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.
Highlights
The fundamental diffraction problem consists of two parts: (i) determining the effects of introducing the diffracting aperture upon the field immediately behind the screen and (ii) determining how it affects the field downstream from the diffracting screen.A “diffraction grating” is an optical element that imposes a “periodic” variation in the amplitude and/or phase of an incident electromagnetic wave.[1]
We have demonstrated that when the grating equation is expressed in terms of the direction cosines of the propagation vectors of the incident beam and the diffracted orders, even wide-angle diffraction phenomena is shift invariant with respect to variations in the incident angle
The paraxial behavior described by Eq (36) above leads to the common misconception that it is impossible to get more than 33.86% of the incident energy into the first diffracted order with a sinusoidal phase grating
Summary
The fundamental diffraction problem consists of two parts: (i) determining the effects of introducing the diffracting aperture (or grating) upon the field immediately behind the screen and (ii) determining how it affects the field downstream from the diffracting screen (i.e., what is the field immediately behind the grating and how does it propagate). The second generalization includes the situation where the light is incident upon the grating at an arbitrary angle θi rather than normal incidence This situation will be taken care of by including the incident angle in the grating equation discussed in Sec. 2, where we will review the planar grating equation and the sign convention for numbering the various diffracted orders. 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.[16]. A summary, statement of conclusions, and an extensive set of references will complete this paper
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