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Abstract
A well designed ring-core fiber can theoretically support numerous orbital angular momentum (OAM) modes with low crosstalk for space-division-multiplexing (SDM) data transmission, which is considered as a promising solution for overcoming the capacity crunch in optical communication network. However, the accumulated chromatic dispersion in OAM-fiber could limit the data speed and transmission distance of communication systems. A potential solution is to insert a dispersion compensation ring-core fiber with opposite-sign of dispersion in the transmission fiber along the fiber link. In this work, we propose a triple ring-core fiber with broadband negative dispersion. A highest negative dispersion of -24.47 ps/(nm·km) at 1550 nm and an average dispersion slope in the C band from -0.182 ps/(nm2·km) to 0.065 ps/(nm2·km) can be achieved to compensate multi-order dispersion. The effects of Ge-doping concentration fluctuation in the high-index ring core and fabrication errors on fiber geometric structures are also investigated. Furthermore, the effective mode area decreases as the widths of high-index rings increase due to the enhanced confinement ability. The designed triple ring-core fiber could offer potential for compensating OAM fiber links with positive dispersions.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the increasing demand for data transmission capability, the pursuit for higher spectral efficiency and faster data rate always exists [1,2]. To satisfy this tendency in optical communication, different multiplexing technologies and high-order modulation technologies have been comprehensively investigated to adequately utilize the multiple photon dimensions, such as time, wavelength, polarization, amplitude and phase [3–6]. Nowadays, additional to the photon dimensions previously mentioned, orbital angular momentum (OAM) has become the next promising dimension to be utilized in multiplexing technology to further increase optical spectral efficiency [7–10]. The OAM beams can be regarded as a complete set of orthogonal basis functions consisting of discrete modes with different spatial phase distributions. The phase distributions of OAM modes are generally described as exp(ilθ), where θ refers to the azimuthal angle and l is an unlimited value counting the number of intertwined helices [6,11].
To build a practical fiber-based OAM communication system, a key requirement is to design a suitable fiber that can support the OAM modes with low crosstalk. Ring-core fiber, which features an annular high-index profile, has been proven its practicability [12–16]. By adjusting the width of the annular high-index area, the ring-core fiber can easily restrain the radially high-order mode, which help tremendously reduce the modal crosstalk. Furthermore, ring core fibers with trench and multi-core structures are proposed or demonstrated to support record-high number of OAM modes [17–20].
Although the transmission properties of OAM modes in the previously reported designs have been taken into consideration, the fiber-based OAM communication system still suffers from the accumulate dispersion as the transmission length extends [21–24]. As the dispersion compensation fiber (DCF) has been extensively investigated and put into commercial application, it exhibits the ability to solve the pulse broadening problem induced by the chromatic dispersion [25,26]. In the past few years, there are several reports on the OAM-based DCF designs that provide highly negative dispersion [22,27]. But the extremely negative dispersion values are achieved with the sacrifice of the dispersive wavelength range, and the parabolic negative dispersion curves potentially increase the complexity of fiber link setting [22]. Thus, a laudable goal would be to provide broadband and flat dispersion design to avoid complex multi-order dispersion compensation management. Furthermore, applications of fiber with broadband and flat dispersion design could be further extended to many others, such as the true time-delay beam former [28], ultrafast optical oscilloscope [29] and optical correlation [30].
In this work, we propose and design a triple ring-core dispersion compensation fiber (TRDCF) for compensating the chromatic dispersion in the OAM-based optical communication link. By utilizing the resonant effect between the Ge-doped ring-shaped areas, a negative dispersion of -18.248 ps/(nm·km) can be achieved at the wavelength of 1550 nm for OAM1,1 mode. With a dispersion slope of -0.1635 ps/(nm2·km) from 1530 to 1565 nm, this design can be used to effectively compensate the dispersion in single ring-core fiber with little residual 2nd order or 3rd order chromatic dispersions in the C-band.
