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Abstract
A novel-hybrid structure polarization-maintaining 19-cell hollow-core photonic bandgap fiber (HC-PBGF) is proposed. Robust single-mode characteristic is achieved by introducing six anti-resonant tubes into the core of 19-cell HC-PBGF. A high birefringence at the level of 10−3 is achieved by adding silicon layers into the y-direction tubes. The higher-order mode extinction ratio (HOMER) is greater than 4.71 × 107, and the high birefringence can be improved to 5 × 10−3. In the waveband from 1530 nm to 1595 nm, the single-mode, high birefringence performance can be effectively maintained even under a tight bending radius of 5 mm.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Polarization-maintaining (PM) fibers [1] can reduce the random birefringence or polarization mode dispersion caused by external disturbances due to their artificially induced high birefringence [2,3], and has been widely applied in many polarization-sensitive systems, such as laser systems [4,5], fiber-optic hydrophone, and fiber-optic gyroscopes (FOGs) [6,7]. Taking advantage of guiding light in the air core, PM hollow-core fibers (PM-HCFs) can realize near-zero dispersion [8], ultra-low nonlinearity, high damage threshold, and low stress/thermal sensitivity [9], which makes them more promising than PM solid-core fibers in the field of high-precision interference sensors, high-power pulse laser systems, and so on [10,11].
The first PM HCF was proposed by K. Saitoh et al. in 2002 [12]. The fiber is a hollow-core photonic bandgap fiber (HC-PBGF) with a high birefringence in the order of 10−3. The high birefringence is achieved by the rhombus-shaped hollow core, which is formed by missing 4-cell air holes. In 2004, X. Chen et al. successfully fabricated a 4-cell HC-PBGF [13]. Experimental results have proved that the birefringence of HC-PBGF can be effectively improved by breaking the circular symmetry of the hollow core. However, HC-PBGF usually requires a large core size to reduce the transmission loss, such as 7-cell, 19-cell, or 37-cell HC-PBGFs. As the core size increases, it becomes very difficult to improve the birefringence by changing the core shape since the energy is more concentrated in the hollow core. Therefore, researchers try to improve the birefringence by changing the core wall thickness. For instance, in 2006, P. Robert et al. proposed a 7-cell HC-PBGF with 4 elliptical nodes on the interface of the core and cladding [14]. The introduction of nodes makes the 7-cell HC-PBGF obtain a birefringence of 10−4. In 2014, M. John et al. proposed a 19-cell PM HC-PBGF, which realized a birefringence of 3 × 10−4 by a tiny difference in the x- and y-direction wall thickness of the hollow core [15]. Nevertheless, it is still a great challenge to further improve the birefringence in HC-PBGF with a large core size. The appearance of hollow-core anti-resonant fibers (HC-ARFs) [16,17] provides another solution for developing PM HCF. HC-ARFs, which feature broadband transmission due to the anti-resonant reflecting optical waveguide (ARROW) guidance mechanism [18,19], are more varied in structure than HC-PBGFs [20]. Intensive study has proved that HC-ARFs are more advantageous in realizing single-mode and low-loss transmission. For instance, the single-ring HC-ARF can realize single-mode characteristic by adjusting the size of anti-resonant tubes [21]. The confinement loss can be significantly reduced by adding extra anti-resonant layers [22,23]. Since the anti-resonant guidance mechanism does not call for a strictly periodic cladding structure, HC-ARFs are more flexible in design. Many novel structures have been proposed, such as the dual hollow-core structure [22], the ice-cream cladding HC-ARF [24], the semicircular cladding HC-ARF structure [25], the epsilon negative-based HC-ARF structure [26], and so on. In terms of PM HC-ARF, the common method to enhance the birefringence is introducing differential wall thickness. For example, in 2015, S.A. Mousavi et al. proposed a PM HC-ARF with a birefringence of 10−4 by introducing differential wall thickness for the outer tubes and nested tubes [27]. Recently, the introduction of silicon layers to enhance birefringence has also been proved to be an effective method. For example, in 2021, M. Selim et al. proposed a PM HC-ARF with hybrid silica/silicon cladding, which achieved a birefringence of 5 × 10−5 [28]. In 2021, X. Zhao et al. proposed a PM HC-ARF with a birefringence of 3 × 10−4 by simultaneously introducing different wall thickness and silicon layers [29]. However, the PM HCFs usually suffer a high bending loss, which seriously restricts its application in FOGs. To tackle these issues, the hybrid structure, which combines a PBG cladding with anti-resonant layers, comes to be an effective solution. Taking advantage of the PBG cladding to enhance the bending resistance performance and taking the anti-resonant layers to modulate the mode characteristics, the hybrid structure has been proved to be more advantageous [31]. Our group has already taken use of the hybrid structure to develop PM HCF and achieved a high birefringence in the order of 10−3 [32].
