Introduction

Surface waves, characterized by their exponential decay in one dimension, exhibit sub-wavelength confinement and are typically found at the metal–insulator interface, where they are referred to as surface plasmon polaritons (SPPs) at visible frequencies1,2. The strong field localization at the interface of materials with differing electromagnetic properties makes these waves highly attractive for applications such as transmission, sensing, and near-field imaging2,3,4,5,6,7. Due to their versatility, surface waves span a broad range of the electromagnetic spectrum, from radio frequencies to terahertz and optical wavelengths5,8. Various materials such as graphene, metasurfaces, and other advanced surface waveguides enable and enhance this wide spectral behavior9,10,11.

Graphene, a two-dimensional material renowned for its unique tunable and adaptable electromagnetic properties, is particularly promising for waveguiding applications12,13,14,15. Its ability to dynamically control surface wave properties through electrostatic or chemical doping makes it ideal for high-performance applications. Furthermore, surface waves can be guided by dielectric materials with varying permittivities, expanding their functionality across a broader range of frequencies16,17,18,19,20,21,22. Building on this, Pendry demonstrated that perforated conductors with subwavelength holes could guide surface waves in the microwave regime, similar to the way metals with negative permittivity guide waves at optical frequencies23. This groundbreaking work led to the development of spoof surface plasmon polaritons (SSPPs), which mimic metal behavior in the optical spectrum and result in guided wave dispersion similar to that of SPPs24,25,26. SSPPs, therefore, enable the manipulation of waves at frequencies previously inaccessible using traditional SPPs, thereby extending their application potential across the microwave and terahertz domains27,28.

In recent years, line waves, which attenuate in two transverse directions as edge modes with strong confinement along the interface between two surfaces, have garnered increasing attention29. Bisharat et al. introduced the concept of line waves generated by the hybridization of transverse magnetic (TM) and transverse electric (TE) surface waves in the microwave regime30. Line waves are typically realized using dual-impedance surfaces with complementary characteristics, wherein the combination of inductive and capacitive surfaces achieves strong wave confinement31. However, Moccia et al. proposed an alternative approach, using purely inductive or capacitive impedance surfaces without requiring duality, allowing for greater flexibility in the wave design32,33. Moreover, gain–loss effects introduced into these structures further enhance wave propagation, enabling applications to amplify or attenuate specific modes.

Graphene-based metasurfaces have become a powerful platform for the implementation of line waves. These metasurfaces enable the development of reconfigurable circuits capable of controlling both the amplitude and phase of the line waves in specified regions. By adjusting the impedance surface values, it is possible to manipulate the spin of the line wave, further expanding its application potential in waveguiding and spintronic devices31. The use of complementary metasurfaces in these structures decouples forward and backward propagating line modes, significantly improving their performance, especially in guiding waves along curved paths. Despite these advancements, complementary metasurfaces lack a bandgap, which leads to the unwanted coupling between line waves and surface waves.

Xu et al. extended these concepts by exploring line wave guidance through parallel plate waveguides with dual-impedance boundaries. This shows that effective wave propagation can be maintained even through out-of-plane bends34. In addition to this, combinations of dual or non-Hermitian impedance surfaces have been shown to guide coupled line waves, wherein two simultaneous line waves coexist, offering enhanced control over wave behavior and potential applications in complex signal processing systems35. One of the primary challenges in implementing line waves lies in achieving duality between two impedance surfaces, which is necessary to ensure that the surface waves have equal momentum. This duality limitation constrains the operational bandwidth of conventional lines.

In this study, we propose an alternative approach: utilizing two inductive-impedance surfaces with equal impedance values and a gap between them to guide quasi-line modes along the edges of the surfaces. Unlike traditional line waves in the terahertz regime, which operate at a single frequency, our structure exhibits a much broader bandwidth and significantly enhanced electric field intensity. The proposed quasi-line mode is also more sensitive to variations in the impedance surfaces, enabling greater tunability and adaptability. Incorporating graphene into our structure further reduces propagation loss, while the use of multilayer graphene instead of a single layer improves control over dissipation loss in the quasi-line mode. This innovative structure supports longer propagation lengths than conventional line waves, with field distribution showing a strong concentration of the mode at the edges of the impedance surfaces. The proposed structure offers considerable potential for practical applications in fields such as sensing, waveguiding, chiral quantum coupling, and optical diodes. The inherent chirality of the quasi-line modes, combined with the reconfigurable nature of the impedance surfaces, makes this structure an attractive candidate for a wide range of next-generation technologies.

