Spatial-division multiplexing in the form of few-mode fibers has captured the attention of researchers since it is an attractive approach to significantly increase the channel capacity. However, the optical components employed in such systems introduce mode-dependent loss or gain (MDL) due to manufacturing imperfections, leading to significant system impairments. In this work the impact of MDL from optical amplifiers in few-mode fibers with either weak or strong mode coupling is analyzed for a 3x136-Gbit/s DP-QPSK mode-division multiplexed transmission system. It is shown that strong mode coupling reduces the impact of MDL in a similar manner as that polarization-dependent loss is reduced in single mode fibers by polarization-mode dispersion.
©2012 Optical Society of America
The continuous demand for higher data rate has fuelled the research to maximize the channel capacity by any possible means. However, transmission over single mode fibers is approaching fundamental limits, also referred to as the nonlinear Shannon capacity limit . Since the physical dimensions of time (time-division multiplexing), polarization (polarization-division multiplexing), quadrature (high-order quadrature amplitude modulation) and frequency (wavelength-division multiplexing) have already been used, exploring the space dimension in the form of space-division multiplexing is emerging as an attractive solution to overcome this capacity limit [2,3]. Both few-mode fibers (FMFs) and multi-core fibers (MCFs) have been recently proposed for exploring the space physical dimension of the fiber [4,5]. In particular, FMFs have gained considerable interest because of their advantage of carrying more data per unit area, power efficiency in terms of optical amplification and higher nonlinearity tolerance with respect to MCFs . In particular, FMFs like hollow-core photonic bandgap fibers are currently being investigated as they present an ultra-low nonlinearity coefficient and lower transmission latency as the light propagates faster than in solid core fibers .
Nevertheless, there are still challenges that need to be addressed for the realization of commercial FMF-based transmission systems. These kind of systems will suffer from increased receiver digital signal processing complexity, due to the significantly longer impulse response [5,8]; complex mode-multiplexing and de-multiplexing structures [5,9,10]; and mode-dependent loss or gain (commonly referred as loss, MDL), which originates from inline optical components like amplifiers and switches . Recent studies show efforts on reducing the MDL from optical amplifiers by tuning the modal pump power and the dopant distribution [12–14]. However, a constant and equal amplifier gain for all the modes over a wide range of input power or frequency has not yet been demonstrated. The impact of MDL and its detrimental effect on the channel capacity at a certain outage probability was shown in  and  through statistical analysis. The frequency dependence of MDL and its relation to group delay was also studied in . However, the relation between MDL and mode coupling has not yet been investigated.
In this paper, the work presented in  is further extended. The impact of MDL in combination with mode coupling is studied on a 3x136-Gb/s dual polarization quadrature phase shift keying (DP-QPSK) mode-division multiplexed system. It is found that for strongly coupled modes, the detrimental effect of MDL on the system performance is lower compared to the case of weakly coupled modes. This effect resembles the beneficial effect of polarization-mode dispersion (PMD) in the presence of polarization-dependent loss (PDL) in dual polarization single mode systems.
2. Simulation setup
Figure 1 shows the simulation setup, which is composed of three parts: transmitter, fiber channel and receiver. In this section, further details of the simulation and channel parameters are described.
As indicated in the transmitter of Fig. 1, six independent signals were generated and multiplexed from three DP-QPSK transmitters to feed the modes LP01, LP11a and LP11b and their two polarizations. As each tributary is modulated with QSPK and a net symbol rate of 25 Gbaud, the total net bit rate results in 300 Gbit/s. Overheads of 24% and 10% were added to account for forward error correction and trainings symbols, resulting in a gross bit rate of 409 Gbit/s. A single channel at a central frequency of 1550 nm was transmitted.
Constant Amplitude Zero Autocorrelation (CAZAC, also known as Frank-Zadoff or Zadoff-Chu sequences) sequences were chosen as training symbols, due to their impulse-like autocorrelation properties . To be able to equalize the channel distortions at the receiver, the training symbol length has to be at least the impulse response length. In this case, the maximum impulse response length is approximately 14.7 ns measured after 1200 km, which corresponds to 500 samples. To preserve orthogonality between the 6 training sequences, each one of them has to be cyclically shifted . For this reason, the training symbols length has to be at least 6 times the maximum impulse response length, which corresponds to 4096 samples (if only power of two sizes are used). In total, the preamble was made to have two identical CAZAC sequences of 4096 samples in order to perform synchronization at the receiver with the Schmidl and Cox method proposed in .
