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Abstract
A novel single-polarization single-mode double-ring hollow-core anti-resonant fiber with two single-polarization regions (1545–1553 nm and 1591–1596 nm) is proposed. Single-polarization guidance is achieved by coupling a polarized fundamental mode and silica mode by using different tube thicknesses. Specifically, when the wavelength is 1550 nm, only a single x-polarized fundamental mode with a low loss of 0.04 dB/m is propagated by a polarization extinction ratio of 17662 and minimum higher-order mode extinction ratio of 393 by optimizing the structural parameters. Furthermore, this fiber also exhibits high-performance bend resistance. The x-polarized FM loss is as low as 0.11 dB/m with single-polarization single-mode guidance when the proposed fiber was bent at a bend radius of 8 cm toward the x-direction.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Polarization effects play a key role in a wide range of polarization-sensitive optical fiber systems, such as, fiber lasers [1], fiber sensors [2], and fiber-optic gyroscopes (FOGs) [3]. Generally, a fiber is not perfectly uniform owing to the imperfections occurring in the fabrication process, such as, distortions, defects, uneven stresses, and environment perturbations, such as, bends and twists. These introduce a random birefringence and result in an unpredictable change in the polarization state of the light [4,5]. To solve this problem, high-birefringence polarization-maintaining fibers (HBPMFs), in which either a stress or an asymmetric form is introduced to produce high birefringence, are applied in many polarization-sensitive optical systems [6–8]. However, the polarization mode dispersion cannot be eliminated by using HBPMFs, because two types of polarized light propagate with different propagation constants through them [9]. Single-polarization fibers (SPFs) propagate light in only one polarization state, whereas the light in another polarization state is either forbidden or suffers a high loss. Thus, compared with HBPMFs, the effects of polarization-mode dispersion can be efficiently eliminated by employing SPFs.
Present SPFs are usually based on solid-core fibers [10], which are limited by material characteristics. Material dispersion and nonlinearity can cause pulse distortions and material damage threshold limits the input power of solid-core fibers. Moreover, stress birefringence is introduced when a solid-core fiber is bent, which causes polarization mode dispersion and a change in the polarization state of the light. Air owns the characteristics of low dispersion, low nonlinearity, high damage threshold, and insensitivity to stress. Hollow-core fibers (HCFs) guide light in the air core, and thus, can effectively reduce the above-mentioned constraints [11,12]. Therefore, HCFs are considered as potential candidates for optical systems requiring low dispersion or nonlinearity (e.g., FOGs). FOG is a type of optical-angle velocity sensor based on the Sagnac effect [3]. At present, commercial FOGs are mainly based on interferometric fiber optic gyroscopes (IFOGs), which enhance the Sagnac effect by using long multi-turn optical fiber coils [13]. Resonant fiber optic gyroscopes (RFOGs) are another type of FOGs that generate an enhanced Sagnac effect by the circulation of light for many turns in an optical ring resonator [14]. Compared with IFOGs, RFOGs have the advantages of small size, light weight, high precision, low cost and, low thermally induced nonreciprocity, and thus, they are considered as important candidates for high-accuracy miniaturization of FOGs [15]. Polarization-maintaining HCFs have been used in RFOGs to decrease the effects of various optical noises [16], such as, the noises caused by the Rayleigh backscatter, Kerr effect, Faraday effect, and Shupe effect. By using hollow-core SPFs instead of polarization-maintaining HCFs, the effects of the polarization state on the accuracy of RFOGs can be further reduced and consequently a higher accuracy can be reached.
Hollow-core photonic bandgap fibers (HC-PBGFs) are a type of HCFs that guide the light via the photonic band gap [11]. To form a photonic band gap, the cladding tubes of the HC-PBGF must be arranged with strict periodicity. Therefore, an HC-PBGF has some disadvantages, such as, complex structure, low design freedom, and narrow bandwidth [17,18]. In recent years, there has been a continuously increasing interest in another type of HCF, i.e., hollow-core anti-resonant fiber (HC-ARF). The guidance mechanism in HC-ARFs is different from that in HC-PBGFs. Both core modes and cladding modes are present in HC-ARFs. At specific wavelengths (resonant wavelengths), the core modes couple with the cladding modes so that light cannot be restricted to propagate in the air core. In comparison, at wavelengths far away from the resonant wavelengths, the core modes do not couple with the cladding modes, and therefore, it can be restricted to propagate in the air core with a small loss [19]. Compared with HC-PBGFs, HC-ARFs have the advantages of a simple structure, flexibility in design, and broad bandwidth [20,21]. Moreover, with single-polarization guidance they can assist in decreasing the polarization noise in RFOGs. In 2016, Mousavi et al. reported single-polarization guidance in a single-ring four-tube HC-ARF with a polarization extinction ratio (PER) of approximately 1000 at 1550 nm [17]. In 2018, Wei et al. achieved single polarization in a single-ring six-tube HCARF with PER of 850 at 1550 nm [22]. However, the single-mode and bend-resistance performances of HC-ARFs were not mentioned in [17] and [22]. Multi-mode guidance can cause modal dispersions, and fibers usually become bent or coiled in FOGs. Therefore, single-mode and bend-resistance performances are important properties of HC-ARFs for RFOGs.
