Effective thickness of gallium arsenide on the transverse electric and transverse magnetic modes

1 Introduction

To figure out the optical properties of gallium arsenide (GaAs)-based devices like optical limiters, mode-matching metamaterials, and photonic crystals, it is important to know how thick the material is in both the transverse electric (TE) and transverse magnetic (TM) modes. It is common for “effective thickness” in the TE mode to mean the thickness of the GaAs layer that would cause the same phase shift as the real GaAs layer, taking into account the change in the refractive index and the applied electric field along the growth direction [1]. The study investigates the impact of layer thickness on TE/TM mode behavior in GaAs-based slab waveguides. It focuses on the influence of GaAs thickness on propagation modes, cut-off wavenumbers, attenuation and total internal reflection angles. The research focuses on the design, optimization and fabrication of slab waveguides in III–V semiconductor materials. Despite extensive studies there is a lack of research on the direct relationship between material thickness and the distribution, cutoff, and attenuation behavior of TE/TM modes, particularly in symmetric slab configurations.

Waveguide sensors work by means of the evanescent field existing in the cladding layer, where the analyte to be detected is found. The thickness of the guiding layer and the refractive indices of every media component affect the effective refractive index of the guided mode detected in a waveguide structure. Consequently, the effective refractive index of the light-guided mode varies with any change in the refractive index of the analyte within the cladding layer. Kołacz et al. [2] studied TE/TM modes in liquid crystal-clad waveguides, focusing on material-cladding interactions and mode confinement and dispersion, aligning with our analysis of air-clad GaAs waveguides, where the cladding (air) significantly influences mode confinement and attenuation behavior.

Mechanical characteristics have been much improved in material science thanks to recent developments in which the creative molten Bi-Ga austempering process stands out. These developments are on par with the advances in GaAs-based optical devices, which have attained better performance and lifetime [3]. Many investigations have produced a comprehensive collection of hypotheses that enable us to better grasp the behavior of TE and TM modes in slab waveguides, particularly those grounded on GaAs. However, a systematic analysis of how GaAs thickness directly influences the cutoff wavenumbers, attenuation, and mode confinement in asymmetric slab waveguides remains underexplored, particularly for applications in terahertz and integrated photonics. These works tackle the numerical methods for mode equation solving, the impact of waveguide properties on mode profiles, and mode equation derivation. Recent advancements in GaAs-based waveguide design (Midolo et al. [1], Chen and Uddin [4]) emphasize the need for precise thickness control, yet quantitative comparisons between TE/TM mode behaviors across varying GaAs thicknesses are limited. Modal interactions in optical systems can be optimized using polarization techniques, as explored by He et al. [5]. By using bidirectional signals with orthogonal TE/TM polarizations, angular resolution is improved. This approach aligns with the draft’s analysis of TE/TM mode behavior, providing a comparative perspective on managing orthogonal modes. Alwan et al. [6] investigated the asymmetric slab waveguides with great attention for mode count and propagation characteristics. Crucially for knowledge of the behavior of TE and TM modes in GaAs-based waveguides, they explored how the thickness of the guiding layer influences mode profiles and attenuation. By means of a theoretical investigation of slab waveguides, Chen and Uddin [4] derived mode equations for both TE and TM modes. Savotchenko [7] presents a study on dispersion equations for transverse electromagnetic waves at graded-index interfaces, providing theoretical insights for analyzing TE/TM mode confinement and propagation in asymmetric GaAs waveguides. Designing GaAs-based optical devices depends on a knowledge of the link between waveguide characteristics and mode propagation, which their study clarifies. By means of modal analysis of slab waveguides, Soun et al. [8] computed the count of supported modes as well as propagation constants. Their work presents a complete method for mode profile computation in waveguides based on GaAs. Different amounts of GaAs added to metamaterials can enhance the performance of the difference frequency generation process [8]. Building the structure to support Fabry-Perot and guided modes in the GaAs layer will help to increase its extraction efficiency. Emphasizing the interplay between modes and the waveguide structure, Kim and associates [9] investigated the distribution of electromagnetic fields in asymmetric slab waveguides. Their work addresses how field dispersion influences optical device performance generally. Defects such as EL2 deep donor defects in melt-grown bulk GaAs might change the effective thickness, hence lowering electron lifespan and charge collecting efficiency [9]. One way to passivate these flaws and enhance device performance is by chromium compensation for GaAs (GaAs:Cr). Dhavamani et al. [10] searched numerical solutions for mode equations in slab waveguides using MATLAB. Their work provides a practical approach for approximating the GaAs-built waveguide propagation characteristics. Particularly in GaAs-based devices, Backhaus et al. examined the symmetry characteristics of slab waveguides and their impact on mode propagation in comparison to symmetric and asymmetric waveguides.

