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We studied a novel long range hybrid tube-wedge plasmonic (LRHTWP) waveguide consisting of a high index dielectric nanotube placed above a triangular metal wedge substrate. Using comprehensive numerical simulations on guiding properties of the designed waveguide, it is found that extreme light confinement and low propagation loss are obtained due to strong coupling between dielectric nanotube mode and wedge plasmon polariton. Comparing with previous studied hybrid plasmonic waveguides, the LRHTWP waveguide has longer propagation length and tighter mode confinement. In addition, the LRHTWP waveguide is quite tolerant to practical fabrication errors such as variation of the wedge tip angle and the horizontal misalignment between the nanotube and the metal wedge. The proposed LRHTWP waveguide could have many application potentials for various high performance nanophotonic components.
© 2016 Optical Society of America
1. Introduction
The integration and miniaturization of photonic components and circuits have been a research focus for many years [1]. But achieving both stronger field confinement and longer range signal transmission are still daunting challenges ahead [2]. Surface plasmon polaritons (SPPs) due to their unique optical properties [3] are seen as the most promising candidate to truly realize nanoscale field confinement and energy transmission in photonic devices [4]. Although SPPs waveguides can break the diffraction limits and lead to nanoscale field confinement [5], high propagation loss introduced by metal is still a major obstacle [6].
In 2008 a hybrid plasmonic waveguide which is consist of a high permittivity semiconductor nanowire embedding in a low permittivity dielectric near a metal surface was proposed [7,8 ]. This new kind of hybrid plasmonic waveguide has sub-wavelength confinement capability and relatively long propagation length. Due to its superior modal properties, lots of integrated photonic components based on hybrid schemes have been presented, including highly efficient nanolasers [9,10 ], resonators [11], compact passive devices [12–14 ], and many other hybridized plasmonic waveguide structures [15–20 ].
In this paper, we proposed and analyzed a novel long range hybrid tube-wedge plasmonic (LRHTWP) waveguide which is composed of a high index dielectric tube embedding on a metallic wedge. The LRHTWP waveguide combines the advantages of dielectric tube waveguide and wedge plasmonic polariton. It has long light propagation length as well as extreme small light confinement ability. In addition, the LRHTWP waveguide is fairly tolerant to geometric errors in practical fabrication process.
2. Geometry and mode properties of the LRHWTP waveguide
Figure 1 shows 3D geometry of the proposed LRHTWP waveguide which has a silicon nanotube separated from a silver metallic wedge by a low index dielectric gap. The whole structure is surrounded by SiO2 cladding. The outer and inner radius of the nanotube is R and r. The gap distance between the nanotube and the metal wedge is set as g. The metal wedge has a tip angle of ɵ and a curvature radius of rw. Numerical simulations are performed by finite element method using COMSOLTM. To mimic the open boundary, eigenmode solver is used with scattering boundary condition. A convergence analysis is performed to ensure that numerical boundaries do not interfere with the solution [7].The modal characteristics of the LRHWTP waveguide are investigated at the telecommunication wavelength λ = 1550 nm. The packing medium in the nanotube is air. The permittivities of air, Si, SiO2 and Ag are, , and , respectively [21].
Electric field distributions of the hybrid mode propagation in LRHTWP waveguide are shown in Fig. 2 , where both 2D field profile and 1D plot along dashed lines are presented. From the picture, it can be clearly seen that the strong hybridization mode coupling between the wedge plasmon polariton at the point end of the silver wedge and dielectric mode supported by the silicon tube is tightly confined at the gap area on both x and y axis. The full width at half maximum of the spot size in x direction is about 20 nm and in y direction is about 5 nm. It is worth to note that there is locally enhanced electric field in the hollow area due to index contrast between the inner and outer regions of the silicon tube. This hybrid plasmonic mode remarkably reduces the propagation loss which will be further discussed in this paper.
Fig. 2 (a) Electric field distributions of the hybrid plasmonic mode supported by the proposed LRHWTP waveguide when R = 120 nm, r/R = 0.5, g = 5 nm, ɵ = 90 deg and rw = 10 nm; (b) The field profile along the X dashed line; (c) The field profile along the Y dashed line.
