**Appendix: Focal length of thermal lens effect**

If part of the core layer is heated by light absorption, the refractive-index gradient in the core is induced by the thermal coefficient of the refractive index *∂n*/*∂T* of the silica glass and the temperature distribution ∆*T* in the core layer. The refractive index *n* of the heated core is expressed as follows:

$$m(r,t) = n\_0 + \frac{\partial n}{\partial T} \Delta T(r,t) \tag{35}$$

where *n*<sup>0</sup> (= 1.46) is the characteristic *n* value of the core layer, *r* is the radial distance from the center of the optical fiber, and *t* is time, respectively.

If we assume that optical power in the optical fiber takes on Gaussian distribution, ∆*T* in the core layer is given by [63], [101]

$$
\Delta T(r,t) = \frac{aP}{4\pi\lambda} \left[ \ln\left(1 + \frac{8Dt}{\omega\_0^2}\right) - \frac{16Dt}{\omega\_0^2 + 8Dt} \frac{r^2}{\omega\_0^2} \right] \tag{36}
$$

where *α* is the absorption coefficient, *P* is the incident optical power, *ω*<sup>0</sup> is the spot size radius of the laser beam, and *λ* is the thermal conductivity of the silica glass, respectively. Parameter *D* = *λ*/*Cpρ*, where *Cp* and *ρ* are the specific heat and dnsity of the silica glass.

Substitution of Eq. (36) into Eq. (35) gives

$$n \cong n\_0 \left[ 1 + \delta \left( \frac{r}{\omega\_0} \right)^2 \right],\tag{37}$$

$$\delta = -\frac{2\alpha P}{4n\_0 \pi \lambda} \left(\frac{\partial n}{\partial T}\right) \frac{8Dt}{\omega\_0^2 + 8Dt}. \tag{38}$$

When refractive index of the core layer takes a radial distribution as shown in Eq. (37), propagating laser beam will be focussed as a result of the thermal lens effect. In this case, the focal length *F* is given by [63], [101]

$$F(t) = \frac{n\_0 \pi \lambda \omega\_0^2 (\omega\_0^2 + 8Dt)}{\alpha P l (\partial n/\partial T)(8Dt)} = F\_\infty \left[1 + \frac{t\_c}{2t}\right],\tag{39}$$

$$F\_{\infty} = \frac{n\_0 \pi \lambda \omega\_0^2}{aPl(\partial n/\partial T)},\tag{40}$$

$$t\_c = \frac{\omega\_0^2}{4D} \tag{41}$$

where *l* is the length of the heating core, where *α* exhibits large value. In typical SM fiber, *λ* = 9.2 W/mK, *Cp* = 788 J/kgK, *ρ* = 2,200 kg/m<sup>3</sup> [5], and *ω*<sup>0</sup> ∼ 4.5 *µ*m. If we insert these values into Eqs. (39)–(41), we obtain *tc* <sup>∼</sup> 0.95 *<sup>µ</sup>*s. This means that *<sup>F</sup>* <sup>∼</sup><sup>=</sup> *<sup>F</sup>*<sup>∞</sup> when *<sup>t</sup>* is 10 *<sup>µ</sup>*<sup>s</sup> or above. If we assume *P* = 2W, *l* ∼ 40 *µ*m, *α* = 5.6 × 10<sup>4</sup> m−<sup>1</sup> [5], and *∂n*/*∂T* = 1.23 × 10−<sup>5</sup> K−<sup>1</sup> [5], *F* is given by
