## 4 Numerical Simulation and Analysis
### Model Characteristics Analysis  
In order to optimize the structure designed in this paper and apply it to the nanolaser, its mode characteristics, gain threshold, and quality factor (Q) were analyzed. The main parameters reflecting the mode characteristics are the effective refraction index (\(n_{\text{eff}}\)), the effective propagation loss (\(\alpha_{\text{eff}}\)), the normalized mode area (\(SF\)), and the confinement factor (\(\Gamma\)). Among them, the imaginary part of the effective index characterizes the propagation loss of the mode. The normalized mode area represents the constraint ability of the mode field. The calculation formula is [41; 42]:  
\[SF=\frac{A_{\text{eff}}}{A_{0}}=\frac{\left(\iint|E|^{2}dxdy\right)^{2}/\left( \iint|E|^{4}dxdy\right)}{\lambda^{2}/4} \tag{3}\]  
where \(A_{\text{eff}}\) is the effective mode area, \(A_{0}\) is the mode area under diffraction limit, and \(\lambda\) is the wavelength in vacuum. The confinement factor, \(\Gamma\), is defined as the ratio of the electric field energy in  
Figure 3: (a) Three-dimensional mode of the nanolaser and the (b) y-z normalized electric field.  
Figure 2: (a) 2D electric field distribution of the nanolaser. (b) Horizontal serif electric field distribution.  
the CdS nanowire to the electric field energy of the entire mode, which is used to characterize the field strength limiting ability of the gain medium nanowire. The calculation formula is:  
\[\Gamma=\frac{W_{\mathrm{s}}}{W} \tag{4}\]  
where \(W_{\mathrm{s}}\) represents the energy stored in the gain medium, and \(W\) represents the total energy of the mode field.  
Figure 4a-d show the variation of the mode characteristics of the nanolaser with the air gap width, \(w_{\mathrm{g}}\), and the distance, \(h_{\mathrm{t}}\), from the bottom of the nanowire to the top of the graphene. As can be seen from Figure 4a,b, when the width of the air gap increases, the effective refraction index decreases gradually; the propagation loss of the mode decreases first and then increases. At \(w_{\mathrm{g}}=35\) nm, the minimum value is 0.0096, and the overall average value is 0.013. This is because when the width of the air slot is small, although the distance from the bottom of the nanowire to the top of the graphene remains unchanged, the overall relative height of the two is large, and the SP mode is mainly concentrated on the metal surface, causing a large loss. As the width of the air gap increases, the relative height decreases, and the SP mode coupling effect of the gain dielectric waveguide is enhanced, and the loss is gradually reduced. When the air gap reaches an optimum value, the coupling effect is no longer significant, and the continued expansion of the air gap causes an increase in loss. As can be seen from Figure 4c,d, the increase in the air gap causes the normalized mode area to gradually decrease and the confinement factor to gradually increase. It can be seen that the normalized mode area is much smaller than 0.1, the minimum value is 0.0032, and the average value is 0.018, which indicates that the hybrid plasma laser structure has a strong ability to constrain the light field and can achieve a deep sub-wavelength constraint; the confinement factor is greater than 0.55, indicating that the mode field energy is mainly located in the gain medium nanowire, and the hybrid plasma waveguide propagates in the nanowire.  
Figure 4: The mode properties change with the air gap, \(w_{\mathrm{g}}\), and the distance, \(h_{\mathrm{t}}\); (**a**) effective index, \(n_{\mathrm{eff}}\); (**b**) effective propagation loss, \(\alpha_{\mathrm{eff}}\); (**c**) normalized mode area, SF; (**d**) confinement factor, \(\Gamma\).  
At the same time, it can be seen that the refraction effective index and the confinement factor gradually decrease, as the distance, \(h_{\rm t}\), from the bottom end of the nanowire to the top of the graphene increases, while the propagation loss and the normalized mode area increase.