1. Introduction

Advances in photonics technology have imposed increasing demands for photonic devices that are small, compact, have low loss, and are highly efficient. Plasmonics presents a new way to meet some of these demands by controlling light in a small fraction of wavelengths by exploiting plasmons, which are couplings of light to the oscillations of conduction-band electrons [1–3]. Due to the free electrons in plasmonic structures (mainly metals), they suffer from high energy dissipations, normally at optical frequencies [4,5]. Also, plasmonic particles (such as metal dimers) only support electric dipoles which result in a single-polarization coupling [6]. To overcome the energy dissipations of plasmonic structures, photonic crystals are ideal replacements. These crystals are artificial dielectric structures in periodic arrangements; because of the periodic nature, light is blocked in photonic bandgaps, which act as mirrors in all directions as a result of the Bragg effect [7–9].

Defects in photonic crystals can localize one or more guiding modes in photonic bandgaps, and enable steering of a light wave in the defects. Defects can be created by removing or modifying elements of the photonic crystals. Another way is to put two different lattices together to create an edge defect mode at the boundary between them; this structure is called a topological photonic crystal. This kind of defect localizes a Dirac-shaped mode inside the dispersion band diagram, and results in propagation of light through the defect [10–13]. Although photonic crystals are low-loss structures, they are large, and can only manipulate light waves that have wavelengths on the order of the lattice constants.

The transition from photonic crystals to all-dielectric metamaterials brings a new era in designing high-efficient, low-loss, compact, and shrank photonic devices. Because of the high-permittivity nature of dielectric elements in metamaterials, electric and magnetic dipoles are excited which in turn support orthogonal polarizations [6]. The other merits of high-refractive-index dielectric elements are their low-loss nature, resulting in high-efficient devices and controlling light in subwavelengths with very low energy dissipations [14–16].

The transition to metamaterials from photonic crystals can use one of two scenarios. In the $r$/$a$-scenario (where $r$ is the radius of particles and $a$ is the lattice constant), the lattice constant is decreased. In the $\varepsilon$-scenario, the permittivity of the elements is increased [17]. The goal of both scenarios is to increase light-matter interactions to increase the ability to control subwavelength light waves. The transition exploits the phenomenon of Mie resonance rather than the Bragg effect. Mie resonance is a phenomenon related to the scattering of light by small particles [18–20]. According to the Mie resonance, the total scattering cross-section of a spherical particle is [21]

In dielectric particles that have high permittivity, the displacement currents create magnetic responses that lead to Mie resonances [14,18]. In high-$n$ particles that are arrayed in the form of periodic structures such as metasurfaces, strong Mie resonances appear as a result of constructive interference of scatterings. In high-$n$ all-dielectric 2D metamaterials, these Mie resonances appear in the form of bandgaps, which represent interesting properties such as immunity to position disordering for the TE₀₁ Mie bandgap that is due to the transition from photonic crystals ($\varepsilon$ = 4) to metamaterials ($\varepsilon$ = 25) [29].

Introducing optical integrated circuits (OICs) by Miller in 1969 has motivated researchers to produce on-chip optical devices. In recent decades, the large capacity of data communications has increased, consequently, the demands for highly efficient and shrank on-chip optical devices to steer and split the light waves have risen for utilizing in high-density OICs. The optical devices are waveguides and power splitters which are the most applicable optical components for OICs. Numerous waveguides and power splitters have been proposed based on the slow light of photonic crystals [30–32], topological photonic crystals [13,33–36], and plasmonic structures [37–40]. Waveguides and power splitters based on the photonic crystals are based on the Bragg grating effect that results in a large footprint. Topological photonic crystal waveguides and power splitters are designed based on the localized Dirac-shaped mode, suffer from the big size. The plasmonic-based waveguides and power splitters suffer from the dissipation powers owning to the lossy plasmons. For overcoming the big footprint and energy dissipations, this paper presents ultra-narrow waveguides and power splitters using 2D germanium (Ge) rods in cubic and hexagonal lattices in air. Owning to the high and constant refractive index of Ge over the mid-infrared region ($2\mu m<\lambda <15\mu m$) [41], it supports Mie resonances, tolerates position disorderings in a form of arrays, and is compatible with CMOS fabrication technology. The proposed waveguides are in straight and $\Omega$ shapes with only two rows of Ge rods in air. The power splitters are in T and Y shapes with two rows of Ge rods in air. These structures are designed considering the TE₀₁ Mie bandgap, which is unaffected by position disordering. To simulate the structures, FDTD module of Lumerical software was used. This paper is organized into three sections. The second section describes theory and design; the third presents conclusions.

