The Pendellösung effect or phenomenon is seen in
diffraction
Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...
in which there is a
beating in the intensity of
electromagnetic waves
In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength, ran ...
travelling within a
crystal lattice
In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystal, crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that ...
. It was predicted by
P. P. Ewald in 1916 and first observed in
electron diffraction
Electron diffraction is a generic term for phenomena associated with changes in the direction of electron beams due to elastic interactions with atoms. It occurs due to elastic scattering, when there is no change in the energy of the electrons. ...
of
magnesium oxide
Magnesium oxide (MgO), or magnesia, is a white hygroscopic solid mineral that occurs naturally as periclase and is a source of magnesium (see also oxide). It has an empirical formula of MgO and consists of a lattice of Mg2+ ions and O2− ions ...
in 1942 by Robert D. Heidenreich and in X-ray diffraction by
Norio Kato
Norio Kato (Japanese: 加藤 範夫, ''Kato Norio'', March 10, 1923 – April 5, 2002) was a Japanese physicist and crystallographer. He was known for his contributions in diffraction topography and the dynamical theory of diffraction, including ...
and
Andrew Richard Lang in 1959.
At the exit surface of a
photonic crystal
A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of Crystal structure, natural crystals gives rise to X-ray crystallograp ...
(PhC), the intensity of the diffracted wave can be periodically modulated, showing a maximum in the "positive" (forward diffracted) or in the "negative" (diffracted) direction, depending on the crystal slab thickness.
The Pendellösung effect in
photonic crystal
A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of Crystal structure, natural crystals gives rise to X-ray crystallograp ...
s can be understood as a beating phenomenon due to the phase modulation between coexisting plane wave components, propagating in the same direction.
This thickness dependence is a direct result of the so-called Pendellösung phenomenon, consisting of the periodic exchange inside the crystal of the energy between direct and diffracted beams.
The Pendellösung interference effect was predicted by dynamical diffraction and also by its fellow theories developed for visible light.
References
Condensed matter physics
Metamaterials
Photonics
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