Pendellösung Explained

The Pendellösung effect or phenomenon is seen in diffraction in which there is a beating in the intensity of electromagnetic waves travelling within a crystal lattice. It was predicted by P. P. Ewald in 1916^[1] and first observed in electron diffraction of magnesium oxide in 1942 by Robert D. Heidenreich^[2] and in X-ray diffraction by Norio Kato and Andrew Richard Lang in 1959.^[3]

At the exit surface of a photonic crystal (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 crystals can be understood as a beating phenomenon due to the phase modulation between coexisting plane wave components, propagating in the same direction.^{[4] [5]}

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.^[6]

The Pendellösung interference effect was predicted by dynamical diffraction and also by its fellow theories developed for visible light.

References

1. Ewald . P. P. . 1916 . Zur Begründung der Kristalloptik . Annalen der Physik . de . 354 . 2 . 117–143 . 10.1002/andp.19163540202. 1916AnP...354..117E .
2. Heidenreich . Robert D. . 1942-09-01 . Electron Reflections in MgO Crystals with the Electron Microscope . Physical Review . en . 62 . 5–6 . 291–292 . 10.1103/PhysRev.62.291 . 1942PhRv...62..291H . 0031-899X.
3. Kato . N. . Lang . A. R. . 1959-10-10 . A study of pendellösung fringes in X-ray diffraction . Acta Crystallographica . 12 . 10 . 787–794 . 10.1107/S0365110X59002262.
4. Savo. S.. Gennaro. E. Di. Miletto. C.. Andreone. A.. Dardano. P.. Moretti. L.. Mocella. V.. 2008-06-09. Pendellösung effect in photonic crystals. Optics Express. EN. 16. 12. 9097–9105. 10.1364/OE.16.009097. 18545621 . 1094-4087. 0804.1701. 2008OExpr..16.9097S . 12456215 .
5. 1983-05-01. Observation of pendellösung effect in polarized neutron scattering from a magnetic crystal. Physica B+C. en. 120. 1–3. 80. 10.1016/0378-4363(83)90343-1. 0378-4363. Baruchel . J. . Guigay . J.P. . Mazuré-Espejo . C. . Schlenker . M. . Schweizer . J. . 1983PhyBC.120...80B .
6. Punegov. Vasily I.. 2021. X-ray Laue diffraction by sectioned multilayers. I. Pendellösung effect and rocking curves. Journal of Synchrotron Radiation. 28. 5. 1466–1475. 10.1107/S1600577521006408. 34475294 . 237398971 . 1600-5775. free.