API Reference

ε0 = 8.8541878176e-12

Free space permittivity.

μ0 = 4pi * 1e-7

Free space permeability.

DielectricSphere(
    radius      = error("missing argument `radius`"), 
    filling     = error("missing argument `filling`")
)

Constructor for the dielectric sphere.

DielectricSphereThinImpedanceLayer(
    radius    = error("missing argument `radius`"),
    thickness = error("missing argument `thickness` of the coating"),
    thinlayer = error("missing argument `thinlayer`"),
    filling   = error("missing argument `filling`")
)

Constructor for the dielectric sphere with a thin impedance layer. For this model, it is assumed that the displacement field is only radial direction in the layer, which requires a small thickness and low conductivity. For details, see for example T. B. Jones, Ed., “Models for layered spherical particles,” in Electromechanics of Particles, Cambridge: Cambridge University Press, 1995, pp. 227–235. doi: 10.1017/CBO9780511574498.012.

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

ex = HertzianDipole(
        embedding   = Medium(ε0, μ0),
        frequency   = error("missing argument `frequency`"),
        amplitude   = 1.0,
        position    = error("missing argument `position`"),
        orientation = SVector{3,typeof(frequency)}(0.0, 0.0, 1.0),
)

LayeredSphere( 
    radii   = error("Missing argument `radii`"), 
    filling = error("`missing argument `filling`")
)

Constructor for the layered dielectric sphere.

LayeredSpherePEC( 
    radii   = error("Missing argument `radii`"), 
    filling = error("Missing argument `filling`")
)

Constructor for the layered dielectric sphere.

Medium(ε, μ)

Homogeneous, isotropic background medium.

PECSphere( 
    radius = error("missing argument `radius`"), 
)

Constructor for the PEC sphere.

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),
)

ex = UniformField(; 
    embedding=Medium(ε0, μ0), 
    amplitude=1.0, 
    direction=SVector{3,typeof(amplitude)}(1.0, 0.0, 0.0)
)

amplitude(sphere, excitation::PlaneWave, quantity::MagneticField, r)

Returns $E₀$, where $E₀$ is the electric field of the incident plane wave. For PEC layers, it returns $0$.

amplitude(sphere, excitation::PlaneWave, quantity::MagneticField, r)

Returns $E₀$, where $E₀$ is the electric field of the incident plane wave.

amplitude(sphere, excitation::PlaneWave, quantity::MagneticField, r)

Returns $H₀/ηᵢ$, where $H₀$ is the magnetic field of the incident plane wave. For PEC layers, it returns $0$.

associatedLegendre(n::T, m::T, ϑ::F)

Compute the normalized associated Legendre polynomials according to the definition by Hansen (A1.25) times m divided by sin(ϑ), but including the Condon-Shortley phase. By using the recursion relations (A1.34a) all corner cases are accounted for correctly.

m * bar(P)n|m|(cos(ϑ)) / sin(ϑ)

ϑ ∈ [0, π] assumed.

cart2sph(vec)

Convert a 3D (3 entry) point from Cartesian to spherical coordinates.

convertSpherical2Cartesian(F_sph, point_sph)

Takes a 3D (3 entry) vector F_sph and converts it from its spherical basis to its Cartesian basis representation.

The location of the vector has to be provided in spherical coordinates by point_sph ordered as $(r, ϑ, φ)$.

derivatieAssociatedLegendre(n::T, m::T, ϑ::F) where {T<:Integer,F<:Real}

Compute the derivatives of the normalized associated Legendre polynomials according to the definition by Hansen (A1.25), but including the Condon-Shortley phase. By using the recursion relations (A1.34b) all corner cases are accounted for correctly.

d bar(P)n|m|(cos(ϑ)) / dϑ

ϑ ∈ [0, π] assumed.

ex = electricRingCurrent(;
        embedding   = Medium(ε0, μ0),
        frequency   = error("missing argument `frequency`"),
        amplitude   = 1.0,
        radius      = error("missing argument `radius`"),
        center      = error("missing argument `center`"),
        orientation = SVector{3,typeof(frequency)}(0.0, 0.0, 1.0),
)

expansion(sphere::Sphere, excitation::PlaneWave, quantity::Field, r, plm, cosϑ, sinϑ, n::Int)

Compute functional dependencies of the Mie series for a plane wave travelling in +z-direction with polarization in x-direction.

field(excitation::FitzgeraldDipole, point, quantity::FarField; parameter::Parameter=Parameter())

Compute the electric far field radiated by a Fitzgerald dipole at given position and orientation at point.

