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Nanofocusing of mid-infrared electromagnetic waves on graphene monolayer
Weibin Qiu, Xianhe Liu, Jing Zhao, Shuhong He, Yuhui Ma, Jia-Xian Wang, and Jiaoqing Pan
Citation: Applied Physics Letters 104, 041109 (2014); doi: 10.1063/1.4863926
View online: http://dx.doi.org/10.1063/1.4863926
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APPLIED PHYSICS LETTERS 104, 041109 (2014)
Nanofocusing of mid-infrared electromagnetic waves on graphene
monolayer
Weibin Qiu,1,2,a),b) Xianhe Liu,1,a) Jing Zhao,1 Shuhong He,1 Yuhui Ma,1 Jia-Xian Wang,1
and Jiaoqing Pan2
1
College of Information Science and Engineering, National Huaqiao University, Xiamen 361021,
Fujian, China
2
Institute of Semiconductors, Chinese Academy of Science, 100083 Beijing, China
(Received 23 September 2013; accepted 19 January 2014; published online 29 January 2014)
Nanofocusing of mid-infrared (MIR) electromagnetic waves on graphene monolayer with gradient
chemical potential is investigated with numerical simulation. On an isolated freestanding
monolayer graphene sheet with spatially varied chemical potential, the focusing spot sizes of
frequencies between 44 THz and 56 THz can reach around 1.6 nm and the intensity enhancement
factors are between 2178 and 654. For 56 THz infrared, a group velocity as slow as 5 105 times
of the light speed in vacuum is obtained at the focusing point. When the graphene sheet is placed
on top of an aluminum oxide substrate, the focusing spot size of 56 THz infrared reduces to 1.1 nm
and the intensity enhancement factor is still as high as 220. This structure offers an approach for
focusing light in the MIR regime beyond the diffraction limit without complicated device geometry
C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863926]
engineering. V
Nanofocusing of electromagnetic (EM) waves in surface
plasmon polariton (SPP) waveguide structures has attracted
intensive attention due to its promising applications in the
fields of sensing,1–3 tip-enhanced Raman spectroscopy,4
extreme ultraviolet (EUV) generation,5 and optical nonlinearity.6 Until now, various nanostructures, such as taped
gaps,7 nanowedges,8 pyramids,9 and cones,10 have been
demonstrated both theoretically and experimentally to
achieve nanofocusing of EM waves. In many reported structures, the geometry is altered gradiently as the plasmon wave
propagates and the EM field is enhanced as the group velocity slows down, achieving nanofocusing eventually.11–14
Though the nanofocusing of mid-infrared (MIR) or farinfrared (FIR) light is important for various applications,
such as probing,15 spectroscopy,16 and chemical analysis,17
metals like gold and silver commonly used in the structures
mentioned above are not ideal for MIR regime due to the
poor field confinement and the intrinsic mismatch between
MIR and the surface plasmon resonance frequencies of these
metals.18,19
Graphene, a 2-dimentional (2D) material composed of a
single carbon atom layer, has caught intensive interest due to
its unique electronic band structure and promising applications in electronic and photonic devices,20–24 metametarials,25 optical cloaking,26 femtosecond lasers,27 nonlinear
optics,28 and solar cells.29 Furthermore, SPP waves can propagate on graphene with supporting substrate30 and possess
outstanding advantages, such as relatively low loss, high
confinement, and most significantly, tunability of chemical
potential.31 In this letter, we numerically analyze nanofocusing of MIR on a graphene monolayer, which has spatially
gradient chemical potential without complicated device
geometry. This would be an effective alternative to tapered
waveguide structures that have more complicated geometry
and higher cost in terms of fabrication.
For analyzing the mechanism of nanofocusing, the system of interest shown in Fig. 1 is air-graphene-air configuration, where the one-atom thick graphene sheet is located in
an XOZ plane. The SPP waves are launched at x ¼ 0 nm,
propagating along positive x direction. It has been pointed
out that as long as the thickness of the material becomes
extremely small compared to the wavelength, the value of
the thickness loses the meaning, i.e., different thicknesses
yield similar propagation properties and are equivalent in
terms of simulation.30,32 In our simulation, we treat the graphene monolayer as a 2D sheet with zero thickness. Thus, in
the configuration, where the zero thickness graphene sheet is
surrounded by air, graphene itself is the boundary. There is
surface current density J ¼ rs E along graphene, where E is
the electric field and rs is the surface conductivity of the graphene sheet. Since it is 2D current existing in the monolayer
graphene sheet, there is obviously no current in y direction.