Quasi-line wave propagation in inductive impedance structures

Figure 1a illustrates the general schematic of the proposed structure, which supports the quasi-line mode. The structure consists of two inductive impedance surfaces separated by a sub-wavelength gap. This gap introduces capacitance between the surfaces, transforming the structure from purely inductive to a hybrid configuration that can support edge modes. In traditional designs, inductive impedance surfaces are not capable of supporting edge modes. However, by introducing this sub-wavelength gap, the proposed design enables the propagation of a new type of electromagnetic wave along the edges of self-inductive impedance surfaces, referred to as quasi-line modes.

Fig. 1
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(a) Schematic of the proposed waveguide featuring inductive impedance boundaries with a sub-wavelength gap supporting quasi-line modes. (b) Magnitude distribution of the quasi-line mode in the central plane between the inductive surfaces, highlighting edge confinement. (c) Field profile distribution for one-dimensional waves under different gap conditions (h = λ0/10 and h = λ0/100), showing enhanced confinement for smaller gaps. (d) Comparison of two-dimensional field distributions for various values of h, demonstrating increased edge confinement as h decreases.

In typical implementations of line waves, both inductive and capacitive impedance surfaces are required to satisfy mirror reflection symmetry36. However, the proposed structure does not adhere to this symmetry. The sub-wavelength gap not only makes the structure nearly two-dimensional but also allows for wave propagation similar to line waves without the need for complementary surfaces. Various types of inductive surfaces can be employed to realize this structure. Among these, graphene and graphene-based metasurfaces are excellent candidates due to their tunable properties, which enable them to support surface waves with sub-wavelength confinement37,38,39,40. Therefore, in this study, graphene-based structures are used to implement the waveguide that supports the quasi-line mode. The surface impedance of isotropic surfaces capable of guiding surface waves with both transverse magnetic (TM) and transverse electric (TE) polarizations is expressed by the following equations:

$${Z}_{s}^{TM}=j\frac{{\eta }_{0}}{{\xi }_{TM}}\text{ and }{Z}_{s}^{TE}=-j{\eta }_{0}\times {\xi }_{TE}$$
(1)

Here, ξTM, and ξTE are positive dimensionless parameters, and η0 represents the impedance of free space41,42.

The electromagnetic (EM) simulations conducted in this study employed the finite element method (FEM), a well-established computational technique recognized for its effectiveness in addressing complex problems involving irregular geometries, inhomogeneous materials, and intricate boundary conditions. The simulations were conducted using ANSYS HFSS (High-Frequency Structure Simulator), Version 2024, a commercial FEM-based software platform selected for its high precision in modeling electromagnetic field distributions. The software was developed by ANSYS, Inc. and is available at https://www.ansys.com. This capability made it particularly suitable for analyzing the quasi-line wave structures examined in this work.

To accurately represent the behavior of six-layer graphene, its equivalent surface impedance was first calculated using a custom MATLAB code. In HFSS, graphene was modeled as a zero-thickness sheet with inductive surface impedance properties, employing the surface impedance boundary condition to ensure an accurate depiction of its electromagnetic characteristics. This integrated approach, combining MATLAB computations and HFSS simulations, enabled precise and computationally efficient modeling of the multi-layer graphene structure.

Two main simulation setups were utilized: Driven-mode and Eigenmode, each tailored to specific aspects of the structure’s behavior.

Driven-mode setup

This configuration focused on source-excited scenarios to analyze wave propagation and transmission characteristics. A three-dimensional computational domain with a height of λ0/2\lambda_0/2λ0/2 along the z-axis was defined to capture spatial variations in the electromagnetic fields. Wave ports were placed at the input and output of the waveguide to serve as excitation and data collection points. These ports were designed with uniform cross-sectional lengths to ensure proper field coupling and impedance matching. The FEM first solved a two-dimensional field problem on the port surface to generate an excitation source for the subsequent full-wave 3D simulation. This approach facilitated a detailed analysis of field interactions and transmission properties.

Eigenmode setup

This setup investigated the intrinsic properties of the quasi-line waves, focusing on their dispersion characteristics. The structure was modeled as two adjacent semi-infinite boundaries characterized by surface impedances (ZTM) to define the conditions for guided quasi-line waves. FEM solved the eigenvalue problem of the system, deriving dispersion relations that describe the phase velocity of the guided waves as a function of frequency. This analysis isolated the natural modes of the structure, independent of external excitation, providing valuable insights into the waveguide’s fundamental behavior.

Both setups complemented each other. The Driven-mode configuration simulated the interaction between an excitation source and guided quasi-line waves under realistic boundary conditions at the wave ports. In contrast, the Eigenmode setup focused on the dispersion characteristics under idealized boundary conditions, offering a deeper understanding of the waveguide’s inherent performance.

Figure 1b shows the field distribution of the quasi-line mode at the central plane between the inductive impedance surfaces. This mode emerges due to constructive interference of surface waves with TM polarization at the boundary between two inductive surfaces, creating a mode that shares characteristics with the TM-type mode on both sides of the surfaces.

