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Abstract
We present a low-loss weakly guiding ring-core fiber for orbital angular momentum (OAM) mode group multiplexing (MGM) transmission. This special fiber design supports 50 radially fundamental modes divided into 13 mode groups with only 0.7% relative refractive index difference between the fiber ring core and cladding. Except the first two groups with 10−5 mode spacing, the other mode groups are separated with each other with effective refractive index difference (Δneff) larger than 10−4, indicating relatively low-level inter-group crosstalk. One can directly use different OAM mode groups for MGM communications without multiple-input multiple-output digital signal processing (MIMO-DSP) technique. Besides, one can employ different OAM modes among the same mode group to carry different data information assisted by small-scale MIMO technique. The target fiber exhibits small and flat dispersion within (14.3, 39.7) ps/nm/km which is comparable to that in the standard single-mode fiber (SMF), and extremely large mode area within (787.9, 841.2) µm2 over the whole C + L band. MIMO equalization complexities for modified small-scale MIMO-DSP assisted intra-group modes multiplexing combined with MIMO-free inter-group modes multiplexing method in both time and frequency domain are much simpler compared to traditional 50×50 MIMO equalization.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
19 December 2019: Typographical corrections were made to Refs. 4 and 23.
1. Introduction
The typical traditional physical dimensions of lightwaves, including amplitude, phase, polarization, frequency/wavelength and time, have almost reached their scalability limits. To overcome the forthcoming capacity crunch, space-division multiplexing (SDM) exploring the space physical dimension, such as multi-core fiber (MCF) possessing multiple cores and few-mode fiber (FMF) supporting multiple modes, has been widely studied in fiber-optic transmission systems [1–3]. In general, linearly polarized (LP) mode or orbital angular momentum (OAM) mode bases are used in fiber mode-division multiplexing (MDM) transmission system [1–13]. In fact, LP modes and OAM modes can be synthetized by fiber vector eigenmodes which are generally four-fold degenerate in the traditional circular-core FMFs and multi-mode fibers (MMFs) [14–22]. As a result, with the increase of the multiplexed mode number and transmission length, inter-modal coupling and crosstalk are almost unavoidable due to fiber intrinsic defects and external perturbations in practical fabrication process and deployment, which makes complex multiple-input multiple-output (MIMO) equalization compulsory to compensate crosstalk and dispersion at the receiver [3]. To simplify MIMO digital signal processing (MIMO-DSP) complexity, ring-core fibers have been proposed and fabricated which can enlarge the mode spacing between HEm+1, n and EHm-1, n modes, thus transmit multiple OAM modes stably [7, 23–33]. Note that each HE and EH mode is still two-fold degenerate.
In view of the fact that the fiber vector eigenmodes in the most commonly used FMF, MMF and ring-core fiber can be divided into several mode groups according to the different effective modal index (neff) values, one can employ different mode groups to carry different data information channels. In detail, the neff values in the same mode group are approximate equal to each other, while different mode groups possess relatively larger neff difference, typically on the order of 10−4. Different mode groups with relatively large neff difference (Δneff) have low-level inter-group crosstalk, showing possible mode-group multiplexing (MGM) communications [18, 19, 30–32]. Besides, when we use different modes among the same mode group with relatively large intra-group crosstalk to carry different data information, small-scale MIMO-DSP technique could be used to mitigate the mode crosstalk [33]. Hence, small-scale MIMO-DSP assisted intra-group modes multiplexing combined with MIMO-free inter-group modes multiplexing can be considered to facilitate the mode multiplexing communications in fibers.
Remarkably, all of the existing fibers somehow have some defects. For example, the conventional graded-index MMF supports numerous radially higher-order modes which are extremely hard to multiplex/demultiplex. Moreover, the mode number in each group would increase with the mode group order, thus requires more complex MIMO-DSP technique for intra-group modes multiplexing. The ring-core fiber usually adopt relatively high-contrast-index to enhance mode spacing, which is quite difficult to manufacture and may result in large fiber loss [7, 23–29]. In this scenario, it would be meaningful to make the trade-off between the multiplexed modes number and MIMO-DSP complexity, while ensuring low fiber loss.
