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Weyl expansion

## Summary

In physics, the Weyl expansion, also known as the Weyl identity or angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. In a Cartesian coordinate system, it can be denoted as[1][2]

${\displaystyle {\frac {e^{-jk_{0}r}}{r}}={\frac {1}{j2\pi }}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }dk_{x}dk_{y}e^{-j(k_{x}x+k_{y}y)}{\frac {e^{-jk_{z}|z|}}{k_{z}}}}$,

where ${\displaystyle k_{x}}$, ${\displaystyle k_{y}}$ and ${\displaystyle k_{z}}$ are the wavenumbers in their respective coordinate axes:

${\displaystyle k_{0}={\sqrt {k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}}}$.

The expansion is named after Hermann Weyl, who published it in 1919.[3] The Weyl identity is largely used to characterize the reflection and transmission of spherical waves at planar interfaces; it is often used to derive the Green's functions for Helmholtz equation in layered media. The expansion also covers evanescent wave components. It is often preferred to the Sommerfeld identity when the field representation is needed to be in Cartesian coordinates.[1]

The resulting Weyl integral is commonly encountered in microwave integrated circuit analysis and electromagnetic radiation over a stratified medium; as in the case for Sommerfeld integral, it is numerically evaluated.[4] As a result, it is used in calculation of Green's functions for method of moments for such geometries.[5] Other uses include the descriptions of dipolar emissions near surfaces in nanophotonics,[6][7][8] holographic inverse scattering problems,[9] Green's functions in quantum electrodynamics[10] and acoustic or seismic waves.[11]

## References

1. ^ a b Chew 1990, p. 65-75.
2. ^ Kinayman & Aksun 2005, p. 243-244.
3. ^ Weyl, H. (1919). "Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter". Annalen der Physik (in German). 365 (21): 481-500. Bibcode:1919AnP...365..481W. doi:10.1002/andp.19193652104.
4. ^ Chew, W. C. (November 1988). "A quick way to approximate a Sommerfeld-Weyl-type integral (antenna far-field radiation)". IEEE Transactions on Antennas and Propagation. 36 (11): 1654-1657. doi:10.1109/8.9724.
5. ^ Kinayman & Aksun 2005, p. 268.
6. ^ Novotny & Hecht 2012, p. 335-338.
7. ^ Ford, G. W.; Weber, W. H. (November 1984). "Electromagnetic interactions of molecules with metal surfaces". Physics Reports. 113 (4): 195–287. Bibcode:1984PhR...113..195F. doi:10.1016/0370-1573(84)90098-X. hdl:2027.42/24649.
8. ^ de Abajo, F. J. García (10 October 2007). "Colloquium: Light scattering by particle and hole arrays". Reviews of Modern Physics. 79 (4): 1267–1290. arXiv:0903.1671. Bibcode:2007RvMP...79.1267G. doi:10.1103/RevModPhys.79.1267. hdl:10261/79230. S2CID 18698507.
9. ^ Wolf, Emil (1969). "Three-dimensional structure determination of semi-transparent objects from holographic data". Optics Communications. 1 (4): 153-156. Bibcode:1969OptCo...1..153W. doi:10.1016/0030-4018(69)90052-2.
10. ^ Agarwal, G. S. (January 1975). "Quantum electrodynamics in the presence of dielectrics and conductors. I. Electromagnetic-field response functions and black-body fluctuations in finite geometries". Physical Review A. 11 (1): 230-242. Bibcode:1975PhRvA..11..230A. doi:10.1103/PhysRevA.11.230.
11. ^ Aki & Richards 2002, p. 189-192.