Diffraction Theory for Polygonal Apertures.

We explain and describe diffraction from polygonal apertures over a wide range of sizes and observation distances. In the first case considered, a small square aperture (2a times 2a, ka << 1, where k = 2pi/lambda is the wavenumber) in a perfectly conducting plane screen of vanishing thickness diffracts a normally incident, linear polarized, monochromatic plane wave. Within the vector framework of Maxwell's equations, we hypothesize a solution for the dominant component of the electric field. Subsequently, by means of an integro-differential equation formulation of the diffraction problem applied to small apertures, we substantiate the solution. The solution represents the first three terms in a more general expansion for the aperture field. Physical intuition and the solutions for circular apertures and slits motivate us to propose this expansion. Numerical calculations validate the solution over most of the aperture except in the close vicinity of the corners of the aperture. This limited expansion does not achieve an accurate description of the field near the corners. In the remainder of the investigation we treat diffraction within the realm of Fourier optics. We develop a Gaussian beam expansion and use it to describe diffraction from a plane-screen corner of arbitrary angle. Two intersecting, coplanar half-plane screens form this corner. For Gaussian illumination of the corner, we consider several opening angles and explain computer-generated plots of the diffracted intensities. In addition we compute the optical transform of a uniformly illuminated triangular aperture and a sector of a circular aperture. Both of these aperture are plane -screen corners, but they have different bounding edges. The triangular aperture transform solution constitutes a basis for describing diffraction by a polygonal aperture. A suitable combination of rotated, elemental triangular apertures can represent the polygon. Hence, the diffraction pattern for the polygon comprises the diffraction patterns of these elemental triangles. A similar decomposition procedure is the key to writing a simple closed-form solution for diffraction by nested polygonal apertures. We present results for the particular case of a regular pentagon; here, the elemental building blocks are isosceles triangles with a grating-like structure. The diffraction patterns of these nested apertures contain interesting, low-intensity features: nested polygons are traced out in the diffraction pattern. Numerical calculations and careful measurements confirm them.