Understanding diffraction grating behavior: including conical diffraction and Rayleigh anomalies from transmission gratings

TLDR

Rigorous electromagnetic codes are widely available, yet most optical engineers overlook nonparaxial parametric diffraction behavior, and this work emphasizes consistent sign conventions, conical diffraction analysis, and the prediction of Rayleigh anomalies using scalar theory. The study reviews a linear systems formulation of nonparaxial scalar diffraction theory and applies it to predict nonparaxial behavior of sinusoidal and square‑wave amplitude gratings under Littrow +1 order conditions. The authors derive and graphically present paraxial efficiencies for five elementary grating types, tabulate and compare them, and then employ a linear systems formulation to predict nonparaxial efficiencies for sinusoidal and square‑wave amplitude gratings.

Abstract

With the wide-spread availability of rigorous electromagnetic (vector) analysis codes for describing the diffraction of electromagnetic waves by specific periodic grating structures, the insight and understanding of nonparaxial parametric diffraction grating behavior afforded by approximate methods (i.e., scalar diffraction theory) is being ignored in the education of most optical engineers today. Elementary diffraction grating behavior is reviewed, the importance of maintaining consistency in the sign convention for the planar diffraction grating equation is emphasized, and the advantages of discussing "conical" diffraction grating behavior in terms of the direction cosines of the incident and diffracted angles are demonstrated. Paraxial grating behavior for coarse gratings (<italic>d</italic> ≫ λ) is then derived and displayed graphically for five elementary grating types: sinusoidal amplitude gratings, square-wave amplitude gratings, sinusoidal phase gratings, square-wave phase gratings, and classical blazed gratings. Paraxial diffraction efficiencies are calculated, tabulated, and compared for these five elementary grating types. Since much of the grating community erroneously believes that scalar diffraction theory is only valid in the paraxial regime, the recently developed linear systems formulation of nonparaxial scalar diffraction theory is briefly reviewed, then used to predict the nonparaxial behavior (for transverse electric polarization) of both the sinusoidal and the square-wave amplitude gratings when the +1 diffracted order is maintained in the Littrow condition. This nonparaxial behavior includes the well-known Rayleigh (Wood's) anomaly effects that are usually thought to only be predicted by rigorous (vector) electromagnetic theory.