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Abstract
Numerical analysis of Dyakonov waves generation has been carried out at uniaxial chiral-plasma planar interface. The extended electromagnetic wave theory is utilized, and an impedance boundary conditions approach is employed to obtain characteristics equation. Effective mode index and attenuation under the different values of collisional frequency, plasma frequency and chirality in the THz frequency range for three cases for uniaxial chiral media are discussed. These results can be used in the field of photonics and integrated optics to fabricate nanophotonic devices in the THz frequency range.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
Corrections
30 January 2024: A typographical correction was made to the acknowledgments.
1. Introduction
The electromagnetic surface waves are produced at the interface between two different materials and exhibit attenuation as they travel away from the interface [1]. The use of surface waves has led to the development of nanophotonic devices for sensing, optical interconnectors, and optical interconnectors for communication [2–8]. Dyakonov surface waves represent a special class of electromagnetic surface waves that propagate along anisotropic media interface. As a matter of fact, the partnering optical materials may be identical (and anisotropic), but they must be oriented differently so that the eigenvectors of their permittivity dyadic do not coincide at all. Unlike metals, dielectric materials tend to be less dissipative. Whenever dissipation is insignificant enough to be ignored, Dyakonov waves, can propagate infinitely long distances. Several studies of Dyakonov waves have been conducted on anisotropic materials, but most of the research is limited to specific orientations of these materials because of the complexity of dealing with them. The most promising anisotropic materials are metamaterials because of their controlled properties rather than naturally occurring crystals [1]. There are certain characteristics that make the Dyakonov wave unique, such as its narrow propagation interval, which is typically a few degrees [4,5]. Additionally, the Dyakonov wave exhibits a broader angular range when encountering interfaces of dissipative or magnetic media, further showcasing its distinct properties. Their unique properties, tunability, non-reciprocity, enhanced light-matter interaction, and surface sensing capabilities make them highly promising for a variety of applications in optics, photonics, and sensing technologies [9,10]. Unlike surface plasmon polaritons (SPPs), which are TM-polarized, Dyakonov waves are hybrids in nature, involving both TE and TM polarizations. For Dyakonov wave, the imaginary part of the permittivity of a dielectric material is often relatively insignificant due to this reason propagation length of Dyakonov wave must be larger than SPPs. The significance of longer propagation lengths extends beyond communication systems [11]. The first theoretical work was proposed by F.M. Marchevsky [12] and M.I. Dyakonov [13] at the interfaces of anisotropic crystals. However, after experimental detection of Dyakonov surface wave has emerged as a promising tool in photonics and integrated optics devices such as optical modulators, plasmonic sensors, and surface-enhanced spectroscopy [4,11]. Uniaxial and biaxial optical materials are extensively used in materials science to form complex structures. The literature on Dyakonov waves has not only focused on their existence at the interfaces of isotropic materials, but also on their presence at the interfaces of other types of optical materials. These include hyperbolic materials, chiral materials, bianisotropic materials, as well as limited media [14]. Different authors used different methods to explore the characteristics of Dyakonov wave. M V Zakharchenko and G F Glinskii analyzed the Dyakonov wave propagation between two biaxial anisotropic media [15]. M.V. Zakharchenko and G.F. Glinskii studied Dyakonov surface wave at the interface of biaxial crystals to obtain maximum value of the angular domain of their existence. Juan J. Miret et. al. studied Dyakonov surface wave with hybrid polarization to enlarge angular range with minimal losses. Sipeng Chen et. al. presented the numerical analysis of Dyakonov surface wave between isotropic media and conductor-backed uniaxial slab at planar interface [16]. Chenzhang Zhou et.al numerically analyzed Dyakonov–Tamm (DT) surface wave propagation at the planar interface between nonhomogeneous uniaxial dielectric and isotropic dielectric [17]. K. Yu. Golenitskii, and A. A. Bogdanov analyzed spectrum and radiative losses of Dyakonov wave in cylindrical waveguides to fabricate nanophotonic devices [18]. B Zhu et.al numerically studied the Dyakonov wave propagation at hexagonal boron nitride and isotropic material for sensing applications