A graphene-assisted all-pass filter for a tunable terahertz transmissive modulator with near-perfect absorption
A graphene-assisted all-pass filter for a tunable terahertz transmissive modulator with near-perfect absorption
thang Q. t ran
Published: xx xx xxxx We proposed an all-pass filter based perfect absorber scheme which also can function as a highly efficient transmissive modulator. We theoretically analyzed the proposed scheme using the temporal coupled mode theory and showed that near-perfect absorption could be achieved with practically modest deviation from the critical coupling condition. We also demonstrated the feasibility of the proposed scheme in a grating-based all-pass filter device with a variable loss implemented by two separate graphene layers, achieving an absorption of ~99.8% and a transmission modulation depth of ~70 dB in a terahertz frequency range. We also numerically investigated the tunability of the designed device.
Recently, terahertz (THz) technology has been developed rapidly relying on the advances in source and
detector development1–4 and there have been huge progress in its applications such as bio-medical imaging, security,
time-domain spectroscopy, and communications5. In order to further develop the applications of the THz
technology, the means to control and modulate the THz wave propagation should be developed, which, however, is
still challenging since it is difficult to find materials responding to THz wave. To solve this problem,
metamaterials, which are artificial structures enabling a variety of exotic electomagnetic properties, and their active control
have been studied widely6–12. Recently, graphene has attracted a lot of interest as the means to enable the active
control of the metamaterials due to its easily controllable permittivity via carrier density variation13–20. Moreover,
an ultra-wide absorption bandwidth of graphene makes it considered as a promising candidate for a THz
absorber. A myriad of studies to enhance the low absorption efficiency of atomically thin monolayer graphene
have been conducted and many kinds of graphene-based THz perfect absorbers have been suggested21–32. In most
of the previously suggested THz perfect absorbers, absorption enhancement is achieved by adopting a reflector as
well as a resonant structure such as the metamaterial or a grating. Two representative types of the reflectors used
in the previously THz perfect absorbers are a metallic mirror21–29 and a distributed Bragg reflector (DBR)30–33.
Due to the aforementioned controllable permittivity of graphene, those perfect absorbers based on graphene can
also work as modulators by introducing a proper capacitive structure to apply gate voltage to graphene, in which,
however, only the reflected wave can be modulated because of the reflectors. So, if a perfect absorber can be
realized without a reflector, a transmissive modulator of a high modulation depth is accompanied. In this work,
we propose the reflectorless near-perfect graphene absorber scheme based on an all-pass filter composed of two
identical gratings and numerically demonstrate ~99.8% absorption and its transmissive modulator operation
with a ~70 dB modulation depth in a THz frequency range. We have theoretically investigated the operation
principle of the proposed scheme using temporal coupled mode theory (TCMT)34. The numerical simulation
was conducted using the rigorous coupled wave analysis (RCWA) method35 -based commercially available tool
(DiffractMOD), and the particle swarm optimization (PSO) method36 was used for the device design.
Theory. We performed the TCMT analysis for two lossy resonators with different loss rates of 1/τL1 and 1/τL2,
but otherwise identical, having the same decay rates to the wave propagation channel 1/τ. The schematic diagram
of the coupled resonators is presented in Fig. 1. The quality factor Q of the two resonators is related to the
resonance frequency (ωo) and the decay rate 1/τ of the resonators through the relationship Q = ω0τ/4. The two
resonators are coupled both indirectly through the propagation channel with a phase retardation of θ and directly
through evanescent coupling with a coupling coefficient of µ. The temporal change of the normalized mode
amplitudes of the two resonators (a1 and a2) can be described as follow:
da1 = jω0 −
1 − τ2 a1 − jμa2 + κs+1 + κs+2,
da2 = jω0 −
1 − τ2 a2 − jμa1 + κs+3 + κs+4,
where δ τ. At resonance (ω = ω0), from (
) and (
), we obtain the approximated complex transmission and
where s+i and s−i are the complex amplitudes of the incoming and outgoing waves, respectively, and κ is the
coupling coefficient from the wave propagation channel to the resonators. Because of energy conservation and time
reversal symmetry constraints, μ is real and κ is given by κ = eiθ 2/τ 34.
