This section describes the direction unit vector coordinates used by far field projections and grating projections.
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The standard 3D far field and grating projection functions calculate the far field profile on a hemispherical surface 1 meter from the simulation. |
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Calculating the field profile on a hemispherical surface creates a minor issue when we try to plot the data on a flat computer screen. The data must be 'flattened' in some way so that we can represent a curved surface on a flat computer screen. By default we plot the far field data as if we look straight down on the hemisphere. |
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The position vectors u1, u2 of the data returned by the far field projections use 'direction unit vector' coordinates. Each unit vector goes from -1 to 1 as shown in the figure to the right.
The far field is calculated at a linearly spaced set of points as measured in the u1, u2 direction unit vector coordinates.
The point u1,u2 = 0,0 corresponds to propagation at normal incidence, in the middle of the hemisphere. |
Coordinate transformations between spherical and direction cosine units are described below.
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Coordinate limits and units $$radius \qquad 0< r \qquad m$$ $$polar\ angle \qquad 0\leq \theta\leq\pi \qquad rad$$ $$azimuthal\ angle \qquad 0\leq \phi\leq2\pi \qquad rad$$ $$unit\ vector \qquad -1\leq u\leq1 \qquad$$ |
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Spherical to direction cosine $$u_{x}=sin(\theta)cos(\phi)$$ $$u_{y}=sin(\theta)sin(\phi)$$ $$u_{z}=cos(\theta)$$ $$u_{z}=\sqrt{1-u_{x}^{2}-u_{y}^{2}}$$ $$u_{x}^{2}+u_{y}^{2}+u_{z}^{2}=1$$ |
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| Direction cosine to spherical
$$r=1$$ $$\theta=acos(u_{z})$$ $$\phi=atan(\frac{u_{y}}{u_{x}})$$ |
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Note: farfieldspherical The farfieldspherical function can be used to interpolate far field data from ux,uy coordinates to spherical coordinates. |
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Note: Performing integrals We typically want to perform integrals in spherical coordinates such as the following $$power=\int\int P(\theta,\phi)R^2 sin(\theta)d\theta d\phi$$ where P is the Poynting vector and R is the radius. The far field projections return the electric field as a function of the variables ux and uy which are the x and y components of the unit direction vector. When changing integration variables from (q,j) to (ux,uy), it can be shown that $$power=\int\int P(\theta,\phi)R^{2}sin(\theta)d\theta d\phi \\=\int\int P(u_{x},u_{y})R^{2}\frac{du_{x} du_{y}}{cos(\theta)} $$ Care must be taken to avoid numerical errors due to the cos(q) term. It's best to use the farfield3dintegrate function to evaluate these integrals. |