Long-haul few mode fiber optic link with differential mode delay compensation

This paper present results of simulation of the few mode fiber optic link with the differential mode delay compensation. There were considered some variants of compensation of differential mode delay including differential mode delay compensation together chromatic dispersion compensation on line optical amplifiers and differential mode dispersion compensation on more density maps.


Model of the long-haul two-mode fiber optic communication line.
Since the purpose of the work was to determine the feasibility of DMD compensating on LOAs, it was suggested that for this, as in [19,[21][22][23][24][25], it suffices to analyse in the two-mode fiber optic link the propagation of the LP 01 and LP 11 modes. In the general case, six modes propagate in such a line. There are the two orthogonally polarized modes for LP 01 mode, the two orthogonally polarized modes for LP 11a mode and the two orthogonally polarized modes for LP 11b mode. As is known, the delay between orthogonally polarized modes is negligible compared to the delay between modes LP 01 and LP 11 . The delay between modes LP 11a and LP 11b is also negligible compared to the delay between modes LP 01 and LP 11 . This allowed us to consider as degenerate as orthogonally polarized modes, as well as modes LP 11a and LP 11b , and restrict ourselves to the analysis of the propagation of two modes LP 01 and LP 11 .
To simulate the fiber optic links under investigation, the algorithm described in detail in [25] has been used. The process of signal propagation in a few-mode optical fiber taking into account the linear and nonlinear coupling of modes, can be represented as a system of nonlinear Schrödinger equations [26,27] where A p -envelope of optical signal in p-th mode in time domain; β 0p , β 1p , β 2p -propagation constant, first derivation of propagation constant and group velocity dispersion for p-th mode, respectively; β 0r , v gr -mean values of propagation constant and group velocity for p-th mode; γnonlinear coefficient; q mp -linear mode coupling coefficient; f lmnp -nonlinear coupling coefficient.
Linear mode coupling coefficient is defined by mode field overlap integral and refractive index perturbation internal and external factors. Nonlinear coupling coefficient is defined by mode fields overlap integral.
This equation can be simplified for particular cases of weak and strong mode coupling. In particular, for the weak mode coupling averaging random variations of the birefringence, the Manakov equation was obtained, differing by the coefficients 8/9 and 4/3 for the terms taking into account the self phase modulation (SPM) and the cross phase modulation (XPM) of the spatial modes, respectively [20]. For the strong mode coupling, when the strength of linear mode coupling is comparable with the coupling of linearly polarized modes, the coefficient of the nonlinear summand in the Manakov equation is determined in summary form over all modes, taking into account the overlap factor of the mode fields [20].
In general form, (1) can describe all regimes of propagation in few-mode fibers. To solve (1), the split-step Fourier method (SSFM) method and its various modifications has been widely used [20,26].
In this paper, we propose to use the following approach: dispersion effects and linear coupling of modes are taken into account independently from nonlinear effects and nonlinear mode coupling.

 A
where -is the operator takes into account attenuation, dispersion effects and linear mode coupling; N -is the operator takes into account nonlinear effects and nonlinear mode coupling.
Neglecting nonlinear operator the system of coupled equations for optical signal Fourier transform can be represented as -is the column matrix Nх1, formed from Fourier transform of optical signal in spatial modes; Ĉis the square matrix NxN taking into account dispersion effects and linear mode coupling.
Matrix coefficients are defined by: where α -the attenuation coefficient; It should be noted that in this case it is assumed that the polarization modes are degenerate. To analyze the influence of mode coupling on the system performance the two mode regime was analyzed. The solution of system (3) for two modes can be found in the form Here parameter  take into account power exchange between modes under propagation. The equal group velocity dispersion was assumed for propagated modes. This obtained equation is similar to solution in [28].
Analyzing presented above equations, one can conclude that there are periodical power exchange between propagated modes with spatial step and coupling strength determined by the parameter θ and q pm . Because of the significant values of Δβ1, spatial step is of the order of millimeters, depending on the optical fiber construction, which leads to the necessity to use the average coupling factor.
At simulation, it is proposed to use the modified symmetric SSFM method written in general form as where operator take into account attenuation and dispersion effects; operator take into distributed and discrete mode coupling; operator N take into account nonlinear effects; Δz is the step size. The step size is determined by the strength of the dispersion and nonlinear effects or by the pulse walk-off in optical channels and should smaller than the correlation length determining the mode coupling strength Δz<<0.37/h , where h is the linear coupling coefficient, 1/ km.
At simulation the distribution of coupling coefficients formed by a random Gaussian process with a specified correlation function corresponding to the propagating regime. The discrete mode coupling taking into account using a unitary coupling matrix in the form where q 1 and q 2 are matrix elements describes mode coupling and satisfies to equation . In this paper, considered two known methods of DMD compensating [29]. One is the modes crossing (Figure 1a) and other is the inserting of delay line (Figure 1b). As noted earlier, DMD compensation were included on linear optical amplifiers. When modes crossing was used, the even number of amplification spans was chosen, and their length was assumed to be same. To compare the merits of the few-mode fiber optic links as in [19,30], the concept of the FEC threshold was used. By the simulation results, the dependences of the BER for the each optical mode channel on the far end from the peak power at the fiber optic link input were built. Based on this curves the dynamic range was estimating, assuming that for BER below the FEC threshold, the signal at the reception is completely restored. The threshold value was assumed to be BER = 4.7 · 10 -3 [30].

