# Separation of Gaussian and Laguerre-Gaussian Laser Modes

For my research project, I will answer the question “Can we design an optical element that physically separates the different modes of Gaussian and Laguerre-Gaussian beams? My project deals with lasers, which are highly focused beams of monochromatic electromagnetic radiation, usually in the form of light. Lasers emit radiation in a few different “modes.” These “modes” correspond to the beam spot’s overall shape and appearance. To analyze the behavior of light, we can shine lasers through optical elements, such as lenses and irises. Different optical elements allow us to manipulate the light differently, perhaps by diffracting it, refracting it, or modifying its properties.

I will focus on Gaussian and Laguerre-Gaussian beams, which are specific types of laser modes. Gaussian beams are characterized by spots whose intensities decrease with distance from the center according to a Gaussian function. Laguerre-Gaussian beams are a sub-category of Gaussian beams. They produce higher order modes, described by the values of p and l.  The p mode indicates the radius of the beam, and the l mode expresses the number of phase changes the beam experiences. When we shine a Laguerre-Gaussian beam through an optical element, we obtain an image of a circle, which allows us to observe these modes. (See image below). I will work with the range of Laguerre-Gaussian modes corresponding to l = 0, while p varies, i.e. the first column in the image below.

Once we focus a beam to a circle, we need to separate the different p and l modes fully in order to analyze the beams closely. To do so, we must use additional optical elements to convert the image of the beam spot from a circle into parallel lines that explicitly indicate these different modes. In the past, researchers have successfully separated a narrow range of modes.[1] In my research, I will build on these results. I will shine Laguerre-Gaussian beams through various optical elements to observe their effects on light behavior. With this new understanding, I plan to create an optical element that will separate more Laguerre-Gaussian modes with high resolution.

Ultimately, the separation of a broader range of modes of a Laguerre-Gaussian beam will profoundly affect the fields of image processing and compression. Speaking broadly, separating the modes of the beam corresponds to decomposing the constituent parts of an image file. For example, this separation will allow us to project the different layers of an image in different directions. This way, we will be able to analyze and manipulate the information an image contains. Ultimately, the goal is to compress an image as much as possible, to be able to store it most efficiently, without sacrificing its quality.

[1] Berkhout, Gregorius C. G., Lavery, Martin P. J, Courtial, Johannes, Beijersbergen, Marco W., & Padgett, Miles. J. (2010). Efficient Sorting of Orbital Angular Momentum States of Light. Physical Review Letters, 105 (October 8), doi: 10.1103/PhysRevLett.105.153601