ABSTRACT for the April 2009 Celebration of Research and Creative Activity Creating Higher-order Mode Laser Beams with an Open Cavity Laser Justin Tian, Marty Cohen and John Noé Laser Teaching Center Department of Physics & Astronomy Stony Brook University Laser beams can only exist in certain specific spatial patterns called modes, which are the discrete solutions of the paraxial wave equation that describes light moving at small angles to some axis. Most lasers are designed to produce only the zero-order or "fundamental" mode. It has cylindrical symmetry and a cross-sectional intensity profile I(r) described by the Gaussian ("bell curve") function exp(-2r^2/w^2). Higher order modes with more complex patterns are also possible, as are superpositions of higher order modes. In the higher order modes of the Hermite-Gaussian (HG) type the Gaussian function of the fundamental mode is multiplied by two polynomial functions that depend on x only and y only, respectively - Hm(x) and Hn(y). These functions are labelled by two integers m and n, which correspond to the degree of the polynomial. A mode with order m (or n) has m (n) nodes or dark bands along the x (y) - direction. It can easily be seen that HG modes have rectangular symmetry. A second complete family of higher-order modes, the Laguerre Gaussian (LG) modes labelled by l and p, is the product of functions of the polar coordinates r and phi and has circular symmetry. Since the phase of an LG mode varies as exp(ilphi), for l > 0 there is a singularity (dark spot) in the center of the beam. These singular LG modes have a twisted wavefront and are often referred to as optical vortices. The first goal of this project is to learn how to create stable and good quality higher-order HG modes with an open-cavity helium neon (HeNe) laser. (In such a laser the low-pressure gas mixture is separated from the atmosphere by a Brewster window and there is an open space between this window and the output coupler mirror which allows objects to be placed in the intracavity beam of the laser.) The method used so far with our laser is to insert a human hair (diameter about 75 microns) into the exposed laser beam. While this method does work, the results are unpredictable and unreliable. We have recently obtained more stable higher-order modes by inserting a glass microscope slide with fine scratches into the cavity. The slide is tilted at Brewster's angle to suppress reflections, and the scratches (created by wiping the glass with sandpaper) are much smaller than the width of a hair, 10 microns or less. We plan to try very fine wires next and to improve the positioning mechanism. The relative phase between adjacent "spots" of an HG mode pattern is pi radians (180 degrees). Recently we have been able to demonstrate this experimentally using a Mach-Zehnder interferometer assembled for this purpose. One arm of the interferometer has a beam expander which creates a reference plane wave. As the interference fringes cross the two spots of an HG(1,0) mode they shift by one-half the fringe separation, as expected. Currently we are setting up a "mode converter" in the interferometer arm that doesn't contain the beam expander, to transform HG modes into LG modes. The interference pattern for a singular LG mode (l > 0) should contain a "fork" or spirals, depending on adjustments of the interferometer.