2. Concept and TRDCF design
Figure 1 shows the fundamental origin and evolution process from DCF to TRDCF. During the past decade, there has been a series of single ring-core fibers proposed for transmitting the OAM modes [12–14,31–35]. Although the supported mode number and other transmitting properties of the former designs are promising, the dispersion for different modes in different wavelengths still accumulates with the propagation distance being extended, which could be one of the limitations to the speed of communication. Thus, a suitable dispersion compensating fiber for OAM modes is required. With the dispersion several times higher than that of the single ring-core fiber in absolute value and dispersion slope, a well-designed and fabricated dispersion compensating fiber can remedy the deformation of the light pulse without causing too much loss and other effects on the transmission. Fiber designs for compensating the fundamental HE1,1 mode have been mature and commercially utilized in step-index fiber-based systems. Based on that, in order to match with the high index area of single ring-core fibers, we propose the triple ring-core dispersion compensating fiber, where the extra high-index ring-shaped areas are introduced for accommodating the OAM mode in different wavelengths.
Figure 2(a) displays the cross section and refractive index profile of the designed TRDCF at 1550 nm. To achieve the broadband negative dispersion, a triple ring-shaped high-index area is specially designed to support the OAM mode. d1, d2, d3, d4, d5, and d6 are the radius of ring-shaped structures on the fiber profile. Δd1 = d6-d5 = d2-d1 and Δd2 = d4-d3 represent the corresponding widths of the high-index regions. The material of the inner core and background cladding is SiO2. The innermost and outermost high-index ring-shaped areas have lower GeO2 doping concentration and the middle ring-shaped area is more heavily doped. n1 and n2 refer to the material refractive indices of Ge-doped SiO2 with different doping concentrations. nSiO2 refers to the refractive index of pure fused silica. The designed structure and material choice can be potentially achieved, as similar fibers have been fabricated in the previous work with the modified chemical vapor deposition (MCVD) method [2,14,36].
Fig. 2. (a) Cross section and refractive index profile of the designed dispersion compensating ring fiber; (b) Normalized intensity and (c) phase distributions of the supported OAM1,1 mode in the designed ring fiber (d1 = 5.5 µm, d2 = 8.6 µm, d3 = 8.9 µm, d4 = 11.4 µm, d5 = 11.7 µm, d6 = 14.8 µm).
In the simulation, the effect of material dispersion is taken under consideration by Sellmeier equations [37,38]. By using the finite element method (FEM), the normalized ring-shaped intensity and helical phase distributions of OAM1,1 mode with an azimuthal 2π phase change are drawn as illustrated in Fig. 2(b) and Fig. 2(c).
Figure 3 shows the basic principle of designing the TRDCF with broadband negative dispersion. Chromatic dispersion origins from the difference in group velocity of the light in different wavelengths. The group velocity of a specific guiding mode can be adjusted by changing the modal intensity distribution in the designed fiber. Here we calculate the normalized intensity distribution of OAM1,1 mode in the designed TRDCF at 1400 nm, 1500 nm, and 1600 nm, respectively. As one can see, the intensity distribution gradually spreads as the wavelength increases, which produces the difference on the group velocity of the OAM1,1 mode, thus forming the broadband negative dispersion covering the C band shown in the third section.
Fig. 3. Refractive index profile and normalized intensity distribution (%) of the OAM1,1 mode at 1400 nm, 1500 nm and 1600 nm with d1 = 5.5 µm, d2 = 8.6 µm, d3 = 8.9 µm, d4 = 11.4 µm, d5 = 11.7 µm, d6 = 14.8 µm.