in optics community [19]. Uniaxial chiral have attained considerable attention in recent years due to their unexceptional properties and potential applications in various fields, including photonics, electronics, and telecommunications. By incorporating small chiral items, like wire spirals, into an anisotropic host medium, we can induce both chirality and anisotropy in the resulting optical material. In uniaxial chiral materials, the rotation of the polarization plane is dependent on the direction of propagation of the light [20–22]. In contrast to other optical materials, uniaxial chiral materials are characterized by more constitutive parameters as described in Eq. (1), which provides extra degree of freedom to control Dyakonov waves. Furthermore, uniaxial chiral simultaneously exhibit positive and negative permittivity. But the degree of chirality of isotropic chiral materials is hard to control once they have been manufactured. The presence of chirality in combination with anisotropy in uniaxial chiral media is expected to give rise to some exceptional characteristics of a proposed structure. The controllable nonhomogeneity of uniaxial chiral mediums opens new avenues for surface-wave research and development that are inaccessible with conventional optical materials. The study of electromagnetic surface waves in plasma medium has garnered significant attention from researchers and plasma scientists in the plasmonics community. The exceptional electromagnetic response exhibited by these waves has led to advancements in various fields including biochemical sensing, spectroscopy, light-trapping devices, optoelectronic devices, and communications. [23–29]. The behavior of electromagnetic waves in plasma medium is influenced by numerous key factors, including the collisional frequency, plasma frequency, and incident wave frequency. These traits play a crucial role in determining the absorption, reflection, and transmission traits of electromagnetic waves in a plasma medium. Plasma shows distinct features from conventional dielectric. One of the key differentiating factors is the density of electrons of plasma medium. Unlike other dielectric materials, the density of electrons in plasma can be tailored to manipulate the permittivity of the material. Another key advantage of plasma in nanophotonic devices is its ability to reduce absorption losses. Traditional dielectric materials, such as silicon or gallium arsenide, often suffer from significant absorption at certain wavelengths, limiting the overall device efficiency. By introducing plasma, these losses can be minimized as plasma serves as a low-loss medium, effectively decreasing the absorption and enabling higher transmission of light. Furthermore, isotropic plasma has a high permittivity, which enables it to store high amount of charge. This high permittivity enables efficient energy storage, making it suitable for applications such as capacitors and batteries.These properties inspire us to study the Dyakonov wave at uniaxial chiral plasma interface which has not been presented formerly.
In this manuscript, Dyakonov waves generation at uniaxial chiral-plasma planar structure has been examined numerically. Normalized propagation constant and attenuation or propagation loss at THz frequency regime. The variation in normalized propagation constant and attenuation under the different values of plasma features (collisional frequency and plasma frequency) and chirality for three cases of uniaxial chiral media are discussed. Numerical results indicate that, plasma features and chirality could be used used to tune the frequency band, normalized propagation constant and attenetion of of Dyakonov wave for proposed structure. The presented results can be used in the field of photonics and integrated optics to fabricate nanophotonic devices in the THz frequency range.
2. Mathematical formulation
In this section, mathematical formulation of both materials i.e., uniaxial chiral and isotropic plasma is discussed. Consider Dyakonov wave is propagating along z-axis as shown in Fig. 1.
The constitutive relation of uniaxial chiral are [30]:
In Eq. (1), $\xi $ represent the chirality parameter which is responsible for electromagnetic coupling, ${\overline{\overline {\textrm I}} _\textrm{t}} = {\mathrm{\hat{e}}_\textrm{x}}{\mathrm{\hat{e}}_\textrm{x}} + {\mathrm{\hat{e}}_\textrm{z}}{\mathrm{\hat{e}}_\textrm{z}}$ is a dyadic vector, ${\varepsilon _0}$ and ${\mu _0}\; $ are the permittivity and permeability of free space, respectively, $\textrm{}{\mathrm{\varepsilon }_\textrm{t}}$, ${\mu _t}$, and ${\mathrm{\varepsilon }_\textrm{z}}$, ${\mu _z}$ are transverse and longitudinal components of uniaxial chiral medium, and ${\mathrm{\hat{e}}_\textrm{x}}$, $\textrm{}{\mathrm{\hat{e}}_\textrm{y}}$, and $\textrm{}{\mathrm{\hat{e}}_\textrm{z}}$ are mutually perpendicular unit vectors in a Cartesian coordinate system.
