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 No�, 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.