3. Mode property
To investigate the impact from the geometric parameter of the TRDCF, we first calculate the chromatic dispersion of the designed TRDCF for different Δd1 as displayed in Fig. 4(a). As one can see, the dispersion uniformly becomes smaller as Δd1 decreases. With d1 = 5.5 µm, Δd1 = 2.9 µm, Δd2 = 2.5 µm, the dispersion of the designed fiber at 1550 nm can reach down to -24.47 ps/nm/km. With smaller Δd1 from 3.1 µm to 3.05 µm, the designed TRDCF monotonously decreases with wavelength. As Δd1 is reduced down to 3 µm, the sign of the dispersion slope starts to get reversed in the C band. To show the curves slope change more specifically, the chromatic dispersion slope has been calculated as illustrated in Fig. 4(b). When Δd1 is within 3.1 µm to 3.05 µm, the designed TRDCF can provide a negative slope of -0.1714 ps/nm2/km at 1550 nm and the slope stays negative in the whole C-band. For the proposed fiber with Δd1 < 3.05 µm, as the sign of the dispersion slope changes within the C band, more complex configuration schemes on the dispersion compensation are needed.
Fig. 4. (a) Chromatic dispersion and (b) dispersion slope of the OAM1,1 mode in the designed TRDCF as a function of wavelength with different Δd1 (d1 = 5.5 µm, Δd2 = 2.5 µm).
Figure 5 illustrates the chromatic dispersion and the dispersion slope of the OAM1,1 mode supported in the designed TRDCF over the C-band for different Δd2. With d1 = 5.5 µm, Δd1 = 3.1 µm, Δd2 = 2.45 µm, the dispersion of the designed fiber at 1550 nm can reach down to -21.86 ps/nm/km and the dispersion becomes smaller as Δd2 decreases. The change on Δd2, which stands for the width change of the middle high-index ring-shaped area, not only can adjust the dispersion value but also enormously influences the dispersion slope as depicted in Fig. 5(b). The dispersion slope increases as Δd2 decreases from 2.7 µm to 2.55 µm. The sign of dispersion slope stays negative when Δd2 ranges from 2.7 µm to 2.5 µm, and the negative dispersion slopes from -0.192 ps/nm2/km to -0.0802 ps/nm2/km at 1550 nm can be achieved.
Fig. 5. (a) Chromatic dispersion and (b) dispersion slope of the OAM1,1 mode in the designed TRDCF as a function of wavelength with different Δd2 (d1 = 5.5 µm, Δd1 = 3.1 µm).
As Δd1 and Δd2 of the TRDCF can both influence the chromatic dispersion value and the slope of the dispersion curves, in order to get the most proper design and determine the tolerance of manufacturing, the contour map of chromatic dispersion for the OAM1,1 mode at 1550 nm with different Δd1 and Δd2 and corresponding dispersion difference from 1530 to 1565 nm are drawn. As one can see in Fig. 6, the dispersion value at 1550 nm gradually gets smaller with the lower Δd1 and Δd2 meanwhile the dispersion variation in the C band changes differently. When d1 = 5.5 µm, Δd1 = 2.9 µm and Δd2 = 2.5 µm, a most negative of -24.47 ps/nm/km at 1550 nm can be provided by the designed fiber and the dispersion value at 1550 nm increases smoothly to -6.45 ps/nm/km as Δd1 raise to 3.1 µm and Δd2 raise to 2.7 µm. As illustrated in Fig. 6(b), when d1 = 5.5 µm, Δd1 = 3.1 µm and Δd2 = 2.55 µm, the dispersion variation in the C band can reach the minimal value of -6.3597 ps/nm/km, which corresponds to the minimal dispersion slope -0.182 ps/(nm2·km). With the values of Δd1 and Δd2 changes from 3.1 µm to 2.9 µm and from 2.55 µm to 2.5 µm respectively, the dispersion variation gradually increases to 2.275 ps/nm/km (d1 = 5.5 µm, Δd1 = 2.9 µm, Δd2 = 2.5 µm). Based on the data above, the balance between the most negative dispersion value and suitable dispersion slope should be considered for practical applications of the designed TRDCF. Previous reports demonstrated that the ring-core fiber can support OAM mode with low dispersion as small as 4 ps/nm/km [35,39], the designed TRDCF can function as a DCF with the 1:5 length ratio in the OAM fiber transmission link.