2. Theory and design

A design used cylindrical Ge rod with $n = 4.13$ in air, with radius $r$ and the length $L\gg r$ (Fig. 1(a)). By illuminating light to the rod, Fabry-Perot (FP) and Mie resonances in the rod appear [28,42]. The resonances are denoted by $b_{l}^{m}$ and $a_{l}^{m}$ (where $m$ and $l$ are integer mode numbers for FP and Mie resonances, respectively). Incident TE polarization excites the $b_{l}^{m}$ resonance and incident TM polarization excites the $a_{l}^{m}$ resonance.

 

Fig. 1. (a), a single Ge rod with a refractive index of 4.13, the radius $r$ and the length of $L$ ($L$ $\gg$ $r$) in air, which is illuminated with a TE polarized plane wave and (b) $x$ and $y$ directions, and total scattering cross sections of the rod versus normalized frequency. (inset: $x$$y$ view of electric field (E) and magnetic field (H) distributions have been shown for each scattering peak in (b).

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In the following, the focus is on the excitation of $b_{l}^{m}$ by illuminating a Ge rod with a TE-polarized light wave in air (Fig. 1(a)). Due to the homogeneity of the rod in the $z$ direction, 2D simulations used the FDTD module of Lumerical software, and the simulation results for the cross-section scatterings show the Mie resonances $b_{1}^{m}$, $b_{2}^{m}$, and $b_{3}^{m}$ (Fig. 1(b)). The excited $b_{1}^{m}$, $b_{2}^{m}$, and $b_{3}^{m}$ resonances show one, two, and four magnetic dipoles, respectively, which are created by circulating polarization currents inside the rod as represented in the $xy$ view of electromagnetic field distributions (Fig. 1(b), inset). The scattering cross-section in the $y$ direction (parallel to the incident electrical field direction) is stronger in of $b_{1}^{m}$ than in $b_{2}^{m}$ and $b_{3}^{m}$; this difference means that putting other rods in periodic arrangements in the direction parallel to the incident electrical field would create strong hotspots of electrical fields between them.

The bond theory explains the creation of electronic bandgaps in electronic crystals and sets out the coupling between the bonding and antibonding states of two adjacent atoms which causes the creation of the lower and upper bound of the electronic bandgaps instead of the periodicity of the lattice. Furthermore, this theory can be used for describing the electronic bandgaps in amorphous semiconductors. The bond theory can be applied somewhat to describe the creation of photonic bandgaps in disordered all-dielectric MMs if the coupling of a dielectric rod’s optical state to the optical state of its adjacent rod is considered [43]. The optical state in a dielectric rod has the form of a quasi-bound state in which the discrete states of each rod’s resonances (bond states) couple the other one’s discrete states, and the same thing happens to the continuous states (antibound states), as a result, the upper and lower bounds of the photonic bandgap (Mie bandgap) of two rods will appear as a unique bandgap. The coupling between states is a reason for the creation of electric or magnetic field coupling between two adjacent rods due to the TE or TM polarized incident light. Owing to the nature of Mie bandgap depending on an individual rod instead of the periodicity of the lattice, the bandgap tolerates a certain level of position disordering [44,45]. For the arrays of high-$n$ dielectric rods which are in touch with illuminating of TE polarized light, the coupling results in electrical field coupling between adjacent rods.

