## 4 Numerical Results:  
### Dielectric Waveguide Facet  
In fact, the waveguides terminated by air were studied using a variety of techniques [19; 20; 21; 22; 23; 24; 25] in TE and TM polarization modes. Figure 1 depicts the geometry of the problem under consideration (akin to waveguide Laser facet), where a three-layer dielectric slab waveguide occupying the half-space z \(<\) 0 is abruptly terminated by air at \(z=0\). The waveguide core index is \(n_{co}\) and width \(h\), is surrounded by a cladding with refractive index \(n_{cl}\). An FFT-based BPM program including the reflection operator of Equation (9) has been implemented in Fortran to calculate the reflected power in TE and TM cases. The numerical values of the FFT-BPM algorithm are as follows: Number of sampling points along x-axis N = 2\({}^{16}\), sampling interval along the \(x\)-axis \(\Delta x=0.0003\cdot\lambda_{o}/n_{ref}\), propagation step-size \(\Delta z=1\) nm. The dielectric waveguide core thickness \(h\) varies from 1 to 900 nm, the core refractive index is \(n_{co}\) = 3.6, the cladding refractive index is expressed as a percentage \(\delta\) of the core refractive index: \(n_{cl}=n_{co}(1-\delta)\) (where \(\delta\) = 3%, and 10%), and the free space wavelength \(\lambda_{o}\) = 0.86 \(\upmu\)m. The fundamental TE and TM modal fields are calculated by solving numerically the corresponding eigenvalue equations [26]. Using the reflection operator formalism of the previous section, we calculated the reflected field at z = 0, and hence the transmitted field is readily evaluated via Equation (17). The power reflectivity is simply calculated as follows [24]:  
\[\textit{Power reflectivity}=\left(\frac{\left|\int_{-\infty}^{\infty}G_{r}(x) H_{i}^{*}(x)dx\right|}{\int_{-\infty}^{\infty}\left|H_{i}(x)\right|^{2}dx} \right)^{2} \tag{19}\]  
Figure 2 compares the results of our method with those obtained by the accurate method of [PERSON] [22] and the plane wave expansion (PWE) [24] at different values of \(\delta\) = 3% and 10%, for both polarization TE (a) and TM (b). The curves of Figure 2 reveal a general good agreement. Figure 1: Schematic diagram of a 2-dimensional waveguide facet terminated by air. The method is then applied to waveguides terminated by antireflection coating. A single layer coating is applied to a dielectric waveguide with \(n_{\omega}=3.6\) and \(n_{cl}=3.564\). The waveguide core width \(h\) is fixed at 1.457 \(\upmu\)m. The coating layer has an index equal to the square root of the TE mode effective index \(n_{e}\), which is equal to 1.8938 at a wavelength \(\lambda_{\rm o}=1.55\)\(\upmu\)m. Figure 3 displays the power reflectivity as a function of the coating thickness \(u\). Theoretically, the reflection from the coating layer is minimized when its thickness is equal to a quarter of the guided wavelength \(\lambda_{\rm g}\). So, we expect a minimum power reflectivity close to quarter the guided wavelength \(\lambda_{\rm g}/4\), where \(\lambda_{\rm g}=\lambda_{\rm o}/n_{\rm e}=1.55/1.8938=0.8185\)\(\upmu\)m. Namely, the optimal coating layer thickness is expected to be close to 0.8185/4 = 0.205 \(\upmu\)m. Figure 3 reveals a minimum reflectivity at \(u=0.206\)\(\upmu\)m which agrees with \(\lambda_{\rm g}/4\) and with previously published results [23, 24]. Figure 2: The power reflectivity curve comparison between our proposed method and Vassallo [15], and plane wave expansion (PWE) techniques [24] at (**a**) transverse electric (TE) polarization case, (**b**) transverse magnetic (TM) polarization case. The extension to multiple dielectric layer is straightforward. Figure 4 shows the power reflectivity as function of the wavelength for high reflection coating with waveguide core width 0.487 \(\upmu\)m.  
The layers thickness is fixed to quarter wavelength. The core refractive index \(n_{\text{co}}=3.6\), the cladding refractive index \(n_{\text{cl}}=3.24\), and the high and low reflection coating material refractive indices are 3.5 and 1.7 respectively.