In this paper, we propose a novel hybrid structure PM HCF based on a 19-cell HC-PBGF. Robustly single-mode operation can be achieved because of the introduction of six anti-resonant tubes in the 19-cell core. A high birefringence up to 5 × 10−3 is realized owing to the extra silicon layers deposited into the two anti-resonant tubes in y-direction. Moreover, numerical analysis results prove that the proposed fiber has excellent bending resistance performance. The robust single mode operation and high birefringence can be maintained under a tight bending radius of 5 mm, and the bending loss is as low as 10−5 dB/m.
2. Structure and principle
Figure 1 illustrates the structure of our proposed PM HCF. The fiber is formed by adding six tubes into the core region of a typical 19-cell HC-PBGF, which aims to modulate mode contents taking use of the anti-resonant effect [30]. High refractive index silicon layers (represented by orange color) with a thickness of ts are added into the two tubes along the y-direction to enhance the mode birefringence [28]. To reduce the difficulty of adding silicon layers, the tubes with silicon layers are slightly larger than the rest ones. As shown in Fig. 1(b), the radii of small tubes and large tubes are represented by R1 and R2 respectively, where R1 is set as 2.32 µm, R2 is set as 2.57 µm. The thickness of these tubes (t1) is identical and set as 0.5 µm. Other structural parameters are set as follows: PBG cladding lattice constant Λ=4 µm, the diameter of cladding air hole D = 0.98Λ, the hexagonal air hole circular angle diameter is 0.55Λ, the wall thickness of the 19-cell core is 0.5 × (Λ-D). The effective refractive index of silicon is set as 3.48 [31,32], the thickness of silicon layer (ts) is set as 0.125 µm.
Fig. 1. Cross section (a), the core region (b) and enlarged view of the silicon layer (c) of the proposed fiber.
To illustrate the design principle of the proposed fiber, we make a comparison between the following four kinds of fibers. It can be seen from Fig. 2(a) that four kinds of fiber labeled as FD#1, FD#2, FD#3, and FD#4, are a typical 19-cell HC-PBGF, a 19-cell HC-PBGF with six tubes, a 19-cell HC-PBGF with both tubes and silicon layers, and a HC-ARF with the same design but without a PBG cladding. Figure 2(b) shows the confinement loss (CL) and the higher-order mode extinction ratio (HOMER) of the four kinds of fibers. The CL can be calculated through CL = 8.686k0Im(neff) (where k0 is the wavenumber in vacuum and neff is the effective refractive index) [33]. The HOMER is defined as HOMER = CLHOM/ CLFM, which represents the ratio of the lowest CL of high-order modes (HOMs) to the highest CL of fundamental modes (FMs). The simulation is carried out by COMSOL Multiphysics, a commercial software based on full-wave vector finite element method (FEM). In the simulation process, a perfectly matched layer (PML) boundary is used to absorb any radiation leaking from the proposed optical fiber. For the bandgap cladding, the maximum mesh sizes in air and silica regions are set as λ/2 and λ/5, respectively. In terms of the anti-resonant tubes and silicon layers, the maximum mesh size is set as λ/15.
Fig. 2. Four kinds of fiber structures (a), the comparison of their confinement loss and HOMER (b), birefringence and PER (c).
It can be seen from Fig. 2(b) that the introduction of anti-resonant tubes can help the HC-PBGF to reduce the CL of FM and improve the HOMER in the meantime. For example, at the wavelength of 1550 nm, the CL of FM is 1.5 × 10−6 dB/m for FD#1, and it reduces to 8.44 × 10−8 dB/m after adding 6 anti-resonant tubes in the core (FD#2). Accordingly, the HOMER is significantly improved to 1.89 × 108, which indicates that a robust single-mode transmission can be realized. Moreover, the introduction of silicon layers not only has a negligible effect on the single-mode performance, but also greatly improves the mode birefringence. It can be seen from Fig. 2(c) that the birefringence is improved from 10−9 to 10−3 when the silicon layers are introduced (FD#3). In addition, compared with FD#4, the main contribution of the PBG cladding in FD#3 is the significant reduction in confinement loss. It can be seen from Fig. 2(b) that the confinement loss can be reduced by 109 orders of magnitude by adding a PBG cladding on FD#4. It is worth noting that the extremely high confinement loss for FD#4 is due to that the core diameter is only 8 µm. Although the confinement loss can be reduced by enlarging the core size or adding nested tubes in a HC-ARF, the birefringence will also reduce at the same time since a lower confinement loss means a smaller spatial overlap between the core mode and cladding structure. In contrast, the introduction of a PBG cladding can significantly reduce confinement loss without affecting the high birefringence performance.
