Supplementary Materials
Graphene plasmonic lens for manipulating energy flow
Guoxi Wang, Xueming Liu,* Hua Lu, and Chao Zeng
State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and
Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China
*Corresponding author: liuxm@opt.ac.cn
1. The model in FDTD simulations
Graphene
(a)
(b)
Dielectirc
y
Si
z
Figure S1 (a) Required smooth effective mode index profile of the graphene sheet (blue dashed line)
and the designed stepped profile (red solid line) in FDTD simulations. (b) Schematic of the proposed
idea of stepped profile of dielectric spacer underneath the graphene sheet in order to construct stepped
effective mode index distributions.
As shown in Fig. S1(a), the graphene sheet is divided into a series of ribbons with different
effective mode indices in our FDTD simulations. To achieve the inhomogeneous effective
mode index patterns, the dielectric spacer is designed as a stepped profile in the calculations,
as shown in Fig. S1(b). It can be found that the whole structure is equivalent to the parallel
connection of a series of capacitors due to the stepped profile of the dielectric spacer. Each
capacitor is a parallel plate configuration and the carrier concentration at the graphene surface
can be obtained from Eq. (4). In addition, the capacitance of each capacitor is different due to
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the different thickness of spacer. So the conductivity patterns of the graphene sheet are
spatially inhomogeneous and the distributions of the effective mode index follow a parabolic
profile.
2. The influence of bias voltage on the pitch of the Selfoc lens
(a)
(b)
Figure S2 (a) Lateral profiles of the effective mode index (a) and parameter ζvol (b) as a function of the
bias voltage. In the calculations, the lateral profile of the substrate thickness is the same as that shown
in Fig. 3 and the incident frequency is 40 THz.
Firstly, we investigate the influence of the external bias voltage on the pitch of the proposed
graphene plasmonic Selfoc lens. As shown in Fig. S2(a), when the external bias voltage varies,
the lateral profile of the effective mode index differs, which is similar to the results shown in
Fig. 5(a). We also define a new parameter as ζvol=nvol(Vg, z)/nvol(Vg0, z), where nvol(Vg, z) is the
effective mode index at the position of z for the bias voltage of Vg. Here, Vg0=20 V is the bias
voltage which determines the original lateral profile of h in Fig. 3. As shown in Fig. S2 (b),
we calculate the parameter ζvol for different bias voltages. For a specific bias voltage, the
parameter ζvol keeps constant, and thus the relative change in effective mode index Δn
(=(n0-n1)/n0) is also unchanged. In addition, the lateral profile of nGP for different frequencies
always has the form of Eq. (5), besides n0 and n1 have new values. Thus, for different bias
voltages, the pitch of the Selfoc lens is still determined by P=2π/g with g=2 2Δn/w. Since Δn
2
is constant, the pitch of the Selfoc lens is unchanged for different bias voltages.
3. FWHM of the output SPP waves for different bias voltages
(a)
(b)
Vg=25 V
(c)
Vg=30 V
(d)
Vg=35 V
Vg=40 V
Figure S3 Simulation results of the Selfoc lens, showing the amplitude of the y component of electric
field (Ey) of GP waves for different bias voltages (a) 25 V, (b) 30 V, (c) 35 V, and (d) 40 V, respectively.
In calculations, the lateral profile of h is the same as that in Fig. 3.
Successively, we study the FWHM of the output SPP waves for different external
voltages when the proposed Selfoc lens is used as an optical coupler. For practical
applications, it is important for the optical coupler to couple light into the devices with
different dimensions. So we will demonstrate that the FWHM of the output SPP waves
can be easily tuned by applying different bias voltages. As shown in Fig. S3, the
wavelength of SPP wave λSPP (=λ0/neff) differs for different bias voltages. When the
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voltage increases, the λSPP also increases. This is because the lateral profile of the
effective mode index decreases with the increase of the bias voltage, which results in
the larger λSPP.
110
FWHM (nm)
105
100
95
90
85
80
20
30
40
Voltage (V)
Figure S4 FWHM of the output SPP waves versus the external bias voltage.
In addition, the FWHM of the output SPP waves is also different for various bias voltages.
It is found that the FWHM of the output SPP waves increases with the increase of the bias
voltage as shown in Fig. S4. When bias voltage changes from 20 V to 40 V, the FWHM of the
output SPP waves can be tuned from 81 nm to 106 nm. Therefore, when the Selfoc lens is
used as an optical coupler, it can couple energy flow into optical devices with different
dimensions.
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