In this paper, we propose a novel double-ring HC-ARF that can achieve single-polarization guidance not only with an ultrahigh PER of 17662 but also with a higher-order mode extinction ratio (HOMER) of 393. Two single-polarization regions are present simultaneously: a single x- and y-polarization region in a wavelength range from 1545 nm to 1553 nm and from 1591 nm to 1596 nm, respectively. The x-polarized fundamental mode (FM) loss is only 0.04 dB/m at the wavelength of 1550 nm. Moreover, this fiber also exhibits a good bend resistance, so that it can maintain single-polarization single-mode guidance for the x-direction bend with a bend radius of 8 cm.
2. Fiber structure and performance
The cross-section of the proposed fiber is shown in Fig. 1(a). To achieve single-mode and single-polarization guidance simultaneously, a double ring structure is employed in a jacket silica tube. Compared with a single-ring circular structure, this double ring structure has the advantages of easy implementation of single-mode guidance and more flexibility in the fiber design. The outer ring silica tubes have same radius R1 and thickness t1. The inner ring silica tubes have two sizes, of which two large tubes (orange color) have radii R2 and thicknesses t2 and four small tubes have the same radius R3 and thickness t1. The semi-minor axis of the core region is represented by Rc.
Fig. 1 Cross-section of the proposed fiber (a) and the electric field distributions of the FM at the wavelengths of (b) 2060 nm and (c) 1050 nm.
At the beginning, structure parameters Rc, R1, R2, R3, t1, and t2 are set with fixed values of 7.7 μm, 14 μm, 6 μm, 4.6 μm, 0.5 μm, and 1 μm, respectively. The finite-element method perfectly matches the layer boundary conditions used to simulate the transmission characteristics of the proposed HC-ARF. For the HC-ARF, the resonant wavelengths can be obtained by [18],
where n is the refractive index of the fiber material, m equals any positive integer, and t is the tube thickness.Using Eq. (1), two resonant wavelengths 1050 nm and 2060 nm are obtained respectively at t1 = 0.5 μm and t2 = 1 μm when m = 1, which are far away from the wavelength of 1550 nm. Figure 1(b) and 1(c) show the electric field distributions of the FM at the wavelengths of 2060 nm and 1050 nm, respectively. It can be seen that the FM at 2060 nm spreads into large tubes in the inner ring, because 2060 nm is a resonant wavelength at t2 = 1 μm and m = 1. In comparison, the FM at 1050 nm spreads into the entire fiber section, because 1050 nm is a resonant wavelength at both t1 = 0.5 μm, m = 1 and t2 = 1 μm, m = 2.
The proposed HC-ARF has two silica modes (SM1 and SM2) in the wall of two large tubes in the inner ring. The silica modes are the cladding modes that are attached to the surfaces of the cladding tubes. From the simulation, we find that SM1 can couple with the y-polarized FM, whereas SM2 can couple with the x-polarized FM. Therefore, a single x- and y-polarization region is formed in the wavelength region near the wavelength at which SM1 couples with the y-polarized FM and SM2 couples with the x-polarized FM, respectively. The confinement loss is obtained by,
where k0 = 2π/λ is the wave vector and Im(neff) is the imaginary part of the effective refractive index.Figure 2(a) shows the effective refractive indices of the x- and y-polarized FMs, SM1, and SM2 and birefringence. It can be seen that there are abrupt changes in the effective refractive index curves of the y- and x-polarized FMs at the wavelengths of 1550 nm and 1594 nm, respectively. The effective refractive indices of SM1 and the y-polarized FM and of SM2 and x-polarized FM are approximately equal at the wavelengths of 1550 nm and 1594 nm, respectively. Therefore, mode coupling between SM1 and the y-polarized FM occurs at the wavelength of 1550 nm and between SM2 and the x-polarized FM occurs at the wavelength of 1594 nm. This can be also observed in the electric field distribution in Fig. 2(b). The birefringence of the proposed fiber is above 10−5, and abrupt changes exist at the wavelengths of 1550 nm and 1594 nm owing to the mode couple.
