Abstract

Step-index fibers (SIFs) with alumina cores were fabricated employing the powder-in-tube technique. The fabricated SIFs have alumina concentrations of up to 32 mol%, which is the highest value reported so far for fibers with core diameters smaller than 25 μm. The mixing mechanisms between alumina and silica during fiber drawing were revealed by energy dispersive X-ray analysis of the neck-down area of the preform. The results of the measurements and simulations indicate that besides diffusion, fluid dynamics between softened silica and alumina powder also play an important role in the resulting alumina and silica concentrations in the fiber. The influence of different drawing parameters on the alumina and silica concentrations of the fibers is also presented.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The production and application of optical fibers with unconventional core compositions has been of high interest in the field of optics and optical materials [1]. To date, fibers with core materials like silicon [25], germanium [6], chromium [7] or alumina [8,9] have been fabricated successfully. These fibers possess special optical and electrical properties, such as producing an electrical signal when light is incident on the fiber facet [10] or when the ambient temperature changes [11], monitoring their own performance and structural integrity [12], detecting sound or ultrasonic signals [13], and suppressing nonlinear effects [8]. Therefore, they can be used in some emerging areas like optoelectronics, mid-infrared power delivery, and non-linear optics [14,15]. Among these specialty optical fibers, alumina core fibers show great potential to suppress or even eliminate stimulated Brillouin scattering (SBS) in high-power narrow-linewidth fiber laser systems [16,17], since the photo-elastic constant of alumina is negative [18].

At present, there are three approaches which are mainly used for the fabrication of such fibers: the molten-core technique [19], high-pressure chemical vapor deposition [20], and the powder-in-tube technique (PIT) [21]. Based the on molten-core technique, alumina core fibers were first fabricated by Dragic et al. [8], which have inserted a sapphire rod into a silica tube. The Brillouin gain coefficient of the fabricated fibers is 100 times lower than that of standard silica-based fibers. However, the alumina concentration of the fabricated fiber decreases rapidly with diminishing core diameter.

In the present work, instead of using a sapphire rod, fibers with alumina cores were fabricated employing the powder-in-tube technique. In the corresponding literature regarding PIT, a change of the material concentration is evident which is mainly attributed to diffusion [7,21,22], but cannot be explained solely by this effect. Therefore, we present a comprehensive analysis of the whole drawing process to give details on the involved mechanisms. For this purpose, the neck-down area of the preform itself and the drawn fibers were analyzed using energy dispersive X-ray spectrometry (EDX). In addition, the preform was drawn using different drawing parameters, including temperature, feeding speed and fiber diameter.

2. Experimental methodology

For the assembly of the preform, a suspension of ethanol and alumina powder, with a particle size of 50 to 60 μm and a purity of 99.99%, was poured into a fused silica tube. This first tube with inner and outer diameters of 3.7 and 25.5 mm, respectively, was later inserted into another fused silica tube with inner and outer diameters of 26.0 and 33.0 mm, respectively. Finally, the assembled preform was dried in an oxygen atmosphere at 900 °C for 4 hours to remove residual ethanol.

Fibers were drawn using the drawing tower available at the IFSW with different drawing temperatures, feeding speeds and cladding diameters, while applying a slight underpressure to remove air from the preform. The targeted nominal cladding diameters of the fibers were 105, 130, 174, and 218 µm, with corresponding nominal core diameters of 12, 15, 20 and 25 µm, respectively. EDX analysis was performed by using a JEOL JSM-6490LV scanning electron microscope. To provide a conductive layer to mitigate any charging effects, all fiber and preform samples were prepared in the same manner by sputtering a thin gold layer onto them using a Hummer JR sputtering machine. Since the EDX analysis can only detect elements itself, instead of measuring the mass concentrations of alumina and silica, the mass concentrations of aluminum and silicon were determined. In addition, refractive index difference (RID) measurements of fiber pieces were also performed by using an IFA-100 optical fiber analyzer to verify the results of the EDX analysis.

3. Results and analysis

3.1 Characterization of the PIT-based step-index fibers

Figure 1 shows the measured aluminum and silicon mass concentrations of fibers with different nominal core diameters which were drawn at a temperature of 2050 °C and at a feeding speed of 0.365 mm/min. The inset in Fig. 1 shows the microscope image of the fiber facet, which was observed by coupling white light into the fiber with a nominal core diameter of 25 µm, and shows a good symmetry and circularity of the core of the fiber. However, the measured maximum aluminum mass concentration of all fibers is only around 50%, which cannot be explained solely by diffusion like stated in the corresponding literature so far on the powder-in-tube process [7,21,22]. In addition, the resulting core diameters, which were measured to be 20.4, 20.2, 26.4 and 30.4 µm, using the second-moment definition [23], are larger than the nominal values of 12, 15, 20 and 25 µm, respectively. These results indicate that during the fiber drawing, alumina migrated from the core to the cladding and silica migrated from the cladding to the core. It should be noted that within the measurement accuracy, the resulting core diameters of the fibers with nominal core diameters of 12 and 15 μm are the same. With the aluminum and silicon mass concentrations, the alumina mole fraction $m{f_{\textrm{A}{\textrm{l}_2}{\textrm{O}_3}}}$ between alumina (Al2O3) and silica (SiO2) is calculated by

$$m{f_{\textrm{A}{\textrm{l}_\textrm{2}}{\textrm{O}_\textrm{3}}}} = {{\left( {\frac{{{m_{\textrm{Al}}}}}{{{w_{\textrm{Al}}}}} \cdot \frac{1}{2}} \right)} / {\left( {\frac{{{m_{\textrm{Al}}}}}{{{w_{\textrm{Al}}}}} \cdot \frac{1}{2} + \frac{{{m_{\textrm{Si}}}}}{{{w_{\textrm{Si}}}}}} \right)}}, $$
where ${m_{\textrm{Al}}}$ is the mass concentration of aluminum, ${w_{\textrm{Al}}}$ is the atomic weight of aluminum, ${m_{\textrm{Si}}}$ is the mass concentration of silicon, and ${w_{\textrm{Si}}}$ is the atomic weight of silicon. The silica mole fraction $m{f_{\textrm{Si}{\textrm{O}_2}}}$ is determined by $1 - m{f_{\textrm{A}{\textrm{l}_2}{\textrm{O}_3}}}$.

 

Fig. 1. Aluminum and silicon mass concentrations of the SIFs with different core diameters which were drawn at a temperature of 2050 °C and at a feeding speed of 0.365 mm/min. (Inset: microscope image of the core area of fiber with a nominal core diameter of 25 µm)

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Therefore, the difference Δ between the refractive index of the core and the cladding of the fiber is calculated by

$$\Delta = m{f_{\textrm{A}{\textrm{l}_\textrm{2}}{\textrm{O}_\textrm{3}}}} \cdot {n_{\textrm{A}{\textrm{l}_\textrm{2}}{\textrm{O}_\textrm{3}}}} + m{f_{\textrm{Si}{\textrm{O}_\textrm{2}}}} \cdot {n_{\textrm{Si}{\textrm{O}_\textrm{2}}}} - {n_{\textrm{Si}{\textrm{O}_\textrm{2}}}}, $$
where ${n_{\textrm{A}{\textrm{l}_2}{\textrm{O}_3}}}$ and ${n_{\textrm{Si}{\textrm{O}_2}}}$ are the refractive indices of alumina and silica, respectively, at a certain wavelength [24].

Figure 2 shows the alumina mole fraction and RID profiles of the fiber with a nominal core diameter of 25 μm. The alumina mole fraction $m{f_{\textrm{A}{\textrm{l}_2}{\textrm{O}_3}}}$ was calculated according to Eq. (1) with the aluminum and silicon mass concentrations obtained from the EDX analysis. The measured refractive index difference ${\mathrm{\Delta }_{\textrm{meas}}}$ was obtained from the IFA-100 optical fiber analyzer measured at a wavelength of 960 nm. Using Eq. (2), the RID profile was calculated and fitted (${\mathrm{\Delta }_{\textrm{fitted}}}$) to the directly measured RID profile by minimizing the value of α which is defined as

$$\alpha = \sum {|{{\Delta _{\textrm{meas}}} - {\Delta _{\textrm{fitted}}}} |}$$
to find the value of ${n_{\textrm{A}{\textrm{l}_2}{\textrm{O}_3}}}$ (fit parameter) while the refractive index of silica ${n_{\textrm{Si}{\textrm{O}_2}}}$ was fixed to 1.4509 [25]. For the data shown in Fig. 2 the SIF with a nominal core diameter of 25 µm, the fit results in a refractive index of alumina ${n_{\textrm{A}{\textrm{l}_2}{\textrm{O}_3}}}$ of 1.6843, which is within the range of previously reported values [8,2628]. Figure 2 also shows that the profile of the fitted RID is in good agreement with the directly measured RID. The measured and calculated data of the SIFs with different core diameters are summarized in Table 1.