Figure 1c presents a comparison of field profiles for quasi-line waves under two different conditions: h = λ0/10 and h = λ0/100. By varying the capacitance between the surfaces, the field intensities of waves guided by the inductive impedance surface can be adjusted, which is not possible in traditional line waves. In the case of h = λ0/100, the electric field intensity of the one-dimensional wave is significantly higher than that of previously studied line waves. This highlights the enhanced confinement and propagation efficiency of the proposed structure for one-dimensional waves. As h decreases, the field distribution increasingly resembles that of a traditional line wave but with enhanced confinement along the edges of the self-inductive surfaces. Figure 1d presents a comparison of the two-dimensional field distributions for two different values of h, illustrating the increasing edge confinement as h decreases. As h is reduced, the field distribution increasingly approximates that of a traditional line wave, exhibiting stronger confinement along the edges of the self-inductive surfaces.

Figure 2a illustrates the field distribution of the quasi-line mode across the cross-section of the waveguide. This mode is formed at the interface between dual inductive and capacitive surfaces with differing values of ξ (i.e., ξTM \(\ne\) ξTE), resulting in an asymmetric mode that shifts toward the TM surface.

Fig. 2
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(a) Quasi-line mode distribution at the interface of dual impedance surfaces with different ξ values, illustrating the asymmetry caused by the shift toward the TM surface. (b) Cross-sectional quasi-line mode distribution for the proposed waveguide structure with a gap of h = λ/10, showing the concentration of the field along the edges. (c) Electric field distribution of a traditional line wave at the interface between inductive and capacitive surfaces with equal ξ values. (d) Field distribution of the one-dimensional wave along the edges of purely inductive impedance surfaces for h = λ/100, demonstrating enhanced confinement compared to traditional line waves.

Even though the proposed waveguide consists solely of inductive surfaces, the asymmetry of the mode is evident, as seen in Fig. 2b. For comparison, Fig. 2c displays the field distribution of a symmetric line wave at the interface between inductive and capacitive surfaces with equal ξ values. On the other hand, Fig. 2d presents the field distribution of a one-dimensional wave guided along the edges of purely inductive impedance surfaces where h = λ0/100. This one-dimensional wave shows superior confinement compared to traditional line waves, which is a significant improvement for edge-guided modes. As expected, the inductive surfaces primarily support a TM mode. However, the introduction of the gap between these surfaces facilitates the propagation of a hybrid mode, which combines characteristics of both TM and quasi-line waves along the edges of the inductive surfaces. Figure 3a illustrates the vector field distribution of the inductive impedance surface, where the electric field varies mainly in the two-dimensional x–z plane. Meanwhile, the magnetic field primarily varies along the y-axis, as shown in Fig. 3b, confirming that the mode propagating on the inductive surfaces retains its TM characteristics.

Fig. 3
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(a) Vector distribution of the electric fields. (b) Vector distribution of the magnetic field, showing how the magnetic field varies primarily along the y-axis while remaining tangential to the inductive surfaces.

Further analysis of the vector field distributions for the quasi-line mode across the waveguide’s cross-section is shown in Fig. 4a. As the wave propagates, the electric field shifts from being perpendicular to one inductive surface to becoming perpendicular to the opposite surface. In contrast, for a line wave propagating between dual inductive and capacitive surfaces, the electric field starts tangential to the capacitive surface before transitioning to a perpendicular orientation at the inductive surface.

Fig. 4
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(a) Vector field distribution of electromagnetic waves in the transverse direction relative to the edges of coupled impedance surfaces, demonstrating how the fields are concentrated along the surface edges. (b) Vector field distribution, showing the propagation of EM waves and the hybrid mode characteristics along the waveguide edges.

In the proposed structure, the magnetic field remains mainly tangential to the inductive surfaces with minimal variation, although more noticeable changes occur around the waveguide’s edges. Figure 4b illustrates the vector electromagnetic field distribution of the quasi-line mode. These electromagnetic field variations mainly occur along the z-axis, with minor fluctuations along the longitudinal x-axis, indicating that the propagating mode is indeed hybrid.

Possible implementations of quasi-line waves

Graphene stands out as an exceptional material for the implementation of the proposed waveguide due to its unique ability to guide surface modes with tunable polarization, particularly transverse magnetic (TM) polarization43. One of the most significant advantages of graphene is the tunability of its chemical potential, which allows for precise control over its surface impedance. This control directly affects the behavior of propagating surface waves, making graphene highly adaptable for various waveguiding applications. Additionally, graphene is known for its ability to support surface plasmon polaritons (SPPs)44,45,46,47. This property further enhances graphene’s usefulness in guiding waves at terahertz frequencies and beyond. The surface impedance of a freestanding graphene sheet can be calculated using the following equation:

$${Z}_{s}= \frac{2}{{\sigma }_{g}}$$
(2)

