Optics and Lasers in Engineering 193 (2025) 109059

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Optics and Lasers in Engineering

journal homepage: www.elsevier.com/locate/optlaseng

High modulation depth, ultra-compact, and tunable electro-optical

modulator based on dielectric-loaded graphene-plasmonic waveguides
Saima Kanwal a , Ali M. Abdulsada b,c , Mir Hamid Rezaei d,*
a
School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai, 200093, China
b
Biomedical Engineering Department, College of Engineering, University of Warith Al Anbiyaa, Karbala, 56001, Iraq
c
Technical Institute of Karbala, Al-Furat Al-Awsat Technical University, Najaf, Iraq
d
Department of Communications and Electronics, School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran

A R T I C L E I N F O A B S T R A C T

Keywords: In this study, we propose a novel ultra-compact, tunable electro-optical (EO) modulator based on dielectric-
3D-FDTD loaded graphene-plasmonic waveguides (DLGPW). By leveraging the exceptional optical and electrical proper­
Electro-optic modulator ties of graphene, the modulator achieves high modulation depth (MD) and extinction ratio (ER) for operation in
Dielectric-loaded graphene-plasmonic wave­
the mid-infrared (MIR) region. The device utilizes surface plasmon polaritons (SPPs) excited at the graphene-
guide (DLGPW)
High modulation depth and extinction ratio
dielectric interface, enabling strong light confinement and low propagation losses. By adjusting the chemical
Surface plasmon polariton (SPP) potential of graphene through an external voltage, the refractive index of graphene is tuned, allowing for precise
Chemical potential tuning control over the resonant wavelength and modulation characteristics. Through detailed numerical simulations
using the three-dimensional finite-difference time-domain (3D-FDTD) method, we demonstrate that the proposed
modulator achieves an MD of 98.69%, an ER of 18.835 dB, and IL of 2.5 dB, with a modulation bandwidth of 1.13
THz. The ultra-compact design and compatibility with CMOS technology make the proposed modulator a
promising candidate for next-generation photonic systems, particularly in applications requiring high-speed and
efficient modulation in MIR spectral ranges.

1. Introduction strong light confinement in subwavelength regions, low losses, and a

tunable refractive index [6]. The refractive index of graphene can be
Plasmonic devices are structures that can confine light in very small easily changed through chemical doping or the application of external
areas, making the overall size of the structure much smaller. These voltage. The combined optical and electrical properties of graphene
structures can excite electromagnetic waves called surface plasmon make it an ideal material for use in optoelectronics [7,8]. So far, various
polaritons (SPPs) at the junction of metal and dielectric, which propa­ optical devices utilizing graphene have been developed, such as
gate along the interface. However, the electromagnetic field is greatly switches [9,10], filters [11,12], Boolean logic gates [13,14], sensors [15,
weakened inside the materials [1–3]. The term "polariton" refers to the 16], and modulators [17–19].
oscillation of the bound electrons of the metal in coupling with exciting An optical modulator alters a constant optical beam that travels
photons. In this case, photons cause surface plasmon excitation, and the through free space or along an optical waveguide [20]. Modulators can
term "plasmon polariton" is used to express the coupling between a adjust various aspects of an optical beam, including its amplitude, phase,
photon and a plasmon [4]. Noble metals, such as gold and silver, can or polarization. Additionally, they are categorized into two types,
serve as plasmonic materials in the visible and near-infrared (NIR) fre­ electro-refractive and electro-absorbing, based on whether they change
quency ranges to excite SPPs. However, these materials are not qualified the real or imaginary part of the refractive index [21,22]. The rapid
for mid-infrared (MIR) to terahertz (THz) frequencies because they advancement of photonic integrated circuits has spurred significant in­
experience high losses [4,5]. Since the advent of graphene, there has terest in high-performance electro-optic (EO) modulators, particularly
been a significant transformation in the fields of electronics and pho­ those leveraging the exceptional optoelectronic properties of graphene.
tonics. Graphene’s outstanding electrical and optical properties have led Over the past decade, various graphene-based EO modulators have been
to its use in various optoelectronic applications. Graphene exhibits explored, spanning waveguide-integrated configurations, single-layer