 

Fig. 2. RID profile measured with the optical fiber analyzer, calculated alumina mole fraction and fitted RID profile from the EDX analysis of the SIF with a nominal core diameter of 25 μm which was drawn at a temperature of 2050 °C and at a feeding speed of 0.365 mm/min.

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Tables Icon

Table 1. Measured and calculated data of the SIFs drawn at a temperature of 2050 °C and at a feeding speed of 0.365 mm/min

From Table 1 it can be seen that the refractive indices of alumina obtained by applying the above fitting procedure to different fibers range between 1.6843 and 1.7274, which is consistent with the previously reported values [8,2628]. Table 1 also shows that for all fibers the maximum mole fractions of alumina are around 32 mol% while the contrast of the refractive index is about 0.08. These values are significantly higher than the ones of fibers made by the traditional MCVD technique which exhibit alumina mole fractions of 6-8 mol% [29,30]. Also using the molten-core technique for fibers with similar core diameters, alumina mole fractions of only up to 26.9 mol% were demonstrated, which resulted in a maximum RID of 0.0596 [8]. To the best of our knowledge, we have obtained the highest alumina mole fraction and refractive index difference in alumina-core SIFs for core diameters ≤ 25 μm.

3.2 Mixing mechanism between alumina and silica during fiber drawing

In order to analyze the mechanisms which are involved in the reduction of the mole fraction of alumina in the area of the fiber’s core from initially 100 to around 32 mol%, numerical simulations were performed. First, the temperature distribution of the preform during fiber drawing was simulated using the finite element method (FEM) implemented in COMSOL Multiphysics. The temperature of the heating-element (graphite) inside the furnace of the drawing tower was set to 2050 °C for the simulation, while the ambient temperature outside of the furnace was set to 25 °C. A laminar air flow around the furnace and the preform was considered. It should be noted that the neck-down area of the preform was simplified to a circular cone in the simulation.

Figures 3(a) and 3(b) show the simulated temperature distribution of the preform during fiber drawing. As seen from Fig. 3(b), the maximum temperature of the preform is 2027 °C which is lower than the actual temperature of the heating element. Since the melting point of the alumina powder is within the range of 2000 to 2030 °C [31], the alumina powder first maintains the powder form when the temperature of the preform is lower than its melting point. The softening point of fused silica lays within the range of 1500 to 1670 °C [32], which is far below the melting point of the alumina powder. This is why a mixture of softened silica and solid alumina powder occurs in the area of the preform where the temperature is lower than the melting point of alumina. As the diameter of the preform is getting smaller in the neck-down area, the temperature of the preform increases. When the temperature has reached the melting point of the alumina powder, the powder melts and diffusion between the alumina and silica occurs.

 

Fig. 3. FEM simulation results of the temperature distribution and the photograph of the preform. (a) Simulation model, (b) temperature distribution in the center of the preform in axial direction, and (c) photograph of the preform after fiber drawing.

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The material concentrations in the neck-down area of the preforms were analyzed, to investigate the mixing mechanism between alumina and silica during the drawing of the fiber in more detail. After the fiber preform was drawn at a temperature of 2050 °C and at a feeding speed of 0.365 mm/min, the preforms’ neck-down area was cut into slices, which were polished, sputtered, and analyzed using EDX. The exact axial position of each slice in the preform could not be determined due to the needed polishing process. Therefore, the diameters of the cladding and the core in the slices were measured as listed in Table 2.

Tables Icon

Table 2. Slices of the neck-down area of the preform

Figure 4 shows the results of the EDX mapping of topmost slice 1. It can be seen that the whole area of the core of slice 1 contains aluminum and silicon, which indicates that the composition of the core has already became a mixture of alumina and silica. In order to have a deeper insight on the mixing mechanisms a FEM model was implemented in COMSOL Multiphysics to simulate the flow of fused silica within the porous alumina powder. The distribution of the porous alumina powder used in this model was taken from Fig. 4. The pressure difference between the inside of the preform and the ambient air was set to -35 mbar, as used in the experiment in order to remove air from the preform. Based on the simulation results shown in Fig. 3, where the temperature of slice 1 was determined to be 1992 °C, the viscosity of fused silica was set to 100 Pa·s in our simulation according to [33].

 

Fig. 4. Mapping result of slice 1 using EDX with a magnification of 30, high-voltage of 25 kV, and working distance of 20.0 mm (the black line indicates the EDX line scan position, the yellow rectangle indicates the area without polishing defects).

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Figure 5 shows the results of the simulation for the flow of fused silica within the porous alumina powder. Considering the mean velocity of $4 \times {10^{ - 6}}$ m/s, the time it takes for the fused silica to flow from the core-cladding boundary to the center of the core is found to approximately be 7 minutes, which indicates that the applied underpressure of -35 mbar plays an important role in the mixing process of alumina and silica. However, this moderate underpressure does not evacuate the preform completely. Air bubbles, which were trapped inside the powder, therefore expand during the drawing of the fiber and cause micro-explosions. These explosions lead to a further mixing of the softened silica and the alumina powder. In this study, we refer to this whole mixing process as the fluid dynamics. It should be noted that the air bubbles shown in Fig. 3(c) are mainly located at the interface between the core and the cladding. We assume that these bubbles are formed during the cooling of the preform, caused by a mismatch between the expansion coefficients of the core and cladding materials.

 

Fig. 5. Results of the simulation of the velocity of fused silica flowing within the porous alumina powder.

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Figure 6(a) shows the EDX line scans of the mass concentrations of aluminum for all slices. All scanning lines cross the center of the fiber’s core area (see black line in Fig. 4). The maximum aluminum concentration of slice 1 is not at 100%, which indicates that the alumina powder has already melted and mixed with silica. The standard deviation σ of the mass concentrations of aluminum was calculated to further analyze the distribution of aluminum in the core area of different slices. It should be noted that to avoid the influence of polishing defects only the areas without polishing defects (for instance, the yellow rectangular area in Fig. 4) were used for this analysis. Figure 6(b) summarizes the standard deviations σ of the mass concentrations of aluminum for all the slices. It can be seen that the standard deviation diminishes with decreasing diameter of the preform which means that the distribution of the mass concentration of aluminum is flattened along the process (see Fig. 6(a)).

 

Fig. 6. Results of the EDX analysis of the slices. (a) Line scans of all slices and (b) standard deviations of the mass concentrations of aluminum in the core area of the slices without polishing defects.

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In order to analyze the flattening of the distribution of the mass concentrations of aluminum in the core area, the diffusion process was calculated according to the diffusion theory using Fick’s law and the Arrhenius equation [34]. It was assumed that diffusion only occurs in radial direction because of the radial symmetry of the fiber’s refractive index profiles. The aluminum concentration of slice 2 was used in the calculation since slice 1 had major polishing defects.

Figure 7 shows the calculated change of the distribution of the mass concentrations of the aluminum in slice 2 for different diffusion times. The calculation shows that the profile of the mass concentration of aluminum is flattened as the diffusion time increases. Hence, the smoothed distribution of the mass concentrations of aluminum in the core area can be attributed to diffusion.

 

Fig. 7. Calculated mass concentrations of the aluminum in slice 2 with a diffusion time T of (a) 0 s (originally measured profile) (b) 30 s (c) 300 s.

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In summary, during the drawing of SIFs with alumina cores which were fabricated using the PIT technique, the alumina powder and the surrounding fused silica first mix due to the above-mentioned fluid dynamics, which causes the evident drop of the concentration of alumina in the core. This effect is apparent when the softened silica mixes with the still solid alumina powder and diffusion only starts to occur when the alumina powder begins to melt. Therefore, the alumina distribution at the top of the neck-down area (see slice 1) shows a strong fluctuation over the core area and flattens during drawing. Finally, in this part of the preform with a flat distribution of the alumina concentration, diffusion continues until the fiber exits the furnace and the temperature drops rapidly.

3.3 Influence of the drawing parameters on the alumina concentration

To show the influence of the drawing parameters on the concentration of alumina in the core, the preform was drawn with different temperatures, feeding rates and fiber diameters. Then, the individual fiber pieces were analyzed using EDX.

