## 4 Discussion  
Recently, our group has demonstrated that the form birefringence related to the so-called nanogratings is quantitatively correlated to the porosity-filling factor of these nanostructures [27]. Control of the porosity was achieved by adjusting the laser pulse energy and the number of pulses/micron; i.e., the overlapping rate [28]. To further explore the reason why thermal stability is improved when irradiated with higher pulse energy in 3D printed silica fiber, the Rayleigh-Plesset (R-P) equation was used to simulate the erasure of pores in the nanostructure during the annealing process, and taking into account the same conditions for a direct comparison between experiment and theory. The equation describes the evolution of a spherical bubble inside an incompressible Newtonian fluid, and it can be expressed as follows:  
\[\frac{\Delta\mathrm{P}}{\rho}=\mathrm{R_{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{  
taken in a unit volume of porous nanolayer (V), so it is obviously related to the number of nanopores (N) with an averaged nanopore radius (R\({}_{\text{pore}}\)) and can be calculated using:  
\[\text{FF}=\text{N}\frac{4\pi\text{R}_{\text{pore}}^{3}}{3\text{V}} \tag{3}\]  
The birefringence B can be determined by the difference between the refractive indices of the ordinary axis (n\({}_{\text{o}}\)) and the extraordinary axis (n\({}_{\text{e}}\)). The following equation illustrates the detail:  
\[\text{B}=\text{n}_{\text{o}}-\text{n}_{\text{e}}=\sqrt{\left[1-\frac{\delta}{ \Lambda}\right]\text{n}_{\text{G}}^{2}+\frac{\delta}{\Lambda}\text{n}_{\text{ pl}}^{2}}-\left[\sqrt{\frac{\left[1-\frac{\delta}{\Lambda}\right]}{\text{n}_{ \text{G}}^{2}}+\frac{\frac{\delta}{\Lambda}}{\text{n}_{\text{pl}}^{2}}} \right]^{-1} \tag{4}\]  
n\({}_{\text{G}}\) represents the refractive index between the porous nanolayers [28], \(\Lambda\) is the average spacing between nanolayers, \(\delta\) is the porous nanolayer thickness, and (\(\Lambda\)-\(\delta\)) is the interlayer thickness. Finally, with the birefringence B calculated, the retardance R can be worked out by the expression R = B \(\times\) L as we defined above. More details about the simulation are provided in our recent work [16]. The conditions used in the simulation were set as: the surface energy (S) = 0.3 J/m\({}^{-2}\); the viscosity n(T) fitted using a VTF (Vogel-Tamman-Fulcher) law for binary GeO\({}_{2}\)-SiO\({}_{2}\) glass with (GeO\({}_{2}\))-10 mol % [30]. Measured annealing curves of the 3D silica printed fiber (pulse energies of 0.6 uJ and 0.4 uJ), and the predicted ones calculated based on the R-P, are shown in Figure 7. We suppose the total birefringence of the nanogratings is composed by the contributions additive and independent. Then, the measured data are normalized relative to the sharp retardance decrease at the highest annealing temperature values that we hypothesized here to be characteristic of the nanopore erasure. In this paper only the final erasure step of the measured retardance R(\(\Delta\)t,T) is investigated. Following the same experimental conditions as in the experiment, the R-P equation was used to simulate the nanopore erasure, and the evolution of the retardance was calculated based the nanopore diameter, the filling factor FF and using the form birefringence equations (Equations (2)-(4)) shown above. Figure 7: Evolution of the normalized retardance during step isochronal annealing experiments for 3D printed silica fiber in various pulse energies (taken from Figure 2b and re-normalized relative to the sharply decreased retardance value) compared with the Rayleigh–Plesset (R-P) equation used to simulate the nanopores’ erasure. As can be seen from experimental and simulation results in Figure 7, two pairs of curves are separated in the figure. The evolution in pulse energy corresponds to the increase in pore size, and each pair is consistent. Improved thermal stability can be observed as pulse energy or pore size increases.