* Corresponding author.
E-mail address: m.h.rezaei@shirazu.ac.ir (M.H. Rezaei).

https://doi.org/10.1016/j.optlaseng.2025.109059
Received 9 March 2025; Received in revised form 15 April 2025; Accepted 3 May 2025
Available online 9 May 2025
0143-8166/© 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
S. Kanwal et al. Optics and Lasers in Engineering 193 (2025) 109059

Table 1
Drude-Lorentz coefficients of Au metal.
k 0 1 2 3 4 5

ωk - 0.415 0.83 2.969 4.304 13.32

fk 0.76 0.024 0.01 0.071 0.601 4.384
Γk 0.053 0.241 0.345 0.87 2.494 2.214

Fig. 1. (a) 3D schematic of the DLGPW. (b) Cross-sectional view (z-y plane) of
the DLGPW. The geometrical parameters along with the electric field distri­ Fig. 2. Real and imaginary parts of the Au permittivity in the wavelength range
bution of the fundamental mode have been presented. of 7–10 μm.

graphene structures, and double-layer graphene designs, each offering regions, research involving these spectra has been relatively limited in
distinct advantages in terms of modulation efficiency, speed, and foot­ recent years. Consequently, there is a significant need to develop effi­
print. Waveguide-integrated graphene modulators have demonstrated cient and compact silicon-based photonic devices that function effec­
remarkable performance by combining plasmonic or dielectric wave­ tively in these domains. To address this gap, we propose a
guides with graphene’s tunable conductivity. For instance, Liu et al. [17] dielectric-loaded graphene-plasmonic waveguide (DLGPW) EO modu­
developed a hybrid graphene-silicon waveguide modulator achieving lator that combines the benefits of graphene’s tunability with strong
broadband operation with a 30 GHz bandwidth, where light-graphene plasmonic confinement in the MIR range (7–10 µm). Unlike graphene
interaction was enhanced via evanescent field coupling. Similarly, Pan nanoribbon-based designs, our modulator employs a continuous gra­
et al. [23] utilized a one-dimensional photonic crystal cavity coupled to phene sheet, eliminating sensitivity to edge defects. Additionally,
a graphene-loaded bus waveguide, achieving a modulation depth (MD) compared to suspended graphene structures, our approach simplifies
of ~12.5 dB at telecommunication wavelengths. These designs benefit fabrication by utilizing a dielectric substrate, ensuring mechanical sta­
from strong light confinement but often face challenges in balancing bility while maintaining high performance.
insertion loss (IL) and modulation depth. Single-layer graphene modu­ The remainder of the paper is structured as follows: Section 2 delves
lators capitalize on the material’s gate-dependent absorption to achieve into the fundamental characteristics of DLGPWs. In Section 3, we
high-speed modulation. Ansell et al. [18] demonstrated a compact introduce the proposed modulator and analyze how geometric param­