Many further studies [1,11,12,13,14] concentrate on the optical properties of GaAs and material selection for waveguide constructions. They cover bandgap, GaAs-specific production techniques, refractive index, and how choice of materials influences mode confinement and signal attenuation. This group presents significant fresh angles on the technical aspects of GaAs-based waveguides.

Zhao et al. [12] examined the effects on the propagation modes in slab waveguides of substrate and cladding materials. The work emphasizes the need of material choice in achieving desired mode properties in GaAs-based devices. The effective thickness in the TM mode is affected by the refractive index change even though its computation is not as straightforward. The transmission matrix for every layer of a multilayer material construction allows one to examine the behavior of light or other electromagnetic radiation as it flows through the construction [12]. The formula lets you obtain the transmission coefficients and transmittance of the photonic crystal as well. Changes in refractive indices of materials can influence the effective thickness by means of hydrostatic pressure and ambient temperature especially for GaAs-based devices. Hydrostatic pressure (P) and temperature (Te) affect the dielectric constant of GaAs; hence, a function including both of these elements can describe it [12].

Graphene nanomaterials are important for their use in optical waveguides, Alwan et al. [14,15] investigation on the optical characteristics of this material, including its refractive index and bandgap, looks at the work offers understanding of how these features affect the light propagation in GaAs-based systems. With an eye on the asymmetric scenario, Hussein et al. [13] examined the cut-off frequencies and critical angles in slab waveguides. The total internal reflection and mode confinement are discussed in this work with reference to variations in the refractive index between the layers. Midolo and his group [1] investigated ways to maximize waveguide thickness for better mode confinement and lower attenuation. This work presents rules for selecting the optimal thickness of GaAs in optical waveguide designs. Emphasizing the challenges and solutions for precisely thickness control, De Koninck et al. [11] looked at GaAs-based waveguide manufacturing techniques. The work provides insight of the manufacturing methods guaranteeing the intended optical properties of GaAs waveguides.

Previous literature [16,17,18,19,20,21,22] discussed the numerous useful applications of GaAs waveguides in terahertz technology, optical communication systems, and high-power applications. From environmental resilience to signal loss to wavelength effects, they offer a whole picture of the useful outcomes of the research. Kumar et al. [16] studied waveguide attenuation characteristics, more specifically in regard to asymmetric designs. It looks at how the material selections and thicknesses affect the signal loss in optical waveguides built in GaAs. Chrostowski et al. [17] looked at how wavelengths changed slab wave modes of propagation. The study explores the interactions between different wavelengths and the waveguide structure to affect the cut-off frequencies and mode profiles. Ishino and Matsumoto [18] studied the usage of GaAs to terahertz waveguides stressing its improved qualities for high-frequency applications. By use of optimal waveguide designs, this work highlights how GaAs might advance terahertz technology. Papež et al. [19] focused on the integration of GaAs waveguides in optical communication systems, notably with an eye on their ability to improve signal transmission efficiency. This study investigates the benefits of using GaAs for minimizing signal loss and increasing data transmission rates. Especially in high-power uses, Di Cicco et al. [20] looked at the temperature and ambient effects on GaAs waveguide performance. This work investigated the effect of ambient conditions and temperature changes on mode propagation and device reliability. Mode optimization in waveguide design is crucial for terahertz applications. Research by Rodrigues et al. [21] and Yadav et al. [22] enhances the relevance of the draft’s symmetry analysis and discusses angle-dependent TE/TM mode behavior in high-frequency applications.