Figure 8 shows the mass concentrations of aluminum in the cores with a nominal diameter of 12 and 25 µm. The fibers were drawn at a high (2200 °C) and low (2050 °C) temperature. It can be seen that the maximum mass concentrations of aluminum in the cores, which were drawn at 2200 °C, are always lower than that of the fibers, which were drawn at 2050 °C. This is explained by two reasons: first, the alumina powder has melted earlier due to the higher temperature and therefore the time in which diffusion between alumina and silica occurs was longer; second, according to the Arrhenius equation [34], the higher temperature increases the diffusion coefficients, leading to a higher diffusion velocity. Hence, a longer diffusion time and a larger diffusion coefficient caused a stronger diffusion between alumina and silica. It should be noted that more diffusion leads to larger diameters of the core according to the diffusion theory. However, the core diameters of the fibers, which were drawn at a temperature of 2200 °C, seem to be smaller than the ones of the fibers that were drawn at a temperature of 2050 °C. Still, the difference between the core diameters of fibers, which were drawn at different temperatures, is small and within the measurement accuracy.

 

Fig. 8. Influence of the temperature on the mass concentrations of aluminum in the cores with diameters of (a) 12 μm and (b) 25 μm. The fibers were drawn with a feeding rate of 0.365 mm/min.

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Figure 9 shows the mass concentrations of aluminum in the cores with different diameters. Taking the accuracy of the EDX device and the measurement itself into account, the maximum mass concentration of aluminum was the same regardless of the diameters of the core. However, as the diameter of the fiber is reduced, the profile of the aluminum concentration becomes more Gaussian. To explain this phenomenon, the diffusion process in the fibers with different core diameters was calculated according to the diffusion theory. From the results shown in Section 3.2 we know that alumina powder and silica mixes with each other in the neck-down area of the preform, leading to a ratio between alumina and silica in the core of the fiber around 1:1. Therefore, in order to consider the mixing of alumina and silica initiated by the fluid dynamics described above, the initial profile of the aluminum mass concentration was simplified by a top-hat shape assuming a maximum mass concentration of aluminum of 50% in the simulation. In addition, the diffusion coefficients and diffusion times used in the calculation were the same for the different fibers.

Figure 10 shows the calculated effect of the diffusion process in the SIFs with different core diameters. It can be seen that due to diffusion the profiles change from top-hat to Gaussian, which is more dominant for smaller core diameters. According to the calculation, the maximum mass concentration of aluminum in the center of the core starts to decrease when the diameter of the core is smaller than a certain value. Here, the diffusion length of silica is longer than the radius of the core of the fiber therefore, the core area is unpurified. In our experiments, however, the maximum of the mass concentrations of the aluminum in the core was almost the same in all investigated SIFs, indicating that the diameters of the cores of all the investigated SIFs were still higher than this critical value. It should be noted that fibers with smaller diameters were drawn with higher drawing speeds and a fixed preform feeding speed. Higher drawing speeds lead to higher drawing tensions and inward forces (from cladding to core). These accelerate the diffusion process, leading to an increase of the diffusion coefficient. Thus, fibers with smaller diameters experience more diffusion than indicated in the simulation.

 

Fig. 9. Influence of the core diameter of the fiber on the mass concentrations of aluminum in the cores of the fibers which were drawn with feeding rates of (a) 0.365 mm/min, (b) 0.550 mm/min and (c) 0.730 mm/min, at a temperature of 2050 °C.

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Fig. 10. Calculated effect of the diffusion in the SIFs with different core diameters with a given diffusion coefficient and diffusion time.

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Based on the experimental data, which is depicted in Fig. 9, the influence of the feeding rate of the preform on the alumina concentrations in the core was analyzed. Figure 11 shows the maximum values of the mass concentrations of aluminum in the SIFs with nominal core diameters of 15, 20, and 25 μm, which were drawn with different preform feeding rates of 0.365, 0.550, and 0.730 mm/min. The difference (df) between the maximum mass concentrations of aluminum in the SIFs with a nominal core diameter of 25 μm, which were drawn at different feeding rates, is 1.8%. For the SIFs with nominal core diameters of 15 and 20 μm, the differences between maximum mass concentrations of aluminum are 6.6 and 5.5%, respectively, which is still within the measurement uncertainty of the EDX device. Thus the influence of the preforms’ feeding speed on the concentrations of alumina and silica is minor. However, higher preform feeding speeds lead to a reduced diffusion time and hence there should be less diffusion.

 

Fig. 11. Influence of the preforms’ feeding rate on the maximum mass concentrations of aluminum in the SIFs with different nominal core diameters. The fibers were drawn at a temperature of 2050 °C.

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To analyze this in more detail, the neck-down area of the preform, which was drawn at a temperature of 2050 °C and at a feeding speed of 0.730 mm/min, was also cut into slices. After polishing, the slices with cladding diameters of 6.97 and 0.30 mm, which correspond to core diameters of 0.8 and 0.034 mm, respectively, were chosen and compared with slices 5 and 6 as stated in Table 2.

Figure 12 shows the EDX scans across the slices with similar core diameters, which were drawn at feeding speeds of 0.365 and 0.730 mm/min. It can be seen that the profiles of the aluminum concentration does not change significantly with different feeding speeds as the differences are within the measurement uncertainty. This indicates that the influence of the feeding rate on the material concentration in the preforms’ neck-down area is minor. We explain this by the fact that the diameter of the preform in the neck-down area is relatively large compared to the fiber itself. Thus, even though the lower feeding rate leads to a longer diffusion time, the absolute diffusion is comparable. When the final fiber diameter is reached the temperature will drop rapidly since the fiber is pulled out of the furnace rather fast ($\ge $ 8 m/min in our experiments). Thus, the influence of the feeding speed on the diffusion process in the fiber part is also negligible.

 

Fig. 12. EDX measurement performed on the preform slices which were drawn at different feeding speeds with core diameters of (a) 0.79 and 0.80 mm, (b) 0.030 and 0.034 mm.

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As a next step, the research will focus on the improvement of the optical performance of the fabricated fibers. Although the inset in Fig. 1 indicates that the fibers have a good circularity and symmetry, at present the transmission losses of the fibers are relatively high and were measured to be around 5 dB/m at a wavelength of 1050 nm. Dragic et al. [8] have shown transmission losses of ca. 0.4 dB/m at a wavelength of around 1 µm by using a single-crystalline sapphire rod as core material in the preform. Their losses were limited by the purity of the used sapphire rod and are roughly one order of magnitude lower than our results. We also suspect that the purity of 99.99% of our alumina powder and the nature of the powder itself, which is much more susceptible to foreign particles, are mainly responsible for the relatively high losses shown in this work. However, further studies will be carried out to explain and address this issue. Additionally, air bubbles have appeared every few tens of meters in the produced fibers. Once the bubbles appear, the fibers cannot propagate light anymore. This shows that further improvement of the involved processes is needed.

4. Conclusion

In summary, step-index fibers with alumina cores were fabricated using the powder-in-tube technique. The fabricated fibers have alumina concentrations of up to 32 mol% which is the highest value reported so far for fibers with core diameters ≤ 25 μm. Energy dispersive X-ray analysis of the neck-down area of preform indicates that other than diffusion also fluid dynamics between the softened silica and the alumina powder play an important role determining the resulting concentrations of alumina and silica. Regarding the influence of the drawing parameters on the concentrations of alumina and silica in the SIFs, high temperatures and small core diameters both lead to a lower alumina concentration in the core area. However, the influence of the feeding speed is negligible. Further research will be devoted to improve the quality of the core and hence the optical performance of the fibers.

Funding

Bundesministerium für Bildung und Forschung (FKZ 13N12767); Deutsche Forschungsgemeinschaft (GSC 262/2); China Scholarship Council (201806840052).

Acknowledgments

The authors would like to thank Katja Schweiger for sputtering the samples at the Institut für Photovoltaik (IPV), University of Stuttgart, Liane Hoster and Johannes Wahl for polishing and measuring the samples at the IFSW, University of Stuttgart, and Laboratoire de Physique des Lasers Atomes et Moécules (PhLAM), IRCICA Research Institute for conducting the refractive index difference characterization. Lingqiang Meng would like to thank the China Scholarship Council (CSC) for supporting his scientific research in Germany.

Disclosures

The authors declare no conflicts of interest.

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18. T. Davis and K. Vedam, “Photoelastic properties of sapphire (α-Al2O3),” J. Appl. Phys. 38(11), 4555–4556 (1967). [CrossRef]  

19. S. Morris and J. Ballato, “Molten core fabrication of novel optical fibers,” Bull. Am. Ceram. Soc. 92, 24–29 (2013).

20. N. F. Baril, R. He, T. D. Day, J. R. Sparks, B. Keshavarzi, M. Krishnamurthi, A. Borhan, V. Gopalan, A. C. Peacock, N. Healy, P. J. A. Sazio, and J. V. Badding, “Confined high-pressure chemical deposition of hydrogenated amorphous silicon,” J. Am. Chem. Soc. 134(1), 19–22 (2012). [CrossRef]  

21. J. Auguste, G. Humbert, S. Leparmentier, M. Kudinova, P. Martin, G. Delaizir, K. Schuster, and D. Litzkendorf, “Modified Powder-in-Tube Technique based on the consolidation processing of powder materials for fabricating specialty optical fibers,” Materials 7(8), 6045–6063 (2014). [CrossRef]  

22. G. Granger, C. Röhrer, G. Kleem, C. Kizler, A. Voß, T. Graf, and M. Abdou-Ahmed, “First fabrication of optical Bragg fibers by a single-step powder in tube process based on the SiO2-Al2O3 system,” presented at Specialty Fiber Processing and Advances in Fabrication I, SPIE Photonics Europe, Brussels, Belgium, 4-7 April, 2016.

