= Solution
The raised-index region has $n_1=1.01(1.50)=1.515$, whereas $n_2=1.50$. Therefore its <numerical aperture> factor is $\sqrt{n_1^2-n_2^2}=0.212662$. With the printed rounded cutoff,
$$
a_c=\frac{2.4(1.5\,\mu\mathrm m)}{2\pi(0.212662)}=2.6942\,\mu\mathrm m,
\qquad
\log_{10}s_c=\frac{20-2.6942}{6.25}=2.76892.
$$
Thus
$$
\boxed{s_c\simeq5.87\times10^2\,\mathrm{mm\,s^{-1}}.}
$$
Slower scanning increases the written radius and makes the guide multimode; faster scanning makes it single-spatial-mode in this idealization, provided a physically meaningful positive core radius remains. Using $j_{0,1}=2.404825\ldots$ gives $s_c\simeq586.2\,\mathrm{mm\,s^{-1}}$. This numerical threshold extrapolates the supplied empirical radius law; its experimental range is not given.
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