Weyl expansion

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 asTemplate:SfnTemplate:Sfn Template:Short description

$\frac{e^{-j{k}_{0}r}}{r}=\frac{1}{j2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}d{k}_{x}d{k}_y{e}^{-j(k_{x}x+k_{y}y)}\frac{e^{-j{k}_z|z|}}{k_z}$,

where $k_x$, $k_y$ and $k_z$ are the wavenumbers in their respective coordinate axes:

$k_0=\sqrt{k_x^2+k_y^2+k_z^2}$.

The expansion is named after Hermann Weyl, who published it in 1919. 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.Template:Sfn

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. As a result, it is used in calculation of Green's functions for method of moments for such geometries.Template:Sfn Other uses include the descriptions of dipolar emissions near surfaces in nanophotonics,Template:Sfn holographic inverse scattering problems, Green's functions in quantum electrodynamics and acoustic or seismic waves.Template:Sfn

See also

• Angular spectrum method
• Fourier optics
• Green's function
• Plane wave expansion
• Sommerfeld identity

Sources

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