Fig. 6. (a) Chromatic dispersion of the OAM1,1 mode at 1550 nm for different Δd1 and Δd2; (b) Dispersion variation from 1530 to 1565 nm of the OAM1,1 mode for different Δd1 and Δd2 (d1 = 5.5 µm).
The dispersion property of the designed TRDCF can be influenced by not only the geometric structures but also the doping concentration of the triple ring-core area. In Fig. 7, where mf1 and mf2 stand for the doping concentration of the middle and bilateral ring-core area respectively, the chromatic dispersion and its corresponding slope are further calculated for the designed TRDCF in the C band. As one can see, the dispersion value gradually decreases while the dispersion slope variation gradually increases with the doping concentration changes from 4.1 mol% to 3.8 mol%. With d1 = 5.5 µm, Δd1 = 3.1 µm, Δd2 = 2.5 µm, mf1 = 3.8 mol%, mf2 = 0.6 mol%, a negative dispersion of -20.69 ps/nm/km can be achieved at 1550 nm. And a negative dispersion slope value at 1550 nm ranging from -0.196 ps/nm2/km to -0.0478 ps/nm2/km can be achieved in the C band by adjusting the doping concentration. For mf1 in the range from 3.9 mol% to 4.1 mol%, the dispersion slope of the designed fiber stays negative in the C band, which brings more flexible choice in manufacturing.
Fig. 7. (a) Chromatic dispersion and (b) dispersion slope of the OAM1,1 mode in the designed TRDCF as a function of wavelength with different mf1 (d1 = 5.5 µm, Δd1 = 3.1 µm, Δd2 = 2.5 µm, mf2 = 0.6 mol%).
Figure 8 depicts the chromatic dispersion and dispersion slope of the designed TRDCF with different mf2. With the mf2 decreasing, both of the dispersion value and dispersion slope decrease. As the mf2 changes from 1.2 mol% to 0.6 mol%, the dispersion at 1550 nm decreases from 5.578 ps/nm/km to -18.249 ps/nm/km and the corresponding dispersion slope varies from 0.00173 ps/nm2/km to -0.171 ps/nm2/km. To perform the dispersion compensation function, the doping concentration of the inner ring core and outer ring core should be controlled to lower than 0.8 mol%. Compared with the alteration caused by the mf1 depicted in Fig. 7, the refractive index of the inner and the outer ring-core area has slighter influence on the dispersion slope, thus can function as the parameter for fine-tuning the slope.
Fig. 8. (a) Chromatic dispersion and (b) dispersion slope of the OAM1,1 mode in the designed TRDCF as a function of wavelength with different mf2 (d1 = 5.5 µm, Δd1 = 3.1 µm, Δd2 = 2.5 µm, mf1 = 3.9 mol%).
In order to show the thorough influence of adjusting the doping concentration of the ring-core area, the contour maps of chromatic dispersion for the OAM1,1 mode at 1550 nm with different mf1 and mf2 and corresponding dispersion difference from 1530 to 1565 nm are drawn as Fig. 9 shows. A relatively low dispersion of -20.69 ps/nm/km is achieved with mf1 = 3.8 mol% and mf2 = 0.6 mol%. Dispersion variation in the C band from -6.494 ps/nm/km to 0.408 ps/nm/km can be offered by modifying the doping concentration of the triple ring-core region. As one can see, the reduction of chromatic dispersion at 1550 nm is mainly caused by the decrease of doping concentration in the inner and outer ring areas, which generates the enlarged effective index difference between lights of different wavelength as light intensity of larger wavelength is gradually shifted from the middle ring to the bilateral rings. The increasing chromatic dispersion difference in the C band as the mf2 decreases is also caused by the same reason.
Fig. 9. (a) Chromatic dispersion of the OAM1,1 mode at 1550 nm for different mf1 and mf2; (b) Dispersion variation from 1530 to 1565 nm of the OAM1,1 mode for different mf1 and mf2 (d1 = 5.5 µm, Δd1 = 3.1 µm, Δd2 = 2.5 µm).