plasmonic modulator where a single graphene layer was coupled to a eters and variations in graphene’s chemical potential influence the
metallic waveguide, enabling an extinction ratio (ER) of >15 dB with transmission spectrum. This section also outlines modulation features,
low power consumption. However, such structures typically suffer from including modulation depth and extinction ratio. Lastly, Section 4 pre­
limited modulation depth due to graphene’s weak single-pass absorption sents the conclusion.
(~2.3% per layer). To mitigate this, resonant structures, such as ring
resonators and Fabry-Perot cavities, have been employed to enhance 2. Characteristics of DLGPWs
light-graphene interaction [20]. Double-layer graphene configurations,
separated by a thin dielectric spacer, have been proposed to enhance The schematic of the DLGPW is illustrated in Fig. 1. A one-atom-thick
modulation efficiency via capacitive tuning. Phare et al. [19] demon­ graphene sheet is placed on a dielectric substrate with a thickness of h2.
strated a double-layer graphene modulator integrated with a silicon A dielectric strip with a width of w and a height of h1 μm is deposited on
nitride waveguide, achieving a 30 GHz bandwidth and improved MD top of the graphene sheet. Additionally, a metallic layer beneath the
through enhanced electrostatic control. These designs benefit from substrate functions as an electrode for voltage application. The charac­
stronger light absorption due to dual graphene layers but require precise teristics of stimulated SPPs in a DLGPW are not affected by the edge
alignment and suffer from increased fabrication complexity. Other shapes of the graphene. Furthermore, the fabrication of this type of
graphene-based modulators have also been designed. For instance, plasmonic waveguide is compatible with CMOS technology and is
Rezaei and Shiri [24] numerically investigate a low-loss resonant EO relatively easy to produce. The graphene sheet used in this waveguide
modulator based on a suspended graphene plasmonic waveguide. They can be fabricated simply through standard lithography processes [26].
achieved an IL of 1.3 dB, an ER of 22 dB, and a modulation bandwidth of For the sake of simplicity, the refractive index of both the substrate
71 GHz in a structure with a footprint of less than 3 μm2. Kim et al. [25] and the dielectric strip is set to nd = 2. Moreover, gold (Au) is selected as
reported an optical modulator consisting of a PhC structure and a the metallic layer. The permittivity of Au is calculated using the well-
waveguide with graphene, using the electromagnetically induced known Drude-Lorentz model, as follows [27]:
transparency-like (EIT-like) transmission with an MD of 98% at around 1
THz. Despite the numerous attractive applications in the MIR to THz

