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Abstract
In this study, a refractive-index guiding single-crystal fiber (SCF) with air–solid cladding was proposed and numerical simulation investigation was carried out. In general, refractive-index guided cladding was constructed through air-holes in the solid material. It resulted in the effective reduction in the number of guided-modes, and the single-mode and few-mode transmission could be realized. The influences of different materials with different refractive indices, cladding structure, and fabrication errors on the confinement loss and effective guided-mode number with the variation in wavelength from 2.5 to 3.2 µm were numerically investigated by the finite element method. Thus, the optimal design of the SCF was successfully obtained. This study may open a new avenue for the design of SCFs and their applications in the fiber lasers and sensors.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Single-crystal fibers (SCFs) possess a number of advantages with respect to both bulk crystals and optical fibers, including high melting point, excellent thermal conductivity, and high laser damage threshold [1]. Moreover, it has high doping concentration, low Brillouin gain coefficient, large aspect ratio, high specific surface area, excellent mechanical properties, and stable chemical properties [2]. Owing to these fascinating advantages, SCFs have been applied for fiber laser gain medium [3–5], platform for supercontinuum generation [6,7], and fiber-optic sensors in high-temperature, high-pressure, and chemically aggressive harsh environments [8–11].
One of the most important limitations of SCF hindering its practical applications is the highly multimodal operation in unclad SCFs. Therefore, it is necessary to utilize the cladding technique to reduce guided-modes in order to realize single-mode or few-mode transmission. Currently, cladded SCFs is based on the total reflection of cladded materials with a lower refractive-index than the single-crystal core, to achieve the refractive-index guiding.
Essentially, the thermal expansion coefficient of the cladding material should be close to that of the core by the realizable fabrication. The proposed cladding techniques include magnetron sputtering [12], sol–gel method [13,14], liquid phase epitaxy [15], co-drawing laser heating pedestal growth [16–22], hydrothermal growth [23], dopant segregation method [24], ion implantation [25–28], and micro-structure cladding [7,29,30].
Some research attempts have also been made to design SCFs with novel structure. For example, Pfeiffenberger et al. (2010) proposed 6-rod bundled single crystal sapphire photonic crystal fibers, which could reduce modal volume but still up to 3496 [8,31]. Cheng et al. (2015) designed a “windmill” shaped cladding single crystal sapphire fiber. The simulation results showed that the number of guided modes was significantly reduced in the “windmill” fiber design. However, practical fabrication methods for this design have not yet been developed [32]. Therefore, currently used cladded SCF still suffers from some disadvantages, such as confinement loss (CL), small bandwidth, limited availability of cladding materials, and requirement of high fabrication accuracy.
In order to break through the current problems of SCFs cladding techniques, the refractive-index guiding SCF with air–solid cladding was proposed and systematically studied. The influences of micro-structured cladding on the transmission performance of SCFs were investigated numerically, and the optimized SCF design was successfully obtained.
2. Design of single-crystal fiber
2.1. Structural design
The refractive-index guiding SCF proposed in this study is shown in Fig. 1. Similar to a photonic crystal fiber (PCF), the single-crystal core is also surrounded with some air-holes in the cladding material. The effective refractive index of the cladding can be reduced by the air-holes; therefore, the alternatives of cladding materials can be significantly expanded from materials with lower refractive-index to higher refractive-index than that of the core layer.
Fig. 1. The structure scheme of refractive-index guiding SCF. D: diameter of the entire fiber, Dcore: diameter of the core, d: diameter of the air-hole, and g: gap between the air-holes and the core.
Herein, the air-ratio is defined to describe the air-holes in cladding, and the value of air-ratio is attained in terms of the ratio of air-hole area Sa to that of the entire cladding Sc. The Sa and Sc represent the area of air region and entire cladding region, respectively.
2.2. Materials
The optical and thermal parameters of the single-crystal core and cladding materials are listed in Table 1. The different materials are described in terms of refractive index in the numerical simulation proposed herein, which indicates that different refractive-index of cladding suggests utilizing different materials.