In this paper, we propose and design a low-loss weakly guiding ring-core fiber for OAM MGM transmission. The relative refractive index difference between the fiber ring core and cladding is set to be 0.7%, which is fully compatible with the current mature fiber manufacture technologies and enables low-loss fiber transmission. Through both semi-analytic and numerical simulation and optimization, the final fiber design supports 50 vector eigenmodes divided into 13 mode groups, where the first mode group only contains two fundamental modes and the other groups each has four modes. Except the first two groups, the Δneff between the other mode groups are all larger than 1×10−4, enabling relatively low-level inter-group crosstalk, thus would be promising for MIMO-free inter-group modes multiplexing. Also, we analyze the mode number, Δneff, neff, differential group delay (DGD), chromatic dispersion (Dλ), effective mode area (Aeff) and nonlinearity (γ) [34, 35] for each OAM mode versus wavelength. Finally, we discuss the MIMO algorithmic complexities based on both adaptive time-domain equalization (TDE) and frequency-domain equalization (FDE).
2. Concept
Figure 1(a) shows the concept of OAM based small-scale MIMO-DSP assisted intra-group modes multiplexing combined with MIMO-free inter-group modes multiplexing in the specially designed weakly guiding ring-core fiber. Lots of multiplexed OAM modes from channel 1 to channel N propagate through the specially designed weakly guiding ring-core fiber. Here the mode group number in the fiber is denoted by M and N = 4×M-2. After demultiplexing, these modes are demodulated separately at the receivers. In order to recover the transmitted data, a 6×6 MIMO-DSP equalization block is applied for the six multiplexed OAM modes in the first and second mode groups, while M-2 sets of 4×4 MIMO-DSP is applied for the other groups to undo coupling effects occurring within the fiber. Figure 1(b) shows the refractive index profile of the specially designed weakly guiding ring-core fiber at 1550 nm. The relative refractive index difference between the fiber ring core and cladding is set to be 0.7%, which is compatible with the current mature fiber manufacture technologies and facilitates low-loss fiber transmission.
Fig. 1. (a) Concept of OAM based small-scale MIMO-DSP assisted intra-group modes multiplexing combined with MIMO-free inter-group modes multiplexing. (b) Refractive index profile of the specially designed weakly guiding ring-core fiber. Mux: multiplexing; Demux: demultiplexing.
3. Fiber design by semi-analytic method
In this work, the radii of the fiber inner and outer ring core are denoted by a1 and a2, respectively, with ratio of inner to outer radii defined as α = a1/a2 and ring width as d = a2 - a1. The cladding radius is fixed at 62.5 µm. The refractive index ratio between the ring core (n2) with inner (n1) and outer cladding (n3) is defined as β1 = n1/n2 and β2 = n3/n2, respectively. Note that n2 keeps larger than n1or n3. And the normalized waveguide frequency at a wavelength λ is defined as ${V_0} = {k_0}{a_2}\sqrt {n_2^2 - n_1^2}$, where k0 = 2π/λ. The eigenvalue equation of supported eigenmodes including TE0m, TM0m, HEvmand EHvm (here the v and m in the subscript refer to azimuthal and radial indices) in the ring-core fiber are expressed as [36]:
TE0m mode:
To simplify the weakly guiding ring-core fiber design, we adopt the pure-SiO2 in both inner and outer cladding, that is, n1 = n3 and β1 = β2. The relative refractive index difference between the ring core and cladding is then defined as Δn= (n2- n1)/n1. When we fix β1 at 0.993, which corresponds to the Δn of 0.7%, that is, n2 = 1.4541 and n1 = 1.444 at 1550 nm, the calculated supported mode number in the weakly guiding ring-core fiber as functions of α and V0 is shown in Fig. 2. The white region in the left is where the fiber supports radially higher-order modes which are hard to multiplex/demultiplex yet. To meet the requirement of mode groups separated while supporting radially fundamental modes as much as possible, we choose the point α = 0.83 and V0 = 17.3 (that is, a1 = 20.8 µm and a2 = 25 µm) in the black dotted circle as the final design. The supported 50 vector eigenmodes are divided into 13 mode groups. Except the first two groups with 10−5 mode spacing, the other mode groups are separated with each other with Δneff larger than 10−4.
