Plane Wave

Setup for the excitation with a plane wave. TODO: more suitable image.

Definition

A plane wave with amplitude $a$, wave vector $\bm k = k \hat{\bm k}$, and polarization $\hat{\bm p}$ (vectors with a hat denote unit vectors) is defined by the field

\[\bm e_\mathrm{PW}(\bm r) = a \hat{\bm p} \, \mathrm{e}^{-\mathrm{j} \bm k \cdot \bm r} \,,\]

where the polarization and wave vector are orthogonal, that is,

\[\bm k \cdot \hat{\bm p} = 0\]

holds.

API

The API provides the following constructor with default values:

SphericalScattering.planeWave — Function

ex = planeWave(;
        embedding    = Medium(ε0, μ0),
        frequency    = error("missing argument `frequency`"),
        amplitude    = 1.0,
        direction    = SVector{3,typeof(frequency)}(0.0, 0.0, 1.0),
        polarization = SVector{3,typeof(frequency)}(1.0, 0.0, 0.0),
)

source

Note

The provided direction and the polarization vectors have to be orthogonal. This is checked during initialization.

Tip

The direction and the polarization vectors are each automatically normalized to unit vectors during the initialization.

Incident Field

The electric field of the plane wave is as given above. The magnetic field is given by

\[\bm h_\mathrm{PW}(\bm r) = \cfrac{a}{Z_\mathrm{F}} (\hat{\bm k} \times \hat{\bm p}) \mathrm{e}^{-\mathrm{j} \bm k \cdot \bm r}\]

with $Z_\mathrm{F} = \sqrt{\mu / \varepsilon}$.

API

The general API is employed:

E  = field(ex, ElectricField(point_cart))

H  = field(ex, MagneticField(point_cart))

Note

The far-field of a plane wave is not defined.

Scattered Field

The scattered field computation follows [1, pp. 347ff].

Note

Internal details of the computations: Following [1, pp. 347ff] the plane wave is initially assumed to travel in positive $z$-axis direction and to have a polarization along the positive $x$-axis. Arbitrary directions and orientations (forming a valid pair) are obtained via rotations.

API

The general API is employed:

E  = scatteredfield(sp, ex, ElectricField(point_cart))

H  = scatteredfield(sp, ex, MagneticField(point_cart))

FF = scatteredfield(sp, ex, FarField(point_cart))

Total Field

API

The general API is employed:

E  = field(sp, ex, ElectricField(point_cart))

H  = field(sp, ex, MagneticField(point_cart))

Note

The total far-field is not defined (since the incident far-field is not defined).

Radar Cross Section

To compute the bistatic radar cross section (RCS) [1, pp. 350ff]

\[\sigma (\vartheta, \varphi) = \lim_{r\rightarrow \infty} \left( 4 \pi r^2 \frac{{|e^\mathrm{sc}|}^2}{{|e^\mathrm{inc}|}^2} \right)\]

the function

σ = rcs(sp, ex, points_cart)

is provided. For the monostatic RCS, the function

σ = rcs(sp, ex)

is provided.