A MATRIX ANALYSIS of the "MIRAGE" OPTICAL-ILLUSION TOY Sriya Adhya, John Noe and Harold Metcalf, Laser Teaching Center, Department of Physics & Astronomy, Stony Brook University One rewarding aspect of studying elementary optics is the many rich connections encountered between optics and diverse areas of mathematics. For example, matrix analysis can be used to describe light rays in the paraxial approximation; model optical resonators and laser beam propagation; and charaterize polarized light. The goal of this project is to analyze the "Mirage," a familiar optical-illusion toy, by creating mathematical models of light rays using matrix analysis. The analysis uses 2x1 vectors to represent the distance of light rays from the axis of symmetry, r, and their inclination angle theta. Optical elements (mirrors and open spaces between them) are represented by 2X2 square matrices, and the combined effect of several elements is found by simply multiplying together the corresponding matrices, in the correct order. The Mirage toy is composed of two identical horizontal concave mirrors, placed one on top of the other. The upper mirror has a large hole in the center allowing light rays to pass through. Light reflected from a small object standing at the center of the bottom mirror is reflected twice between the two concave mirrors before passing through the hole and forming the floating "real" image. The image appears to float in space and can be enlarged with a magnifying glass or reflected in a mirror. Additional, weaker, images can be observed by displacing the top mirror upwards. In order to apply matrix analysis we need to know the separation of the two mirrors and their focal length, which is related to their radius of curvature. We determined by careful measurements and by making templates that the mirrors have a parabolic shape that can be well approximated by a circle with radius R = 6.35 inches. The focal length is 1/2 this value, and the mirrors are exactly one focal length apart. With this information the matrix analysis can be used to predict the positions, sizes and orientations (reversed or not) of the main image and the additional weaker images that occur when the two mirrors are separated. This analysis and measurements related to the additional images are underway.