Different nanotube and metallic wedge structures are simulated with 5 nm fixing gap distance and 10 nm metal tip curvature radius. The corresponding results demonstrated in Fig. 3 indicate that the electric fields are tightly confined in the gap area between the nanotube and the metallic wedge when the hollow area and the wedge tip angle are small [e.g. Figures 3.(a), 3(d) and 3(g)]. For a fixed tip angle, by increasing the ratio between inner and outer radius more electric fields will spread to the cladding causing gradually weakened mode confinement [e.g. Figures 3(a), 3(b) and 3(c)]. These results are consistent with the observations in low index contrast micro-tube based hybrid structure [18]. It is also found that when increasing tip angle with a fixed ratio value, the mode distribution is slightly increased [e.g. Figures 3(b), 3(e) and 3(h)].
Fig. 3 Electric field distributions of the hybrid plasmonic mode supported by the proposed LRHWTP waveguide: (a) R = 120 nm, g = 5 nm, r/R = 0.2, rw = 10 nm, ɵ = 60 deg; (b) R = 120 nm, g = 5 nm, r/R = 0.5, rw = 10 nm, ɵ = 60 deg; (c) R = 120 nm, g = 5 nm, r/R = 0.8, rw = 10 nm, ɵ = 60 deg; (d) R = 120 nm, g = 5 nm, r/R = 0.2, rw = 10 nm, ɵ = 90 deg; (e) R = 120 nm, g = 5 nm, r/R = 0.5, rw = 10 nm, ɵ = 90 deg; (f) R = 120 nm, g = 5 nm, r/R = 0.8, rw = 10 nm, ɵ = 90 deg; (g) R = 120 nm, g = 5 nm, r/R = 0.2, rw = 10 nm, ɵ = 120 deg; (h) R = 120 nm, g = 5 nm, r/R = 0.5, rw = 10 nm, ɵ = 120 deg; (i) R = 120 nm, g = 5 nm, r/R = 0.8, rw = 10 nm, ɵ = 120 deg.
In Fig. 4 , we demonstrate the field profiles of hybrid plasmonic modes with different outer radius and gap distance, where the ratio between inner and outer radius and tip angle of metal wedge are fixed at 0.5 and 90 deg. The electric fields confinement is getting better when the outer radius of the dielectric tube becomes larger [e.g. Figures 4(a), 4(b) and 4(c)]. It is because the increasing outer radius of the tube weakens the effect of the hollow section. The field confinement in the gap area is enhanced while the propagation loss increases as a price.
Fig. 4 Electric field distributions of the hybrid plasmonic mode supported by LRHWTP waveguide: (a) R = 60 nm, g = 5 nm, r/R = 0.5, rw = 10 nm, ɵ = 90 deg; (b) R = 120 nm, g = 5 nm, r/R = 0.5, rw = 10 nm, ɵ = 90 deg; (c) R = 180 nm, g = 5 nm, r/R = 0.5, rw = 10 nm, ɵ = 90 deg; (d) R = 60 nm, g = 10 nm, r/R = 0.5, rw = 10 nm, ɵ = 90 deg; (e) R = 120 nm, g = 10 nm, r/R = 0.5, rw = 10 nm, ɵ = 90 deg; (f) R = 180 nm, g = 10 nm, r/R = 0.5, rw = 10 nm, ɵ = 90 deg; (g) R = 60 nm, g = 15 nm, r/R = 0.5, rw = 10 nm, ɵ = 90 deg; (h) R = 120 nm, g = 15 nm, r/R = 0.5, rw = 10 nm, ɵ = 90 deg; (i) R = 180 nm, g = 15 nm, r/R = 0.5, rw = 10 nm, ɵ = 90 deg.
