A novel single-polarization single-mode hollow-core photonic bandgap fiber with thin slab waveguide (TSW) was designed and simulated. Single-polarization guidance is achieved by the high loss of a polarized fundamental mode (FM) induced by mode coupling with a higher-order TE/TM mode of TSW and low loss of another polarized FM. We achieve a polarization loss ratio ∼ 46.9 dB, birefringence Δn ∼ 2.4 × 10−4, loss ∼ 35.9 dB/km and minimum higher-order mode extinction ratio > 15 dB. Moreover, the performance could be maintained when the guidance wavelength λ = 1.44 ∼ 1.56 µm and bending radius rc > 9 mm. The proposed model will be suitable for application as resonator sensing paths of miniaturized resonator fiber optic gyroscopes, high-performance interferometers, fiber lasers, frequency metrology, quantum communications, and laser-based gas sensing, etc.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Hollow-core fibers (HCFs) are advanced microstructure fibers with low dispersion, low nonlinearity, high damage threshold, and low stress/thermal sensitivity [1,2]. Owing to these advantages, HCFs can be implemented as resonator sensing paths in resonator fiber optic gyroscopes (RFOGs) to reduce the Kerr-induced drift, thermal polarization instability and so on [3–6]. Hollow-core photonic-bandgap fibers (HC-PBFs) were first used in RFOGs with a random walk of 0.055°/s1/2, a long-term drift with a standard deviation of 0.5°/s, and a peak-to-peak variation of 2.5°/s over 1 h . However, the multimodal operation, relatively high loss, and high backscattering [1,3] of HC-PBFs reduce the resonator finesse and sensitivity [7,8]. In recent years, hollow-core antiresonant fibers (HC-ARFs), including nested anti-resonant nodeless fibers (NANFs), and Kagome fibers, have also been used as resonator sensing paths of RFOGs [4,9]. Hollow-core RFOGs using single-mode (SM), low-loss, and low-backscattering NANFs exhibit a long-term stability of 0.05°/h, which represents an improvement by a factor of three over any prior hollow-core RFOGs, and by a factor of 500 over any prior results at integration times longer than 1 h . In addition, HC-ARFs have excellent polarization purity characteristics, even without high-birefringence (Hi-Bi), because the field at the core boundary has excellent spatial regularity with no surface modes and exceptionally low field overlap with the glass . This is beneficial for the application of HC-ARFs in RFOGs.
Miniaturization is the key development goal and advantage of RFOGs, owing to the relatively shorter fiber length compared to interferometric fiber optic gyroscopes. However, it is difficult to realize the miniaturization of RFOGs using HC-ARFs, including NANF and Kagome fiber. This is because of the high bending loss (BL) of these fibers [11–13], arising from the large core size and bending-sensitivity of the antiresonant reflecting optical waveguide principle and inhibited-coupling guidance. In contrast, the HC-PBF based on photonic bandgap (PBG) guidance with a relatively small core size has a low BL , which is beneficial for the miniaturization of RFOGs. Therefore, studies on HC-PBFs for miniaturized RFOGs are desired.
Single-polarization single-mode (SPSM) fibers transmit light in only one polarized fundamental mode (FM), while another orthogonal polarized fundamental mode and other higher-order modes (HOMs) are forbidden or suffer high losses. Therefore, resonator sensing paths using SPSM fibers, with propagation loss ∼ 0.05 dB/m, undesired FM loss ∼ 8 dB/m, and polarization loss ratio (PLR, defined as the ratio of the high-loss of polarized FM and low-loss of polarized FM) ∼ 22 dB, could reduce the temperature-dependent polarization-fluctuation drift, polarization noise, and effects of polarization mode dispersion, thus increasing the long-term stability of RFOGs [14,15]. However, practical SPSM fibers are almost solid core [16–21], and SPSM HCFs [22–24] are still in the design phase.
