Characterization of LDPC-coded orbital angular momentum modes transmission and multiplexing over a 50-km fiber

Andong Wang; Long Zhu; Shi Chen; Cheng Du; Qi Mo; Jian Wang

doi:10.1364/OE.24.011716

1. Introduction

Driven by the increasing demand of supporting higher data rate to end-customers, more multiplexing dimensions are desired to be combined to overcome the impeding installed capacity exhaustion of current single-mode fibers (SMF). Recently, mode-division multiplexing (MDM) has been under intense investigation as a potential scheme of space-division multiplexing (SDM) for high capacity optical communications without adding extra spectral band [1]. In MDM, multiple independent data channels can be located on different spatial modes, and the orthogonality facilitates efficient (de)multiplexing and low inter-modal crosstalk among multiple modes [2]. There are several types of spatial modal basis sets that can be used in such MDM systems, such as linearly polarized (LP) modes [3, 4], orbital angular momentum (OAM) modes [5], and vector modes [6]. MDM using multimode fibers (MMFs) or few-mode fibers (FMFs) based on linearly polarized (LP) modes [7–9] has attracted much attention in the last few years. Besides using LP modes in FMF as one modal basis, OAM modes as another modal basis to represent spatial modes, featuring a helical phase front of exp(ilφ), where l is topological charge number and φ is azimuthal angle, have also shown great potential of MDM both in free-space [5] and fiber-based optical communications [10–12].

However, LP modes based MDM systems have to face some formidable challenges such as mode coupling and differential mode group delay (DMGD) during propagation in a long-haul fiber. The energy of a given data symbol launched into a particular mode spreads out into adjacent symbol time slots as a result of mode coupling and DMGD [11]. To relieve inter-mode crosstalk and simplify processing complexity, various specially designed fibers for MDM have been reported [13, 14]. Computationally intensive digital signal processing (DSP) algorithms have also been widely reported to solve the crosstalk in LP based MDM over long-haul FMF [15]. Similar to the instability of LP modes during propagation over long-haul fiber, OAM modes will also suffer differential mode delay (DMD) and modal crosstalk in long-distance fiber. A lot of theoretical and experimental efforts have been devoted for OAM transmission in fibers [10–12, 16–18]. Recently, different kinds of OAM fibers supporting multiple OAM modes featuring low-crosstalk have already been designed for multiple OAM modes fiber transmission [19–21]. Remarkably, most of the transmission distance in previous OAM transmission in fibers works is around 1 km, and the mode properties and propagation effects of OAM modes in long-distance fiber (e.g. 50 km) has not yet been reported so far. In this scenario, a laudable goal would be to give a comprehensive characterization of OAM modes transmission and multiplexing over long-distance fiber.

In this paper, we characterize the OAM modes (OAM₊₁ and OAM_-1) transmission over a 50-km FMF. We focus on the influence of DMD over the 50-km fiber. As OAM modes of an optical fiber can be equivalently described in a modal basis of eigen modes, it is found that in long-distance OAM transmission the DMD among the four eigen modes significantly influences the successful reception of the information carried by the OAM modes. A detailed comparison is made between 1-Gbaud, 2-Gbaud, 5-Gbaud and 10-Gbaud quadrature phase-shift keying (QPSK) signals over 50-km OAM mode transmission. Furthermore, OAM mode transmission experiments over 10-km FMF are also demonstrated to compare with the 50-km OAM mode transmission. Both of the theoretical and experimental results show that the mode coupling and DMD may limit the capacity of OAM-based long-distance fiber transmission. Here we employ low-density parity-check (LDPC) codes to mitigate the influence of mode coupling and DMD on long-distance fiber transmission. At last, we demonstrate OAM multiplexing and transmission with LDPC-coded 1-Gbaud QPSK signals over a 50-km FMF link.