EM field components for region $x > 0$:
Here, ${q_1}$ and ${q_2}$ are wavenumbers for uniaxial chiral medium. The remaining EM fields components can be derived from [30].
EM field components for isotropic plasma medium are given as:
Here, ${C_3}$, and ${C_4}$ are amplitude constants, ${k_p}$ is the wavenumber of isotropic plasma medium, ${k_p} = \sqrt {{\beta ^2} - {\omega ^2}{\varepsilon _p}{\mu _0}} $, ${k_0} = \omega \sqrt {{\mu _0}{\varepsilon _0}} $, ${\varepsilon _p}$ is the permittivity of isotropic plasma medium, ${\varepsilon _p} = 1 - \frac{{{\omega _p}^2}}{{{\omega ^2} + j\nu \omega }}$, ${\omega _p}$ and $\nu $ are plasma frequency and collisional frequency respectively as reported in [31]. By using the above EM field equation for uniaxial chiral and plasma medium the following boundary conditions are applied to obtain the characteristics equation:
3. Results
In this section, analytical results are elucidated by using characteristics Eq. (17) and some interesting features of Dyakonov wave at uniaxial chiral plasma interface are explored. The variation in effective mode index and attenuation under the different plasma parameters such as collisional frequency and plasma frequency versus incident wave frequency are analyzed. Since, in the realm of physics, the plasma frequency is intricately linked to the number density of charged particles within a plasma. The higher the number density, the higher the plasma frequency, and vice versa.
This fundamental relationship is vital in understanding the collective behavior of plasmas and has significant implications in both theoretical and applied aspects of plasma physics. Chirality parameter is also used to control the Dyakonov wave in THz frequency range for three different cases of uniaxial chiral medium.
Case I
In this case, we have set the parameter values as: ${\mu _t} = {\mu _z} = {\mu _0}$, ${\mu _1} = 1$, ${\varepsilon _t} ={-} 2{\varepsilon _0}$, ${\varepsilon _z} = 1.1{\varepsilon _0}$, $\xi = 0.1$, $\nu = 6\; THz$ and ${\omega _p} = 1\; THz$. In optics community, effective mode index $Re\left( {\frac{\beta }{{{k_0}}}} \right)$, is crucial parameter to fabricate nanophotonic devices. Figure 2 exhibits the influence of collisional frequency on effective mode index and attenuation as the function of incident wave frequency. In Fig. 2(a), as collisional frequency increases, band gap starts squeezing and characteristics curves moved towards low wave frequency region. Higher incident frequency leads to higher effective mode index with broader band gap. Furthermore, slope of variation is higher for lower value of collisional frequency. Since plasma temperature is affected by increasing collisional frequency. When collisions occur more frequently, particles gain and lose energy at a faster rate, leading to an overall increase in the average kinetic energy, or temperature, of plasma. In response to increasing temperature, plasma becomes more energetic and dynamic. As a result of this increased energy, plasma can absorb more waves. Moreover, higher collisional frequencies also increase in scattering of the oscillation mode leading to decrease in effective mode index. The variation in attenuation versus incident wave frequency under the different values collisional frequency is plotted in Fig. 2(b). Obviously, higher collisional frequency leads to higher attenuation and lower cut-off frequency in proposed frequency range. Furthermore, after 6 THz unphysical region starts which is not crucial for practical aspects in optics community. The collisional frequency of an isotropic plasma has a direct impact on the attenuation for proposed structure. As the collisional frequency increases, the attenuation also increases, leading to a reduction in the amplitude and intensity of the waves. This relationship is important to consider in various plasma applications, ranging from communication systems to fusion devices. The impact of plasma frequencies on effective mode index and attenuation versus incident wave frequency is plotted in Fig. 3. Since plasma frequency is directly related to the density of electrons in plasma medium. As the density of electrons increases, the plasma frequency increases as well. Figure 3(a) analyzed the variation in effective mode index for four different values of plasma frequencies indicated by black thick, red thick, blue thick, and green thick characteristic curves. As plasma frequency of plasma medium increases, cut-off frequency starts increasing but effective mode index increases. Moreover, band gap shows broadening for higher plasma frequencies. It’s of peculiar interest to that highest slope variation occurs at highest plasma frequency. In addition, the relationship between plasma frequency of plasma medium and effective mode index is a considerable factor in the field of optics and photonics