## 4 Numerical Results:
### Butt-Coupling Between Plasmonic and Dielectric Waveguide  
To validate further our proposed method, we considered the fundamental TM\({}_{0}\) mode incident from a plasmonic waveguide upon a single mode dielectric waveguide as depicted in Figure 5.  
Figure 6 illustrates the evolution of the total propagated magnetic field as it crosses the junction plane at \(Z=20\) nm, and ultimately evolves to the characteristic shape of the fundamental TM\({}_{0}\) mode of the symmetric silicon guide. The parameters of the two guides assume the following values:  
Number of sampling points along x-axis N = 218, sampling interval along the x-axis \(\Delta x=0.1\) nm, propagation step-size \(\Delta z=1\) nm, the propagation distance in the plasmonic and dielectric guides equals 20 and 100 nm respectively. The dielectric waveguide core thickness \(W_{\text{d}}=220\) nm, MIM waveguide core thickness \(W_{\text{p}}=42\) nm, Silicon refractive index \(n_{\text{co}}=3.477\), metal refractive index (silver) \(n_{\text{cl}}=0.397-\text{j}\,11.4\), and the free space wavelength \(\lambda_{\text{o}}=1.55\) um. The contour plot depicted in Figure 7, shows the expansion of the confined light inside the plasmonic guide as it crosses the  
Figure 5: Butt-coupled metal-insulator-metal (MIM) and dielectric waveguides. Figure 6: Evolution of the magnetic field \(\text{H}_{\text{y}}\) of the fundamental mode in the plasmonic guide as it propagates towards the silicon guide where the symmetric TM\({}_{0}\) mode is well established after almost 100 nm. junction plane z = 0, and after almost 100 nm the characteristic TM\({}_{0}\) mode of the dielectric guide is well established as shown in Figure 6.  
The integration and merging of tightly confined waveguides, like plasmonic guides, and thin-film dielectric waveguides will be an essential technology for the development of extremely compact photonic components. Thus, numerical modeling and optimization of the power transmission efficiency is a key factor to increase device density in the implementation of photonic integrated circuits. Accordingly, we considered the butt-coupled plasmonic-MIM waveguide of Figure 5. Upon the incidence of the TM\({}_{0}\) plasmonic mode from the MIM guide on the interface plane z = 0, a backward reflected field is generated. The total transmitted field (sum of the incident and reflected) constituting the guided, radiation and evanescent components is directly calculated from Equation (17). And the power transmission efficiency \(\eta\) is readily obtained from Equation (19) except \(G_{r}\) is replaced by \(H_{t}\), that is:  
\[\eta=\left(\frac{\left|\int_{-\infty}^{\infty}H_{t}(x)H_{t}^{*}(x)dx\right|}{ \int_{-\infty}^{\infty}\left|H_{t}(x)\right|^{2}dx}\right)^{2} \tag{20}\]  
The power transmission efficiency \(\eta\) of such coupler as function of the plasmonic guide core width \(W_{p}\) has been studied by [27], using a combined mode-matching and FDFD (finite-difference frequency domain). Figure 8 reveals good agreement between our results and those calculated by the (FDFD). The numerical values of the BPM parameters are as follows: Number of sampling points along x-axis \(N=2^{16}\) with sampling interval \(\Delta x\) = 0.46 nm, propagation step-size \(\Delta z\) = 0.75\(\Delta x\), propagation distances in the plasmonic and dielectric guides = 20 and 130 nm respectively, hence the total propagation distance =150 nm. The dielectric waveguide core thickness is kept constant \(W_{\text{d}}\) = 300 nm, the plasmonic waveguide core thickness \(W_{\text{p}}\) varies from 5 to 200 nm, Silicon refractive index \(n_{\text{co}}\) = 3.477, metal refractive index (silver) \(n_{\text{cl}}\) = 0.397 - j 11.4, and the free space wavelength \(\lambda_{\text{o}}\) = 1.55 \(\upmu\)m.  
The FFT-BPM gives a maximum power transmission when \(W_{\text{p}}\cong\) 42 nm., which is the same width predicted by the FDFD.