2. Characterization of OAM modes transmission in FMF

Figure 1 shows the refractive index profiles and the image of the employed 50-km and 10-km FMF in the experiment. It is clear that the 50-km FMF has a graded index profile, while the 10-km FMF has a step index profile. In order to mitigate the effect of modal dispersion, a graded refractive index with trench that minimizes the group velocity spread is employed in the 50-km FMF. In this work, we mainly focus on the characterization of OAM modes (OAM₊₁ and OAM_-1) transmission over FMFs. The graded-index 50-km FMF has a 25.87-µm core diameter and supports six eigen modes in total ( $H E_{11}^{e v e n}$ , $H E_{11}^{o d d}$ , $T E_{01}$ , $T M_{01}$ , $H E_{21}^{e v e n}$ , $H E_{21}^{o d d}$ ), while the step-index 10-km FMF has a 18.53-µm core diameter and supports similar six eigen modes to 50-km FMF, as shown in Fig. 2. Those six eigen modes are divided into two mode groups with relatively large effective refractive index $N_{e f f}$ difference between mode Group 1 and mode Group 2 (>1.6 $\times 10^{- 3}$ ). Spatial modes in FMF can be divided into different mode groups. Mode groups are determined by the propagation constants. The propagation constants of the modes in the same mode group are more similar to each other than those in different mode groups [22]. One can get X polarized OAM₊₁ ( $O A M_{+ 1}^{X}$ ) and OAM_-1 ( $O A M_{- 1}^{X}$ ) modes and Y polarized OAM₊₁ ( $O A M_{+ 1}^{Y}$ ) and OAM_-1 ( $O A M_{- 1}^{Y}$ ) modes through proper linear combinations of eigen modes in the mode Group 2 (e.g. LP₁₁ group) [23, 24]. OAM modes are given by the following equations:

\begin{array}{l} O A M_{+ 1}^{X} = T M_{01} + i T E_{01} - H E_{21}^{o d d} + i H E_{21}^{e v e n} \\ O A M_{- 1}^{X} = T M_{01} - i T E_{01} - H E_{21}^{o d d} - i H E_{21}^{e v e n} \\ O A M_{+ 1}^{Y} = T M_{01} + i T E_{01} + H E_{21}^{o d d} - i H E_{21}^{e v e n} \\ O A M_{- 1}^{Y} = T M_{01} - i T E_{01} + H E_{21}^{o d d} + i H E_{21}^{e v e n} \end{array}

In general, modes in different mode groups with relatively large effective refractive index difference (>1.6

\times 10^{- 3}

) will experience significantly less mode coupling as compared to modes in the same mode groups. Consequently, when we selectively excite one mode in Group 2, the excited mode is not easily couple into Group 1. However, mode coupling within Group 2 will be much stronger than that between Group 1 and Group 2.

Fig. 1 (a) Relative refractive index profile of a graded-index 50-km FMF. (b) Relative refractive index profile of a step-index 10-km FMF. (c) Image of a 50-km FMF spool. (d) Image of a 10-km FMF spool.

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Fig. 2 (a) Fiber properties of the 50-km FMF. (b) Fiber properties of the 10-km FMF.

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In order to clarify the factors that may cause the signal degradation in long-distance OAM transmission, we calculate the chromatic dispersion ( $D_{λ}$ ) and differential mode delay (DMD) of the each supported mode at 1550 nm for the two kinds of FMFs in Fig. 2. The chromatic dispersion and DMD are expressed as follows [24]:

D_{λ} = - \frac{λ}{c} \times \frac{d^{2} N_{e f f}}{d λ^{2}},

τ_{p} = - \frac{λ^{2}}{2 π c} \times \frac{d β_{p}}{d λ},

where

c

is the light velocity and

λ

is the wavelength in vacuum,

N_{e f f}

is the effective refractive index, and

β_{p}

is the phase constant of the

p

th fiber eigen mode. The DMD values between eigen modes and fundamental modes (

H E_{11}

) are shown in Fig. 2. The calculated chromatic dispersion are ~20.2 ps/(nm·km) and ~21.4 ps/(nm·km) for 50-km fiber and 10-km fiber, respectively. The chromatic dispersion can be electronically compensated in the offline processing. In this work, we only use the modes in Group 2 for data transmission. Thus, we only consider the DMD between the four eigen modes in Group 2.

