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Abstract
In this paper, we propose and design a multi-orbital-angular-momentum multi-ring air-core fiber, which has seven high-index rings with each ring supporting 62 radially fundamental OAM modes across C and L bands (from 1530 nm to 1625 nm), i.e. 434 OAM modes in total. The designed fiber features >4×10−4 intra-ring modal indices difference for OAM modes with the same topological charge l in a ring across the C and L bands. Moreover, it can keep <−52 dB crosstalk between the OAM modes in the adjacent rings at 1550 nm, and <−24 dB crosstalk across C and L bands after 100-km fiber propagation. This kind of seven-air-core-ring fiber would be a robust candidate for transmitting efficient OAM modes and boosting the capacity of optical fiber communications systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A critical issue in optical fiber communication systems of the modern era is how to meet the expeditiously growing data traffic requirement. In order to satisfy the explosively-increasing data bandwidth demands, intensive research efforts have been performed to multiplex the optical signals into various dimensions such as time, wavelength, amplitude, phase, polarization, and space [1,2]. Space division multiplexing (SDM), as a newly introduced multiplexing technique, has attracted extensive popularity for its tremendous increment in the communication capacity using single-mode fibers (SMFs). SDM technology could be implemented using multicore fibers, or multimode (few mode) fibers [3–8]. For multicore fibers, each fiber core carries a separate channel, and these different channels can be thus be multiplexed in a single fiber. For multimode fibers, the channels are carried by different modes in the same high-index core region of the fiber. Besides, multicore fiber carrying lots of modes should be carefully designed or else it may suffer large intra-ring modal crosstalk. Fibers with multiple cores carrying multiple modes (i.e. multimode multicore fibers) are also achieved with elaborate design to further increase capacity of the next-generation communication systems and networks.
Several types of mode bases multiplexed in optical fibers have been reported. Among them, linearly polarized (LP) modes are extensively used because they are relatively easy to generate [9–13]. However, they require multiple-input multiple-output (MIMO) technology with high complexity to mitigate the crosstalk among channels. Orbital angular momentum (OAM) mode is one of the most promising candidates to be exploited in hybrid multimode/multicore fibers because it couples less than LP mode does, lessening the need for MIMO processing [14]. It is characterized by the rotation of the phase front of the light beam and the beams carrying OAM modes are often called vortex beams. The phase of an OAM beam has the form exp(jlφ), where j is an imaginary unit and φ is the azimuthal coordinate [15–18]. Noted that l is the value of topological charge which can be infinite theoretically, thus, leading the quantity of OAM-carrying beams to infinite in principle. Since OAM modes with different topological charges are orthogonal to each other while propagating coaxially, they can be used in mode-division multiplexing (MDM) system and they are also compatible with wavelength-division multiplexing (WDM) technique.
A new type of fiber which has an air core and a high-index ring layer has been proposed and experimentally demonstrated [19–23]. It can commendably guide OAM modes due to its enhanced stability of OAM mode. The ring-shaped layer makes it suitable for supporting and preserving OAM beam with an annular intensity profile. Moreover, the improved material index contrast between the high-index ring and air core regions makes it possible for supporting more OAM modes. For multicore multimode fiber supporting OAM modes, S. Li et al. [24] designed a compact trench-assisted multi-ring fiber supporting 342 OAM modes. C. Chen et al. [25] designed a multi-ring micro-structured fiber featuring ultra-high-density and low-level crosstalk. However, its manufacturing process is complex and hard to achieve experimentally. Moreover, it is still urgently needed to design the multicore multimode fibers that support OAM modes by investigating some OAM mode transmission related parameters in these fibers.
In this paper, we propose and design a multi-orbital-angular-momentum multi-ring air-core fiber. The new type of fiber satisfies the following conditions to support OAM modes: (1) it has seven high-index Ge-doped rings with each ring supporting 62 radially fundamental OAM modes across C and L bands (from 1530 nm to 1625 nm), i.e. 434 OAM modes in total. The ring-core design properly fits the doughnut electric field distributions of the OAM light beam. Furthermore, the large material index difference and multi-ring design can potentially support numerous OAM modes for highly-dense SDM. (2) It has large modal effective index difference Δneff beyond 10−3 for most OAM modes and 4×10−3 for all OAM modes with same l in a ring within the C and L bands, which can split the near-degeneracy modes, avoid crosstalk of intra-ring modes and thus provide stable mode transmission. (3) The designed fiber has small inter-ring interference, featuring <−52 dB crosstalk at 1550 nm and <−24 dB crosstalk across C and L bands after 100-km fiber propagation.