4. Fiber design by numerical method
Furthermore, we also use full-vector finite element method to determine the weakly guiding ring-core fiber structure and analyze the OAM modes properties. Similarly, by sweeping the three parameters, including Δn (from 0.5% to 1%), a1 (from 12 µm to 24 µm) and d (from 1 µm to 6 µm), we eventually select the point Δn = 0.7%, a1 = 20.8 µm and a2 = 25 µm as the target fiber structure size. Table 1 lists all 50 radially fundamental vector modes and OAM modes and Δneff between adjacent mode groups in the target weakly guiding ring-core fiber at 1550 nm. Each $HE_{lm}^{e/o}$ or $EH_{lm}^{e/o}$ mode is two-fold degenerate, comprising even and odd modes. Each $OAM_{ {\pm} l,m}^{L/R}\; (l > 0)$ is four-fold degenerate including polarization and rotation, while $OAM_{0,m}^{L/R}$ is two-fold degenerate including only polarization. Note that here l and m in the subscript represent the topological charge number and the number of radial concentric rings in the intensity profile of the OAM mode, respectively. As all 50 modes supported in the proposed fiber are radially fundamental, m is a constant and equals one. R and L in the superscript refer to right and left handed circular polarization, respectively. Obviously, the mode group separation increases with group number. Except the first two mode groups with 2.72×10−5 separation, the Δneff between the other mode groups are all larger than 1×10−4. Specially, Fig. 3 illustrates the synthetic process and phase distributions of the x-component electric field of all 50 circularly polarized OAM modes. One can see that each $OAM_{ {\pm} l,m}^{L/R}$ mode can be obtained by proper combining of $HE_{l + 1,m}^{even}$ and $HE_{l + 1,m}^{odd}$, $EH_{l - 1,m}^{even}$ and $EH_{l - 1,m}^{odd}$, or TM01 and TE01 with a relative π/2 phase shift [16, 17].
Table 1. Vector mode and OAM mode groups in the designed weakly guiding ring-core fiber at 1550 nm
Fig. 3. Spatial phase distributions of the x-component electric field of the supported 50 OAM modes in the designed weakly guiding ring-core fiber.
Remarkably, it has been concluded earlier [23] that the fiber designs whose refractive index profiles mirror that of the mode itself would maximize Δneff while also yield large refractive index gradients and modal field gradients. Figure 4(a) shows the refractive index profile and E-field intensity of the TE01 mode in the target weakly guiding ring-core fiber at 1550 nm. One can see that the two profiles are highly overlapped. Besides, the corresponding gradients of the refractive index and E-field amplitude of the TE01 mode are shown in Fig. 4(b). The both gradients peak at the two core-cladding boundaries which eventually enhances the Δneff. Similarly, the refractive index profile, E-field intensity and the corresponding gradients of the HE13,1 mode at 1550 nm are shown in Figs. 4(c) and 4(d), which also reveal high consistency.
Fig. 4. (a) Refractive index profile and E-field intensity and (b) the corresponding gradients of the refractive index and E-field amplitude of the TE01 mode at 1550 nm. (c) Refractive index profile and E-field intensity and (d) the corresponding gradients of the refractive index and E-field amplitude of the HE13,1 mode at 1550 nm.
Finally, we investigate the broadband characteristics of the target weakly guiding ring-core fiber over the whole C + L band (1530 to 1625 nm), as shown in Fig. 5. The cut-off wavelengths of four highest-order modes ($HE_{13,1}^{e/o}$ and $EH_{11,1}^{e/o}$) in the 13th mode group are 1595 nm. The minimum Δneff (min(Δneff)) between different mode groups (except the first two groups) are all above 1×10−4 as shown in Fig. 5(a). Additionally, Fig. 5(b) shows the neff of all vector modes versus wavelength. One can clearly see that the neff values in the same mode group (in same color) are approximately equal, while the 2nd to 13th mode groups possess relatively large Δneff. Actually, we only present calculated results for one of the two-fold degenerate vector modes (i.e. one of the $HE_{lm}^{e/o}$ modes or one of the $EH_{lm}^{e/o}$ modes) in this section due to the fact that degenerate vector modes have similar neff, DGD and Dλ values. Moreover, degenerate OAM modes also have similar Aeff and γ, thus we only provide Aeff and γ for two of the four-fold degenerate $OAM_{ {\pm} l,m}^{L/R}$ modes (i.e. one of the $HE_{lm}^{e/o}$ related OAM modes and one of the $EH_{lm}^{e/o}$ related OAM modes). The simulation results show that all vector modes corresponding to related OAM modes feature flat DGD (between OAM0,1 mode and OAMl,m mode) sitting in (117.3, 2.0×104) ps/km over the whole C + L band, as shown in Fig. 5(c). The Dλin Fig. 5(d) within (14.3, 39.7) ps/nm/km is comparable to that in the standard single-mode fiber (SMF) (17 ps/nm/km). Figure 5(e) shows that the Aeff for all OAM modes in 13 mode groups increase with wavelength and are extremely large within (787.9, 841.2) µm2 over the whole C + L band. The Aeff values are above eight times of that in SMF of the ITU-T recommendation. Moreover, the γ which is inversely proportional to the Aeff is displayed in Fig. 5(f). Accordingly, the γ values are sitting in (0.1156, 0.1235) km-1W-1. Obviously, four-fold degenerate $OAM_{ {\pm} l,m}^{L/R}$ modes exhibit alike neff, DGD, Dλ, Aeff and γ values.