in this equation, the factor of 2 accounts for the two-sided nature of graphene, where waves can propagate on both surfaces. It is important to note that ξ =|η₀/Im(Z_s)| for graphene. The surface conductivity of graphene, denoted by σg, plays a crucial role in determining the surface impedance and, thus, the waveguiding characteristics of the material. The optical surface conductivity of graphene can be described using Kubo’s formula, which accounts for two types of conductivity: intraband conductivity and interband conductivity. Together, these two components define the overall response of graphene to electromagnetic waves at different frequencies and energy levels29,48,49:

$${\sigma }_{g} (\omega , {E}_{f}, \tau ,\text{ T}) = {\sigma }^{intra} (\omega , {E}_{f}, \tau ,\text{ T}) +{\sigma }^{inter} (\omega , {E}_{f}, \tau ,\text{ T})$$
(3)

The intraband conductivity (σintra) is primarily influenced by the movement of charge carriers within the same energy band and is given by the following expression:

$${\sigma }^{intra}= \frac{- j{e}^{2}}{\pi {\hbar}^{2} (\omega -\frac{j}{\tau } )} \left\{{E}_{f}+2{k}_{B}Tln [\text{exp}\left(\frac{-{E}_{f}}{{k}_{B}T}\right)+1]\right\}$$
(4)

Here, Ef is the Fermi energy, T is the temperature (typically 300⁰ K), e is the elementary charge, \(\hbar\) is the reduced Planck constant, \(\omega\) is the angular frequency, and \(\tau\) represents the relaxation time, which is often set to 10–12 s.

The interband conductivity (σinter) results from transitions between different energy bands and can be expressed as:

$${\sigma }^{inter}= \frac{{e}^{2}}{4 \hbar} (\frac{1}{2}+\frac{1}{\pi }\text{arctan}\left(\frac{\omega \hbar-2{E}_{f}}{2{k}_{B}T}\right)+\frac{j}{2\pi }\text{ln}[\frac{{\left(\omega \hbar+2{E}_{f}\right)}^{2}}{{\left(\omega \hbar -2{E}_{f}\right)}^{2}+ {\left(2{k}_{B}T\right)}^{2}}])$$
(5)

In these formulas, σintra accounts for the intraband transitions—essentially the free electron movement within the same band, while σinter represents interband transitions, where electrons move between different energy bands.

Figure 5a illustrates the real component of graphene’s surface conductivity as a function of frequency for various values of Ef. The real part of conductivity determines dissipation losses within the material, which contribute to the real part of the surface impedance. By adjusting the Fermi energy Ef, the dissipation and wave propagation characteristics can be controlled. This tunability allows engineers to fine-tune the material’s response to different frequencies, making graphene ideal for specific applications where loss minimization is critical. Figure 5b displays the imaginary component of graphene’s surface conductivity, which is responsible for the material’s reactive behavior, such as whether it behaves inductively or capacitively. At low frequencies, when σg is negative, graphene exhibits inductive impedance, allowing it to guide surface waves with TM polarization. As the frequency increases and the sign of σg changes, the surface impedance of graphene becomes capacitive, enabling it to support surface waves with transverse electric (TE) polarization.

Fig. 5
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(a) The real part of graphene’s surface conductivity as a function of frequency for various Fermi energy (Ef) values, illustrating how the real component influences dissipation losses and surface impedance. (b) The imaginary part of graphene’s surface conductivity as a function of frequency for different Ef values, showing the transition from inductive to capacitive behavior, which enables the tuning of waveguiding properties depending on the frequency.

This frequency-dependent switching between inductive and capacitive behavior makes graphene an excellent candidate for guiding edge modes with both TM and TE polarizations, depending on the design requirements.

While graphene offers significant flexibility in waveguiding due to its ability to switch between TM and TE polarizations, doped graphene (graphene modified with additional carriers) possesses both capacitive and inductive impedances. However, it is unable to guide waves efficiently30. This limitation arises because the momentum of surface waves in graphene does not align simultaneously with both TM and TE polarizations, which is a requirement for traditional line waveguiding.

In contrast, the proposed quasi-line wave design eliminates the need to satisfy the stringent conditions of duality and momentum equality between surface waves. This design leverages purely inductive surfaces, allowing for quasi-line wave propagation across a broad frequency range. Such a configuration is ideal for applications that require high flexibility in terms of frequency response while still achieving efficient wave steering and guiding.

Quasi-line mode implementation using graphene and multi-layer structures

Figure 6a presents the schematic of the proposed waveguide configuration, which consists of two graphene sheets separated by a narrow gap. A 4 µm thick SiO₂ substrate is positioned between the freestanding graphene layers to provide structural stability and maintain the desired separation. It is essential to highlight that the surface impedance of graphene can exhibit inductive characteristics when the chemical potential is tuned correctly. This tunability allows for the realization of a quasi-line mode that can operate across a broad frequency spectrum, offering significant flexibility in various applications.