To realize the single-polarization (SP) characteristic of HC-PBFs, the polarization-maintain (PM) characteristic with Hi-Bi should be realized for HC-PBFs. In general, the 7-cell HC-PBF with triangular lattice of air-holes (TLH) cladding and an approximately regular rounded dodecagon core exhibits no PM characteristics, due to the C6v symmetry structure, as shown in Fig. 1(a). To induce Hi-Bi in HC-PBF, structural parameters are generally adjusted to degenerate from the C6v symmetry to a C2v symmetry. For example, the 4-cell  and elliptical 7-cell [26,27] PM HC-PBFs exhibit Hi-Bi through shape birefringence from the C2v symmetry shape of the core. Moreover, by changing the core wall thickness in one direction [28,29], the mode coupling between surface modes and a polarized FM could increase the difference between the two modes of the effective refractive index neff of FMs, thus inducing Hi-Bi. Moreover, methods such as introducing partially collapsed holes in the cladding  are used to realize Hi-Bi. However, PM HC-PBFs generally do not exhibit SP characteristics. Both polarized FMs have relatively low confinement losses (CLs), because light is limited to the core by the complete PBG cladding structure. One method to induce SP characteristics in PM HC-PBFs is to reduce the air-filling fraction of a row of airholes in the cladding to change the PBG spectral ranges along the corresponding direction. This leads to a polarization dispersion loss (PDL) of ∼ 3000 dB/m at 1500 nm, but SM characteristics are not obtained . Another method to realize SP characteristics (PDL ∼ 79 dB/m) is by changing the number of cladding layers in different directions, but this leads to a high-loss of ∼ 1.6 dB/m, which is not suitable for applications .
To realize SM characteristics in HC-PBFs, the HOMs should be forbidden or suffer high losses. The 7-cell and 19-cell HC-PBFs with large cores are multimode fibers, because some HOMs with relatively low loss are located in the PBG range . There are two common methods to realize SM characteristics in HC-PBF. One is by decreasing the core size to shift out the HOMs from the PBG range, such as in the 3-cell HC-PBF , but the FMs also suffer high loss in this approach. The other is by adding the defect holes in the cladding to introduce mode coupling between the HOMs and defect holes modes , such that the HOMs suffer high loss. However, the design and fabrication of HC-PBFs with defect holes is complicated and unmanageable. Thus, to the best of our knowledge, a practical design for relatively low-loss SPSM HC-PBFs has not been reported until now.
A thin slab waveguide (TSW) is a common waveguide structure. When the thickness t for silica TSWs in air is smaller than the cut-off thickness tcut-off of the higher-order TE/TM modes, the higher-order TE/TM modes of the TSW are radiation modes, and suffer high loss and neff < nair . If a TSW structure with a certain t < tcut-off is introduced in an HC-PBF, one polarized FM would suffer high loss by coupling with a higher-order radiation TE/TM mode in the same electric field direction and at similar neff, while another polarized FM without mode coupling of higher-order radiation TE/TM maintains low loss. In addition, the HOMs could also suffer high losses by coupling with higher-order TE/TM modes. In this way, the SPSM HC-PBFs with Hi-Bi, high PLR factor, and low loss can be realized.
In this paper, we propose a novel and practical SPSM HC-PBF with TSW, which has a PLR of ∼ 46.9 dB, birefringence Δn ∼ 2.4×10−4, loss ∼ 35.9 dB/km, and minimum HOM extinction ratio (HOMER) > 15 dB. In the wavelength range of 1.44 ∼ 1.56 µm, the proposed SPSM HC-PBF exhibits Δn > 1.8×10−4, PLR > 45 dB, and loss < 43 dB/km. Moreover, we achieve BL < 42 dB/km at a bending radius rc > 9 mm. In other words, the SPSM HC-PBF with TSW has excellent SPSM and low BL performance, which might be suitable for resonator sensing paths in miniaturized RFOGs, high-performance interferometers, fiber lasers, frequency metrology, quantum communications, laser-based gas sensing, etc.