3. Numerical analysis and discussion
3.1 Impact of structural parameters
Since the single-mode and high birefringence characteristics are achieved by the anti-resonant tubes and silicon layers, the impact of structural parameters of anti-resonant tubes and silicon layers are analyzed in this section. Figure 3 illustrates the impact of tube size on the CL and HOMER at the wavelength of 1.55 µm. It can be seen that the CL of HOM increases with the tube radius (R1), which leads to the significant improvement of HOMER. This is because the increase in tube size can enhance the coupling of core HOMs with tube modes. When R1 varies from 2.1 µm to 2.45 µm, the CL of FM keeps in the order of 10−6 dB/m while the CL of HOM increases from 10−2 dB/m to 104 dB/m, the corresponding HOMER also improves from 105 to 1010, especially when R1 is larger than 2.32 µm, the CL of HOM will be higher than 1 dB/m, indicating a robust single-mode transmission.
Figure 4 shows the impact of silicon layer thickness (ts) on the CL and birefringence. It can be seen that the birefringence increases slowly with the increase of ts, but when ts exceeds 0.132 µm, the birefringence suddenly drops from 5.7 × 10−3 to 10−5. Meanwhile, the CL of y-polarized FM increases from 10−7 dB/m to 10−5 dB/m.
Fig. 4. Confinement loss and Birefringence as functions of the thickness of silicon layer ts at the wavelength of 1.55 µm.
To explain why inflection points appear in the curves, the mode field distribution of FM, cladding mode (CM), and their effective refractive index curves are given. As shown in Fig. 5(a), the effective refractive index curve of the x-polarized FM has no obvious variation with the increase of ts. However, the effective refractive index curve of the y-polarized FM drops down when ts reaches 0.132 µm. That is because the y-polarized FM suddenly changed when the silicon thickness reaches 0.132 µm. The evolution of y-polarized FM can be directly observed in Fig. 5(b) (represented by red arrow). It can be concluded that when the silicon layer is thin (less than 0.13 µm), the mode field will expand toward the cladding tubes under the modulation of silicon layers, which also results in a high birefringence. However, as the silicon layer thickness increase, the modulation effect will disappear and the thick silicon layers only act as high refractive index absorption layers to introduce an additional loss. Therefore, in our design, the thickness of silicon layers is set as 0.125 µm.
Fig. 5. Effective refractive index (a), and mode field distribution (b) with the change of ts at the wavelength of 1.55 µm.
Figure 6 illustrates the confinement loss and birefringence characteristics of the designed fiber in the waveband from 1345 nm to1650 nm. It can be seen that the CL of FM is maintained below 10−3 dB/m while that of HOM is in the order of 104 dB/m. The HOMER keeps above 107, and even up to 1010 when the wavelength ranges from 1530 nm to 1595 nm, meanwhile the birefringence also reaches the highest level (6.12 × 10−3). Therefore, the proposed fiber can realize a wideband single-mode, high birefringence characteristics, and the operation waveband reaches 65 nm (from 1530 nm to 1595 nm). Figure 6(c) shows the mode field distribution of the FM and HOMs at the wavelength of 1550 nm. It can be seen that the HOMs all suffer a CL as high as thousands of dB/m while the x-polarized FM is well-maintained with a CL as low as 1.78 × 10−6 dB/m. Due to the introduction of silicon layers in the y-direction tubes, y-polarized FM expands to the tubes and leads to a high birefringence. It is worth noting that the CL of y-polarized FM is lower than the x-polarized FM although the mode field distribution of the y-polarized FM has distortions, this is because the high-index silicon layers enhance the confinement ability of y-polarized FM.
Fig. 6. Confinement loss (a), birefringence and HOMER (b) as a function of the wavelength, (c) mode field distribution at the wavelength of 1.55 µm.
3.2 Impact of bending
The impact of bending on single-mode and birefringence is analyzed in this section. For a bent fiber, the effective refractive index profile can be equivalent to a straight fiber with the following expression [34]:
where neq(x,y) is the equivalent refractive index distribution of a bent fiber, n(x,y) is the refractive index distribution of a straight fiber, R is the bending radius, and θ is the angle between the bending direction and + x axis. As shown in Fig. 7, θ=0° and θ=90° represents the bending direction along + x and + y axis, respectively.Figure 8 shows the CL and birefringence curves under different bending radii. The legend ‘x-bend’ or ‘y-bend’ represents the bending direction along the + x axis or + y axis, respectively. It can be seen that there is no significant difference in CL and birefringence along with two opposite directions of x or y (solid and dashed lines are substantially coincident). Taking the ‘x-bend’ as an example, the bending loss of x-polarized FM remains below 10−5 dB/m for R ≥ 5 mm, and the bending loss of y-polarized FM remains below 10−6 dB/m for R ≥ 3 mm. To ensure the stable transmission of both polarization states of FM, the critical bending radius is set as 5 mm. Figure 8(b) shows the mode field distribution of FMs and HOMs at the critical bending radius. It can be seen that there is no obvious deformation for the FM under the tight bending radius (5 mm). On the contrary, the HOMs will obviously leak to the cladding, indicating that the single-mode characteristic can be improved at a bent state.
