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Abstract
As the dimension of orbital angular momentum (OAM) is orthogonal to the other degrees of freedom for photon, such as wavelength, it can be utilized to further increase data capacity in the wavelength division multiplexing (WDM) systems. However, the non-zero dispersion-shifted fiber (NZDSF) for the OAM mode has not yet been investigated or even proposed. In this work, we propose and design a ring fiber with low chromatic dispersion for the HE2,1 mode, which can serve as NZDSF for its corresponding OAM1,1 mode. A low dispersion of 3.3 ps/(nm·km) at 1550 nm and <2.9 ps/(nm·km) dispersion variation from 1530 to 1565 nm for the OAM1,1 mode is achieved in simulation, which satisfies the standard of the ITU-T G.655.C. The designed fiber with ring width from 1.5 µm to 3.5 µm can support the OAM1,1 mode within the C-band, and a large effective area of about 646 µm2 is obtained. We also note that the fiber with larger inner radius and ring width are more tolerant to the perturbations, such as fiber ellipticity and bending. In the fiber-based optical communication systems, the designed ring fiber could be used as a candidate for supporting OAM modes with low dispersion and reduced nonlinear effects.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the continuous development of the communication society, the emerging services have brought an increasing demand for data transmission capacity. In order to make full use of different dimensions of a photon, researchers have been widely exploring various multiplexing technologies, including wavelength-division multiplexing (WDM), polarization-division multiplexing (PDM), and space-division multiplexing (SDM) [1–3]. Orbital angular momentum (OAM) modes, as a special type of orthogonal modal basis set with special doughnut-shaped intensity distribution and theoretically infinite topological states, have attracted widespread interest in the field of optical communications [3–7]. In the area of optical fiber communications, the corresponding optical ring fiber designs have already gained encouraging achievements in preserving the OAM beams’ ring-shaped intensity profile and reducing the mode crosstalk [8–16]. Furthermore, multifarious applications have been achieved using OAM beams, such as micromanipulation [17,18], imaging [19,20], and sensing [21,22].
The helical phase front of an OAM beam can be described as exp (ilθ), where θ refers to the azimuthal angle and l is an unlimited value counting the number of intertwined helices [3]. As the number of OAM states is theoretical infinite and different OAM states can be efficiently separated because of the inherent orthogonality, it is possible to increase the spectral efficiency and data capacity of fiber-based communication systems by multiplexing information with this additional degree of freedom [3,23,24]. Moreover, multiplexing of OAM modes have no connection with wavelength, indicating that they could be used in the wavelength-division multiplexing (WDM) technique to improve system capacity [25–27].
However, one of the major limitation of OAM-based optical fiber communication links is the significant performance degradation induced by chromatic dispersion and nonlinearity from the transmission medium optical fiber [28,29]. Digital signal processing (DSP) and dispersion compensating fibers (DCF) can be used for reducing the signal distortion due to the chromatic dispersion [30,31]. However, the correcting ability of DSP is proportional to the power consumption, and the use of DCF will not only increase the loss but also lead to the complexity and the cost of the system [32]. On the other hand, considering the demand of overcoming four wave mixing (FWM) in the dense wavelength-division multiplexing (DWDM) systems, dispersion is needed to avoid phase matching conditions [32]. As a result, the chromatic dispersion in the optical fiber should be within a moderate range which is large enough to restrain the nonlinear effects, but also small as much as possible to reduce the dispersion-induced signal quality penalty [33–35]. The corresponding standard is given by ITU-T G.655, and the optical fiber is named as non-zero dispersion-shifted fiber (NZDSF). Because the NZDSF is specially designed with the zero-dispersion wavelength (ZDW) away from the 1550-nm telecommunications band, FWM and the other nonlinear effects are minimized [36]. Therefore, it can accommodate more WDM channels than the dispersion-shifted fiber (DSF) in the optical fiber communications systems [37]. Recently, OAM-based optical fiber communication is one of the hot research topics in the optical communications society [5]. However, to the best of our knowledge, the NZDSF for OAM mode has not yet been even proposed.
