A Triangle Hybrid Plasmonic Waveguide with Long Propagation Length for Ultradeep Subwavelength Confinement

Abstract

Facing the problems of ohmic loss and short propagation length, the application of plasmonic waveguides is limited. Here, a triangle hybrid plasmonic waveguide is introduced, where a cylinder silicon waveguide is separated from the triangle prism silver waveguide by a nanoscale silica gap. The process of constant optimization of waveguide structure is completed and simulation results indicate that the propagation length could reach a length of 510 μm, and the normalized mode area could reach 0.03 along with a high figure of merit 3150. This implies that longer propagation length could be simultaneously achieved along with relatively ultra-deep subwavelength mode confinement due to the hybridization between metallic plasmon polarization mode and silicon waveguide mode, compared with previous study. By an analysis of fabrication errors, it is confirmed that this waveguide is fairly stable over a wide error range. Additionally, the excellent performance of this is further proved by the comparison with other hybrid plasmonic waveguides. Our work is significant to manipulate light waves at sub-wavelength dimensions and enlarge the application fields, such as light detection and photoelectric sensors, which also benefit the improvement of the integration of optical devices.

1. Introduction

Surface plasmon polaritons (SPPs) are a special mode of electromagnetic surface waves that exist at the metal–medium interface. Due to its unique surface wave characteristics, it can confine light waves to regions with a spatial size much smaller than its free-space wavelength [1,2,3,4]. Hence the use of SPPs to control and manipulate light waves at sub-wavelength dimensions has become one of the international research hotspots [5,6]. In addition, SPPs have broad application prospects in many fields such as optical storage, optical sensing, and negative refractive index materials [7,8,9,10,11,12,13]. In the field of integrated optics, the unique advantage of SPPs is that they can break the diffraction limit, which helps to reduce the size of optoelectronic devices [14,15,16].

With the growing interconnection density, it is difficult for traditional electronic interconnections to fully meet the requirements of rapidly developing communication technologies due to the problems of energy loss and crosstalk. Integrated optics has become a research hotspot because of its unique properties, such as low loss. However, the accuracy of traditional photolithography is limited by the properties of materials and the law of light diffraction. Therefore, in order to achieve the miniaturization of photonic devices, nano-photonics has received a lot of attention, which is dedicated to controlling the interaction of light and matter in the deep sub-wavelength range. SPPs are a special mode of electromagnetic surface waves, excited by the coupling of the incident photon and electron plasma oscillation in the conductor. In the direction perpendicular to the interface between medium and metal, the attenuation length of the light field in the medium can be less than half of the incident light wavelength. Therefore, since the field confinement scale of SPP can break the diffraction limit, SPPs are considered to be a key to controlling and manipulating optical signals at the sub-wavelength scale.

Waveguides are the basic element that connects different devices in the integrated optical path. The size of conventional waveguides is restricted by the light diffraction limit and the relative refractive index of the single-mode waveguide [17]. Compared with conventional waveguides, for SPP waveguides, energy can be transmitted in the sub-wavelength structure and it is possible to pass through the bends and T-shaped structures [15,18,19]. Hence, they are conducive to the convergence and guidance of light fields and benefit the development of highly integrated photonic circuits [20,21]. In recent years, a variety of surface plasmon waveguides have been studied, such as metal slot waveguide structures and V-groove structures. They usually have a strong field confinement ability, but at the same time, because of the ohmic loss generated by the metal, the propagation distance becomes shorter, which is a drawback that cannot be neglected [22,23,24].

The hybrid plasmonic waveguide (HPW) is constructed on the basis of the plasmonic waveguide, including a high refractive index region, a metal region and a low refractive index region, in which the refractive index difference at the interface between the metal and the medium mainly determines the strength of the mode confinement. It is the pursuit of hybrid plasmonic waveguides to significantly improve the transmission loss while maintain a highly restricted mode [25,26,27].

To circumvent the above problems and achieve the goal of enhancing the hybrid waveguide performance, an excellent cylinder-triangle hybrid plasmonic waveguide is proposed, including a high refractive index dielectric nanostructure placed very close to the metal-dielectric surface. In our work, the structure of the HPW is proposed first and next the theoretical analysis method is briefly explained. Then, by analyzing assessment criteria, including propagation length, the normalized mode area and figure of merit, unique properties such as extraordinary long propagation length of proposed HPW are demonstrated clearly. Finally, comparisons with other waveguides are given. Because of the hybridization between the metal SPPs mode and silicon waveguide mode, simulation results indicate that longer propagation length could be simultaneously achieved along with relatively ultra-deep subwavelength mode confinement in this HPW, compared with previous HPWs [28,29]. Additionally, on account of a significant performance enhancement of HPW, our work is beneficial to enlarge the applicable area of HPWs.