23. V. S. Pugachev, Probability Theory and Mathematical Statistics for Engineers (Oxford University, 1984).

24. P. Brocos, A. Pinerio, R. Bravo, and A. Amigo, “Refractive indices, molar volumes and molar refractions of binary liquid mixtures: concepts and correlations,” Phys. Chem. Chem. Phys. 5(3), 550–557 (2003). [CrossRef]  

25. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55(10), 1205–1208 (1965). [CrossRef]  

26. I. H. Malitson and M. J. Dodge, “Refractive-index and birefringence of synthetic sapphire,” J. Opt. Soc. Am. 52(12), 1377 (1962). [CrossRef]  

27. S. Unger, A. Schwuchow, J. Dellith, and J. Kirchhof, “Codoped materials for high power fiber lasers-diffusion behavior and optical properties,” Proc. SPIE 6469, 646913 (2007). [CrossRef]  

28. J. Kirchhof, S. Unger, B. Knappe, and J. Kobelke, “Interaction of alumina with other codopants in the preparation of active fiber materials,” Materials Research Society Fall Meeting, Boston, MA/USA, paper Za034 (1994).

29. K. Nassau, J. W. Shiever, and J. T. Krause, “Preparation and properties of fuses silica containing alumina,” J. Am. Ceram. Soc. 58, 461 (1975). [CrossRef]  

30. J. R. Simpson and J. B. Macchesney, “Optical fibers with an Al2O3-doped silicate core composition,” Electron. Lett. 19(7), 261–262 (1983). [CrossRef]  

31. R. F. Geller and P. J. Yavorsky, “Melting point of alpha-alumina,” J. Res. Natl Bur. Stand. 34(4), 395–401 (1945). [CrossRef]  

32. H. L. Watson, “Some properties of fused quartz and other forms of silicon-dioxide,” J. Am. Ceram. Soc. 9, 511–534 (1926). [CrossRef]  

33. A. Massaro, Photonic crystals: introduction, applications and theory, 1st ed. (InTech Press, 2012

34. K. Lyytikäinen, S. T. Huntington, A. L. G. Carter, P. McNamara, S. Fleming, J. Abramczyk, I. Kaplin, and G. Schötz, “Dopant diffusion during optical fibre drawing,” Opt. Express 12(6), 972–977 (2004). [CrossRef]  

References

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  1. K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).
    [Crossref]
  2. F. A. Martinsen, B. K. Smeltzer, M. Nord, T. Hawkins, J. Ballato, and U. J. Gibson, “Silicon-core glass fibres as microwire radial-junction solar cells,” Sci. Rep. 4(1), 6283 (2015).
    [Crossref]
  3. J. Ballato, T. Hawkins, P. Foy, R. Stolen, B. Kokuoz, M. Ellison, C. McMillen, J. Reppert, A. M. Rao, M. Daw, S. Sharma, R. Shori, O. Stafsudd, R. R. Rice, and D. R. Powers, “Silicon optical fiber,” Opt. Express 16(23), 18675–18683 (2008).
    [Crossref]
  4. C. Hou, X. Jia, L. Wei, S. Tan, X. Zhao, J. D. Joannopoulos, and Y. Fink, “Crystalline silicon core fibres from aluminium core preforms,” Nat. Commun. 6(1), 6248 (2015).
    [Crossref]
  5. N. Healy, M. Fokine, Y. Franz, T. Hawkins, M. Jones, J. Ballato, A. C. Peacock, and U. J. Gibson, “CO2 laser-induced directional recrystallization to produce single crystal silicon-core optical fibers with low loss,” Adv. Opt. Mater. 4(7), 1004–1008 (2016).
    [Crossref]
  6. J. Ballato, T. Hawkins, P. Foy, B. Y. Kokuz, R. Stolen, C. McMillen, N. K. Hon, B. Jalali, and R. Rice, “Glass-clad single-crystal germanium optical fiber,” Opt. Express 17(10), 8029–8035 (2009).
    [Crossref]
  7. Y. Huang, J. Wang, K. Chu, T. Lin, W. Wang, T. Chou, S. Yeh, S. Huang, and W. Cheng, “Fabrication and characteristic of Cr-doped fibers employing Powder-in-Tube technique,” in Optical Fiber Communication Conference/ National Fiber Optic Engineers Conference 2011, OSA Technical Digest(CD), paper OWS1. Optical Society of America, New York, 2011.
  8. P. Dragic, T. Hawkins, P. Foy, S. Morris, and J. Ballato, “Sapphire-derived all-glass optical fibres,” Nat. Photonics 6(9), 627–633 (2012).
    [Crossref]
  9. P. Dragic, M. Cavillon, and J. Ballato, “The linear and nonlinear refractive index of amorphous Al2O3 deduced from aluminosilicate optical fibers,” Int. J. Appl. Glass Sci. 9, 421–427 (2018).
    [Crossref]
  10. A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Lare-scale optical field measurements with geometric fibre constructs,” Nat. Mater. 5(7), 532–536 (2006).
    [Crossref]
  11. M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-sensing fiber devices by multimaterial codrawing,” Adv. Mater. 18, 845–849 (2006).
    [Crossref]
  12. M. Bayindir, O. Shapira, D. S. Hinczewski, J. Viens, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Integrated fibres for self-monitored optical transport,” Nat. Mater. 4(11), 820–825 (2005).
    [Crossref]
  13. S. Egusa, Z. Wang, N. Chocat, Z. M. Ruff, A. M. Stolyrov, D. Shemuly, F. Sorin, P. T. Rakich, J. D. Joannopoulos, and Y. Fink, “Multimaterial piezoelectric fibres,” Nat. Mater. 9(8), 643–648 (2010).
    [Crossref]
  14. G. Tao and A. F. Abouraddy, “Multimaterial fibers,” Int. Int. J. Appl. Glass Sci. 3, 349–368 (2012).
    [Crossref]
  15. J. Ballato and P. Dragic, “Rethinking optical fiber: new demands, old glasses,” J. Am. Ceram. Soc. 96(9), 2675–2692 (2013).
    [Crossref]
  16. M. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007).
    [Crossref]
  17. L. G. Nielsen, S. Dasgupta, M. D. Mermelstein, D. Jakobsen, S. Herstrom, M. E. V. Pedersen, E. L. Lim, S. Alam, F. Parmigiani, D. Richardson, and B. Palsdottir, “A silica based highly nonlinear fibre with improved threshold for stimulated Brillouin scattering,” Proc. Euro. Conf. Opt. Commun. Paper Tu.4.D.3 (2010).
  18. T. Davis and K. Vedam, “Photoelastic properties of sapphire (α-Al2O3),” J. Appl. Phys. 38(11), 4555–4556 (1967).
    [Crossref]
  19. S. Morris and J. Ballato, “Molten core fabrication of novel optical fibers,” Bull. Am. Ceram. Soc. 92, 24–29 (2013).
  20. N. F. Baril, R. He, T. D. Day, J. R. Sparks, B. Keshavarzi, M. Krishnamurthi, A. Borhan, V. Gopalan, A. C. Peacock, N. Healy, P. J. A. Sazio, and J. V. Badding, “Confined high-pressure chemical deposition of hydrogenated amorphous silicon,” J. Am. Chem. Soc. 134(1), 19–22 (2012).
    [Crossref]
  21. J. Auguste, G. Humbert, S. Leparmentier, M. Kudinova, P. Martin, G. Delaizir, K. Schuster, and D. Litzkendorf, “Modified Powder-in-Tube Technique based on the consolidation processing of powder materials for fabricating specialty optical fibers,” Materials 7(8), 6045–6063 (2014).
    [Crossref]
  22. G. Granger, C. Röhrer, G. Kleem, C. Kizler, A. Voß, T. Graf, and M. Abdou-Ahmed, “First fabrication of optical Bragg fibers by a single-step powder in tube process based on the SiO2-Al2O3 system,” presented at Specialty Fiber Processing and Advances in Fabrication I, SPIE Photonics Europe, Brussels, Belgium, 4-7 April, 2016.
  23. V. S. Pugachev, Probability Theory and Mathematical Statistics for Engineers (Oxford University, 1984).
  24. P. Brocos, A. Pinerio, R. Bravo, and A. Amigo, “Refractive indices, molar volumes and molar refractions of binary liquid mixtures: concepts and correlations,” Phys. Chem. Chem. Phys. 5(3), 550–557 (2003).
    [Crossref]
  25. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55(10), 1205–1208 (1965).
    [Crossref]
  26. I. H. Malitson and M. J. Dodge, “Refractive-index and birefringence of synthetic sapphire,” J. Opt. Soc. Am. 52(12), 1377 (1962).
    [Crossref]
  27. S. Unger, A. Schwuchow, J. Dellith, and J. Kirchhof, “Codoped materials for high power fiber lasers-diffusion behavior and optical properties,” Proc. SPIE 6469, 646913 (2007).
    [Crossref]
  28. J. Kirchhof, S. Unger, B. Knappe, and J. Kobelke, “Interaction of alumina with other codopants in the preparation of active fiber materials,” Materials Research Society Fall Meeting, Boston, MA/USA, paper Za034 (1994).
  29. K. Nassau, J. W. Shiever, and J. T. Krause, “Preparation and properties of fuses silica containing alumina,” J. Am. Ceram. Soc. 58, 461 (1975).
    [Crossref]
  30. J. R. Simpson and J. B. Macchesney, “Optical fibers with an Al2O3-doped silicate core composition,” Electron. Lett. 19(7), 261–262 (1983).
    [Crossref]
  31. R. F. Geller and P. J. Yavorsky, “Melting point of alpha-alumina,” J. Res. Natl Bur. Stand. 34(4), 395–401 (1945).
    [Crossref]
  32. H. L. Watson, “Some properties of fused quartz and other forms of silicon-dioxide,” J. Am. Ceram. Soc. 9, 511–534 (1926).
    [Crossref]
  33. A. Massaro, Photonic crystals: introduction, applications and theory, 1st ed. (InTech Press, 2012
  34. K. Lyytikäinen, S. T. Huntington, A. L. G. Carter, P. McNamara, S. Fleming, J. Abramczyk, I. Kaplin, and G. Schötz, “Dopant diffusion during optical fibre drawing,” Opt. Express 12(6), 972–977 (2004).
    [Crossref]

