Representing Antenna Fields with Spherical Mode Expansions

One of the most popular ways to represent an antenna field is by a spherical mode expansion.

In AntennaFieldRepresentations.jl, spherical expansions are represented by a struct SphericalWaveExpansion{P, H, C} which is a subtype of AntennaFieldRepresentation{P, C}. The type parameters have the following meaning

Constructors for a SphericalWaveExpansion

To generate a SphericalWaveExpansion, use one of the following constructors:

SphericalWaveExpansion{P, H, C}(coefficients::H, wavenumber <: Number) where{P <: PropagationType, C <: Complex, H<: AbstractSphericalCoefficients{C}}

SphericalWaveExpansion(::P, coefficients::AbstractVector{C}, wavenumber) where{P <: PropagationType, C}

The input arguments for the costructors are

• P <: PropagationType : Can be Radiated, Absorbed, or Incident
• coefficients: Coefficient vector. Can be either a struct of SphericalCoefficients{C}, a struct of FirstOrderSphericalCoefficients{C}, or an AbstractVector{C}
• wavenumber : Wavenumber $\omega = 2\pi \, f$

The AbstractSphericalCoefficients{C} Type

As described in more detail in the theory section, a spherical mode expansion is characterized by the expansion coefficents $\alpha_{s \ell m}$ with $s \in \{1,2\}$, $\ell = 1,\ldots, L$, and $m = -\ell, \ldots, \ell$.

In the numerical implementation, the coefficients $\alpha_{s \ell m}$ are stored in a vector, where the triple index $s \ell m$ is mapped to a single consecutively running index $j$ by the function sℓm_to_j(s,ℓ,m). The inverse map from a single index to the triple index $s \ell m$ is implemented in the function j_to_sℓm(j). This behaviour is implemented in implementations of the abstract type AbstractSphericalCoefficients{C}. Instances of an AbstractSphericalCoefficients{C} can be accessed like an AbstractVector{C} with a single index j or with a triple index (s,ℓ,m), where j = 2 * (ℓ * (ℓ + 1) + m - 1) + s as in the following example

Example:

julia> using AntennaFieldRepresentations

julia> sph_coefficients= SphericalCoefficients(ComplexF64.(collect(1:16)))
16-element SphericalCoefficients{ComplexF64}:
  1.0 + 0.0im
  2.0 + 0.0im
  3.0 + 0.0im
  4.0 + 0.0im
  5.0 + 0.0im
  6.0 + 0.0im
  7.0 + 0.0im
  8.0 + 0.0im
  9.0 + 0.0im
 10.0 + 0.0im
 11.0 + 0.0im
 12.0 + 0.0im
 13.0 + 0.0im
 14.0 + 0.0im
 15.0 + 0.0im
 16.0 + 0.0im

julia> println(sph_coefficients[9]) # Here the coefficient vector is accessed with a single index
9.0 + 0.0im

julia> println(sph_coefficients[1, 2, -1]) # Here the same element is accessed with a triple index
9.0 + 0.0im

julia> sph_coefficients[1, 2, -1] = 111.0im; # setindex!() is also defined

julia> println(sph_coefficients[9])
0.0 + 111.0im

Besides the general SphericalCoefficients{C}, also the type FirstOrderSphericalExpansion{C} exists in AntennaFieldRepresentations.jl. A FirstOrderSphericalExpansion{C} indicates that all spherical expansion coefficients $\alpha_{s \ell m}$ with $|m| \neq 1$ are equal to zero. First-order spherical expansions do play a special role in the development of fast algorithms for spherical wave expansions and, thus, have their own representation in AntennaFieldRepresentations.jl. When a FirstOrderSphericalExpansion{C} is generated from an AbstractVector{C}, all elements which do not correspond to a triple index with $|m|=1$ are ignored.

Use the following constructors to generate a struct of AbstractSphericalCoefficients{C}

SphericalCoefficients(v::AbstractVector{C}) where{C <: Number}

FirstOrderSphericalCoefficients(v::AbstractVector{C}) where{C <: Number}

FirstorderSphericalCoefficients(x::SphericalCoefficients)

julia> fo_sph_coefficients= FirstOrderSphericalCoefficients(sph_coefficients)
16-element FirstOrderSphericalCoefficients{ComplexF64}:
  1.0 + 0.0im
  2.0 + 0.0im
  0.0 + 0.0im
  0.0 + 0.0im
  5.0 + 0.0im
  6.0 + 0.0im
  0.0 + 0.0im
  0.0 + 0.0im
  0.0 + 111.0im
 10.0 + 0.0im
  0.0 + 0.0im
  0.0 + 0.0im
 13.0 + 0.0im
 14.0 + 0.0im
  0.0 + 0.0im
  0.0 + 0.0im

Spherical Expansion Examples