When a 2D periodic arrangement of Ge rods in air is excited by TE₀₁ bandgap mode, which is the result of $b_{1}^{m}$ scattering resonance, the electrical field localizations show stronger coupling between the rods in the row (I) than in rows (II), (III), (IV), and (V) (Fig. 2(a)). An incident plane wave which is located at the bottom of the structure propagates from the bottom to the top of the periodic structure and is recorded at the top of the structure (2(a, c)). The scattering resonance induces the formation of a circulating electrical field inside each rod, and this field generates an inward or outward magnetic field inside the rod, and also creates electrical field couplings between adjacent rods (Fig. 2(a, c)). The circulating electric field inside each rod of row (I) travels clockwise, so the direction of the coupled electric field is from $+x$ to $-x$, and scattering in the $y$ direction is small, so electric fields couple to row (II) and generate an electric field that circulates anticlockwise in each rod of row (II) and thereby results in an electric-field coupling from $-x$ to $+x$; and this process also occurs for the other rows. In each rod, the clockwise circulating electric field generates an outward magnetic field, and the anticlockwise circulating electric field generates an inward magnetic field. The electrical field coupling among the rods does not circulate, so no outward and inward magnetic fields appear among them (Fig. 2(a, c)). Because of the circulating nature of the electric field inside the rods, each of them develops a small circular region that has no localized electric field and that is not located exactly at the center of the rod (Fig. 2(a, b)). This region is a result of scattering of a small portion of light in the $y$ direction (Fig. 2(b), dashed blue line at $x = 0$). The normalized electric field versus $y$ axis shows that the electrical field coupling decreases exponentially from row (I) to row (V) at a line of $x = 0.5a$ (Fig. 2(b), solid black). The normalized electrical field at $x = 0$ shows the small localization of electrical field inside the rods, that decreases from the rows (I) to (V) (Fig. 2(b), dashed blue). The simulation results (Fig. 2(c, d)) indicate that the normalized magnetic field over $y$ axis for $x = 0$ shows that a magnetic field is strongly localized in the rods, and diminishes exponentially from row (I) to (V), but the normalized magnetic field at $x = 0.5a$ (Fig. 2(b), solid black) shows approximately no magnetic coupling between rods.

 

Fig. 2. Excitation of TE₀₁ mode by a plane wave in a 2D periodic array of Ge rods in air. (a) and (c) show the electric and magnetic-field distributions, respectively. (b) and (d) represent normalized electric and magnetic fields versus $y$ for (a) and (b), respectively, for two lines of $x=0$ and $x=0.5a$. Incident plane wave with TE polarization which is located at the bottom of the structure, propagates from $-y$ to $+y$ direction.

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TE₁₁ and TE₂₁ bandgap modes and a Bragg bandgap mode also appear in the periodic structure of 2D Ge rods in air. The TE₀₁, TE₁₁, and TE₂₁ bandgap modes are effects of Mie scattering resonances $b_{1}^{m}$, $b_{2}^{m}$, and $b_{3}^{m}$, respectively. The Bragg bandgap mode is a result of the periodic lattice (Fig. 3). The TE₀₁ bandgap is stronger than the others and is immune to position disordering caused by the strong electric field coupling between adjacent rods. In the periodic arrangement, the radius of the Ge rods is $r = 0.3a$ (Fig. 2).

 

Fig. 3. (a) Transmission spectrum of cubic structure with $\eta$ = 0 (black), 10 (red), and 20% (blue), respectively. (b) Transmission spectrum of hexagonal structure for $\eta$ = 0 (black), 10 (red), and 20% (blue), respectively. Insets of (a) and (b): unit cells of Ge rods in cubic and hexagonal structures.