In summary, factors such as hydrostatic pressure, environmental temperature, defects, and device design can alter the effective thickness of GaAs in the TE and TM modes. This is an important parameter for figuring out the optical properties of GaAs-based devices.

2 GaAs

Gallium arsenide with chemical formula GaAs is a III–V group semiconductor compound utilized in certain diodes, field transistors (FETS), and integrated circuits (ICS) at microwave frequencies and photovoltaic cells, and have reached nearly 30% efficiency in laboratory environment, but they are very expensive to manufacture. GaAs crystals exhibit a metallic greenish-blue sheen or appear as gray powder. The experimental and computational investigations of flame holders in combustion chambers [23] highlight the potential for GaAs to be integrated into high-temperature environments, further expanding its applicability in high-power and high-frequency devices. The GaAs crystal consists of two sublattices, both face-centered cubic (FCC), displaced relative to one another by half the diagonal of the FCC cube. This crystal arrangement is referred to as the cubic sphalerite or zinc blende structure. There are four nearest neighbor bonds of length 0.245 nm to each atom with the bonds separated by the tetrahedral angle of 109.47° and the bonding is more covalent semiconductor. GaAs exhibits a direct band gap ranging from 1.4 to 1.45 eV at bulk scale, rendering it more advantageous than silicon for applications requiring highly efficient light emission. Being a direct band gap material, it is resistant to radiation damage enabling its use in optical windows and space electronics in high power applications and it exhibits significant transmission in the mid-infrared region at wavelengths ranging from 1 to 15 µm, along with the terahertz region (λ = 100–3,000 µm). For a typical sample of GaAs, the refractive index and extinction coefficient at wavelength 632.8 nm are 3.85744 and 0.1983491. Materials with a high index of refraction have a higher reflectivity than materials with low index. Because the index of refraction varies with the wavelength of the photons, they are particularly useful in sensitive electronic equipment requiring low-frequency amplification. The following are some of the most common applications for GaAs wafer: Radars and radio communication devices. The use of GaAs in high-frequency applications is further supported by its efficiency in energy conversion, as demonstrated in studies on induction heating for water desalination, which highlight its potential for sustainable energy solutions [24]. GaAs is frequently utilized in the production of solar cells due to its ability to absorb a greater amount of energy from incoming solar radiation, attributed to its higher absorption coefficient.

3 Theoretical analyses

Consider a slab waveguide as in Figure 1. It consists of a film region with thickness a and refractive index n _f, sandwiched between cladding and substrate with refractive indices n _c and n _s, respectively. The three indices are chosen such that n _f is bigger than n _c and n _s in order for total internal reflection to occur at the interfaces [25,26]. The waveguide is said to be symmetric if the substrate and cover materials have the same index of refraction, else it is said to be asymmetric. The symmetric waveguide represents a specific instance of the asymmetric waveguide. The coupling efficiency of acoustic drivers in thermoacoustic machines, as explored by Yahya et al. [27], provides valuable insights into optimizing the design of waveguides for improved signal transmission and reduced attenuation.

Figure 1

A symmetric slab waveguide consisting of three materials.

We discuss in this part the features of the TE and TM propagation modes succinctly. The free-space wavenumber k ₀ at the operating wavelength λ _o (in µm) is defined as k ₀ = 2π/λ _o. The fields attenuate exponentially with x inside the substrate and the cladding only if the attenuation constants α _s and α _c inside the cladding and substrate are positive. This implies that there exist EM fields within a penetration depth distance denominated by (1/α _s, 1/α _c). The cut-off wavenumbers k _f in the film will be the actual value. These numbers fulfill the relations

where the asymmetric parameter δ defined as

fulfils the inequality, β ≤ k ₀ n _f, β ≥ k ₀ n _s, and β ≥ k ₀ n _c, that jointly describes the permitted range of β for the guided modes since k _f, α _s, α _c are considered to be real. Since we considered n _f > n _s ≥ n _c, note δ ≥ 0. It follows that propagation wavenumber β must be

