Low-loss nanowire and nanotube plasmonic waveguide with deep subwavelength light confinement and enhanced optical trapping forces

With the rapid development of the micro/nano fabrication technology, the semiconductor nanowires and nanotubes with size and dimensions controllable realize wide applications in nanophotonics. In this talk, we propose two kinds of hybrid plasmonics waveguides, one is consisting of nanowires, another is consisting of nanotubes. By employing the simulating with different geometric parameters, the basic waveguiding properties, including the effective mode area, the propagation length, the mode character and the optical trapping forces can be achieved. Compared with previous plasmonic waveguide with plane metal substrate, current plasmonics waveguides with ease of fabrication have the advantage of long propagation length and effectively optical trapping of nanoparticles with deep subwavelength light confinement, which may be very useful for nanophotonic integrated circuits, nanolasers and biosensing.


Introduction
With the nanometer fabrication technology rising, the semiconductor nanowires/nanotube with size and dimensions controllable realize wide applications in nanophotonics, such as waveguides, sensors, photodetectors, and lasers [1,2] . However, confined to the diffraction limited, the nanowire/nanotube waveguide is difficult to achieve deep subwavelength optical scale. Breaking the shackles of the diffraction limited and manipulating photons under deep subwavelength are very important for the next-generation integrated photonic devices. On one hand, the subwavelength optical confinement can enhance the interaction between light and matter. On the other hand, the strong optical confinement can enhance optical field strength and gradient of light field, which will highly enhance the optical force in the nanoscale region [3][4][5] .
Surface plasmon polaritons (SPPs) are a coherent oscillation of free charges in the surface of metals in resonance with the incident light wave, which have the distinguishing capabilities of enhancing the local electric field intensity as well as of confining the optical energy within the nanoscale domain [6][7][8][9][10][11] . As one of the most promising candidates for nanophotonic circuits, SPPbased nanoscale optical devices have attracted enormous interest in recent years. However, SPP based waveguides suffer huge propagation loss because of the intrinsic ohmic loss. It is difficult to balance between these two opposites to strong optical confinement and propagation loss. In recent report, a novel hybrid plasmonic waveguide consisting of a dielectric waveguide on one side of a metal surface has been proposed by R.F.Oulton et al [12] . It is able to support long range propagation length, while maintaining an ultra-small mode size due to the coupling between the dielectric cylindrical waveguide mode and the SPP mode at the metaldielectric interface [13][14][15][16][17][18][19] .
In this article, we propose the nanowire/nanotube hybrid waveguide by replacing the metal film with a metal nanowire. Through theory simulation and analysis, we can draw a conclusion that the hybrid mode in this structure can achieve deep-subwavelength mode, while maintaining propagation distance as well as that in the SPP. Moreover, the combination of ultra-low loss and strong coupling strength between the dielectric waveguide mode and SPP mode would lead to a much larger optical force [5] . On the whole, the proposed structure provides an excellent platform for nanoscale optical devices, such as nanolasers and nanotweezers.