Figure 10 shows the effective mode area and nonlinear coefficient of the OAM1,1 mode under different structure parameters. All the effective mode area and nonlinear coefficient data below are calculated by the generalized full vectorial model reported before [40,41]. As Fig. 10(a) shows, an effective mode area of 1448 µm2 for the TRDCF with Δd1 = 2.9 µm and Δd2 = 2.5 µm, and 640 µm2 for the lowest one with Δd1 = 3.1 µm and Δd2 = 2.7 µm. The corresponding nonlinear coefficient of the designed fiber with Δd1 = 2.9 µm and Δd2 = 2.5 µm can reach down to 7.52 × 10−5 /W/m, and the maximal nonlinear coefficient of the fiber with Δd1 = 3.1 µm and Δd2 = 2.7 µm is 1.71 × 10−4 /W/m. The modal confinement is strengthened as the fiber structure parameter Δd1 and Δd2 increase, therefore the effective mode area decreases as these parameters increase.
Fig. 10. (a) Effective mode area and (b) nonlinear coefficient of the OAM1,1 mode supported in the designed fiber with different Δd1 and Δd2 (d1 = 5.5 µm, mf1 = 3.9 mol%, mf2 = 0.6 mol%).
The OAM modes can be seen as the combination of the even and odd fiber eigenmodes for HEm,l or EHn,l. For practical use, the fiber faces the fabrication errors and birefringence such as fiber ellipticity and bending. These effects would influence the propagation of the even and odd modes, and finally cause the distortion of the OAM mode. Figure 11 depicts the effective refractive index difference (Δneff) between the even and odd HE2,1 modes caused by the fiber ellipticity and bending, which shows the anti-disturbance capability of the OAM1,1 mode in the designed TRDCF. As Fig. 11(a) shows, the designed fiber with smaller high-index region shows more tolerance for the fiber ellipticity. As the fiber ellipticity changes from 0% to 2%, the effective refractive index difference is well controlled under 2 × 10−5. For the influence of the fiber bending, different from the fiber ellipticity, we note that the designed fiber with larger high-index region shows more tolerance to the fiber bending. The effective refractive index difference is controlled under 7 × 10−5 with the fiber bending radius from 600 mm to 200 mm. As the structure of the designed TRDCF is more sensitive to the fiber bending, larger high-index region may be considered for better robustness.
Fig. 11. Effective refractive indices difference of the HE2,1 mode as a function of the (a) fiber ellipticity and (b) bending radius with different Δd1 and Δd2 (d1 = 5.5 µm, mf1 = 3.9 mol%, mf2 = 0.6 mol%).
4. Conclusion
In summary, the triple ring-core fiber supporting OAM mode is designed for serving as DCF. A most negative dispersion of -24.47 ps/(nm·km) at 1550 nm in the C band can be achieved with the designed TRDCF (d1 = 5.5 µm, Δd1 = 2.9 µm, Δd2 = 2.5 µm, mf1 = 3.9 mol%, mf2 = 0.6 mol%). And flat negative dispersion slopes down to -0.182 ps/(nm2·km) (d1 = 5.5 µm, Δd1 = 3.1 µm, Δd2 = 2.55 µm, mf1 = 3.9 mol%, mf2 = 0.6 mol%) can be offered by adjusting the doping concentration and fiber geometric parameters. The nonlinear effects can be effectively weakened as the effective mode area is more than 600 µm2. According to the DCF reported before, the negative dispersion value of TRDCF is also limited by the resonance effect between multi high-index regions, which is usually optimized for one OAM mode. Furthermore, there is a tradeoff between the negative dispersion value and the dispersive wavelength range. Compared with the recent works on DCF for OAM mode, the designed TRDCF provides a relatively linear negative dispersion over wide bandwidth, which has better potential to be used in the OAM-based optical fiber communication systems.
Funding
National Key Research and Development Program of China (2018YFB0703500); Key Technologies Research and Development Program of Tianjin (20YFZCGX00440); Shaanxi Key Laboratory of Deep Space Exploration Intelligent Information Technology (2021SYS-04).
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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