Behavior of TE and TM propagation modes in nanomaterial graphene using asymmetric slab waveguide

1 Introduction

Numerous studies have examined graphene’s potential use in waveguide applications. Graphene is a two-dimensional material with exceptional optical and electronic properties. This review focuses on recent research utilizing asymmetric slab waveguides to investigate the behavior of transverse electric/transverse magnetic (TE/TM) propagation modes in the nanomaterial graphene. Several studies that have contributed to our understanding of this topic are discussed.

Based on a bent asymmetric-slab waveguide, Cao et al. [1] proposed an ultra-compact and fabrication-tolerant polarization rotator. A thin layer of graphene was sandwiched between two dielectric layers with varying refractive indices. Shen et al. [2] looked at TM modes in a slab waveguide that had a core made of a multilayer structure of graphene and dielectrics and walls made of different common materials. More interestingly, their study found that graphene significantly alters the propagation characteristics of TM modes.

Yang et al. [3] examined the transmission characteristics of asymmetric slab waveguides with left-handed materials (LHM). They investigated the transmission characteristics of asymmetric slab waveguides with LHMs and proposed a new dielectric slab waveguide structure with LHM cover and substrate. The researchers discover unusual properties in the asymmetric LHM slab waveguide, such as the absence of fundamental and first-order modes and double degeneracy of higher-order modes. Glytsis [4] offered the basic concepts of slab dielectric waveguides and elaborated on their significance in optical communication, sensing, and imaging applications.

Using the transfer matrix method, Salman and Yasser [5] analyzed guided modes in a slab waveguide with a central anisotropic metamaterial layer. They showed that mode properties can be controlled by varying the permittivity tensor of the anisotropic metamaterial layer, the metamaterial’s thickness, and the refractive index. The study by Taya et al. [6] looked at how TE/TM propagation modes work with polarized light in two slab waveguide structures that are made of LHMs and have three layers. The results show that LHM layers have an effect on the confinement factor. To change these parameters, the effective index of TE/TM modes and the thickness of the LHM core layer could be changed. Using double asymmetric discontinuities in leaky wave structures. To prevent open-stopbands, Baccarelli et al. [7] presented an innovative approach. The authors demonstrate that high directivity and gain can be obtained with the open stopband effectively blocked by the two asymmetric discontinuities.

In their work, He and Li [8] investigated the differences between graphene waveguides and their TM and TE surface plasmon modes. They found that the TM surface plasmon mode has a higher effective index and a lower confinement factor than the TE surface plasmon mode. The choice of these modes has a significant impact on the waveguides’ performance. Raghuwanshi [9] provided an analysis of dielectric optical waveguides to show how the waveguide’s structure impacts wave propagation and to what extent it influences the propagation characteristics.

Bai et al. [10] explored TE and TM propagation modes in graphene and proposed a TE/TM mode switch using monolayer graphene and metal-dielectric resonances. They demonstrated that the resonant behavior of the device can be modified by adjusting the graphene layer thickness. Liu et al. [11] created an efficient and compact TE-TM polarization converter on a silicon-on-insulator platform. Chaharmahali et al. [12] focused on controlling terahertz waves using graphene-based metamaterials. They proposed a tunable ultrabroadband terahertz metamaterial absorber and filter using disk graphene patterns, demonstrating their effectiveness for practical applications such as filters and sensors at terahertz spectra.

A study by Evseev et al. [13] looked at surface plasmon-polaritons in a symmetric dielectric waveguide with active graphene plates. The results showed that graphene strengthens the connection between plasmons and waves. Kumar et al.’s study [14] on metamaterials’ impact on evanescent fields in a grounded slab metamaterial waveguide structure showed that metamaterial choice significantly affects the evanescent field. Talebi [15] explored optical modes in slab waveguides and highlighted the significance of the magneto-electric effect in the design of waveguides. Werra et al. [16] studied TE resonances in graphene-dielectric structures, noting that the influence of the dielectric layer’s thickness and dielectric constant can have a big effect on how the TE mode behaves when it is resonating.

Alwan et al. [17,18] studied the impact of polymer thickness and frequency on the TE mode using slab waveguides. They proposed a method for high data rates for optical carriers made of graphene. They showed how the presence of graphene significantly increases the data rate of optical carriers. Graphene modifies the confinement factor and effective index of the TE and TM modes. Controlling these parameters involves adjusting graphene’s Fermi energy. The behavior of the TE and TM modes in a silicon nanomaterial slab waveguide was the main focus of the study conducted by Alwan et al. [19]. On the TE and TM modes, the silicon nanomaterial layer had a major effect. The ninth TE mode was found to be weakly coupled to the substrate. By adjusting the permittivity tensor of the nanomaterial silicon layer, one could modify the effective index and confinement factor of the TE and TM modes.

