1 ~\&~ |y| \leq 1 & \mathrm{~for~square~waveguide}
\end{cases}
\\
0 & \mathrm{elsewhere}
\end{cases}.$$ $V_0$ is the height of the potential and determines the number of linear waveguide modes.

Considering the specific case of silica-made waveguides, typical values for the parameters are $n_b=1.4$, $|n-n_b| \leq 9 \times 10^{-3}$, $n_2 = 3\times 10^{-16} ~\mathrm{cm}^2/\mathrm{W}$ at a vacuum wavelength of $\lambda_0=790$ nm, and those values do not vary much when we choose any $\lambda_0$ in the range from visible to near-infrared. The laser-induced-damage-threshold (LIDT) for synthetic fused silica is about $10 ~\mathrm{J}/\mathrm{cm}^2$ for nanosecond pulses (according to Eq. (1) and Table 2 in Ref. [@kuzuu_ao_1999]) corresponding to $\sim 10^{10} ~\mathrm{W}/\mathrm{cm}^2$, up to which the model should be valid. For shorter, femtosecond pulses this damage threshold is $1000$ times higher, but since we consider “CW” beams the lower value for nanosecond pulses is relevant. It is justified to neglect temporal effects on beam propagation, because the dispersion length is already of the order of kilometers for pulse durations of a few tens of picoseconds, much longer than propagation distances considered in this work. Throughout this paper, we choose $V_0=1000$, and find that $r_0 \approx 25.0 ~\mu \mathrm{ m}$ through $r_0=\sqrt{\left|n_bV/\left[2k_0^2(n-n_b)\right]\right|}$, which nicely corresponds to 