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Abstract
We demonstrated the fabrication of bandwidth tunable ultra-broadband mode converters based on CO2-laser inscribed long-period fiber gratings (LPFGs) and helical long-period gratings (HLPGs) in a two-mode fiber (TMF). The simulation and experimental results show that there is a dual-resonance coupling from LP01 to LP11 core mode at the dispersion turning point. The mode converters based on the TMF-LPFG and TMF-HLPG provide a 10-dB bandwidth of ∼300 nm and ∼297 nm, respectively, which covers O + E+S + C band. The 1st order orbital angular momentum (OAM) mode based on TMF-LPFG was generated by adjusting the polarization controllers (PCs), while the 1st order OAM mode can be generated directly by the TMF-HLPG. When the twist rate is varied from -36 rad/m ∼ 36 rad/m, the tunable range of the 10-dB bandwidth is ∼52 nm and ∼91 nm for the LPFG and HLPG mode converters, respectively. The ultra-broadband mode converter can be adopted as a bandwidth tunable mode converter, which can be applied in ultra-broadband mode-division-multiplexing transmission systems and optical fiber sensing systems based on few-mode fibers.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Mode division multiplexing (MDM) technology has attracted lots of attention, which can utilize orthogonal linearly polarized (LP) modes and orbital angular momentum (OAM) modes as transmission channels to expand communication capacity [1–3]. An important device in MDM system is mode converter, which could realize mode conversion from the fundamental (LP01) mode to high order core modes or OAM modes. The OAM beams with helical phase-fronts are a linear combination of the vector modes or degenerated modes with a π⁄2 phase difference. Many implement methods of mode converters have been reported, such as photonic lanterns [4], fiber couplers [5], fiber Bragg gratings (FBGs) [6], and long-period fiber gratings (LPFGs) [7]. The LPFG based mode converter has the advantages of small size, low cost and flexible fabrication. It can also be flexibly adopted in various systems, such as all fiber laser device [8]. Zhao et al. [9] realized mode conversion from LP01 to high order modes (LP11, LP21, LP02) and demonstrated the generation of the 1st, 2nd order OAM modes based on the CO2 laser inscribed LPFG in a four-mode fiber. Zhang and Wei et al. [10–11] generated the ± 1st order optical vortex beams and 2nd order OAM modes using cascaded acoustically grating in a few-mode fiber (FMF). In order to cover the broadband wavelength channel of optical communication, the mode converter with wide bandwidth would be required. K. Rottwitt et al. proposed a chirped microbend LPFGs to increase the bandwidth of mode conversion between LP01 to LP11 mode with a conversion efficiency of 99% [12]. K. S. Chiang et al. proposed an ultra-broadband mode converter based on a length apodized waveguide long-period grating in the C + L band [13]. Y. Zhao et al. [14] demonstrated an ultra-broadband fiber mode converter based on the length apodized phase-shifted LPFGs with a 10-dB bandwidth of 182 nm. Y. Guo et al. [15] proposed a broadband generation of OAM beams based on a LPFG written in a two-mode fiber (TMF) utilizing a dual-resonance coupling mechanism. They generated OAM modes using a polarization controller (PC) to realize a π⁄2 phase difference. Recently, a simple OAM mode generator based on helical long-period gratings (HLPGs) was proposed [16–20]. The HLPGs were characterized by a π⁄2 phase difference between the degenerated modes because of the helical refractive index (RI) modulation along the fiber axis, which can be used to generate directly OAM modes. Cao et al. [18] generated the 1st order OAM based on a chiral LPFG in a TMF. H. Zhao et al. [19] generated the ± 2nd order OAM modes using chiral LPFGs in an FMF with a 1-dB bandwidth of 10 nm. Y. Zhang et al. [20] generated the ± 1st order OAM modes in an FMF using chiral LPFGs with a 10-dB bandwidth of 25 nm. K. Ren et al. [21] simulated the mode coupling of the HLPGs written in single mode fiber at turning point for the broadband OAM mode conversion.