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Reducing the size of the OICs’ components results in the inevitable cost fabrication imperfections [29,46–52]. The fabrication imperfection in the metamaterials normally appears in the form of position disordering. Using the definitions in Ref. [29], the position of each rod ($x^{i},y^{i}$) is $x^{i}=x_{0}^{i}+\sigma U_{x}$ and $y^{i}=y_{0}^{i}+\sigma U_{y}$, where ($x_{0}^{i},y_{0}^{i}$) and $\sigma$ are the original position of the rod and the strength of the disordering, respectively, $U_{x}$, $U_{y}$ are random variables between -1 and 1. Also using these definitions, the disorder parameter of $\eta$ is defined as the ratio of $\sigma$ to lattice constant $a$. The TE₀₁ bandgap mode shows the disorder immunity of $\eta =20\%$ for the 2D Ge metamaterials in cubic and hexagonal lattices (Fig. 3). Exploiting this bandgap mode provides freedom to design subwavelength structures in the form of waveguides and power splitters.

2.1 Waveguides

TE₀₁ mode is not sensitive to position disordering, and the strong electric field couplings only happen for the first row of rods; furthermore, the other rows have very weak interactions between light and matter (Fig. 3). The normalized bandwidth of the TE₀₁ mode is wider than those of the TE₁₁ and TE₂₁ modes, so TE₀₁ mode is suitable to localize a broad defect mode.

To exploit the strong couplings between rods in just the first row, straight and $\Omega$-shaped waveguides with positional disordering $\eta =10\%$ were evaluated (Fig. 4(a–h)). The defect-free structures (Fig. 4(a, e)) are called A type. Straight (Fig. 4(b)) and $\Omega$-shaped (Fig. 4(f)) with ten rows of Ge rods are called B type. Straight (Fig. 4(c)) and $\Omega$-shaped (Fig. 4(g)) waveguides with four rows of Ge rods are called C type. Straight (Fig. 4(d)) and $\Omega$-shaped (Fig. 4(h)) waveguides with two rows of Ge rods are called D type. In the rest of the paper, incident plane and Gaussian waves are depicted in orange rectangle and bell shapes, respectively. The plane waves are illuminated to A type and Gaussian waves are applied to B, C, and D types which propagate from $-y$ to $y$ direction (Fig. 4).

 

Fig. 4. Structure of the 2D Ge-based waveguides with $\eta$ = 10%. (a) and (e): structures with no defects; (b)–(d) straight waveguides with ten, four, and two rows of Ge rods in air. (f)–(h): structures of $\Omega$ shape waveguides with ten, four, and two rows of Ge rods in air. Orange rectangle and bell shapes show plane and Gaussian electromagnetic waves, respectively, which propagate from $-y$ to $y$ direction. Blue: air, red: Ge rods.

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Transmission spectra (Fig. 5) were obtained for structures A, B, C, and D of straight and $\Omega$-shaped waveguides and for position disordering $\eta$ = 0, 10, and 20%. The structures that had no defects (Fig. 5, black dashed curves) show the immunity of the TE₀₁ bandgap to position disordering over the range $0.2893<\frac {a}{\lambda }<0.33$ (where $\lambda$ is wavelength). When a line defect was created in either the straight or the $\Omega$ shape, guided mode developed in the immune bandgap. For $\eta = 0$, structures with no disordering (Fig. 5(a, d)), structure B shows a localized mode (solid red lines) in the bandgap with slow oscillations that are due to the rows of rods surrounding the defects in the straight and $\Omega$ shapes; i.e., due to the constructive and destructive interferences between the refractions and reflections of light among the many rods in structure B, the transmission spectra have some slow oscillations. When the number of rows was reduced to four, the number of refractions and reflections decreased, so the number of oscillations decreased (Fig. 5(a, d), dotted green lines). When the number of rows was reduced to two, the number of oscillations also decreased (Fig. 5(a, d), dot-dashed blue lines); the transmission amplitudes were $\sim$ 90% around the normalized frequency of $\frac {a}{\lambda }=0.327$ (solid black circles) for both straight and $\Omega$-shaped waveguides with ten, four, and two rows of Ge rods (B, C, and D structures) in air.

