Diffraction -- from Eric Weisstein's World of Physics

Diffraction is a phenomenon by which wavefronts of propagating waves bend in the neighborhood of obstacles. Diffraction around apertures is described approximately by a mathematical formalism called scalar diffraction theory. Diffraction problems are among the difficult encountered in optics, and exact rigorous solutions are quite rare. The first such rigorous solution was found by Sommerfeld (1896). Variants of this problem dealing with line sources, point sources, and generalization to a wedge instead of a plane were solved exactly by Carslaw (1899), Macdonald (1902), and Bromwich (1916). Mie (1908) rigorously solved scattering by a sphere having finite dielectric constant and finite conductivity.

Certain diffraction problems yield integral equations that can be solved exactly using the method of Wiener and Hopf (Titchmarsh 1937, p. 339), leading to exact solutions by Copson (1946) and others (Carlson and Heins 1947; Heins and Carlson 1947; Heins 1948; Levine and Schwinger 1948; Miles 1949; Bouwkamp 1954; Born and Wolf 1999, p. 557). Levine and Schwinger (1948, 1949) have used variational methods to calculate exactly the power diffracted through certain apertures (Born and Wolf 1999, p. 557). Other rigorous solutions have been found to a small number of other mostly two-dimensional diffraction problems (Born and Wolf 1999, pp. 370 and 556-592).

where a is the "radius" of the aperture,

is the wavelength, and R is the distance from aperture, qualitatively different types of diffraction occur. In particular,

produces a type of diffraction known as Fraunhofer diffraction, while

produces Fresnel diffraction.

Born, M. and Wolf, E. "Elements of the Theory of Diffraction" and "Rigorous Diffraction Theory." Chs. 8 and 11 in Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th ed. Cambridge, England: Cambridge University Press, pp. 370-458 and 556-592, 1999.

Bouwkamp, C. J. "Diffraction Theory." Rep. Prog. Phys. 17, 35-100, 1949.

Bromwich, T. J. I'A. "Diffraction of Waves by a Wedge." Proc. London Math. Soc. 14, 450-468, 1916.

Carlson, J. F. and Heins, A. E. "The Reflection of an Electromagnetic Plane Wave by an Infinite Set of Plates, I." Quart. J. Appl. Math. 4, 313-329, 1947.

Carslaw, H. S. "Some Multiform Solutions of the Partial Differential Equations of Physical Mathematics and their Applications." Proc. London Math. Soc. 30, 121-161, 1899.

Copson, E. T. "On an Integral Equation Arising in the Theory of Diffraction." Quart. J. Math. 17, 19-34, 1946.

Heins, A. E. "The Radiation and Transmission Properties of a Pair of Semi-Infinite Parallel Plates--I." Quart. J. Appl. Math. 6, 157-166, 1948.

Heins, A. E. "The Radiation and Transmission Properties of a Pair of Semi-Infinite Parallel Plates--II." Quart. J. Appl. Math. 6, 215-220, 1948.

Heins, A. E. and Carlson, J. F. "The Reflection of an Electromagnetic Plane Wave by an Infinite Set of Plates, II." Quart. J. Appl. Math. 5, 82-88, 1947.

Levine, H. and Schwinger, J. "On the Theory of Diffraction by an Aperture in an Infinite Plane Screen. I." Phys. Rev. 74, 958-974, 1948.

Levine, H. and Schwinger, J. "On the Theory of Diffraction by an Aperture in an Infinite Plane Screen. II." Phys. Rev. 75, 1423-1432, 1949.

Macdonald, H. M. Electric Waves. Cambridge, England: Cambridge University Press, 1902.

Miles, J. W. "On the Diffraction of an Electromagnetic Wave through a Plane Screen." J. Appl. Phys. 20, 760-771, 1949.

Sommerfeld, A. "Mathematische Theorie der Diffraction." Math. Ann. 47, 317-374, 1896.