Fig. 1. (a) Power distribution along a fiber span in forward propagation, (b) constant step size distribution backpropagation, and (c) logarithmic step size distribution backpropagation
In a constant step size (CSS) distribution, the step size is uniform for all steps across the span of the virtual fiber. For such schemes, it is required to choose a very small step size, typically less than 100m, to control the generation of fictitious four-wave mixing (FWM) artifacts (24). Alternatively, by evaluating the smallest step, which allows a 10% accuracy on all in-band FWM artifacts, a constant step size of up to 400m can be chosen (26). The selection of such a small constant step size significantly compromises the computational speed of the SSFM calculation. On the other hand, a large constant step size generates fictitious FWM tones during SSFM calculation, degrading the accuracy of nonlinear phase shift compensation.
Logarithmic Step size SSFM
Variable step sizes where the step size increases and power decreases are usually preferred to the constant step size. The logarithmic step size (LSS) distribution is an emerging non-uniform distribution technique proposed to accurately estimate the nonlinear distortion in fewer steps and suppress the generation of fictitious FWM tones. This method aims to keep the average nonlinear phase shift after each step constant (27). For a fiber span of length L , attenuation coefficient α , the nth step size is given by (24).
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where N is the number of steps per span for SSFM calculation.
In (28), the slope coefficient of the conventional logarithmic distribution has been chosen as 1 to reduce the relative global error. A modified logarithmic step size (MLSS) has also been introduced. The slope coefficient is modified as an optimized attenuation adjusting factor to control the difference in step sizes of adjacent steps for optimal performance (29). A generalized logarithmic step size distribution scheme has been proposed to consider high symbol rates and the number of steps by optimizing the base for the step size calculation and the nonlinear coefficient scale factor (27).
Principle of Binary Logarithmic Step size
For a small number of steps, the conventional logarithmic step-size method, which computes sizes using the natural logarithm (ln), results in a significant variation of the signal waveform over one step and does not produce optimal SSFM calculation. We present a binary logarithmic step size (BLSS) for implementing the DBP, where the step size selection of the conventional LSS is optimized using the binary logarithm (log2). Also, the optimized factor introduced by (29) for adjusting the attenuation coefficient is considered. The proposed BLSS algorithm for SSFM step size selection is given by 4:
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where k is the attenuation adjusting factor.
The modification using the binary logarithm better approximates the nonlinear distortions for compensation with higher accuracy than the natural logarithmic step size technique. The proposed technique assumes that the optimised target function is highly multimodal, with a large number of local optima in terms of step size. This can make it difficult for traditional optimization methods to converge to the global optimum. To address this, the technique utilizes a binary logarithmic step size distribution. The binary aspect allows the algorithm to explore the complex search space efficiently by allowing for coarse and fine-grained searches. On the other hand, the logarithmic scale enables the algorithm to move swiftly across the search space, which helps to overcome the issue of getting stuck in local optima.
This approach of controlling the variation of the signal waveform over one step using a binary logarithmic step size is particularly useful in digital backpropagation. Here, when a small number of steps per span is used, and the attenuation factor k is optimized, it helps to ensure that the difference between one step and the next is not too substantial for nonlinear compensation. The modification produces optimal step size distribution while maintaining a low computational overhead.
Fig. 2 compares the variation of adjacent step sizes for different compensation techniques. We use the optimized adjusting factor k= 0.4 determined by . For an 80 km fiber span, the step sizes for 10 steps using different selection techniques are given in Table 1.