2. Fiber structure and stacking method
As shown in Fig. 1(a), there is no TSW structure in the 7-cell HC-PBF with TLH cladding, which has an operational wavelength range of approximately 1550 nm. The detailed structure parameters are the inter-hole pitch Λ ∼ 3.88 µm, cladding airhole diameter dcl ∼ 3.788 µm, cladding airhole fillet diameter Dc ∼ 2.15 µm, fillet diameter and core boundary thickness of hexagonal air holes near the core D1 ∼ 845 nm and t1 ∼ 89 nm, respectively, fillet diameter and core boundary thickness of pentagonal air holes near the core D2 ∼ 445 nm and t2 ∼ 151 nm, respectively, fillet diameter of dodecagonal core D3 ∼ 3.16 µm, and the core diameter in the x and y directions Dcore-x ∼ 11.69 µm and Dcore-y ∼ 11.43 µm, respectively. The simulated loss of the 7-cell HC-PBF with no TSW was ∼ 25 dB/km. The middle row of silica capillaries was replaced with silica rods to obtain the 7-cell HC-PBF with the TSW structure, as shown in Fig. 1(b), which has t ∼ 4 µm. However, the proposed fiber suffers a high-loss of ∼ 3669.5 dB/m, low PLR of ∼ 1.82 dB, and low Δn ∼ 3×10−5, because the light escapes from core through the thick TSW.
In addition to the TLH cladding, the PBG cladding structures of HC-PBF also have square lattice (SL) [33,34] and triangular lattice of rods (TLR) cladding [35,36], as shown in Fig. 2(a). The TLH and TLR have C6v symmetry, while the SL has C4v symmetry. The TLH has no TSW structure, whereas the SL and TLR have TSW structures. In Fig. 2(a), the TSW structure in the SL is divided by the periodic vertical silica struts with period ∼ Λ. The TSW structure in the TLR is divided by the periodic 60° and 120° silica struts with period ∼ √3 Λ. Therefore, the SL and TLR structures could be introduced in the TLH structure to build the TSW structure in the TLH cladding.
The TLH and TLR are both triangular lattice structures, but their unit cells are respectively based on air holes and silica rods, as shown in Figs. 2(a) and (b). As shown in Fig. 2(c), the stack methods of silica capillaries are also periodic triangular arrangements, but the red shadow region in the TLR closes during the drawing process to form the silica rods. Therefore, it is difficult to combine them for practical fiber fabrication. By comparison, the stack method of silica capillaries in the SL is a periodic square arrangement, and the blue silica rods are stacked only on the sides of the stacking structure to stabilize the large capillaries.
In particular, for TLH and SL, the unit cells are air holes surrounded by silica rods and struts, while the stacking structure has a row of capillaries. Therefore, it is practical to introduce an SL structure in the TLH to form the TSW structure. As shown in Fig. 3(a), the stacking method of the two middle row capillaries in the HC-PBF stacking structure is changed from a triangular arrangement to a square arrangement. A row of silica rods on both sides of the stacking structure are added between two rows of capillaries with a square arrangement to stabilize the entire stacking structure. The ten capillaries are moved out in the middle of the stacking structure to form the core defect, and reserved on both sides of the stacking structure to stabilize the entire stacking structure. The TSW structure appears through the close of the gap in the cladding, similar to the SL cladding structure of the HC-PBF shown in Fig. 2(a). The core region was kept open by precise and real-time individual gas pressure control of the core region during fabrication.
Finally, the HC-PBF with TSW model is established as shown in Fig. 3(b), which is a C2v symmetric fiber structure. The typical structural parameters are similar to those of the 7-cell HC-PBF with no TSW structure, except for Dcore-y ∼ 13.73 µm. The TSW structure is located in the two middle rows of cladding-rounded pentagon air holes with a square arrangement beside the core. By adjusting the value of t, the SPSM characteristic is realized by the high loss of one polarized FM and HOMs that couple with the TE/TM modes in the TSW, and low loss is obtained for another polarized FM without mode coupling.
As shown in Fig. 3(c), when t ∼ 0.1 µm, mode coupling occurs between the x-polarized FM (FMx) and one TM mode in the TSW, whereas the y-polarized FM (FMy) is strongly limited in the core, because there is no mode coupling between FMy and any TE mode in TSW. In this case, the CL of FMy is ∼ 3.25 dB/km, and the surface scattering loss (SSL) of FMy is ∼ 28.75 dB/km. Moreover, the CL of FMx is ∼ 1.32×106 dB/km, and the SSL of FMx is ∼ 42.77 dB/km. Thus, the proposed model has good SPSM characteristics with PLR ∼ 46 dB, Δn ∼ 1.9 ×10−4 and HOMER > 20 dB.