Table 1. Parameters of the single-crystal core and cladding materials.
The lutetium oxide (Lu2O3) crystal exhibits some excellent properties, such as low phonon energy, greater chemical stability, and high damage threshold which make it a promising laser crystal material candidate for high-power solid laser [33]. Therefore, during the simulation process, the Lu2O3 crystal was used as core material, and lanthanide glass with a refractive index similar to that of Lu2O3 single crystal was selected as the cladding material.
3. Results and discussions
3.1. Numerical method
In cylindrical coordinates, Maxwell’s equations for electromagnetic fields in optical fibers with invariant index profiles along the z-direction can be decomposed into longitudinal and transverse components by using following Eq. (4) [34]:
where ξ denotes the E or H field and the subscript t and z denote the transverse and longitudinal components, respectively, and ω is the angular frequency, β is the propagation constant. A special boundary condition of the computational domain, a perfect matched layer (PML), is applied to calculate β. After the propagation constant β is obtained, the effective refractive index (neff) and CL can be calculated by using the following formula:In this study, the neff and CL of the optical fiber were estimated via the implementation of the finite element method (FEM). Cylindrical PML was used in the outermost layer of the fiber, which could theoretically absorb without reflecting any type of wave traveling toward the boundaries. The mesh used herein was a free triangular shape mesh with the density of λ/4.
The guided-modes number (Nm) of step-index optical fiber can be approximated by using V parameter:
For the SCF structure proposed herein, it was necessary to adopt different methods to calculate the number of effective guided-modes. Herein, the following two criteria were used to determine whether the mode was effective: (1) the neff difference between the higher order mode and the fundamental mode is less than 0.006. (2) The CL of the mode is less than 10−3 dB/km. Then, the mode can be judged as an effective guided-mode when the two conditions are met at the same time.
For example, for the two structures shown in Fig. 2, when the wavelength varies from 2.5 to 3.2 µm, the parameters and results calculated according to formula (7) are as follows: Structure 1: D1 = 80 µm, d1 = 25 µm, N1 = 6, g1 = 0 µm, V = 25.989–21.658, NA = 0.6893, and Nm = 68–22; Structure 2: D2 = 80 µm, d2 = 16 µm, N2 = 10, g2 = 5 µm, V = 22.585–18.820, NA = 0.5990, and Nm = 52–36. Moreover, the Dcore was fixed at 30 µm in the entire simulation process.
Fig. 2. Structure scheme and the normalized intensity profiles of the first 4 modes for the SCF structure 1 and 2: (a): D1 = 80 µm, d1 = 25 µm, N1 = 6, g1 = 0 µm; (b): D2 = 80 µm, d2 = 16 µm, N2 = 10, g2 = 5 µm.
Herein, the neff and CL of the four modes were calculated for the two structures and the results are shown in Fig. 3, respectively. The results show that three modes of structure 1 meet the judgment criteria and four modes of structure 2 meet the criteria. Therefore, the number of effective guided-mode is 3 and 4 for structure 1 and 2, respectively. Simultaneously, the normalized intensity profiles of the modes are shown in Fig. 2.
Fig. 3. The (a) neff and (b) CL of the first 4 fundamental and high-order modes for the SCF structures 1 and 2. D1 = 80 µm, d1 = 25 µm, N1 = 6, g1 = 0 µm; D2 = 80 µm, d2 = 16 µm, N2 = 10, g2 = 5 µm.
3.2. Numerical results
The neff and CL for the fundamental mode of SCF with ncore = 1.89, ncladding = 1.9, N = 6, d = 25 µm, g = 0 µm, and air-ratio = 0.6818, when the wavelength increased from 2.5 to 3.2 µm are shown in Fig. 4.
Fig. 4. The neff and the CL for the fundamental mode of SCF with ncore = 1.89, ncladding = 1.9, N = 6, d = 25 µm, g = 0 µm, and air-ratio = 0.6818 when wavelength is increased from 2.5 to 3.2 µm.
