Design and Analysis of a Hybrid Plasmonic Waveguide with Chiral Metamaterial and Graphene Interfaces

Abstract

To obtain a long propagation length and a small effective mode area, a new hybrid plasmonic waveguide model is proposed. We design a hybrid plasmonic waveguide composed of chiral metamaterials, silver metal at its center, and surrounded by a graphene interface using the COMSOL environment based on the finite element method. The interface thickness is 1.02nm, where three monolayer graphene were used. The waveguide properties can be achieved with the range of radius 5-60 nm, chemical potential 0.1-1 eV, and refractive index of metamaterial is -3 within the wavelength used. Moreover, we obtained a long propagation length, a small effective mode area, high power confinement, and lower loss. The results showed that the presence of graphene positively affects the waveguide properties, while increasing the chirality parameter, K, does not have a good effect. At the chemical potential 1eV, the propagation length will be 2.45mm in the absence of chirality, while it equals 0.4mm at K=1 with the used wavelength 1.55μm.

Index Terms Hybrid plasmonic waveguide; chiral metamaterial; graphene.

I. INTRODUCTION

In contemporary photonics and nanotechnology, the need to improve the efficiency and performance of optical devices has prompted extensive research into innovative materials and architectures [¹]. Among them, plasmonic waveguides have emerged as intriguing possibilities for a variety of applications due to their ability to confine and modify light at subwavelength levels [²]. Harnessing the unique features of plasmonic materials, such as metals, and combining them with other sophisticated materials, such as metamaterials and graphene, opens new possibilities for controlling and manipulating light [³]. The design and study of hybrid plasmonic waveguides, particularly those containing chiral metamaterials and graphene interfaces, has received a lot of interest in recent years [⁴]. Many efforts are directed toward the complexities of such a hybrid waveguide system, examining its fundamental concepts, design issues, and performance characteristics [⁵ ]-[⁸].

By combining the electromagnetic properties of plasmonic materials with the tunability and versatility provided by chiral metamaterials and graphene, this hybrid waveguide has the potential to revolutionize a wide range of photonic applications, from communication systems to sensing devices and beyond [⁹]. Plasmonic waveguides, defined by their capacity to sustain surface plasmon polaritons (SPPs), have distinct benefits above typical dielectric waveguides. These benefits result from the close confinement of electromagnetic fields to the metal-dielectric interface, which allows for highly localized energy concentration and substantial field augmentation [⁹], [¹⁰]. As a result, plasmonic waveguides enable light confinement at the subwavelength scale, opening the door for smaller photonic devices with greater functionality [¹¹]. Despite its promise, plasmonic waveguides encounter obstacles such as significant propagation losses and short propagation lengths, owing to the ohmic losses inherent in metallic materials [¹²]. To address these issues and maximize the potential of plasmonic waveguides, researchers have investigated hybrid techniques that use complementary materials and architectures to tune the waveguide's characteristics and overcome inherent constraints. Using chiral metamaterials is a viable approach to improving the performance of plasmonic waveguides [¹³]. Chiral metamaterials have unique optical characteristics due to their inherent chirality, defined by a lack of mirror symmetry. These materials can modify the polarization state of light in novel ways, allowing for polarization control and manipulation beyond what is possible with natural materials [¹⁴]. Researchers can increase the degrees of freedom for manipulating light propagation by adding chiral metamaterial layers into plasmonic waveguides, such as polarization-dependent routing and increased light-matter interactions. Furthermore, incorporating graphene into hybrid plasmonic waveguides adds another level of utility and tenability [¹⁵].

Graphene, a two-dimensional substance of a single sheet of carbon atoms organized in a honeycomb lattice, has excellent optical and electrical characteristics [¹⁶]. Notably, its variable conductivity via electrical gating enables dynamic control over the propagation properties of plasmonic waves [²], [⁸]. By varying the carrier density in graphene, researchers may actively modify the dispersion properties, loss characteristics, and even the phase velocity of plasmonic modes, providing unprecedented flexibility in waveguide design and operation [¹⁷]. Against this backdrop, the hybrid plasmonic waveguide with chiral metamaterial and graphene interfaces appears as an appealing platform for pushing the boundaries of nanophotonics research. By combining the strengths of plasmonic materials, chiral metamaterials, and graphene, this hybrid system has the potential to unlock new capabilities in light manipulation, modulation, and sensing [¹⁹]. However, designing and analyzing such complicated waveguide systems presents significant obstacles and necessitates a multidisciplinary approach incorporating electromagnetic theory, materials science, and nanofabrication methods [²⁰].