2. Fiber design
2.1 Fiber structure
Figure 1(a) shows the cross section and the refractive index profile of the proposed multi-ring air-core fiber. For easy identification, the cores in the seven-air-core fiber are numbered into Core#1–7, representing the serial number of different cores. Silica (SiO2) is typical material for optical fibers because its high transparency leads to very low material loss for fiber around the 1550 nm telecom wavelength. Here, the cladding diameter D of the designed fiber is made of silica around 125 µm, the same as a standard single-mode optical fiber (SMF). To obtain higher refractive index, we design the doped silica ring with Germania (GeO2). Ge-doped material is often less lossy than the other kinds of glasses with higher refractive indices, making it usable for km-long modal transmission [3]. The proposed fiber is composed of seven air-cores and each air-core with 7-µm radius (r1) is surrounded by a Ge-doped ring with 1.5-µm ring width (Δr), which is thin enough to maintain the radially single-mode condition. The ring pitch (Λ) represents the core-to-core distance. During the fiber design and optimization, we balance large intra-ring modal effective index difference and sufficiently low inter-ring crosstalk to achieve ideal performance for the designed fiber. All numerical analyses and simulated calculations of the designed fiber are performed by the finite element method (FEM).
From the view of practical application, Ge-doped fiber has many excellent physical properties, such as long-term structural stability, high mechanical strength, low chemical activity, low-enough nonlinear optical characteristics and so on. Meanwhile, the GeO2-based silica fiber has the advantages of achievable manufacturing process and the fibers using same material have been manufactured in practice [26].
2.2 OAM states in designed fiber
The solution of vector Helmholtz equations for the perfect circular symmetric waveguides is formed from a series of eigenmodes including TE, TM, HE and EH. OAM mode can be described as a linear combination of two fiber eigenmodes having the same propagation constants:
where l is the mode topological charge, |l| is the mode order, m is the radial index and the superscripts + and − signs indicate the circular polarization of the OAM dominant component, right and left, respectively. Here, we design the fiber supporting only radially fundamental mode so that all m=1. It's worth noting that the topological charge value l can be a zero, positive, or negative value, representing for no helix (Laguerre–Gaussian beam), clockwise or counterclockwise phase helices, respectively. Consequently, for |l| = 1, OAM0,1 composed by HE1,1 mode that is not strictly speaking OAM because their phase fronts are planar other than helical.As an example, Fig. 2 shows the simulated distributions of the normalized intensity and the phase for OAMl,1 mode in the air-core-ring fiber which are obtained by full-vector finite-element-method (FEM). These OAM modes (l = 1, 2, 11, 11, 16, 16) are composed by the coherent superposition of the even and odd fiber eigenmodes of HE2,1, EH1,1, HE12,1, EH10,1, HE17,1, and EH15,1, respectively. The profiles indicate that the ring-shaped intensity distribution is well confined and the phase distribution of the OAMl,1 mode has a 2lπ change azimuthally.
Fig. 2. Intensity and phase distributions of OAMl,1 (l = 1, 2, 11, 11, 16, 16) modes in a single ring-core region.
2.3 Supported OAM number and mode properties
We investigate the number variations of the supported OAM modes as a function of different ring width at 1550-nm wavelength in the designed multi-ring-core fiber with different mole fraction of GeO2 (100%, 80% and 50%) as shown in Fig. 3(a). The OAM mode number monotonically increases with the ring width, which is mainly because of the larger space to support more fiber eigenmodes. In addition, more OAM modes could be supported with the higher mol% GeO2 in SiO2 due to large effective index difference between the fiber materials. However, the fiber’s optical loss will also grow up as the higher mole fractions of GeO2. According to the previous research, the corresponding optical-loss of the fiber with 51, 75, and 97 mol.% GeO2 are about 10, 40 and 242 dB/km at 1550 nm, respectively [26]. Consequently, the trade-off between the material loss and the supported mode numbers should be considered on the basis of fiber’s different applications.
Fig. 3. OAM mode number supported in the fiber as a function of (a) the ring width (Δr) and (b) wavelength with different mole fraction of GeO2.