Fig. 8. (a) Confinement loss and birefringence for x-bend and y-bend as functions of bending radius R, (b) mode field distributions for x-bend and y-bend with the critical bending radius R = 5 mm at λ=1.55 µm.
Under the critical bending radius, the impact of bending orientation on CL and birefringence is investigated. It can be seen from Fig. 9 that the bending direction has little effect on the CL and birefringence. When the bending orientation angle varies from 0° to 90°, the CL for x-polarized and y-polarized FM is maintained at 10−5 dB/m and 10−6 dB/m, respectively, the CL for HOM is as high as 103 dB/m, and the birefringence keeps above 5.4 × 10−3. The results show that the structure is not sensitive to the bending direction, although the symmetry of the structure is broken by the introduction of silicon layers in the y-direction tubes.
Fig. 9. Confinement loss and birefringence for different θ with the critical bending radius R = 5 mm at the wavelength of 1550 nm.
As mentioned in section 3.1, the optimum operation bandwidth for the proposed fiber is from 1530 nm to 1595 nm. Thus, we analyze the bending characteristics in this waveband. It can be seen from Fig. 10 that in the wavelength from 1530 nm to 1595 nm, the birefringence keeps nearly constant when the fiber is bent to the critical bending radius (5 mm). In terms of the loss characteristic, the increase in CL introduced by bending for x-polarized FM is more obvious than for y-polarized FM. For example, at the wavelength of 1550 nm, the CL of x-polarized FM increases from 1.78 × 10−6 dB/m to 2.62 × 10−5 dB/m when the fiber is bent to 5 mm, while the CL of y-polarized FM only increases from 3.78 × 10−7 dB/m to 8.76 × 10−7 dB/m. Meanwhile, the CL of HOM also significantly increases from 78.6 dB/m to 5.75 × 103 dB/m, making the fiber robustly single-mode transmission. The above analysis all proves that the fiber has good bending resistance, and the bandwidth of single-mode and high birefringence is 65 nm.
3.3 Tolerance discussion
In terms of fabrication, the proposed hybrid structure which combines the PBG cladding and anti-resonant tubes can be fabricated by the stack-and-draw method [35], and the silicon layers can be deposited using a high-pressure chemical vapor deposition method [36]. However, during the fabrication process, the deviation of structural parameters is unavoidable. Therefore, structural tolerance is discussed in this section.
First of all, the tolerance of the thickness of the silicon layers (ts) is discussed since the silicon layers play a key role in achieving high birefringence. Figure 11 shows the CL and birefringence characteristics when there is a ± 5% variation of the silicon layer thickness (ts). The initial value of ts is set as 0.125 µm. It can be seen that in the waveband from 1530 nm to 1595 nm, the CL of FM fluctuates below 10−5 dB/m while the CL of HOM fluctuates above 100 dB/m when ts fluctuates by ±5%. The HOMER keeps above 107. The birefringence will decline at the short-wavelength when ts fluctuates +5%, but it still maintains in the order of 10−3. Therefore, it can be concluded that ±5% variation of silicon layer thickness (ts) has a neglectable impact on the single-mode and high birefringence characteristics.
In addition, the tolerance of the tube thickness (t1) is also taken into consideration since the tube thickness is the most important parameter that needs to be precisely controlled during the fabrication process. It can be seen from Fig. 12 that ±5% variation of tube thickness will neither affect the single-mode nor high birefringence characteristics, but it is still worth noting that if the fluctuation of tube thickness is above +5%, the CL of HOM will be reduced to less 1 dB/m, which means that the HOM cannot be suppressed anymore.
4 Conclusion
In this paper, a hybrid structure PM HC-PBGF with anti-resonant tubes and silicon layers is proposed and numerically analyzed. Numerical analysis results demonstrate that the introduction of six anti-resonant tubes can help the 19-cell HC-PBGF to realize single-mode operation. Moreover, adding silicon layers into the y-direction tubes can significantly enhance the birefringence without affecting the single-mode characteristics. In the wavelength from 1530 nm to 1585 nm, the proposed fiber can achieve robust single-mode operation with a HOMER higher than 4.71 × 107. Meanwhile, the birefringence is up to the level of 10−3. The robust single-mode and high birefringence characteristics can be effectively maintained under a tight bending radius of 5 mm. Tolerance analysis results indicate that ±5% variation of silicon layer thickness and tube thickness will not affect the single-mode and high birefringence characteristics. This work provides a novel approach to the development of PM HCF.
Funding
National Natural Science Foundation of China (12174022).
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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