Analysis of Electromagnetic Wave Propagation in Carbon Nanotube-Coated Metamaterials in Terms of Backward Electromagnetic Waves

Abstract

This article explores the propagation behaviors of electromagnetic waves within a metamaterial structure composed of three distinct layers, nano, micro, and macro, arranged from the outermost to the innermost section. The outermost layer, which serves as the focus of this investigation, consists of carbon nanotubes. The second layer, positioned just behind the outermost coating, exhibits micro properties and features a graded structure in terms of nonlocal characteristics and material property parameters. Therefore, the analyses conducted in this micro layer are grounded in nonlocal theory. The nonlocal constant is set at values of $\eta$: 0.7, $\eta$: 0.5, and $\eta$: 0.25, with investigations carried out using a nano-graded approach. Additionally, this micro layer is configured in a material-graded manner concerning its property parameters, defined as $D$: 0.1, $D$: 0.3, and $D$: 0.7, respectively. In the micro layer, a nano-graded approach achieves the highest frequencies of electromagnetic wave propagation when the material property parameter $D$ is set at 0.5 and the nonlocal constant $\eta$ is 0.25. In contrast, the lowest frequencies of electromagnetic wave propagation are observed when the material property parameter $D$ is 0.1, and the nonlocal constant $\eta$ is 0.5. The innermost layer of the metamaterial structure is characterized by macro properties. Notably, unlike many other studies, this research specifically examines the behavior of backward electromagnetic waves, rather than traveling waves, within the context of the aforementioned metamaterial properties. The amplitude values of the reflected waves, particularly those corresponding to the backward electromagnetic waves delineated in this study, exhibit a reduction as they propagate through the metamaterial components.

1. Introduction

The rapid advancements in optical and optoelectronic technology, a significant milestone in the field, have ushered in a new era of innovation and improvement in satellite and communication technologies [1,2,3,4]. This progress is crucial for our understanding and application of electromagnetic wave propagation in metamaterials. The different optical and electromagnetic properties of advanced materials such as photonic crystals, waveguides, and metamaterials, the movements of electromagnetic waves during electromagnetic wave propagation in these structures, and the characteristics they display are among the remarkable elements in research [5,6,7,8,9,10,11]. Based on this, in the studies conducted, the optical and dielectric properties of materials in photonic crystals, waveguides, and metamaterials are of great importance [12,13]. Permeability (μ) and permittivity (ε) come first among these properties. In particular, metamaterials are divided into different media depending on permeability and permittivity, and these different media have different material property parameters within themselves [14,15,16]. In linear, homogeneous, and isotropic media, travelling electromagnetic waves, namely, incident electromagnetic waves, encounter transmission and reflection events [17,18,19]. Here, the electromagnetic waves that travel between the media are directly affected by each medium’s permittivity and permeability values during the transitions [20,21]. With the invention of carbon nanotubes, the number of studies on nanostructures has also increased [22]. In addition, nonlocal theory has been developed to more easily examine the electromagnetic wave propagation behavior occurring in these nanostructures theoretically [23,24]. This theory reveals that atoms that are not neighboring each other, i.e., nonlocal, can also be affected by each other [25,26,27]. Today, carbon nanotube-based metamaterials are widely used, especially in studies on stealth technology [28,29]. In aviation, the outer part of these structures is specially shielded with radar-absorbent material (RAM paint) containing carbon nanotubes. As a result, these structures become invisible to the radar when exposed to electromagnetic waves [30,31,32,33]. The sources that provide the best understanding of the electromagnetic wave propagation behaviors occurring in these structures and that guide in this direction are also included in the literature [34,35]. When these studies are examined, it is concluded that the electromagnetic wave propagation equation is obtained by solving Maxwell’s equations. Depending on this equation, the behaviors of the transmitted and reflected electromagnetic waves can also be determined [36].

This study examines the behavior of electromagnetic wave propagation by analytically solving Maxwell’s equations in one dimension, specifically along the x-axis. Numerical methods can be employed to investigate electromagnetic wave propagation in two-dimensional structures. These structures can be divided into nodes or elements, allowing for the necessary analyses to be conducted [37,38].