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S. Kanwal et al. Optics and Lasers in Engineering 193 (2025) 109059

Fig. 3. (a) Real and (b) imaginary components of the graphene surface conductivity for different μc values in the wavelength range of 7–10 μm.

K
∑ fk ω2p
ε(ω) = εr,∞ + (1)
k=0
ω − ω2 + jωΓk
2
k

The parameter εr,∞ represents the dielectric constant at infinite fre­

quencies, while ωp denotes the plasma frequency. Additionally, the pa­
rameters fk, ωk, and Γk refer to the strength, resonant frequency, and
damping frequency of the kth oscillator, respectively. The Drude-
Lorentz model enhances the original Drude model by incorporating
the effects of bound electrons, utilizing damped harmonic oscillators to
characterize small resonances observed in the spectral response of
metals. By including the Lorentz term, the range of validity for the Drude
model is expanded. The coefficients of the Drude-Lorentz model for Au
metal, taken from Ref [28], are presented in Table 1. Fig. 2 illustrates the
real and imaginary components of the permittivity of Au across the
wavelength range of 7 to 10 μm.
Two-dimensional materials like graphene are usually characterized
by their surface conductivity. The surface conductivity of graphene (σ g)
is expressed by the Kubo formula with parameters of angular frequency
(ω), chemical potential (μc), scattering rate (Γ), and temperature (T) as
follows [29]: Fig. 4. Distribution of the electric field |E| of the fundamental mode of the
⎡ ⎛ ⎞⎤ DLGPW at different (a) wavelengths for μc = 1 eV and (b) graphene’s chemical
ie2 kB T μc μ
− c ⎠⎦ ie2 2|μc | − (ω + i2Γ)ℏ potentials at λ = 8.5 μm.
σg = 2 ⎣ + 2ln 1 + e
⎝ k B T + ln
πℏ (ω + i2Γ) kB T 4πℏ 2|μc | + (ω + i2Γ)ℏ
method. In this paper, we have used the Ansys Lumerical, FDTD pack­
(2)
age, version 2020 [31], to simulate our structures. Perfectly matched
Also, kB, ħ, and e are the Boltzmann’s constant, the reduced Planck’s layers (PMLs) have been used to prevent light reflections. PML layers
constant, and the electron charge, respectively. The complex permit­ absorb the incident light hitting the simulation boundaries [32]. A local
tivity of graphene (εg) is calculated using the following equation [14]: mesh with fine mesh sizes has been used to accurately take into account
σg the effect of the graphene sheet. Moreover, the “mode” source was
εg = εr + i (3) selected as the light source. This source provides excitable SPP modes in
ε0 ωΔ
the designed waveguide, regardless of how the SPPs are excited. Fig. 4
where εr is the permittivity of the dielectric layer underlying graphene, shows the dependence of the stimulated fundamental mode on the
ε0 is the free space permittivity, and Δ is the effective thickness of gra­ wavelength and graphene’s chemical potential. Fig. 4(a) illustrates the
phene. A change in the surface conductivity of graphene results in a distribution of the electric field within a DLGPW with geometric pa­
change in the permittivity and, consequently, the optical properties of rameters of w = h1 = h2 = 100 nm and a graphene chemical potential of
graphene. At the room temperature of T = 300◦ K and with a scattering μc = 1 eV at three distinct wavelengths of λ = 7 μm, 8.5 μm, and 10 μm.
rate of Γ = 0.11 meV, the graphene surface conductivity is determined This visualization is crucial for understanding how the electric field
by the chemical potential at any given wavelength. Fig. 3 demonstrates behaves in such waveguides, which are significant in the field of
the real and imaginary components of the σ g for different μc values in the nano-photonics and plasmonics due to their ability to confine light at
wavelength range of 7–10 μm. Graphene behaves like a noble metal and subwavelength scales. At λ = 7 μm, the electric field distribution appears
can support the transverse-magnetic (TM) mode of SPPs Of spps when to be more tightly confined, indicating stronger plasmonic effects at this
the imaginary component of the σ g is positive [4,30]. Therefore, higher wavelength. At λ = 8.5 μm, the confinement of the electric field distri­
μc values represent more metallic behavior. Fig. 3 reveals the important bution is decreased, suggesting a balance between field intensity and
point that both the real and imaginary components of the surface con­ spatial extent. At λ = 10 μm, the electric field distribution is less
ductivity of graphene increase with increasing chemical potential and confined compared to the shorter wavelengths. This indicates weaker
wavelength. plasmonic effects and a broader spatial distribution of the field. More­
The waveguide characteristics of the DLGPW are investigated using over, Fig. 4(b) depicts the distribution of the electric field for λ = 8.5 μm
the three-dimensional finite-difference time-domain (3D-FDTD) at three different chemical potentials of μc = 0.4 eV, 0.7 eV, and 1 eV.

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S. Kanwal et al. Optics and Lasers in Engineering 193 (2025) 109059