Given the strong mode coupling within Group 2 described above, the DMD between the four eigen modes might be reduced significantly. A possible explanation for such phenomena is as follows. When modes strongly couple to each other, every independent data stream has equal probability to travel on the fast or slow modes, increasing the chances that all data streams will have similar amounts of delay [25]. In addition, the value of DMD may slightly change along the fiber due to the variation of fiber parameters during the fiber fabrication process. The statistics of DMD in Group 2 could be theoretically analyzed. For N eigen modes in Group 2 (N = 4), the DMD for the i^th eigen mode in Group 2 can be defined as

Δ t_{i} = t_{i} - \frac{1}{N} \sum_{j = 1}^{N} t_{j}

where

t_{i}

is the DMD between the i^th eigen mode (i = 1, 2…N) and the fundamental mode (

H E_{11}

). The average DMD (ADMD) between N eigen modes in mode Group 2 can be defined as

\bar{Δ t} = \frac{1}{N} \sum_{i = 1}^{N} | Δ t_{i} |

The calculated ADMD between N eigen modes are 2.78 ps/km and 3.82 ps/km for 50-km fiber and 10-km fiber, respectively. The average mode delays of the entire 50-km and 10-km fiber links are estimated to be ~139.12 ps and ~38.20 ps, respectively.

Figure 3 illustrates the concept and principle of strong mode coupling in long-distance OAM transmission. When an OAM mode (e.g. $O A M_{+ 1}^{X}$ ) transmits in an FMF, the OAM mode can be divided into four eigen modes. These eigen modes propagate with different group velocities along the FMF, and mode coupling and inter symbol interference (ISI) spreading over multiple symbols may occur. Considering the strong coupling of four eigen modes in Group 2, the DMD in real FMFs could be reduced significantly. To analyze the influence of DMD in real case, we define the ADMD, which is expressed in Eq. (5), to simulate and evaluate the practical situation. Remarkably, with the increase of transmission distance, the system performance is getting worse due to the influences of mode coupling and DMD, which are the main limiting factors for OAM modes transmission in long-distance FMF.

Fig. 3 Concept and principle of strong mode coupling in long-distance OAM transmission.

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3. Experimental setup

Figure 4 shows the experimental setup of LDPC-coded OAM modes transmission and multiplexing over 50-km fiber. At the transmitter, a light at 1550 nm is modulated with QPSK signals. We investigate the dependence of system performance of long-distance OAM transmission on Baud-rate by using four different system configurations, operating at 1-Gbaud, 2-Gbaud, 5-Gbaud and 10-Gbaud, respectively. To mitigate the influence of DMD and mode coupling for long-distance fiber OAM transmission and multiplexing, the pseudorandom binary sequences (PRBS) are encoded using LDPC codes before mapping to QPSK signals. An arbitrary waveform generator (Tektronix AWG 70002) is used to generate in-phase (I) and quadrature (Q) drive voltages of IQ modulator. Then, the LDPC-coded QPSK signal is pre-amplified by an erbium-doped optical fiber amplifier (EDFA). The amplified signal light is converted into two OAM modes using two spatial light modulators (SLM1 and SLM2) and combined by a polarization beam splitter (PBS). The SLMs employed in the experiment are Holoeye PLUTO phase-only SLMs based on reflective liquid crystal on silicon (LCOS) microdisplays enabling 0-2π phase modulation at 1550 nm. These SLMs have a spatial resolution of 1920 x 1080 pixels and a small pixel pitch size of 8 μm. After that, the light passes through two lenses 4f system (L1 (f = 200 mm)). Then the light is focused by a 10X objective lens (OL1) with a focal length f = 20 mm and coupled into the FMF. We use an inline few mode fiber polarization controller (FMF-PC) to adjust the output mode of FMF in agreement with the input mode. After 50-km FMF transmission, the light is collimated by a 20X objective lens. Then the light passes through L1 and L2 (f = 100 mm). The third spatial light modulator (SLM3) is used to convert the output OAM modes back to Gaussian-like mode. Then the converted Gaussian-like mode is coupled into single mode fiber. After amplified by a low-noise EDFA, the resulting signal is fed into a coherent receiver. Both lasers at the transmitter and local oscillator (LO) laser at the receiver have a linewidth of ~1 kHz. The real-time sampling oscilloscope (Keysight DSA-Z 204A) operating at 80 GS/s stores the electrical waveforms for offline processing and LDPC soft-decoding. The number of LDPC decoder iterations is set to 50. Note that bulk chromatic dispersion compensation is performed prior to off-line processing.

