Optical Propagation Loss （OPL） is the core problem of Silicon-On-Insulator （SOI） optical waveguide devices， and the main source of the OPL is the inevitable Sidewall Roughness （SWR） caused by the waveguide fabrication processing ^［1-2］. With the development and applications of SOI optical waveguide photonic integrated devices in the past few years， enormous research efforts in this area have been made in the calculation of waveguide OPL， an obvious trend is that the original two-dimensional （2-D） waveguide structure is moved to the understanding to the three-dimensional （3-D） roughness distribution， including the paper we just published， the calculation model is gradually improving the Payne-Lacey （PL） model ^［3］.

The currently accepted theoretical model for the relationship between the waveguide SWR and the light guided mode is the PL theoretical model that was proposed by PAYNE F P and LACEY J P R in 1994 ^［4］. Since then， the PL theoretical model has been widely employed to study the Optical Propagation Loss （OPL） and some improvements have been carried out ^［5-7］. Typically， in 2005， BARWICZ T and HAUS H A carried out their theoretical work based on the 3-D interaction between the polarized Poynting vector and the Vertical Shape of the Field （VSF） so that they could give rise to more detailed simulations of the OPL values of both the low and high index-contrast waveguides ^［5］. In 2006， POULTON C G et al. gave a more powerful explanation that the OPL caused by the SWR is from the two conversions： the conversion from a guided-mode to a radiation mode and the conversion from the radiation mode to a leaky mode， so that they could accurately compute the electric fields with the Finite Difference Time-domain （FDTD） method ^［6］. Especially for the SOI waveguides， in 2008， SCHMID J H et al. employed the non-uniform waveguide boundaries to study the scattering loss ^［7］， and in 2009 YAP K P et al. studied the correlation among the light scattering loss， the sidewall roughness and channel width of the SOI waveguides ^［8］. Since 2019， we have introduced the Confocal Laser Scanning Microscope （CLSM） into the measurement technology of waveguide SWR and obtained the 3-D distribution properties of silicon dioxide and SOI waveguides ^{［3，9-10］}. In our previous work， both the improved theoretical model and the latest roughness test results show that the correlation length determined by the waveguide structure plays a key role in causing the light propagation loss， but not much attention has been paid to it.

In this article， the improved theoretical model of defining the OPL coefficient on the SWR is exploited where the components of the SOI waveguide SWR at horizontal and vertical direction are introduced. Then， the measured values of 3-D distributed waveguide SWR are obtained with the CLSM technique. As a result， with such a combinative model， the dependences of the optical propagation loss coefficient of the waveguide on the SWR distribution and the width are systematically simulated. Finally， the great agreements between the numerical simulation results based on the measured SWR values and the experimental results are achieved.

Fig. 1 shows a schematic 3-D configuration of a straight waveguide in which the waveguide core has a refractive index of $n_{\mathrm{1}}$ and a channel width of 2d ， and the cladding material has a refractive index of $n_{\mathrm{2}}$， and the effective index of the guided-mode is $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$. If the wavelength of the light wave in air is $\lambda$， with the wave number $k_{\mathrm{0}}=\mathrm{2}\mathrm{\pi }/\lambda$ and the Finite Difference Processing of Beam Propagation Method （FD-BPM）， the effective index $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ of a single guided-mode is obtained， and further based on the propagation constant of this guided-mode $\beta =k_{\mathrm{0}}\cdot N_{\mathrm{e}\mathrm{f}\mathrm{f}}$， three dimensionless parameters $h$， $V$ and $p$ of guided-mode defined by the PL model can be defined as ^［3-4］

$\left(h,V,p\right)=\left(\begin{array}{ccc}d\sqrt[]{n_{\mathrm{1}}^{\mathrm{2}}{k}_{\mathrm{0}}^{\mathrm{2}}-{\beta }^{\mathrm{2}}},& k_{\mathrm{0}}d\sqrt[]{n_{\mathrm{1}}^{\mathrm{2}}-n_{\mathrm{2}}^{\mathrm{2}}},& d\sqrt[]{{\beta }^{\mathrm{2}}-n_{\mathrm{2}}^{\mathrm{2}}{k}_{\mathrm{0}}^{\mathrm{2}}}\end{array}\right)$（1）

where $h$ and $p$ are the parameters defining the guided mode field in the literature on optical waveguides ^{［11， 12］}， $V$ is the product of three elements， $\sqrt[]{n_{\mathrm{1}}^{\mathrm{2}}-n_{\mathrm{2}}^{\mathrm{2}}}$ is the numerical aperture of a symmetric planar waveguide， $d$ is half width of the waveguide core， leading to the other dimensionless parameters as

$\left(\Delta ,x,\gamma \right)=\left(\begin{array}{ccc}(n_{\mathrm{1}}^{\mathrm{2}}-n_{\mathrm{2}}^{\mathrm{2}})/(\mathrm{2}{n}_{\mathrm{1}}^{\mathrm{2}}),& p(L_{\mathrm{c}}/d)p(L_{\mathrm{c}}/d),& (n_{\mathrm{2}}V)/(n_{\mathrm{1}}p\sqrt[]{\Delta })\end{array}\right)$（2）

