The Rayleigh methods used in the theory of reflection by a grating have been investigated numerically. The errors made in the power balance and the integrated-square errors made in the fulfillment of the boundary conditions are considered. Only the integrated-square error is a sufficient check for the convergence of our numerical results. The validity of the so-called Rayleigh hypothesis has been confirmed numerically. The minimization of these integrated-square errors leads to a method of general validity; in this method the results are always convergent, and the errors made in the power balance are of the same order as the integrated-square errors in the boundary conditions.
© 1981 Optical Society of America
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