Results of simulation
In this paper, the simulation was performed for an optical signal with quadrature phase modulation (QPSK) with a symbol bit rate 28.5 Gbit/s with pulse shape 50%RZ, providing an information bit rate of 114 Gbit/s. The regeneration section consists from 10x100 km amplifying sections with total length of 1000 km. The average value of the delivery length of the optical cable was assumed equal to 5 km. An erbium optical amplifier was used for ideal compensation of optical path loss with 5 dB noise factor. The noise of the optical amplifier was modeled by additive white Gaussian noise and added to the signal at the end of each amplifier section. The two-mode fiber used for simulation supports the propagation of two modes LP 01 and LP 11 at the optical signal wavelength under the condition of equal mode excitation. The parameters of two-mode fiber are represented in Table 1.  a, b show the simulation results for case of differential mode delay compensation and chromatic dispersion compensation on LOAs. As shown by calculations, for the case of the chromatic dispersion compensation on the LOAs, of the DMD compensation on the LOAs causes an insignificant, but still an increase in the dynamic range. The increment was about 0.15 dBm. This suggests that DMD compensation and the chromatic dispersion compensation on denser maps can yield a significantly larger gain. Especially, if we take into account the effect of chromatic dispersion compensation on LOAs and the experience of group delay management in mode-division multiplexing systems [29].
To reveal the influence of the compensation card density, the dependences of the error probability on the input power of the signals for a number of distances between the DMD compensation points on  fig. 3a, b are examples of the curves obtained for the LP 01 mode. For the LP 11 mode, the curves have a similar character. As the calculations showed, the DMD compensation gives any noticeable effect when the distances between the compensators are less than the delivery length of a cable. This does not allow us to speak about the effectiveness of its use on long-haul fiberoptic links, but it does suggest that the use of fibers with reduced DMD is promising. a) b) Figure 2. The dependence of the BER on the peak power at the input of a two-mode fiber-optic link with compensation of chromatic dispersion on the LOAs for the LP 01 (a) and LP 11 (b) modes (1without compensation DMD; 2 -DMD compensation on LOAs by modes crossing; 3 -DMD compensation on the LOAs with the delay line inserting). a) b) Figure 3. The dependence of the BER on the peak power at the input of a two-mode fiber-optic link with a compensation of chromatic dispersion on the LOAs and with a DMD compensation for the LP 01 mode for some distances between points of compensation (a -compensating by modes crossing; b -compensating by inserting of delay line).

Conclusion
The presented simulation results for a twomode fiber optic link with DMD compensation showed that a fairly noticeable effect in terms of increasing the dynamic range of the system and increasing the power transmitted through the optical fiber takes place at distances between compensators less than the delivery length of an optical cable. But even with distances between compensation points equal to 1 km, the dynamic range increases by less than 1 dBm. This does not allow to speak about the effectiveness of this measure for long-haul fiber optic links, but it makes it possible to consider the use of fibers with reduced DMD as promising for them. Of course, these findings are preliminary and require verification based on modeling and experimentation.