Fig. 5. (a) Distribution of the electric field |E| for the first six lowest-order
modes of the DLGPW with w = 300 nm, h1 = h2 = 100, μc = 0.7 eV, and λ =
8.5 μm. (b) Real part of Neff versus the width of the DLGPW for μc = 1 eV at λ =
8.5 μm. Fig. 6. (a) 3D schematic of the proposed EO modulator. (b) Top view (y-x
plane) of the dielectric strips and the metal resonators with the geometrical
parameters. (c) Cross-sectional view (z-y plane) of the modulator along with
This representation is fundamental for elucidating the influence of
how to apply external voltage to the device.
graphene’s chemical potential on the spatial distribution of the electric
field. From a visual perspective, it is clear that at μc = 0.4 eV, the electric
the real part of Neff as a function of the waveguide width at a wavelength
field distribution is strongly confined within the waveguide. At μc = 0.7
λ = 8.5 μm for μc = 1 eV and with the same aforementioned geometrical
eV, the electric field distribution becomes less confined compared to the
parameters. It is seen that the waveguide acts as a single-mode wave­
μc = 0.4 eV case, and at μc = 1 eV, the electric field distribution is even
guide for widths up to w = 160 nm, and for higher widths, it is a multi-
more weakly confined. This indicates that at lower chemical potentials,
mode waveguide. It is worth noting that the number of stimulated
the plasmonic effects are stronger, resulting in a more confined SPP
modes decreases with an increase in the graphene’s chemical potential.
mode. Physically, the change in the distribution of the electric field is
attributed to the alteration in the effective refractive index (Neff) of the
3. Proposed DLGPW-based EO modulator
waveguide, which results from variations in the μc of graphene. As μc
changes, the surface conductivity and, consequently, the permittivity of
Fig. 6 shows the proposed EO modulator, which comprises four
graphene are affected, leading to a change in the Neff of the waveguide.
dielectric strips and two metal resonators positioned on a graphene
The values of the real and imaginary components of Neff, shown in the
sheet. Apart from the input and output strips, two of the strips serve as
boxes above the electric field distributions, both decrease as μc increases.
coupling waveguides, where the stimulated mode resonates within this
The decline in the real part of Neff leads to reduced confinement of light
region alongside the metal resonators. The geometrical parameters of
within the waveguide. Meanwhile, the decrease in the imaginary part
the modulator are the width of the strip waveguides (w), the length of
results in lower waveguide losses, increasing the propagation length of
the coupling waveguides and the resonators (L2), the coupling region
SPPs O. spps within the waveguide.
between the input/output waveguides and the coupling waveguides
As previously mentioned, increasing the λ and the μc have a similar
(L1), the gap between the input/output waveguides and the coupling
effect, leading to increases in the real and imaginary components of the
waveguides (g1), the gap between the coupling waveguides and the
σ g. Fig. 4 confirms this subject, showing that the confinement of the
resonators (g2), the height of the dielectric strips and the resonators (h1),
electric field within the waveguide decreases as the λ and the μc are
and the height of the dielectric substrate (h2). An external voltage is used
increased.
between the metal resonator and the metallic layer beneath the dielec­
Apart from the wavelength and chemical potential of graphene, the
tric substrate to adjust the chemical potential of the graphene sheet.
width of the dielectric strip significantly affects the number of excited
The effects of geometrical parameters, including L1, g1, g2, and L2, on
modes in the waveguide. In Fig. 5(a), the electric field distribution of the
the transmission spectrum of the modulator in the wavelength range of
fundamental mode and higher-order modes of the waveguide at λ = 8.5
7–10 μm are illustrated in Fig. 7. The initial values of the parameters are
μm for w = 300 nm is shown. The other parameters of the waveguide are
as follows: w = h1 = h2 = 100 nm, L1 = 1 μm, g1 = 20 nm, g2 = 0, L2 = 3
h1 = h2 = 100 nm and μc = 0.7 eV. It is observed that from the fourth
mode and above, electromagnetic energy leaks out of the waveguide
μm, and μc = 0.7 eV. In all subsequent simulations, the values of the
parameters w, h1, and h2 remain fixed. Fig. 7(a) demonstrates how
region, and a large portion of the optical power is transmitted at the
transmission varies with different values of the geometrical parameter
junction boundary of the graphene sheet and air. Fig. 5(b) demonstrates
L1, which is adjusted to three distinct levels: 0.6 μm, 0.8 μm, and 1 μm.

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S. Kanwal et al. Optics and Lasers in Engineering 193 (2025) 109059

Fig. 7. Transmission spectrum of the proposed modulator for different values of geometrical parameters, including (a) L1, (b) g1, (c) g2, and (d) L2.