3. Theoretical analysis of mode coupling
To realize mode coupling between FMs and higher-order radiation TE/TM modes in TSW, the same directions of electric fields, similar neff and effective mode overlap are necessary. For an ideal silica TSW in air, when t < tcut-off, the higher-order TE and TM modes are high-loss radiation modes, as shown in Fig. 4(a). The neff and period length d of the higher-order TE/TM modes are influenced by t and the resonance order n. In Fig. 4(b), the practical TSW structure in HC-PBF is divided by periodic vertical silica struts with period Λ. The higher-order resonance TE/TM modes exist only when Eq. (1) is satisfied, as shown in Fig. 4(b). Otherwise, they would be destroyed by periodic vertical silica struts.
Since the eigen equations of the TSW are transcendental equations, it is difficult to obtain analytical solutions for neff and d. The numerical solutions of neff and d for different values of t and n were obtained by finite element mode simulation. The results are shown in Fig. 4(c), where the points on the lines show the neff and d values of the TE/TM modes for different values of n and t. The lines connect points with the same t for the TE and TM modes. The single points are d values satisfying in Eq. (1) when N = 1, 2, 3, and 4, with neff of core modes in the fiber. According to the results for the HC-PBF with TSW, the mode coupling between TM modes and FMx occurs when t = 0.07 ∼ 0.092 µm or approximately equals ∼ 0.2 µm. Similarly, the mode coupling between the TE modes and FMy occurs when t = 0.15 ∼ 0.2 µm, 0.3 ∼ 0.45 µm, or 0.55 ∼ 0.65 µm.
To verify the validity of the theoretical analysis and simulated results shown in Fig. 4, the structure of the HC-PBF with TSW is simplified to HC-ARF with TSW, as shown in Fig. 5(a). The core diameter Dcore ∼ 10 µm is similar to the Dcore of HC-PBF. The thickness of the silica antiresonant rings is tsilica ∼ 0.372 µm, while the thickness of the air antiresonant rings is tair ∼ 3.25 µm . Therefore, the period of antiresonant cladding Λac ∼ 3.622 µm is similar to Λ ∼ 3.88 µm in the HC-PBF with TSW. As shown in Fig. 4(c), the points of FMs of HC-ARF with TSW are nearly located in the same regions of the points of FMs of HC-PBF with TSW. Therefore, for both HC-ARF with TSW and HC-PBF with TSW, the ranges of t existing mode couplings between the TE/TM modes and FMs are similar. As shown in Fig. 5(b), when t = 0.092 µm, FMy has no mode coupling, whereas FMx exhibits mode coupling, which is the same as in Fig. 3(c).
The simulated dependences of the CLs of FMx and FMy on t are shown in Fig. 5(c). The gray and red regions are the estimated ranges of mode coupling, as shown in Fig. 4(c). The CLs of FMx and FMy increase in the gray and red regions, respectively, owing to mode coupling with the TM or TE modes. Therefore, the validity and accuracy of the mode-coupling model in Fig. 4 are verified. The slight biases between the loss peaks and estimated ranges arise from the difference in the TE/TM modes neff between the ideal TSW and practical TSW with periodic vertical struts in the HC-ARF model.
The simulated dependences of Δn, PLR, and minimum CL of FMs with different t values are shown in Fig. 5(d). The desired Hi-Bi, high PLR and low CL of FMs is observed at t = 0.07 ∼ 0.13 µm. Thus, the HC-PBF model with TSW is optimized when t = 0.07 ∼ 0.13 µm.
