Spherical Modes

Definition

The spherical modes are defined following the conventions of [5], however adapted to the time convention $\mathrm{e}^{\,\mathrm{j}\omega t}$.

TE Modes

The functions

\[\begin{aligned} \bm{f}_{1mn}^{(c)}(r,\vartheta, \varphi) = \cfrac{1}{2\pi} \cfrac{1}{n(n+1)}{\left(-\frac{m}{|m|}\right)}^m &\left(\mathrm{h}_n^{(c)}(kr)\frac{\mathrm{j}m\bar{\mathrm{P}}_n^{|m|}(\cos\vartheta)}{\sin\vartheta}\mathrm{e}^{\mathrm{j}m\varphi} \bm{e}_\vartheta \right. \\ &~~\left. -\mathrm{h}_n^{(c)}(kr)\cfrac{\mathrm{d} \bar{\mathrm{P}}_n^{|m|}(\cos\vartheta) }{\mathrm{d}\vartheta} \mathrm{e}^{\mathrm{j}m\varphi} \bm{e}_\varphi ~\right) \end{aligned}\]

form the $\mathrm{TE}_{mn}$ modes, where $\mathrm{P}_n^m$ denote the associated Legendre polynomials and $\mathrm{h}_n^{(c)}(x)$ denote the spherical Hankel functions.

TM Modes

The functions

\[\begin{aligned} \bm{f}_{2mn}^{(c)}(r,\vartheta, \varphi) = \cfrac{1}{2\pi} \cfrac{1}{n(n+1)}{\left(-\frac{m}{|m|}\right)}^m \Bigg(&\cfrac{n(n+1)}{kr}\mathrm{h}_n^{(c)} \bar{\mathrm{P}}_n^{|m|}(\cos\vartheta) \mathrm{e}^{\mathrm{j}m\varphi} \bm{e}_r \Bigg. \\ &+ \cfrac{1}{kr} \cfrac{\mathrm{d}\left( kr \mathrm{h}_n^{(c)}(kr)\right)}{\mathrm{d}(kr)} \cfrac{\mathrm{d} \bar{\mathrm{P}}_n^{|m|}(\cos\vartheta) }{\mathrm{d}\vartheta} \mathrm{e}^{\mathrm{j}m\varphi} \bm{e}_\vartheta \\ &\Bigg. \frac{1}{kr} \cfrac{\mathrm{d}\left( kr \mathrm{h}_n^{(c)}(kr)\right)}{\mathrm{d}(kr)}\cfrac{\mathrm{j}m\bar{\mathrm{P}}_n^{|m|}(\cos\vartheta)}{\sin\vartheta}\mathrm{e}^{\mathrm{j}m\varphi} \bm{e}_\varphi \Bigg) \,\text{.} \end{aligned}\]

form the $\mathrm{TM}_{mn}$ modes.

API

The API provides the following constructors with default values:

ex = SphericalModeTE(;
        embedding   = Medium(ε0, μ0),
        frequency   = error("missing argument `frequency`"),
        amplitude   = 1.0,
        m           = 0,
        n           = 1,
        c           = 1,
        center      = SVector{3,typeof(frequency)}(0.0, 0.0, 0.0),
        orientation = SVector{3,typeof(frequency)}(0.0, 0.0, 1.0),
)

ex = SphericalModeTM(;
        embedding   = Medium(ε0, μ0),
        frequency   = error("missing argument `frequency`"),
        amplitude   = 1.0,
        m           = 0,
        n           = 1,
        c           = 1,
        center      = SVector{3,typeof(frequency)}(0.0, 0.0, 0.0),
        orientation = SVector{3,typeof(frequency)}(0.0, 0.0, 1.0),
)

Incident Field

The electric field is given by

\[\bm e^\mathrm{TE,inc}_{mn} = k \sqrt{Z_\mathrm{F}} \bm{f}_{1mn}^{(1)}\]

and

\[\bm e^\mathrm{TM,inc}_{mn} = k \sqrt{Z_\mathrm{F}} \bm{f}_{2mn}^{(1)}\]

The magnetic field is given by

\[\bm h^\mathrm{TE,inc}_{mn} = \cfrac{\mathrm{j} k}{\sqrt{Z_\mathrm{F}}} \bm{f}_{1mn}^{(1)}\]

and

\[\bm h^\mathrm{TM,inc}_{mn} = \cfrac{\mathrm{j} k}{\sqrt{Z_\mathrm{F}}} \bm{f}_{2mn}^{(1)}\]

API

The general API is employed:

E  = field(ex, ElectricField(point_cart))

H  = field(ex, MagneticField(point_cart))

FF = field(ex, FarField(point_cart))

Scattered Field

Matching incoming and outcoming waves to fulfull the boundary condition $\bm e_\mathrm{tan} = \bm 0$ yields the scattering coefficients

\[\xi_\mathrm{TE} = -\cfrac{\mathrm{H}^{(2)}_{n + 0.5}(k a)}{\mathrm{H}^{(1)}_{n + 0.5}(k a)}\]

and

\[\xi_\mathrm{TM} = -\cfrac{\mathrm{H}'^{(2)}_{n + 0.5}(k a)}{\mathrm{H}'^{(1)}_{n + 0.5}(k a)}\]

where $\mathrm{H}^{(\nu)}_{n}(x)$ denotes the Hankel function of $\nu$-th kind and $n$-th order.

The scattered fields $\bm e^\mathrm{sc}$ are then given by

\[\bm e_\mathrm{TE}^\mathrm{sc} = \xi_\mathrm{TE} k \sqrt{Z_\mathrm{F}} \bm{f}_{1mn}^{(2)}\]

and

\[\bm e_\mathrm{TM}^\mathrm{sc} = \xi_\mathrm{TM} k \sqrt{Z_\mathrm{F}} \bm{f}_{2mn}^{(2)} \,.\]

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))