In this paper, we demonstrated the fabrication of bandwidth tunable ultra-broadband mode converters based on CO2-laser inscribed TMF-LPFGs and TMF-HLPGs. The LP01 to LP11 mode converter based on the TMF-LPFG provides a 15-dB bandwidth of ∼249 nm and a 10-dB bandwidth of ∼300 nm. The 10-dB bandwidth of the mode converter can be tuned from 287 nm to 339 nm with the twist rate varying from -36 rad/m to 36 rad/m. The LP01 to LP11 mode converter based on the TMF-HLPG provides a 15-dB bandwidth of ∼224 nm and a 10-dB bandwidth of ∼297 nm. The 1st order OAM mode was generated directly by the TMF-HLPG. The 10-dB bandwidth of the mode converter can be tuned from 246 nm to 337 nm with the twist rate varying from -36 rad/m to 36 rad/m. The proposed ultra-broadband mode converter can be adopted in the broadband MDM system.
2. Principle and simulation
In the experiment, the fiber used is a two-mode step index fiber. The transverse RI profile of the TMF was measured by a RI profiler (S14, Photon Kinetics), as shown in Fig. 1(a). The TMF has a core diameter of 11 µm and a cladding diameter of 125 µm, the RI difference between cladding and core of the fiber is 0.005, which can support LP01 and LP11 modes. The cut-off wavelength of the fiber is 1700 nm. We calculated the grating period and effective RI of the modes corresponding to different resonance wavelengths by using COMSOL Multiphysics based on the finite element method (FEM), as shown in Fig. 1(b). The dependence of the grating pitches for mode conversion on resonance wavelength shows a non-monotonic parabolic trend, which proves that there exists a dispersion turning point. A higher-order mode that satisfies phase matching conditions can be generated around the turning point, which implies that two resonance wavelengths correspond to the same order mode coupling with the same grating period [22]. According to the coupled-mode theory [23], the resonance wavelength of the gratings satisfies the phase-matching condition,
Fig. 1. (a) The transverse RI profile of TMF measured by S14; (b) Calculated RIs of modes and dependence of the calculated grating pitches for mode conversion on resonance wavelength.
We did the simulation of the transmission spectra of TMF-LPFG based on the Rsoft software. The transmission spectra of the TMF-LPFGs with different grating periods were simulated at an index modulation depth Δn = 0.00057, respectively. Figure 2(a) shows the transmission spectra of the TMF-LPFG with different grating periods. When the period of the LPFG decreases from 620 µm to 611 µm, the two resonance dips move close to each other until the two resonance dips have merged together as one dip. When the grating period is 614 µm, a maximum 10-dB bandwidth of 277.5 nm can be achieved, which covers the wavelength range from 1198.5 nm to 1476 nm. The mode field distributions of the LPFG with a period of 614 µm were calculated at several wavelengths. Figure 2(b) shows the simulated transmission spectrum of the TMF-LPFG and corresponded mode field distributions at different wavelength. The results indicate that an ultra-broadband TMF-LPFG mode converter couples the light from the fundamental mode to LP11 mode over the bandwidth of more than 277 nm.
Fig. 2. (a) The simulated transmission spectra of the TMF-LPFGs with different grating pitches; (b) The simulated transmission spectrum and mode distribution of the TMF-LPFGs with a period of 614 µm.
