Polarization-Current-Based FDTD Near-to-Far-Field Transformation

arXiv:0903.0663v1 [physics.optics] 4 Mar 2009 Polarization-Current-Based FDTD Near-to-Far-Field Transformation Yong Zeng and Jerome V. Moloney Arizona Center for mathematical Sciences, University of Arizona, Tucson, Arizona 85721 ∗Corresponding author: zengy@acms.arizona.edu A new near-to-far-field transformation algorithm for three-dimensional finite-different time-domain is presented in this article. This new approach is based directly on the polarization current of the scatterer, not the scattered near fields. It therefore eliminates the numerical errors originating from the spatial offset of the E and H fields, inherent in the standard near-to-far-field transformation. The proposed method is validated via direct comparisons with the analytical Lorentz-Mie solutions of plane waves scattered by large dielectric and metallic spheres with strong forward-scattering lobes. c⃝2021 Optical Society of America The grid-based finite-difference time-domain (FDTD) method is one of the most popular Maxwell solvers, which has been proven to be efficient, stable and easy-to-implement [1]. Due to the computational resource limitation, a FDTD simulation truncates the open boundary to a spatial domain adjacent to the scatterer. The near-to-far-field (NTFF) transformation, therefore, is routinely employed to obtain the far-zone information such as antenna scattering patterns and nanocavity radiation patterns [1–8]. The standard NTFF (S-NTFF) transfor- mation, introduced in the early 1980s, is based on the vector Kirchhoffintegral relation [1]. The scattered fields in the far zone are calculated through an integration of the near-zone fields over a virtual closed (Huygens) surface completely enclosing the scatterer [9]. To ac- complish this we need compute, via FDTD and discrete Fourier transformation, the scattered E and H fields tangential to the fictitious surface. However, the spatial and temporal offset between E and H field, a character of the Yee update scheme [1], may result in unacceptable numerical errors [10–12]. For instance, Ref. [13] demonstrated that the accuracy of the S- NTFF is unacceptable when calculating the backscattering from strongly forward-scattering objects, and the relative error may be as high as two orders of magnitude at short wave- length. To improve its numerical accuracy, at least two different modifications have been proposed, including discarding information in the forward-scattering region [13] and using geometric mean in place of arithmetic mean [14]. 1 In classical electrodynamics, the optical response of medium made up of a large number of atoms or molecules originates from the perturbed motions of the charges bound in each molecule. The molecule charge density is distorted by the external electromagnetic fields, and further produces an electric polarization P in the medium [9]. Consequently in this article we proposal using the polarization current, rather than the scattered near fields, to derive the far-zone information. The proposed approach is validated via comparison with rigorous analytical solutions of plane wave scattered by large dielectric and metallic spheres. A detailed discussion regarding its advantages and disadvantages is also presented. In the absence of sources, the Maxwell equations are read as ∇· B = 0, ∇× E = −∂B ∂t , ∇· D = 0, ∇× H = ∂D ∂t . (1) Assuming the following constitutive relations D = ǫ0E+P as well as B = µ0H (the scatterer is therefore nonmagnetic), we arrive an inhomogeneous Helmholtz wave equation for the vector potential A in the Lorentz gauge [9] ∇2A −1 c2 ∂2A ∂t2 = −µ0J(t), (2) where J is the polarization current defined as J = ∂P/∂t [15]. With a time dependence e−iωt understood, Eq. (2) becomes ∇2A + k2A = −µ0J(ω), (3) with k = ω/c being the vacuum wave number. A formal solution of the above equation, with the help of free-space Green function, can be written as [9] A(r) −A(0)(r) = µ0 4π Z v J(r′)eik|r−r′| |r −r′|dr′. (4) The left-hand side represents the scattered wave with A(0) representing the incident wave. To obtain the field in the radiation zone, it is sufficient to approximate the numerator with [9] |r −r′| ≈r −n · r′, (5) while in the denominator |r −r′| ≈r. Here n is a unit vector in the direction of r. The far-zone vector potential of the scattered field is therefore expressed as lim r→∞A(r) ≈µ0eikr 4πr Z v J(r′)e−ikn·r′dr′ ≡µ0eikr 4πr p. (6) It behaves as an outgoing spherical wave while depends on the polar and azimuth angles (θ,φ) in spherical polar coordinate. The scattered fields in the radiation zone, keeping only the leading order, are further given as B = ikµ0 4πr eikrn × p, E = cB × n. (7) 2 500 600 700 800 900 1000 10 -9 10 -8 10 -7 10 -6 10 -5

Scattering magnitude Wavelength [nm] Mie Forward Backward

FDTD (d=1µm) FDTD (d=3µm) Fig. 1. Forward and backward scattering magnitudes of 1-µm and 3-µm diameter dielectric spheres with ǫr = 1.21. Both numerical FDTD solution and rigorous Lorentz-Mie solution are presented. The cell size em

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