![]() ![]() In the limit of a large number of very narrow slits, the primary interference maxima (diffraction orders) become narrower and narrower, with more and more ( n − 2) small secondary maxima in between them. The Fraunhofer diffraction pattern of an array of equally spaced narrow slits is illustrated as the number of slits is increased: (a) two slits, (b) three slits, (c) five slits, and (d) eleven slits. Figure 2 illustrates the one-dimensional profile of the Fraunhofer diffraction pattern of an array of slits as we progress from two slits (Young’s interference pattern) to three slits, to five slits, and to eleven slits. ![]() The angular width of this interference maximum depends upon the number of slits making up the grating. Every point P in the focal plane of the lens that satisfies this condition will exhibit a primary maximum. It is clear to see that there will be constructive interference only in those discrete directions where the optical path difference from adjacent slits is an integral number of wavelengths (i.e., phase differences in multiples of 2 π). Consider the cylindrical Huygens’ wavelet produced at each narrow slit when the grating is illuminated by a normally incident plane wave as shown in Fig. 1. This would be a binary amplitude grating (completely opaque or completely transparent). One example of a diffraction grating would be a periodic array of a large number of very narrow slits. Other applications include acousto-optic modulators or scanners. Diffraction gratings are thus widely used as dispersive elements in spectrographic instruments, 2 – 5 although they can also be used as beam splitters or beam combiners in various laser devices or interferometers. 1 It thus produces, through constructive interference, a number of discrete diffracted orders (or waves) which exhibit dispersion upon propagation. 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).Ī “diffraction grating” is an optical element that imposes a “periodic” variation in the amplitude and/or phase of an incident electromagnetic wave. 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. 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. ![]() Paraxial diffraction efficiencies are calculated, tabulated, and compared for these five elementary grating types. ![]() Paraxial grating behavior for coarse gratings ( d ≫ 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. 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. 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. ![]()
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