Fig. 5. (a) Mode number, min(Δneff) between different mode groups (except the first group), (b) neff, (c) DGD, (d) Dλ, (e) Aeff and (f) γ for each mode versus wavelength.
Remarkably, the neff of guided modes in the fiber is sitting between the index of fiber core and cladding, which limits the total mode spacing between adjacent fiber vector modes. For example, the target fiber we present exhibits 1.01×10−2 index difference between the ring core (n1 = 1.444) and cladding (n2 = 1.4541), and the accumulated Δneff mainly existing between adjacent mode groups equals 5.78×10−3, while the fundamental mode and highest-order mode are 3.86×10−3 and 4.6×10−4 away from the fiber core and cladding, respectively. Obviously, to enlarge the mode group separation, one way is to reduce the supported mode number by decreasing the ring width or position, and the other is to increase the ring index. For example, under the 0.7% weakly guiding condition, with certain ring size the fiber is able to support 26 radially fundamental modes divided into seven mode groups with Δneff >10−4 among the lower-order four mode groups and >10−3 among the higher-order three mode groups, which benefits much more stable MGM transmission.
5. MIMO algorithmic complexities
There are two OAM multiplexing approaches that could be considered for the designed low-loss weakly guiding ring-core fiber. One is to use different OAM mode groups to carry different data information channels. Different OAM mode groups (except the first two groups) with relatively large Δneff have low-level inter-group crosstalk, showing possible OAM MGM communications without MIMO. Another is to employ different OAM modes among the same mode group to carry different data information assisted by small-scale MIMO-DSP technique. In this section, we discuss the MIMO algorithmic complexities of the second multiplexing approach based on both TDE and FDE.
To recover transmitted signals from what is received in different mode channels at the receiver, an ideal linear filter matrix W(n) which is an inverted transfer matrix (H(n)-1) can be applied. Hence, the signal y(n) obtained by convolution of the received signal x(n) and linear equalizer matrix W(n) is believed to have undone the channel crosstalk. The input-output relation can be expressed as [37]:
According to Eq. (17), the equalization can be implemented in both time domain and frequency domain. The estimation of filter coefficients is generally obtained by using the blind or data-aided (DA) method. The blind equalization method can determine the filter coefficients based on the received signal. However, it normally features slower convergence rate and worse performance for low optical signal-to-noise ratio (OSNR). DA equalization method is to insert a known training sequence in the preamble of each data frame. The computational load of DA estimation is much lower than blind estimation [38], thus in this section, we focus on the comparison of MIMO equalization algorithmic complexities based on DA equalization method, and the analyses below are all about the mostly experimentally adopted least-mean-square (LMS) method. In detail, when we use LMS algorithm of TDE, the computational complexity corresponds to the number of complex multiplications per symbol per mode. Assuming that the multiplexed channel number is Ds, the mode transfer matrix H(n) is Ds× Ds matrix, the symbol rate (or baud rate) is B, the sampling rate is Rs (samples/symbol), the total modal group delay (MGD) of the entire link is τMGD = Δτ·L, where Δτ is the differential modal group delay (DMGD) and L is the fiber length, the tap length of each finite impulse response (FIR) filter is given by [37]: To obtain every output symbol per mode, we need DsNTap multiplications. And to update the filter coefficients, we need other DsΔτLB multiplications. Hence, the total complexity to recover all symbols per mode of TDE regardless of carrier phase recovery can be expressed as: Meanwhile, the total computational complexity to recover all symbols at once by MIMO equalization algorithms is given by:Compared to TDE, FDE can further reduce the complexity of MIMO equalization algorithms by replacing complex convolution with multiplication computations [39]. That is, the filter coefficients updating is converted from time domain to frequency domain, while both the received signal and