Figure 3(b) illustrates the supported OAM mode number in seven-air-core-ring fiber with 7-µm air-core radius and 1.5-µm ring width as a function of wavelength under different mole fraction of GeO2. The shade regions of light cyan, green, light yellow, orange, and magenta represent for five popular (O, E, S, C, and L) optical communication bands. Besides, the solid line means that radially single-mode condition can be maintained and the dash dot line represents for appearing radially high-order mode. One can see that more than 400 radially fundamental OAM modes can be supported across C + L bands for 100 mol% GeO2 with radially single-mode condition. Also from the figure, lower doped mole fraction brings about less supported OAM mode number, but a wider applicable wavelength range for single-radial mode condition.
Figure 4(a) illustrates the effective refractive indices (neff) of some OAM modes supported in the designed fiber with 100 mol% GeO2 as a function of wavelength. The effective refractive indices of the modes decrease as the wavelength increases and the maximum modes of HE and EH supported by the fiber across C and L bands can be up to HE17,1 and EH15,1, respectively. The electric field distributions at 1625 nm of two highest eigenmodes show that they still experience good confinement in ring-core region. To avoid power change of mode group (i.e. mode coupling) and support OAM modes stably, the effective index difference Δneff between the vector eigenmodes should be >10−4, generally on the order of 10−3. Effective refractive index differences of OAM mode with same topological charge value l (i.e. HEl+1,1 and EHl−1,1 mode) in a ring are shown in Fig. 4(b) for the designed fiber with 100 mol% GeO2 across the C and L bands. We can see that the Δneff monotonously decreases with the increase of mode order and the decrease of wavelength. The closest modes are HE17,1 and EH15,1 with a separation of 4.08×10−4 at a wavelength of 1625 nm. For the mode with |l| ≤14, the Δneff is larger than 10−3 across the whole C and L band. The large Δneff of a ring in the designed fiber can effectively split the near-degeneracy and prevents the formation of LP-like modes, which could further increase the transmission distance without complicated multiple-input multiple-output digital signal processing (MIMO-DSP) based on mode-group multiplexing (MGM) [27,28].
Fig. 4. (a) Effective refractive index, (b) neff difference, (c) chromatic dispersion, and (d) effective mode field area of different OAM modes in the seven-air-core-ring fiber as a function of wavelength.
Figure 4(c) shows the dispersion characteristics of the eigenmodes in the corresponding structure of the fiber. Here, we calculate the chromatic dispersion by considering both the material dispersion and the waveguide dispersion of the guided mode. As seen in Fig. 4(c), the dispersion of HE2,1 and EH1,1 mode show great property, featuring low dispersion and little dispersion variations (<0.253 ps/nm/km and 1.526 ps/nm/km) across C and L bands. The variation of chromatic dispersion is found to increase as mode order and it is about 10.94 ps/nm/km and 23.13 ps/nm/km for EH15,1 and HE17,1 across all C and L bands. The effective mode field area (Aeff) of the OAM modes supported in each core of the seven-air-core-ring fiber is shown as a function of wavelength in Fig. 4(d). Here, the Aeff is defined as [29],
2.4 Mode purity
For fibers with large material index contrast, the solution of the waveguide characteristic equation will result in impure eigenmodes. Spin angular momentum (right/left circular polarization, $\; {\hat{\sigma }^ \pm }$) will couple with OAM mode because of the high electric fields at the air and Ge-doped glass boundary, which can form spin-orbit interaction in the designed fiber. The transverse components of the fiber OAM modes, with respect to cylindrical coordinates (r, ϕ), can be written as [30–32]
The normalized power weight (Pl) of a specific OAM state with topological charge number of l can be calculated as
To estimate the effect of spin-orbit interaction, we calculate the purity of some supported OAM modes as an example. Figure 5(a) presents the purity value of the $\textrm{OAM}_{2,1}^ + $, $\textrm{OAM}_{4,1}^ + $, $\textrm{OAM}_{6,1}^ + $, $\textrm{OAM}_{8,1}^ + $, and $\textrm{OAM}_{10,1}^ + $ composed by EH1,1, EH3,1, EH5,1, EH7,1, and EH9,1 for 60%, 80%, and 100% GeO2 mole fraction. The OAM mode purity of the $\textrm{OAM}_{10,1}^ + $ mode is beyond 0.9 for the designed fiber with three different mole fractions, and the maximum purity value of 0.945 can be found for 60% mole fraction. For $\textrm{OAM}_{2,1}^ + $ modes, the purity values are lowest around 0.58 at 1550 nm. From Fig. 5(b), we can see the mode purity value of the $\textrm{OAM}_{1,1}^ + $, $\textrm{OAM}_{3,1}^ + $, $\textrm{OAM}_{5,1}^ + $, $\textrm{OAM}_{7,1}^ + $, $\textrm{OAM}_{9,1}^ + $, and $\textrm{OAM}_{11,1}^ + $ composed by HE2,1, HE4,1, HE6,1, HE8,1, HE10,1, and HE12,1. The lowest purity value is from HE2,1 in the air-core fiber with 100% mole fraction, which are about 0.645. For the OAM order larger than 7, all the purity of OAM mode composed by HE eigenmodes could be higher than 0.9, which also indicates that the HE-supported OAM modes have higher purity compared to EH-supported OAM modes with the same value of l. also from Fig. 5(a) and Fig. 5(b), usually the higher-order modes are purer because of the less confinement of electrical fields in the ring-core area.