2018 (1)

P. Dragic, M. Cavillon, and J. Ballato, “The linear and nonlinear refractive index of amorphous Al2O3 deduced from aluminosilicate optical fibers,” Int. J. Appl. Glass Sci. 9, 421–427 (2018).
[Crossref]

2016 (1)

N. Healy, M. Fokine, Y. Franz, T. Hawkins, M. Jones, J. Ballato, A. C. Peacock, and U. J. Gibson, “CO2 laser-induced directional recrystallization to produce single crystal silicon-core optical fibers with low loss,” Adv. Opt. Mater. 4(7), 1004–1008 (2016).
[Crossref]

2015 (2)

F. A. Martinsen, B. K. Smeltzer, M. Nord, T. Hawkins, J. Ballato, and U. J. Gibson, “Silicon-core glass fibres as microwire radial-junction solar cells,” Sci. Rep. 4(1), 6283 (2015).
[Crossref]

C. Hou, X. Jia, L. Wei, S. Tan, X. Zhao, J. D. Joannopoulos, and Y. Fink, “Crystalline silicon core fibres from aluminium core preforms,” Nat. Commun. 6(1), 6248 (2015).
[Crossref]

2014 (2)

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).
[Crossref]

J. Auguste, G. Humbert, S. Leparmentier, M. Kudinova, P. Martin, G. Delaizir, K. Schuster, and D. Litzkendorf, “Modified Powder-in-Tube Technique based on the consolidation processing of powder materials for fabricating specialty optical fibers,” Materials 7(8), 6045–6063 (2014).
[Crossref]

2013 (2)

J. Ballato and P. Dragic, “Rethinking optical fiber: new demands, old glasses,” J. Am. Ceram. Soc. 96(9), 2675–2692 (2013).
[Crossref]

S. Morris and J. Ballato, “Molten core fabrication of novel optical fibers,” Bull. Am. Ceram. Soc. 92, 24–29 (2013).

2012 (3)

N. F. Baril, R. He, T. D. Day, J. R. Sparks, B. Keshavarzi, M. Krishnamurthi, A. Borhan, V. Gopalan, A. C. Peacock, N. Healy, P. J. A. Sazio, and J. V. Badding, “Confined high-pressure chemical deposition of hydrogenated amorphous silicon,” J. Am. Chem. Soc. 134(1), 19–22 (2012).
[Crossref]

P. Dragic, T. Hawkins, P. Foy, S. Morris, and J. Ballato, “Sapphire-derived all-glass optical fibres,” Nat. Photonics 6(9), 627–633 (2012).
[Crossref]

G. Tao and A. F. Abouraddy, “Multimaterial fibers,” Int. Int. J. Appl. Glass Sci. 3, 349–368 (2012).
[Crossref]

2010 (1)

S. Egusa, Z. Wang, N. Chocat, Z. M. Ruff, A. M. Stolyrov, D. Shemuly, F. Sorin, P. T. Rakich, J. D. Joannopoulos, and Y. Fink, “Multimaterial piezoelectric fibres,” Nat. Mater. 9(8), 643–648 (2010).
[Crossref]

2009 (1)

2008 (1)

2007 (2)

M. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007).
[Crossref]

S. Unger, A. Schwuchow, J. Dellith, and J. Kirchhof, “Codoped materials for high power fiber lasers-diffusion behavior and optical properties,” Proc. SPIE 6469, 646913 (2007).
[Crossref]

2006 (2)

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Lare-scale optical field measurements with geometric fibre constructs,” Nat. Mater. 5(7), 532–536 (2006).
[Crossref]

M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-sensing fiber devices by multimaterial codrawing,” Adv. Mater. 18, 845–849 (2006).
[Crossref]

2005 (1)

M. Bayindir, O. Shapira, D. S. Hinczewski, J. Viens, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Integrated fibres for self-monitored optical transport,” Nat. Mater. 4(11), 820–825 (2005).
[Crossref]

2004 (1)

2003 (1)

P. Brocos, A. Pinerio, R. Bravo, and A. Amigo, “Refractive indices, molar volumes and molar refractions of binary liquid mixtures: concepts and correlations,” Phys. Chem. Chem. Phys. 5(3), 550–557 (2003).
[Crossref]

1983 (1)

J. R. Simpson and J. B. Macchesney, “Optical fibers with an Al2O3-doped silicate core composition,” Electron. Lett. 19(7), 261–262 (1983).
[Crossref]

1975 (1)

K. Nassau, J. W. Shiever, and J. T. Krause, “Preparation and properties of fuses silica containing alumina,” J. Am. Ceram. Soc. 58, 461 (1975).
[Crossref]

1967 (1)

T. Davis and K. Vedam, “Photoelastic properties of sapphire (α-Al2O3),” J. Appl. Phys. 38(11), 4555–4556 (1967).
[Crossref]

1965 (1)

1962 (1)

1945 (1)

R. F. Geller and P. J. Yavorsky, “Melting point of alpha-alumina,” J. Res. Natl Bur. Stand. 34(4), 395–401 (1945).
[Crossref]

1926 (1)

H. L. Watson, “Some properties of fused quartz and other forms of silicon-dioxide,” J. Am. Ceram. Soc. 9, 511–534 (1926).
[Crossref]

Abdou-Ahmed, M.

G. Granger, C. Röhrer, G. Kleem, C. Kizler, A. Voß, T. Graf, and M. Abdou-Ahmed, “First fabrication of optical Bragg fibers by a single-step powder in tube process based on the SiO2-Al2O3 system,” presented at Specialty Fiber Processing and Advances in Fabrication I, SPIE Photonics Europe, Brussels, Belgium, 4-7 April, 2016.

Abouraddy, A. F.

G. Tao and A. F. Abouraddy, “Multimaterial fibers,” Int. Int. J. Appl. Glass Sci. 3, 349–368 (2012).
[Crossref]

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Lare-scale optical field measurements with geometric fibre constructs,” Nat. Mater. 5(7), 532–536 (2006).
[Crossref]

M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-sensing fiber devices by multimaterial codrawing,” Adv. Mater. 18, 845–849 (2006).
[Crossref]

M. Bayindir, O. Shapira, D. S. Hinczewski, J. Viens, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Integrated fibres for self-monitored optical transport,” Nat. Mater. 4(11), 820–825 (2005).
[Crossref]

Abramczyk, J.

Aichele, C.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).
[Crossref]

Alam, S.

