*2.1. Analysis for the PL Model*

An equivalent rectangular 3-D vision of an SOI waveguide is shown in Figure 1a, where *n*<sup>1</sup> and *n*<sup>2</sup> are the refractive indices of the core and cladding layers, respectively, and 2*d* is the width of the waveguide. The SWR of a waveguide was caused by the inductively coupled plasma (ICP)-etching, as shown in Figure 1b, then as shown in Figure 1c, the distributions of the SWR at the vertical and horizontal directions were probably different from each other.

**Figure 1.** Sidewall roughness (SWR) of silicon-on-insulator (SOI) waveguide: (**a**) the schematic three-dimensional (3-D) equivalent rectangular waveguide with SWR; (**b**) the ICP-etching to a waveguide sidewall; (**c**) the distributions of the SWR at x- and z-coordinate.

In accordance with the 3-D simulations of radiation mode, the three linear polarization states (*x*, *y*, and *z*) could cause the different profiles of the radiation mode [8,9]. The exponential form of the PL model was used to define a space distribution Equation (1) for an interest. The power spectral density of the roughness was of interest in the optical scattering loss, then the Fourier transform of the autocorrelation function of the roughness was expressed to be Equation (2) [8,9]:

$$R(\mu) \approx \sigma^2 \exp(-|\mu|/L\_c) \tag{1}$$

$$R(\xi) \approx 2\sigma^2 L\_c / (1 + L\_c^2 \xi^2) \tag{2}$$

where σ is the root-mean-square (rms) roughness, *Lc* is the correlation length of the roughness with the assumption that there is no correlation between the two sidewalls, *u* is the space variable of a waveguide sidewall, and ξ is the space-frequency of the power spectral function of roughness. If the wavelength of the light wave in air was λ, with the wavenumber as *k*<sup>0</sup> = 2π/λ and the finite difference processing of beam propagation method (FD-BPM), we obtained the effective index *Neff* of a single guided-mode, and further with the propagation constant of this guided-mode β = *k*<sup>0</sup> · *Neff* , we cited three dimensionless parameters *h*, *V*, and *p* of guided-mode defined by the PL model as [9]

$$h = d\sqrt{n\_1^2 k\_0^2 - \beta^2}, \; V = k\_0 d\sqrt{n\_1^2 - n\_2^2} \text{ and } p = d\sqrt{\beta^2 - n\_2^2 k\_0^2} \tag{3}$$

where *h* and *p* are the very popular parameters, defining the guided mode field in the literature on optical waveguides [1,2]. *<sup>V</sup>* is the product of three elements: the numerical aperture *n*2 <sup>1</sup> <sup>−</sup> *<sup>n</sup>*<sup>2</sup> <sup>2</sup> of a symmetric planar waveguide, *d* is the radius (or half a width) of the waveguide core, and *k*<sup>0</sup> is the wave number in the air. Further, we obtained the dimensionless parameters as:

$$
\Delta = (n\_1^2 - n\_2^2) / (2n\_1^2), \; \mathbf{x} = p(\mathbf{L}\_\mathbf{c}/d), \; \mathbf{y} = (n\_2 V) / (n\_1 p \sqrt{\Delta}) \tag{4}
$$

Consequently, for the two-SWR-induced optical scattering loss, with the above definitions for the guided-mode profile defined by Equations (3) and (4) and a combination of the PL model (1)–(4), the Yap improvement for the optical loss coefficient dependence on the SWR was expressed as [12]

$$\log \text{l}(TE/TM) = \frac{4.34 \sigma\_{2D}^2}{\sqrt{2} d^4 \mathfrak{g}\_{TE/TM}} \text{g}(V) \cdot f\_{\varepsilon}(\mathbf{x}, \mathbf{y}) \tag{5}$$

where σ2*<sup>D</sup>* is the SWR defined in the 2-D form, the loss coefficient is in dB/cm, and the functions *g*(*V*) and *fe*(*x*, γ) are defined by

$$g(V) = \frac{h^2 V^2}{1 + p^2} \text{ and } f\_\epsilon(\mathbf{x}, \mathbf{y}) = \frac{\left[\left((1 + \mathbf{x}^2)^2 + 2\mathbf{x}^2 \mathbf{y}^2\right)^{1/2} + 1 - \mathbf{x}^2\right]^{1/2}}{\left[\left(1 + \mathbf{x}^2\right)^2 + 2\mathbf{x}^2 \mathbf{y}^2\right]^{1/2}} \tag{6}$$

The model defined by Equations (5) and (6) might be referred to as a Yap-form PL model.