Fig. 2 Effective refractive indices and birefringence (a) and electric field distributions (b) of the x-polarized and y-polarized FMs and two SMs; the losses in the x-polarized and y-polarized FMs (c) and corresponding PERs and HOMERs (d) of these modes.
Figure 2(c) displays the losses in the x- and y-polarized FMs. Owing to the mode coupling, there is a decrease in the peaks of the y- and x-polarized FMs at the wavelengths of 1550 nm and 1594 nm, respectively. In this paper, PER is defined as the loss ratio of the y-polarized FM to the x-polarized FM and HOMER is defined as the minimum loss ratio of the higher-order modes or cladding modes to the lower-loss FM. It can be regarded as single-polarization guidance if PER is higher than 100 (single x-polarization) or lower than 0.01 (single y-polarization) and regarded as single-mode guidance if HOMER is higher than 100.
As shown in Fig. 2(d), there are two single-polarization regions (marked by gray) of a single x-polarization region within the wavelength range from 1545 nm to 1553 nm and a single y-polarization region within the wavelength range from 1591 nm to 1596 nm. Especially, it is not only single-polarization guidance but also single-mode guidance within the wavelength range from 1545 nm to 1553 nm. PER reaches up to 17662, and the loss in the x-polarized FM is only 0.04 dB/m at the wavelength of 1550 nm. The wavelength of 1550 nm is the target wavelength of the proposed fiber, thus only the first single-polarization single-mode region is analyzed in the next sections.
In this structure, the thickness of the two large tubes in the inner ring (radius of R2) is different from that of the other tubes. Although the different tube thicknesses increase the complexity of the structure and fabrication difficulty, the birefringence caused by asymmetric structure with same tube thickness is very small in HC-ARF [17,19]. Therefore, the difference in tube thickness is an effective way to achieve a high birefringence or PER [17,22].
Figure 3 presents the loss ratios of several higher-order modes and cladding modes to the x-polarized FM at the wavelength of 1550 nm. CM1, CM2, and CM3 are the fundamental-like modes in the outer ring tubes (radius of R1), large tubes of the inner ring (radius of R2), and small tubes of the inner ring (radius of R3), respectively. It can be seen from Fig. 3 that the minimum loss ratio is 393 at the wavelength of 1550 nm. Therefore, the proposed fiber exhibits a good single-mode performance at this wavelength.
From the above analysis it can be inferred that in the proposed fiber, single x-polarization guidance is achieved by making the effective refractive index of the y-polarized FM in the core and SM1 to be approximately equal at the wavelength of 1550 nm. The values of R3 and Rc can directly impact the effective refractive index of the y-polarized, whereas that of R1 has some effect on the effective refractive index of SM1. This implies that there exist combinations of (R1, R3, Rc) that approximately equalize the effective refractive indices of the y-polarized FM in the core and SM1 at the wavelength of 1550 nm. Therefore, all R1, R3, and Rc are considered to achieve the most optimized combination. Figure 4 exhibits PER as a function of Rc and R3 at different R1 ranging from 9 μm to 15 μm at the wavelength of 1550 nm. PER can reach a maximum value of 17662 when (R1, R3, Rc) are (14 μm, 4.6 μm, 7.7 μm).
Fig. 4 PER as function of R3 and Rc at the wavelength of 1550 nm for R1 = 9 μm (a), 10 μm (b), 11 μm (c), 12 μm (d), 13 μm (e), 14 μm (f), and 15 μm (g).