This paper proposes a novel graphene-based hybrid plasmonic waveguide with a graphene layer surrounding a silver metal layer at its center. Silicon is utilized in addition to silica, which comprises the remainder of the model. A circular chiral metamaterial is encapsulated in a silicon isosceles triangle. This structure aims to generate a hybrid TM mode. Through it, the waveguide parameters are calculated, the most important of which are the propagation length and the normalized effective mode area. Controlling the waveguide parameters and the presence of the above materials aims to maximize the propagation length and minimize the normalized effective mode area.

II. THE PROPOSED WAVEGUIDE

The proposed HPW consists of two vertically opposite isosceles silicon triangles; the triangle base length is W, and the other sides have a length (H/2-g). The distance between the two bases is H, while θ represents the angle of the vertex. The waveguide height is determined from the angle of the vertex and the coefficients of the guide in the form $H=2[g+0.5W\cot (\theta /2)]$, where g is the distance from waveguide center the vertex of triangle. In the middle of the height, we put a layer of silver with a thickness H₂ and length W. The silver layer is covered on both sides with graphene (three-layered) with a thickness of 1.02nm. The areas between the graphene and the silicon will be filled with silica. Next, we create a virtual circle-shaped chiral metamaterial in each silicon triangle. The radius of the metamaterial circle is r and the distance from its center to the base is H₃. This configuration will generate a gap on both sides of the silver area between the silicon triangles and the graphene layers of thickness T, in which the hybrid TM mode will be generated. Fig. 1 obtains the proposed HPW, and Table I presents the simulation parameters and expressions.

TABLE I
SIMULATION PARAMETERS AND EXPRESSIONS.

Parameter	Values/expression	Parameter	Values/expression
W	400nm	g	$0.5H_2+1.02nm+T$
T	(1,2,….12) nm	H3	H/4
H2	(2,3,…..12)nm	H	$2[g+0.5W\cot (\theta /2))]$
θ	(30,40,……, 120) deg	Chirality factor K	0,0.5,1,1.5,2,2.5
r	(5,10,20,….,60) nm	Chemical potential EF	(0.01,….,1) eV
nmeta	-3	Graphene thickness	1.02nm

Fig. 1
Structure of the proposed HPW.

III. WAVEGUIDE METERS AND MATERIALS

The real effective index, propagation length, figure of merit (FOM), and effective mode area are the parameters that can describe the properties of the graphene-based hybrid plasmonic waveguide [²¹]. The effective mode area dictates the optical field constraint power of the waveguide. The light is more concentrated inside the waveguide when the effective mode area is smaller. The effective mode area represents the ratio of total electromagnetic energy to maximum energy density. Averaging the energy density over the cross-section is required for determining the definition of effective mode area, which is a numerical measurement [⁷]

(1) $$A_{eff}={\left[{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}W(x,y)dxdy}\right]}^2{\left[{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{[W(x,y)]}^{2}dxdy}\right]}^{-1}$$

where W(x,y) is the energy density of the waveguide. The normalized modal area A_m is defined as $A_{eff}/A_o$, where A_o represents the diffraction-limited area in free space, $A_o={\lambda }^2/4$ [²²]. Propagation length L_p has an essential role in defining the mode feature of a plasmonic waveguide. The term propagation length refers to the distance at which the mode energy attenuates to 1/e of the initial value is calculated by [²²]

(2) $$L_p=\frac{\lambda }{4\pi Im(n_{eff})}$$

where $Im\left\{n_{eff}\right\}$ is the imaginary part of the effective mode index of the waveguide and λ is the wavelength. Plasmonic waveguides can be quantitatively measured using FOM, which can also help to balance attenuation against mode confinement. The definition of FOM, in this case, is the ratio of the propagation length to the diameter of A_eff, which is the effective mode size [²³]

(3) $$\text{FOM}=\frac{L_p}{2\sqrt{A_{eff}/\pi }}=\frac{\lambda }{4Im(n_{eff})\sqrt{\pi A_{eff}}}$$

The materials that comprise the current waveguide include metals like silver, insulators like silica, and chiral metamaterial. To have a deeper comprehension of certain traits or attributes of some substances, consider the following: a wide range of physical phenomena can be effectively studied with metamaterials [²⁴]. A chiral metamaterial prevents particles from being superimposed on their mirror images. Chiral media respond differently to a left circularly polarized (LCP) wave and a right circularly polarized (RCP) wave because of the medium's inherent chiral asymmetry. The chirality parameter K measures the effect of cross-coupling between the electric and magnetic fields. The following constitutive relations control the electromagnetic wave propagation in such a chiral system [²⁵]

(4a) $$\overrightarrow{D}={\epsilon }_o{\epsilon }_{r}E-j\sqrt{{\epsilon }_o{\mu }_o}KH$$

(4b) $$\overrightarrow{B}={\mu }_o{\mu }_{r}H+j\sqrt{{\epsilon }_o{\mu }_o}KE$$

where $\overrightarrow{E},\overrightarrow{H},\overrightarrow{D}$ and $\overrightarrow{B}$ are the electric field, magnetic field, electric displacement, and magnetic flux density, respectively. The permeability of the vacuum is μ_o, the ε_o is the vacuum permittivity, and the chiral medium's relative permittivity and permeability are ε_r and μ_r, respectively.

A lot of people think that metals are excellent candidates for plasmonic applications. Silver is among the best plasmonic materials because of its lowest optical loss in the visible and NIR spectral ranges [²⁶]. The production of noble metals in various nanostructures, including metal rings, nanowires, and nanotubes, has also been accomplished [²⁷]. We describe the silver permittivity using the Drude-Lorentz model, which is known to be reasonably accurate in the wavelength range of (0.2-2)μm [²⁸]

(5) $$\epsilon (w)=1-\frac{w_p^2}{w^2+iw\gamma }+{\displaystyle \sum _{n=1}^5\frac{f_n{w}_n^2}{w_n^2-w^2-iw{\gamma }_n}}$$

Where the summation refers to the number of oscillations,n refers to the resonant modes, $w_p=2\pi \times 2002.6THz$ is the plasma frequency of bulk silver, $\gamma =2\pi \times 11.61THz$ is the damping constant, and w the angular frequency. The γ_n is the effect of damping, f_n indicates the weighting coefficients and w_n corresponds to the resonance frequencies (see Table II).

TABLE II
LORENTZ - DRUDE MODEL PARAMETERS FOR SILVER [¹⁴].

n	$w_n(THz)/2\pi$	${\gamma }_n(THz)/2\pi$	fn
0	---	11.5907	0.845
1	197.3	939.62	7.9247
2	1083.5	109.22	0.5013
3	1979.1	15.71	0.0133
4	4392.5	221.49	0.8266
5	9812.1	584.91	1.1133

It is possible to enhance the properties of other materials by utilizing graphene, the most significant substance in the known world. Furthermore, the graphene layer atop the silver metal shields the silver film and dielectric layer [¹⁵], [²⁹]. For a single layer of graphene, the surface conductivity is given by [¹¹]

(6a) $${\sigma }_{\mathrm{int}ra}=\frac{j{e}^2{k}_B{T}_o}{\pi {\hslash }^2\left(w+2j\Gamma \right)}\left[\frac{E_F}{k_B{T}_o}+2\ln \left(1+e^{-E_F/k_B{T}_o}\right)\right]$$

(6b) $${\sigma }_{\mathrm{int}er}=\frac{j{e}^2}{4\pi \hslash }\ln \left[\frac{2\left|E_F\right|-(w+2j\Gamma )}{2\left|E_F\right|+(w+2j\Gamma )}\right],k_B{T}_o\ll E_F$$

Where the total conductivity is $\sigma ={\sigma }_{\text{intra}}+{\sigma }_{\text{inter}}$. In the above equations, the chemical potential is E_F, $\hslash$ is the reduced Planck constant, Γ is the scattering rate with $\tau =1/2\Gamma$ being the carrier relaxation lifetime, k_B is the Boltzmann constant, T_o is the temperature, and e is the electron charge. We consider tri-layer graphene to be present on the substrate for graphene sheets. As a result, the number of layers $N<6$, which belongs to few-layer graphene, can behave as a superposition of a single sheet. Furthermore, $\Delta =0.335nm$ is the effective thickness of a monolayer of graphene. The total thickness of graphene sheets $t=N\Delta$ [¹⁶]. The formula below can be used to determine the permittivity of graphene in the mid-infrared frequency range [¹¹]

(7) $${\epsilon }_g=1+\frac{j\sigma (w)}{w{\epsilon }_{o}t}$$

The Sellmeier relation provides the refractive index of the silica layer as [¹⁴]

(8) $$n(\lambda )=\sqrt{1+\frac{A_1{\lambda }^2}{{\lambda }^2-B_1^2}+\frac{A_2{\lambda }^2}{{\lambda }^2-B_2^2}+\frac{A_3{\lambda }^2}{{\lambda }^2-B_3^2}}$$

This equation is valid for the wavelength lying between (0.21 and 2.2)μm. The Sellmeier dispersion coefficients $A_1,A_2,A_3,B_1,B_2$ and B₃ have certain numeric values, as in Table III.

TABLE III
VALUES OF THE SELLMEIER COEFFICIENTS OF SILICA [¹⁴].

Coefficient	Value	Coefficient	Value (μm2)
A1	0.6961663	B1	0.00467915
A2	0.4079462	B2	0.0131206
A3	0.8974794	B3	97.93400254

IV. RESULTS AND DISCUSSION

The properties of the resulting waveguide mode are affected not only by the thickness of the gap itself but also by the thickness of silver, the radius of the metamaterial circle, the angle of the vertex, and others. The width of the waveguide is a constant that does not change in this simulation, while the waveguide's height is a function of the vertex's angle.

A. Field Distribution and Mode Configuration

This section will present the hybrid mode (TM) configuration and corresponding electric field distribution in the y-direction at specific values of proposed structure parameters. Fig. 2(a)-(f) shows the electric field distribution and configuration of the TM hybrid plasmonic mode of the structure when K=0, r=30nm, $\theta ={90}^{\text{o}},H_2=4nm$. Three different values for T are 3, 5, and 7nm. This aims to study the effect of increasing the gap thickness on the proposed waveguide. The mode is confined in the gap region between silicon material (the vertex of the triangle) on the one hand and the metal surface and graphene on the other. In other words, it is seen that due to the strong hybridization of the plasmonic mode and the dielectric mode, the electric field is greatly enhanced in the low-index silica layer. With the increase of T, power is more confined in the gap. However, it leads to a decrease in the field distribution $|E{|}^2$ in the y-direction and vice versa. Fig. 3(a)-(f) shows the electric field distribution $|E{|}^2$ and configuration of the TM hybrid plasmonic mode of the structure at different circle radii in the waveguide filled with chiral metamaterial where r=10, 30, and 50 nm, respectively. The figure clearly shows an increase in the radius of the circle. Increasing r means increasing the presence of metamaterial. This has a positive effect in increasing the electric field when the values of the parameters of the proposed waveguide are K=0, T=3nm, $\theta ={90}^{\text{o}},H_2=4nm$.

Fig. 2
The mode configuration of hybrid TM in y-direction at $\theta ={90}^{\text{ο}}\text{,}{H}_2=4nm,K=0,r=30nm$ for (a)$T=3nm$, (b) $T=5nm$ and (c) and the corresponding field distributions in (d),(e) and (f), respectively. $T=7nm$

Fig. 3
The mode configuration of hybrid TM in y-direction at $\theta ={90}^{\text{ο}},H_2=4nm,K=0,r=30nm$ for: a) $r=10nm$, b) $r=30nm$ and c) and the corresponding field distributions in (d),(e) and (f), respectively. $r=50nm$

B. Waveguide Parameters

This section investigates the real effective index, propagation length, normalized mode area, and FOM of the supported hybrid mode. It aims to maximize field confinement with a long propagation length. Fig. 4 shows the modal properties of a hybrid plasmonic waveguide as a function of the chemical potential of graphene interfaces at r=40nm, T=3nm, $\theta ={90}^{\text{o}},H_2=4nm$. The different values of chiral parameters were adopted, which are K=0, 1, 2. The severe change in the real effective index appears at chemical potential equals 0.51eV, as in Fig. 4(a). The longest propagation length in Fig. 4(b) equals 2.45mm when K=0, r=30nm, $\theta ={90}^{\text{o}},H_2=4nm$. The minimum effective mode area in Fig.4(c) is $A_m=0.011$ when chemical potential equals 0.51eV for the different K values. Finally, the maximum value of FOM in Fig. 4(d) is 2900 for K=0. When K increases, the real effective index is less but higher when the chemical potential is close to 0.51eV. In case K=0, we obtained the longest propagation length and the least A_m compared to cases K=1 or 2. The best performance of the proposed waveguide is when the chiral parameter is as low as possible.

Fig. 4
Dependence of a) real effective index, b) propagation length, c) normalized area, and d) FOM on the chemical potential of the graphene interfaces for many cases of K, where r=40nm, T=3nm,$\theta ={90}^{\text{ο}},H_2=4nm$.

Fig. 5 shows the dependence of modal properties on the chemical potential of graphene interfaces when K=1.2, T=3nm, $\theta ={90}^{\text{ο}},H_2=4nm$ and the radius of the chiral metamaterial circle takes the values$(35,50,60)nm$, respectively. For r=35nm, we note that the real effective index in Fig. 5(a) starts from 1.88 but drops to 1.3 as soon as E_F reaches 0.51eV, rises until its maximum value is 2.15, and then decreases again. At other r values, the mode propagates in the same behavior, but the maximum value of $\mathrm{Re}(n_{eff})$ is lower. For E_F =1eV, the propagation length in Fig. 5(b) is 0.41mm at r=35nm and 0.12mm at r=5nm but 0.07mm at r=60nm. For different r values, the lowest value of A_m in Fig.5(c) is when $E_F=0.51eV$, but when E_F is higher or lower than this value, A_m increases with the increase in the radius. According to this data, we can say that case r=35nm gives the best result. For E_F =1eV, the maximum value of FOM in Fig. 5(d) is 4900 at r=35nm, while FOM equals 800 at r=60nm. The results show that the small chiral metamaterial radius causes the better performance of the proposed waveguide, so we select the optimal radius of the chiral metamaterial as 35nm or less.

Fig. 5
Dependence of a) real effective index, b) propagation length, c) normalized mode area, and d) FOM on the chemical potential of the graphene interfaces for many cases of r, where $\theta ={90}^{\text{ο}},T=3nm,H_2=4nm,K=1.2$.

Fig. 6 shows the dependence of TM hybrid mode properties on the chemical potential of graphene interfaces of the proposed structure when T=3nm, r=40nm, $H_2=4nm$, K=1.2, and the vertex angle of the silicon material $\theta ={60}^{\text{o}},{90}^{\text{o}},{120}^{\text{o}}$. Fig. 6(a) illustrates that the real effective index drops when $E_F=0.51eV$. Then, it increases with the increase of the chemical potential at $\theta ={60}^{\text{o}}$ but decreases again when $\theta ={90}^{\text{o}},{120}^{\text{o}}$. The real effective index increases with a decreasing vertex angle of silicon. Correspondingly, the propagation length, as in Fig. 6(b), also sees an increase with the expansion of the mode area when $E_F>0.55eV$ as in Fig. 6(c). The best result is generally when the angle is small, where the loss is low, and the propagation length is long. In general, from observing the FOM, as in Fig. 6(d), of the $\theta ={120}^o$ case, which reaches approximately 500, we can say that the case $\theta ={120}^{\text{o}}$ does not give the best results, as in the case $\theta ={60}^{\text{o}}$, in which the FOM reaches 14000. Fig. 7 shows the dependence of TM hybrid mode properties on the chirality parameter of the proposed structure when $\theta ={90}^{\text{ο}},T=3nm,H_2=4nm$ for many values of r. In Fig. 7(a), the real effective index decreases with increasing the chiral parameter in the case r=40nm. The real effective index rises slightly and then decreases with the increase of the chiral parameter at r=50nm. When r=60nm, the real effective index increases dramatically with increasing K. In Fig. 7(d), with the increase K, the propagation length decreases gradually and A_m increases. In general, this shows us that increasing the chirality parameter and the radius has a negative effect on the proposed structure. Therefore, it is better to choose the lowest values.

Fig. 6
Dependence of a) real effective index, b) propagation length, c) normalized mode area, and d) FOM on the chemical potential of the graphene interfaces for many cases of θ, where $r=40nm,T=3nm,H_2=4nm,K=1.2$.

Fig. 7
Dependence of a) real effective index, b) propagation length, c) normalized mode area, and d) FOM on chirality factor for many cases of r, where $\theta ={90}^{\text{o}},T=3nm,H_2=4nm$.

Fig. 8 shows the properties of the model for the proposed design as a function of the chiral parameter when $\theta ={90}^{\text{ο}},H_2=4nm$, and r=55nm. In this figure, three different values for the thickness of the gap between the triangle vertex of silicon and graphene were taken, which are T=(2,4,6)nm. In Fig. 8(a), the effective refractive index increases with increasing K but decreases when the values of K are very large for different values of T. In Fig. 8(b)-(d), we notice that both. A_m increases while FOM and the propagation length decrease with the increase of K. With increasing T, the propagation length and A_m increase, and the real effective index decreases, while FOM does not change for all values of T.

Fig. 8
Dependence of a) real effective index, b) propagation length, c) normalized mode area, and d) FOM on chirality factor for many cases of T, where $\theta ={90}^{\text{ο}},r=55nm,H_2=4nm$.

Fig. 9 shows the properties of the TM hybrid mode of a hybrid plasmonic waveguide when $\theta ={90}^{\text{o}},H_2=4nm,$K=1 as a function of the radius of the chiral metamaterial layer. The thickness of the gap between silicon and graphene takes three different values: T=(4,5,6)nm. In Fig. 9(a)-(d), real effective index, the FOM, and propagation length decrease with increasing r. On the contrary, it increases with the increase of r. The proposed model performs best when r=5nm in the case of T=6nm. Increasing T gave longer propagation length and higher FOM with a small radius compared to lower values of T. An increase in T leads to an expansion of the effective mode area and a decrease in the real effective index. Fig. 10 shows the model properties for the proposed structure as a function of the radius of the chiral metamaterial layer when $\theta ={90}^{\text{o}}$, K=1, and T=3nm. The thickness of the silver layer takes the values $H_2=(4.5,5,5.5)nm$. In Fig. 10(a)-(d), there is a decrease in the real effective index, propagation length and FOM and an increase in A_m with an increase in r. But this increase and decrease varies with the increase in the thickness of the silver layer, where A_m decreases with the increase of H₂. We notice that propagation length and FOM decrease with the increase of H₂ when r is small, but the opposite occurs when r=25nm or greater than this value. Fig. 11 shows the change in the model properties of the proposed waveguide with the thickness of the silver layer at different values of the chiral parameter with $\theta ={90}^{\text{o}}$, r=40nm, and T=3nm. In Fig. 11(a) , we notice that the real effective index increases linearly with the thickness of the silver layer when K=0. When K=1 or 2, the $\mathrm{Re}(n_{eff})$ increases by a lesser extent than in the K=0 case. In Fig. 11(b) and (c), the propagation length and FOM decrease for K cases; they return to rise again in the case of K=1 or 2 with the increase of H₂. The propagation length and FOM are less than in the case of K=0. In Fig. 11(d), with the increase, we also noticed a A_m decrease at K=0. An A_m increases and then decreases with the increase of H₂ at K=1 or 2. We conclude that the best result (real effective index, propagation length, and FOM are high) is when the chiral property is absent.

Fig. 9
Dependence of a) real effective index, b) propagation length, c) normalized mode area and d) FOM on r for many cases of T, where $\theta ={90}^{\text{ο}},K=1,H_2=4nm$.

Fig. 10
Dependence of a) real effective index, b) propagation length, c) normalized mode area, and d) FOM on r for many cases of H₂, where $\theta ={90}^{\text{ο}},T=3nm,K=1$.

Fig. 11
Dependence of a) real effective index, b) propagation length, c) normalized mode area and d) FOM onH₂ for many cases of K, where $\theta ={90}^{\text{ο}},$T=3nm, r=40nm.

Fig. 12 shows the change in the properties of the TM hybrid mode with the thickness of the silver layer of the proposed waveguide when $\theta ={90}^{\text{o}},$K=1, r=40nm, and T takes three values 4, 5, and 6 nm. In Fig. 12(a)-(d), we notice that all the properties of the mode have the same behavior for different values of T. The real effective index and FOM are larger while A_m and propagation length are less when the gap thickness is less. The real effective index increases with the increase of the thickness of the silver layer, and the opposite occurs for A_m. As for propagation length and FOM, it decreases and then increases with the increase in the thickness of the silver layer.

Fig. 12
Dependence of a) real effective index, b) propagation length, c) normalized mode area and d) FOM on H₂ for many cases of T, where $\theta ={90}^{\text{ο}},$K=1, r=40nm.

Fig. 13 shows the dependence of the properties of the model for the proposed structure on the thickness of the silver layer when T=3nm, K=1, and r=40nm with the triangle vertex angle of the silicon material $\theta ={70}^{\text{o}},{90}^{\text{o}},{120}^{\text{o}}$. In Fig. 13(a) and (b), with the increase of H₂, we notice that the real effective index increases while propagation length decreases and then increases. In Fig.13(c), the A_m is decreases with the increase of H₂. In Fig. 13(d), the FOM starts with a sharp decrease and then increases gradually with the increase of H₂. The FOM when $\theta ={70}^{\text{o}}$ is much greater than FOM when $\theta ={90}^{\text{o}},{120}^{\text{o}}$. In addition, in the case of great angles, the loss is greater, and the effective mode area is smaller. The waveguide generally gets the best FOM, propagation length, and real effective index when the angle is small. Fig. 14 shows the dependence of the properties of the TM hybrid mode on the thickness of the gap between the graphene and the triangular vertex of the silicon material with K=0, 1.5, and 2.5. The other parameters are r=40nm, $\theta ={90}^{\text{ο}},H_2=4nm$. In Fig. 14(a) and (b), we notice that the real effective index decreases and propagation length increases with increasing T. In Fig. 14(c), an A_m increases with the increase T. In Fig. 14(d), the FOM increases linearly with T, but when K=1.5 or 2.5 is less than the case of K=0. The FOM, propagation length, and the real effective index decrease when the chiral parameter is strong, while the effective mode area increases. This indicates that the strong chiral parameter does not positively affect the proposed structure.

Fig. 13
Dependence of a) real effective index, b) propagation length, c) normalized mode area and d) FOM on H₂ for many cases of θ, where T=3nm, K=1, r=40nm.

Fig. 14
Dependence of a) real effective index, b) propagation length, c) normalized mode area and d) FOM on T for many cases of K, where $\theta ={90}^o,H_2=4nm,r=40nm$.

Fig.15 shows the dependence of the model properties of the proposed waveguide as a function of the thickness of the gap between the triangular vertex of the silicon material and the graphene material when K=1, $H_2=4nm,$r=40nm with $\theta ={30}^{\text{o}},{60}^{\text{o}},{90}^{\text{o}}$. In Fig.15(a)-(d), with the increase of T and $\theta ={30}^o$, we notice that the real effective index, propagation length, and FOM increase while the effective mode area decreases, which must correspond to a high confinement of power. When $\theta ={60}^{\text{o}}$, the real effective index decreases with the increase of T, the other properties increase. For the angle is $\theta ={90}^{\text{o}}$, FOM decreases and then increases with the increase of T, but the real effective index decreases. As for propagation length and A_m, they increase with the increase in T. At $\theta ={30}^{\text{o}}$, we get the best performance of the proposed structure compared to other angles, and with the increase of T. This means that increasing the angle has a negative effect on the propagation length and FOM, thus affecting the proposed model in general.

Fig. 15
Dependence of a) real effective index, b) propagation length, c) normalized mode area, and d) FOM on T for many cases of θ, where $H_2=4nm,K=1,r=40nm$.

Table IV shows a comparison of the propagation length and the area of the normalized mode with several available researches.

TABLE IV
COMPARISON OF PROPAGATION LENGTH AND MODAL AREA.

Model	Maximum propagation length (mm)	Normalized mode area
Proposed HPW	2.45	0.03
Ref.[14]	1.2	0.035
Ref.[20]	0.45	0.03
Ref.[30]	0.2	0.05

V. CONCLUSIONS

We obtain the best performance of the proposed hybrid plasmonic waveguide when the radius, chiral parameter and gap are as low as possible and silver thickness and chemical potential are as high as possible. We get the longest propagation length, which is equal to 3mm when Fermi energy is 1eV and the chirality factor value is one or two at the angle 60. In the absence of the chiral property, we can achieve a propagation length longer than this by using small angles for the vertex of the triangle of silicon. Also, the FOM can reach higher than 3400 when the thickness of the gap is large, and the vertex angle is small. In the absence of the chiral property, the normalized mode area is smaller, and the FOM is maximum when the thickness of the silver layer is the minimum.

REFERENCES

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Publication Dates

• Publication in this collection
  22 Nov 2024
• Date of issue
  2024

History

• Received
  25 Feb 2024
• Reviewed
  19 Apr 2024
• Accepted
  26 Sept 2024