In this work, we propose and design a non-zero dispersion-shifted ring fiber (NZDSRF) that can support optical OAM modes [38]. By carefully adjusting the fiber structure, the designed fiber with 39-µm core diameter and a 1.5-µm ring width can theoretically provide a low dispersion (3.296 ps/(nm·km)) at 1550 nm and small dispersion variation (<2.831 ps/(nm·km) in total) for OAM1,1 mode across C-band from 1530 to 1565 nm. Moreover, the designed fiber has a large effective area of about 646 µm2, which could diminish nonlinear effects and thus upgrade the DWDM network performance. This new type NZDSRF designed could be an ideal candidate for reducing loss, dispersion, and nonlinear effects in optical WDM fiber communication systems carrying OAM modes.
2. Concept and NZDSRF design
Figure 1 depicts the major motivation of designing a NZDSRF for OAM mode. In the traditional communication systems building over optical fiber, single mode fiber (SMF) and dispersion compensating fiber (DCF) were implemented as different segments to balance the tradeoff between the fiber dispersion and nonlinearity. NZDSF could instead provide an alternative solution, which can significantly simplify the fiber link management. In the field of OAM-based optical fiber communication, the fundamental dispersion management scheme is the same. In the OAM-based optical fiber communications link, the fundamental dispersion management scheme is the same. The corresponding single ring fiber (SRF) [8] and the dispersion compensating ring fiber (DCRF) [13] with ring-shaped high index profile that can better support OAM modes have been proposed to serve the same function as traditional SMF and DCF. As DWDM technology can significantly boost the data transmission capacity of optical communication systems, we believe the corresponding NZDSRF design for OAM mode is also worth exploring.
Fig. 1. NZDSRF-based OAM communication system simplifies the traditional OAM-based fiber communication system using the SRF and DCRF combination.
Figure 2 depicts the cross section of the designed NZDSRF and the refractive index profile at 1550 nm. In order to better guide the OAM modes, a specially designed fiber with a high-index ring region is used to avoid the radially high-order modes. r1, r2, and r3 are the radius for different regions in the designed fiber and Δr represents the width of the ring-shaped high-index region. nSiO2 and nGe-doped refers to the refractive indices of the pure fused silica and the Ge-doped SiO2 at 1550 nm. To better tailor the dispersion property, a 9 mol% Ge-doped SiO2 is thus chosen as the material of the high-index ring region. We choose pure fused silica for the cladding and the standard 125-µm for the fiber cladding diameter. This designed structure and material choice is practicable, because of the mature fabrication process of Ge-doped optical fiber. The corresponding fiber preform fabrication has been reported in the previous work by using the modified chemical vapor deposition (MCVD) [39]. In the simulation, the effect of material dispersion is considered by using Sellmeier equations for the materials above [40,41].
3. Mode property
By modeling with the finite element method (FEM), we can obtain the normalized intensity and phase distributions of OAM1,1 mode supported in the designed fiber with different parameters as indicated in Fig. 3. As shown in the color bar, the normalized ring-shaped intensity distributions are constituted by the even and odd eigenmodes. One can see that the NZDSRF with larger r1 features extended intensity profile. The phase distributions at the bottom of the figure indicates that the OAM1,1 mode has a 2π phase change azimuthally.
Fig. 3. Normalized intensity and phase distributions of the supported OAM1,1 mode in the Ge-doped ring fiber (r1 = 13.5, 19.5, 29.5 µm, Δr = 1.5, 2.5 µm).
Figure 4 depicts the number of supported OAM modes and the effective refractive indices of the supported modes in the designed ring-shaped fiber. As Fig. 4(a) shows, the number of supported OAM modes increases with the thickness of the high-index region in the designed fiber. For the C and L bands, there are up to 13 OAM modes that can be supported in the designed fiber. Since the mode coupling can be the major limiting factor for the communication link performance, we further investigate the crosstalk between the different orders of OAM modes. The effective refractive index difference between the HE2,1 mode that constitutes the OAM1,1 mode and the HE3,1 mode that constitutes the OAM2,1 mode is 1.54×10−4. The effective refractive index difference between the higher-order OAM modes is even larger than 1.54×10−4 as one can see in Fig. 4(b). Thus, the level of mode crosstalk between OAM modes with different orders is relatively low, as the effective refractive index difference maintains more than 10−4 [42]. As Fig. 4(c) shows, the dispersion property of different modes supported in the designed NZDSRF is within the range of 2 ps/(nm·km) to 14 ps/(nm·km). A minimum dispersion is achieved for the HE2,1 mode, and therefore we focus on the fluctuation of the dispersion property for HE2,1 mode caused by different geometric parameters.