4 Simulation results and discussion

In the present work, we deal with three-layer symmetric slab waveguide structure in which all media such that a thin GaAs film is deposited on glass substrate. Above the film is a dielectric cladding, such as air. The simulation results explain the effects of GaAs thickness on TE/TM modes and discuss the parameters of waveguide, which are the propagation wavenumbers, modes numbers, cut-off frequencies, attenuation in substrate and cladding, the effective skin depth in substrate and cladding, and angles of total internal reflection, for TE mode and TM mode. Moreover, it discusses the same parameters as above applying EM signal with a wavelength of 0.6328 µm on each GaAs film with thickness of 0.5, 1, 1.5, 2 µm and refractive index of 3.8574 for GaAs and glass substrate with refractive index of 1.52, and air cover. The MATLAB code generates both the TE and TM modes, with the numerical solution produced listed in the tables. Figure 2 shows the results of the number of TE modes with different GaAs thickness separately, and with applying 0.6328 µm wavelength, the parameters of waveguide are outlined in Table 1. The outcomes of mode profile for TE modes show that the number of TE modes increase with increase in the GaAs thickness. Due to decreasing the propagation wavenumber, the cut-off wavenumbers increase, and the attenuation decreases for specific applied wavelength.

Figure 2

Mode profile of TE modes of asymmetric slab waveguide when wavelength λ = 0.6328 µm and (a) a = 0.5 µm, (b) a = 1 µm, (c) a = 1.5 µm, and (d) a = 2 µm.

This study examines a three-layer symmetric slab waveguide structure, comprising a thin GaAs film deposited on a glass substrate. A dielectric cladding, such as air, is positioned above the film. The simulation results elucidate the impact of GaAs thickness on TE/TM modes and examine the waveguide parameters, including propagation wavenumbers, mode numbers, cutoff frequencies, attenuation in substrate and cladding, effective skin depth in substrate and cladding, and angles of total internal reflection for TE mode and TM mode. Additionally, it examines the previously mentioned parameters by applying an electromagnetic signal with a wavelength of 0.6328 µm to GaAs films of thicknesses 0.5, 1, 1.5, and 2 µm, utilizing a reference active index of 3.8574 for GaAs, a refractive index of 1.52 for the glass substrate, and air as the cover medium. The MATLAB code generates the TE and TM modes, with the numerical solutions presented in the tables. Figure 2 illustrates the number of TE modes corresponding to varying thicknesses of GaAs, utilizing a wavelength of 0.6328 µm. The parameters of the waveguide are detailed in Table 1. The results indicate that the quantity of TE modes increases as the thickness of GaAs is augmented. As the propagation wavenumber decreases, the cut-off wavenumbers increase, resulting in a decrease in attenuation for a specific applied wavelength. Compared to the findings by Alwan et al. [29] where dielectric slab configurations were analyzed, our symmetric design presents lower cutoff frequencies for the different thickness and frequency. This contrast underscores the impact of slab symmetry on modal behavior.

This implies that the outside electric and magnetic fields are less confined. The confinement of the wave in the guide layer is improved and the evanescent fields in the surrounding reduce when the propagation wavenumber increases the film thickness. The thickness and optical parameters of the guiding layer define the modal propagation wavenumber of a given mode essentially. The parameters of waveguide are outlined in Tables 1 and 2. The simulation results show that the fundamental mode (m = 0) has the largest incidence angle θ _m and lowest penetration depth in substrate and cladding and the highest mode has the incidence angle very close the critical angle and greatest penetration depth. These findings align with Soun et al. [8] who reported similar trends in mode confinement for silicon waveguides, but our results demonstrate a 15% higher attenuation reduction in GaAs due to its superior refractive index contrast. Li et al. [30] explored TM modes in spoof plasmonic structures, examining confinement mechanisms and TM attenuation and cut-off behavior in GaAs waveguides under varying thicknesses. Compared to the findings by Alwan et al. [6] where asymmetric slab configurations were analyzed, our symmetric design presents lower cutoff frequencies for the same thickness levels. This contrast underscores the impact of slab symmetry on modal behavior. The influence of material thickness on mechanical and thermal properties, as analyzed in the context of cold rolling of aluminum alloys [31], underscores the importance of precise thickness control in GaAs waveguides to achieve optimal mode confinement and propagation characteristics. Both TE and TM modes are guided modes that can be realized in a symmetric slab waveguide. The TE mode of propagation is characterized by a non-propagating electric field. There is no component of a magnetic field along the propagation axis in the TM mode. Because of this, TM mode and TE mode start acting similarly.

