*2.2. Three-Dimensional Model for the SWR-caused Optical Scattering Loss*

In the roughness-improved PL model, the correlation length of sidewall roughness *Lc* is a paramount important parameter [9]. Before 2000, for a low index-contrast waveguide, such as silica-waveguide, *Lc* was observed to be a few micrometers, and even the values less than 500 nm were ever exploited, and meanwhile, the optical scattering loss was determined to be a forward-scattering process, where *Lc* ≈ 1/2β that matched with the maximum attenuation [14]. In contrast, Barwicz and Haus studied the 3-D optical scattering process for both the high and low index-contrast waveguides with respect to three *Lc* values as 1, 50, and 150 nm [9]. In the Barwicz-Haus's 3-D theory, the optical scattering loss is thought to be caused by a radiative mode coupled from a guided mode, meanwhile, for straight roughed waveguides, the phase-matching condition between the guided and the radiated modes allows only a narrow range of spectral frequencies of roughness to produce radiation loss, then the estimated correlation length *Lc* value is in a range of 1/(β + *n*2*k*0) < *Lc* < 1/(β − *n*2*k*0). Consequently, for both the TE-like and TM-like modes, the SWR-induced optical scattering loss was related to both the *y*- and *z*-components of roughness, so the dependences of the optical intensity loss coefficients α3*D*(*TE*/*TM*) on the SWR σ3*D*(*TE*/*TM*) could be expressed as

$$\alpha\_{\rm 3D}(TE/TM) = 4.34 \left( \frac{\sigma\_{\rm 3D}^2(TE/TM)}{\sqrt{2}d^4 \beta\_{TE/TM}} g(V) \cdot f\_\epsilon(\mathbf{x}, \mathbf{y}) \right) \tag{7}$$

Hence, the new model Equation (7) was the combination of the Yap-form PL model and the 3-D SWR distribution. In this model, the optical loss coefficient was in dB/cm.