Fig. 4 Experimental setup for LDPC-coded orbital angular momentum (OAM) modes transmission and multiplexing over 50-km fiber. PBS: polarization beam splitter; HWP: half-wave plate; QWP: quarter-wave plate; PC: polarization controller; EDFA: erbium-doped fiber amplifier; AWG: arbitrary waveform generator; I/Q Mod.: in-phase/quadrature modulator; OC: optical coupler; SMF: single-mode fiber; FMF: few-mode fiber; FMF-PC: few-mode fiber polarization controller; Pol.: polarizer; SLM: spatial light modulator; Col.: collimator; VOA: variable optical attenuator; LO: Local oscillator.

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4. Experimental results

We first characterize the transmission performance of OAM modes in two FMFs. The measured transmission loss of the OAM mode group (Mode Group 2), which we used in experiment, are 0.24 dB/km for 50 km grade-index FMF and 0.22 dB/km for 10 km step-index FMF. The complex phase patterns employed in the experiment for generating OAM₊₁ and OAM_-1 modes are plotted in Fig. 5(a), which are fork patterns for pure generation of OAM modes. Figure 5(b) shows the measured intensity profiles of the generated input OAM modes. By using a reference Gaussian beam with the same polarization to interfere with the OAM modes, one can tell the topological charge value of the OAM modes. The measured interferograms of the input OAM modes are shown in Fig. 5(b). After 10-km FMF propagation, the output modes are recorded by a camera (HAMAMATSU InGaAs Camera C10633). An inline few mode fiber polarization controller (FMF-PC) is used to adjust the output mode of FMF in agreement with the input mode. By appropriately adjusting the FMF-PC, one could get the corresponding output OAM modes with high quality and compensate the mode coupling of the two OAM modes [11, 22]. The measured output OAM intensity profiles, interferograms and demodulated Gaussian-like intensity profiles are shown in Fig. 5(c). The measured doughnut-shape intensity profiles and demodulated Gaussian-like intensity profiles for the OAM transmission through a 50-km FMF link under four different baud rates of 1-Gbaud, 2-Gbaud, 5-Gbaud and 10-Gbaud are shown in Fig. 6. One could see from Fig. 6 that the demodulated intensity of outer ring increases with the Baud rate, which might be ascribed to the influences of DMD and mode coupling within the four eigen modes for long-distance fiber transmission.

Fig. 5 Experimental results for the OAM generation and OAM transmission in a 10-km FMF. (a) Complex phase patterns for generating OAM₊₁ and OAM_-1 modes. (b) Measured intensity profiles and interferograms for input OAM modes. (c) Measured doughnut-shape intensity profiles, interferograms and demodulated Gaussian-like intensity profiles for output OAM modes carrying 10-Gbaud QPSK.

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Fig. 6 Measured doughnut-shape intensity profiles and demodulated Gaussian-like intensity profiles for the OAM transmission in a 50-km FMF under four different baud rates: 1-Gbaud, 2-Gbaud, 5-Gbaud and 10-Gbaud.