1.2 Theoretical analysis

A common part used in integrated optics is the three-layer asymmetric dielectric waveguide depicted in (Figure 1) [21].

Figure 1

Asymmetric dielectric slab waveguide consisting of three layers.

A thin 2a-thick dielectric film is deposited on a substrate, with a cladding of air above. Total internal reflection propagation occurs when refractive indices n _f > n _s ≥ n _c [22].

The expression for the free space wavenumber at the designated operating frequency ω or f in Hertz is by k ₀ = ω√μ ₀ c ₀ = 2πf/c ₀ = 2π/λ ₀. The fields’ t and z dependence is considered to follow the standard exp j(ωt − βt) pattern. Adequate positive decay constants (α _s and α _c) are required in both the substrate and cladding to achieve exponential field attenuation with x. Transverse wavenumbers jα _s and −jα _c are corresponding transverse wavenumbers, while real-valued transverse wavenumber k _f within the film is satisfied [23,24]:

(1) k f 2 = k 0 2 n f 2 − β 2 α s 2 = β 2 − k 0 2 n s 2 α c 2 = β 2 − k 0 2 n c 2 ⇒ k f 2 + α s 2 = k 0 2 ( n f 2 − n s 2 ) k f 2 + α c 2 = k 0 2 ( n f 2 − n s 2 ) ( 1 + δ ) = k 0 2 ( n f 2 − n c 2 ) α c 2 − α s 2 = k 0 2 ( n f 2 − n s 2 ) δ = k 0 2 ( n s 2 − n c 2 ) ,

where a asymmetry parameter δ is

(2) δ = n s 2 − n c 2 n f 2 − n s 2 .

The acceptable range of β for guided mode is defined by the inequalities, k ₀ n _s ≤ β ≤ k ₀ n _f, and β ≥ k ₀ n _c, assuming that k _f, α _s, and α _c are real.

(3) n c ≤ n s ≤ β k 0 ≤ n f .

2 Numerical results and discussion

This section provides an explanation and discussion of the behavior of the TE/TM propagation modes in the MATLAB scripts for nanomaterial graphene. A thin graphene film is applied to silica substrate. On top of the film is a dielectric coating, like air. Graphene nanomaterial was incorporated into an asymmetric slab waveguide in order to assess its optical characteristics and performance. Graphene nanomaterial, with unique characteristics and waveguide parameters such as propagation wavenumbers, modes numbers, cut-off frequencies, decay wavenumbers, transverse wavenumbers, decay wavenumbers, skin depth, and angles of total internal reflection, is utilized in various applications such as asymmetric dielectric slabs, which are the key components of modern integrated optics. The study used an EM wave with a wavelength of 1.55 µm to propagate on a graphene film with a refractive index of 2.75 and a silica substrate with a refractive index of 1.45. The MATLAB application generated both TE/TM modes, with the ten TE modes exhibiting weak binding to the substrate side as a result of a small decay wavenumber and a cutoff frequency that is approaching to the operating frequency.

Figure 3 shows that nine number of TM modes. Note that guide modes within asymmetric slab waveguide include TE/TM modes. The propagation mode (TE) lacks an electric field component in the propagation direction, while the mode (TM) lacks a magnetic field component in the same direction. As a result, TE and TM modes behave similarly, indicating that asymmetric slab waveguides are well suited for efficient and low loss propagation. The transverse profiles for TE and TM modes are plotted in Figures 2 and 3; the magnetic and electric fields have the typical setup. It oscillates within the slab and becomes evanescent in the surrounding media. The transverse magnetic modes and transverse electric modes are influenced by the structure of graphene, silica substrate, and its refractive indices of substrate and nanomaterial graphene film.

Figure 2

Mode profile for TE modes.

Figure 3

Mode profile for TM modes.

Figure 4 shows the propagation wavenumbers and cut-off frequencies for each TE/TM mode with a constant thickness. The cut-off frequency is directly proportional to the number of TE/TM modes, whereas the propagation wavenumber is inversely proportional. The graphene film thickness and cut-off frequency are fixed for each mode. Higher-order modes propagate with smaller propagation wavenumber values, while lower-order modes propagate with smaller values. The propagation wavenumber is determined by nanomaterial characteristics and wavelengths. A guide TE mode has an equal nearly propagation constant to a TM mode of the same order. The propagation constant has frequency dependence due to the guide’s structure and nanomaterial dispersion. Polarization dispersion also exists due to different propagation constants for TE/TM modes.

