3.1. Expressions for the Regularized Principal Diagonal Dyad pd∞
 The function erfc(ν) denotes the complementary error function [Abramowitz and Stegun, 1965] of argument β± = RnE ± ik/(2E). The adopted Ewald parameter E is [Kustepeli and Martin, 2000]
which is chosen based on optimizing the total number of necessary terms in both the scalar spatial and spectral series, since with this choice they both exhibit the same Gaussian convergence rate. Note that the spatial 1/R singularity is fully represented by the n = 0 term of the spatial sum.
 In (19) a prime (f′) denotes a derivative of f in (17) with respect to its argument Rn, whose expression is given in Appendix B.
 Then, we accordingly split the dyadic GF in (3) as
where “type” is either spectral or spatial. The first term of the dyad in (21) is proportional to the scalar GF which is given in (15) for the spectral and in (16) for the spatial type. The terms ∇∇Gspectral∞ and ∇∇Gspatial∞ are derived based on (15) and (16) leading to
and n = Rn/Rn. Furthermore, f″ and f‴ are the second and third derivatives of f in (17) with respect to its argument Rn, respectively, whose expressions are given in Appendix B. Similarly to what discussed for the scalar GF, here ∇∇ spatial∞(r0, r0, kB) = [∇∇Gspatial∞(r, r0, kB) − ∇∇G(r, r0)]. In its computation we need to evaluate the limit because both ∇∇Gspatial∞ and ∇∇G have singular terms at r = r0 of the kind 1/R0p with p = 1, 2, 3. After performing the limit for r r0 (see details in the Appendix A) one has
3.2. Expressions for the Regularized Antidiagonal Dyad ad∞
 Similarly to what done for the principal diagonal dyad in the previous section, also the dyadic GF in (7) is split into the hybrid sum of spectral and spatial dyadic terms
through the Ewald method, where
and “type” is either spectral or spatial. The terms ∇Gspectral∞ and ∇Gspatial∞ are derived based on (15) and (16) leading to
3.3. Convergence of the Scalar GFs and of the Principal and Antidiagonal Blocks of the Dyadic GF
 As stated in section 3.1, the Ewald method allows for obtaining series with Gaussian convergence rate, so only a handful of summation terms are needed to achieve convergence. In order to verify this, we define the relative error as
with Kspectral either Gspectral∞(r0, r0, kB), or ∇Gspectral,i∞(r0, r0, kB), or ∇∇Gspectral,ij∞(r0, r0, kB) and spatial either spatial∞(r0, r0, kB), or ∇ spatial,i∞(r0, r0, kB), or ∇∇ spatial,ij∞ (r0, r0, kB), and i, j = x, y, z indicating the vector and dyad components. In (31), Kspectralexact and spatialexact are evaluated with a sufficiently large number of terms to achieve high numerical accuracy, whereas KspectralN and spatialN are evaluated with n1 = n2 = n3 = 0, ±1, …, ±N terms only, as was done in Capolino et al. [2005, 2007]. As shown in section 2, the two principal blocks in (12) are proportional to one regularized principal diagonal dyad pd∞ whereas the two antidiagonal blocks are proportional to one regularized antidiagonal dyad ad∞. This fact implies that it is sufficient to check the convergence rate of pd∞ and ad∞ for a given wave vector kB.
Figure 3. Relative error Err in (31) of the dyadic quantities in the legends for kB = kB (sin θ + cos θ ) with θ = 30° for the 3D lattice of PbTe microspheres analyzed in section 4 at 25 THz.
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 Note that, as expected, scalar spectral and spatial components in Figures 2 and 3 converge very rapidly, with about the same rate (i.e., same number of summation terms N). The same applies to the shown dyadic quantities. For example, in Figures 2 and 3, Err < 10−8 for all the series when using N = 2. Note that even N = 1 would already guarantee a relative error Err < 10−3.
Figure 5. Frequency behavior of the magnitude of the electric (a1) and magnetic (b1) Mie dipole coefficients for (a) a PbTe microsphere with radius 1 μm and (b) a TiO2 microsphere with radius 52 μm.
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 Last, from a numerical point of view, one can notice that still in the provided numerical examples only a relatively small number of summation terms (N = 2) are needed to achieve convergence when using the Ewald method proposed here, with respect to the fully spatial (singular when computed at the dipole location, as in the present case) or the fully spectral (which needs an infinite number of terms to reconstruct the singularity when computed at the dipole location, as in the present case) counterparts.