2.2 Fiber channel
The FMF channel  was modeled as a linear channel, excluding fiber nonlinear effects. The fiber profile design corresponds to a depressed cladding index profile as the one shown in . The purpose of introducing a trench around the graded index core is to avoid the guidance of higher order modes and at the same time to decrease the differential mode delay (DMD), since the core area is smaller and therefore the modes are more confined in the core. Although in this case nonlinearities are not modeled, the core diameter was chosen to be the largest reported . The FMF parameters (see Table 1 ) were optimized to achieve a low DMD of approximately 9 ps/km. The FMF model takes into account the inter- and intra-modal dispersion, linear mode coupling, attenuation and polarization-mode dispersion, as sources of signal degradation. In Table 1 Aeff, D, S, α, V and PMDc, stand for effective mode area, intra-modal dispersion and slope, attenuation, normalized frequency and PMD coefficient, respectively.
As described in , modal coupling was emulated by introducing a periodic core displacement as shown in Fig. 2 . The core displacement is a way of representing the bended core boundary in a discrete manner. In between each one of the sections a constant, but random, displacement of the core center position and constant coupling coefficients are assumed. In this case, the sections are 200 m long to ensure that the accumulated modal dispersion is much smaller than the pulse duration . The core displacement is assumed to vary uniformly from zero to a maximum value depending on the strength of the mode coupling. In this work two fiber scenarios are studied: weak and strong mode coupling.
To generate strong mode coupling the maximum allowed core displacement was 4.7% of the core radius. To emulate weakly coupled modes the maximum allowed core displacement was 0.6% of the core radius. To visualize the mode coupling strength, a Gaussian pulse of 30-ps width was transmitted through one of the polarization of the LP01 mode. After 80 km the induced average mode crosstalk between LP01 and the degenerate modes LP11a and LP11b is −1.25 dB for the modes which are strongly coupled and −20.6 dB for the weakly coupled modes. These crosstalk levels were the result of averaging the crosstalk of 200 channel realizations. Figure 3 shows the average power over all channel realizations for LP01 (power average of the polarizations) and LP11 (power average of LP11a, LP11b and their polarizations) as a function of the transmission distance. As Fig. 3 depicts, in the case of strongly coupled modes the power distributes among all the modes much faster with the distance than in the weakly coupled modes case. At 80 km the fiber with strong coupling achieves almost equal power for all the modes, while in the weak coupled case at 80 km there is still a difference of 23 dB between LP01, and LP11a and LP11b.
As illustrated in Fig. 1, the signals are transmitted through 80 km x Nspans of FMF. G01 and G11 are the amplifiers gains for LP01 and LP11, respectively. To generate MDL, a gain offset Goffset = G01 – G11 was introduced. LP11 was chosen to have a lower gain compared to LP01, since that is what typically has been observed in the transmission experiments reported to date [12,14]. G01 and G11 are assumed to be constant over the frequency and the transmission link.
As depicted in Fig. 1, a lumped noise model was employed, where the amplified spontaneous emission noise (ASE) or additive white Gaussian noise was added at the input of the receiver. No laser phase noise and no imperfections from the mode multiplexing and de-multiplexing to/from the FMF have been introduced.
The receiver digital signal processing consists of four stages: dispersion compensation, synchronization, channel estimation and equalization. The dispersion compensation and the minimum mean square error (MMSE) equalization correspond to an extended version of the multiple-input multiple-output (MIMO) equalizer described in . As mentioned before the frame synchronization is implemented using the training symbols and the Schmidl and Cox method. Channel estimation is performed in the time domain by correlating the received and transmitted training symbols. The estimated impulse response is converted into the frequency domain to obtain the estimated channel transfer function, which is subsequently inverted by means of the MMSE method. As the length of the signal is considerably long the overlap-save method is used for block wise frequency domain equalization .
The generated strong and weak mode coupling can be also distinguished from the estimated impulse responses as shown in Fig. 4 . The figure illustrates the most representative impulse responses out of a total of 36 after 80 km of transmission. In Fig. 4 it can be seen that the impulse responses from one input mode to the same output mode ((a), (d), (e) and (h)) are composed of a main peak, representing the light that is sent through the mode and is received in the same mode. In the case of strong mode coupling it can be observed these figures have besides the main peak also additional artifacts resulting from the random mode coupling of power to the other modes. The difference between strong and weak mode coupling can be also observed from Figs. 4(b), 4(c), 4(f), and 4(g). They represent the light that couples from LP11a to LP01, and vice versa. As shown in Fig. 4, the coupling power is larger in (b) and (c), than in (f) and (g).
3.1 MDL penalty
To evaluate the impact of MDL different values of Goffset were considered. The results are shown in Fig. 5 , in which the mean optical signal-to-noise (OSNR) penalty from the required OSNR at bit error rate of 1e-3 over the three spatial modes is illustrated for weak and strong mode coupling as a function of the transmission distance. Since the channel transfer function is a random variable due to the random mode coupling and PMD, the results of 200 channel realizations are represented with each one of the markers in the figure. The dashed and the solid lines show the average over these 200 channel realizations.