It was found that neff decreased linearly with the increase of wavelength. The fluctuation of CL was within two orders of magnitude, and the minimum CL appeared at wavelength of 2.85 µm. The guided-modes number was determined to be 3. Furthermore, the results also indicated that the light can be guided by a cladding material with a refractive index higher than that of the core due to the air-holes induced index-guiding.
The relationship between the number of air-holes (N) and CL of fundamental mode is shown in Fig. 5.
Fig. 5. The CL of fundamental mode with the number of air-holes. ncore = 1.89, ncladding = 1.9, g = 5 µm, wavelength is 2.8 µm.
When g = 5 µm, the refractive indices of single-crystal core and cladding material are 1.89 and 1.90, respectively. The diameter of the cladding is constant; therefore, the maximal N that can be accommodated varies with different air-hole diameters. The results show that the larger the d, the faster the decrease in CL with the increase of air-hole number. The guided-modes number is in the range of 2–4. The larger the value of N, the smaller the CL, and the larger the number of guided-modes. A mutual restriction exists between the guided-modes number and CL, which requires comprehensive consideration and optimization design. The optimal structure can be obtained from Fig. 5, that is, N = 10 and d = 15 µm, with the fundamental mode CL being 6.22844 × 10−10 dB/m, and the guided-modes number is 4.
Furthermore, the CL of fundamental mode with air-hole diameter is illustrated in Fig. 6.
Fig. 6. The fundamental mode CL with the air-hole diameter. ncore = 1.89, ncladding = 1.9, g = 5 µm, wavelength is 2.8 µm.
When N = 8 and 10, the fundamental mode CL decreases with the increase of d with a constant N due to the reduction of effective refractive-index of cladding material caused by the increase of air-ratio. According to formula (1), the air-ratio is related not only to the diameter of air-holes, but also to the number of air-holes. When the number of air-holes is small, the regulation effect of air-hole diameter on the air-ratio is not obvious, thus when N = 6, the change of CL with air-holes diameter is not obvious. Therefore, fiber structure with more air-holes should be selected to achieve lower CL. Finally, the optimal structure with N = 10 and d = 16 µm can be obtained from Fig. 6. The fundamental mode CL is 6.91332 × 1010 dB/m, and the guided-modes number is 4.
In order to analyze the influences of the air-ratio, the fundamental mode CL and neff with air-ratio were obtained as shown in Fig. 7. The neff decreases monotonically with the increase of air-ratio. Obviously, the CL decreases with the increase of air-ratio when air-ratio < 0.47. However, for air-ratio > 0.47, the CL increases with the increase of air-ratio, thus indicating that the air-ratio should be set reasonably. The minimal CL is 4.60 × 10−12 dB/m with the air-ratio of 0.4655 and N = 10, d = 16 µm.
Fig. 7. The fundamental mode CL and neff with air-ratio when ncore = 1.89, ncladding = 1.9, N = 10, d = 16 µm, g = 5 µm, wavelength is 2.8 µm.
The fundamental mode CL and guided-modes number for different structures of SCFs with the equal air-ratio of 0.47 are presented in Table 2. In the case of a constant air-ratio, lower CL is obtained by the larger air-hole number. This is attributed to the fact that the air and the solid materials in the cladding get completely mixed due to the more air-holes; as a result, the refractive index of the cladding is reduced, and the refractive-index guiding effect is enhanced. When the ratio is 0.47, structures of all SCFs can achieve few-mode transmission. The lowest CL and the smallest guide-modes number can be obtained by the SCF structure with the hexagonal array of 90 air-holes.
Table 2. The CL and the guided-modes number for different SCF structure with the air-ratio of 0.47.
However, considering the difficulty in fabrication of the fiber with similar performances, the structure of single ring of 10 air-holes was considered as the optimal structure, which could achieve low loss and few-mode transmission, and the fabrication was relatively simple.