Changes in L1 can result in shifts in the resonant wavelengths or alter­ Here, P0 and P1 represent the optical power in the logical "0″ state
ations in the bandwidth of the transmission dips. For instance, the (OFF) and logical "1" state (ON), respectively. When the transmission
modulator exhibits a transmission of − 18.15 dB at the resonant wave­ value decreases from − 2.5 dB to − 19.91 dB, the ER becomes 17.41 dB,
length of λ = 8.36 μm for L1 = 0.6 μm. and the MD reaches 98.18%.
The insertion loss (IL) parameter can be calculated using the The parameter g2 specifies the resonance intensity, where a lower g2
following equation [33]: value leads to stronger resonances, as demonstrated in Fig. 7(c). With
parameters set as L1 = 0.6 μm, g1 = 10 nm, L2 = 3 μm, and μc = 0.7 eV,
Pin
IL(dB) = 10log (4) changing g2 from 0 to 10 nm results in a resonance wavelength shift of
Pout
20 nm, yielding ER and MD values of 16.94 dB and 97.97%, respectively.
where Pin and Pout represent the optical power at the input and output These changes, although measurable, are relatively minor. However, for
ports, respectively. The IL can be determined from the transmission larger variations in g2, the modulator’s performance is significantly
spectrum, where the difference between unity transmission and the affected. For example, setting g2 to 20 nm leads to two dips in the
transmission at the output port indicates the IL. Consequently, an IL of transmission spectrum at wavelengths of λ = 7.37 μm and λ = 8.39 μm.
about 2.5 dB is obtained for the proposed modulator. At λ = 8.39 μm, the ER and MD values are 14.3 dB and 96.28%,
The parameter g1 indicates the optical power exchanged between the respectively.
input/output waveguides and the coupling waveguides. Fig. 7(b) ex­ The impact of changes in L2 on the modulation characteristics of the
plores how the value of g1 affects the transmission spectrum for L1 = 0.6 proposed modulator is discussed. Fig. 7(d) illustrates the transmission
µm, g2 = 0, L2 = 3 µm, and μc = 0.7 eV. It is noted that lower values of g1 spectrum of the modulator for L2 values ranging from 2.6 μm to 3.4 μm
result in more significant dips in the transmission spectrum. For in increments of 200 nm. In this scenario, the length of the coupling
instance, when g1 is set to 10 nm, a notable dip is observed, with a region (L1) remains constant at 1 μm, while other parameters are set to g1
transmittance of − 19.91 dB at a wavelength of λ = 8.06 µm. If g1 is = g2 = 10 nm and μc = 0.7 eV. The optimal modulation behavior occurs
increased to 20 nm, two dips are recorded with transmission values of when L2 is equal to 2.6 μm, resulting in an ER of 18.835 dB and an MD of
− 17.2 dB and − 17.36 dB at wavelengths of λ = 7.6 µm and λ = 8.48 µm, 98.69% at a wavelength of λ = 7.92 μm. While high ER and MD values
respectively. Thus, changing g1 from 10 to 20 nm decreases resonance are achieved for L2 = 2.8 μm, the wider bandwidth associated with this
intensity by 2.55 dB. Furthermore, a lower transmission value at the setting is less desirable for modulation applications.
resonant wavelength corresponds to higher ER [34] and MD [35] values. As an EO modulator, the resonant wavelength can be adjusted by
changing the chemical potential of the graphene sheet. The design of the
P1 modulator features a tunable capacitor formed between the graphene
ER(dB) = 10log (5)
P0 sheet and the metallic layer located beneath the substrate. By applying
an external voltage to the metal resonator on the graphene sheet and the
P1 − P0
MD(%) = × 100 (6) underlying metallic layer, we can alter the carrier concentrations within
P1
this capacitor. The relationship between the chemical potential of gra

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S. Kanwal et al. Optics and Lasers in Engineering 193 (2025) 109059

output side. The metallic resonators function as perfect mirrors, which

leads to multimode interference (MMI) over the entire length of L2.
To calculate the modulation bandwidth of the modulator, it is
necessary to find several parameters, including the total capacitance of
the device (CT), consisting of the gate capacitance (CG) and quantum
capacitance (CQ), the graphene sheet resistance (Rg), and the contact
resistance (Rc). The gate capacitance is CG = 0.32 fF, which is obtained
by multiplying the area of the modulator’s active region (L × weff ≈ 3 μm
× 0.6 μm) by the per-unit capacitance (Cp). The quantum capacitance is
in series with the CG which is expressed by [37]:
[ ( )]
2e2 kB T μ
CQ = ln 2 1 + cosh c (8)
π(ℏυF ) 2
kB T