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Here we study the performance dependence of long-distance OAM modes transmission on DMD by using four different baud rates of 1-Gbaud, 2-Gbaud, 5-Gbaud and 10-Gbaud. The DMD values of two FMFs have been calculated in the above section, showing ADMD in Group 2 of 139.12 ps for the 50-km FMF and ~38.20 ps for the 10-km FMF. The periodic duration (d) of one symbol in the four configurations are 1000 ps, 500 ps, 200 ps and 100 ps, respectively. Here, we define the ratio (R) between the ADMD and the duration of one symbol as below:

R = \frac{\bar{Δ t}}{d}

Figure 7 plots measured BER performance for 1-Gbaud LDPC-coded OAM mode transmission. The two utilized LDPC codes (LDPC(16200, 9720) and LDPC(16200, 13320)) are based on the standard of digital video broadcasting-satellite second generation (DVB-S.2) LDPC code with lengths of 16200 bits. The code rate of the two utilized LDPC codes are 0.6 and 0.82, respectively. In the single OAM transmission case, it is observed that the coding gains at BER = 10⁻⁵ of LDPC(16200, 9720) and LDPC(16200, 13320) are about 4.5 dB and 3.5 dB, and BER curves for OAM₊₁ transmission with the two LDPC coding exhibit the well-known “waterfall” (BER curves drop dramatically with the increasing of OSNR after LDPC coding). It shows that the influence of DMD is negligible because the ratio (R) for 1-Gbaud OAM transmission in 50-km FMF is only 0.14, which is relatively small.

Fig. 7 Measured BER performance for 1-Gbaud LDPC-coded OAM modes transmission over 50-km FMF. Two LDPC codes, i.e. LDPC(16200, 9720) and LDPC(16200, 13320) are adopted.

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Figure 8 plots measured BER performance for 2-Gbaud, 5-Gbaud and 10-Gbaud LDPC-coded OAM modes transmission over the 50-km FMF. The BER performance for 2-Gbaud LDPC-coded OAM modes transmission is shown in Fig. 8(a). One can see obvious “error floor” at about 10⁻⁴ in BER curve for OAM₊₁ transmission without LDPC coding, and the lower code rate LDPC code ((LDPC(16200, 9720)) outperforms LDPC(16200, 13320) by ~1 dB. When uncoded BER hits to “certain value”, one can observe that BER curves start to drop dramatically, which means LDPC start to work efficiently. The “certain value” of LDPC(16200, 9720) and LDPC(16200, 13320) are about 0.12 and 0.07, respectively. In Fig. 8(b), we demonstrate the BER performance for 5-Gbaud and 10-Gbaud LDPC(16200, 9720) coded OAM modes transmission. LDPC coding can still operate at 5-Gbaud, and the coding gain of LDPC(16200, 9720) for 5-Gbaud OAM₊₁ is larger than 14 dB, while at 10-Gbaud, we can observe obvious “error floor” phenomenon both for OAM₊₁ transmission with and without LDPC coding. It is apparent that the ratio (R) of 5-Gbaud signal is 0.7. Thus, the BER performance of 5-Gbaud OAM transmission without LDPC cannot reach the enhanced forward error correction (EFEC) threshold (2x10⁻³). Furthermore, this ratio (R) is larger than 1 at 10-Gbaud signal. So the decoding failure of LDPC occurs. As increasing the transmission distance, the increase of the accumulated DMD leads to serious crosstalk between eigen modes. When the ratio of DMD to duration of one symbol reaches a certain limit, most of the energy of a given symbol launched into a particular mode spreads out into adjacent symbol time slot, thus severe inter-symbol interference (ISI) occurs and the BER performance degrades dramatically even with LDPC coding.

