**5.1 DBP algorithms and numerical model**

Fig. 9, illustrates the different SSFM algorithms used in this study for a span compensated by 4 DBP-steps. The backward propagation direction is assumed from the left to the right, as the dashed arrows show. For the constant step-size scheme, step size remains the same for all steps, while for the logarithmic step-size scheme, step size increases with decreasing power. The basic principle is well known from the implementation of SSFM to calculate signal propagation in optical fibers, where adaptive step size methods are widely used. As signal

Fig. 9. Schemes of SSFM algorithms for DBP compensation. S: Symmetric-SSFM, A: Asymmetric-SSFM, and M: Modified-SSFM. The red-dotted curves show the power dependence along per-span length..

power exponentially decays along each fiber span, the step size is increased along the fiber. If backward propagation is regarded, the high power regime locates in the end of each span, illustrated in Fig. 1 by the red dotted curves and the step size has to be decreased along each backward propagation span.

Note that the slope coefficient for logarithmic step-size distribution (see section 3.2.4 of this chapter) has been chosen as 1 to reduce the relative global error according to (Jaworski, 2008). The solid arrows in Fig. 9 depict the positions for calculating the non-linear phase. For the symmetric scheme, the non-linearity calculating position (NLCP) is located in the middle of each step. For the asymmetric scheme, NLCP is located at the end of each step. For the modified scheme, NLCP is shifted between the middle and the end of each step and the position is optimized to achieve the best performance (**?**). In all schemes, the non-linear phase was calculated by *φNL* = *γDBP* · *P* · *Leff* , where the non-linear coefficient for DBP *γDBP* was optimized to obtain the best performance. All the algorithms were implemented for DBP compensation to recover the signal distortion in a single-channel 16-QAM transmission system with bit rate of 112Gbps (28Gbaud). In this simulation model, we used an 20x80km single mode fiber (SMF) link without any inline dispersion compensating fiber (DCF). SMF has the propagation parameters: attenuation *α*=0.2dB/km, dispersion coefficient *D*=16ps/nm-km and non-linear coefficient *α*=1.2 km−1W−1. The EDFA noise figure has been set to 4dB and PMD effect was neglected.

#### **5.2 Simulation results**

16 Will-be-set-by-IN-TECH

HNLF 1

WDM coupler

filter pump 1

Fig. 8. Block diagram of optical backward propagation module (OBP) (Kumar et al., 2011).

propagation) and one-span OBP (per-span backward propagation), respectively.

**5. Analysis of step-size selection in 16-QAM transmission**

dispersion compensation fibers (DCFs) and non-linear compensation by using HNLFs, as shown in Fig. 8. In this article the technique is evaluated for 32QAM modulation transmission with 25G-symbols/s over 800km fiber. The transmission reach without OBP (but with the DCF) is limited to 240km at the forward error correction limit of 2.1x10−3. This is because the multilevel QAM signals are highly sensitive to fiber non-linear effects. The maximum reach can be increased to 640km and 1040km using two-span OBP (multi-span backward

This technique is still in the early stages of development. As DCF in the OBP module can add additional losses and limit the performance of backward propagation algorithm, as a matter of fact we have to keep launch power to the DCF low so that the non-linear effects in the DCF

In this section we numerically review the system performances of different step-size selection methods to implement DBP. We apply a logarithmic distribution of step sizes and numerically investigate the influence of varying step size on DBP performance. This algorithm is applied in a single-channel 16-QAM system with bit rate of 112Gbit/s over a 20x80km link of standard single mode fiber without in-line dispersion compensation. The results of calculating the non-linearity at different positions, including symmetric, asymmetric, and the modified (**?**) schemes, are compared. We also demonstrate the performance of using both logarithmic step sizes and constant step sizes, revealing that use of logarithmic step sizes performs better than constant step sizes in case of applying the same number of steps, especially at smaller numbers of steps. Therefore the logarithmic step-size method is still a potential option in terms of improving DBP performance although more calculation efforts are needed compared with the existing multi-span DBP techniques such as (Ip et al., 2010; Li et al., 2011). Similar to the constant step-size method, the logarithmic step-size methods is also applicable to any kind of

Fig. 9, illustrates the different SSFM algorithms used in this study for a span compensated by 4 DBP-steps. The backward propagation direction is assumed from the left to the right, as the dashed arrows show. For the constant step-size scheme, step size remains the same for all steps, while for the logarithmic step-size scheme, step size increases with decreasing power. The basic principle is well known from the implementation of SSFM to calculate signal propagation in optical fibers, where adaptive step size methods are widely used. As signal

Dispersion compensating

can be ignored.

modulation formats.