3. Fabrication of LPFGs
3.1 Fabrication of LPFGs
We fabricated the LPFGs in the TMF using a CO2-laser (CO2-H10, Han’s laser). The average power and pulse frequency of a laser are about 1.0 W and 5.0 kHz, respectively. The CO2-laser was focused on the fiber by a lens with a spot diameter of ∼50 µm. Both sides of TMF were spliced with single-mode fiber (SMF). A supercontinuum source (NKT Photonics) and an optical spectrum analyzer (OSA, AQ6370D, YOKOGAWA) were used to monitor the transmission spectra of the TMF-LPFG. The LPFGs were fabricated with several periods from 611 µm to 618 µm at intervals of 1 µm. The period number is 20. The transmission spectra of the gratings are shown in Fig. 3(a). With the period of the grating decreasing, two resonance dips move towards closely from two opposite directions. When the period reduces to 611 µm, two resonance dips merge together, forming a flat bottom resonance dip. When the grating period is 614 µm, the TMF-LPFG has the widest 10-dB bandwidth, which is 300 nm. Figure 3(b) shows the experimental and simulation results for the resonance wavelength of two dips. The experimental results are well consistent with simulation results. Figure 3(c) shows the transmission spectra of the TMF-LPFG with a period number of 15, 20 and 40, respectively, when the grating period is 614 µm. It can be seen that the LPFG with a less grating period number has wider bandwidth, which is consistent with the previous research [24]. The suitable period number of the TMF-LPFG can be selected to write a LPFG with both wide bandwidth and high coupling efficiency. In our experiments, the period of 614 µm and the period number of 20 is the most perfect parameters for the fabrication of the LPFG with wide bandwidth.
Fig. 3. (a) The experimental transmission spectra of the TMF-LPFGs with different grating periods; (b) The simulation and experimental results of the resonance wavelength of two dips; (c) The transmission spectra of TMF-LPFGs with a period of 614 µm and period number of 15, 20 and 40, respectively.
We observed the mode field distributions at several wavelengths by the experimental setup shown in Fig. 4(a). The light from a tunable laser (81600B, Agilent) was divided into two paths by an optical coupler. One path was used as a reference beam to interfere with the other beam which launched into a LPFG to generate OAM beams. A PC (PC1) is used to adjust the polarization of input light, and another PC (PC2) is used to adjust the phase of the vector modes to realize π⁄2 phase difference. We monitored the mode intensity and interference pattern with a CCD (InGaAs, Model C10633-23, Hamamatsu Photonics) at the output of the TMF. By adjusting the PC1, the intensity distributions of the vector modes at the output were generated. The 1st order OAM modes were generated at a corresponding wavelength by adjusting the PCs to obtain a π⁄2 phase difference. The interference patterns interfered with a Gaussian reference beam were captured. The mode intensity distributions and interference patterns are shown in Fig. 4(b). The proposed mode converters can generate the 1st order OAM mode over the entire 10-dB bandwidth wavelength range. The operating wavelength of the tunable laser is 1454 nm∼1640 nm. Therefore, the wavelength of the observed OAM modes started at 1454 nm. The coupling efficiencies of modes are different at the different resonance wavelengths due to the different grating contrast. Therefore, the purity of the OAM modes is a little different at different wavelengths. The other characteristics of the OAM modes are also a little different at different wavelengths. The results indicated that an ultra-broadband mode converter was realized by the TMF-LPFG.
Fig. 4. (a) Experimental setup to observe the mode distribution; (b) Output mode distribution and interference patterns captured at different wavelengths for the ultra-broadband mode converter.