L. G. Nielsen, S. Dasgupta, M. D. Mermelstein, D. Jakobsen, S. Herstrom, M. E. V. Pedersen, E. L. Lim, S. Alam, F. Parmigiani, D. Richardson, and B. Palsdottir, “A silica based highly nonlinear fibre with improved threshold for stimulated Brillouin scattering,” Proc. Euro. Conf. Opt. Commun. Paper Tu.4.D.3 (2010).

Amigo, A.

P. Brocos, A. Pinerio, R. Bravo, and A. Amigo, “Refractive indices, molar volumes and molar refractions of binary liquid mixtures: concepts and correlations,” Phys. Chem. Chem. Phys. 5(3), 550–557 (2003).
[Crossref]

Arnold, J.

M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-sensing fiber devices by multimaterial codrawing,” Adv. Mater. 18, 845–849 (2006).
[Crossref]

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Lare-scale optical field measurements with geometric fibre constructs,” Nat. Mater. 5(7), 532–536 (2006).
[Crossref]

Auguste, J.

J. Auguste, G. Humbert, S. Leparmentier, M. Kudinova, P. Martin, G. Delaizir, K. Schuster, and D. Litzkendorf, “Modified Powder-in-Tube Technique based on the consolidation processing of powder materials for fabricating specialty optical fibers,” Materials 7(8), 6045–6063 (2014).
[Crossref]

Badding, J. V.

N. F. Baril, R. He, T. D. Day, J. R. Sparks, B. Keshavarzi, M. Krishnamurthi, A. Borhan, V. Gopalan, A. C. Peacock, N. Healy, P. J. A. Sazio, and J. V. Badding, “Confined high-pressure chemical deposition of hydrogenated amorphous silicon,” J. Am. Chem. Soc. 134(1), 19–22 (2012).
[Crossref]

Ballato, J.

P. Dragic, M. Cavillon, and J. Ballato, “The linear and nonlinear refractive index of amorphous Al2O3 deduced from aluminosilicate optical fibers,” Int. J. Appl. Glass Sci. 9, 421–427 (2018).
[Crossref]

N. Healy, M. Fokine, Y. Franz, T. Hawkins, M. Jones, J. Ballato, A. C. Peacock, and U. J. Gibson, “CO2 laser-induced directional recrystallization to produce single crystal silicon-core optical fibers with low loss,” Adv. Opt. Mater. 4(7), 1004–1008 (2016).
[Crossref]

F. A. Martinsen, B. K. Smeltzer, M. Nord, T. Hawkins, J. Ballato, and U. J. Gibson, “Silicon-core glass fibres as microwire radial-junction solar cells,” Sci. Rep. 4(1), 6283 (2015).
[Crossref]

J. Ballato and P. Dragic, “Rethinking optical fiber: new demands, old glasses,” J. Am. Ceram. Soc. 96(9), 2675–2692 (2013).
[Crossref]

S. Morris and J. Ballato, “Molten core fabrication of novel optical fibers,” Bull. Am. Ceram. Soc. 92, 24–29 (2013).

P. Dragic, T. Hawkins, P. Foy, S. Morris, and J. Ballato, “Sapphire-derived all-glass optical fibres,” Nat. Photonics 6(9), 627–633 (2012).
[Crossref]

J. Ballato, T. Hawkins, P. Foy, B. Y. Kokuz, R. Stolen, C. McMillen, N. K. Hon, B. Jalali, and R. Rice, “Glass-clad single-crystal germanium optical fiber,” Opt. Express 17(10), 8029–8035 (2009).
[Crossref]

J. Ballato, T. Hawkins, P. Foy, R. Stolen, B. Kokuoz, M. Ellison, C. McMillen, J. Reppert, A. M. Rao, M. Daw, S. Sharma, R. Shori, O. Stafsudd, R. R. Rice, and D. R. Powers, “Silicon optical fiber,” Opt. Express 16(23), 18675–18683 (2008).
[Crossref]

Baril, N. F.

N. F. Baril, R. He, T. D. Day, J. R. Sparks, B. Keshavarzi, M. Krishnamurthi, A. Borhan, V. Gopalan, A. C. Peacock, N. Healy, P. J. A. Sazio, and J. V. Badding, “Confined high-pressure chemical deposition of hydrogenated amorphous silicon,” J. Am. Chem. Soc. 134(1), 19–22 (2012).
[Crossref]

Bartelt, H.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).
[Crossref]

Bayindir, M.

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Lare-scale optical field measurements with geometric fibre constructs,” Nat. Mater. 5(7), 532–536 (2006).
[Crossref]

M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-sensing fiber devices by multimaterial codrawing,” Adv. Mater. 18, 845–849 (2006).
[Crossref]

M. Bayindir, O. Shapira, D. S. Hinczewski, J. Viens, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Integrated fibres for self-monitored optical transport,” Nat. Mater. 4(11), 820–825 (2005).
[Crossref]

Bierlich, J.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).
[Crossref]

Borhan, A.

N. F. Baril, R. He, T. D. Day, J. R. Sparks, B. Keshavarzi, M. Krishnamurthi, A. Borhan, V. Gopalan, A. C. Peacock, N. Healy, P. J. A. Sazio, and J. V. Badding, “Confined high-pressure chemical deposition of hydrogenated amorphous silicon,” J. Am. Chem. Soc. 134(1), 19–22 (2012).
[Crossref]

Bravo, R.

P. Brocos, A. Pinerio, R. Bravo, and A. Amigo, “Refractive indices, molar volumes and molar refractions of binary liquid mixtures: concepts and correlations,” Phys. Chem. Chem. Phys. 5(3), 550–557 (2003).
[Crossref]

Brocos, P.

P. Brocos, A. Pinerio, R. Bravo, and A. Amigo, “Refractive indices, molar volumes and molar refractions of binary liquid mixtures: concepts and correlations,” Phys. Chem. Chem. Phys. 5(3), 550–557 (2003).
[Crossref]

Carter, A. L. G.

Cavillon, M.

P. Dragic, M. Cavillon, and J. Ballato, “The linear and nonlinear refractive index of amorphous Al2O3 deduced from aluminosilicate optical fibers,” Int. J. Appl. Glass Sci. 9, 421–427 (2018).
[Crossref]

Chen, X.

Cheng, W.

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Rice, R. R.

Richardson, D.

L. G. Nielsen, S. Dasgupta, M. D. Mermelstein, D. Jakobsen, S. Herstrom, M. E. V. Pedersen, E. L. Lim, S. Alam, F. Parmigiani, D. Richardson, and B. Palsdottir, “A silica based highly nonlinear fibre with improved threshold for stimulated Brillouin scattering,” Proc. Euro. Conf. Opt. Commun. Paper Tu.4.D.3 (2010).

Röhrer, C.

G. Granger, C. Röhrer, G. Kleem, C. Kizler, A. Voß, T. Graf, and M. Abdou-Ahmed, “First fabrication of optical Bragg fibers by a single-step powder in tube process based on the SiO2-Al2O3 system,” presented at Specialty Fiber Processing and Advances in Fabrication I, SPIE Photonics Europe, Brussels, Belgium, 4-7 April, 2016.

Ruff, Z. M.

S. Egusa, Z. Wang, N. Chocat, Z. M. Ruff, A. M. Stolyrov, D. Shemuly, F. Sorin, P. T. Rakich, J. D. Joannopoulos, and Y. Fink, “Multimaterial piezoelectric fibres,” Nat. Mater. 9(8), 643–648 (2010).
[Crossref]

Ruffin, A. B.

Sazio, P. J. A.

N. F. Baril, R. He, T. D. Day, J. R. Sparks, B. Keshavarzi, M. Krishnamurthi, A. Borhan, V. Gopalan, A. C. Peacock, N. Healy, P. J. A. Sazio, and J. V. Badding, “Confined high-pressure chemical deposition of hydrogenated amorphous silicon,” J. Am. Chem. Soc. 134(1), 19–22 (2012).
[Crossref]

Schötz, G.

Schuster, K.

J. Auguste, G. Humbert, S. Leparmentier, M. Kudinova, P. Martin, G. Delaizir, K. Schuster, and D. Litzkendorf, “Modified Powder-in-Tube Technique based on the consolidation processing of powder materials for fabricating specialty optical fibers,” Materials 7(8), 6045–6063 (2014).
[Crossref]

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).
[Crossref]

Schwuchow, A.

S. Unger, A. Schwuchow, J. Dellith, and J. Kirchhof, “Codoped materials for high power fiber lasers-diffusion behavior and optical properties,” Proc. SPIE 6469, 646913 (2007).
[Crossref]

Shapira, O.

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Lare-scale optical field measurements with geometric fibre constructs,” Nat. Mater. 5(7), 532–536 (2006).
[Crossref]

M. Bayindir, O. Shapira, D. S. Hinczewski, J. Viens, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Integrated fibres for self-monitored optical transport,” Nat. Mater. 4(11), 820–825 (2005).
[Crossref]

Sharma, S.