2. Simulation Model and Method

2.1. Simulation Model

The schematic geometry of the proposed hybrid triangle plasmonic waveguide is shown in Figure 1, where a cylinder high-index semiconductor silicon waveguide (Si) is separated from the triangle prism metal silver waveguide (Ag) by a nanoscale low-index dielectric silicon dioxide gap of height $h$. The vertex angle of the triangle silver metal is noted as $\alpha$ and the curvature radius of $\alpha$ is noted as $rw$. The radius of the cylinder Si waveguide is defined as $r$. In the proposed waveguide, the continuity of the displacement field at the material interfaces and the effective optical capacitance effect of the gap region could lead to the strong electric field component in the gap [25,30]. SPP mode and silicon dielectric mode are strongly hybridized, and the light could be confined within the nanoscale silicon dioxide gap [31,32].

We simulate the 2D model of the proposed waveguide by using COMSOL Multiphysics software 5.4 in order to acquire the electric field distribution of the cross-section and to find the relationship between model propagation properties of the proposed hybrid waveguide and waveguide size. In the simulation, using the physics-controlled meshing in COMSOL to control the mesh size, the perfect electric conductor is applied to all four sides of the waveguide region to simulate the boundary condition in the mode analysis and to search for effective transmissive modes of the hybrid waveguide. TM mode is analyzed and the convergence analysis is applied to ensure that the meshing and boundary conditions employed in the models are reasonable. The size of the simulation domain is 600 nm × 600 nm. In our simulation, waveguide characteristics are analyzed at $\lambda$ = 1550 nm. In this case, silicon has a relative permittivity of 12.25(${\epsilon }_1$), SiO₂ of 2.25, and Ag of −129 + 3.3i(${\epsilon }_2$) [22,33].

The proposed waveguide could be fabricated based on the complementary metal-oxide-semiconductor (CMOS) technology step by step. First, a triangular V-groove is etched on the SiO₂ layer and is used to deposit silver by the focused ion beam (FIB) method. Then, a nanoscale silicon dioxide layer is deposited above the silver triangular prism as a thin dielectric gap between the triangular silver and cylinder silicon by plasma enhanced chemical vapor deposition (PECVD) method [34]. Next, the well-known vapor-liquid-solid (VLS) mechanism, using a metal catalyst, could be used to grow the silicon nanowire [35] and the silicon nanowire will be transformed and placed at the right above peak triangular silver in the next step. Last, the silicon dioxide cladding is deposited.

2.2. Method

The model properties of the hybrid waveguide could be evaluated in terms of normalized effective mode area $A_{eff}/A_0$, propagation length $L_m$, figure of merit $FOM$, and the imaginary part of the modal effective refractive index $Im\left(n_{eff}\right)$.

The effective model area $A_{eff}$ is expressed by Equation (1) [25], which represents the ability of light confinement of the hybrid waveguide and can be defined as the ratio of the total electromagnetic energy to the maximum energy density. The electromagnetic energy in the metallic region is defined as Equation (2) [25], related with the energy density $W\left(r\right)$.

$$A_{eff}=\iint W\left(r\right)dA/\max \left(W\left(r\right)\right)$$

(1)

$$W\left(r\right)=\frac{1}{2}Re\left\{\frac{d\left[\omega \epsilon \left(r\right)\right]}{d\omega }\right\}{\left|E\left(r\right)\right|}^2+\frac{1}{2}{\mu }_0{\left|H\left(r\right)\right|}^2$$

(2)

where $E\left(r\right)$, $H\left(r\right)$, $\epsilon \left(r\right)$, $\omega$, and ${\mu }_0$ are the electric field, magnetic field, dielectric permittivity, angular frequency, and vacuum magnetic permeability, respectively. $A_0$ is the diffraction-limited mode area and is defined as Equation (3) in free space [25]:

$$A_0={\lambda }^2/4$$

(3)

The propagation length is given by the following equation [22] and is relative to $\left(n_{eff}\right)$:

$$L_m=\lambda /\left[4\pi Im\left(n_{eff}\right)\right]$$

(4)

where $\lambda$ is the light wavelength and is equal to 1550 nm.

$FOM$ is defined as Equation (5) and is related to both propagation length and effective model area [22,36]. It is usually considered as a comprehensive measurement for waveguide characteristics.

$$FOM=\frac{L_m}{2\sqrt{\frac{A_{eff}}{\pi }}}$$

(5)

3. Results

Here, the classical control variable method is used to find the relationship between the structure parameters of the hybrid waveguide and the model properties. As shown in Figure 1b, parameters such as the radius of the cylinder silicon waveguide $r$, the vertex angle of the triangle silver metal wedge $\alpha$, and the curvature radius of the vertex angle $rw$, the height of the gap between cylinder silicon and triangle silver $h$, are all studied. The fabrication tolerance of this waveguide is also analyzed.