3.2 Characteristics of LPFGs
The twist sensing characteristics of the TMF-LPFG were investigated experimentally. The period of the grating is 614 µm and the period number is 14. The 10-dB bandwidth of the grating is 315 nm. The LPFG is set in the center of the two translation stages and both ends of the fiber are fixed with two rotary grippers. The twist rate of the gratings can be altered by rotating the fiber rotary grippers [25]. The twist rate is calculated by
where T is the twist rate of the LPFG, α is the rotation angle of the rotary grippers, L is the distance between the rotary grippers. In the experiment, the L is 17.5 cm. The LPFG is rotated clockwise and anticlockwise from 0∼360° and 0∼-360° at an interval of 40°, respectively, and the twist rate is varying from -36 rad/m to 36 rad/m in the step of 4 rad/m. With the twist rate increasing from -36 rad/m to 36 rad/m, the resonance dips shift towards the opposite wavelength directions. The resonance wavelength shift has a good linear relationship with the twist rate, as illustrated in Fig. 5(a). The torsion sensitivities of the two resonance dips of the mode converter were calculated to be -0.180 nm/(rad/m) and 0.267 nm/(rad/m), respectively. Figure 5(b) shows that the 10-dB bandwidth of the mode converter increases with the twist rate. Therefore, the tunable ultra-broadband mode converter can be implemented by adjusting the twist rate of the LPFG. The tunable range of 10-dB bandwidth is 52 nm. The temperature characteristics of the TMF-LPFG are investigated with the temperature increasing from room temperature to 120 °C with a step of 10 °C. As shown in the Fig. 5(c), the resonance wavelength shift has a good linear relationship with the temperature increasing. The sensitivities of the two resonance dips were 233.5 pm/°C and -71.6 pm/°C, respectively. Figure 5(d) indicates that the 10-dB bandwidth of the mode converter can be tuned by controlling the temperature. The tunable range of the 10-dB bandwidth was 26 nm.Fig. 5. (a) The twist characteristics of the TMF-LPFG with the twist rate variation; (b) The dependence of 10-dB bandwidth of the mode converter on the twist rate; (c) The temperature characteristics of the TMF-LPFG with the temperature variation; (d) The dependence of 10-dB bandwidth of the mode converter on the temperature; (e) The strain characteristics of the TMF-LPFG with the strain variation; (f) The bending characteristics of the TMF-LPFG with curvature variation.
We studied experimentally the strain and bending characteristics of the TMF-LPFG, respectively. Figure 5(e) shows the transmission spectra of the TMF-LPFG with strain variation from 0 to 1339.2 µε. The strain sensitivities of the two resonance dips are 0.3 pm/µε and -2.3 pm/µε, respectively. The maximum wavelength shifts of the mode converter with strain variation are 0.9 nm and 1 nm, respectively. Figure 5(f) presents the bending response of the grating spectrum with the curvature increasing from 0 to 3.869 m-1. The bending sensitivities of the two resonance dips are 0.7 nm/m-1 and -0.75 nm/m-1, respectively. The maximum wavelength shifts of the mode converter with the curvature variation are 2.4 nm and 2.7 nm, respectively. Because the TMF-LPFG with the periods of 614 µm is not exactly at the turning point (the two dips of the grating would merge into one dip at the turning point), the proposed TMF-LPFG shows lower sensitivities to strain and bending in our experiments. When the grating is bent, both the effective mode index and physical length of the grating will vary simultaneously. Because the core modes are in the neutral layer, the bending sensitivity of the TMF-LPFG is lower than that of cladding modes for conventional LPFGs. For the CO2 laser inscribed LPFG, the asymmetric index modulation could be induced across the fiber section due to the high dosage of one-side CO2 laser irradiation, which leads to the directional bending sensitivity. In our experiments, the LPFGs were inscribed by lower CO2 laser irradiation. Therefore, the gratings have negligible extra insertion loss and much lower asymmetric index modulation. In the experiments, the bending sensitivity of the mode converter doesn’t change with the direction of bending. Compared with the twist and temperature characteristics, the resonance wavelength of the grating is less sensitive to strain and bending variation.
