Here, we show the derivation of the weak form of the vector-valued Helmholtz equation for our boundary conditions. As a starting point, recall that the general variational form of the vector Helmholtz equation is given by:
To simplify modelling of far-field radiation, we used a coordinate transformation as shown in Frei, 2015, where we applied a coordinate transform x→αtanhx,y→βtanhy to mathematically extend the domain to infinity without needing to construct a larger grid, where, α and β are respectively the half-length and half-width of the domain bounding box. Physically, such a coordinate transform approximately replicates an open boundary that allows outgoing electromagnetic waves to radiate outwards in all directions, allowing the radiative boundary conditions to be imposed without needing to discretize an infinite domain.
However, the direct application of the coordinate transformation results in significant mathematical complexities due to the dependence of the representation of the variational form on the choice of coordinates. Thus tensors will be used instead to give expressions of physical laws that are invariant under coordinate transformations. From this point on, all expressions of the weak form will be given in tensors unless otherwise specified. All tensors are assumed to be within Euclidean space where upper and lower indices are equivalent, that is, Ai=Ai. The Einstein summation convention is assumed, in which repeated indices are implicitly summed over, and all indices take the numerical values of i=1,2,3 unless otherwise specified.
To begin, the weak form may be expressed in tensor notation as:
The old coordinates are denoted xi=x=(x,y), and new coordinates denoted xk=x′=(x′,y′) where x′=x′(x) and y′=y′(y). A change of variables was applied on the weak form xi→xk. On the first integral term, the Kronecker delta was used to relabel indices from i→k, by the relationships Φi=δikΦk,Ei=δkiEk, from which one can substitute into the first integral term to obtain:
The two Kronecker deltas are contracted over both of their indices, so δikδki=δii=δ11+δ22+δ33=3. Therefore the first term further simplifies to 3k2∫ΩΦkEkdA. For the second term, the chain rule ∂j=∂xj∂xk∂xk∂=∂jxk∂k may be used to rewrite derivatives in terms of the new coordinates xk. After substitution one finds:
Altogether, after substitution of all simplified terms and ordering the terms such that the quasi-linear term is second due to software requirements, one finally obtains the weak form in the transformed coordinates:
When applying boundary conditions to the boundary integral, the only contribution is that of the radiative boundary condition, which reduces to the constant Dirichlet boundary condition E∣∂ΩB=ϵ and thus:
In this section, we show the domain parametrization used for our model. To start, the Laplacian in a generalized Euclidean coordinate transformation xi→xj takes the form:
Which is then substituted into the definition of the Laplacian ∇2=∂k∂k and distributed to obtain the result in (17). Using this expression, with the coordinate transforms x′=αtanhx, y′=βtanhy, the transformed Laplacian becomes:
The geometry of the simulation was parametrized by four constants, the simulation volume width D and length L, the primary reflector radius R1, secondary reflector radius R2, opening gap radius b (where the opening is located at the rear of the primary reflector dish), and the primary reflector anchor point d1. The primary collector and secondary mirror were respectively parametrized as follows: