Introduction

The rapid advancement of semiconductors and integrated circuits in microelectronics offers a solid foundation for many applications, most notably in communication and information processing. Despite this, further innovation in the field of information transmission is always constrained by the efficiency of electronic components because electrons are inherently capable of carrying less data. Photonics may offer an alternative solution to this problem, as photons can carry more information and travel faster. Most recently, it has been demonstrated that the trade-off between capacity and size can be minimized quite effectively by making use of surface plasmon polaritons. The fundamental concept here is that electromagnetic waves, which are normally propagated within a dielectric waveguide, can propagate along the metal surface through surface plasmon-polariton (SPP) phenomena [1,2,3,4,5,6]. The subwavelength mode confinement of the surface plasmon polariton (SPP) mode makes plasmonic waveguides more flexible in size than dielectric waveguides. In this context, SPP has the potential to be a crucial technology for next-generation nanophotonic devices [7]. The major disadvantage of current plasmonic waveguides is that they have very serious propagation loss particularly if the mode confinement scales down to the subwavelength range [8]. In general, all SPP modes have this fundamental drawback: stronger confinement causes the field to move closer to the metal, which leads to serious energy losses, resulting in a shorter propagation length [9]. To accomplish tight mode confinement through plasmonic waveguides with relatively low propagation losses, novel plasmonic waveguide structures are needed in the optic community. In the optical sector, there is an increasing demand for plasmonic materials to overcome these challenges [10].

Recently, graphene revolutionized the optical sector for developing photonic devices due to its extraordinary optical and electrical properties [11,12,13,14,15,16]. The zero-band gap structure of graphene has made it a promising alternative candidate for overcoming the deficiencies of metallic-based optoelectronic devices and an ideal candidate for developing broadband saturable absorbers. It has also been reported that graphene’s electron mobility reaches up to 200,000 cm2/Vs, which could enable the design of high-performance modulators and photodetectors. Furthermore, graphene-based photonic devices have a remarkable potential for biochemical sensing applications due to their large surface area and high adsorption capacity. By tailoring the carrier density in graphene, the adsorbed chemical molecules can also modify the optical traits of photonic devices [17, 18].

Chiral material that exhibits nondirectional chirality is known as a UAC material [19]. Interestingly, the process of manufacturing uniaxial chiral mediums is very convenient and easy to do [20]. As a standard procedure, UAC materials are produced by the immersion of small chiral items (such as wire spirals) in an anisotropic medium [21]. Vapor deposition processes can be used to produce such a medium in a very straightforward manner [22]. UAC mediums are anticipated many fascinating features by virtue of their chirality coupled with anisotropy. In this manuscript, the numerical analysis of SPPs at UAC-metal-UAC in the visible spectrum is analyzed. Three cases of UAC medium are studied to demonstrate the influence of chirality and core width on effective mode index as the function of wave frequency.

Plasma is the highly ionized state of gas comprising electrons, ions, and neutral particles. Optic researchers have taken a keen interest in studying electromagnetic surface wave traits in a plasma medium. Numerous fields have taken advantage of the unexceptional response of electromagnetic waves in a plasma medium, including biochemical sensing, spectroscopy, light-trapping devices, optoelectronic devices, and communications. In the plasma medium, EM surface waves are affected by numerous factors such as collision frequency, plasma frequency, and operating frequency. Electromagnetic waves in plasma are affected by these properties, which determine their absorption, reflection, and transmission attributes [23,24,25,26]. Compared to conventional dielectrics, plasma shows distinct characteristics. Among them is the plasma’s permittivity which can be controlled by altering the number density of electrons. Plasma also offers the advantage of reducing absorption losses in nanophotonic devices. The efficiency of nanophotonic devices is limited by absorption problems in traditional dielectrics, such as silicon and gallium arsenide. This loss can be minimalized by introducing plasma, since plasma acts as a low-loss medium, resulting in a lower absorption rate and a higher light transmission rate [27]. Thus, the literature review motivated us to conduct the study of graphene-loaded SPP waveguides surrounded by uniaxial chiral and plasma layers.

Methodology

A planar graphene-loaded waveguide structure surrounded by UAC and plasma layer is shown in Fig. 1. Here, we present the characteristic equation for the proposed waveguide structure. Let us consider the SPPs propagating along the z-axis while attenuating along the x-axis.

Fig. 1
figure 1

Schematic configuration of graphene-loaded SPP waveguide surrounded plasma uniaxial chiral and plasma layer

Full size image

The EM waves in the UAC medium are described by the following constitutive relations.

$$\genfrac{}{}{0pt}{}{{\varvec{D}}=\left[{\upvarepsilon }_{\text{t}}{\overline{\overline{\text{I}}}}_{\text{t}}+{\upvarepsilon }_{\text{z}}{\widehat{\text{e}}}_{\text{z}}{\widehat{\text{e}}}_{\text{z}} \right].\mathbf{E}-j{\xi }_{c}\sqrt{{\mu }_{0}{\varepsilon }_{0}}{\widehat{e}}_{z}{\widehat{e}}_{z}.\mathbf{H}}{{\varvec{B}}=\left[{\mu }_{t}{\overline{\overline{\text{I}}}}_{\text{t}}+{\mu }_{z}{\widehat{\text{e}}}_{\text{z}}{\widehat{\text{e}}}_{\text{z}} \right].\mathbf{H}-j{\xi }_{c}\sqrt{{\mu }_{0}{\varepsilon }_{0}}{\widehat{e}}_{z}{\widehat{e}}_{z}.\mathbf{E}}$$
(1)

Here, \(\xi\) represents the chirality parameter, and \({\varepsilon }_{0}\) and \({\mu }_{0}\) describe the permittivity and permeability of free space, respectively.

$${E}_{z}={K}_{1} {e}^{-{q}_{1}x}+{K}_{2} {e}^{-{q}_{2}x}$$
(2)
$${H}_{z}=\frac{j{\alpha }_{1, 2}}{{\eta }_{t}} ({K}_{1} {e}^{-{q}_{1}x}+{K}_{2} {e}^{-{q}_{2}x})$$
(3)

Here, \({q}_{1}\) and \({q}_{2}\) are the wavenumbers of the UAC medium. Other EM field components can be derived from [27, 28]:

$${E}_{t}=\left(-\frac{j\beta {\nabla }_{t}}{{{q}_{1}}^{2}}-\frac{{\nabla }_{t}{\text{k}}_{t}{\alpha }_{1}}{{{q}_{1}}^{2}}{\widehat{e}}_{z}\right){K}_{1} {e}^{-{q}_{1}x}+\left(-\frac{j\beta {\nabla }_{t}}{{{q}_{2}}^{2}}-\frac{{\nabla }_{t}{\text{k}}_{t}{\alpha }_{1}}{{{q}_{2}}^{2}}{\widehat{e}}_{z}\right){K}_{2} {e}^{-{q}_{2}x}$$
(4)
$${{q}_{\text{1,2}}}^{2}=\frac{{\lambda }^{2}}{2}\left[\frac{{\mu }_{z}}{{\mu }_{t}}+\frac{{\varepsilon }_{z}}{{\varepsilon }_{t}}+\sqrt{{\left(\frac{{\mu }_{z}}{{\mu }_{t}}-\frac{{\varepsilon }_{z}}{{\varepsilon }_{t}}\right)}^{2}+\frac{4{{\xi }^{2}\mu }_{z}{\varepsilon }_{z}}{{\mu }_{t}{\varepsilon }_{t}}}\right]$$
(5)

Here,

$${\lambda }^{2}={\beta }^{2}-{\omega }^{2}{{\mu }_{t}\varepsilon }_{t}$$
(6)
$${\alpha }_{1}=\left(\frac{{q}_{1}^{2}}{{\lambda }^{2}}-\frac{{\varepsilon }_{z}}{{\varepsilon }_{t}}\right) \frac{\sqrt{{{\mu }_{t}\varepsilon }_{t}}}{\xi \sqrt{{{\mu }_{z}\varepsilon }_{z}}}$$
(7)
$${\alpha }_{2}=\left(\frac{{q}_{2}^{2}}{{\lambda }^{2}}-\frac{{\varepsilon }_{z}}{{\varepsilon }_{t}}\right)\frac{\sqrt{{{\mu }_{t}\varepsilon }_{t}}}{\xi \sqrt{{{\mu }_{z}\varepsilon }_{z}}}$$
(8)
$${\eta }_{t}=\sqrt{{\varepsilon }_{t}/{\mu }_{t}}$$
(9)

According to Kubo formula, the graphene conductivity is given as follows:

$$\sigma \left({\mu }_{c},\tau ,\text{T}\right)=\frac{j{e}^{2}\left(\omega -j{\tau }^{-1}\right)}{\pi {h}^{2}}\times \left[\frac{1}{{\left(\omega -j{\tau }^{-1}\right)}^{2}}{\int }_{0}^{\infty }\times {\xi }_{n}\left(\frac{\sigma {f}_{d}\left({\xi }_{n},{\mu }_{c},T\right)}{\sigma {\xi }_{n}}-\frac{\sigma {f}_{d}\left(-{\xi }_{n},{\mu }_{c},T\right)}{\sigma \xi }\right)d{\xi }_{n}-{\int }_{0}^{\infty }\times \frac{{f}_{d}\left(-{\xi }_{n},{\mu }_{c},T\right)-{f}_{d}\left({\xi }_{n},{\mu }_{c},T\right)}{{\left(\omega -j{\uptau }^{-1}\right)}^{2}-4{\left(\nicefrac{{\xi }_{n}}{\hslash}\right)}^{2}}d{\xi }_{n}\right]$$
(10)

where \(j, {\xi }_{n}, \hslash {f}_{d},{\mu }_{c},\tau ,\text{T},\) \(e,\) and \(\omega\) represent the imaginary unit, energy, reduced Plank’s constant Fermi–Dirac distribution, chemical potential, relaxation time, temperature, charge on electron, and operating frequency, respectively.

The EM fields for the plasma medium are given below:

$${E}_{z}={K}_{3} {e}^{{k}_{p}x}$$
(11)
$${H}_{z}={K}_{4} {e}^{{k}_{p}x}$$
(12)
$${E}_{y}=-\frac{j\omega {\mu }_{0}{K}_{4} {e}^{{k}_{p}x}}{{k}_{p}}$$
(13)
$${H}_{y}=-\frac{j\omega {\varepsilon }_{p}{K}_{3} {e}^{{k}_{p}x}}{{k}_{p}}$$
(14)

Here, \({k}_{p}\) is the wavenumber in a plasma medium \({k}_{p}=\sqrt{{\beta }^{2}-{\omega }^{2}{\varepsilon }_{p}{\mu }_{0}}\) [29] and \({\varepsilon }_{p}\) is the permittivity of isotropic plasma medium \({\varepsilon }_{p}=1-\frac{{\omega }_{p}^{2}}{{\omega }^{2}+i\omega v}\). \({\omega }_{p}\) and \(v\) represent the plasma frequency and collisional frequency of plasma medium, respectively [30].

$$\widehat{x}\times \left[{H}_{1}-{H}_{2}\right]=\sigma E$$
(15)
$$\widehat{x}\times \left[{E}_{1}-{E}_{2}\right]=0$$
(16)

The above boundary conditions are used to obtain the following characteristic equations:

$$q_1q_2(\alpha_1-\alpha_2)\eta_t\mu_0\omega(k_p\sigma-j\varepsilon_p\omega)+{kt}^2\alpha_1(\alpha_1-\alpha_2)\alpha_2k_p\eta_t(-jk_p+\mu_0\sigma\omega)+kt(q_2\alpha_1-q_1\alpha_2)(k_p^2\eta_t^2\sigma-jk_p(\alpha_1\alpha_2\mu_0+\eta_t^2(\varepsilon_p-\mu_0\sigma^2))\omega+\varepsilon_p\eta_t^2\mu_0\sigma\omega^2)=0$$
(17)

Results

Here, the characteristics of Eq. (17) are used to elucidate the numerical results of graphene-loaded SPPs surrounded by UAC and plasma layers. Graphene supports the terahertz (THz) frequency regime [6, 12, 31,32,33]. To investigate the properties of the proposed waveguide structure, two cases of UAC medium are analyzed. The THz spectral region is studied with respect to the normalized propagation constant (NPC) under different graphene and plasma parameters. The NPC is Re(\(\frac{\beta }{{k}_{0}}\)) where \(\beta\) is the propagation constant and \({k}_{0}\) is the wavenumber in free space.

Case I

In this case, we have set the parameters as follows: \({\mu }_{t}={\mu }_{z}={\mu }_{0}\), \({\varepsilon }_{t}=-2{.3\varepsilon }_{0}\), \({\varepsilon }_{z}=-0.1{\varepsilon }_{0}\), \(\xi =1.1\), \(\nu =4\times {10}^{6}\text{Hz}\), \({\omega }_{p}=1 \text{THz}\), \(T=300 K\), \({\mu }_{c}=0.2\text{eV}\), and \(\tau =2 \text{ps}\). To investigate the influence of chemical potential, a number of graphene layers on NPC by using Eq. 17 are analyzed in Fig. 2a,b. The operating frequency increases from 0 to 12 THz. In Fig. 2a, chemical potential varies from \({\mu }_{c}=0.2\)–\(0.4eV\) as shown in red, black, and blue characteristic curves. It can be noted that the variation in NPC significantly varies with chemical potential. Lower values of chemical potential lead to higher NPC as reported in [6, 12, 31,32,33,34,35,36]. Due to the increasing chemical potential, the energy levels of the graphene lattice become more crowded, resulting in increased scattering and a slower propagation rate. Furthermore, the energy gap between graphene’s conduction and valence bands increases with increasing chemical potential. As a result of this energy gap, electrons change their behavior, causing disturbances such as photons and electrons to propagate more slowly. In Fig. 2b, the impact of the number of graphene layers on NPC versus wave frequency is analyzed by using the characteristics of Eq. 17. In a recent study, it was demonstrated that a single layer of graphene absorbs only 2.3% of the incident light [37]. The NPC begins to decrease as the number of graphene layers increases [12, 33, 38, 39]. Due to the stronger interactions between electrons in graphene layers, the scattering of electrons in the graphene layer also becomes stronger, resulting in reduced transmission through the graphene layer. Consequently, a reduction in transmission results in a decrease in wave velocity and a decrease in the NPC. Further analyses of the influence of plasma parameters on the NPC are depicted in Fig. 3a,b. In Fig. 3a, the variation in NPC under the different plasma frequencies is analyzed in the THz wave frequency range. As plasma frequency increases, the cutoff frequency and NPC both increase. A higher plasma frequency leads to an increase in the number of excited plasma particles, thereby producing a stronger coupling with the wave. Thus, there is a greater effective wave velocity, resulting in a larger NPC. In isotropic plasma, plasma frequency plays a vital role in the development of plasmonic-based devices. Furthermore, plasma electrons oscillate at certain frequencies in resonance with the external electromagnetic field, leading to significant enhancements in the scattering properties. Thus, it is possible to design and fabricate plasmonic devices with diverse functionality by exploiting this resonance phenomenon. Figure 3b depicts the impact of collisional frequency on NPC in a certain wave frequency band. In this context, wave frequency extends from 0 to 12 THz. It can be clearly seen that higher collisional frequency values reflect lower NPC as reported in [27, 30]. When the particles collide with each other at a higher collisional frequency, they can transfer energy to the wave, effectively dampening it. Consequently, this damping effect reduces the amplitude and slows down the propagation of the EM surface wave. Figure 4 depicts the impact of chirality on the NPC. Obviously, lower chirality values lead to higher NPC.

Fig. 2
figure 2

Effect of the chemical potential and number of graphene layers on NPC for case I

Full size image
Fig. 3
figure 3

Effect of the plasma frequency and collisional frequency on NPC for case I

Full size image
Fig. 4
figure 4

Effect of the chemical potential and number of graphene layers on NPC for case I

Full size image

Case II

In this case, we have set the parameters as follows: \({\mu }_{t}={\mu }_{z}={\mu }_{0}\), \({\varepsilon }_{t}=-2{.3\varepsilon }_{0}\), \({\varepsilon }_{z}=0.1{\varepsilon }_{0}\), \(\xi =1.1\), \(\nu =4\times {10}^{6}\text{Hz}\), \({\omega }_{p}=1 \text{THz}\), \(T=300 K\), \({\mu }_{c}=0.2\text{eV}\), and \(\tau =2 \text{ps}\). Figure 5a presents the variation in NPC under the different chemical potentials. The characteristic curves are observed to exhibit unphysical regions with increasing wave frequencies, and this is of no practical importance to the plasmonic industry. Furthermore, as chemical potential grows, the wave frequency band becomes narrow and characteristics curves shift toward the low-frequency region. Moreover, higher chemical potential leads to lower NPC as reported in [40, 41]. With increased chemical potential, graphene’s valence and conduction bands become more distant from each other, resulting in the formation of an energy gap. In addition, this energy gap alters electron behavior, which impacts the propagation speed of wave-like disturbances, such as photons and electrons. The variation in NPC versus frequency for different numbers of graphene layers is depicted in Fig. 5b. The cutoff frequency decreases with increasing layers of graphene. Furthermore, it is important to note that with an increasing number of graphene layers, the frequency band becomes narrow and characteristics curves are shifted towards lower NPC. Multilayer graphene exhibits this behavior due to the reduced mobility of carriers. Additionally, in graphene, electron and hole mobility decreases because of increased scattering and interactions with the lattice. Consequently, the wave’s amplitude decreases, resulting in a decrease in the NPC. It is imperative to understand the NPC to tailor graphene’s properties for various applications, including electronics, photonics, and nanophotonics. The variation in NPC for various plasma frequencies is depicted in Fig. 6a. As plasma frequency grows, NPC starts increasing and characteristics curves are shifted towards a high-frequency region as reported in [42]. Due to the increasing plasma frequency, the charged particles become more mobile and responsive to the EM waves. Consequently, the effective refractive index increases and the propagation constant increases as well. It is important to select the appropriate plasma frequency of an isotropic plasma to design and develop plasmonic devices. The variation in NPC for different collisional frequencies is analyzed in Fig. 6b. In this context, the EM wave frequency band increases from 0 to 5 THz and collisional frequency increases from \(\upsilon =2\times {10}^{6} \text{Hz}\) to \(\upsilon =4\times {10}^{6} \text{Hz}\) as indicated by red, black, and blue characteristic curves. It can be noted that the unphysical region vanished after the 1.2 THz frequency regime. In response to an increase in collisional frequency, both cutoff frequency and NPC increase. As the collisional frequency of an isotropic plasma increases, charged particles interact more intensely and frequently with each other. Furthermore, collisions between plasma particles occur more frequently, resulting in the exchange of energy and momentum. As a result of this transfer process, the wave’s phase velocity is affected resulting in an increase in NPC. The influence of chirality versus EM wave frequency on the NPC is depicted in Fig. 7. Interestingly, higher chirality values are associated with higher NPCs and higher EM wave frequencies. Based on the results of the analysis above, the chirality parameter strongly depends upon the frequency of the EM waves. Furthermore, the EM wave frequency band starts squeezing for a higher chirality value. Additionally, NPC increases with the increase of the chirality parameter as reported in [33, 43, 44]. Consequently, NPC can be modulated by varying the chirality parameter.

Fig. 5
figure 5

Influence of the chemical potential and number of graphene layers on NPC for case II

Full size image
Fig. 6
figure 6

Influence of the plasma frequency and collisional frequency on NPC for case II

Full size image
Fig. 7
figure 7

Effect of the chemical potential and number of graphene layers on NPC for case II

Full size image

Conclusion

The theoretical model was developed for a graphene-loaded waveguide structure surrounded by uniaxial chiral and plasma layers. For two types of uniaxial chiral medium, NPC parameters can be tuned by tuning graphene properties, including the chemical potential and the number of graphene layers, in addition to plasma properties, such as the plasma frequency, collisional frequency, and chirality parameters. It is concluded that case I support high frequency as compared to case II. The plasma frequency of isotropic plasma plays a crucial role in enabling the development of plasmonic-based devices. When plasma electrons oscillate at certain frequencies in resonance with the external electromagnetic field, their scattering properties are significantly enhanced. This resonance phenomenon can be exploited to design and fabricate plasmonic devices with various functionalities. The numerical results reflect that the presented study can be used to fabricate modulator plasmonic devices ranging from sensing and imaging to communication in the THz frequency regime.