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.