 

Fig. 5. Transmission spectra of the structures A, B, C, and D for different position disordering. (a)–(c) transmission spectra for straight waveguides with disorderings $\eta$ = 0, 10, and 20%, respectively. (d)–(f) transmission spectrums for $\Omega$-shaped waveguides with disorderings of $\eta$ = 0, 10, and 20%, respectively.

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For the straight waveguides with position disordering $\eta = 10\%$ , the transmission amplitudes were $\sim$80, $\sim$90, and $\sim$75% at the normalized frequency $\frac {a}{\lambda }=0.327$ for structures B, and C, and D, respectively (Fig. 5(b)). For the $\Omega$-shaped waveguides with the position disordering of $\eta = 10\%$, the transmission amplitudes for structures B, C, and D were $\sim$90, $\sim$90, and $\sim$50%, respectively, at the $\frac {a}{\lambda }=0.327$. The transmission spectra for structure D (just two rows of Ge rods) (Fig. 5(b, e), in dot-dashed blue) were 75% for the straight waveguide and 50% for the $\Omega$-shaped; this reduction occurs because disordering of the structure increases the distance between some rods, so the electric field coupling between them weakens and the light wave travels freely between them. Also, because of the sharp $90^{\circ }$ bend in $\Omega$-shaped waveguides, the transmission is less in the $\Omega$-shaped D-type structure than in the straight D-type structure. With increasing of position disordering to $\eta = 20\%$, the transmission amplitudes are weaker than in the structures with position disordering of $\eta = 0\%$ (Fig. 5(c)), and 10% (Fig. 5(f)); the difference occurs because the electric field coupling between adjacent rods decreases, and this change results in the change of phases, and in turn changes the normalized guiding frequencies.

Electromagnetic power distributions at $\frac {a}{\lambda }=0.327$ for the both straight and $\Omega$ shapes waveguides consist of two rows of rods (D type), show high coupling between adjacent rods in the structures when $\eta$ = 0, that results in high confinement of electromagnetic powers in the waveguides (Fig. 6(a, d)). As disordering is increased to $\eta$ = 10 and 20%, the coupling distance between adjacent rods decreases. Consequently, light waves flee between rods with increased adjacent distances as obvious from Fig. 6. In this figure, Gaussian electromagnetic waves are located at the bottom of the structures, propagate from bottom and are collected at the top of the structures.

 

Fig. 6. Electromagnetic power distributions of the structure D for different position disordering. (a)–(c) power distributions for straight waveguides with disordering of $\eta$ = 0, 10, and 20%, respectively. (d)–(f) power distributions for $\Omega$ waveguides with disordering of $\eta$ = 0, 10, and 20%, respectively. Incident waves are Gaussian in orange color which are located at the bottom of the structures with the normalized frequency of $\frac {a}{\lambda }=0.327$ that propagate from $-y$ to $+y$.

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In 2010, 2013, 2017, and 2021 Lin et al. [53], Askari et al. [54], Terada et al. [55], and Shiratori et al. [56] proposed straight waveguides with efficiencies of 95, 98, 94, and 97 % using 12, 44, 16, and 20 rows of air holes inside silicon slabs. Shalaev et al. showed an $\Omega$-shaped waveguide using topological photonic crystals with many unit cells consist of two triangular air holes inside a silicon slab with the transmission of about 100 % [57]. In the present study, the straight and $\Omega$-shaped waveguides show the transmission efficiency of 90% with just two rows of Ge-rods in the air. Although the transmission efficiencies of the waveguides are a little less than the previous studies, however, they are the narrowest investigation between the mentioned works.

2.2 Power splitters

This section presents Y-shaped power splitters in hexagonal (Fig. 7(a–d)), and T-shaped power splitters in cubic structures (Fig. 7(e–h)). The hexagonal (Fig. 7(a)) and cubic (Fig. 7(e)) lattices with no defects and with no position disordering ($\eta = 0\%$) are called A type. Y-shaped (Fig. 7(b)) and T-shaped (Fig. 7(f)) line defects in hexagonal and cubic structures are called B type. Y-shaped (Fig. 7(c)) and T (Fig. 7(g)) shapes with four rows of rods are called C type. Y-shaped (Fig. 7(d)) and T (Fig. 7(h)) shapes with two rows of rods are called D type. The plane incident sources are illuminated to the A types and Gaussian waves are illuminated to the B, C, and D types (Fig. 7). The incident waves are located at the left and bottom parts of the structures which propagates from $-x$ to $x$ and $-y$ to $y$ directions and are monitored at the right and up parts of the structures as shown in Fig. 7(a–d) and 7(e–h), respectively.