3. Bend properties
The bend characteristics of fibers are important in practical applications. However, they vary with the bend direction owing to the anisotropy of the fibers. To compare the bend resistance of at different bend directions, we defined the critical bending radius as the minimum bend radius that can keep single-polarization single-mode guidance for a certain bend direction. In simulation, typically, a bent fiber is equivalent to a straight fiber having an equivalent refractive index distribution expressed as [23]:
where (x, y) are the coordinates with the origin at the center of the fiber, neq(x, y) is the equivalent refractive index distribution of the bent fiber, n(x, y) is the original refractive index distribution of the straight fiber, and R is the bending radius toward the x direction, and θ is the angle between the bend axis and + x axis, which is shown in Fig. 5(a). The critical bend radius and x-polarized FM loss at the wavelength of 1550 nm are shown at different bend directions in Fig. 5(b). And the PER, HOMER, and birefringence at the wavelength of 1550 nm with different bend directions are shown in Fig. 5(c). It can be seen that the critical bend radius for θ varying from 0° to 20° is 8 cm and increases to 46 cm for θ = 90°. The birefringence is above 10−5 and the x-polarized FM loss is smaller than 0.21 dB/m at the critical bend radius. The x-polarized FM loss is 0.11 dB/m with PER of 407 and HOMER of 152 at bend radius of 8 cm.Fig. 5 Bend degree of θ (a), critical bend radius and x-polarized FM loss (b) for different θ, and PER, HOMER, and birefringence (c) at different θ with the critical bend radius at the wavelength of 1550 nm
Figure 6(a) presents the x-polarized FM loss and birefringence and Fig. 6(b) presents the PER and HOMER in 1540–1560 nm for straight condition and bend radius of 8 cm and 15 cm with θ = 0°. It can be seen that bend can cause the increase of x-polarized FM loss and decrease of PER and HOMER. With bend radius of 8 cm, the single-polarization single-mode region is located in the wavelength range from 1547 nm to 1552 nm. The bandwidth decrease to 5 nm from 8 nm of straight condition. Birefringence for the wavelength without mode couple changes slightly and that for the mode-couple wavelength changes significantly because bend result in the variation of mode-couple wavelength.
Fig. 6 The x-polarized FM loss and birefringence (a), and PER and HOMER (b) in 1540–1560 nm for straight condition and bend radius of 8 cm and 15 cm with θ = 0°.
4. Fabrication tolerances
In 2018, a conjoined tube HC-ARF has been fabricated successfully [24], whose structure is similar to our proposed fiber. In fabrication process, size fluctuation and rotation of tubes may occur unintentionally [25]. In this section, we analyzed the fabrication tolerances on the performance of the fiber. Figure 7(a) shows the rotation of four small tubes of the inner ring (radius of R3). Rotation direction shown by black arrows and rotation angle is represented by α. Figure 7(b), (c), and (d) show the x-polarized FM loss, birefringence, PER and HOMER 1540–1560 nm for ideal condition and α = ± 5°. It can be seen that loss increases for α = − 5° and decrease slightly for α = + 5°. Birefringence for the wavelength without mode couple increases with an increase of α. Rotation of the four tubes causes a move of single-polarization region in wavelength. PER is 263 with HOMER of 227 for α = + 5° and PER is 1592 with HOMER of 362 for α = − 5° at the wavelength of 1550 nm.
Fig. 7 Rotation of four tubes with degree of α (a), x-polarized FM loss (b), birefringence (c), and PER and HOMER (d) for ideal condition and α = ± 5°.
Figure 8 presents x-polarized FM loss, birefringence, PER and HOMER for ideal condition and ± 10% variation of diameter of large tubes (R1). It can be seen that loss increases with + 10% variation of diameter and changes a little with −10% change of diameter. The change of large tube diameter causes a move of single-polarization region in wavelength. PER is 236 with HOMER of 134 for + 10% variation of diameter and PER is 2574 with HOMER of 403 for – 10% variation of diameter at the wavelength of 1550 nm.
Fig. 8 The x-polarized FM loss (a), birefringence (b), and PER and HOMER (c) for ideal condition and diameter change of ± 10%.
5. Discussion
By using the mode coupling between the y-polarized FM and SM1, single x-polarization guidance at a wavelength of 1550 nm can be implemented with the proposed double-ring structured HC-ARF. This single-polarization guidance can be also achieved at other wavelengths or in other structures by using only the coupling of a single-polarized FM core mode with a silica mode, without allowing the coupling with other polarized FMs. Here, we discuss the single-polarization guidance at other wavelengths or in other similar structures.