Fig. 6
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(a) Quasi-line mode propagating between the edges of graphene structures, demonstrating an inductive electromagnetic response. (b) Imaginary component and (c) real component of the surface impedance of a six-layer graphene structure, plotted as a function of frequency for various ξ values, with the graphene layers modeled as an equivalent impedance surface. (d) The relationship between ξ values and frequency, showing the influence of ξ on the existence and bandwidth of the propagating mode.

To mitigate the effects of dissipation losses, we advocate for the use of multi-layer graphene structures in the waveguide design50. In this approach, multiple layers of graphene are stacked together to form an effective impedance surface. The combined surface impedance is modeled as an equivalent impedance in our simulations, allowing for more accurate predictions of wave behavior. Unlike metallic metasurfaces, which typically exhibit fixed electromagnetic responses, graphene’s tunable impedance enables dynamic control over wave propagation characteristics. This flexibility is particularly important in designing waveguides that must operate across a range of frequencies.

In the case of single-layer graphene, Ef is limited by the material properties and cannot be elevated beyond a certain threshold. However, with multi-layer graphene, the Fermi level can be proportionally increased as more layers are added. This scaling of Ef allows for enhanced control over the total conductivity and surface impedance. The total surface conductivity (\({\sigma }_{g}^{tot}\)) and impedance (\({Z}_{s}^{tot}\)) for a constant Ef value can be expressed as follows:

$${\sigma }_{g}^{tot}= {\sigma }_{g}^{1}+\dots +{\sigma }_{g}^{n}=n{\sigma }_{g}$$
$${Z}_{s}^{tot}= \frac{2}{n{\sigma }_{g}}$$
(6)

where n is the number of graphene layers, as the number of layers increases, Ef can be adjusted to a desired level, allowing for greater control over the waveguide’s performance50.

Figure 6b illustrates the imaginary component of graphene’s surface impedance as a function of frequency for various Ef values. Across a broad frequency spectrum, graphene exhibits inductive characteristics, making it a highly advantageous material for use in open waveguides that support quasi-line modes. In Fig. 6c, the real component of the surface impedance is shown for different Ef values at τ = 1 ps, and T = 300 K. Increasing the Fermi energy reduces the real part of the surface impedance, thereby lowering propagation losses—a critical factor for efficient waveguiding in practical applications.

Figure 6d presents the parameter ξ, which relates to graphene’s surface impedance as a function of frequency. Full-wave simulations indicate that when ξ < 1, no propagating mode exists. By adjusting ξ, it is possible to enhance the bandwidth of the waveguide. However, the use of multi-layer graphene structures limits the range of ξ, causing it to approach unity at lower frequencies. Selecting appropriate Ef values is, therefore, critical for controlling the range in which quasi-line modes can emerge and propagate efficiently.

Guiding traditional line waves along curved paths often results in significant wave scattering, primarily due to the lack of band gaps in the supporting metasurfaces51. This scattering effect, especially pronounced at sharp bends, leads to performance degradation, as highlighted in36.

Figure 7a illustrates how conventional line waves struggle to maintain stability along such bends, exhibiting inadequate performance under these conditions. In contrast, the quasi-line mode offers superior performance along bent paths, particularly when coupled with transverse magnetic ™ surface waves, as demonstrated in Fig. 7b. The quasi-line mode’s ability to handle bends with minimal scattering is largely attributed to the principle of universal spin-momentum locking, a phenomenon commonly observed in surface waves. This principle ensures that the mode experiences significant transverse attenuation, which stabilizes the wave propagation, even in complex geometries52,53.

Fig. 7
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(a) Electric field (E-field) distribution of a conventional line wave propagating along a bent path at f = 8 THz, showing significant scattering and instability. (b) E-field distribution of the quasi-line wave along the same bent path and frequency, demonstrating improved stability and reduced scattering, highlighting the quasi-line wave’s superior performance in maintaining wave confinement and propagation through curved geometries.

The quasi-line mode, capable of coupling with transverse magnetic surface waves, demonstrates outstanding performance even along bent paths. This behavior is due to the phenomenon of universal spin-momentum locking, which is typically observed in surface waves54. To further explore this behavior, we employed point sources placed in the intermediate region between two impedance surfaces to observe the effects of spin-momentum locking. These point sources were designed to be chiral, meaning they possess either spin angular momentum or orbital angular momentum55. By altering the chirality of these sources, we demonstrated that it is possible to control the direction of the excited quasi-line mode, as depicted in Fig. 8. This directional control not only enhances the versatility of the quasi-line mode but also opens up potential applications in chiral quantum processes, where the precise manipulation of wave direction is critical for advanced functionalities.