Consequently， for both the TE-like and TM-like modes， the SWR induced optical scattering loss is related to both the y- and z-components of roughness， so the dependences of the optical intensity loss coefficients ${\alpha }_{\mathrm{3}\mathrm{D}}(\mathrm{T}\mathrm{E}/\mathrm{T}\mathrm{M})$on the 3-D SWR ${\sigma }_{\mathrm{3}\mathrm{D}}(\mathrm{T}\mathrm{E}/\mathrm{T}\mathrm{M})$ can be expressed as ^［3］

${\alpha }_{\mathrm{3}\mathrm{D}}(\mathrm{T}\mathrm{E}/\mathrm{T}\mathrm{M})=\mathrm{4.34}\left(\frac{{\sigma }_{\mathrm{3}\mathrm{D}}^{\mathrm{2}}(\mathrm{T}\mathrm{E}/\mathrm{T}\mathrm{M})}{\sqrt[]{\mathrm{2}}{d}^{\mathrm{4}}{\beta }_{\mathrm{T}\mathrm{E}/\mathrm{T}\mathrm{M}}}g(V)\cdot f_e(x,\gamma )\right)$（3）

where ${\alpha }_{\mathrm{3}\mathrm{D}}(\mathrm{T}\mathrm{E}/\mathrm{T}\mathrm{M})$ is the SWR defined in the 2-D form and the loss coefficient is in dB/cm， the functions $g(V)$ is defined as

$g(V)=\frac{h^{\mathrm{2}}{V}^{\mathrm{2}}}{\mathrm{1}+p^{\mathrm{2}}}$（4）

and $f_{\mathrm{e}}(x,\gamma )$ is defined as

$f_{\mathrm{e}}(x,\gamma )=\frac{\{[(\mathrm{1}+x^{\mathrm{2}}{)}^{\mathrm{2}}+\mathrm{2}{x}^{\mathrm{2}}{\gamma }^{\mathrm{2}}{]}^{\mathrm{1}/\mathrm{2}}+\mathrm{1}-x^{\mathrm{2}}{\}}^{\mathrm{1}/\mathrm{2}}}{[(\mathrm{1}+x^{\mathrm{2}}{)}^{\mathrm{2}}+\mathrm{2}{x}^{\mathrm{2}}{\gamma }^{\mathrm{2}}{]}^{\mathrm{1}/\mathrm{2}}}$（5）

where transferring functions， $f_{\mathrm{e}}(x,\gamma )$ is used to determine the correlation length $L_{\mathrm{c}}$ with the parameters $x$ and $\gamma$ at the peak value.

The numerical simulation for the function $f_{\mathrm{e}}(x,\gamma )$ is shown in Fig. 2， where the peak position indicates the maximum autocorrelation value. So， the corresponding values of $x$ and $\gamma$ play the role of making $f_{\mathrm{e}}(x,\gamma )$ have the maximum autocorrelation effect， and then the corresponding $L_{\mathrm{c}}$ value calculated from Eq. （2） is the correlation length value of this waveguide.

With Eq. （4）， the numerical simulation results of two key variables $x$ and $\gamma$ of the function are obtained as shown in Figs. 3（a） and 3（b）， respectively. From Fig. 3（a）， we find that the variable $x$ is strongly dependent on the refractive index of the SOI waveguide core， but it is weakly dependent of the rib width. Illustratively， all the five curves are very close to one another， namely， when the core refractive index is changed from 3.43 to 3.48， almost all the five $x$ values are decreased from 1.691 3 to 1.688 5. For the simulation results of variable $\gamma$ are almost the same as the case of $x$. For instance， $\gamma$ is also weakly dependent of the rib width， so all the five curves are very close to one another， namely， when the core refractive index is changed from 3.43 to 3.48， almost all the five $\gamma$ values are decreased from 0.673 to 0.662. The comparison between Figs. 3（a） and 3（b） also shows that corresponding to the effective index change， the relative change of $x$ is only 0.17%， while the relative change of $\gamma$ reaches 1.63%. In summary， $\gamma$ has the much stronger dependence on the refractive index of SOI waveguide core than $x$， but they are almost independent of the rib width.

With the simulation results of $x$ and $\gamma$ shown in Figs. 3（a） and 3（b）， respectively， with Eq. （4）， the plot of function $f_{\mathrm{e}}(x,\gamma )$ versus variable $x$ for each waveguide structure is carried out and then the value of the correlation length $L_{\mathrm{c}}$ of this waveguide structure is obtained. Further， for the possible waveguide structures that can create single-mode operations， the corresponding distributions of $L_{\mathrm{c}}$ are obtained with respect to five values of channel width $W_{\mathrm{r}}$ as shown in Fig. 4.