 

Fig. 7. Structure of the 2D power splitters based on Ge rods in the air with $\eta$ = 0%. (a) and (e) hexagonal and cubic structures with no defects, respectively. (b) and (f) Y and T shapes splitters in hexagonal and cubic lattices, respectively. (c) and (d) Y splitters with four and two rows of Ge rods in a hexagonal lattice; (g) and (h) T splitters with four and two rows of Ge rods in a cubic lattice. Blue: air; red: Ge. The orange bell and rectangular shapes are incident Gaussian and plane waves which propagate from $-x$ to $x$ for (a–d) and propagate from $-y$ to $y$ for the structures (e–h).

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In the transmission spectra, at one output branch of the Y-shaped (Fig. 8(a)) and T-shaped (Fig. 8(b)) power splitters, the A-type structures with no defects show a strong bandgap (solid black) with different bandwidths. Creation of either Y or T defects yielded a guided mode localized in a bandgap that has a few oscillations that are due to the reflections and refractions of light between the rods of the whole lattice; these oscillations decrease as with a decrease in the number of rows of the waveguides (Fig. 8). The Y-shaped waveguides with the B and C types showed high transmittance 47% at $\frac {a}{\lambda }=0.319$, and the D-type showed transmittance of 46% at $\frac {a}{\lambda }=0.327$. At $\frac {a}{\lambda }=0.327$, the refractions and reflections of light between rods in structures B and C interfere destructively with the guided mode, so the transmittance amplitudes decreased to 30%, but at $\frac {a}{\lambda }=0.327$ the structure D with just two rows of rods does not show any reflections and refractions interferences, so the transmittance amplitude is higher than in structures B and C (Fig. 8(a)). The transmittance amplitudes of the T-shaped power splitters (Fig. 8(b)) of B, C and, D structures were all 47% at $\frac {a}{\lambda }=0.327$ this is the result of the structure being cubic.

 

Fig. 8. Transmission spectrum for the Y and T splitters for A, B, C, and D structures. (a) and (b) Y and T splitters, respectively.

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In the D structure, the electromagnetic power distributions for Y-shaped and T-shaped power splitters (Fig. 9) show the high confinement of electromagnetic waves in the waveguides. The structures have no positional disordering, so couplings are strong between adjacent rods in types of structure; therefore, small and highly-efficient all-dielectric power splitters can be obtained with just two rows of Ge rods. The Gaussian waves are located at the left and bottom which propagate from $-x$ to $x$ and $-y$ to $y$ and monitored at the right and top of the structures as shown in Fig. 9(a and b), respectively.

 

Fig. 9. Electromagnetic power distributions for the Y and T splitters for the D structure. (a) Y-shaped and (b) T-shaped splitters for the D structure at $\frac {a}{\lambda }=0.327$. Incident waves are Gaussian in an orange color which are located at the left and bottom and propagate from $-x$ to $x$ and $-y$ to $y$ for (a) and (b), respectively

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There are some studies about the inverse design of optical power splitters based on silicon photonics consist of air holes inside silicon slabs. In 2019, Tahersima et al. designed power splitters with different splitting ratios consist of $20 \times 20$ air holes and an efficiency of 90% [58]. In 2020, Ma et al. proposed a $1\times 2$ power splitter with 177 air holes and a transmission efficiency of 70% [59]. In 2020, Ma et al. designed a $1 \times 2$ power splitter with $20 \times 15$ air holes and a transmission efficiency of 80% [60]. The present paper proposed two types of Y- and T-shaped power splitters based on the two rows of Ge rods (49 rods) in the air with an efficiency of more than 92% which indicates highly efficient power splitters.