In this work, we assumed that the all-pass filter conditions for the lossless case was satisfied, which are μ = 2/τ
and θ = π/237. Assuming s+4 = 0, and using the steady state condition with a time harmonic incident wave
s+1 = s+1e jωt, we obtained the complex transmission and reflection coefficients of the system as follow:
s−4 = j
2 + τ
+ j(ω − ω0)τ( τ + ττL2 ) + (ω − ω0)2τ 2
+ j(ω − ω0)τ 2 + ττL2 + j(ω − ω0)τ
s−1 = j
2τ( 1 −
2 + τ
+ j(ω − ω0)τ 2 +
+ j(ω − ω0)τ
), it is found that perfect absorption (r = t = 0) is obtained only when τL1 = τL2 = τ/2, which implies
the balance between the loss rate and the total leakage (coupling to the propagation channel) rate in each
resonator and is the same as the critical coupling (perfect absorption) condition in the lossy one-port resonator with a
reflector38. So, the perfect absorber based on the lossy all-pass filter can be understood as the variation of the lossy
one-port resonator-based perfect absorber: the reflector is eliminated by placing another identical resonator with
a mirror inversion symmetry. In another words, two resonators excited with the phase retardation of θ = π/2 is
equivalent to placing a 100% mirror at the distance of λ/4 from one resonator (halfway between two
However, designing a system of two resonators with identical loss rate is quite difficult to achieve in
practice. Therefore, we would like to analyze how a small deviation from the perfect absorption condition affect the
absorption performance. Assuming the loss rate of the two resonators are given by
τ 2 + 2τ −τ δ2
2 + 2τ + δ2
(2τ + δ2 )(τ2 − δ2 )
Therefore, when δ τ, the transmission and the reflection can be approximated as T = t 2
R = r 2 4δτ2 , respectively. Since the transmission shows the fourth power dependence on the relative loss rate
difference (δ/τ) near the perfect absorption condition, a very low transmission value can be obtained even for
modest deviation from the perfect absorption condition, enabling a high transmission modulation depth with
variation of the loss rate or the resonance frequency. The resulting absorption (A = 1 − T − R 1 − 4τ2
shows quadratic dependence on the relative loss rate difference (δ/τ) near the perfect absorption condition.
Figure 2 shows the transmission and the reflection spectra calculated by (
) with (
) for several values of δ/τ
around the critical coupling condition (τL1 = τL2 = τ/2). The case far away from the critical coupling condition
(τL1 = τL2 = τ), corresponding to the case that the loss is the half of the critical coupling condition and δ = 0, is
also plotted for reference, which is represented by the orange curve. Note that when the loss difference between
two resonators is introduced under the non-critical coupling condition such as τL1 = τL2 = τ, the minimum
transmission value rather increases slightly because the absorption decreases with the critical coupling condition
further ruined by the loss imbalance. So, we conservatively adopted the δ = 0 case for reference. For non-zero δ
around the critical coupling condition, the all-pass filter condition is slightly broken due to the difference between
two resonators, resulting in non-zero reflection. However, the reflection is acceptably small, which is ~−30 dB at
the resonance even for δ = 0.1τ. Fig. 2(b) is the enlarged graph corresponding to the region represented by the
dashed box in Fig. 2(a). As expected, a reasonable 20% loss difference (δ = 0.1τ) around the critical coupling
condition still resulted in quite low transmission of ~−50 dB at the resonance frequency. So, if we can somehow
change the losses of resonators from τL1, 2 = τ ± δ/2 to τL1, 2 = τ/2 ± δ/2, a transmission modulation depth of
~40 dB can be achieved. For 4% loss difference (δ = 0.02τ), the minimum transmission of ~−80 dB and a
modulation depth of ~70 dB can be achieved. If the loss variation is increased by choosing the smaller loss close to zero
(τL1, 2 ~ ∞) for the high transmission state (reference), the larger modulation depth can be obtained in each
case. The estimated absorptions at the resonance frequency are ~99.9% and ~99.99% for the moderate loss
difference values of 20% and 4%, respectively.