In this study, the propagation behaviors of electromagnetic waves in a metamaterial structure consisting of three distinct layers, nano, micro, and macro, arranged from the outermost to the innermost section, are examined. The outermost layer, which is the primary focus of this investigation, comprises carbon nanotubes. The second layer, located just behind this outer layer, has micro-level characteristics and features a graded structure, especially concerning nonlocal properties and material property parameters $\left(D\right)$. In the metamaterial structure, the range of the material property parameters is specified as 0.01 to 1. For the micro components of the metamaterial, the material property parameters are carefully selected within the range of 0.1 to 0.7 [20]. The first difference between this study and other studies in the literature is that the frequencies of the electromagnetic wave propagation and electromagnetic wave propagation behaviors occurring in the nano and micro-sections of the metamaterial structure are examined with the help of nonlocal effects. The second is that while the electromagnetic wave propagation occurring in the structure is examined, the backward waves of the reflected waves are also examined in detail. This study explores a metamaterial structure in which reflected electromagnetic waves travel unidirectionally (along the x-axis) across its macro, micro, and nano components. Notably, the micro section is formulated in either nanoscale or graded material forms, which sets this research apart from other studies in the field. By addressing these critical aspects, this work fills a notable gap in the existing literature and contributes valuable insights to the topic.

2. Materials and Methods

This study delves into the propagation behaviors of electromagnetic (EM) waves within a metamaterial structure that consists of three distinct layers, each characterized by unique properties: nano, micro, and macro. These layers are organized sequentially from the outermost to the innermost section of the structure. The primary focus of this investigation is the outermost layer, which is composed of carbon nanotubes known for their exceptional electrical, optical, and material properties.

Figure 1 illustrates the propagation of an electromagnetic wave through a three-layer metamaterial structure comprising distinct layers. The initial layer, made of carbon nanotubes, serves as the first interface for the incident wave. As the electromagnetic wave travels from this nanoscale layer, it enters the intermediary micro layer, where various electrical and magnetic material properties may modify its characteristics. Finally, the wave progresses into the macro layer of the metamaterial, which is designed to manipulate the electromagnetic wave further before it exits the structure. This sequential transition between the different layers highlights the complex interactions and potential applications of metamaterials in controlling electromagnetic wave propagation behaviors. A strong, cohesive bond is observed among the components of the metamaterial structure under examination. This investigation maintains the assumption that the electromagnetic wave interacts with each component perpendicularly at its interface.

In a source-free, linear, homogenous, and isotropic medium, Maxwell’s curl equations are as follows [34]:

$$\nabla ·\overrightarrow{E}={\displaystyle \frac{\rho }{ϵ}},$$

(1)

$$\nabla ·\overrightarrow{H}=0,$$

(2)

$$\nabla \times \overrightarrow{E}=-i\Omega \mu \overrightarrow{H},$$

(3)

$$\nabla \times \overrightarrow{H}=i\Omega ϵ\overrightarrow{E}.$$

(4)

where $i=\sqrt{-1}$, $\overrightarrow{E}$ is the electromagnetic field vector, $\overrightarrow{H}$ is the magnetic field vector, $\mu$ is permeability, and $ϵ$ is permittivity. $\rho$ is the electric charge density, and the value of $\rho$ is zero here. $\Omega$ denotes the frequency of the electromagnetic wave propagating in the medium. Here, by solving Maxwell’s equations, the equation of electromagnetic wave propagating along the x-axis and depending on position–time is obtained as follows:

$${\displaystyle \frac{1}{\mu ϵ}}{\displaystyle \frac{{\partial }^{2}H}{{\partial x}^2}} - {\displaystyle \frac{{\partial }^{2}H}{{\partial t}^2}}=0.$$

(5)

where $D={\displaystyle \frac{1}{\mu ϵ}}$ represents the material property parameter.

