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where E z
ð H z Þ is the incident electric-field (magnetic-field), and E z
ð H z Þ is the
scattered electric-field (magnetic-field).
7.3.3 Volume Integral Equation Method
As a rigorous solution to Maxwell's equations, a volume integral equation (VIE)
method [ 59 ] is developed to characterize the optical properties of 3D OSCs.
Considering nonmagnetic optical materials with an arbitrary inhomogeneity pro-
file, the VIE can be written as
Þ i 0 xl 0 Z
v
J ð r Þ
i 0 xe ð r Þ e 0
E i ð r Þ¼
G ð r ; r 0 Þ J ð r 0 Þ dr 0
ð 7 : 15 Þ
ð
and
J ¼ i 0 xP ¼ i 0 xe e 0
ð
Þ E
ð 7 : 16 Þ
where i 0 is the imaginary unit, E i ð r Þ is the incident electric field of the light, e ð r Þ is
the position-dependent permittivity of the inhom og eneous materials, J is the
volumetric polarization current to be solved, and G ð r ; r 0 Þ is the dyadic Green's
tensor in free space. The widely adopted approach for solving the VIE is the
discrete dipole approximation (DDA) method [ 73 ]. Due to the hypersingularity of
the Green's tensor and spurious discontinuity of the tangential E-field induced by
the scalar (piecewise constant) basis functions, the DDA method cannot accurately
characterize the subwavelength plasmonic physics [ 74 ] and breaks down in the
multilayered device structure with high-contrast metallic nanostructures. Here, we
develop an alternate algorithm to bypass the difficulties. In our model, the
polarization currents are expanded using the roof-top vector basis functions [ 75 ],
and thus the continuity of normal current is naturally satisfied at the material
interfaces. Furthermore, the hypersingular Green's tensor is smoothened by using
the finite-difference approximation.
From the VIE ( 7.15 ), the scattered electric field generated by the volumetric
polarization current J can be written as
E s ð r Þ¼ i 0 xl 0 Z
v
G ð r ; r 0 Þ J ð r 0 Þ dr 0
ð 7 : 17 Þ
Considering the Cartesian coordinate system, we use the short notation ð u 1 ; u 2 ; u 3 Þ
instead of ð x ; y ; z Þ , then we have
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