the error are still in the time domain. In detail, an equivalent half-symbol-spacing FIR filter in frequency domain is realized by the use of even and odd sub-equalizers. Similarly, to obtain NTap/2 output symbols per mode per block, we need DsNTap multiplications. And to update both even and odd equalizers, we need other DsNTap multiplications. Besides, the total number of (inverse) fast Fourier transform (FFT/IFFT) per mode is 4 + 2Ds, including two FFT for the input signals, a pair of FFT/IFFT for filter coefficients updating, and 2Ds FFT/IFFT for gradient constraint in the gradient estimation block, in which Nlog(N)/2 (N refers to complex numbers) complex multiplications are required to execute each FFT. Hence, the total complexity to recover all symbols per mode of FDE can be expressed as [37]:
Meanwhile, the total computational complexity to recover all symbols at once by MIMO equalization algorithms is given by: It can be clearly seen that the complexity of TDE and FDE scale linearly with the MGD of the link and the number of modes. Although MIMO equalization algorithms can undo the multiplexed channel crosstalk in long-haul MDM transmission system, the required computational load is incredible. There are two main methods that can reduce the computational load for SDM MIMO equalization signal processing. One is to reduce DMGD, and the other is to make the mode transfer matrix sparse. Taking the second method as an example, assuming that a specially designed fiber supports 6 eigenmodes divided into 2 mode groups (one with 2 fundamental modes and another with 4 first-order modes), the mode crosstalk mainly exists within the same mode group while the inter-group crosstalk is negligible. The mode transfer matrix can be expressed as Eq. (23) below:As for the designed low-loss weakly guiding ring-core fiber design supporting 50 eigenmodes divided into 13 mode groups, the Δneff between the 2nd to 13th mode groups are all larger than 1×10−4, enabling MIMO-free inter-group modes multiplexing, while the first two groups possess insufficient Δneff. Hence, the traditional 50×50 MIMO equalization is simplified to 11 sets of 4×4 MIMO equalization and a 6×6 MIMO equalization. Assuming that the DMGD among the same and different mode groups are Δτd and Δτ, respectively, according to the simulation results, Δτ/Δτd ≈ 346. When we consider MIMO equalization in the time domain, the modified MIMO equalization need 212(Rs+1)ΔτdLB complex multiplications in total according to Eq. (20), while in comparison, the traditional 50×50 MIMO equalization need 2500(Rs+1)ΔτLB complex multiplications, which is around 4080 times larger than the modified one. Similarly, when we consider MIMO equalization in the frequency domain, the modified MIMO equalization needs 624log2(RsΔτdLB) + 848 complex multiplications in total according to Eq. (22), while in comparison, the traditional 50×50 MIMO equalization need 5200log2(RsΔτLB) + 10000 complex multiplications, which is around 8 times larger than the modified one.
6. Conclusion
In summary, a low-loss weakly guiding ring-core fiber supporting 50 radially fundamental modes divided into 13 mode groups is designed and analyzed. The min(Δneff) between different mode groups (except the first two groups) are above 10−4 over the whole C + L band with extremely small and flat dispersion and nonlinearity, thus can be compatible with existing WDM technique. This kind of fiber design could be considered for low-loss small-scale MIMO-DSP assisted intra-group modes multiplexing combined with MIMO-free inter-group modes multiplexing transmission.
Funding
National Key R&D Program of China (2018YFB1801803); National Natural Science Foundation of China (61761130082, 11574001, 11774116); Royal Society-Newton Advanced Fellowship; National Program for Support of Top-notch Young Professionals; Yangtze River Excellent Young Scholars Program; Natural Science Foundation of Hubei Province (2018CFA048); Key R&D Program of Guangdong Province (2018B030325002); Open Fund of IPOC Beijing University of Posts and Telecommunications (IPOC2018A002); Program for Huazhong University of Science and Technology Academic Frontier Youth Team (2016QYTD05); Fundamental Research Funds for the Central Universities (2019kfyRCPY037).
Disclosures
The authors declare no conflicts of interest.
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