The electromagnetic wave equation for wave propagation in the nth medium is as follows:

$${\displaystyle \frac{1}{{\mu }_n{ϵ}_n}}{\displaystyle \frac{{\partial }^2{H}_n}{{\partial x}^2}} - {\displaystyle \frac{{\partial }^2{H}_n}{{\partial t}^2}}=0.$$

(6)

The index n refers to the properties of the first medium in the three-layer metamaterial structure, while index (n + 1) denotes the properties of the medium located in the medium after the nth medium. The electromagnetic wave propagation behavior within the macro section of the metamaterial structure was analyzed using the equations established up to Equation (6). In accordance with the nonlocal theory, the equation representing the electromagnetic wave propagation occurring in the nth medium at the nano and micro scales of the metamaterial structure is formulated as follows [23,24]:

$$D_n{\displaystyle \frac{{\partial }^2{H}_n}{{\partial x}^2}} - \left(1 - {\left(e_{0}a\right)}^2{\displaystyle \frac{{\partial }^2}{{\partial x}^2}}\right){\displaystyle \frac{{\partial }^2{H}_n}{{\partial t}^2}}=0.$$

(7)

where $a$ represents the internal characteristic length and $e_0$ refers to a constant. Additionally, the nonlocal coefficient incorporates the term of $e_{0}a$. Thus, the nonlocal coefficient is expressed as $\eta =e_{0}a$.

$w_n\left(x,t\right)$ represents the wave propagation field of the incident electromagnetic wave propagating in the x-axis direction within the nth medium.

$w_{n+1}\left(x,t\right)$ refers to the wave propagation field of the electromagnetic wave transmitted in the (n + 1)th medium.

An electromagnetic wave that travels from the nth medium to the (n + 1)th medium along the x-axis was subjected to a comprehensive analysis, focusing on three specific scenarios. These scenarios include the incident electromagnetic wave, which is the wave approaching the boundary between the two media; the reflected electromagnetic wave, which is the portion of the wave that is reflected back into the nth medium upon encountering the boundary; and the transmitted electromagnetic wave, which is the portion of the wave that successfully passes through into the (n + 1)th medium. This detailed examination is systematically presented in Equations (8) and (9), which provide a mathematical framework for understanding the interactions between the waves and the different media involved:

$$w_n\left(x,t\right)=W_n{e}^{-i\left({\mathrm{k}}_{n}x-\Omega t\right)}+W_n{e}^{-i\left({-\mathrm{k}}_{n}x-\Omega t\right) },$$

(8)

$$w_{n+1}\left(x,t\right)=W_{n+1}{e}^{-i\left({\mathrm{k}}_{n+1}x-\Omega t\right)}.$$

(9)

where $W_{inc}:W_n{e}^{-i\left({\mathrm{k}}_{n}x-\Omega t\right)}$ refers to the incident electromagnetic wave, $W_{ref}:W_n{e}^{-i\left({-\mathrm{k}}_{n}x-\Omega t\right)}$ denotes the reflected electromagnetic wave, and ${W_{tra}:W}_{n+1}{e}^{-i\left({\mathrm{k}}_{n+1}x-\Omega t\right)}$ represents the transmitted electromagnetic wave [27,36]. The Ω values refer to the electromagnetic wave propagation frequencies. k_n and k_n+1 are wavenumbers in the nth and (n + 1)th mediums, respectively. In Equations (10) and (11), the transmission and reflection coefficients are expressed as follows, respectively:

$$T+\Gamma =1,$$

(10)

$$\Gamma ={\left({\displaystyle \frac{{\mathrm{k}}_n - {\mathrm{k}}_{n+1}}{{\mathrm{k}}_n+{\mathrm{k}}_{n+1}}}\right)}^2.$$

(11)

3. Results

As illustrated in Figure 2a and Figure 3a, the micropart of the metamaterial structure is divided into three sections for analysis. Figure 2b and Figure 3b depict the micropart in both nano-grade and material-grade forms. Specifically, Figure 2 represents the metamaterial structure with the micro part in nano-graded form.

Figure 3 depicts a five-part metamaterial structure, with the micro part in nano-graded form and divided into three sections.

The dispersion relation values for various scenarios were derived by solving Equation (7) from the preceding section, as illustrated in Figure 4. Figure 4 presents the dispersion relation for the micro part of the five-part metamaterial structure under two distinct scenarios. The first scenario examines the micro part when it is material-graded, showing how variations in material composition influence the dispersion relation. The second scenario focuses on the micro part when it is nano-graded, illustrating the effects of nanoscale variations in material properties on the dispersion relation. When this change in dispersion relation values is examined, it is concluded that the electromagnetic wave propagation frequency values increase with the increase in the value of the material property parameter ($D$), but as the nonlocal constant ($\eta$) increases, the electromagnetic frequency values ($\Omega$) decrease.