To better demonstrate the effect of the hollow design on the field distribution and propagation properties, we compare the LRHTWP waveguide with nanotube hybrid plasmonic (NHP) waveguide [18] and hybrid wedge plasmonic (HWP) waveguide [22]. The geometry schemes of the two waveguides are shown in Fig. 5 . The NHP waveguide is consist of a high index dielectric nanotube separated from a metal surface by very thin low index dielectric gap. The HWP waveguide is consist of a high index dielectric cylinder embedding on a metallic wedge surrounded by low index cladding. It is necessary to introduce two important characteristic parameters for intuitive comparisons, the propagation length and the normalized modal area. The effective mode index Neff is described as Neff = Nre + iNim, where Nre and Nim represent the real and imaginary part, respectively. The propagation length L is calculated by L = λ/4πNim, so it is closely related to the imaginary part of the effective index which is easy to understand due to the imaginary part of the index represents energy loss. The normalized modal area of the waveguide is calculated as Aeff/A0, where A0 is the diffraction-limited mode area in free space and defined as λ2/4. Aeff is the effective mode area which can be calculated as:
In Eq. (2), E(r) and H(r) are the electric and magnetic fields, is the electric permittivity and is the vacuum magnetic permittivity.Fig. 5 Schematic illustration of the nanotube based hybrid plasmonic waveguide(a) [18] and hybrid wedge plasmonic waveguide(b) [22].
Figure 6 illustrates the evolution of the propagation length and the normalized modal area for the three different wave guides. The tip angle of the metallic wedge is 90 deg with a 10 nm curvature radius and the ratio between the inner radius and the outer radius of the dielectric tube in NHP waveguide and LRHTWP waveguide is 0.8.
Fig. 6 Dependence of hybrid plasmonic modal properties on gap distance and radius of the dielectric cylinder for NHP waveguide, HWP waveguide and LRHTWP waveguide: (a) and (c) propagation length; (b) and (d) normalized modal area; the radius of the dielectric cylinder and the outer radius of the tube in these waveguides are the same as 120 nm. The inset pictures are electric field distributions of hybrid modes.
From Fig. 6, we can find that the LRHTWP waveguide has much lower propagation loss compared with the HWP waveguide while the modal area is still constrained. The introduction of hollow part filled with low refractive index medium in LRHTWP waveguide enhances the ability to couple surface plasmon polariton to the dielectric waveguide. It significantly reduces the modal propagation loss. Although NHP waveguide has longer propagation length, the modal area is almost two orders of magnitude higher than LRHTWP waveguide which can be clearly seen from the inset pictures in Fig. 6. It is because the metallic wedge structure has much stronger constraints on the hybrid mode than metal plate. Furthermore, for these three kinds of waveguides as shown in Figs. 6(a) and 6(b), the propagation length increases slightly with increasing gap distance while the normalized modal area is strongly dependent on the gap distance. This unique property can be used to tune the modal area without losing much propagation distance.
3. Processing method and fabrication error tolerance
The proposed LRHTWP waveguide has a dielectric nanotube and a metallic wedge. At present, the main feasible methods to fabricate a metallic wedge structure are wet etching, focused ion beam (FIB) and three dimensional electron beam lithography (3D EBL). The wet etching method is based on KOH etching of silicon, deposition of metal and mold release [23]. Alhough this chemical method is low cost and convenient, it has a big issue: the fabricated wedge angle is fixed as 70.5 deg directly related to the crystalline orientation of silicon. FIB has high fabrication precision [24], but it is not able to fabricate a smooth-walled silver wedge and it is very time-consuming and labor-intensive. Furthermore, the incorporation of Ga+ contamination into the silver surface needs to be considered. 3D EBL could be a promising solution which achieves gradient structures through grayscale lithography and dose modulates exposure [25–27 ]. The obtained wedge of photoresist (PMMA in Fig. 7 ) can be used as a mask for reactive ion etching to form a metallic wedge. A thin SiO2 layer is deposited on the wedge metal as the low index gap by atomic layer deposition (ALD) due to its good conformability. The dielectric nanotube can be formed by chemical synthetic methods [28–30 ]. The combination of the metallic wedge and silicon nanotube is operated in the vacuum chamber of a dual beam FIB system which has nanomanipulator module as well as gas injection system (GIS). A nanoscale needle integrated with three dimensional electric displacement platform and rotating platform (called nanomanipulator in Fig. 7) is moved closely to the silicon nanotube and a small amount of metal platinum is deposited at the joint by GIS, then the nanotube is stuck on the needle and can be manipulated according to process requirements. After the nanotube is positioned right above the metallic wedge, the platinum knot for fixing function is erased by FIB. The nanotube is then covered by SiO2 cladding to finally make the LRHTWP waveguide. The process flow is shown in Fig. 7.