As Fig. 5 shows, the OSNR penalty increases with both distance and Goffset. In addition, it can be observed that for weak mode coupling the OSNR penalty is higher than in the strong coupling case. Allowing an OSNR penalty of 1 dB the strongly coupled modes provide 120 km and 580 km more reach in comparison to the weakly coupled modes for a Goffset of 1 dB and 2 dB, respectively. For a Goffset of 0.5 dB, no OSNR penalties were observed at 1 dB until 1200 km.
As observed in Fig. 3, during propagation, a power leakage from one mode to another takes place due to the mode coupling. The amount of power leakage depends on the strength of the mode coupling. When strongly coupled modes propagate, the power leaks faster compared to when weakly coupled modes propagate. This power exchange averages the loss per mode, which reduces the overall impact of MDL on the system performance. The tendency of the loss per mode to average with the mode coupling is similar to the case of PDL in combination with PMD . In the best case, in presence of PDL and two polarizations, PMD rotates both polarizations so they are equally attenuated. In the worst case, PDL attenuates only one of the polarizations. For spatial-division multiplexing in form of few-mode fiber the best case would be when the modes have equal loss and all the inline optical components have the same gain or attenuation; the worst case would take place when there is no mode coupling and some of the modes are more attenuated than the others, thus, the modes with the lowest gain would be the most corrupted by the added noise.
Although the strong mode coupling reduces the OSNR penalty in the system, it is worth to note that not having mode coupling is desirable for two main reasons. First, the spatial modes propagate independently, which would avoid MIMO equalization schemes higher than 2x2. Secondly, if DMD is not sufficiently low, DMD compensated transmission [5,9] would be an option to keep the impulse response short. In this kind of systems mode coupling would introduce artifacts in the impulse response, which would make the impulse response to expand with the propagation through the negative and positive DMD fibers. Here, the mode coupling is avoided from inline optical components, as it originates not only in the fiber itself [9,10].
3.2 MDL variation monitoring
The total accumulated MDL over the link was investigated in the frequency domain. The MDL was calculated as described in . As shown in Eq. (1), MDL is the ratio of the maximum and the minimum of the square root of the eigenvalues λi2 of HH* or the singular values λi of H. Here, i denotes the spatial and polarization mode number, H the estimated channel transfer function and * the transpose conjugate.
Figure 6 depicts the MDL for 160 km (2 amplifier stages) and 400 km (5 amplifier stages) as a function of the frequency for Goffset equal to 2 dB. In Fig. 6, 200 curves per each distance and level of mode coupling are plotted, each corresponding to a channel realization. Figure 6 shows that for 160 km the MDL for weak and strong mode coupling is nearly constant over frequency and their values do not differ much from each other.
On the other hand, at the transmission distance of 400 km, the MDL is still constant for weak mode coupling. However, for the case of strong mode coupling, the MDL results being frequency dependent, which causes it to randomly fluctuate between approximately 4 and 9 dB, which is lower than the almost constant 10-dB MDL for the weak mode coupling.
The average MDL over frequency as a function of the distance was also studied. Figure 7 illustrates how the MDL varies up to 1200 km for weakly and strongly coupled modes for a Goffset of 2 dB. The variance of the MDL from 200 channel transfer function realizations are indicated in the figure by means of the vertical bars.
Figure 7 shows for the case of weak mode coupling, that the MDL average increases linearly with the distance. On the other hand, in the case of strong mode coupling the average MDL increases with the square root of the transmission distance. Similarly to Fig. 6, it can also be seen in Fig. 7 in the weakly coupled case very small MDL variations. In contrast, the MDL in the strongly coupled case has larger ranges of variations, which increases with the distance or with the modal coupling. For Goffset of 0.5 dB and 1 dB the average MDL scales in the same manner as in Fig. 7.
The impact of mode-dependent loss in a 3x136-Gb/s DP-QPSK mode-division multiplexed system was investigated in the presence of weak and strong mode coupling. By means of lumped noise simulations, it was shown that high mode coupling has the benefit of reducing the overall MDL and therefore the induced OSNR penalty. Thanks to high mode coupling and the rapidly power interchange among the modes, the loss per mode averages over all modes, leading to a higher MDL tolerance compared to the weak coupling propagation. This effect can be compared with the beneficial effect of PMD in presence of PDL.
We showed as well that the MDL per amplifier stage should be below 1 dB in case of strongly coupled modes to keep the received OSNR penalty below 1 dB after a transmission distance of 1200 km, a very challenging specification for the design of FMF amplifiers.
This work has been partially supported by the European Communities Seventh Framework Programme under grant 258033 (MODE-GAP).
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