In recent years, surface plasmons are introduced to break diffraction limit of waveguide, whose size must be larger than the half wavelength of the optical field in all three dimensions . In this way, plasmonic waveguide can reach subwavelength optical confinement  by using the surface plasmon polaritons, which are Transverse Magnetic (TM) polarized surface wave propagating along metal-dielectric interfaces . Therefore, various types of plasmonic waveguide have been presented, such as metal-insulator-metal (MIM) , long-range SPP (LSPP) , metallic nanowire  waveguides and hybrid plasmonic waveguide . However, the previous reports on plasmonic waveguide almost achieved at visible and infrared region because of the lower metal absorption . From the developing trend of the lasing, the main direction goes forward to short wavelength , such as ultraviolet band. It benefits to increase storage density of optical information and bandwidth of optical communication  . In addition, the ultraviolet resonance Raman spectroscopy is an important means to detect biological molecules . In this paper, we propose a novel hybrid plasmonic waveguide at ultraviolet band. The structure shows tight field confinement and long propagation distance, and reaches deep sub-wavelength- scale. The design has promising potential for application in nanolaser, plasmonic systems and biological detection.
2. Structural Design and Simulation
Figure 1.(a) Geometry of the proposed waveguide; (b) Normalized electric field distribution of the fundamental hybrid plasmonic mode of the proposed structure; (c) and (d) Normalized electric field distribution along the horizontal and vertical dashed lines in (b) ().
The geometry of the proposed waveguide is shown in Figure 1(a). The structure consists of a Al metal ridge, a
low-index MgF2 dielectric layer, a SiO2 layer and a high-index GaN nanowire. The width of the metal layer is 300 nm, and its height is 100 nm. The length of the GaN nanowire L is 30 μm, its radius is r. The radius of the metal ridge is fixed at 35 nm. The thickness of the MgF2 dielectric layer is 5 nm. At the working wavelength of 370 nm, the refractive indices of Al, MgF2, SiO2 and GaN are 0.38829 + 4.3466i, 1.3856, 1.46 and 2.65, respectively  .
Figure 1(b) shows the electric field distribution of the fundamental hybrid plasmonic mode of the proposed structure, where the geometric parameters are chosen as, ,. The field enhancements in the horizontal and vertical directions are shown in Figure 1(c) and Figure 1(d), respectively. By introducing a metal ridge and the filmy MgF2 dielectric layer into the design, the electric field energy is concentrated in a tiny area.
To quantify the mode properties, we introduce the indices of the mode scaling factor () and the propagation distance (D) . The is calculated as the ratio of the effective mode area to the diffraction-limited mode area. It represents the confinement ability for the mode field which is expressed  as
Here, the effective mode area and the diffraction-limited mode area are expressed by using the following formulas :
In the above expressions, E is the electric field intensity of the hybrid mode and is the working wavelength. The propagation distance (D) is defined as 
where is the mode propagation constant.
The Figure 2 shows the effect of the nanowire radius(r) on the performances of the structure. Obviously, the mode scaling factor firstly decreases before increaseing while the propagation distance declines with the nanowire radius increasing. When the nanowire radius approaches the metal rib radius, the mode scaling factor achieves the minimum value 0.0194. In the case, the effective mode area is. The larger nanowire radius leads to larger propagation distance, but larger mode scaling factor. So it is meaningful to define the figure of merit (), which is given by 
The lager indicates better performance of a waveguide. The Figure 3 shows the with the nanowire radius increasing firstly increases before decreasing. The maximum value 28.56 mm is obtained at. In addition, it indicates the of the present structure is larger than that of previous structure with a metal plate , owing to the metal ridge and the filmy MgF2 dielectric layer.
The pump threshold is the minimal value the gain reaches when achieving lasing action. It is related to the nanowire length L and the end facet reflectivity R. The R  is expressed by the following equation:
The lasing threshold is calculate  by
where, is the refractive index of the gain nanowire, and is the enhancement part of the modal effective index. Figure 4 shows that the pump threshold of the proposed waveguide increases with enlarging r. Its minimum is.
Figure 2. The SF and D of the fundamental hybrid plasmonic mode with different r.
Figure 3. The FOM of the fundamental hybrid plasmonic mode of present structure and previous structure with different r.
Figure 4. The pump threshold of the fundamental hybrid plasmonic mode of present structure with different r.
We present a new type of ultraviolet waveguide based on surface plasmons which attains deep-subwavelength scale and has long propagation distance. By using the COMSOL Multiplicity software, we investigate the light field distribution, and analyze the effect of the radius of the gain medium nanowire on the properties and the lasing threshold. The results show that the larger nanowire radius causes the better performance of the proposed waveguide. So we can select the optimal radius of the nanowire as 80 nm. In this case, the SF, the D and the threshold are 0.0359, and, respectively. Compared to the previous ultraviolet waveguide with a metal plate, the performance of the present structure with a metal redge is improved significantly with the same geometric parameters. The designed structure offers a new idea for the high-density photonic integrated devices, such as the deep-subwavelength-scale ultraviolet nanolaser.
We acknowledge support from the Hundred-Talent Program of Hebei Province and the Natural Science Foundation of Hebei Province in China.
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