Shemuly, D.

S. Egusa, Z. Wang, N. Chocat, Z. M. Ruff, A. M. Stolyrov, D. Shemuly, F. Sorin, P. T. Rakich, J. D. Joannopoulos, and Y. Fink, “Multimaterial piezoelectric fibres,” Nat. Mater. 9(8), 643–648 (2010).
[Crossref]

Shiever, J. W.

K. Nassau, J. W. Shiever, and J. T. Krause, “Preparation and properties of fuses silica containing alumina,” J. Am. Ceram. Soc. 58, 461 (1975).
[Crossref]

Shori, R.

Simpson, J. R.

J. R. Simpson and J. B. Macchesney, “Optical fibers with an Al2O3-doped silicate core composition,” Electron. Lett. 19(7), 261–262 (1983).
[Crossref]

Smeltzer, B. K.

F. A. Martinsen, B. K. Smeltzer, M. Nord, T. Hawkins, J. Ballato, and U. J. Gibson, “Silicon-core glass fibres as microwire radial-junction solar cells,” Sci. Rep. 4(1), 6283 (2015).
[Crossref]

Sorin, F.

S. Egusa, Z. Wang, N. Chocat, Z. M. Ruff, A. M. Stolyrov, D. Shemuly, F. Sorin, P. T. Rakich, J. D. Joannopoulos, and Y. Fink, “Multimaterial piezoelectric fibres,” Nat. Mater. 9(8), 643–648 (2010).
[Crossref]

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Lare-scale optical field measurements with geometric fibre constructs,” Nat. Mater. 5(7), 532–536 (2006).
[Crossref]

Sparks, J. R.

N. F. Baril, R. He, T. D. Day, J. R. Sparks, B. Keshavarzi, M. Krishnamurthi, A. Borhan, V. Gopalan, A. C. Peacock, N. Healy, P. J. A. Sazio, and J. V. Badding, “Confined high-pressure chemical deposition of hydrogenated amorphous silicon,” J. Am. Chem. Soc. 134(1), 19–22 (2012).
[Crossref]

Stafsudd, O.

Stolen, R.

Stolyrov, A. M.

S. Egusa, Z. Wang, N. Chocat, Z. M. Ruff, A. M. Stolyrov, D. Shemuly, F. Sorin, P. T. Rakich, J. D. Joannopoulos, and Y. Fink, “Multimaterial piezoelectric fibres,” Nat. Mater. 9(8), 643–648 (2010).
[Crossref]

Tan, S.

C. Hou, X. Jia, L. Wei, S. Tan, X. Zhao, J. D. Joannopoulos, and Y. Fink, “Crystalline silicon core fibres from aluminium core preforms,” Nat. Commun. 6(1), 6248 (2015).
[Crossref]

Tao, G.

G. Tao and A. F. Abouraddy, “Multimaterial fibers,” Int. Int. J. Appl. Glass Sci. 3, 349–368 (2012).
[Crossref]

Unger, S.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).
[Crossref]

S. Unger, A. Schwuchow, J. Dellith, and J. Kirchhof, “Codoped materials for high power fiber lasers-diffusion behavior and optical properties,” Proc. SPIE 6469, 646913 (2007).
[Crossref]

J. Kirchhof, S. Unger, B. Knappe, and J. Kobelke, “Interaction of alumina with other codopants in the preparation of active fiber materials,” Materials Research Society Fall Meeting, Boston, MA/USA, paper Za034 (1994).

Vedam, K.

T. Davis and K. Vedam, “Photoelastic properties of sapphire (α-Al2O3),” J. Appl. Phys. 38(11), 4555–4556 (1967).
[Crossref]

Viens, J.

M. Bayindir, O. Shapira, D. S. Hinczewski, J. Viens, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Integrated fibres for self-monitored optical transport,” Nat. Mater. 4(11), 820–825 (2005).
[Crossref]

Voß, A.

G. Granger, C. Röhrer, G. Kleem, C. Kizler, A. Voß, T. Graf, and M. Abdou-Ahmed, “First fabrication of optical Bragg fibers by a single-step powder in tube process based on the SiO2-Al2O3 system,” presented at Specialty Fiber Processing and Advances in Fabrication I, SPIE Photonics Europe, Brussels, Belgium, 4-7 April, 2016.

Walton, D. T.

Wang, J.

M. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007).
[Crossref]

Y. Huang, J. Wang, K. Chu, T. Lin, W. Wang, T. Chou, S. Yeh, S. Huang, and W. Cheng, “Fabrication and characteristic of Cr-doped fibers employing Powder-in-Tube technique,” in Optical Fiber Communication Conference/ National Fiber Optic Engineers Conference 2011, OSA Technical Digest(CD), paper OWS1. Optical Society of America, New York, 2011.

Wang, W.

Y. Huang, J. Wang, K. Chu, T. Lin, W. Wang, T. Chou, S. Yeh, S. Huang, and W. Cheng, “Fabrication and characteristic of Cr-doped fibers employing Powder-in-Tube technique,” in Optical Fiber Communication Conference/ National Fiber Optic Engineers Conference 2011, OSA Technical Digest(CD), paper OWS1. Optical Society of America, New York, 2011.

Wang, Z.

S. Egusa, Z. Wang, N. Chocat, Z. M. Ruff, A. M. Stolyrov, D. Shemuly, F. Sorin, P. T. Rakich, J. D. Joannopoulos, and Y. Fink, “Multimaterial piezoelectric fibres,” Nat. Mater. 9(8), 643–648 (2010).
[Crossref]

Watson, H. L.

H. L. Watson, “Some properties of fused quartz and other forms of silicon-dioxide,” J. Am. Ceram. Soc. 9, 511–534 (1926).
[Crossref]

Wei, L.

C. Hou, X. Jia, L. Wei, S. Tan, X. Zhao, J. D. Joannopoulos, and Y. Fink, “Crystalline silicon core fibres from aluminium core preforms,” Nat. Commun. 6(1), 6248 (2015).
[Crossref]

Wondraczek, K.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).
[Crossref]

Yavorsky, P. J.

R. F. Geller and P. J. Yavorsky, “Melting point of alpha-alumina,” J. Res. Natl Bur. Stand. 34(4), 395–401 (1945).
[Crossref]

Yeh, S.

Y. Huang, J. Wang, K. Chu, T. Lin, W. Wang, T. Chou, S. Yeh, S. Huang, and W. Cheng, “Fabrication and characteristic of Cr-doped fibers employing Powder-in-Tube technique,” in Optical Fiber Communication Conference/ National Fiber Optic Engineers Conference 2011, OSA Technical Digest(CD), paper OWS1. Optical Society of America, New York, 2011.

Zenteno, L. A.

Zhao, X.

C. Hou, X. Jia, L. Wei, S. Tan, X. Zhao, J. D. Joannopoulos, and Y. Fink, “Crystalline silicon core fibres from aluminium core preforms,” Nat. Commun. 6(1), 6248 (2015).
[Crossref]

Adv. Mater. (1)

M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-sensing fiber devices by multimaterial codrawing,” Adv. Mater. 18, 845–849 (2006).
[Crossref]

Adv. Opt. Mater. (1)

N. Healy, M. Fokine, Y. Franz, T. Hawkins, M. Jones, J. Ballato, A. C. Peacock, and U. J. Gibson, “CO2 laser-induced directional recrystallization to produce single crystal silicon-core optical fibers with low loss,” Adv. Opt. Mater. 4(7), 1004–1008 (2016).
[Crossref]

Adv. Opt. Technol. (1)

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).
[Crossref]

Bull. Am. Ceram. Soc. (1)

S. Morris and J. Ballato, “Molten core fabrication of novel optical fibers,” Bull. Am. Ceram. Soc. 92, 24–29 (2013).

Electron. Lett. (1)

J. R. Simpson and J. B. Macchesney, “Optical fibers with an Al2O3-doped silicate core composition,” Electron. Lett. 19(7), 261–262 (1983).
[Crossref]

Int. Int. J. Appl. Glass Sci. (1)

G. Tao and A. F. Abouraddy, “Multimaterial fibers,” Int. Int. J. Appl. Glass Sci. 3, 349–368 (2012).
[Crossref]

Int. J. Appl. Glass Sci. (1)

P. Dragic, M. Cavillon, and J. Ballato, “The linear and nonlinear refractive index of amorphous Al2O3 deduced from aluminosilicate optical fibers,” Int. J. Appl. Glass Sci. 9, 421–427 (2018).
[Crossref]

J. Am. Ceram. Soc. (3)

J. Ballato and P. Dragic, “Rethinking optical fiber: new demands, old glasses,” J. Am. Ceram. Soc. 96(9), 2675–2692 (2013).
[Crossref]