4. Parameter optimization and performance simulation
The simulated HC-PBF with TSW model shown in Fig. 3(b) was established to optimize the key parameter t. The simulated dependence of Δn, PLR, and CL of FMy with different values of t is shown in Fig. 6(a). Δn > 1.9× 10−4, PLR > 50 dB and CL of FMy < 0.006 dB/m when t = 0.08 ∼ 0.11 µm are similar to the performance in the green region in Fig. 5(d). For the HC-PBF, the SSL is primary and non-negligible. The results considering the SSL are also shown in Fig. 6(a). Δn > 1.2×10−4, PLR > 45 dB and total loss (TL) of FMy < 0.04 dB/m when t = 0.09 ∼ 0.125 µm, which indicates well SP characteristics. Moreover, the optimized t is chosen as 0.093 µm, where Δn = 2.4×10−4, PLR = 46.9 dB, loss of FMs = 0.0359 dB/m. The spectral dependences of Δn, PLR, and TL of FMy are shown in Fig. 6(b). The SP characteristics are maintained when λ = 1.44 ∼ 1.56 µm, with Δn > 1.8× 10−4, PLR > 45 dB and TL of FMy < 0.043 dB/m. Therefore, the HC-PBF with TSW has low loss, Hi-Bi, and high PLR SP characteristics. Thus, the proposed model meets the requirements for realizing miniaturized RFOGs, in addition to other potential applications.
The 7-cell HC-PBF is a multimode fiber that can guide the LP11 modes, LP02 modes, in addition to FMs. The existence of four degenerate LP11 modes with relatively low loss increases the noise of the RFOG using 7-cell HC-PBF. However, the four LP11 modes are no longer degenerate under the C2v symmetry of HC-PBF with TSW, as shown in Fig. 7(a). The differences in the mode location and electric field directions lead to differences in neff and loss for these LP11 modes. The simulated dependences of neff, loss, and HOMER of these LP11 modes and FMy with different t are shown in Figs. 7(b), (c), and (d). The LP11ey, LP11ox, and LP11oy modes suffer high losses, due to mode coupling with the TE/TM modes in TSW according to Fig. 4(c), but the LP11oy mode exhibits low loss when t < 0.13 µm. This is because the LP11oy mode, which has low field overlap with TSW, is difficult to couple with the TE/TM modes in TSW. However, the strong mode coupling between LP11oy and LP11ey is unavoidable in practice, because their neff and mode locations are almost equal, as shown in Figs. 7(a) and (b). Therefore, the HC-PBF with TSW has SM characteristics because of the high loss of the LP11 modes. The minimum HOMER > 15 dB when t = 0.093 µm is beneficial for miniaturized RFOGs and other applications.
The HC-PBF with TSW has different bending characteristics in two orthogonal bending directions, because of the C2v symmetry structure. The simulated dependence of the BLs of FMy in the x and y bending directions with different rc values is shown in Fig. 8. FMy suffers a high loss when rc decreases in the x-bending direction, due to mode coupling with the surface mode, and maintains a low loss in the y-bending direction. However, BLs < 42 dB/km at rc > 9 mm in the two orthogonal bending directions, and the difference in anisotropic BLs in different bending directions is negligible. Therefore, the HC-PBF with TSW is suitable for miniaturized RFOG in practice, even though it has anisotropic bending characteristics.
5. Other designs
In this section, we describe the SPSM and loss performance of HC-PBFs with TSW with different numbers of core cells. The 24-cell, 14-cell, 4-cell, and 18-cell or 6-cell with diamond or rectangle shape HC-PBFs with TSW were constructed by moving out the respective locations and number of core cells. The capillary stacking structures and models are shown in Fig. 9. The cladding parameters and t in these models are the same as the optimized parameters of 10-cell HC-PBFs with TSW in Section 4. Since the SPSM is almost unaffected by the surface modes in the model, the optimization of the core wall thickness and round corners are ignored. According to the simulated electric field distributions of FMs, mode coupling between the FMx and TM mode in TSW is observed, while FMy uncouples with the TE mode in TSW in these designs. However, the mode coupling in the 24-cell HC-PBF with TSW is weak, due to the larger neff of FMx and lower effective mode overlap. Therefore, the validity of the theoretical analysis in Section 3 was further verified.