The point and the returned field are in Cartesian coordinates.

field(excitation::HertzianDipole, point, quantity::FarField; parameter::Parameter=Parameter())

Compute the electric far field radiated by a Hertzian dipole at given position and orientation at point.

The point and the returned field are in Cartesian coordinates.

field(excitation::PlaneWave, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field of a plane wave.

The point and the returned field are in Cartesian coordinates.

field(excitation::PlaneWave, point, quantity::FarField; parameter::Parameter=Parameter())

Throw error since the far-field of a plane wave is not defined.

field(excitation::PlaneWave, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the magnetic field of a plane wave.

The point and the returned field are in Cartesian coordinates.

field(excitation::PlaneWave, quantity::Field; parameter::Parameter=Parameter())

Compute the electric field of a plane wave.

field(excitation::SphericalMode, quantity::Field; parameter::Parameter=Parameter())

Compute the electric field of a spherical mode.

field(excitation::SphericalModeTE, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field of a TE spherical mode.

field(excitation::SphericalModeTE, point, quantity::FarField; parameter::Parameter=Parameter())

Compute the electric far-field of a TE spherical mode.

field(excitation::SphericalModeTE, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field of a TM spherical mode.

field(excitation::SphericalModeTE, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric far-field of a TM spherical mode.

field(excitation::Dipole, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field radiated by a Hertzian dipole at given position and orientation at point.

The point and the returned field are in Cartesian coordinates.

field(excitation::Dipole, point, quantity::MagneticField; parameter::Parameter=Parameter())

Compute the magnetic field radiated by a Hertzian dipole at given position and orientation at point.

The point and the returned field are in Cartesian coordinates.

field(excitation::Dipole, quantity::Field; parameter::Parameter=Parameter())

Compute the electric field radiated by a magnetic/electric ring current at some position and with some orientation.

field(excitation::ElectricRingCurrent, r, ϑ, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field radiated by an electric ring current placed in origin at point (r, ϑ)

field(excitation::ElectricRingCurrent, r, ϑ, quantity::FarField; parameter::Parameter=Parameter())

Compute the (electric) far-field radiated by an electric ring current placed in origin at point (r, ϑ)

field(excitation::ElectricRingCurrent, r, ϑ, quantity::MagneticField; parameter::Parameter=Parameter())

Compute the magnetic field radiated by an electric ring current placed in origin at point (r, ϑ)

field(excitation::RingCurrent, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field radiated by a magnetic/electric ring current at some position and with some orientation.

field(sphere::Sphere, excitation::Excitation, quantity::Field; parameter::Parameter=Parameter())

Compute the total field in the presence of a sphere for a given excitation.

field(sphere::Sphere, excitation::PlaneWave, quantity::Field; parameter::Parameter=Parameter())

Descriptive error for the total far-field in the presence of a sphere for an incident plane wave.

field(sphere::Sphere, excitation::SphericalMode, quantity::Field; parameter::Parameter=Parameter())

Descriptive error for the total far-field in the presence of a sphere for an incident spherical mode.

field(excitation::UniformField, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field of a uniform field.

The point and the returned field are in Cartesian coordinates.

field(excitation::UniformField, point, quantity::ScalarPotential; parameter::Parameter=Parameter())

Compute the scalar potential of a uniform field.

field(excitation::UniformField, quantity::Field; parameter::Parameter=Parameter())

Compute the field or potential of a uniform field.

getFieldType(excitation::ElectricRingCurrent, quantity::Field)

Do nothing.

getFieldType(excitation::FitzgeraldDipole, quantity::Field)

Exchange electric and magnetic field for a magnetic current + apply duality relations.

getFieldType(excitation::HertzianDipole, quantity::Field)

Do nothing.

getFieldType(excitation::MagneticRingCurrent, quantity::Field)

Exchange electric and magnetic field for a magnetic current.

getFieldType(excitation::SphericalMode, quantity::ElectricField)

Do nothing.

getFieldType(excitation::SphericalModeTE, quantity::MagneticField)

Exchange TE and TM. Other prefactor: im / ZF.