Fig. 8 Measured BER performance for 2-Gbaud, 5-Gbaud and 10-Gbaud LDPC-coded OAM modes transmission over 50-km FMF. (a) 2-Gbaud LDPC-coded OAM modes transmission. Two LDPC codes, i.e. LDPC(16200, 9720) and LDPC(16200, 13320) are adopted. (b) 5-Gbaud and 10-Gbaud LDPC-coded OAM modes transmission.

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In order to further verify the influence of DMD in long-distance OAM mode transmission, we also demonstrate 10-Gbaud LDPC-coded OAM modes transmission over 10-km FMF, as shown in Fig. 9, from which one can clearly visualize favorable transmission performance. The coding gain of LDPC(16200, 9720) is about 7.8 dB. The calculated ratio (R) of 10-Gbaud OAM transmission in 10-km FMF is 0.38, which is still low enough to enable OAM mode transmission. However, it is 3-dB worse compared to the 2-Gbaud OAM transmission in 50-km FMF for the larger ratio (R). Figure 9 indicates improved tolerance to mode coupling and DMD with the decrease of transmission distance. The obtained experimental results shown in Figs. 7-9 confirm that mode coupling and DMD could limit the transmission capacity and transmission distance of fiber-based OAM modes transmission.

Fig. 9 Measured BER performance for 10-Gbaud LDPC-coded OAM modes transmission over 10-km FMF.

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Moreover, we demonstrate OAM multiplexing using LDPC-coded 1-Gbaud QPSK signals in 50-km FMF. The measured BER performance for 1-Gbaud LDPC-coded OAM modes multiplexing is plotted in Fig. 10. When compared to the single OAM mode transmission case shown in Fig. 7, the coding gain of LDPC(16200, 9720) in OAM multiplexing for demultiplexed OAM₊₁/OAM_-1 increases to about 5.4 dB/5.6 dB, respectively, which could be due to the extra 7-dB crosstalk between two transmitted OAM modes in OAM (de)multiplexing.

Fig. 10 Measured BER performance for 1-Gbaud LDPC-coded OAM modes multiplexing over 50-km FMF.

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5. Conclusions

In summary, we present a comprehensive characterization of two OAM modes (OAM₊₁ and OAM_-1) transmission and multiplexing over a 50-km fiber. In order to evaluate the performance dependence of long-distance transmission link on inner mode group DMD and mode coupling, we compare OAM modes transmission experiments by using four different baud rates of 1-Gbaud, 2-Gbaud, 5-Gbaud and 10-Gbaud, respectively. LDPC codes are employed to mitigate the influence of mode crosstalk and DMD for long-distance fiber OAM modes transmission. We also demonstrate a 10-km fiber OAM　transmission for comparison. The obtained experimental results show that the QPSK signal can be successfully transmitted with OAM modes in FMF when the ratio (R) between the ADMD and the duration of one symbol is less than 0.4. Moreover, we demonstrate OAM multiplexing transmission experiments for LDPC coded 1-Gbaud QPSK signals over 50-km FMF. Both of the theoretical and experimental results show that inner mode group DMD and mode coupling may limit the capacity of long-distance fiber OAM transmission, which could be improved by adopting LDPC codes.

Acknowledgments

This work was supported by the National Basic Research Program of China (973 Program) under grants 2014CB340004 and 2014CB340003, the National Natural Science Foundation of China (NSFC) under grants 11274131, 11574001 and 61222502, the Program for New Century Excellent Talents in University (NCET-11-0182), the Wuhan Science and Technology Plan Project under grant 2014070404010201, the seed project of Wuhan National Laboratory for Optoelectronics (WNLO), and the open fund of state key laboratory of advanced optical communication systems and networks.

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Characterization of LDPC-coded orbital angular momentum modes transmission and multiplexing over a 50-km fiber

Abstract

1. Introduction

2. Characterization of OAM modes transmission in FMF

3. Experimental setup

4. Experimental results

5. Conclusions

Acknowledgments

References and links

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Optics Express