The non-zero components of the EM field are Hz, Ex, Ey. The
electrical field component perpendicular to the graphene
sheet, Ey, is discontinuous and anti-phase, while Ex parallel
a)
FIG. 1. Schematic of a monolayer graphene sheet located in the XOZ plane.
SPP waves are launched at x ¼ 0 nm and propagate along positive x
direction.
Weibin Qiu and Xianhe Liu contributed equally to this work.
Author to whom correspondence should be addressed. Electronic
addresses: [email protected] and [email protected]
b)
0003-6951/2014/104(4)/041109/5/$30.00
104, 041109-1
C 2014 AIP Publishing LLC
V
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Qiu et al.
Appl. Phys. Lett. 104, 041109 (2014)
to graphene is continuous and in-phase. Hz is also discontinuous due to the existence of the surface current. The dispersion relation of the SPP wave propagating along the
graphene sheet surrounded by air is given by
b ¼ k0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ð2=g0 rs Þ2 ;
(1)
where b is the complex propagation constant along graphene,
k0 is the propagation wave number in free space, rs is the
surface conductivity of graphene sheet, and g0 (377 X) is the
impedance of the air.30,32 Note that we only consider
transverse-magnetic surface waves.33 rs has the contributions
from both interband electron transition and intraband
electron-photon scattering, i.e., rs ¼ rinter þ rintra . rinter and
rintra can be expressed as
"
#
e2
2jlc j hðx þ is1 Þ
ln
(2)
rinter ¼ i
4ph
2jlc j þ hðx þ is1 Þ
and
rintra
e2 k B T
lc
lc
þ 2ln exp þ1 ;
¼i 2
kB T
p
h ðx þ is1 Þ kB T
(3)
where s is momentum relaxation time, x is radian frequency,
lc is chemical potential, T is temperature, h is the reduced
Planck constant, and kB is the Boltzmann constant.26 In this
letter, we choose the temperature to be 300 K and s ¼ 0:5 ps,
which is conservative for practical graphene.26,34 It should
be noted that the chemical potential lc can be modified
locally by doping35,36 and biased gate voltage,37,38 which is
the particular advantage of graphene over metals commonly
used. In general, the complex wavenumber obtained from
Eq. (1) can be expressed as the sum of the real part and the
imaginary part b ¼ b0 þ ib00 , where b0 denotes the wavenumber, while b00 describes the loss property of SPP waves.
Larger b0 implies shorter wavelength and stronger localization of EM waves.
The wavenumber b0 and the group velocity of SPP
waves on graphene sheet with spatially homogeneous lc are
shown in Fig. 2(a) as a function of frequency f. Here, we
define that for a certain chemical potential, the critical frequency fc is the frequency that gives equal real part and
imaginary part of the complex wavenumber, i.e., b0 ¼ b00 . In
other words, the SPP wave is fully damped. The curve of the
inset of Fig. 2(b) shows critical frequency corresponding to
different chemical potentials. As the frequency approaches
the critical frequencyfc for a certain chemical potential, b0
increases drastically. Since 2p=b0 is the wavelength of the
SPP wave and implies the spatial extension of EM waves,
the confinement of the EM waves along graphene is very
strong and nanofocusing effect is expected around this frequency fc . For example, the critical frequencies for chemical
potentials of 0.12 eV and 0.40 eV are 48.1 THz and 160.9
THz, respectively. Furthermore, the group velocity vg
becomes very low around the critical frequency. Ideally, if b0
goes to infinite as the frequency increases, zero group velocity of the SPP wave would be achieved and the wavelength
FIG. 2. (a) The dispersion relation and group velocity of SPP waves propagating along freestanding graphene with homogeneous chemical potential.
(b) The wavenumber and group velocity of SPP waves as a function of
chemical potential. The inset in (b) shows the critical frequency as a function of chemical potential. The green regime indicates that SPP waves are
allowed and the red regime indicates that SPP waves are fully damped or not
supported. For each chemical potential, calculation is not performed for frequencies higher than its critical frequency.
of the SPP wave would shrink to zero as well, which means
that the EM field is concentrated on a singularity point on
the graphene sheet. Of course, there is loss accompanying
wave propagation and such ideal case is hardly achievable.
Similar behavior will occur when the frequency is kept
fixed and the chemical potential is gradually reduced, which
means that the SPP wave of a certain frequency propagates
along a graphene sheet with spatially decreasing gradient
chemical potential. This scenario is indicated by the blue trajectory in the inset of Fig. 2(b). The green regime means
b0 > b00 and the red regime means b0 < b00 or SPP waves are
not supported. As the chemical potential decreases, b0
becomes larger, the confinement of the light field becomes
stronger, and the group velocity slows down. Together with
the slowing down of group velocity, the intensity accumulates and becomes increasingly higher, which is essentially a
result of energy conservation.39 As the point follows the trajectory and approaches the red regime in the inset of Fig.