The radius of Si waveguide $r$ is analyzed first when the gap $h$ is 15 nm and the vertex angle of the triangle metal wedge $\alpha$ is 60°. Pre-simulation results indicate that the electric field will spread into the silicon waveguide and the cladding area if the radius of the cylinder silicon waveguide r is larger than 60 nm, so the variation range of $r$ is from 25 nm to 60 nm with a different curvature radius of the triangle metal wedge $rw$ (10 nm, 15 nm) in order to achieve the strong light confinement in the silicon dioxide gap. Furthermore, a narrow gap region could prevent the mostly electric field from spreading into the core of cylinder silicon waveguide and silver, confining the light within the nanoscale gap region, so the height of gap region h is fixed at the small size of 15 nm. In Figure 2a, line curve indicates the propagation length $L_m$ as shown in the left Y-axis, and dash curve indicates the imaginary part of the modal effective refractive index $Im\left(n_{eff}\right)$, as shown in the right Y-axis. From Figure 2a, we can see that $Im\left(n_{eff}\right)$ increases with the increase in $r$, which implies gradually strong confinement in the HPW. Additionally, the variation trend of $A_{eff}/A_0$ in Figure 2b is monotone, decreasing when $r$ is becoming larger and larger, indicating that the electric field distribution is more concentrated, which is simultaneously inconsistent with the hint of $Im\left(n_{eff}\right)$ tendency in Figure 2a. Meanwhile, $L_m$ continues to decline from 430 μm to 290 μm, approximately when $rw$ is equal to 15 μm in Figure 2a, the maximal of which is preponderant enough compared to those of the dielectric-loaded plasmonic waveguides and with much stronger confinement. $rw$ is focused because the influence of which on $L_m$, $Im\left(n_{eff}\right)$, $A_{eff}/A_0$, and $FOM$ should not be overlooked. With the same $r$, comparing the results of $L_m$ and $A_{eff}/A_0$ at $rw$ = 10 nm and $rw$ = 15 nm, respectively, we can find that for larger $rw$, $L_m$ and $A_{eff}/A_0$ are both increasing. It implies that increasing the curvature radius is beneficial to improve the propagation length, but not to mode confinement. Additionally, due to the change of contact area, the smaller the $rw$, the bigger the $FOM$ from Figure 2c.

Then, we consider the influence of the gap height $h$ between silver and silicon on the waveguide model properties at $r$ = 25 nm and $\alpha$ = 60°. In the proposed waveguide, the light could be confined dramatically within the nanoscale silicon dioxide gap if the parameter’s sizes are fixed suitably. A narrow silicon dioxide gap region could prevent the mostly electric field from spreading into the core of cylinder silicon waveguide and silver, so the small size of the gap region is simulated. Based on the simulation results of $r$, the maximum of the normalized effective mode area could reach 0.032, so in the simulation of h, the range of h is fixed from 5 nm to 10 nm in order to reduce the normalized effective mode area, enhancing the high-confinement performance of this waveguide. The dependence of model properties on gap height $h$ at different curvature radius $rw$ is shown in Figure 3. Because of a gradually weakened mode hybridization at larger $h$, $Im\left(n_{eff}\right)$ decreases, and $A_{eff}/A_0$ increases as shown in Figure 3a,b. According to Equation (3), $L_m$ grows up as well. From Figure 3c, we could see that the trend of $FOM$ has not changed significantly, indicating that $h$ has little effect on $FOM$.

The structure of the proposed hybrid waveguide also changes with the angle of the vertex angle $\alpha$ of triangular silver. As the angle increases, silver becomes larger in the structure when the whole height of silver is determined. Therefore, the influence of the angle $\alpha$ on the waveguide is analyzed next. Figure 4a–c are the 2D electric field distribution of the proposed waveguide under different conditions. Among them, for the same $\alpha$ in Figure 4a,b, it can be seen that there is almost no difference in the electric field and for different $\alpha$ in Figure 4a,c, there is only a slight difference. Figure 5 indicates the relationship between mode properties and vertex angle of triangle silver with different rotation angles. In Figure 5a, line curve indicates the propagation length $L_m$ as shown in the left Y-axis, and dash curve indicates the imaginary part of the modal effective refractive index $Im\left(n_{eff}\right)$ as shown in the right Y-axis. From Figure 5a,b, it is observed that both $L_m$ and $FOM$ increase, and $Im\left(n_{eff}\right)$ decreases with a larger angle, while $\alpha$ is in the range of 20–60°. This implies that for a range of angles, the mode characteristics, including transmission length and figure of merit, are improved, which results from the hybridization enhancement.