Ref. [29] describes the robustness of TE₀₁ Mie bandgap under position disorderings which employs the transition of photonic crystals ($n = 2$) to all-dielectric metamaterials ($n = 5$). Also, they studied linear and point defects for the application of all-dielectric waveguides and cavities in disordered arrays of all-dielectric rods. I have presented some novelties in this paper rather than Ref. [29] such as explanations in details of physics behind the creation of Mie bandgaps using Mie scattering for an isolated rod, designing ultra-narrow straight and $\Omega$-shaped waveguides with just only two rows of rods with the efficiency of more than 90% under $\eta =0\%$ and the efficiency of more than 75 and 50% for $\eta =10\%$. The other significant comparison is designing ultra-narrow Y and T-shaped power splitters consist of two Ge rods in air under $\eta = 0\%$ which reveals the transmission efficiency of more than 92%. In addition, this paper utilized a real material of Ge with a refractive index of 4.13 in the middle infrared region which is compatible with the current CMOS fabrication technology. In the end, the effect of temperature on the D type structures of waveguides and power splitters is studied.

2.3. Effect of temperature

An important factor influencing the refractive index of semiconductors is temperature. Growing the temperature increases the interaction among free electrons, holes, and recombined electron-hole pairs with phonons and linear expansion of the lattice constant which leads the electronic bandgap of semiconductor to decrease. Having declined the width of the bandgap, increases the refractive index, consequently. To put it in a vivid picture, increasing the temperature increases the refractive index of the semiconductors [61–64]. In this study, Fig. 10 clearly illustrates the variation of the transmission over temperature for the straight and $\Omega$-shaped waveguides, and also, Y- and T-shaped splitters are considered under position disordering of $\eta =0, 10, \textrm{and}\; 20\%$. For simulating the structures, the incident wavelength is set to $\lambda =2 \mu m$, $a = 0.654 \mu m$ and $r = 0.3a$, and it is worth to mention that the temperature dependence data of Ge is collected from Ref. [64].

 

Fig. 10. Transmission versus temperature of the D-type waveguides under different position disordering. (a) transmission versus temperature for straight and $\Omega$-shaped waveguides and Y- and T-shaped power splitters under $\eta = 0\%$. (b) and (c) transmission versus temperature for straight and $\Omega$-shaped waveguides, respectively under position disorderings $\eta = 10\%$ (blue color) and $\eta = 20\%$ (red color).

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As it has been seen from Fig. 10(a), transmissions versus temperature over the interval [-193, 26]$^\circ$ C for the D-type waveguides and splitters do not change under the circumstance of $\eta = 0\%$. Fig. 10(b and c) reveal a small variation of transmission over the temperature of the D structure for straight and $\Omega$-shaped waveguides, respectively, under $\eta = 10\%$ (blue color) and $\eta = 20\%$ (red color).

3. Conclusion

This study has presented and analyzed all-dielectric ultra-narrow high-efficient straight and $\Omega$-shaped waveguides, and Y-shaped and T-shaped power splitters composed of Ge rods in air. These optical components were designed to exploit the TE₀₁ Mie-resonant bandgap, which is immune to a positional disordering of $\eta = 20\%$. The straight and $\Omega$-shaped waveguides, and T-shaped power splitters use Ge rods in a cubic lattice. The Y-shaped power splitters uses Ge rods in a hexagonal lattice. At the normalized frequency of $\frac {a}{\lambda }=0.327$, the transmittance results with no positional disordering for two rows of Ge rods were 90% for waveguides and $>40\%$ for power splitters (for each branch). The transmittances with four rows of straight and $\Omega$-shaped waveguides with disordering of $\eta = 10\%$ were $\sim$ 90% at $\frac {a}{\lambda }=0.327$. The results show that all-dielectric ultra-narrow low-loss 2D optical power splitters and waveguides are good candidates for use in future OICs.