Fig. 5. OAM mode purity based on (a) EH eigenmodes and (b) HE eigenmodes as a function of mode order for different GeO2 mole fraction.
3. Inter-ring crosstalk
To effectively reduce the core-to-core coupling and suppress the inter-ring crosstalk, we optimize the ring pitch of the designed fiber. The coupling length is defined as the shortest propagation distance of a mode power to be fully transferred from the input core to another core, which can be used to analyze the coupling behavior and inter-ring crosstalk in the multi-ring fiber. The crosstalk (XT) from the mode in one adjacent ring is the ratio of the received power from one core to that of another core at the receiver end [34], which can be expressed by Eq. (5):
where P and P’ are the output power from the input core and that from the neighboring core, respectively. The worst total crosstalk is to the central ring-core (core#1), which can be calculated by 10log10(6P’/P). Moreover, the total crosstalk of the other cores (core#2–7) is 10log10(3P’/P). The normalized power transfer between two identical rings can be evaluated by sin2[πz/(2Lc)], where Lc and z are the coupling length and propagation length [35,36]. Futhermore, the Lc between the identical rings is calculated by the mathematical Lc = π/(βa − βb), where βa and βb are the propagation constants of the even and odd modes, respectively.Figure 6(a) shows the theoretical coupling length at 1550 nm. It is confirmed that from Fig. 6(a), the coupling length of 104 km is achieved when Λ is 40 µm, while the coupling length between the core with Λ=20 µm could be less than 10 m. The fiber with larger ring pitch shows longer coupling length, which is mainly due to larger separation between adjacent rings thus leading to less mode overlap and power transfer. To further evaluate the performance of the proposed seven-air-core-ring fiber, the crosstalk between the adjacent rings as a function of Λ is depicted in Fig. 6(b) with 100-km propagation distance. When Λ is 40 µm, the inter-ring crosstalk in adjacent rings can be maintained lower than −50 dB. Moreover, we also notice in Fig. 6(b) that the inter-ring crosstalk of the low-order mode is smaller than that of the high-order mode owing to tight mode confinement in the ring region.
Fig. 6. (a) Coupling length and (b) crosstalk for different OAM modes in the seven-air-core-ring fiber as a function of the ring pitch Λ under a propagation length of 100 km at 1550 nm.