to fabricate nanophotonic devices. To investigate the impact of plasma frequency on attenuation versus incident wave frequency is analyzed in Fig. 3(b). in contrast to Fig. 3(a), attenuation at lower values of plasma frequency. Hence, the change in plasma frequency will obviously affect incident frequency and attenuation of propagation loss. Effective mode index, under the different values of chirality Versus incident wave frequency is depicted in 4a. Incident frequency band extends from 0 to 2 THz. As chirality increases cut-off frequency also increases and characteristics curve show higher effective mode index. Unphysical regions vanish after some higher frequencies as indicated by black thick, red thick, blue thick, and green thick characteristics curves. The highest slope of variation occurs at $\xi = 0.4$. As chirality increases, the band gap starts broadening in the proposed frequency range. Frequency dependence attenuation is analyzed in Fig. 4(b). lowest attenuation could be achieved at highest value of chirality.
Case II
In this case, we have set the parameter values as: ${\mu _t} = {\mu _z} = {\mu _0}$, ${\mu _1} = 1$, ${\varepsilon _t} = 2{\varepsilon _0}$, ${\varepsilon _z} = 1.1{\varepsilon _0}$, $\xi = 0.1$, $\nu = 6\; THz$ and ${\omega _p} = 1\; THz$. To investigate the variation in effective mode index and attenuation versus incident wave frequency under the different values collisional frequency is analyzed in Fig. 5. In Fig. 5(a) it can be clearly seen that slope variation is smaller at higher value of collisional frequency and characteristics curves are moved towards low frequency region. Variation in characteristics curves is smaller at lower incident frequency. It’s of peculiar importance interest to note that bandgap starts squeezing for higher collisional frequencies. The further tunability trait suggests that Dyakonov wave can be engineered by collisional frequency of plasma for proposed structure. Figure 5(b) presents the frequency dependent attenuation for four different values of collisional frequency. Higher collisional frequency of plasma medium leads to lower attenuation for proposed case. Besides that, it’s of peculiar interest to note that after 0.2 THz frequency, characteristics curves level off mean unphysical region started that has no practical importance in the optics community. The variation in effective mode index and attenuation under the different values of plasma frequencies versus incident wave frequency is plotted in Fig. 6. To illustrate the effective mode index under the different values of plasma frequencies is analyzed in Fig. 6(a). Obviously, as plasma frequency increases, incident frequency also increases but the band gap remains same for proposed structure. When the plasma frequency is low, the electromagnetic wave can easily propagate through the plasma with minimal interaction. However, as the plasma frequency increases, the plasma particles start to respond more strongly to the EM wave, causing a decrease in the effective mode index. So, a higher effective mode index could be achieved with a lower value of plasma frequency. Figure 6(b) describes the influence of plasma frequency on attenuation as the function of incident wave frequency. Attenuation or propagation loss is valid up to certain frequency range, after some higher frequencies characteristics curves are level off which reflect unphysical region. This increase in attenuation occurs because the plasma frequency determines the ability of the plasma to support electromagnetic wave propagation. At lower plasma frequencies, the plasma behaves more like a dielectric material, allowing the waves to pass through with minimal attenuation. However, as the plasma frequency increases, the plasma becomes more conductive, leading to increased absorption and scattering of the waves, resulting in higher attenuation. Thus, lower attenuation could be achieved at lower plasma frequencies. To study the factor of effective mode index and attenuation under the different value of chirality versus incident wave frequency is depicted in Fig. 7. In Fig. 7(a) as chirality increases band gap also increases but wave frequency decreases. Additionally, effective mode of index tends to increase with the increases if chirality. Furthermore, slope of variation is smaller at lower values of chirality and chirality is more dominant at higher incident frequencies. . Besides that, it is of particular importance to note, that up to $\xi = 0.2$ band gap exhibit very small variation. In other words, larger band gap exhibited at higher values of chirality. The variation in attenuation for different values of chirality is revealed in Fig. 7(b). Clearly, at lower frequencies characteristics curves exhibited unphysical region. Attenuation can be tuned by tunning chirality of proposed structure.