2(b), the chemical potential goes sufficiently low and the
loss of the SPP waves becomes higher and higher at the
same time. Therefore, nanofocusing effect actually depends
on the competition between energy accumulation and loss
during propagation. As long as the energy accumulation rate
far exceeds energy loss rate, nanofocusing effect can be
expected. Additionally, if the launched wave has several frequency components, different frequencies would be focused
at different positions on a graphene sheet with appropriately
designed chemical potential distribution.
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Appl. Phys. Lett. 104, 041109 (2014)
To numerically justify the discussion above, we conducted finite element method simulation using COMSOL
Multiphysics Ver 4.3b to compute the light field of SPP
waves propagating along a graphene monolayer with gradient chemical potential lc . The graphene sheet is in the XOZ
plane at y ¼ 250 nm and SPP waves are launched at
x ¼ 0 nm. The distribution of chemical potential lc over the
graphene sheet is a function of position. In our simulation, it
is defined to be
lc ¼ lc0 ½1 expð0:015ðx 1000ÞÞ;
(4)
where lc0 ¼ 0:4 eV is the chemical potential at x ¼ 0 nm.
The purpose of choosing this distribution is nothing more
than having a spatially rapidly varied chemical potential.
The frequencies of simulated SPP waves are 44, 47, 50, 53,
and 56 THz. The corresponding wavelength in free space is
6.81 lm, 6.38 lm, 6.00 lm, 5.66 lm, and 5.35 lm. The evolution of jEj during propagation at y ¼ 250 nm, i.e., along the
graphene sheet, is shown in Fig. 3(a) and the values have
been normalized with the norm of electric field at the launching point jE0 j. The intensity enhancement factor R
¼ jEpeak j2 =jE0 j2 reaches 654 for 56 THz and even 2178 for
44 THz. The full width at half maximum (FWHM) of the focusing spot of 56 THz is 1.6 nm. It is obvious that other frequencies have spot size on the same order, which indicates
that nanofocusing effect is achieved for different frequencies
FIG. 3. (a) The evolution of electric field norm around the focusing spots
along the graphene sheet at y ¼ 250 nm. The values are normalized by the
electric field norm at the launching point. The inset shows how the chemical
potential varies near the focusing spots. (b) The time averaged energy density distribution near the focusing spots. The values are normalized by the
time averaged energy density at the launching point. If five frequencies are
launched together, they will be focused at different positions. The inset in
(b) is a magnification of the focusing spot of 44 THz.
on the same structure. The five distinct focusing spots of the
frequencies are clearly shown in Fig. 3(b), where the time
averaged energy density of each frequency is normalized by
the time averaged energy density at the launching point.
Since the position of the focusing spot depends on the chemical potential distribution, it can be tuned by modifying the
distribution of chemical potential. Meanwhile, the intensity
enhancement factor R might be changed consequently as
well. Passing the focusing spot, the SPP waves vanish due to
EM loss.9 Before the SPP waves reach the focusing spot, the
local chemical potential is far above hx=2, rinter has a small
real part and the loss is dominated by intraband electronphoton scattering, thereby bringing very limited loss to the
propagation of SPP waves. However, when the chemical
potential continues decreasing and reaches around hx=2, the
contribution of loss from interband electron transition indicated by Eq. (2) becomes more significant and eventually
dwarfs the counterpart from intraband electron-photon scattering,26,27 rendering the SPP waves to vanish. In a way similar to defining critical frequency, the critical chemical
potential lc for a certain frequency can be defined as the
chemical potential that gives b0 ¼ b00 . It is worthwhile to
point out that the position of the focusing point is determined
by the location of the criticallc , while the size of the focusing point is fully controlled by how rapidly lc approaches
the critical value. Practically, one should keep the chemical
potential far above hx=2 to avoid propagation loss and then
reduce it to the critical value as rapidly as possible at the
desired position to achieve nanofocusing. Under the chemical potential distribution governed by Eq. (4), the group velocity remains almost unchanged before the chemical
potential falls down rapidly. Take 56 THz as an example. At
the FWHM of the focusing spot, the group velocity drastically slows down to 5 105 times the light speed in vacuum, effectively enabling energy accumulation in the
vicinity of the focusing spot.