Considering the chemical potential of 0.7 eV, the quantum capaci­

tance of CQ = 16.54 µF/cm2 is obtained. Due to the very high value of
CQ, this capacitance is neglected. Therefore, the total capacitance of the
Fig. 8. Transmission spectrum of the proposed EO modulator for various values device is equal to the gate capacitance (CT = CG). Utilizing the high
of the chemical potential of graphene. carrier mobility of a graphene sheet allows us to overlook the carrier
transition time in graphene when compared to the RC time constant,
phene and the external voltage can be expressed as follows [23]: which arises from the series contact resistance combined with the ohmic
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ resistance. It has been reported that the resistance of the graphene sheet
( )̅
Cp |Vext | ranges from 100 to 300 Ω/sq, while the contact resistance falls between
μc = ℏvF π n0 + (7)
e 100 and 1000 Ω.μm [24,38,39]. For our highly doped graphene, if we
consider the graphene sheet resistance to be Rg = 200 Ω/sq and the
Here, n0 is the intrinsic carrier concentration of graphene, νF = 106 contact resistance to be Rc = 600 Ω.μm, the overall ohmic resistance of
m/s is the Fermi velocity, and Cp = ε0εr/t = 0.0177 μF/cm2 is the the device is estimated to be around 440 Ω, using the following equation
effective per-unit capacitance, where ε0 is the free space permittivity, εr [40]:
is the permittivity of the dielectric layer beneath the graphene sheet, and
t is the dielectric spacer thickness [36]. Fig. 8 illustrates the transmission 2Rc Rg × weff
R= + (9)
spectrum of the modulator for various values of graphene’s chemical L L
potential. This figure shows a sensitivity of Δλres/Δμc = 7 nm/meV, Therefore, a modulation bandwidth of 1.13 THz is achieved for the
where Δλres denotes the variation in the resonance wavelength and Δμc proposed modulator using the f3-dB = 1/2πRCT. The equivalent circuit
indicates the changes in graphene’s chemical potential. model of the modulator is illustrated in Fig. 10. Additionally, Table 2
To better understand the performance of the modulator, we examine compares the modulator parameters, including wavelength, extinction
the distribution of the magnitude (|E|) and the x-component (Ex) of the ratio, insertion loss, bandwidth, and length, with those of other
electric field at two resonant and non-resonant wavelengths. Figs. 9(a)
and (c) illustrate the field distributions at the resonant wavelength of λres
= 7.92 μm. At this wavelength, the propagating mode is scattered due to
destructive interference that occurs in the coupling region on the output
side. Conversely, Figs. 9(b) and (d) demonstrate that high transmission
of − 2.75 dB at the non-resonant wavelength of λnon-res = 8.25 μm results
Fig. 10. Equivalent circuit model of the proposed modulator.
from constructive interference occurring in the coupling region on the

Fig. 9. Distribution of the magnitude of the electric field (|E|) at the (a) resonant wavelength of λres = 7.92 μm and (b) non-resonant wavelength of λnon-res = 8.25 μm.
Distribution of the x-component of the electric field (Ex) at the (c) resonant wavelength of λres = 7.92 μm and (d) non-resonant wavelength of λnon-res = 8.25 μm.

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S. Kanwal et al. Optics and Lasers in Engineering 193 (2025) 109059

Table 2
Comparison between the results of the proposed modulator with those of other graphene-based plasmonic waveguides.
Ref. Modulator structure λ (μm) ER (dB) IL (dB) Bandwidth (GHz) L (μm)

[40] Hybrid plasmonic modulator with graphene and Al2O3 3.3–3.7 – 0.53 156.49 (GHz) 40
[41] Graphene-HfO2-ITO stack embedded in silicon 1.55 14.6 2.98 256.9 1
[42] Electro-absorption modulator based on graphene-on-silicon slot 1.55 28 1.28 117 120
[43] Dual-slot hybrid plasmonic electro-absorption modulator 1.55 – 1.89 – 10
[44] Plasmon-enhanced graphene all-optical modulator 1.55 3.5 6.2 100 GHz 12
[45] Graphene-insulator-graphene plasmonic modulator 5 3 0.1 – 0.8
[24] Suspended graphene plasmonic waveguides 7–11 22 13 71 3
This work Dielectric-loaded graphene plasmonic waveguides 7–10 18.835 2.5 1130 3

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