage is proportional to power). Once the voltages matched, I replaced the wide fiber with the desired narrow one and repeated these adjustments. This entire process took over a day. Luckily, it was meant just to be a learning experience because somehow, one mirror got slightly shifted, which threw off the entire setup! I definitely got a good sense of how alignment works though, and we decided to move onto my main task.

To get some background knowledge on the LSU project, I attended a presentation by a grad student, Savannah, with whom I’d be working. She described recent research that used machine learning, specifically neural networks, to describe a open-quantum systems, including optical ones. These systems are notoriously cumbersome to model mathematically, but this research could reduce the amount of storage needed for this large data.

With this knowledge in mind, we returned to the lab. Savannah showed me how to take the images of the laser modes. We directed a laser to a spatial light modulator (SLM), which reflected the light back at a camera. The SLM modifies the beam and projects it onto a phase mask so that the properties corresponding to the different modes are visible on the camera. The SLM is connected to Matlab, so we can easily input the desired l and p values and their percentages, thus obtaining an image.

After getting the hang of the procedure, I was on my way to collecting the data. Soon, however, my images stopped matching the theoretical ones. It took us all of Thursday ad most of Friday tweaking the setup and the code before the images started looking authentic. I had to start over a few times, but it made for a helpful learning process. This weekend, Savannah will check if we need to modify the code further, but so far, I’m making progress. If all goes well, I will finish with the data collection next week. For now, here are some of the more exciting images I’ve taken:

|-1,0> combined with |0,0> at 50%-50%

|-1,0> combined with |1,0> at 50%-50%

|-1,0> combined with |1,1> at 50%-50%