Numerical methods for solving two transcendental equations which appear in Fraunhofer diffraction

Abstract

The intensity distribution in a Fraunhofer single-slit diffraction pattern is described by transcendental equations of the form x=C sinx and x=C tanx. The first equation appears in the solution for the width of the central maximum of the pattern. The second results from an attempt to locate the positions of the secondary maxima. Students who are used to explicit (closed) solutions of equations are often distressed by the apparent need for trial-and-error methods. Graphical solutions give a good ’’feel’’ for the situation, but are not satisfying to those who would like greater precision. Numerical methods, which rely only on introductory calculus, can be easily applied. Algorithms employing pocket calculators are presented.