From a practical point of view, graphene may be supported by a substrate rather than remain isolated in free
space. We apply the same principle of nanofocusing to this
case though the substrate can introduce quantitative difference to the dispersion relation. The jHj field loses the symmetry along y direction, albeit the jEj field keeps symmetric
over the air and the substrate. According to eigen SPP mode
analysis, the EM energy of SPP waves is dominated by electric field. The time averaged EM energy density is given by
hwi ¼ ðl0 lr jHj2 þ e0 er jEj2 Þ=2. Thus, the energy stored in
the substrate at the focusing point is approximately er;sub
times that in the air above the graphene sheet. In order to
demonstrate the ability of graphene sheet on a substrate to
focus SPP waves, aluminum oxide film is used as the substrate in the simulation and 56 THz is the frequency as an
example. The chemical potential distribution remains the
same as the freestanding graphene case. Aluminum oxide
sits below y ¼ 250 nm and the graphene sheet is still in the
XOZ plane at y ¼ 250 nm. The refractive index of aluminum
oxide is approximately 1.48 and the material absorption is
negligible.40 Figure 4(a) reveals the |E| distribution of 56
THz MIR focused at x ¼ 971 nm with FWHM of 1.1 nm and
the intensity enhancement factor R is around 220, showing
that the mechanism of nanofocusing is still valid. Also as
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Qiu et al.
Appl. Phys. Lett. 104, 041109 (2014)
In summary, we proposed and numerically analyzed
nanofocusing of MIR in monolayer graphene with gradient
chemical potential distribution. Nanofocusing was dominated
by the competition between energy accumulation and loss
during propagation. Intensity enhancement factors obtained
on isolated freestanding graphene were between 2178 and
654 for frequencies between 44 THz and 56 THz. For a graphene sheet resting on aluminum oxide substrate, the intensity enhancement factor could still reach 220 for 56 THz. The
proposed structure was planar structure without any tapers,
wedges, grooves, or cones, which were commonly studied in
other literatures. This is an effective alternative to achieving
nanofocusing of MIR without complicated and expensive device geometry engineering. Last but not least, the nanofocusing effect can be tuned by tailoring the chemical potential
distribution and there would be a broad span of applications,
including sensing, nanoimaging, single molecule detection,
optical information process, and ultrahigh density optical
storage.
FIG. 4. (a) The electric field norm around the focusing spot of 56 THz light.
The values are normalized by the electric field norm at the launching point.
(b) The time averaged energy density distribution near the focusing spots.
The value is normalized by the time averaged energy density at the launching point. Below y ¼ 250 nm is the aluminum oxide substrate with a refractive index of 1.48. The graphene sheet is in the plane of y ¼ 250 nm. The
intensity enhancement factor is 220. Pronounced nanofocusing effect still
occurs, though weaker than freestanding case.
The authors are grateful to the support by the National
Science and Technology Major Project under Grant No.
2011ZX02708, the National 863 Project under Grant No.
2012AA012203, the Opened Fund of the State Key
Laboratory on Integrated Optoelectronics under Grant No.
IOSKL2012KF12.
1
expected in Fig. 4(b), the time averaged energy density in
the substrate is er;sub times as high as that in the air. Though
the intensity enhancement factor R is now lower than 654 for
the same frequency in the freestanding case, it is still considerable enhancement. The reason for the drop of intensity
enhancement factor is that the existence of the substrate
underneath changes the loss property of SPP waves. If low
loss is desired for practical application purpose, low refractive index material is preferable. When it comes to fabricating a real structure, several other issues might also be taken
into account. First, rough substrate surface usually introduce
extra light scattering, thereby enhancing the attenuation of
the SPP waves. It is favorable to use substrate with high surface quality. Second, if doping is used for realizing gradient
chemical potential, a reasonably high doping level should be
carefully chosen. Usually dopants may scatter electrons associated with SPP wave propagation and consequently reduce
relaxation time s. It may not be recommended to have
extremely high doping level. Third, if the chemical potential
is controlled by a gate, the electrostatic design should meet
two requirements simultaneously, effective control over the
chemical potential and weak influence on the propagation of
SPP waves. In addition, the theory behind our simulation is
basically Maxwell’s theory. In practice, there might be other
effects involved as well, such as heating effect from nanofocusing and nonlinear optical effect. Understanding the coupling of these effects with nanofocusing is challenging and
significant for practical applications. However, as the very
first attempt to demonstrate the possibility of using graphene
to focus MIR, we have shown that this structure might be a
feasible way to trap, focus, and manipulate MIR light.
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