In fact, in the process of manufacture, there are much craft inaccuracies existing, so it is easy to understand the actual accuracy is different from the theory and the fabrication tolerance should be considered as well. Here, we mainly assess the impact of fabrication tolerance on the waveguide properties with two factors: the triangular silicon rotation angle $\theta$ and lateral deviation $d$, and study them separately. As shown in Figure 4c, $\theta$ is defined as the triangular silicon rotation angle that will lead to the misalignment between the cylindrical silicon and triangular silver. As shown in Figure 6, the lateral deviation of triangle silver with respect to the cylinder silicon is denoted as $d$.

In Figure 5a, solid lines with different colors correspond to the changes in $L_m$ for different rotation angles $\theta$ and we could see that there is only a minor discrepancy between them. It indicates that although the angle is rotated by 15 degrees, $L_m$ does not fluctuate much. Additionally, in Figure 5b, $FOM$ rises when the rotation angle is nonzero, which results from the fact that even if the triangle is rotated by a certain angle, there are little structure changes for the unique cylinder-triangle model. Based on the above analysis, it is concluded that the proposed model is robust against the possible fabrication errors.

Figure 7 shows the influence of lateral derivation d on mode properties. The results in Figure 7 indicate that when $d$ increases from 0 to 30 nm, $L_m$ increases from 510.83 μm to 523.56 μm, $A_{eff}/A_0$ increases from 0.03407 to 0.04004, and $FOM$ decreases from 3164.7 to 2992. Here, the fluctuation ratio is defined as the ratio of the difference between the initial value and the final value to the initial value. The calculation above specifies that the fluctuation ratio of $L_m$, $A_{eff}/A_0$, and $FOM$ caused by lateral deviation are 0.02, 0.17 and 0.05, respectively, which indicates the properties of this waveguide are fairly stable over a wide misalignment range.

To further verify the high performance of the proposed waveguide, we also do some research about the other two models: cylinder-rectangle and double cylinder, as shown in Figure 8a,b. Compared with the other waveguides, the contact area between silver and silicon dioxide of the proposed triangular waveguide is smaller, which is beneficial to reduce the propagation loss and enhance the hybridization of the SPP mode and silicon mode, thereby significantly improving the confinement performance in the gap region. As shown in Figure 9, compared with the other two waveguides, it is obvious that the electric field distribution of the cylinder-triangle waveguide is more concentrated. Meanwhile, Figure 10 indicates that the figure of merit, $FOM$, of the proposed cylinder-triangle waveguide has excellent performance and the mode area of it is the smallest.

Besides, in

Table 1. Comparisons between the presented hybrid plasmonic waveguide and other waveguides with different structure.

Type	Proposed Hybrid Waveguide	Hybrid Tube-Triangle Plasmonic Waveguide	Triangle Hybrid Plasmonic Waveguide	Hybrid Wedge Plasmonic Polariton Waveguide
schematic
$L_m$(μm)	250–525	25–85	30–190	10–50
$FOM$	2300–4500	300–800	153–4600	120–200

, in striking contrast to other high-performance waveguides which have been studied previously in [23,24], we could see that, compared with the hybrid tube-triangle plasmonic waveguide and the hybrid wedge plasmonic polariton waveguide, the proposed waveguide demonstrates a much longer propagation length $L_m$ (1 order of magnitude larger) and a higher figure of merit $FOM$ (5–7 and about 20 times higher, respectively). Furthermore, the propagation length $L_m$ of the proposed waveguide is 2–8 times longer than the triangle hybrid plasmonic waveguide when the figure of merit, $FOM$, is much larger than the triangle hybrid plasmonic waveguide. It can be obtained that $L_m$ in the proposed waveguide is much larger than other waveguides when it achieves a higher $FOM$, indicating that the long propagation length and the high figure of merit could be achieved simultaneously in the proposed waveguide.

4. Conclusions

Based on the hybridization between metallic plasmon polarizations mode and silicon waveguide mode, a hybrid plasmonic waveguide consisting of a cylinder silicon nanowire located above a sliver triangle structure is designed. Through a constant optimization of waveguide structure parameters including the height $h$ of silica gap, the vertex angle and curvature radius of the triangle metal $\alpha$ and $rw$, and the radius of Si waveguide $r$, simulation indicates that the propagation length could reach a length of 370 μm, the normalized mode area could reach to 0.025 along with a high figure of merit 3000. This achieved the pursuit of HPW to significantly enhance the transmission length while maintaining a highly restricted mode simultaneously. With the consideration of fabrication errors including rotation angle and lateral deviation, it is demonstrated that the fluctuation of properties is small, which confirms this waveguide is fairly stable over a wide error range. In addition, comparisons between the present waveguide and other high-performance HPW are researched, and the excellent performance of this is further proved by the abovementioned results.