Fig. 4. (a) Number of supported OAM modes in the designed NZDSRF with different fiber parameters as a function of wavelength; (b) Effective refractive indices of the supported modes in the designed NZDSRF (r1 = 19.5 µm, Δr = 1.5 µm); (c) Chromatic dispersion of the supported modes in the designed NZDSRF (r1 = 19.5 µm, Δr = 1.5 µm).
Figure 5 depicts the dispersion of the OAM1,1 mode at 1550 nm in the proposed fiber with varied r1 from 5.5 µm to 51.5 µm and the dispersion difference in the C-band from 1530 to 1565 nm of the OAM1,1 mode supported in the designed fiber, respectively. As one can observe in Fig. 5(a), the ring fiber with 1.5-µm width can provide a low chromatic dispersion of <4 ps/(nm·km). As ITU-T G.655 specifies, the G.655.C attributes require that the chromatic dispersion coefficient from 1530 nm to 1565 nm should be within 1 to 10 ps/(nm·km), and the difference between the maximum and minimum dispersion value should be less than 5 ps/(nm·km) [36]. Figure 5(b) shows that the NZDSRF designed in this paper satisfy the G.655.C standard and can provide near-zero flattened dispersion with slope less than 0.085 ps/(nm2·km).
Fig. 5. (a) Chromatic dispersion of the OAM1,1 mode at 1550 nm for different r1 and Δr; (b) Dispersion difference from 1530 to 1565 nm of the OAM1,1 mode for different r1 and Δr.
We further investigate the supported wavelength range for OAM1,1 mode under different ring fiber parameters. The cut-off wavelength of OAM1,1 mode given in Table 1 shows that the fiber with ring width from 1.5 µm to 3.5 µm can support OAM1,1 mode over the whole C band.
Table 1. Cut-off Wavelength of the OAM1,1 mode.a
Moreover, system-level simulation is performed to show the advantages of the designed NZDSRF in optical fiber communication. Figure 6 depicts eye-opening penalty for 10-Gb/s optical on-off keying (OOK) signal transmission in the traditional SMF and the designed NZDSRF as a function of propagation distance. As the designed NZDSRF has lower chromatic dispersion and larger effective mode area than the ones of SMF, the optical signal quality is significantly improved as one can observe that the eye diagram in the NZDSRF is much clearer.
Fig. 6. Eye-opening penalty for 10-Gb/s optical on-off keying (OOK) signal transmission in the traditional SMF and the designed NZDSRF as a function of propagation distance with eye diagram comparison at the distance of 20 km, 60 km, and 100 km.
Figure 7(a) displays the dispersion profile of the OAM1,1 mode with different parameters. With the modification of the fiber structure parameters, the zero-dispersion wavelength changes from 1430 nm (r1 = 47.5 µm, Δr = 2.5 µm) to 1500 nm (r1 = 19.5 µm, Δr = 1.5 µm). From 1530 nm to 1565 nm, the dispersion varies from 1.67 ps/(nm·km) to 4.50 ps/(nm·km) for r1 = 19.5 µm and Δr = 1.5 µm, while from 5.83 ps/(nm·km) to 8.12 ps/(nm·km) for r1 = 47.5 µm and Δr = 2.5 µm. Figure 7(b) shows the effective mode area of the NZDSRF under different structure parameters. It is about 646 µm2 for the ring fiber with r1 = 19.5 µm and Δr = 1.5µm, and approximately 1100 µm2 for the other one with r1 = 47.5 µm and Δr=2.5 µm. Figure 7(c) depicts nonlinear coefficient of the supported OAM1,1 mode in the designed NZDSRF with parameters mentioned above. The nonlinear coefficient γ is calculated by using the following equation in [43]:
Fig. 7. (a) Chromatic dispersion, (b) effective mode area, and (c) nonlinear coefficient of the OAM1,1 mode as a function of wavelength for different fiber structure parameters.
According to the simulation results above, the chromatic dispersion characteristics of the designed ring fiber meets the requirement of standard NZDSF. Furthermore, its large effective mode area and small nonlinear coefficient make it an ideal transmission medium for OAM-based optical fiber communication systems.
4. Fiber ellipticity and bending
The OAM modes can be regarded as the composition of the even and odd fiber eigenmodes for HEm,l or EHn,l. The difference in the effective refractive indices between the even and odd eigenmodes can impact the OAM mode field distribution as it propagates. Thus, as the stress and twist situations change the profile of the optical fiber and causes the birefringence, therefore affects the propagation of the even and odd modes. Consequently, the influence of these perturbations should be considered when the fiber for OAM modes is designed. In the figures below, the effective refractive index difference (Δneff) caused by the fiber ellipticity and bending between the even and odd HE2,1 modes is further studied, which can show the anti-disturbance capability of the designed fiber.