The loss in the TCMT modeling is the summation of all kinds of losses such as material absorptions and
scattering/radiation losses of the resonators:
where the background loss represents all the fixed losses that we cannot control. In the ideal case of no
background loss, the variable loss can be changed from zero to 2/τ to achieve a maximum transmission modulation
depth. If there is an unwanted background loss, the minimum loss cannot be zero, so that the minimum
transmission value of the high state will somewhat decrease, but the low transmission state can be the same as the ideal
case by choosing the variable loss to make the total loss 2/τ. The high transmission state (reference) considered
in Fig. 2 corresponds to the case of 1/τbackgroud = 1/τ, in which still a considerably high modulation depth can
be achieved because the proposed transmissive modulator scheme is based on the low state of the near perfect
absorption resulted from the critical coupling condition. Note that the performance (or the minimum
transmission value) of the low state relies on the balance between the total loss and the leakage, not the absolute value of
2/τ, which is associated to the quality factor of the resonator (Q = ωοτ/4). The only constraint for the high
modulation depth with near perfect absorption is that the background loss should be smaller than 2/τ. Our TCMT
modeling reveals that a background loss rate of 1/τ can achieve a considerably high modulation depth of ~70 dB
for a 4% loss difference between two resonators in the proposed scheme.
n ear-perfect absorption and transmission modulation in grating based structure. Based
on the theory described in the previous section, we designed a terahertz near-perfect absorber based on a
graphene-assisted all-pass filter. The schematic diagram of the device is presented in Fig. 3, where two identical
gratings form the all-pass filter, and the graphene layers are added to introduce loss. The-all pass filter consists
of two gratings with thickness tg, a period P, and a fill factor FF (=wg/P), formed on a Si layer of thickness tSi
and separated by an air gap with a distance d. The gratings made of Si or PDMS were considered in this work.
One device with Si gratings was designed, having a quality factor of ~103 and the other with PDMS gratings was
designed to have a much higher quality factor of ~105. The direct coupling coefficient and the phase retardation
in this device are controlled by d and a lateral shift distance s39. Two graphene layers separated by a thin SiO2 layer
with a thickness tgate are embedded in SiO2. The graphene absorbing layers and the all-pass filter are separated by a
distance tbuffer. To tune the loss via variation of the complex permittivity of graphene, a gate voltage Vgate is applied
between the two graphene layers. The complex permittivity of graphene was calculated using the Kubo formula40
with following assumptions: the thickness of graphene was 0.34 nm, the Fermi velocity VF was 106 m·s−1, and
the mobility μ was 1000 cm2·V−1·s−1. The permittivity of Si and SiO2 were assumed to be 3.41672 and 2.12692,
In our design, the structural parameters denoted in Fig. 3 were optimized first to realize the all-pass filter
without the graphene layers at an operating wavelength around 50 µ m, and then, re-optimized to minimize
the transmission at the resonance with the graphene layers, obtaining the near-perfect absorber design. Note
that the introduction of the graphene layers not only adds loss but also changes the all-pass filter condition
slightly. The numerical simulation was conducted using the RCWA method35 -based commercially available tool
(DiffractMOD), and the PSO method36 was used for the device design. In our design process, the grating fill
factor and the gap between the graphene layers were fixed at FF = 0.5 and tgate = 7 nm. When the chemical potential
(Fermi level) of graphene is 0.1521 eV, the optimal design parameters for the device with Si grating were found
as follows: tg = 0.593 µ m, P = 18.775 µ m, tSi = 2.67 µ m, d = 1.486 µ m, s = 7.703 µ m, and tbuffer = 8 µ m. In the rest
of this paper, these parameters are used unless otherwise indicated. Because tbuffer is much larger than d, the loss
rates of the two resonators due to the presence of the graphene layers would be very close, causing the structure
to operate in the near-perfect absorber mode. One can easily decrease both τL1 and τL2 simultaneously simply
by increasing tbuffer, and there should be a proper tbuffer where the near-perfect absorption condition is satisfied.
Therefore, simply by changing tbuffer, one should be able to find a point where nearly perfect absorption occurs
without modifying parameters of the all-pass filter structure. Likewise, one could also change both τL1 and τL2
simultaneously by changing the gate voltage applied between the graphene layers, which causes the change of the
chemical potential of graphene.