Fig. 8
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One-way propagation of the quasi-line mode along the boundaries between impedance surfaces, demonstrating directional wave confinement and unidirectional waveguiding. The quasi-line mode remains confined to the edges, ensuring efficient propagation with minimal scattering.

In the next phase of our study, we explore whether the proposed quasi-line waveguide could be implemented using other types of metasurfaces. It is important to note that the proposed waveguide can be constructed using inductive impedance surfaces across a wide range of frequencies, from microwave to optical wavelengths. This versatility makes the design highly adaptable to different electromagnetic environments. One novel approach we propose is the use of a graphene-patch metasurface to facilitate the quasi-line mode. The surface impedance of a graphene patch can be tuned between inductive and capacitive states, depending on its chemical properties. This tunability allows the metasurface to support quasi-line waves effectively.

When the surface impedance is inductive, the graphene patch promotes the propagation of surface waves, enabling the realization of the quasi-line waveguide. This configuration shows great potential for applications in various fields, including optical communication, where the controlled propagation of light along specific paths is essential. Additionally, it can be used in nanophotonics, where miniaturized optical devices rely on precise waveguiding.

Figure 9a presents the schematic of the proposed waveguide design, where two graphene patch structures are placed adjacent to one another, separated by a subwavelength gap of 4 µm. The region between these metasurfaces is filled with a SiO₂ substrate, which provides mechanical support and impacts the wave propagation characteristics. It is important to note that the presence of the gap is essential for supporting the quasi-line mode. Without this gap, the edge mode cannot exist. When the surface impedance of the graphene patch exhibits inductive behavior, a quasi-line mode emerges at the edges of the non-dual metasurfaces. This edge mode behaves similarly to a traditional line wave, with the electromagnetic fields being highly concentrated along the edges of the metasurfaces, allowing for strong wave confinement. For surface wave propagation along the graphene patch, the surface impedance (Zs) can be defined using the following expression56,57,58:

$$Z_{s} \approx \frac{D}{\omega \sigma } j\frac{\pi }{{2\omega \varepsilon_{0} \varepsilon_{r} Dln\left( {\csc \left( {\frac{{\pi \left( {D - w} \right)}}{2D}} \right)} \right)}}$$
(7)

where D = 1 μm is the period of the structure, \(\omega\)=0.8 μm represents the width of the patch, and \({\varepsilon }_{r}\)=3.9 is the dielectric constant of the SiO₂ substrate material. This equation models the surface impedance of the graphene patch and helps predict the waveguiding behavior. Figure 9b shows the surface impedance of the graphene patch for a Fermi energy of Ef = 4, where a four-layer graphene patch is employed to increase the chemical potential and reduce dissipation losses. This multi-layer configuration allows for enhanced wave propagation by minimizing energy loss and optimizing impedance matching. It is noteworthy that the surface impedance of the structure exhibits capacitive characteristics at low frequencies, gradually transitioning to inductive behavior as the frequency increases. This frequency-dependent transition plays a critical role in enabling the proposed waveguide to operate effectively at higher frequencies.

Fig. 9
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(a) Schematic of the proposed quasi-line wave waveguide consisting of adjacent graphene patches with a subwavelength gap. (b) Surface impedance of the graphene patch as a function of frequency, highlighting the transition between capacitive and inductive behaviors. (c) Electric field (E-field) distribution of the quasi-line mode when the graphene patch exhibits inductive impedance at f = 50 THz, demonstrating strong edge confinement. (d) E-field distribution when the graphene patch is capacitive at f = 3 THz, showing the inability of capacitive surfaces to support the quasi-line mode.

Figure 9c depicts the electric field distribution at the operational frequency of f = 50 THz, where the metasurfaces exhibit inductive impedance. The inductive nature of the surfaces enables the quasi-line mode to propagate along the edges with high field concentration, ensuring robust wave confinement.

However, it is crucial to emphasize that the use of purely capacitive impedance surfaces to construct the quasi-line waveguide prevents the propagation of any mode when a gap is present between the structures. Figure 9d illustrates the electric field distribution at a frequency of f = 3 THz, where the surfaces on both sides of the waveguide exhibit capacitive impedance. As evident from the figure, purely capacitive surfaces do not support the quasi-line mode, highlighting the necessity of inductive surfaces for wave propagation in this design. This property can be exploited to create reconfigurable waveguides by using capacitive surfaces to selectively filter quasi-line waves, allowing for dynamic control over the waveguide’s functionality. Graphene-based metasurfaces are particularly well-suited for these types of reconfigurable circuits due to their tunable surface properties. Figure 10 showcases a field distribution configuration involving three inductive impedance surfaces. This setup facilitates the propagation of coupled quasi-line waves, allowing for easy control over wave propagation in the various waveguide arms by adjusting the surface impedance. This capability is particularly valuable in applications where flexible wave control and routing are required, such as in optical communications, photonic circuits, and nanophotonics.