The influences of the cladding materials were investigated further. The relationship between CL and refractive index of cladding material for three structures is presented in Fig. 8. The refractive index of core is a constant at 1.89, and the three SCF structure parameters are illustrated in detail in Table 3. Obviously, an increase in CL is observed with the increase of refractive index of cladding for both the structures with N = 10. However, when the air-ratio is smaller, the CL increased rapidly. For N = 20 and air-ratio = 0.6146, the increase in CL occurs only when the cladding refractive-index is above 2.25. If the CL limit is set as 0.001 dB/m, the range of refractive-index of cladding material with different structures can be obtained from Fig. 9, and the structure parameter is presented in Table 3. The above-mentioned results show that the selection range of refractive index of cladding materials can be further expanded by the SCF structure with high air-ratio.
Fig. 8. The fundamental mode CL with the refractive index of cladding material when air-ratio is 0.4700, 0.5254, 0.6146, respectively, and ncore = 1.89, ncladding = 1.9–2.3.
Fig. 9. (a) Structure scheme and the (b) fundamental and (c) 2nd modes normalized intensity profiles of the SCF with changed air-holes. ncore = 1.89, ncladding = 1.9, N = 10, d = 16 µm, and g = 5 µm, Δx = 0.5 µm, wavelength is 2.8 µm.
Table 3. The parameters of different SCF structures with different air-ratio and cladding refractive index range (Fig. 8).
3.3 Fabrication errors
In the practical applications, it is necessary to analyze the influences of fabrication errors on SCFs. In fact, the fabrication errors are different due to the different fabrication methods. However, in this study, mainly the geometric errors were utilized to describe the fabrication errors. The influences of the position and diameter errors of air-holes were analyzed in detail. The position errors were defined as Δx, the diameter error as Δd, and the number of changed air-holes as Ne.
The SCF structure scheme, the normalized intensity profiles of fundamental and 2nd modes in the SCF with changed air-holes are illustrated in Fig. 9. The red air-holes may be moved during the fabrication. Furthermore, the CL with Δx of changed air-holes is shown in Fig. 10. The Ne is 2, 3, and 5, respectively, and the structural parameters are as follows: ncore = 1.89, ncladding = 1.9, N = 10, d = 16 µm, and g = 5 µm. The Δx fluctuates in the range of 0–3.5 µm.
Fig. 10. The CL with position errors of changed air-holes. The number of changed air-holes are 2, 3, and 5, respectively. ncore = 1.89, ncladding = 1.9, N = 10, d = 16 µm, and g = 5 µm, Δx = 0.5 µm, wavelength is 2.8 µm.
The results show that the movement of the 2 air-hole has no obvious effect on CL. However, when Ne = 3 and 5, the CL is increased by Δx, and the CL increases more rapidly when Ne = 5 than that when Ne = 3. This is mainly attributed to the fact that the fundamental mode is coupled with the cladding mode due to the Δx of the changed air-holes. The greater the Δx, the more serious the coupling between the fundamental mode and the cladding, as shown in Fig. 10.
The CL remains less than 10−8 dB/m when Ne = 2, with the maximal Δx = 3.5 µm. In the case of Ne = 3 and 5, when Δx ≥ 1.75 µm, the CL exceeds 10−6 dB/m, thus the maximum Δx tolerance is considered as 1.75 µm.
Similarly, the effect of diameter errors of air-holes on SCFs was discussed herein. The structural parameters are as follows: ncore = 1.89, ncladding = 1.9, N = 10, g = 5 µm, and the original air-hole diameter without error is d0 = 16 µm. The red air-holes indicate that the affected air-holes during the fabrication and the number of changed air-holes are 2, 3, and 5, respectively. The Δd fluctuates in the range of 0–3.5 µm. The SCF structure, the fundamental and 2nd normalized intensity profiles, and the CL with Δd of air-holes are shown in Figs. 11 and 12, respectively.