Considering the practical processing conditions of the LRHTWP waveguide, we analyzed its fabrication error tolerance. Although the dimension size of the dielectric nanotube [28–30 ] and the gap distance can be controlled with high precision [31], strictly accurate alignment between the nanotube and the metal wedge is as difficult as the precise control of the tip radius of the metal wedge. It is necessary to figure out the impact of the fabrication errors to test the structure feasibility. In Fig. 8 , the effects of the tip radius and the misalignment between the nanotube and the metal wedge on the propagation length and the normalized modal area are shown at varied tip angles. When δx increases from 0 nm to 20 nm, the changes on propagation length are only ~1% while the modal area keeps small fluctuations (~3% for ɵ = 90 deg, ~5% for ɵ = 60 deg and ~7% for ɵ = 120 deg). Furthermore, Fig. 8(c) depicts that the imperfection of the curvature radius of the metal wedge has little impact (<2%) on the propagation length. Figure 8(d) shows when the wedge curvature radius increases from 8 nm to 20 nm the changes in the modal area are only 8%, 4% and 5% for 60, 90 and 120 deg, respectively. It is needed to notice that when the nanotube is not parallelly aligned with the wedge, the error angle we discussed is relatively small which means the shape of the cross section remains nearly unchanged. Considering that the gap distance can be controlled with high precision [31], the effect of the non-parallel between the nanotube and the wedge can be approximated as the misalignment in horizontal direction.
Fig. 8 Dependence of the propagation length on (a) horizontal misalignment δx (rw = 10 nm) and (c) the metal tip curvature radius rw; dependence of the normalized modal area on (b) horizontal misalignment δx (rw = 10 nm) and (d) the metal tip curvature radius rw when R = 120 nm, g = 5 nm, r/R = 0.8
When comparing the HWP waveguide with the proposed LRHTWP waveguide, we find that for the LRHTWP waveguide the misalignment in horizontal direction and variation of the tip radius have less influence on the propagation length (HWP: δx 2%, rw 3%; LRHTWP: δx 1%, rw 2%) and modal area (HWP: δx 5%, rw 5%; LRHTWP: δx 3%, rw 4%) than HWP. These results demonstrate that the LRHTWP waveguide has better fabrication error tolerance than HWP waveguide.
The obtained results reveal that the fabrication errors in geometry parameters such as misalignment in horizontal direction and variation of the tip radius have little impact on propagation length and modal area. The LRHTWP waveguide is fairly tolerant to these fabrication errors mentioned above.
4. Conclusion
In this paper, we present a novel LRHTWP waveguide consisting of a high index dielectric nanotube embedding on a metallic wedge surrounded by low index medium. A comprehensive numerical study is conducted and the corresponding results reveal that the proposed waveguide has low propagation loss and extreme light confinement. In addition, further evaluations in detailed fabrication errors such as inaccuracies of the wedge tip curvature radius and horizontal misalignment between the nanotube and the metal wedge have been investigated. The obtained results show that the proposed structure is fairly tolerant to these geometric fabrication errors. This novel configuration of LRHTWP waveguide has great application prospects in numerous functional nanophotonic components including various passive devices and active components such as plamon nanolasers and plasmonic resonators.
Acknowledgments
The financial support to this work by University of Science and Technology of China start up funding, the National Key Basic Research Program of China (No.2013CBA01703) and Project of Natural Science Foundation of Anhui Province of China (No.1508085MA06).
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