**5.1 DBP algorithms and numerical model**

fiber (DCF) 3dB

coupler

(Non-linear compensation stage)

Bandpass

WDM coupler HNLF 2

pump 2 x.pol Bandpass filter

y.pol data out

Fig. 10, compares the performance of all SSFM algorithms with varying number of steps per span. In our results, error vector magnitude (EVM) was used for performance evaluation of received 16-QAM signals. Also various launch powers are compared: 3dBm (Fig. 10(a)), 6dBm (Fig. 10(b)) and 9dBm (Fig. 10(c)). For all launch powers the logarithmic distribution of step sizes enables improved DBP compensation performance compared to using constant step sizes. This advantage arises especially at smaller number of steps (less than 8 steps per span). As the number of steps per span increases, reduction of EVM gets saturated and all the algorithms show the same performance. For both logarithmic and constant step sizes, the modified SSFM scheme, which optimizes the NLCP, shows better performance than symmetric SSFM and asymmetric SSFM, where the NLCP is fixed. This coincides with the results which have been presented in **?**. However, the improvement given from asymmetric to modified SSFM is almost negligible when logarithmic step sizes are used, which means

non-optimized *γDBP* (Fig. 12(c)), and with optimized *γDBP* (Fig. 12(d)). The optimized value is 1.28(km−1W−1). With optimization of *γDBP*, the constellation diagram can be rotated back

<sup>43</sup> Digital Backward Propagation:

Fig. 11. (a) Required number of steps per span at various launch powers for different SSFM

Fig. 12. Constellation diagrams of received 16-QAM signals. (a) constant step size with non-optimized *γDBP*, (b) constant step size with with optimized *γDBP*, (c) logarithmic step sizes with non-optimized *γDBP* and (d) logarithmic step sizes with optimized *γDBP*.

which further reduces the computational efforts for DBP algorithms

We studied logarithmic step sizes for DBP implementation and compared the performance with uniform step sizes in a single-channel 16-QAM transmission system over a length of 20x80km at a bit rate of 112Gbit/s. Symmetric, asymmetric and modified SSFM schemes have been applied for both logarithmic and constant step-size methods. Using logarithmic step sizes saves up to 50% in number of steps with respect to using constant step sizes. Besides, by using logarithmic step sizes, the asymmetric scheme already performs nicely and optimizing non-linear calculating position becomes less important in enhancing the DBP performance,

The authors gratefully acknowledge funding of the Erlangen Graduate School in Advanced Optical Technologies (SAOT) by the German National Science Foundation (DFG) in the

algorithms, and (b) Step-size distribution and average power in each step.

A Technique to Compensate Fiber Dispersion and Non-Linear Impairments

completely.

**5.3 Conclusion**

**6. Acknowledgement**

framework of the excellence initiative.

the NLCP optimization reveals less importance and it is already sufficient to calculate the non-linearity at the end of each step if logarithmic step sizes are used. On the other hand, at higher launch powers, EVM increases and the saturation of EVM reduction happens toward larger number of steps. Note that with 9dBm launch power, the EVM cannot reach values below 0.15 (BER=10−3) even if a large number of steps per span is applied.

Fig. 10. EVM of all SSFM algorithms with varying number of steps per span for (a) 3dBm, (b) 6dBm and (c) 9dBm.

