Modeling a Systematic Procedure for Laser Alignment Daniel Minkin, Portledge School, Locust Valley, NY; Simone Agha, Herricks High School, New Hyde Park, NY; Harold Metcalf and John Noé, Laser Teaching Center, Department of Physics and Astronomy, Stony Brook University. Beam alignment is a constant concern in optics. It is the often daunting task of steering a laser beam so that it passes exactly through two arbitrary points in space, or equivalently, is precisely collinear with another laser beam. This "walking the beam" technique is used, for example, when aligning an interferometer or coupling light into an optical fiber. In our project, an iterative process of aligning a beam was investigated conceptually and mathematically. The process uses reflections from two mirrors to change the horizontal and vertical position and angle of the beam. Each mirror is attached to a precision mount with separate horizontal and vertical adjusting knobs; thus, there are four separate parameters to optimize. The process consists of two similar steps which, when repeated in the correct order, are guaranteed to bring the beam into alignment. Step 1 is to use the first or "position" mirror to center the beam on the first point in space; step 2 uses the second or "angle" mirror to center the beam on the second point. The horizontal and vertical alignment steps are independent of each other; our analysis considers just one plane. We derived equations that describe the resulting offset distances from the two points at each step in terms of the mirror angles. These equations were then used to numerically model the process in Quattro Pro, a spreadsheet program. By manipulating the starting values of the equations, we were able to change the rate of convergence. The fastest convergence is achieved when the angle mirror is relatively close to the first pinhole. Plots that we created make this visually apparent and are useful for teaching the correct procedure. We then considered an alternative procedure which requires "overshooting" the desired target point. While potentially faster, it requires considerable intuition and expertise to ensure that the process will not diverge. Once again, our plots made it visually clear how this alternative beam walking procedure can be more efficient, but also less reliable. We would like to thank the Simons Foundation for funding this research.