For approximately equalizing the effective refractive indices of a single-polarized FM core mode and silica mode at other wavelengths, an effective approach is to cause the effective refractive indices of the y-polarized FM and SM1 to be nearly invariable at wavelengths other than 1550 nm. We select some common wavelengths in optical communication, such as, 980 nm, 1060 nm, and 1310 nm, and a mid-infrared wavelength of 3000 nm. The material of the tubes is silica for achieving optical communication wavelengths and ZBLAN glass for mid-infrared wavelengths owing to its low material absorption in the mid-infrared region [26]. The index of ZBLAN glass is 1.482 in terms of the Sellmeier formula [27].
We first enlarge this structure with an enlargement factor of λ/λ0, where λ is the target wavelength and λ0 is the original wavelength of 1550 nm. We then adjust the value of t2 to achieve single-polarization guidance at the target wavelength. The related parameters and results are presented in Table. 1. It can be seen that single-polarization single-mode guidance at common optical communication wavelengths, such as, 980 nm, 1060 nm, and 1310 nm, and mid-infrared wavelength of 3000 nm can be implemented only by adjusting the structural parameters.
Table 1. Related Parameters and Results at Other Wavelengths
Next we discuss the achievement of single-polarization guidance in different multi-layered structures. A multi-layer split cladding fiber was reported to be fabricated successfully [28,29]. Therefore, it is relevant to discuss the single-polarization guidance occurring in a multi-layer split cladding fiber having the same structure. Therefore, we analyzed double-layer and triple-layer split cladding HC-ARFs, whose structures are displayed in Fig. 9. The core radius is 10 μm, and the radius of all the cladding silica tubes is 6 μm. The wall thickness of the normal cladding tubes (blue tubes) is 0.5 μm. Two thick cladding tubes are marked in orange. The thicknesses of both the thickened tubes and numerical results are presented in Table 2 at the operation wavelength of 1550 nm. As can be seen from Table 2, we can achieve single-polarization guidance in double-layer and triple-layer split cladding HC-ARFs by adjusting the wall thicknesses of both the thickened tubes. Figure 10 exhibits the electric field distributions of the x-and y-polarized FMs and silica mode in double- and triple-layer split cladding HC-ARFs at the wavelength of 1550 nm. It can be seen clearly from Fig. 10 that the y-polarized FM in the double- and triple-layer split cladding HC-ARFs couples with the silica mode, whereas the x-polarized FM does not couple with the silica mode at the wavelength of 1550 nm, which confirms the single-polarization guidance.
Table 2. Thickened Thickness and Results
Fig. 10 Electric field distributions of the x-polarized FM (a), y-polarized FM (b), and silica mode (c) in double-layer split cladding HC-ARF and x-polarized FM (d), y-polarized FM (e), and silica mode (f) in the triple-layer split cladding the HC-ARF.
Figure 11 presents x-polarized FM loss, PER, HOMER, and birefringence for double- and triple-layer split cladding HC-ARFs. It can be seen that there is a single-polarization region of 1548–1551 nm in double-layer split cladding HC-ARF and a single-polarization region of 1550–1555 nm in triple-layer split cladding HC-ARF. But the HOMER of these two fibers is very low, which confirms that these two fiber are not single-mode guidance in this wavelength range. The birefringence of these two fiber is above 10−5.
Fig. 11 The x-polarized FM loss, PER, HOMER (a), and birefringence (b) of double-layer split cladding HC-ARF and the x-polarized FM loss, PER, HOMER (c), and birefringence (d) of triple-layer split cladding HC-ARF.
6. Conclusion
In conclusion, we proposed a novel single-polarization single-mode double-ring HC-ARF. Two single-polarization regions, one each in the wavelength region from 1545 nm to 1553 nm and from 1591 nm to 1596 nm, were achieved by using different tube thicknesses. An ultrahigh PER of 17662 was achieved at a wavelength of 1550 nm with a low x-polarized FM loss of 0.04 dB/m and minimum HOMER of 393. The bend-resistance performance of the proposed fiber was good. The x-polarized FM loss was as low as 0.11 dB/m with single-polarization single-mode guidance when the proposed fiber was bent at a bend radius of 8 cm toward the x direction. The proposed fiber could be a potential candidate for RFOGs because of its hollow-core structure, single-polarization single-mode guidance, and good bend resistance performance. In addition, we verified the structure flexibility and universality of the proposed fiber by implementing single-polarization at different wavelengths and in a multi-layer split cladding fiber that was already fabricated successfully.
Funding
National Natural Science Foundation of China (NSFC) (61775014) and Key Project of Hebei Natural Science Foundation (F2018202323).
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