Figure 8 illustrates Δneff between the odd and even fiber eigenmodes for HE2,1 as a function of the fiber ellipticity with different inner radius (r1) and different ring width (Δr). One can see that the designed ring fiber with larger inner radius and smaller ring width has higher tolerance to the impact from the fiber ellipticity up to 2%. As the simulation results show, for the fiber with dissimilar inner radius, the Δneff between the odd and even fiber eigenmodes can be up to 3.36×10−5. It becomes less sensitive to the fiber ellipticity with the increase of inner radius of the NZDSRF. For the designed fiber with different ring width, as the ellipticity changes from 0% to 2%, the Δneff between the odd and even fiber eigenmodes also raises up to 3.462×10−5.
Fig. 8. (a) Effective refractive indices of fiber eigenmodes for HE2,1 as a function of the fiber ellipticity with different inner radius (r1) and (b) different ring width (Δr).
Figure 9 shows Δneff between the odd and even fiber eigenmodes for HE2,1 as a function of the fiber bend radius with different fiber parameters. We note that with the increase of inner radius of the designed fiber, the anti-disturbance capability for fiber bending decreases. Meanwhile, the width of the high-index ring gives little change in Δneff as the bend radius reduces from 350 to 50 mm. The fiber with r1 = 17.5 µm, Δr = 1.5 µm and 50-mm bend radius has the maximum Δneff of 2.2×10−4 for the HE2,1 modes.
Fig. 9. Effective refractive index difference of the HE2,1 modes as a function of the fiber bend radius with different inner radius (r1) and (b) different ring width (Δr).
In summary, the designed fiber with larger inner radius is more tolerant to the fiber ellipticity and more sensitive to the disturbance of fiber bending, while it is more sensitive to ellipticity for NZDSRF with larger ring width. Thus, proper parameters should be selected to meet various demands.
The temporal walk-off effect can be affected by the Δneff during the propagation of the even and odd eigenmodes for HE2,1. The walk-off length is described by two parameters, 2π walk-off length (L2π) and 10-ps walk-off length (L10ps), which can be expressed as below:
where Δt refers to the temporal walk off time, λ represents the wavelength, and c stands for the light velocity in vacuum [8,45]. $n_{eff}^{even}$ and $n_{eff}^{odd}$ are the effective refractive indices of the even and odd eigenmodes. Figure 10 shows the walk-off length of the even and odd eigenmodes for HE2,1 as functions of fiber ellipticity and bend radius with different fiber structure parameters. The influence caused by the Δneff between the even and odd modes is displayed as the decrease of 2π walk-off length and 10-ps walk-off length. For the same ellipticity, the temporal walk-off effect can be reduced as Δr decrease. Furthermore, as one can see in Fig. 10(b) that the 10-ps walk-off length can reach the level of 106 m, the influence caused by bending can be significantly inhibited by reducing the r1 parameter.Fig. 10. 2π and 10-ps walk-off lengths for different OAM modes in the ring fiber as functions of (a) fiber ellipticity and (b) fiber bend radius.
As the ring-shaped fiber may suffer the power leakage from the high-index ring-core region due to fiber bending, we further investigate the bending loss for the designed fiber. Figure 11 shows the bending loss for different fiber parameters as a function of fiber bend radius. The power leakage decreases when Δr is increased, and the bending loss of the designed fiber with Δr larger than 2.5 µm can reach down to the 0.02 dB/km when the bend radius is larger than 200 mm.
5. Conclusion
In summary, the ring-shaped fiber supporting OAM mode is designed for serving as NZDSF. Different dispersion value in the C band can be achieved by adjusting geometric features of the NZDSRF. The nonlinear effects can be effectively reduced in the structure proposed during transmission by maintaining moderate dispersion in the C band, which satisfies the standard of the ITU-T G.655.C. It can be inferred that by properly choosing the materials and the geometric parameters, different kinds of NZDSRF design can be discovered to meet various needs for communication systems.
Funding
National Key Research and Development Program of China (2019YFB1803700); Key Technologies R&D Program of Tianjin (20YFZCGX00440); Fundamental Research Funds for the Central Universities (63201178).
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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