Figure 4(a) shows the calculated transmission (solid lines) and reflection (dashed lines) spectra of the designed
device for various chemical potentials of graphene. When Ef = 0.1521 eV, the near-perfect absorption is achieved:
the transmission dip is ~−70 dB is at the resonance and the reflection is ~−30 dB, resulting in an absorption of
~99.84%. When Ef = 0 eV, which corresponds to the case of almost the same and relatively low losses for two
resonators, the reflection is as low as ~−50 dB, but the transmission is quite high, so that we obtain low absorption.
This implies that we can achieve a transmission modulation depth of ~70dB by changing the chemical potential
from Ef = 0 eV (the high transmission state) to Ef = 0.1521 eV (the low transmission state). When the chemical
potential is varied slightly around the optimal value, the resonance frequency variation is negligible, but the
minimum transmission increases considerably due to the broken critical coupling condition resulting from the
loss rate change. One can see that the minimum transmission increases up to ~30 dB for Ef = 0.16 eV. When the
chemical potential is increased to 0.3 eV, the loss of the graphene layers decreases way below the perfect
absorption condition and the transmission becomes rather high with low absorption. At the same time, the reflection
also increases due to the enhanced conducting property of highly doped graphene layers while the relative
difference in the loss rates of two resonators is kept about the same. Since the variation of the chemical potential
changes both the real and the imaginary parts of the graphene permittivity, the shift of the resonance (minimum
transmission) frequency is observed when Ef varies. Figure 4(b) shows the phases of the transmitted waves for
various chemical potentials. The abrupt phase change at resonance is observed when the near-perfect absorption
condition is achieved (Ef = 0.1521 eV), indicating very strong resonance of the all-pass filter despite the presence
of resonator losses. Figure 4(c) shows the comparison between the numerical (RCWA) calculation and the TCMT
modeling given in (2a), where one can see excellent agreement. The parameters used in the TCMT modeling are
listed in Table 1. As the graphene chemical potential varies, the resonance frequency (ωo) and the loss rate (1/τL)
change as expected. Whereas, it appears that the leakage rate of the resonator (1/τ) is mainly determined by the
grating structure, not by the graphene chemical potential for tbuffer = 8 µ m. The relative loss rate difference (δ/τ)
is not affected by the chemical potential change either, which seems to be closed related to the location of the
graphene layers (tbuffer).
We also investigated the resonance frequency tunability of our device. Figure 5(a,b) show the calculated
transmission spectra on linear and logarithmic scales, respectively, for various air gap distances (d) with all the other
parameters kept the same. Quite reasonable performance of ~−30 dB minimum transmission can be achieved
while the operating frequency is fine tuned. Figure 5(c,d) show the phase of the transmission with and without
the graphene layers. One can see that this resonance frequency tunability is inherent to the grating based all-pass
filter structure, and not a consequence of the near-perfect absorber design presented in this work.
Figure 6 shows the transmission modulation performance for various tbuffer, where the trade-off between
the insertion loss and the chemical potential change required for the full transmission modulation. When the
graphene layers are closer to the all-pass filter, the loss is higher and thus, the insertion loss for the high
transmission state is higher (Ef = 0). At the same time, the optimal graphene loss coefficient to satisfy the critical coupling
condition decreases because the field confinement in the graphene layers increases while the quality factor of
the all-pass filer remains the same. Note that the location of the graphene layers does not affect the quality factor
of the all-pass as mentioned earlier. Then, the chemical potential change for the full transmission modulation
becomes smaller. Note that the loss of graphene increases as the chemical potential increases for a wavelength
longer than ~10 μm40. So, in terms of the insertion loss, the larger tbuffer is desirable, which, in return, increase
the chemical potential variation required for the full transmission modulation. If the insertion loss below 1.0 dB
(that is, T = ~80% at Ef = 0 eV) is required, according to our investigation, the optimal condition appears to be
tbuffer = ~8 μm and the low transmission state of Ef = ~0.1521 eV.
In this work, we have provided the theoretical framework of a lossy all-pass filter based near-perfect absorber.
Based on this, we have designed a graphene-assisted all-pass filter, which is composed of two identical
gratings, for a tunable THz transmissive modulator with near-perfect absorption, and numerically demonstrated an
absorption of ~99.8% and a transmissive modulation depth of ~70 dB via graphene chemical potential variation
of ~0.15 eV.