2.Modal properties of the nanowire plasmonic waveguide
The nanowires hybrid waveguide geometry shown in Fig.  1, which consists of two cylindrical nanowires with a small gap distance h. A high-permittivity semiconductor nanowire is Si with the relative permittivity εs = 12.25. Another one is silver with diameter of D = 200 nm and permittivity of εm = -129+3.3i at λ = 1550 nm. The surrounding dielectric layer is a low-permittivity dielectric of SiO2 (εd = 2.25). In the follow study of nanowires hybrid waveguide, we vary the semiconductor nanowire diameter, d, and the gap distance, h, to adjust the mode field distribution, the propagation distance, Lm, and the mode area, Am/A0. The modal properties are investigated by the finite-element method using COMSOL 4.2. Figure 1. The geometry of the nanowire hybrid surface plasmonic waveguide. Two cylindrical nanowires with a small gap distance h. Figure 2 shows the dependence of the normalized modal area, Am/A0, the propagation length, Lm, and the distributions of electromagnetic energy density on different dielectric diameter, d, and gap distance, h. In this paper, we defined the propagation length and modal area as same as that in [20] . The propagation length is given by where neff is the effective index of the hybrid mode. λ is the wavelength of light transmission.
The effective mode area, Am, is given as follows Here, the W(r) is the energy density for the hybrid plasmonic waveguide (per unit length along the direction of propagation). Due to the dispersion and material loss, the W(r) can be calculated by |E(r)| and |H(r)| are the electric and magnetic fields, respectively. ε(r) is the permittivity, and μ0 is the vacuum magnetic permeability.
The normalized effective mode area is defined as the ratio of the effective mode area and the diffractionlimited area of vacuum, The normalized effective mode area was used for consistently quantifying the mode confinement, and A < 1, indicates the confinement of the hybrid plasmonic mode is subwavelength.
As shown in Fig. 2(a), with a certain value of diameter of dielectric nanowire, the strong confinement in hybrid plasmonic waveguide results in an ultra-small mode area down to 0.0019(λ/2) 2 . As shown in Fig. 2(b), the hybrid plasmonic mode can travel through a long distance, which is more than 24 μm. Figs In this case the propagation loss of the electromagnetic wave in the waveguide is low, but the localization of the dielectric-like mode is weak. What is interesting is that, for the hybrid mode, the normalized modal area and the propagation length are improved remarkably.  In order to gain a deeper understanding, the dependence of the effective index of the hybrid mode nhyb on the diameter of dielectric nanowire, d, and gap distance, h, is shown in Fig. 3(a). The hybrid mode (coloured lines) is beyond that of the pure dielectric mode (black solid line). When the diameter of dielectric nanowire, d, is fixed, the effective index will increase with reducing the gap distance, h. For a fixed gap distance, h, the mode's effective index increases as the diameter of dielectric nanowire, d, increases. This is because that the waveguide mode has a relationship between the diameter of dielectric nanowire and gap distance. A mode character, |a+(d, h)| 2 , is employed to describe the superposition of the dielectric waveguide mode and the SPP mode [12] . Then it can be used to evaluate the degree to which the guided mode is dielectric-like or SPP-like.   (Fig. 3(b)). A larger d and h lead to a larger mode character. The hybrid mode is a dielectric-like waveguide mode. On the other hand, with the decrease of the diameter of dielectric nanowire, a smaller mode character can be achieved, then the hybrid mode is a more SPP-like mode. For a moderate side length, dm, the hybrid mode displays both the characteristics of the dielectric and the SPP mode (|a+(d, h)| 2 = 0.5). This case is corresponded to the condition ndie(d) = nspp , indicating that polarization charge and plasma oscillations move in phase and maximize the effective optical capacitance of the waveguide [12] . Figure 4 schematically shows the geometry of the nanotube hybrid surface plasmonic waveguide, where a high index semiconductor nanotube was embedded in a low index dielectric and placed on a metal nanowire with a small gap distance, h. In this section, the relative permittivity of material and mode character are defined as the same as that in nanowire hybrid surface plasmonic waveguide. In the following, we defined the diameter of the nanotube outer surface and inner surface as d and di, respectively. Figs. 5(a, b) show the propagation length and modal area of the nanotube hybrid mode on the diameter of the Si nanotube outer surface, d, for the different gap distance h. For a fixed diameter of the nanotube inner surface, di = 50 nm, the modal area and the propagation length are decline at first and then gradually increased. We can see that there exists a point with d = 150 nm, that both of the modal area and propagation length is minimum. According to the coupled mode theory, in this point, the coupling strength gets its maximum value and the dielectric nanotube mode and SPP mode satisfy the phase-matched condition ndie = nspp. Then, we will further explore the changed the inner surface diameter of nanotube to confirm the influence of the air hole on field enhancement. As shown in the Figs. 5(c, d), the outer surface diameter of nanotube is fixed of 200 nm, and the diameter of air hole is from 0 nm to 180 nm. Certain monotonous variation of the modal area and the propagation length curve appear with the different size of air hole, di. The modal area and propagation length can be increase by enhancing the diameter of air hole, di, for a fixed gap distance, h. This can be explained that the optical confinement in hybrid mode couples is weaken gradually as di increase.