Fig. 11(a) shows the required number of steps per span to reach BER=10−<sup>3</sup> at various launch powers for different SSFM algorithms. It is obvious that more steps are required for higher launch powers. Using logarithmic distribution of step sizes requires less steps to reach a certain BER than using uniform distribution of step sizes. At a launch power of 3dBm, the use of logarithmic step sizes reduces 50% in number of steps per span with respect to using the A-SSFM scheme with constant step sizes, and 33% in number of steps per span with respect to using the S-SSFM and M-SSFM schemes with constant step sizes. The advantage can be achieved because the calculated non-linear phase remains constant in every step along the complete. Fig. 11(b) shows an example of logarithmic step-size distribution using 8 steps per span. The non-linear step size determined by effective length of each step, *Leff* , is represented as solid-square symbols and the average power in corresponding steps is represented as circle symbols. Uniformly-distributed non-linear phase for all successive steps can be verified by multiplication of *Leff* and average power in each step resulting in a constant value. Throughout all simulations the non-linear coefficient for DBP *γDBP* was optimized to obtain the best performance. Fig. 12 shows constellation diagrams of received 16-QAM signals at 3dBm compensated by DBP with 2 steps per span. The upper diagrams show the results of using constant step sizes with non-optimized *γDBP* (Fig. 12(a)), and with optimized *γDBP* (Fig. 12(b)). The lower diagrams show the results of using logarithmic step sizes with non-optimized *γDBP* (Fig. 12(c)), and with optimized *γDBP* (Fig. 12(d)). The optimized value is 1.28(km−1W−1). With optimization of *γDBP*, the constellation diagram can be rotated back completely.

Fig. 11. (a) Required number of steps per span at various launch powers for different SSFM algorithms, and (b) Step-size distribution and average power in each step.

Fig. 12. Constellation diagrams of received 16-QAM signals. (a) constant step size with non-optimized *γDBP*, (b) constant step size with with optimized *γDBP*, (c) logarithmic step sizes with non-optimized *γDBP* and (d) logarithmic step sizes with optimized *γDBP*.

#### **5.3 Conclusion**

18 Will-be-set-by-IN-TECH

the NLCP optimization reveals less importance and it is already sufficient to calculate the non-linearity at the end of each step if logarithmic step sizes are used. On the other hand, at higher launch powers, EVM increases and the saturation of EVM reduction happens toward larger number of steps. Note that with 9dBm launch power, the EVM cannot reach values

Fig. 10. EVM of all SSFM algorithms with varying number of steps per span for (a) 3dBm, (b)

Fig. 11(a) shows the required number of steps per span to reach BER=10−<sup>3</sup> at various launch powers for different SSFM algorithms. It is obvious that more steps are required for higher launch powers. Using logarithmic distribution of step sizes requires less steps to reach a certain BER than using uniform distribution of step sizes. At a launch power of 3dBm, the use of logarithmic step sizes reduces 50% in number of steps per span with respect to using the A-SSFM scheme with constant step sizes, and 33% in number of steps per span with respect to using the S-SSFM and M-SSFM schemes with constant step sizes. The advantage can be achieved because the calculated non-linear phase remains constant in every step along the complete. Fig. 11(b) shows an example of logarithmic step-size distribution using 8 steps per span. The non-linear step size determined by effective length of each step, *Leff* , is represented as solid-square symbols and the average power in corresponding steps is represented as circle symbols. Uniformly-distributed non-linear phase for all successive steps can be verified by multiplication of *Leff* and average power in each step resulting in a constant value. Throughout all simulations the non-linear coefficient for DBP *γDBP* was optimized to obtain the best performance. Fig. 12 shows constellation diagrams of received 16-QAM signals at 3dBm compensated by DBP with 2 steps per span. The upper diagrams show the results of using constant step sizes with non-optimized *γDBP* (Fig. 12(a)), and with optimized *γDBP* (Fig. 12(b)). The lower diagrams show the results of using logarithmic step sizes with

6dBm and (c) 9dBm.

below 0.15 (BER=10−3) even if a large number of steps per span is applied.

We studied logarithmic step sizes for DBP implementation and compared the performance with uniform step sizes in a single-channel 16-QAM transmission system over a length of 20x80km at a bit rate of 112Gbit/s. Symmetric, asymmetric and modified SSFM schemes have been applied for both logarithmic and constant step-size methods. Using logarithmic step sizes saves up to 50% in number of steps with respect to using constant step sizes. Besides, by using logarithmic step sizes, the asymmetric scheme already performs nicely and optimizing non-linear calculating position becomes less important in enhancing the DBP performance, which further reduces the computational efforts for DBP algorithms