Figure 7(a) displays the adjacent inter-ring crosstalk of four typical OAM modes, the lowest and highest ones composed by HE and EH modes (HE2,1, HE17,1, EH1,1, and EH15,1 related OAM modes), in adjacent rings as a function of the seven-ring-air-core fiber length at 1550-nm wavelength with a ring pitch Λ of 40 µm. For a 1-km propagation length, the crosstalk of adjacent rings can be held less than −90 dB for OAM modes while the crosstalk gradually increases with longer transmission distance. Shorter propagation length can definitely bring lower inter-ring crosstalk, which also indicates some possible use for short-distance optical fiber communications, such as optical access networks. Furthermore, the dark gray dash line represents for the worst total crosstalk of core#1 and the gray line represents for the worst total crosstalk of the other cores (core#2–7). The worst total crosstalk at 1550 nm is −45.00 dB for the central ring-core after propagating through a 100-km designed fiber. Figure 7(b) plots the inter-ring crosstalk of these OAM modes in adjacent rings as a function of wavelength from 1530 nm to 1625 nm across C and L bands under 100-km propagation length. We can see that the crosstalk monotonously grows with the wavelength. The highest inter-ring crosstalk of the seven-air-core-ring fiber is −24.74 dB at wavelength 1625 nm for EH15,1 related OAM mode and it can be achieved for the lowest crosstalk −67.35 dB at wavelength 1530 nm for the same mode. For the low-order OAM modes formed by HE2,1 and EH1,1, inter-ring crosstalk is relatively low about −42 dB at the wavelength of 1625 nm. The worst total crosstalk after 100-km propagation length is −17 dB for the EH15,1 related OAM mode of central ring-core at 1625 nm.
Fig. 7. Crosstalk for several typical OAM modes of the seven-ring air-core fiber with 40-µm ring pitch as a function of (a) fiber length and (b) wavelength.
4. Discussion of dimensional change for actual fiber production
As mentioned above, we set the parameter of the designed fiber as r1=7µm, Δr=1.5µm, D=125 µm, which are selected as optimization solution with the only radially fundamental order modes, more supported OAM mode number and smaller possible intra-ring/inter-ring crosstalk. The optimized configuration is obtained by the control variable method. Here, we will discuss the influence of fiber’s dimensional change in actual production.
As shown in Fig. 8(a), the larger ring width could increase the mode purity and reduce the inter-ring crosstalk. When the fluctuation range of ring width is within 0.05 µm (1.5µm ±0.05 µm), it has little impact on mode purity and crosstalk. Additional simulations in Fig. 8(b) reveals that Δneff are much larger than 10−4 as varying the thickness of ring. Further inspection of the results in Fig. 8(b) shows that the supported OAM mode number across C and L bands could be maintained larger than 400 when the fluctuation range of the ring width is within 0.1 µm (1.5µm ±0.1 µm). From these simulated results, we can see that the seven-ring-air-core fiber allows the existence of the acceptable errors and thus, it will reduce the difficulty of the fiber manufacturing due to the slightly lower accuracy requirement.
Fig. 8. Influence of the varying ring width on (a) crosstalk and mode purity, (b) neff difference and total OAM mode number with fixed ring pitch.
Table 1 shows the performance of the designed fiber with different mole doping concentration, including the highest eigenmodes, supported OAM mode number across C and L bands, the worst adjacent inter-ring crosstalk, and the closest Δneff of OAM mode with the same l at 1550 nm. Higher doped mole fraction designed fiber features tighter light confinement, more supported modes and lower inter-ring crosstalk, which is suitable for highly-dense SDM communication transmission but may pay the cost of relatively large loss. Seven-ring-air-core fiber with lower doping concentration sacrifices supported mode number but can be further considered to longer length transmissions. Moreover, the parameters of proposed fiber with lower Ge-doping concentration such as ring pitch and ring width, could be considered to further optimize to achieve better fiber’s properties.
Table 1. Designed fiber property with different germanium mole fraction at 1550 nm
5. Conclusion and perspective
We propose a new design strategy for a large mode count OAM seven-air-core-ring fiber with low intra-ring and inter-ring modal crosstalk. We also analyze the characteristic of dispersion and mode purity for this designed fiber. With the optimal Λ=40 µm, over 400 OAM modes are confirmed with >4×10−4 intra-ring modal indices difference and <−24 dB crosstalk at 1550 nm that satisfying the requirements for mode division multiplexing in the C + L bands. Moreover, the chromatic dispersion of all modes varies less than 25 ps/nm/km and the mode purity ranges from 0.58 to over 0.95. While preliminary, the proposed fiber has the potential to be used in WDM and MDM communications to create communication links with high capacity. For future application of multi-OAM multi-ring fiber, the whole OAM-communication system, should also be developed to support the SDM scheme. The corresponding OAM system components including transmitter, multiplexer, amplifier, switch, de-multiplexer, and receiver, are still under investigation to constitute a complete OAM transmission system.
Funding
National Key Research and Development Program of China (2019YFB1803700); Key Technologies Research and Development Program (20YFZCGX00440); Fundamental Research Funds for the Central Universities (11874226, 63191511, 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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