4.The enhance optical trapping forces in the nanowire and nanotube plasmonic waveguide
Next, we analyzed the optical trapping forces in the proposed hybrid plasmonic waveguide. The optical trapping forces are proportional to the gradient of the electrical field distribution, which can be described as [21] 2 2 where U is the optical trapping potential, ns is the refractive index of environmental medium (εs = ns 2 ), and α is the polarizability of the nanoparticle. For a dielectric sphere of radius r and dielectric permittivity of εn, the polarizability can be written as The strong optical confinement of the hybrid mode in the gap region, will enhance the optical trapping forces and enable to trap single nanoparticle. Figure 6 shows the system of the optical trapping with a dielectric nanoparticle. In this paper, we used a polystyrene nanoparticle with diameter of 5 nm and refractive index np = 1.59 for studying the optical trapping forces. The environmental medium surrounding the waveguide is water with refractive index of ns = 1.33 [22] . According to Eq. (7), the optical trapping force, at the position where the optical field gradient become largest, was maximized. Figure 7(a) shows the nanowire optical trapping force dependence on D = 200 nm, d = 200 nm, for h = 10, 20, and 30 nm, respectively. As the gap distance reduces, the optical trapping force increases. At h = 10 nm, the maximum of optical trapping force per unit input optical power is 1187 fN/W for the proposed hybrid plasmonic mode. The optical trapping potential profiles along x direction with trapping force integration ( Fig. 7(a)) was shown in Fig. 7(b). We define the zeropoint potential as the particle is 100 nm away from the waveguide center (x = -100 nm). The optical trapping potential Ux is around 10.1 KBT/W right underneath the waveguide at h = 10 nm, which means that the kinetic energy of the optical trapping force was 10.1 times larger than the kinetic energy of Brownian motion (KBT = 4.1 × 10 −21 J at room temperature, where KB is the Boltzmann constant and T is absolute temperature [22] ). In this case, when the particle moves across the x = 0 point, the particle can be trapped effectively in the gap region. potential along x-direction. In a simulation, we set up the D = 200 nm, d = 200 nm and h =10 nm. As the inner surface diameter of nanotube increases, the optical trapping force decreases. The optical trapping force gets its maximum value at the inner surface diameter of nanotube di = 0 nm. As mentioned above, the effective index of the nanotube decreases monotonously along with the inner surface diameter increasing. As a result, the effective index mismatch between the dielectric mode and the SPP mode, the coupling strength become weaken gradually. What is interesting is that, for the nanowire/nanotube hybrid mode, the optical force can be significantly larger than that of previous hybrid plasmonic structures [5] . In summary, the combination of low loss and strong coupling strength between the dielectric waveguide mode and SPP mode would lead to a large averaged optical force per unit propagation length along the waveguide.

Conclusion
We have proposed a hybrid plasmonics waveguide system consisting of nanowire/nanotube. Simulation results based on the FEM method, the basic waveguide properties, including the effective mode area, the propagation length, the mode character and the optical trapping forces, can be achieved. Compared with previous plasmonic waveguide with plane metal substrate, current plasmonics waveguides with ease of fabrication have the advantage of long propagation length and effectively optical trapping of nanoparticles with deep subwavelength light confinement. Additionally, we discussed the effect of the size of nanowire and nanotube air hole on the modal properties of the hybrid plasmonics waveguide. In future experimental, the nanowire/nanotube hybrid waveguide can be fabricated easily by current nanofabrication technology and be useful for nanophotonic integrated circuits, nanolasers and biosensing.