In Table 1, the transmitted electromagnetic waves in the five-segment metamaterial structure are represented as follows:

-

The EM wave traveling from Part 1 to Part 2 is denoted as 21.

-

The EM wave traveling from Part 2 to Part 3 is indicated by 32.

-

The EM wave traveling from Part 3 to Part 4 is represented by 43.

-

The EM wave traveling from Part 4 to Part 5 is labeled 54.

Additionally, the backward electromagnetic wave that first reflects from Part 5 to Part 4 and then from Part 4 to Part 3 is represented by 4534. The backward EM wave reflected from Part 4 to Part 3 and then from Part 3 to Part 2 is denoted as 3423. Lastly, the backward wave that reflects from Part 3 to Part 2 and then from Part 2 to Part 1 is represented by 2312.

Furthermore, the energy values of the electromagnetic waves are expressed as percentages in Table 1.

Table 1. Energies of electromagnetic waves.

Electromagnetic Waves	Nano-Graded (%)	Material-Graded (%)
21	92.310	66.942
32	0	8.264
43	0	3.966
54	0	0.648
4534	0	0.544
3423	0.087	0.457
2312	0.065	0.151

Table 1. Energies of electromagnetic waves.

Electromagnetic Waves	Nano-Graded (%)	Material-Graded (%)
21	92.310	66.942
32	0	8.264
43	0	3.966
54	0	0.648
4534	0	0.544
3423	0.087	0.457
2312	0.065	0.151

Figure 5 illustrates the propagation of traveling transmitted electromagnetic waves.

Figure 6 depicts the propagation of backward electromagnetic waves as they approach the macro section of the five-part metamaterial structure and reflect back from that point.

4. Discussion

In Figure 4, the relationship between the frequencies of electromagnetic wave propagation and the nonlocal coefficient (η) is clearly depicted. As the nonlocal coefficient increases, a noticeable decrease in the frequencies of electromagnetic wave propagation is observed. This trend suggests that the nonlocal interactions within the material significantly influence the behavior of electromagnetic waves. Conversely, the figure also highlights that with an increase in the material property parameter ($D$), the values of electromagnetic frequencies tend to rise. This indicates that certain intrinsic characteristics of the material can enhance the frequency of electromagnetic waves, potentially leading to more efficient wave propagation.

Figure 5 and Figure 6 provide a comprehensive analysis of the dynamics of electromagnetic wave propagation across different segments of the metamaterial structure over time. Specifically, in Figure 6a, the data reveal that in the nano-graded segment of the five-part metamaterial structure, there are no observable backward waves. This absence of backward wave propagation suggests a unique behavior characteristic of this segment, which may be attributed to its specific material properties. Furthermore, during the course of electromagnetic wave propagation through the entirety of the five-part metamaterial structure, a significant observation is made regarding the amplitudes of any backward electromagnetic waves. With time, these amplitudes are seen to decrease progressively as the waves travel through the material, indicating that the metamaterial structure effectively attenuates these backward waves, potentially leading to a more directed and efficient propagation of forward waves.

5. Conclusions

This study presents a unique approach compared to previous research by specifically investigating the properties of backward electromagnetic wave propagation within a five-segment metamaterial structure. In this context, the analysis incorporates nonlocal theory to understand the behaviors of electromagnetic waves as they traverse the nano parts of the metamaterial, specifically in segments referred to as Part 1 through Part 4. In contrast, for the macro component of the metamaterial structure, identified as Part 5, the nonlocal coefficient is set to zero, simplifying the analysis for that segment. This distinction is crucial as it highlights the differing wave propagation characteristics between the nanoscale and macroscale segments of the metamaterial.

Future research could build upon these methods to investigate the electromagnetic wave propagation phenomena within more complex two-dimensional nanostructures. Furthermore, it would be beneficial to include considerations of the non-homogeneous and non-isotropic material properties of the structures being examined. By integrating these aspects, the research can attain a more comprehensive understanding of the intricate electromagnetic interactions that take place within metamaterials, thereby contributing to the advancement of the field.