getFieldType(excitation::SphericalModeTM, quantity::MagneticField)

Exchange TE and TM. Other prefactor: im / ZF.

impedance(sp::Sphere, ex::Excitation, r)

Returns the wavenumber at radius r in the sphere sp. If this part is PEC, a zero medium is returned.

impedance(sp::Sphere, r)

Returns the impedance of the sphere sp at radius r. If this part is PEC, a zero medium is returned.

layer(sp::Sphere, r)

Returns the index of the layer, r is located, where 1 denotes the inner most layer.

layer(sp::LayeredSphere, r)

Returns the index of the layer, r is located, where 1 denotes the inner most layer.

ex = magneticRingCurrent(;
        embedding   = Medium(ε0, μ0),
        frequency   = error("missing argument `frequency`"),
        amplitude   = 1.0,
        radius      = error("missing argument `radius`"),
        center      = error("missing argument `center`"),
        orientation = SVector{3,typeof(frequency)}(0.0, 0.0, 1.0),
)

medium(sp::Sphere, ex::Excitation, r)

Returns the medium of the sphere sp at radius r. If this part is PEC, a zero medium is returned. The argument r may be a vector of positions.

numlayers(sp::Sphere)

Returns the number of layers.

permeability(sp::Sphere, ex::Excitation, r)

Returns the permeability of the sphere sp at radius r. If this part is PEC, a zero permeability is returned. The argument r may be a vector of positions.

permittivity(sp::Sphere, ex::Excitation, r)

Returns the permittivity of the sphere sp at radius r. If this part is PEC, a zero permittivity is returned. The argument r may be a vector of positions.

phiCutPoints(ϕ; r=1.0, resolution=1)

Convenience function returning points (in Cartesian coordinates) on a phi cut at a distance 'r' and a resolution in degrees.

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),
)

prefac(m::T, n::T)

Prefactor for both types of spherical vector wave functions.

Note the factor (-m / abs(m))^m is now part of the associated Legendre polynomials via the Condon-Shortley phase.

rcs(sphere::Sphere, excitation::Excitation, point_cart; parameter::Parameter=Parameter())

RCS only defined for plane waves, so far.

rcs(sphere::Sphere, excitation::Excitation; parameter::Parameter=Parameter())

RCS only defined for plane waves, so far.

rcs(sphere::Sphere, excitation::PlaneWave, point_cart; parameter::Parameter=Parameter())

Compute the bistatic radar cross-section (RCS).

rcs(sphere::Sphere, excitation::PlaneWave; parameter::Parameter=Parameter())

Compute the monostatic radar cross-section (RCS): the bistatic RCS solely for the incident direction of the plane wave.

rotate!(excitation::Excitation, vectors_list; inverse=false)

Determine rotation matrix and perform rotation for a general excitation.

The points are assumed to be in Cartesian coordinates.

The vectors_list IS modified (overwritten).

rotate(excitation::Excitation, vectors_list; inverse=false)

Determine rotation matrix and perform rotation for general excitations.

The points are assumed to be in Cartesian coordinates.

The vectors_list is NOT modified.

rotationMatrix(excitation::PlaneWave)

Determine rotation matrix for a plane wave excitation.

rotationMatrix(excitation::Union{Dipole,RingCurrent})

Determine rotation matrix for a dipole or a ring-current excitation via the Rodrigues formula.

scatterCoeff(sp::DielectricSphereThinImpedanceLayer, ex::UniformField)

Compute the expansion coefficients for the thin impedance layer case and a uniform static field excitation.

scatterCoeff(sphere::PECSphere, excitation::ElectricRingCurrent, n::Int, ka)

Compute scattering coefficient for electric ring current.

scatterCoeff(sphere::PECSphere, excitation::FitzgeraldDipole, n::Int, ka)

Compute scattering coefficient for Fitzgerald dipole.

scatterCoeff(sphere::PECSphere, excitation::HertzianDipole, n::Int, ka)

Compute scattering coefficient for Hertzian dipole.

scatterCoeff(sphere::PECSphere, excitation::MagneticRingCurrent, n::Int, ka)

Compute scattering coefficient for magnetic ring current.

scatterCoeff(sphere::PECSphere, excitation::PlaneWave, n::Int)

Compute scattering coefficients for a plane wave travelling in +z-direction with polarization in x-direction.

