by electrical gating [10,11,28,45]. Further, ω is the angular
frequency of the incident light, and τ is the carrier relaxation
time, which satisfies the relationship τ μE
F
∕ev
2
F
. Unless
otherwise specified, we fix E
F
at 0.6 eV, which is a relatively
conservative value compared with the highest experimentally
feasible values (≥ 0.8eV)[48,49]. The permittivity of
graphene is modeled by an anisotropic dielectric tensor consid-
ering its single-atom thickness. Its z component is set to ε
zz
2.5 on the basis of the dielectric constant of graphite, and
the in-plane components are ε
xx
ε
yy
2.5 iσω∕ε
0
ωt
[10,50,51], where ε
0
is the vacuum permittivity, and
t 1nmis the thickness of graphene. The value of t is rea-
sonable here owing to the large difference between the thick-
ness and the width of the graphene. The simulated results show
excellent agreement for t 0.34, 0.5, and 1 nm, provided that
the mesh is sufficiently fine. We emphasize that in this work we
focus exclusively on the optical properties of the layered system;
thus, we assume that the GNRs are homogeneously doped in
each layer and have uniform distributions of the Fermi energy
E
F
over their surfaces.
3. RESULTS AND DISCUSSION
A. PIT with Single Transparency Window
In recent years, the nanophotonics community has focused on
GNRs as one of the most important building blocks in the nano-
structured graphe ne family because they are relatively easy to
obtain experimentally and they can confine optical fields below
the diffraction limit by supporting localized plasmons (mainly
Fabry–Pérot-like standing wave resonances) [11,47,51]or
propagating plasmons [40,52]. These resonances depend criti-
cally on the ribbon width and optical properties, the control of
which at the atomic scale is a major challenge. In this section,
we exploit the properties of GNRs and the plasmonic coupling
between them to demonstrate an extraordinary PIT effect.
In Fig. 2, we show the results of electromagnetic simulations
of the setup shown in Fig. 1 under excitation by a normally
incident plane wave with θ 0° (that is, with the electronic
field polarized perpendicular to the ULGNRs). Two transmis-
sion dips (absorption peaks) are clearly visible at 4.55 and
3.71 μm. The one at the longer resonant wavelength is dom-
inant for transmission as low as 3.89% (or absorption reaching
30.71%), whereas that at the shorter resonant wavelength is
characterized by a transmission dip of 8.86% (or an absorption
peak of 41.68%), indicating that these two modes are very
strongly coupled under external incidence, as shown in
Figs. 2(a) and 2(b). In Fig. 2(c), we also calculated the delay
times at the two sharp notches, which reach −1.39 and
−1.12 ps, respectively, indicating fast light propagation in the
system. Note that, because the fast light effects occur at the
transmission minima, the better fast light performance comes
at the cost of lower light transmission through the system,
making the fast light effect meaningless.
To understand the physical mechanisms behind this PIT,
we first analyze the system with only the ULGNRs.
Figure 3(a) shows the results of numerical simulations of trans-
mission with only the ULGNRs (upper panel) and only the
LLGNRs (lower panel). It can be concluded that plasmon
oscillations can be strong ly excited when the incident wave
is polarized perpendicular to the transverse ribbon direction.
Under this condition, the GNRs operate in the bright mode.
Plasmons cannot be excited when the polarization of the inci-
dent wave is par allel to the GNRs because of a strong momen-
tum mismatch. In this case, they can be treated as the dark
mode. Excitation of plasmons in GNRs can be understood
by considering the induced charge density ρ
ind
or the induced
dipole/multipole moment p
ind
[see Appendix A, Eqs. (A14)
and (A17)]. As these parameters are both proportional to
the external incidence E
ext
, they will reach their maximum and
minimum when they are perpendicular and parallel to the
GNRs, respectively. Note that these conclusions are in agree-
ment with experimental results [11,53,54 ]. For the two-layer
system, because the ribbons are oriented perpendicular to each
other, the case with only LLGNRs is the same as that with only
the upper layer, except for a polarization angle difference of 90°.
It is indeed this difference that causes the different excitation
efficiencies of the two layers; because the incident waves are
polarized perpendicular to the ULGNRs and parallel to the
LLGNRs (θ 0°), plasmons in the ULGNRs can be excited
directly and therefore represent the bright mode. Conversely,
the LLGNRs represent the dark mode but can be induced
by the bright mode. This bright–dark mode interaction
causes PIT.
This coupling mechanism can be ruled out by examining
the electric field distributions and the corresponding E
z
com-
ponents at the two transmission dips, which are plotted in
Figs. 3(c), 3(f), 3(i), and 3(l). According to the spatial distri-
butions of the E
z
components parallel to the x − y plane, the
mode at 4.55 μm shows an antiphase resonance within the two
layers. This mode is called the quasi-asymm etric mode (QAM)
because it results from the out-of-plane nature of the structure.
The mode at 3.71 μm shows the in-phase resonance, which is
called the quasi-symmetric mode (QSM). In addition, these
Fig. 2. (a) Transmission and (b) absorption spectra of the structure
with normal incidence and polarization angle θ 0°. Solid green
curves and symbols represent the analytical and numerical results,
respectively. (c) Transmission phase (left vertical axis) and delay time
(right vertical axis) of the spectra shown in (a) and (b).
694 Vol. 6, No. 7 / July 2018 / Photonics Research
Research Article