Diffraction Grating Efficiency Calculations

Several techniques have been developed to calculate grating efficiencies, most of which have two characteristics in common: they employ Maxwell’s equations whose boundary conditions are applied at the corrugated grating surface, and their difficulty in implementation varies in rough proportion to their accuracy. In this section only a brief mention of these techniques is provided – more details may be found in Petit, Maystre, and Loewen and Popov.

Grating efficiency calculations start with a description of the physical situation: an electromagnetic wave is incident upon a corrugated surface, the periodicity of which allows for a multiplicity of diffracted waves (each in a different direction, corresponding to a unique diffraction order as described in Diffraction Orders). Efficiency calculations seek to determine the distribution of the incident energy into each of the diffraction orders.

Scalar theories of grating efficiency lead to accurate results in certain cases, such as when the wavelength is much smaller than the groove spacing (d << λ); the vectorial nature of optical radiation (manifest in the property of polarization) is not taken into account in this formalism.

Vector or electromagnetic theories can be grouped into two categories. Differential methods start from the differential form of Maxwell’s equations for TE (P) and TM (S) polarization states, whereas integral methods start from the integral form of these equations. Each of these categories contains a number of methods, none of which is claimed to cover all circumstances.

Both differential and integral methods have been developed and studied extensively, and both have been implemented numerically and thoroughly tested against a wide variety of experimental data. Some of these numerical implementations are commercially available.

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