4. Fabrication of HLPGs
4.1 Fabrication of HLPGs
We fabricated the HLPGs in the TMF by irradiating and rotating the fiber along the axis of the fiber to obtain helical RI modulation using the CO2 laser. The scanning cycles of the laser were 3 cycles. The power of the laser was 1.0 W. The schematic diagram of the experimental set-up is shown in Fig. 6. Both ends of the TMF were spliced with the SMFs. One side of the fiber is fixed by a rotating gripper controlled by a step motor, the other side is placed into a horizontal groove and kept relaxed. When the CO2 laser irradiating the fiber, the fiber was rotated at a constant rate with a rotatable fiber holder and moved simultaneously with a constant speed along the fiber axis by the translation stage. During the process, the fiber rotating and stage movement was controlled by a program. There is a helical RI modulation along the fiber axis. The period of the HLPGs is the screw pitch of the helical structure. It can be expressed as [17]
where v is the speed of the moving stage, and s is the rotatory speed of the fiber rotates. The period can be changed by adjusting the speed of stage movement or rotates of the fiber rotator. We fabricated the HLPGs with different periods. The transmission spectra of the HLPGs are shown in Fig. 7(a). When the period of the HLPG is 608 µm, the single resonance dip appears at the wavelength of 1351 nm. For the HLPGs with a period bigger than 608 µm, there were two resonance wavelengths corresponding to the same one period. And with the periods increasing, the 10-dB bandwidth of the gratings becomes wider. The experimental result is well consistent with the result calculated by the COMSOL software. The widest 10-dB bandwidth is 297 nm when the grating period is 611 µm, which covers the wavelength range from 1211 nm to 1508 nm. In our experiment, the grating period was 611 µm and the grating length was 1.222 cm (period number was 20). The rotated speed of the grating is 58.9°/s. The moving speed of the experimental platform during fabricating the grating is 200 µm/s. Figure 7(b) shows the transmission spectrum of the fabricated TMF-HLPG.Fig. 7. (a) The experimental transmission spectra of the TMF-HLPGs with different grating periods; (b) The transmission spectrum of the TMF-HLPG with a period of 611 µm.
We observed the mode field distributions and OAM modes with the similar experimental setup shown in Fig. 4(a). No PCs are needed in the interference beam. The output near-field mode distributions captured at different wavelengths were illustrated in Fig. 8. A doughnut-shaped intensity profile was observed. Because of the helical structure along the fiber axis, the HLPGs are characterized by a π⁄2 phase difference with horizontal and vertically polarized directions between the vector modes. We utilized the Gaussian beam interfered with the OAM beams from the output to obtain the interference patterns, as shown in Fig. 8. The ultra-broadband LP01-LP11 mode converter was fabricated based on dual-resonance coupling by the TMF-HLPG. The 1st order OAM modes can be generated directly by the HLPGs.
Fig. 8. Output near-field mode distribution and OAM beams captured at different wavelengths for the ultra-broadband mode converter.
4.2 Characteristics of HLPGs
The sensing characteristics of the TMF-HLPG were investigated experimentally including twist, temperature, strain, and bending. As shown in the Fig. 9(a), the resonance wavelength shift has a good linear relationship with the twist rate. The torsion sensitivities of the two resonance dips of the TMF-HLPG were calculated to be -0.525 nm/(rad/m) and 0.841 nm/(rad/m), respectively. Figure 9(b) displays that the 10-dB bandwidth of the mode converter increases with the twist rate. The bandwidth tunable ultra-broadband mode converter can be implemented by adjusting the twist rate of the HLPG. Thanks to the higher twist sensitivity of the TMF-HLPG, the tunable range of the 10-dB bandwidth was 91 nm, which is much wider than that of the TMF-LPFG. The temperature characteristics of the TMF-HLPG are investigated with the temperature increasing from room temperature to 120 °C with a step of 10 °C. The two resonance wavelengths shift towards opposite wavelength direction, which is consistent with the reported results [26]. As shown in the Fig. 9(c), the resonance wavelength shift has a good linear relationship with the temperature increasing. The sensitivities of the two resonance dips were 188.2 pm/°C and -12.2 pm/°C, respectively. Figure 9(d) shows that the 10-dB bandwidth of the mode converter can be tuned by controlling the temperature. The tunable range of the 10-dB bandwidth was 18 nm, which is smaller than that of the TMF-LPFG. Figure 9(e) shows the transmission spectra of the TMF-HLPG and the resonance wavelength shift with strain variation. The strain sensitivities of the two resonance dips are 0.08 pm/µε and -1.1 pm/µε, respectively. The maximum wavelength shifts of the mode converter with strain variation are 0.3 nm and 1.4 nm, respectively. Figure 9(f) shows the transmission spectra of the TMF-HLPG and the resonance wavelength shifts with the curvature variation. The maximum wavelength shifts of the mode converter with the curvature variation are 3 nm and 3 nm, respectively. These indicate the TMF-HLPGs based mode converters have lower sensitivities strain and bending.