Note that the dependences of $L_{\mathrm{c}}$ on the two waveguide parameters： the rib width and the core refractive index are inverse to the cases of the two variables $x$ and $\gamma$. Namely， $L_{\mathrm{c}}$ has no big change with the core refractive index， but all the five lines of $L_{\mathrm{c}}$ values are dramatically different from one another， namely， it strongly dependent on the rib width of the SOI waveguide. Eqs. （1） ~ （3） show that the OPL coefficient ${\alpha }_{\mathrm{3}\mathrm{D}}$ is related to both $L_{\mathrm{c}}$ and $W_{\mathrm{r}}$ （i.e.， 2$d$）， so these two parameters of waveguide： $n_{\mathrm{1}}$ and $\mathrm{2}d$ have great impacts upon the SWR-caused OPL.

With the fabricated device sample， the reconstructed images of CLSM measurements for an SOI waveguide sidewall roughness could further be obtained. During measuring the SWR distributions， the chip of SOI waveguides is rotated by a quasi-90° angle and then measured the sidewall profile of the waveguide with CLSM. As a result， the average horizontal and the average vertical values of SWR are 22 nm and 23 nm， respectively. As defined in our previous work， the SWR values at both TE- and TM-mode are measured. In this definition of roughness， the $\mathrm{P}{\mathrm{S}}_{\mathrm{z}}$， which is given by the CLSM measurement in unit of μm， indicates the average roughness value in the scanned range of CLSM measurement and it can be taken as a one-dimensional （1-D） roughness ${\sigma }_{(\mathrm{1}a)}$， then the 2-D average roughness ${\sigma }_{\mathrm{2}\mathrm{D}}(\mathrm{2}a)$ and the 3-D average roughness ${\sigma }_{\mathrm{3}\mathrm{D}}(\mathrm{3}a)$ are defined as ^［3］

${\sigma }_{\mathrm{i}\mathrm{s}\mathrm{o}}(\mathrm{2}D)=(\mathrm{1}/m){\displaystyle \sum _{j=\mathrm{1}}^m}\mathrm{P}{\mathrm{S}}_z$（6）

${\sigma }_{\mathrm{a}\mathrm{n}\mathrm{i}}(\mathrm{3}D)=(\mathrm{1}/n){\displaystyle \sum _{j=\mathrm{1}}^n}{\sigma }_{\mathrm{i}\mathrm{s}\mathrm{o}}(\mathrm{2}D)$（7）

For the waveguide structure of fabricated device sample， with the theoretical model defined by Eqs. （1）~（7）， the dependences of OPL coefficient on the SWR for both TE- and TM-mode are simulated as shown in Fig. 5. Note that the OPL coefficient caused by the SWR quickly increases with SWR and the TE-mode always bigger than the TM-mode. For instance， at the above measured values， 22 nm and 23 nm for the horizontal and vertical distributions， the OPL values for both the TE- and TM-mode are 5.0 dB/cm. So， a high-quality SOI waveguide， the SWR should be controlled lower than 20 nm. In addition， the difference of the OPL values between the TE- and TM-mode also quickly increases.

In experiments for verifying the mean OPL value， the two ends of the waveguide chip are polished to form a Fabry-Perot （F-P） cavity. Then， a response of the optical resonance output to the temperature change is obtained as shown in Fig. 6. Since the F-P response curve to temperature change contains two wave periods， so， as labelled in Fig. 6， two periods： Period-1 and Period-2 are used to calculate the OPL values with the popular format as defined by Eq. （8）^［13］.

$\alpha =\frac{\mathrm{4.34}}{l_{\mathrm{w}\mathrm{g}}}\mathrm{l}\mathrm{n}\left[\frac{\mathrm{1}-\sqrt[]{T_{\mathrm{m}\mathrm{i}\mathrm{n}}/T_{\mathrm{m}\mathrm{a}\mathrm{x}}}}{\mathrm{1}+\sqrt[]{T_{\mathrm{m}\mathrm{i}\mathrm{n}}/T_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\cdot {\left(\frac{N_{\mathrm{e}\mathrm{f}\mathrm{f}}+\mathrm{1}}{N_{\mathrm{e}\mathrm{f}\mathrm{f}}-\mathrm{1}}\right)}^{\mathrm{2}}\right]$（8）

where l_wg represents the length of the waveguide ^［11］， and T_max and T_min represent the maximum and minimum values of transmission of resonance curve. Consequently， the OPL values are obtained to be 4.218 4 and 4.312 5 dB/cm for the two periods， leading to an average OPL value of 4.265 dB/cm. This result agrees with the calculated values in Fig. 5 for either TE-mode or TM-mode. Nevertheless， compared with the measured values of the similar scaled SOI waveguides in literature published by VLASOV V A et al. in 2002 and BOEUF F et al. in 2016 ^{［14， 15］}， the measured results obtained in this article are higher than the high-quality SOI waveguides should have. So， the fabrication technique for the samples used in this work still needs to be improved.

As a conclusion， through self-consistent modeling and numerical simulations of the two key factors in the guided mode of an SOI waveguide： the correlation length and the SWR， the ideal simulation results are obtained. Based on the research on the surface roughness of SOI waveguides， we can introduce the 3-D distribution of SWR and the interaction effect between the optical field of guided mode and the SWR to create accurate theoretical model for defining the optical loss coefficient dependence on the waveguide structure and SWR.