Fig. 11. (a) Structure scheme and the (b) fundamental and (c) 2nd modes normalized intensity profiles of the SCF with diameter errors of air-holes. ncore = 1.89, ncladding = 1.9, N = 10, d0 = 16 µm, and g = 5 µm, wavelength is 2.8 µm, Δd = 0–3.5 µm, Ne = 2, 3, 5.
Fig. 12. The confinement loss with diameter errors (Δd) of air-holes. The number of changed air-holes are 2, 3, and 5, respectively. ncore = 1.89, ncladding = 1.9, N = 10, d0 = 16 µm, and g = 5 µm, wavelength is 2.8 µm, Δd = 0–3.5 µm.
Figure 11 demonstrates that the effective guided-modes number decreases from 4 to 1 or 2, and the mode field of the fundamental mode gets coupled with the cladding with the increase of the Δd. Therefore, single-mode can be achieved by increasing the Δd; however, this also leads to the increase in the CL at the same time. CL increases with the Δd, as shown in Fig. 12. However, when Ne is 2, 3, and 5, the change trend is basically the same, indicating that SCF is more sensitive to the Δx.
The CL remains less than 10−6 dB/m when Δd < 1.2 µm, but when Δd > 1.2 µm the CL increases obviously, thus the maximum Δd tolerance is considered as 0–1.2 µm.
Fabrication of SCF with air-hole cladding analyzed in this study is potentially feasible by the method involving SCF growth and the co-drawing laser-heated pedestal growth (LHPG), with the improved engineering design. However, the structure of SCF designed can eliminate some influences of fabrication tolerances, and minimize the difficulty of fabrication.
3.4 Discussions
Based on the above-mentioned analysis of fiber properties and fabrication errors, the optimal structure and fabrication error range of refractive-index guiding SCF were obtained. The results of this study were compared with those of other related literature studies as presented in Table 4.
Table 4. Comparison of this study with other literature studies
The table summarizes that the refractive-index guiding SCF proposed herein has lower CL and larger guided-mode number. Owing to the significant difference of melting point and thermal expansion coefficient between the single crystal and the glass, it is not suitable to use the traditional hot drawing method to fabricate the refractive-index guiding SCF. The fiber can be obtained by first preparing the cladding with air holes, and then inserting the single crystal core prepared in advance into the cladding. Undeniably, a lot more systematic explorations are still demanded to investigate and improve the fabrication method, which will be pursued in the future.
4. Conclusion
In this study, a refractive-index guiding SCF with air–solid cladding was proposed and investigated. First, a method to determine the effective guided-mode number of SCFs was proposed. The mechanism of refractive-index light-guiding was confirmed. Then, the influences of structural parameters, including air-hole number, air-hole diameter, and air-ratio of cladding on CL and guided-mode number of SCF were studied in detail, and the optimal structure was successfully obtained. The results showed that the increase of air-holes diameter, air-holes number, and air-ratio resulted in the decrease in the CL, but increase in the number of guided-modes at the same time. For the optimal SCFs structure: N = 10, d = 16 µm, air-ratio = 0.47, CL is 4.60 × 10−12 dB/m and the number of guided-mode is 4. Further, the influences of the refractive index of cladding material were analyzed. It was found that the refractive index range of cladding material could be larger with the increase in the air-ratio of cladding, which could enlarge the select range of the cladding material. Finally, the impact of the fabrication error including the air-hole position error and the diameter error on the SCFs was discussed in detail, and the fabrication error range of the SCFs was obtained, which can provide references for the practical fabrication. This study presents a valuable technical reference and inspiration for the cladding technique and experimental fabrication of cladded SCFs.
Funding
Beijing Municipal Natural Science Foundation (4192047); National Natural Science Foundation of China (61875064); Fundamental Research Funds for the Central Universities (2019JBM345).
Acknowledgements
The authors acknowledge the financial support from the Fundamental Research Funds for the Central Universities (2019JBM345), the Beijing Natural Science Foundation (4192047), and the National Natural Science Foundation of China (61875064).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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