Spherically Sampled Data Structures

In AntennaFieldRepresentations.jl, several data structures represent functions $\bm{f}(\vartheta, \varphi)$ which are sampled on a sphere at certain sampling points $(\vartheta_i, \varphi_i)$. Examples for spherically sampled data structures are PlaneWaveExpansions or SphericalFieldSamplings.

A natural choice for the sampling points $(\vartheta_i, \varphi_i)$ is to choose a tensor product of 1D samplings along $\vartheta$ and $\varphi$, resepectively, i.e., by defining $N_\vartheta$ sampling points $\vartheta_m$ along $\vartheta$ and $N_\varphi$ sampling points $\varphi_n$ along $\varphi$, we end up with $N_\vartheta \times N_\varphi$ pairs $(\vartheta_m, \varphi_n)$ which define the sampling points on the sphere.

Internally, the data can be stored in a matrix^[1], where the $m$th row corresponds the angle $\vartheta_m$ and the $n$th coloumn corresponds to the angle $\varphi_n$. The $m, n$- entry of the matrix therefore corresponds to the function value $\bm{f}(\vartheta_m, \varphi_n)$.

Different choices for the distribution of sampling locations may have application specific benefits or drawbacks. To enable the users to use the most suitable choice for their problem at hand, several different sampling distributions are implemented in AntennaFieldRepresentations.jl. The values of the sampling angles are defined by a SphereSamplingStrategy object.

Each spherically sampled data structure contains a SphereSamplingStrategy object such that we know which sampling locations $(\vartheta_m, \varphi_n)$ correspond to the matrix entries.

The following SphereSamplingStrategys are implemented in AntennaFieldRepresentations.jl:

`RegularθRegularϕSampling`

Objects with a RegularθRegularϕSampling are sampled along $\varphi$ with $N_\varphi$ equally distributed samples in $0\leq \varphi < 2\pi$ and along $\vartheta$ with $N_\vartheta$ equally distributed samples in $0\leq \varphi \leq \pi$.

We have

$\vartheta_m = (m-1)\, \Delta\vartheta$ with $m=1,\ldots,N_\vartheta$
$\varphi_n = (n-1)\, \Delta\varphi$ with $n=1,\ldots,N_\varphi$

The sampling steps $\Delta\vartheta = 2\pi / J_\vartheta$ and $\Delta\varphi = 2\pi / J_\varphi$ must be whole-number fractions of $2\pi$ to guarantee that the sampling step between the two samples at the end of a $\varphi$-ring or a $\vartheta$ ring is the same as everywhere else. This is enforced during the construction of a RegularθRegularϕSampling as the whole-number divisors $J_\vartheta$ and $J_\varphi$ are the input arguments for the constructor.

To generate a RegularθRegularϕSampling struct, use the following constructor

RegularθRegularϕSampling(Jθ::Integer, Jϕ::Integer)

Note

You can define a RegularθRegularϕSampling with $N_\vartheta$ samples along $\vartheta$ by either specifying $J_\vartheta = 2 N_\vartheta$ or $J_\vartheta = 2 N_\vartheta + 1$. The difference is that $\Delta \vartheta$ is different for the two choices, such that the last $\vartheta$-sample will either coincide with $\vartheta_{N_\vartheta} = \pi$ or with $\vartheta_{N_\vartheta} = \pi - \Delta \vartheta /2$.

Note

A RegularθRegularϕSampling has always multiple samples at the North Pole (one sample per $\varphi$-value). However, only RegularθRegularϕSamplings with even values for $J_\vartheta$ also feature samples at the South Pole. Odd values for $J_\vartheta$ lead to no samples at the South Pole.

While local and even global interpolation is easy and efficient with a RegularθRegularϕSampling, the drawback is that evaluating spherical integrals is more costly as compared to a GaussLegendreθRegularϕSampling.

`GaussLegendreθRegularϕSampling`

Objects with a GaussLegendreθRegularϕSampling are sampled along $\varphi$ with $N_\varphi$ equally distributed samples in $0\leq \varphi < 2\pi$ and $N_\vartheta$ $\vartheta$-samples on a Gauß-Legendre based grid with $0< \vartheta < \pi$, according to

$\vartheta_k=\text{arccos}(-x_k)$

where $x_k$ are the roots of the $N_\vartheta$th Legendre polynomial

$\varphi_n = (n-1)\, \Delta\varphi$ with $n=1,\ldots,N_\varphi$

The sampling choice naturally leads to an accurate integration rule over the complete Ewald sphere

\[\oiint f(\hat{\bm{k}}) \, \mathrm{d}^2 \hat{\bm{k}} = \int \limits_{0}^{2\pi} \int \limits_{0}^{\pi} \, f(\vartheta, \varphi) \, \sin{\vartheta}\, \mathrm{d}\vartheta \mathrm{d} \varphi \approx \sum \limits_{i=1}^{N_\varphi} \sum \limits_{k=1}^{N_\vartheta} \dfrac{2\pi}{N_\varphi} w_k\, f(\vartheta_k, \varphi_i)\]

where $w_k$ are the Gauß-Legendre quadrature weights. The calculation of the Gauß-Legendre quadrature weights $w_k$ and quadrature nodes $x_k$ is efficiently implemented (to compute the $N_\vartheta$-point Gauß quadrature nodes and weights to 16-digit accuracy takes $\mathcal{O}(N_\vartheta)$ time ^[2]) in the Julia package FastGaussQuadrature.jl.

To generate a GaussLegendreθRegularϕSampling struct, use the following constructor

GaussLegendreθRegularϕSampling(Nθ::Integer, Nϕ::Integer)

where Nθ corresponds to the stored number of samples in $\vartheta$-direction and Nϕ corresponds to the number of samples in $\varphi$ direction.

The two different SphereSamplingStrategys are illustrated below for reference.

Illustration of different SphereSamplingStrategys. Grid points are marked by orange dots and the blue lines correspond to a RegularθRegularϕSampling for reference.

Left: RegularθRegularϕSampling with Jϕ = Jθ = 16, Right: GaussLegendreθRegularϕSampling with Nϕ = 16 and Nθ=8.

1or multiple matrices, if the function $\bm{f}(\vartheta, \varphi)$ is a vector function containing e.g., data for two polarizations
2I. Bogaert, Iteration-free computation of Gauss–Legendre quadrature nodes and weights, SIAM J.ournal on Scientific Computing, Vol. 36, No. 3, pp. A1008–A1026, 2014 DOI. 10.1137/140954969