Figure 4

Higher propagation wavenumbers from TE/TM cut-off frequencies in graphene.

Electromagnetic waves propagating through graphene nanomaterial attenuate in the direction of propagation. Electric and magnetic fields vary sinusoidally inside the slab, decaying exponentially outside. Upper TE and TM modes have less attenuation wavenumbers than other modes, reducing decay wavenumbers in cladding and substrate parameters as shown in Figures 5 and 6. As wavelength increases, layer decay wavenumbers move to infinite, indicating rapid decay of electric and magnetic fields outside the slab region. This results in a rapid decay of evanescent waves at angles far above the critical angle (21.32°). Transverse wavenumbers within the slab increase in the number of modes, and the cut-off condition occurs when propagation wavenumbers match the transverse wavenumbers inside the slab, resulting in propagation wavenumbers becoming equivalent to those of the surrounding medium.

Figure 5

Cladding decay wavenumbers with higher TE/TM modes.

Figure 6

Wavenumbers of substrate decay associated with greater TE/TM modes.

Skin depth refers to the decay of electromagnetic waves within graphene nanomaterial. The attenuation wavenumber of the electromagnetic field increases exponentially with the transverse distance, indicating the existence of electromagnetic field within a skin depth, as shown in Figure 7. Increase the number of TE/TM modes also enhances the skin depth in the cladding and substrate at a certain wavelength. Upper TE/TM modes commute longer distances in cladding and substrate than lower-order modes. Therefore, graphene nanomaterials with excellent optical characteristics and low absorption loss are ideal for optoelectronics applications.

Figure 7

Substrate/cladding skin depth and wavelength-specific TE and TM modes.

The concept involves using asymmetric dielectric slabs for total internal reflection of EM waves. Lossless dielectric slab waveguides experience scattering and absorption due to impurities. Figure 8 illustrates the total internal reflection angles and cut-off radius for the m-th mode at 1.55 µm wavelength and constant graphene thickness. The cut-off radius increases with more TE and TM modes, and internal reflection angles decrease. This means that the increased number of TE/TM modes leads to decreased total internal reflection angles. Higher modes exhibit a bigger cut-off radius and penetrate deeper into graphene nanomaterials. Internal reflection angles exceed critical angles for substrate (31.82°) and cladding (21.32°), except for the tenth TE mode (33.23°). The tenth TE mode’s internal reflection angle is near critical due to the small decay parameter and near operating frequency.

Figure 8

Total internal reflection angles, TE and TM modes, and the mth mode cutoff radius.

Figure 9 displays the angles and wavenumbers for total internal reflection in graphene. It includes TE and TM modes at 1.55 µm wavelength. Figure 9 shows that the number of modes increases with decreasing propagation wavenumbers and the angles of internal reflection decrease. This implies that the angles and propagation wavenumbers reduce with higher modes in graphene nanomaterial. There are two critical angles (31.82°, 21.32°) connected with the internal reflection at the lower and upper interfaces. Critical angles do not depend on EM wave polarization; phase shifts depend on polarization; therefore, it should be noted that the angles of the TE and TM modes in an asymmetric dielectric slab are greater than both critical angles. Consequently, the wave within the slab undergoes total internal reflection at both interfaces and becomes confined within the slab. An asymmetric situation with one cladding close to the slab and another lower than the slab yields the highest slab thickness.

Figure 9

Total internal reflection angles, TE/TM modes, and propagation wavenumbers.

The asymmetric configuration allows one cladding to have a high refractive index, allowing higher modes to escape, while the other cladding can have a low refractive index, enhancing optical confinement. The consequences indicate that the asymmetric slab waveguide may be more advantageous for specific implementations due to its high-power confinement properties in the slab region. Furthermore, the study explores the use of asymmetric planar waveguides in optical fiber communication systems and their potential in wavelength division multiplexing optical networks (WDM).

3 Conclusion

The properties of TE/TM modes in an asymmetric three-layer slab waveguide structure were investigated in this study. Note that nanographene is regularly temptingly ascribable to its high non-linear influences and low net absorption in the broad-band optical band; therefore, it is an excellent choice for high rendition, high rapidity, and broad-band optoelectronic and electronic equipment. As an additional result of this effort, field profiles of the TE/TM modes were demonstrated so that the properties of nanographene material for various parameters could be comprehended. The study explores the use of asymmetric planar waveguides in optical fiber communication systems and their potential for WDM optical network applications. Future work will explore various materials with diverse layers to determine their usage, comparing their parameters with standardization to determine the best material for protection and radiation protection.