Figure 3 explains that the number of TM modes with different GaAs thickness. Note that the TM modes within a symmetric slab waveguide behave similar to TE modes in Figure 2, also all parameters of waveguide outlined in Table 2 behave similar to parameters of waveguide in Table 1. The index of refraction is the change in angle in the light that pass through the GaAs material. Since n _s > n _c in general, θ _s > θ _c. On the basis of these two critical angles, the possible ranges of the incident angle θ _m exist θ _s < θ _m < 90°, the light is confined in the guiding layer and propagation along the Zig-Zag path Table 3. The parameters of the waveguide are typically established based on the cutoff of the guided modes. When the incident angle θ _m reaches the critical angle θ _s, the light ceases to be confined within the guiding layer.

Figure 3

Mode profile of TM modes of asymmetric slab waveguide when wavelength λ = 0.6328 µm and (a) a = 0.5 µm, (b) a = 1 µm, (c) a = 1.5 µm and (d) a = 2 µm.

Table 3 shows that the mode spacing reduces for TE/TM modes while the number of modes increases with increasing GaAs thickness. The optical carrier source supplied carriers of TE modes and carriers of TM modes with frequency spacing, after dividing the carriers utilization optical demultiplexer.

Computation of the vector error norm of the characteristic for TE/TM modes allows us to verify the convergence simultaneously for all modes. This error aligns with the assumed tolerance, specifically regarding the vector error norm for TE/TM modes across varying thicknesses of GaAs material.

An advantage of the GaAs possesses a wide band gap, allowing it to operate at higher frequencies and temperature, they are suitable for high-performance communication devices that deal with high-frequency signals. Therefore, GaAs semiconductor is widely used in communication devices such as cellular phones, base stations, and satellite communications. GaAs is frequently utilized in the production of solar cells due to its higher absorption coefficient, which allows it to absorb more energy from incident solar radiation. GaAs is more costly than silicon since GaAs are much rarer and harder to get. But its efficiency justifies the higher cost.

5 Conclusion

This study examines the effects of increasing the thickness of GaAs on TE/TM modes utilizing a symmetric three-layer slab waveguide structure. The number of modes increases incrementally with an increase in thickness and the generation of a subtending angle of incidence. The film’s thickness influences the quantity of modes or angles through which waves propagate. Each mode possesses distinct parameters. At a constant thickness of GaAs and an applied wavelength of 0.6328 µm, the propagation wavenumber and attenuation value within the slab decrease as the number of TE/TM modes increases. Conversely, the cut-off wavenumber and cut-off frequency increase, while the angles of internal reflection decrease. This results in an increased penetration depth. Increasing the thickness of GaAs results in a decrease in the cut-off wavenumber and cut-off frequency for a specific mode, while simultaneously increasing the propagation wavenumber, attenuation, and angles of internal reflection. Each mode in a quantified GaAs thickness possesses characteristics that can be utilized for specific applications. GaAs exhibits significant nonlinearity and minimal net absorption across a wide optical band. This occurs because total internal reflection is observed for θ _s < θ _m < 90°, with ranges of 88.18°–23.68° for all TE Mode and 88.14°–25.96° for all TM Mode. Consequently, it is a superior option for high performance, rapid response, and wideband optoelectronic and electronic devices. The prospects for GaAs in device technology appear favorable. It is anticipated that it will remain in use for high-speed computing devices and high-power applications. Advancements in GaAs technology will facilitate the creation of new devices that exhibit enhanced performance and efficiency.