Fig. 9. (a) The twist characteristics of the TMF-HLPG with the twist rate variation; (b) The dependence of 10-dB bandwidth of the mode converter on twist rate; (c) The temperature characteristics of the TMF-HLPG with the temperature variation; (d) The dependence of 10-dB bandwidth of the mode converter on the temperature; (e) The strain characteristics of the TMF-HLPG with the strain variation; (f) The bending characteristics of the TMF-HLPG with curvature variation.
5. Conclusion
In conclusion, we have demonstrated experimentally and theoretically bandwidth tunable ultra-broadband mode converters based on the LPFGs and HLPGs written in a TMF at the dispersion turning point. The fabricated LP01-LP11 mode converter based on TMF-LPFGs provides a 10-dB bandwidth of ∼300 nm. The 1st order OAM mode based on TMF-LPFG was generated by adjusting the PCs. The LP01-LP11 mode converter based on TMF-HLPGs provides a 10-dB bandwidth of ∼297 nm. The 1st order OAM mode was generated directly by the fabricated TMF-HLPG. Because of the high sensitivities of torsion, the ultra-broadband mode converters can be acted as bandwidth tunable mode converters. When the twist rate is varying from -36 rad/m to 36 rad/m, the 10-dB bandwidth tunable ranges of the TMF-LPFG and TMF-HLPG are ∼52 nm and ∼91 nm, respectively. The mode converters can be applied in multichannel mode conversion in the broadband MDM systems.
Funding
National Natural Science Foundation of China (61875117).
Disclosures
The authors declare no conflicts of interest.
References
1. J. Vuong, P. Ramantanis, Y. Frignac, M. Salsi, P. Genevaux, D. F. Bendimerad, and G. Charlet, “Mode coupling at connectors in mode-division multiplexed transmission over few-mode fiber,” Opt. Express 23(2), 1438–1455 (2015). [CrossRef]
2. L. Wang, R. M. Nejad, A. Corsi, J. Lin, Y. Messaddeq, L. Rusch, and S. LaRochelle, “Linearly polarized vector modes: enabling MIMO-free mode-division multiplexing,” Opt. Express 25(10), 11736–11748 (2017). [CrossRef]
3. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013). [CrossRef]
4. S. G. Leon-Saval, A. Argyros, and J. Bland-Hawthorn, “Photonic lanterns,” Nanophotonics 2(5), 429–440 (2013).
5. R. Ismaeel, T. Lee, B. Oduro, Y. Jung, and G. Brambilla, “All-fiber fused directional coupler for highly efficient spatial mode conversion,” Opt. Express 22(10), 11610–11619 (2014). [CrossRef]
6. L. Wang, P. Vaity, B. Ung, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Characterization of OAM fibers using fiber Bragg gratings,” Opt. Express 22(13), 15653–15661 (2014). [CrossRef]
7. Y. Zhao, Y. Liu, L. Zhang, C. Zhang, J. Wen, and T. Wang, “Mode converter based on the long-period fiber gratings written in the two-mode fiber,” Opt. Express 24(6), 6186–6195 (2016). [CrossRef]
8. Y. Zhao, T. Wang, C. Mou, Z. Yan, Y. Liu, and T. Wang, “All-fiber vortex laser generated with few-mode long-period gratings,” IEEE Photonics Technol. Lett. 30(8), 752–755 (2018). [CrossRef]
9. Y. Zhao, Y. Liu, C. Zhang, L. Zhang, G. Zheng, C. Mou, J. Wen, and T. Wang, “All-fiber mode converter based on long-period fiber gratings written in few-mode fiber,” Opt. Lett. 42(22), 4708–4711 (2017). [CrossRef]