Fig. 10
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(a) Schematic of the proposed quasi-line wave waveguide featuring adjacent graphene patches with a subwavelength gap. (b) The surface impedance of the graphene patch as a function of frequency, demonstrating the shift from capacitive to inductive impedance. (c) Electric field (E-field) distribution of the quasi-line mode when the graphene patch exhibits inductive impedance, showing efficient edge confinement and wave propagation. (d) E-field distribution when the graphene patch is capacitive, where the quasi-line mode is not supported, illustrating the impact of surface impedance on mode propagation.

Eigenmode analysis

The incorporation of graphene at the boundaries of the waveguide significantly enhances its bandwidth, prompting a detailed analysis of the dispersion characteristics under these conditions. Figure 11a presents the normalized electric field distribution across various frequencies in the transverse direction. It is expected that the mode will decay more rapidly at higher frequencies due to increased confinement at these frequencies or, similarly, for smaller values of the parameter ξ. In other words, the lateral concentration of the field intensifies at elevated frequencies, indicating stronger mode confinement. This field distribution, particularly along the edges of the two impedance surfaces, demonstrates a substantial field concentration, which is characteristic of the quasi-line mode behavior.

Fig. 11
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(a) Electric field intensity distribution perpendicular to the edges of the impedance surfaces at different frequencies, highlighting enhanced mode confinement at higher frequencies. (b–d) Variation of the confinement factor and figure of merit (propagation length) of the quasi-line mode with frequency for different Fermi energy levels: (b) Ef = 0.6, (c) Ef = 1, and (d) Ef = 1.4. The plots demonstrate the impact of increasing Fermi energy on wave confinement and propagation characteristics.

Figure 11b illustrates the dispersion relation for the quasi-line mode at a Fermi energy of Ef = 0.6, where β/K0 exhibits a nonlinear variation (red curve). Notably, the propagation length of the proposed structure, defined as the ratio Re(β)/Im(β) (blue curve), is significantly greater than that of the line mode supported by a graphene metasurface. This emphasizes the enhanced wave confinement and superior propagation efficiency of the quasi-line mode configuration.

Figure 11c,d depict the dispersion characteristics of the mode for Ef = 1 and Ef = 1.4, respectively. In these cases, the propagation length of the structure increases proportionally with frequency55. Moreover, the proposed waveguide offers a broad and tunable bandwidth, unlike the traditional line wave implementation in the terahertz domain, which is typically restricted to a single operational frequency. Figure 11c,d further reveal the influence of increasing Fermi energy Ef on the quasi-line mode’s bandwidth and performance. As Ef increases, both field confinement and waveguide efficiency are noticeably improved, demonstrating the enhanced capability of the quasi-line mode configuration to support efficient wave propagation and broader operational bandwidths. It is worth noting that the real part of the graphene surface impedance remains relatively stable across the working frequency range, resulting in a nearly constant ratio of Im(β)/K0.

As the frequency increases, the rise in Re(β/K0) leads to a corresponding enhancement in the propagation length of the waveguide, making it highly efficient at higher frequencies. Furthermore, the graphene parameters, including the Fermi energy Ef, relaxation time τ, and the number of graphene, have a direct influence on the real part of the surface impedance. This allows for adjustments in the propagation length, enabling fine-tuning of the waveguide’s performance.

In comparison to the line wave implementation utilizing graphene-based metasurfaces in the terahertz frequency range, the proposed quasi-line waveguide demonstrates remarkable improvements in propagation length. Additionally, it is important to note that as Ef increases, both the bandwidth and the propagation length of the structure also increase proportionally, further enhancing the system’s performance at higher values of Ef.

Practical implementation and fabrication of multilayer graphene-based waveguide

A precise and well-established fabrication process can be employed to realize the proposed multilayer graphene-based waveguide, leveraging advanced deposition and transfer techniques to ensure optimal performance. The process begins with the deposition of a 4 µm thick SiO₂ layer onto a silicon wafer using plasma-enhanced chemical vapor deposition (PECVD). This layer provides mechanical stability and serves as a spacer to maintain the required separation between graphene layers for effective wave propagation.

High-quality graphene can be synthesized using chemical vapor deposition (CVD) on a copper or nickel foil, which acts as a catalyst. Once synthesized, the graphene can be transferred onto the prepared SiO₂ substrate using a wet transfer method. This involves coating the graphene with polymethyl methacrylate (PMMA) for protection, etching away the metal catalyst, and carefully placing the graphene on the substrate. The PMMA can then be removed using solvents like acetone, leaving a clean graphene layer.