K. Nassau, J. W. Shiever, and J. T. Krause, “Preparation and properties of fuses silica containing alumina,” J. Am. Ceram. Soc. 58, 461 (1975).
[Crossref]

H. L. Watson, “Some properties of fused quartz and other forms of silicon-dioxide,” J. Am. Ceram. Soc. 9, 511–534 (1926).
[Crossref]

J. Am. Chem. Soc. (1)

N. F. Baril, R. He, T. D. Day, J. R. Sparks, B. Keshavarzi, M. Krishnamurthi, A. Borhan, V. Gopalan, A. C. Peacock, N. Healy, P. J. A. Sazio, and J. V. Badding, “Confined high-pressure chemical deposition of hydrogenated amorphous silicon,” J. Am. Chem. Soc. 134(1), 19–22 (2012).
[Crossref]

J. Appl. Phys. (1)

T. Davis and K. Vedam, “Photoelastic properties of sapphire (α-Al2O3),” J. Appl. Phys. 38(11), 4555–4556 (1967).
[Crossref]

J. Opt. Soc. Am. (2)

J. Res. Natl Bur. Stand. (1)

R. F. Geller and P. J. Yavorsky, “Melting point of alpha-alumina,” J. Res. Natl Bur. Stand. 34(4), 395–401 (1945).
[Crossref]

Materials (1)

J. Auguste, G. Humbert, S. Leparmentier, M. Kudinova, P. Martin, G. Delaizir, K. Schuster, and D. Litzkendorf, “Modified Powder-in-Tube Technique based on the consolidation processing of powder materials for fabricating specialty optical fibers,” Materials 7(8), 6045–6063 (2014).
[Crossref]

Nat. Commun. (1)

C. Hou, X. Jia, L. Wei, S. Tan, X. Zhao, J. D. Joannopoulos, and Y. Fink, “Crystalline silicon core fibres from aluminium core preforms,” Nat. Commun. 6(1), 6248 (2015).
[Crossref]

Nat. Mater. (3)

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Lare-scale optical field measurements with geometric fibre constructs,” Nat. Mater. 5(7), 532–536 (2006).
[Crossref]

M. Bayindir, O. Shapira, D. S. Hinczewski, J. Viens, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Integrated fibres for self-monitored optical transport,” Nat. Mater. 4(11), 820–825 (2005).
[Crossref]

S. Egusa, Z. Wang, N. Chocat, Z. M. Ruff, A. M. Stolyrov, D. Shemuly, F. Sorin, P. T. Rakich, J. D. Joannopoulos, and Y. Fink, “Multimaterial piezoelectric fibres,” Nat. Mater. 9(8), 643–648 (2010).
[Crossref]

Nat. Photonics (1)

P. Dragic, T. Hawkins, P. Foy, S. Morris, and J. Ballato, “Sapphire-derived all-glass optical fibres,” Nat. Photonics 6(9), 627–633 (2012).
[Crossref]

Opt. Express (4)

Phys. Chem. Chem. Phys. (1)

P. Brocos, A. Pinerio, R. Bravo, and A. Amigo, “Refractive indices, molar volumes and molar refractions of binary liquid mixtures: concepts and correlations,” Phys. Chem. Chem. Phys. 5(3), 550–557 (2003).
[Crossref]

Proc. SPIE (1)

S. Unger, A. Schwuchow, J. Dellith, and J. Kirchhof, “Codoped materials for high power fiber lasers-diffusion behavior and optical properties,” Proc. SPIE 6469, 646913 (2007).
[Crossref]

Sci. Rep. (1)

F. A. Martinsen, B. K. Smeltzer, M. Nord, T. Hawkins, J. Ballato, and U. J. Gibson, “Silicon-core glass fibres as microwire radial-junction solar cells,” Sci. Rep. 4(1), 6283 (2015).
[Crossref]

Other (6)

Y. Huang, J. Wang, K. Chu, T. Lin, W. Wang, T. Chou, S. Yeh, S. Huang, and W. Cheng, “Fabrication and characteristic of Cr-doped fibers employing Powder-in-Tube technique,” in Optical Fiber Communication Conference/ National Fiber Optic Engineers Conference 2011, OSA Technical Digest(CD), paper OWS1. Optical Society of America, New York, 2011.

L. G. Nielsen, S. Dasgupta, M. D. Mermelstein, D. Jakobsen, S. Herstrom, M. E. V. Pedersen, E. L. Lim, S. Alam, F. Parmigiani, D. Richardson, and B. Palsdottir, “A silica based highly nonlinear fibre with improved threshold for stimulated Brillouin scattering,” Proc. Euro. Conf. Opt. Commun. Paper Tu.4.D.3 (2010).

J. Kirchhof, S. Unger, B. Knappe, and J. Kobelke, “Interaction of alumina with other codopants in the preparation of active fiber materials,” Materials Research Society Fall Meeting, Boston, MA/USA, paper Za034 (1994).

G. Granger, C. Röhrer, G. Kleem, C. Kizler, A. Voß, T. Graf, and M. Abdou-Ahmed, “First fabrication of optical Bragg fibers by a single-step powder in tube process based on the SiO2-Al2O3 system,” presented at Specialty Fiber Processing and Advances in Fabrication I, SPIE Photonics Europe, Brussels, Belgium, 4-7 April, 2016.

V. S. Pugachev, Probability Theory and Mathematical Statistics for Engineers (Oxford University, 1984).

A. Massaro, Photonic crystals: introduction, applications and theory, 1st ed. (InTech Press, 2012

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Figures (12)

Fig. 1.
Fig. 1. Aluminum and silicon mass concentrations of the SIFs with different core diameters which were drawn at a temperature of 2050 °C and at a feeding speed of 0.365 mm/min. (Inset: microscope image of the core area of fiber with a nominal core diameter of 25 µm)
Fig. 2.
Fig. 2. RID profile measured with the optical fiber analyzer, calculated alumina mole fraction and fitted RID profile from the EDX analysis of the SIF with a nominal core diameter of 25 μm which was drawn at a temperature of 2050 °C and at a feeding speed of 0.365 mm/min.
Fig. 3.
Fig. 3. FEM simulation results of the temperature distribution and the photograph of the preform. (a) Simulation model, (b) temperature distribution in the center of the preform in axial direction, and (c) photograph of the preform after fiber drawing.
Fig. 4.
Fig. 4. Mapping result of slice 1 using EDX with a magnification of 30, high-voltage of 25 kV, and working distance of 20.0 mm (the black line indicates the EDX line scan position, the yellow rectangle indicates the area without polishing defects).
Fig. 5.
Fig. 5. Results of the simulation of the velocity of fused silica flowing within the porous alumina powder.
Fig. 6.
Fig. 6. Results of the EDX analysis of the slices. (a) Line scans of all slices and (b) standard deviations of the mass concentrations of aluminum in the core area of the slices without polishing defects.
Fig. 7.
Fig. 7. Calculated mass concentrations of the aluminum in slice 2 with a diffusion time T of (a) 0 s (originally measured profile) (b) 30 s (c) 300 s.
Fig. 8.
Fig. 8. Influence of the temperature on the mass concentrations of aluminum in the cores with diameters of (a) 12 μm and (b) 25 μm. The fibers were drawn with a feeding rate of 0.365 mm/min.
Fig. 9.
Fig. 9. Influence of the core diameter of the fiber on the mass concentrations of aluminum in the cores of the fibers which were drawn with feeding rates of (a) 0.365 mm/min, (b) 0.550 mm/min and (c) 0.730 mm/min, at a temperature of 2050 °C.
Fig. 10.
Fig. 10. Calculated effect of the diffusion in the SIFs with different core diameters with a given diffusion coefficient and diffusion time.
Fig. 11.
Fig. 11. Influence of the preforms’ feeding rate on the maximum mass concentrations of aluminum in the SIFs with different nominal core diameters. The fibers were drawn at a temperature of 2050 °C.
Fig. 12.
Fig. 12. EDX measurement performed on the preform slices which were drawn at different feeding speeds with core diameters of (a) 0.79 and 0.80 mm, (b) 0.030 and 0.034 mm.

Tables (2)

Tables Icon

Table 1. Measured and calculated data of the SIFs drawn at a temperature of 2050 °C and at a feeding speed of 0.365 mm/min

Tables Icon

Table 2. Slices of the neck-down area of the preform

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

m f A l 2 O 3 = ( m Al w Al 1 2 ) / ( m Al w Al 1 2 + m Si w Si ) ,
Δ = m f A l 2 O 3 n A l 2 O 3 + m f Si O 2 n Si O 2 n Si O 2 ,
α = | Δ meas Δ fitted |

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