scatterCoeff(sphere::PECSphere, excitation::SphericalModeTE, n::Int, ka)

Compute scattering coefficients for a spherical TE mode travelling towards the origin.

scatterCoeff(sphere::PECSphere, excitation::SphericalModeTM, n::Int, ka)

Compute scattering coefficients for a spherical TM mode travelling towards the origin.

scatterCoeff(sphere::LayeredSpherePEC{LN,LD,LR,LC}, excitation::UniformField{FC,FT,FR}, r) where {LN,LD,LR,LC,FC,FT,FR}

Scatter coefficients according to Sihvola and Lindell. However, the radii of the shells are ordered from inside to outside as depicted in the documentation.

scatterCoeff(sphere::LayeredSphere{LN,LR,LC}, excitation::UniformField{FC,FT,FR}, r) where {LN,LR,LC,FC,FT,FR}

Scatter coefficients according to Sihvola and Lindell. However, the radii of the shells are ordered from inside to outside as depicted in the documentation.

scatterCoeff_of_layer(sphere, r, coeffs)

Given a tuple of scattering coefficients coeffs, it returns the two scattering coefficients of the layer to which r belongs.

scatteredfield(sphere::DielectricSphere, excitation::UniformField, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field scattered by a Dielectric sphere, for an incident uniform field with polarization in given direction.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::DielectricSphere, excitation::UniformField, point, quantity::ScalarPotential; parameter::Parameter=Parameter())

Compute the scalar potential scattered by a dielectric sphere, for an incident uniform field with polarization in given direction.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::DielectricSphereThinImpedanceLayer, excitation::UniformField, point, quantity::DisplacementField; parameter::Parameter=Parameter())

Compute the displacement field D = ε * E.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::DielectricSphereThinImpedanceLayer, excitation::UniformField, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field scattered by a dielectric sphere with a thin coating, where the displacement field in the coating is only in radial direction. We assume an an incident uniform field with polarization in the given direction.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::DielectricSphereThinImpedanceLayer, excitation::UniformField, point, quantity::ScalarPotentialJump; parameter::Parameter=Parameter())

Compute the jump of the scalar potential for a dielectric sphere with a thin coating, where the displacement field in the coating is only in radial direction. We assume an an incident uniform field with polarization in the given direction.

More precisely, we compute the difference Δ = Φi - Φe, where Φi is the potential on the inner side and ϕe the exterior potential.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::DielectricSphereThinImpedanceLayer, excitation::UniformField, point, quantity::ScalarPotential; parameter::Parameter=Parameter())

Compute the scalar potential scattered by a dielectric sphere with a thin coating, where the displacement field in the coating is only in radial direction. We assume an an incident uniform field with polarization in the given direction.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::PECSphere, excitation::ElectricRingCurrent, quantity::FarField; parameter::Parameter=Parameter())

Compute the electric far-field scattered by the PEC sphere, where the ring current is placed along the z-axis.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::PECSphere, excitation::FitzgeraldDipole, quantity::FarField; parameter::Parameter=Parameter())

Compute the electric far-field scattered by a PEC sphere, where the dipole is placed along the z-axis at z0.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::PECSphere, excitation::HertzianDipole, quantity::FarField; parameter::Parameter=Parameter())

Compute the electric far-field scattered by a PEC sphere, where the dipole is placed along the z-axis at z0.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::PECSphere, excitation::MagneticRingCurrent, quantity::FarField; parameter::Parameter=Parameter())

Compute the electric far-field scattered by the PEC sphere, where the ring current is placed along the z-axis.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::PECSphere, excitation::SphericalModeTE, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field scattered by a PEC sphere, when excited by a spherical mode travelling towards the origin.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::PECSphere, excitation::SphericalMode, quantity::Field; parameter::Parameter=Parameter())

Compute the electric field scattered by a dipole at some position and orientation.

scatteredfield(sphere::PECSphere, excitation::Dipole, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field scattered by a PEC sphere, where the dipole is placed along the z-axis at z0.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::PECSphere, excitation::Dipole, point, quantity::MagneticField; parameter::Parameter=Parameter())

Compute the magnetic field scattered by a PEC sphere, where the dipole is placed along the z-axis at z0.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::PECSphere, excitation::Dipole, quantity::Field; parameter::Parameter=Parameter())

Compute the field scattered by a PEC sphere excited by a dipole at some position and orientation.

scatteredfield(sphere::PECSphere, excitation::RingCurrent, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field scattered by the PEC sphere, where the ring current is placed along the z-axis.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::PECSphere, excitation::RingCurrent, quantity::MagneticField; parameter::Parameter=Parameter())

Compute the magnetic field scattered by the PEC sphere, where the ring current is placed along the z-axis.