The simulated results of the SPSM and loss performance in HC-PBFs with TSW with different locations and numbers of core cells are shown in Table 1. Compared with 10-cell HC-PBFs with TSW, the 24-cell HC-PBF with TSW exhibits low loss but poor SPSM performance. The 14-cell, 18-cell with diamond or rectangle shape HC-PBFs with TSW exhibit high loss induced by surface mode and poor SM characteristics, because there is no mode coupling between LP11oy and LP11ey. The 6-cell with diamond shape has good SM characteristics but with high loss and low PLR. The 6-cell with rectangle shape has good SPSM characteristics but very high loss. Moreover, the poor SPSM and loss performance of the 4-cell are due to the strong coupling of FMs with the surface modes and TE/TM modes, as shown in Fig. 9. In brief, the 10-cell HC-PBF with TSW is preferred for application in miniaturized RFOGs, compared with other designs.
6. Fabrication tolerances
In this section, we describe the influence of fabrication tolerances on the performance of HC-PBFs with TSW. As shown in Fig. 10(a), the key parameters of TSW are t, Dctsw, and tver, where Dctsw is the fillet diameter of the TSW, and tver is the thickness of the periodic vertical silica struts. They influence the performance of the HC-PBFs with TSW by changing the neff and loss of higher-order TE/TM modes in TSW. During optimization design, Dctsw and tver were set to 445 nm and 93 nm, respectively, which are equal to D2, and the thickness of the cladding silica struts. However, it is difficult to accurately control them during fiber fabrication. Therefore, the influences of Dctsw and tver on Δn, PLR, HOMER, and loss of FMy are shown in Fig. 10(b) and (c). Δn, HOMER, and loss of FMy decrease with Dctsw and tver when Dctsw > 0.3 µm and tver > 0.073 µm. PLR continues to increase with Dctsw, but is approximately constant when tver > 0.073 µm. In Section 4, the optimized range of t is reported as 0.09 ∼ 0.125 µm when Δn > 1.2×10−4, PLR > 45 dB, HOMER > 14.6 dB and loss of FMy < 0.04 dB/m. Similarly, the performance with −30% and +30% changes of Dctsw are Δn > 1.6×10−4, PLR > 43 dB, HOMER > 14.3 dB and loss of FMy < 0.067 dB/m, while the performances with −20% and +20% changes of tver are Δn > 2.3×10−4, PLR > 44.7 dB, HOMER > 13.8 dB and loss of FMy < 0.055 dB/m. Therefore, the low-loss SPSM HC-PBF with TSW is tolerant to variations in t, Dctsw, and tver of TSW. This robustness facilitates its fabrication.
In summary, we presented a detailed study of a novel and practical SPSM HC-PBF with TSW. The TSW structure is introduced by changing the stacking method of the two middle row capillaries from triangular to square arrangements in the capillary stacking structure of HC-PBF. Then, the theoretical analysis of mode coupling is discussed to explain the mechanism of the SPSM characteristics in the HC-PBF with TSW. The SPSM mechanism is based on the mode coupling theory, in which one polarized FM or HOMs suffer high loss by coupling with higher-order radiation TE/TM modes in the same electric field direction and for similar neff, while another polarized FM without mode coupling maintains a low loss. The SPSM mechanism was verified using a simplified model of HC-ARF with TSW. Finally, parameter optimization and performance simulation were studied in detail. The results show a PLR of ∼ 46.9 dB, Δn of ∼ 2.4×10−4, loss of ∼ 35.9 dB/km, and minimum HOMER > 15 dB. Moreover, well performance could be maintained at λ = 1.44 ∼ 1.56 µm and rc > 9 mm. In addition, other designs of HC-PBFs with TSW with different numbers of core cells and the influence of fabrication tolerances on the performance of HC-PBFs with TSW were also described in detail. In summary, the SPSM HC-PBF with TSW has excellent low-loss, Hi-Bi, high PLR, high HOMER, and low BL performance. Thus, the proposed model might be suitable for resonator sensing paths of miniaturized RFOGs, and other applications such as high-performance interferometers, fiber lasers, frequency metrology, quantum communications, laser-based gas sensing. Moreover, the considerable decrease in polarization noise and Kerr-induced error is well suited for practical applications. The design of TSW in HC-PBF can also be extended to fiber devices of HC-PBF such as polarizers and SP couplers.
National Natural Science Foundation of China (61935002).
The authors declare that there are no conflicts of interest related to this article.
Data underlying the results presented in this paper may be available from the corresponding author upon reasonable request.
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