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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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