The point and the returned field are in Cartesian coordinates.

field(excitation::ElectricRingCurrent, quantity::Field; parameter::Parameter=Parameter())

Compute the electric field radiated by an electric ring current at some position and orientation

scatteredfield(sphere::PECSphere, excitation::UniformField, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field scattered by a PEC sphere, for an incident uniform field with polarization in the given direction.

The point and returned field are in Cartesian coordinates.

scatteredfield(sphere::PECSphere, excitation::UniformField, point, quantity::ScalarPotential; parameter::Parameter=Parameter())

Compute the scalar potential scattered by a PEC sphere, for an incident uniform field with polarization in the given direction.

The point and returned field are in Cartesian coordinates.

scatteredfield(sphere::Sphere, excitation::PlaneWave, point, quantity::Field; parameter::Parameter=Parameter())

Compute the electric field scattered by a PEC or dielectric sphere, for an incident plane wave travelling in +z-direction with E-field polarization in x-direction.

The point and the returned field are in Cartesian coordinates.

scatteredfield(sphere::Sphere, excitation::PlaneWave, quantity::Field; parameter::Parameter=Parameter())

Compute the electric field scattered by a PEC sphere, for an incident plane wave.

scatteredfield(sphere::Sphere, excitation::PlaneWave, quantity::Field; parameter::Parameter=Parameter())

Compute the electric field scattered by a sphere, for an incident uniform field.

scatteredfield(sphere::LayeredSpherePEC, excitation::UniformField, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field scattered by a layered dielectric sphere with PEC core, for an incident uniform field with polarization in the given direction using Sihvola and Lindell, 1988, Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers.

In contrast to Sihvola and Lindell the radii of the shells are ordered from inside to outside as depicted in the documentation.

The point and returned field are in Cartesian coordinates.

scatteredfield(sphere::LayeredSpherePEC, excitation::UniformField{FC,FT,FR}, point, quantity::ScalarPotential; parameter::Parameter=Parameter())

Compute the scalar potential scattered by a layered dielectric sphere with PEC core, for an incident uniform field with polarization in the given direction using Sihvola and Lindell, 1988, Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers

In contrast to Sihvola and Lindell the radii of the shells are ordered from inside to outside as depicted in the documentation.

The point and returned field are in Cartesian coordinates.

scatteredfield(sphere::LayeredSphere, excitation::UniformField, point, quantity::ElectricField; parameter::Parameter=Parameter())

Compute the electric field scattered by a layered dielectric sphere, for an incident uniform field with polarization in the given direction using Sihvola and Lindell, 1988, Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers.

In contrast to Sihvola and Lindell the radii of the shells are ordered from inside to outside as depicted in the documentation.

The point and returned field are in Cartesian coordinates.

scatteredfield(sphere::LayeredSphere, excitation::UniformField, point, quantity::ScalarPotential; parameter::Parameter=Parameter())

Compute the scalar potential scattered by a layered dielectric sphere, for an incident uniform field with polarization in the given direction using Sihvola and Lindell, 1988, Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers.

In contrast to Sihvola and Lindell the radii of the shells are ordered from inside to outside as depicted in the documentation.

The point and returned field are in Cartesian coordinates.

sph2cart(vec)

Convert a 3D (e entry) point from spherical to Cartesian coordinates.

patternPoints(;r=1.0, resolution=5)

Convenience function returning points (in Cartesian and spherical coordinates) on a spherical grid at a distance 'r' and a resolution in degrees.

thetaCutPoints(ϑ; r=1.0, resolution=1)

Convenience function returning points (in Cartesian coordinates) on a theta cut at a distance 'r' and a resolution in degrees.

translate(points, translation::SVector{3,T})

Translate the points in the direction of the translation vector.

All inputs are assumed to be in Cartesian coordinates.

wavenumber(sp::Sphere, ex::Excitation, r)

Returns the wavenumber at radius r in the sphere sp. If this part is PEC, k=0 is returned.