A geometric algebra reformulation of geometric optics

Sugon, Quirino M.; McNamara, Daniel J.

doi:doi:10.1119/1.1621029

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American Journal of Physics -- January 2004 -- Volume 72, Issue 1, pp. 92

A geometric algebra reformulation of geometric optics

Quirino M. Sugon, Jr. and Daniel J. McNamara

Ateneo de Manila University, Loyola School of Science and Engineering, Physics Department, Loyola Heights, Quezon City, Philippines

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We present a tutorial on the Clifford (geometric) algebra Cl_3,0 and use it to reformulate the laws of geometric optics. This algebra is essentially a Pauli algebra, with the Pauli sigma matrices interpreted as unit rays or vectors. In this algebra, the exponentials of imaginary vectors act as vector rotation operators. This property lets us rewrite the laws of reflection and refraction of light in geometric optics in exponential form. The reformulated laws allow easy translation of symbols to words and to diagrams. They also are shown to be equivalent to standard vector formulations. These coordinate-free laws can be shown to simplify the analysis of geometric optics problems such as the tracing of meridional and skew rays in lenses and optical fibers.© 2004 American Association of Physics Teachers.