Case III
In this case, we have set the parameter values as: ${\mu _t} = {\mu _z} = {\mu _0}$, ${\mu _1} = 1$, ${\varepsilon _t} = 2{\varepsilon _0}$, ${\varepsilon _z} ={-} 1.1{\varepsilon _0}$, $\xi = 0.1$, $\nu = 6\; THz$ and ${\omega _p} = 1\; THz$. Figure 8 demonstrates the influence of collisional frequency on effective mode index and attenuation. In Fig. 8(a), band gap starts squeezing with the increment of collisional frequency and characteristics curves starts moving towards low frequency region. It is of peculiar importance to note that a higher effective mode index could be achieved at higher collisional frequencies for the proposed case. By controlling collisional frequency, researchers and engineers can manipulate the properties of plasma mediums, enabling precise control of light at the nanoscale. To demonstrate the behavior collisional frequency on attenuation or propagation loss is plotted in 8b. As collisional frequency increases, characteristics curves are shifted towards low frequency region and band gap starts squeezing. The slope of variation is smaller at higher values of collisional frequency. Furthermore, characteristics curves overlapped at lower incident frequencies and variation starts after some higher frequencies. Frequency dependent effective mode index and attenuation under the different values of plasma frequencies is revealed in Fig. 9. Figure 9(a) presents the influence of plasma frequencies on effective mode index. Clearly, as plasma frequency increases incident frequency also increases, and effective mode index decreases which reflect lower confinement of EM wave for proposed case. The increase in plasma frequency leads to a decrease in the effective mode index due to the phenomenon of plasma dispersion. This behavior is a consequence of the interaction between electromagnetic waves and the collective oscillation of charged particles in a plasma medium. The variation in attenuation versus incident wave frequency under the different values of plasma frequencies is plotted in Fig. 9(b). It can be noted that, as plasma frequency increases band gap and attenuation also increase, and characteristics curves are moved towards low frequency region. When the plasma frequency increases, it implies that the density of free electrons in the plasma also increases. This higher electron density leads to a stronger response of the plasma to electromagnetic waves. As a result, the wave experiences more absorption and scattering within the plasma, leading to increased propagation loss. The variation in effective mode index and attenuation for four different values of chirality is analyzed in Fig. 10. In figure 10(a), as chirality increases, incident frequency, band gap and effective mode index increases. Higher effective mode index could be achieved at higher values of chirality for proposed case. The attenuation versus incident wave frequency under different values of chirality is plotted in figure 10(b). Attenuation can be tailored by tailoring the chirality of proposed structure.
4. Concluding remarks
The analysis of Dyakonov wave at uniaxial chiral and isotropic plasma has been carried out and following conclusions are drawn:
- 1. Dyakonov surface wave travels at uniaxial chiral-isotropic plasma planar structure in THz frequency range.
- 2. Impact of collisional frequency, plasma frequency and chirality on effective mode index and attenuation are analyzed.
- 3. In contrast to case I, the effective mode index increased with in increasing collisional frequency for case II and case III and vice versa for attenuation.
- 4. Case II and case III show a high effective mode index for lower values of plasma frequencies while case I shows highest effective mode index for higher values of plasma frequency. By exploiting the changes in the plasma frequency in the presence of analytes, nanophotonic sensors can achieve highly sensitive detection of chemical and biological species.
- 5. Effective mode index increased with increasing chirality in proposed three cases.
- 6. It is also worth noting that, chirality parameter is very sensitive for case I and case II.
- 7. The presented analytical numerical results can be used in the field of photonics and integrated optics to fabricate nanophotonic devices in the THz frequency range.
Funding
King Saud University (RSP2024R416).
Acknowledgments
This work was supported by the Researchers Supporting Project number (RSP2024R416), King Saud University, Riyadh, Saudi Arabia.
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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