Comparison between the differential and integral methods used to solve the grating problem in the H_|| case

The usual differential method for solving the grating problem in the H|| case is shown to be unable to predict the efficiencies of blazed gratings in a reliable manner. Its predictions are compared with those obtained using the integral method developed by Maystre, a reliable method that has shown its validity over a wide range of applications. The efficiencies of sinusoidal gratings as a function of angle of incidence are calculated by both methods for two values of the groove-height-to-period ratio. For 0.05 (low modulations) both formalisms yield similar results, but for 0.2 only a qualitative agreement is observed. The differential method is shown to involve an approximation valid only for low-modulated surfaces, a fact that accounts for the observed discrepancies. As a self-consistency test, the fulfillment of the electromagnetic boundary conditions is checked by calculating the jumps of the field components, which should be continuous at the grating surface.