In our design of the transmissive modulators in the previous section, the gratings were assumed to be made
of Si for the convenience of simple design, which has a modest quality factor of ~103. A grating structure with a
quality factor of ~103 was fabricated and a ~80% guided-mode resonance (GMR) reflection peak at λ = 742.5 nm
was experimentally demonstrated41, corresponding to a background loss of ~0.25/τ. This implies that the
constraint for the high modulation depth with near perfect absorption (a background loss smaller than 2/τ) can be
met in a practical grating with a quality factor of ~103. Moreover, ~95% GMR reflection peak of ~700 quality
factor at λ = ~1550 nm (corresponding to a background loss of ~0.06/τ) was fabricated42, where authors claimed
that the high reflection peak was achieved via fabrication process improvement. This implies that the dominant
source of the background loss in gratings is not the material loss, but the scattering loss due to the imperfect
fabrication. Assuming an extremely advanced fabrication technology development, a device with a much higher
quality factor can be designed with a modified grating structure, as well. As an example, we designed the device
with the gratings made of PDMS (n = 1.5290). When the chemical potential of graphene is 0.148 eV, the optimal
design parameters were found as follows: tg = 0.63001 µ m, P = 19.92114 µ m, tSi = 2.49017 µ m, d = 1.50903 µ m,
s = 8.327 µm, and t buffer = 22.2 µm. Due to the decreased leakage rate, the graphene layers are located farther away
from the all-pass filter to satisfy the critical coupling condition. As seen in Fig. 7, the designed transmissive
modulator shows a much higher quality factor close to 105.
In the fabrication of the proposed device, the realization of two identical gratings is quite important. One
possible way to realize two identical gratings is to halve a single large grating and put them together facing each
other. The Si grating structure can be fabricated on a Si substrate and put on the SiO2 (glass) substrate after the Si
substrate is thinned down to the designed thickness. One of the SiO2 (glass) substrate should be prepared to have
two graphene layers, which can be conducted with a standard graphene transfer process and SiO2 depositions
using an atomic layer deposition (ALD) and a low-pressure chemical vapor deposition (LPCVD) processes.
The proposed transmissive modulator with a high modulation depth can be used in terahertz
communications. Since a signal-to-noise ratio after passing through a modulator is directly related to the modulation depth
of the modulator, our high modulation depth device will be quite useful in terahertz communication
applications. Besides, the resonance frequency of our device can be sensitively tuned by the air gap distance change,
which implies the refractive index change in the gap will also change the resonance frequency. This feature can
be exploited for bio-chemical sensor applications. As shown in Fig. 7, the quality factor of our device can be
achieved, which will result in a highly sensitive sensor.
Our proposed scheme can be applied to other types of all-pass filter realized by photonic crystal43, ring
resonator44, or planar types45, potentially opening a new class of devices capable of operating in a wide variety of
environments. Since the critical coupling condition is relaxed, as long as the all-pass filter condition can be achieved,
by tuning the loss rate, we can potentially achieve near-perfect absorption at any wavelength desired.
To theoretically investigate the operation principle of the proposed scheme, we used the temporal coupled mode
theory (TCMT)34. The numerical simulation was conducted using two-dimensional RCWA (a commercial
software, DiffractMOD)35, where more than 300 harmonics were applied to guarantee accuracy near the resonant
frequency. For the optimal device design, the particle swarm optimization (PSO)36 method was applied. In all
RCWA calculations, the complex permittivity of graphene (εg) was calculated using Kubo formulation based on
the local random phase approximation for various Ef40, assuming graphene thickness of 0.34 nm, Fermi velocity
of 106 m/s, and mobility of 0.1 m2/Vs.
This work was supported by the National Research Foundation of Korea (NRF-2017R1A2A2A05001226).
T.Q.T. and S.K. proposed the graphene perfect absorber scheme, T.Q.T. and S.L. completed the numerical
simulations and the theoretic modeling, and S.K. supervised the simulations and the analytical model
development. All the authors discussed the results and contributed to the writing of the manuscript.
Competing Interests: The authors declare no competing interests.
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