For multilayer configurations, the transfer process can be repeated, stacking additional layers with precise alignment. This multilayer approach enhances tunability and reduces dissipation losses by increasing the Fermi energy and improving surface impedance control. After stacking, thermal annealing at 300 °C in an inert atmosphere (e.g., nitrogen or argon) can be performed to enhance adhesion, remove residual contaminants, and ensure uniform electrical properties across the graphene layers.

This fabrication process combines scalability, precision, and tunability, making it highly suitable for both research and practical applications. The resulting waveguide offers significant potential in terahertz sensing, high-speed communication, and reconfigurable photonic circuits, providing robust wave confinement and efficient propagation. Recent advancements in graphene technologies confirm the feasibility of this approach, marking a significant step toward the practical realization of graphene-based waveguides for next-generation electromagnetic applications.

Graphene’s unique tunable properties have made it a versatile material in the design of advanced metasurfaces. Several methods have been developed to adjust its electromagnetic properties, offering flexibility for a wide range of applications. Among these, electrostatic doping stands out as the most practical approach, where applying a gate voltage dynamically adjusts graphene’s Fermi energy, enabling real-time control over surface conductivity. This technique has proven effective in applications requiring precise and reversible tuning, particularly in terahertz metasurfaces and waveguiding systems.

Chemical doping provides a more permanent means of modifying graphene’s carrier concentration through chemical treatments. Although less flexible for dynamic tuning, it ensures stable and predictable adjustments, making it suitable for applications requiring long-term consistency. Additionally, optical excitation offers rapid modulation by creating photo-induced carriers through laser irradiation, allowing for transient adjustments of graphene’s impedance. Temperature variation, while less common, affects carrier mobility and concentration, demonstrating graphene’s adaptability to environmental changes55,59,60. For the proposed quasi-line waveguide, electrostatic doping is particularly advantageous due to its compatibility with electronic platforms and ability to tune surface impedance dynamically. By adjusting the Fermi energy, the waveguide’s bandwidth, field confinement, and propagation characteristics can be optimized in real time, enabling high adaptability for applications such as terahertz communication, biosensing, and optical circuits. These tuning capabilities underline graphene’s critical role in advancing reconfigurable metasurfaces and next-generation waveguiding technologies.

Discussion and conclusion

In this study, we presented a novel waveguide structure designed to support quasi-line modes, offering a groundbreaking approach to guiding surface waves along inductive impedance surfaces. The proposed structure consists of two inductive impedance surfaces separated by a subwavelength gap, a critical feature enabling the existence of quasi-line modes. This gap facilitates strong field confinement along the edges of the waveguide, significantly enhancing its performance, particularly along curved paths and in complex geometries where traditional waveguides often struggle.

A key advantage of the proposed design is its adaptability, achieved through the modification of graphene layers and their chemical potential. This tunability provides precise control over surface impedance and wave propagation characteristics, ensuring flexibility for a wide range of applications. By incorporating graphene and graphene-based metasurfaces, the waveguide effectively manages dissipation losses, achieving efficient wave propagation over extended distances. The ability to dynamically tune the impedance values of the waveguide further expands the operational bandwidth, providing unparalleled versatility across a broad frequency range.

This innovative design demonstrates substantial potential for advanced sensing technologies, such as biosensors, where the integration of graphene enhances sensitivity and functionality. The principles underlying the quasi-line waveguide structure offer similar benefits for improving performance in biosensing, imaging, and related applications. Moreover, the waveguide’s ability to guide unidirectional modes with a clearly defined propagation direction opens up exciting possibilities for optical communication, photonic circuits, and quantum technologies.

The proposed quasi-line waveguide offers a highly versatile and tunable platform with applications spanning nanophotonics, terahertz communication, and precision sensing. Its ability to manipulate surface impedance, optimize bandwidth, and minimize dissipation losses—combined with the robustness of unidirectional mode propagation—makes it a promising candidate for next-generation waveguiding solutions. From a theoretical perspective, the study introduces several unique physical insights that pave the way for future research. A notable contribution is the discovery that subwavelength gaps between inductive surfaces can transition these surfaces from supporting only surface waves to enabling edge-mode propagation. This transition highlights new opportunities for manipulating field confinement and propagation dynamics, particularly in the terahertz spectrum.

The incorporation of multilayer graphene in the design further amplifies the waveguide’s capabilities. By reducing propagation loss and enhancing tunability, multilayer graphene enables broader bandwidths and dynamic control over operational frequencies, offering a level of performance that surpasses conventional waveguiding structures.

In conclusion, the proposed quasi-line waveguide structure not only addresses key limitations in traditional waveguide designs but also provides a robust foundation for advancing electromagnetic applications. The novel concepts of quasi-line modes and the utilization of inductive surfaces with subwavelength gaps establish a new paradigm in waveguiding technologies. These findings inspire further exploration into edge-mode waveguides, topological photonics, and reconfigurable photonic systems, laying the groundwork for significant advancements in both theoretical and applied electromagnetics.