10. W. Zhang, L. Huang, K. Wei, P. Li, B. Jiang, D. Mao, F. Gao, T. Mei, G. Zhang, and J. Zhao, “High-order optical vortex generation in a few-mode fiber via cascaded acoustically driven vector mode conversion,” Opt. Lett. 41(21), 5082–5085 (2016). [CrossRef]
11. K. Wei, W. Zhang, L. Huang, D. Mao, F. Gao, T. Mei, and J. Zhao, “Generation of cylindrical vector beams and optical vortex by two acoustically induced fiber gratings with orthogonal vibration directions,” Opt. Express 25(3), 2733–2741 (2017). [CrossRef]
12. S. M. Israelsen and K. Rottwitt, “Broadband higher-order mode conversion using chirped microbend long period gratings,” Opt. Express 24(21), 23969–23976 (2016). [CrossRef]
13. W. Wang, J. Wu, K. Chen, W. Jin, and K. S. Chiang, “Ultra-broadband mode converters based on length-apodized long-period waveguide gratings,” Opt. Express 25(13), 14341–14350 (2017). [CrossRef]
14. Y. Zhao, Z. Liu, Y. Liu, C. Mou, T. Wang, and Y. Yang, “Ultra-broadband fiber mode converter based on apodized phase-shifted long-period gratings,” Opt. Lett. 44(24), 5905–5908 (2019). [CrossRef]
15. Y. Guo, Y. Liu, Z. Wang, H. Zhang, B. Mao, W. Huang, and Z. Li, “More than 110-nm broadband mode converter based on dual-resonance coupling mechanism in long-period fiber gratings,” Opt. Laser Technol. 118, 8–12 (2019). [CrossRef]
16. C. D. Poole, C. D. Townsend, and K. T. Nelson, “Helical-grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9(5), 598–604 (1991). [CrossRef]
17. L. Zhang, Y. Liu, Y. Zhao, and T. Wang, “High sensitivity twist sensor based on helical long-period grating written in two-mode fiber,” IEEE Photonics Technol. Lett. 28(15), 1629–1632 (2016). [CrossRef]
18. X. Cao, Y. Liu, L. Zhang, Y. Zhao, and T. Wang, “Characteristics of chiral long-period fiber gratings written in the twisted two-mode fiber by CO2 laser,” Appl. Opt. 56(18), 5167–5171 (2017). [CrossRef]
19. H. Zhao, P. Wang, T. Yamakawa, and H. Li, “All-fiber second-order orbital angular momentum generator based on a single-helix helical fiber grating,” Opt. Lett. 44(21), 5370–5373 (2019). [CrossRef]
20. Y. Zhang, Z. Bai, C. Fu, S. Liu, J. Tang, J. Yu, C. Liao, Y. Wang, J. He, and Y. Wang, “Polarization-independent orbital angular momentum generator based on a chiral fiber grating,” Opt. Lett. 44(1), 61–64 (2019). [CrossRef]
21. K. Ren, M. Cheng, L. Ren, Y. Jiang, D. Han, Y. Wang, J. Dong, J. Liu, L. Yang, and Z. Xi, “Ultra-broadband conversion of OAM mode near the dispersion turning point in helical fiber gratings,” OSA Continuum 3(1), 77–87 (2020). [CrossRef]
22. Q. Liu, K. S. Chiang, and K. P. Lor, “Dual resonance in a long-period waveguide grating,” Appl. Phys. B 86(1), 147–150 (2006). [CrossRef]
23. W. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009). [CrossRef]
24. S. Ramachandran, Z. Wang, and M. Yan, “Bandwidth control of long-period grating-based mode converters in few-mode fibers,” Opt. Lett. 27(9), 698–700 (2002). [CrossRef]
25. C. Jiang, Y. Liu, Y. Zhao, C. Mou, and T. Wang, “Helical long-period gratings inscribed in polarization maintaining fibers by CO2 Laser,” J. Lightwave Technol. 37(3), 889–896 (2019). [CrossRef]
26. X. Shu, L. Zhang, and I. Bennion, “Sensitivity characteristics of Long-Period Fiber Gratings,” J. Lightwave Technol. 20(2), 255–266 (2002). [CrossRef]
