**1. Introduction**

Higher and higher integration density is the long-standing goal of the optical integrated circuits in the modern optical communications systems [1,2]. Since the beginning of the 21st century, the silicon-on-insulator (SOI) waveguide-based photonic integrated circuits (PIC) have shown unprecedented potential. Therefore, as the main source of optical propagation loss (OPL) of waveguides, the surface roughness-caused optical scattering loss has been attracting much research, and several theoretical models for defining the relations between the optical propagation losses and the sidewall roughness (SWR) of a waveguide are proposed. Illustratively, the average optical loss of SOI strip waveguides is around 0.24 dB/mm [3–5], while that of the small area square SOI waveguides is around 1.3 dB/mm [6]. Thus, it is of paramount importance to be able to quantitatively predict the optical loss on the magnitude and distribution property of waveguide SWR so that we can realize the expected performance specifications of PIC devices.

In the study of the mechanisms of the optical propagation loss caused by waveguide SWR, Marcuse proposed the earliest theory in 1969 in which the optical power ratio at the nonuniform

boundary was considered [7]. In 1994, Payne and Lacey proposed a theoretical model for defining the relation between the optical scattering loss and sidewall roughness of a waveguide based on a combination of the spectral density of the nonuniform edge of the waveguide, and the autocorrelation of roughness caused wave scattering, so it was then commonly accepted and referred to as the Payne-Lacey (PL) model [8]. However, both the Marcuse model and PL model have a similar shortcoming that the light wave at the waveguide boundary is not specified, and the waveguide channel is simplified to be a two-dimensional (2-D) structure, where the SWR is assumed to have a uniform distribution. Since the beginning of the 21st century, there has been a strong attempt to extend the PL model into the three-dimensional (3-D) structures and consider the other elements causing the optical loss of waveguide in addition to the SWR [9–12]. In 2005, Barwicz and Haus 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 the more detailed simulations of the OPL values of both the low and high index-contrast waveguides [9]. In 2006, Poulton et al. gave a more powerful explanation as that the OPL caused by the SWR was 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 [10]. Especially, for the SOI waveguides, in 2008, Schmid et al. employed the non-uniform waveguide boundaries to study the scattering loss [11], and, in 2009, Yap et al. first led the optical scattering loss of SOI waveguides to the synchronous dependences the SWR and the channel width [12]. Until 2019, we first published the measurement metrology of the waveguide SWR with a confocal laser scanning microscopy (CLSM) technique [13].

In this article, we theoretically investigated the SWR property of dielectric waveguides by analyzing its components in the horizontal and vertical direction. Then, we measured the 3-D distribution of waveguide SWR with the CLSM technique and constructed a theoretical/experimental ensemble model for defining the SWR-caused optical propagation loss. This model was an extended version of the PL model with the 3-D distribution of SWR. As a result, with such a combinative model, we simulated the dependences of the optical propagation loss coefficient of the waveguide on the SWR distribution and the width. Finally, we analyzed the theoretical and experimental results.
