Transmission method of improved fiber nonlinearity tolerance for probabilistic amplitude shaping

Wei-Ren Peng; An Li; Qing Guo; Yan Cui; Yusheng Bai

doi:10.1364/OE.400549

1. Introduction

Providing up to ∼1-dB linear shaping gain, probabilistic amplitude shaping (PAS) has been proposed as a feasible solution to the implementation of probabilistic shaping (PS) with the modern binary and systematic forward error correction (FEC) codes with bit metric decoding [1–3]. To enhance further the transmission performance, shaping with enhanced fiber nonlinearity tolerance [4–10], offering nonlinear benefit in addition to or at the expense of original linear shaping gain, is considered as one of the most promising solutions. Within this topic, the finite-blocklength shaping, which adopts relatively short shaping blocklength and keeps the shaped amplitude sequence from the amplitude shaper (AS) to the transmitter output (i.e. bypassing the bit/symbol inter-leaver), has been demonstrated to provide noticeable nonlinearity benefit with relatively low complexity [9–11]. In [11], the authors have illustrated two possible amplitude-to-QAM pairing methods, i.e. inter-distribution-matcher (inter-DM) pairing and intra-DM pairing, and showed that intra-DM pairing results in a higher effective signal-to-noise ratio (SNR) after transmission. In [11] the authors have also explained that, the nonlinear benefit of finite blocklength shaping stems from the temporal property of the shaped sequence that avoids long runs of symbols with identical amplitude.

Given that both polarizations are responsible for contributing fiber nonlinearity distortions, a further optimization for the finite blocklength shaping would involve both polarizations. For this purpose, the above-mentioned temporal property for improving nonlinearity tolerance can be generalized from 1 dimension (1D) to 4D (including the in- and quadrature-phase tributaries of both polarizations, i.e. Xi, Xq, Yi, and Yq), which suggests that, any shaping that can produce sequence of avoiding long runs of 4D symbols with identical “intensity” may enjoy better nonlinear tolerance than the previous finite blocklength shaping.

In this paper, for finite-blocklength PAS we propose a super-symbol method that improves the fiber nonlinearity tolerance [12]. On the basis of the previous finite-blocklength shaping, every shaped block is divided into four equal-sized subblocks which are transmitted parallelly by the four tributaries of Xi, Xq, Yi, and Yq. Since the energy of each shaped block is now confined in a duration which is only 1/4 (1/2) that of the previous finite blocklength method with inter-DM pairing (intra-DM pairing), the proposed method can improve its nonlinear tolerance to that of shaping with only 25% (50%) blocklength while keeping the good linear shaping gain of the 100% blocklength. To explain its enhanced nonlinearity tolerance, we introduce a new figure of merit for indicating the nonlinear performance of a sequence, i.e. the running digital sum (RDS), and link this metric to the width of spectral dip of its intensity waveform which directly relates to the variance of fiber nonlinearities. To verify its enhanced nonlinearity tolerance, we numerically set up a dispersion-unmanaged transmission system that conveys 25-GHz-spaced, 26x 22.5-GBaud dual-polarized PAS-64QAM signals over 12 spans of 80-km SSMF. The new method provides an ∼0.15 dB gain over the finite-blocklength method with intra-DM pairing, an ∼0.26 dB gain over the finite blocklength method with inter-DM pairing, and an ∼0.44 dB gain over the traditional method, all with a blocklength at 200.

The contributions of this paper are threefold. First, for a long yet feasible blocklength of shaping, which in general is a trade-off between complexity and back-to-back (b2b) performance, the proposed method can improve the fiber nonlinearity tolerance without sacrificing b2b performance. Second, we provide a simple and intuitive explanation for the nonlinear benefit of the proposed method. A new figure of merit RDS, taking account of both the intensity of each 4D symbol and the shuffling level of the sequence amplitude, is provided as a nonlinear performance indicator for any sequence. Third, a feasible architecture to implement the proposed method, which is also applicable to both the previous finite blocklength methods, is offered which makes the proposal more promising for its use in the near-future optical transponders.

2. Working principles

In this section we will first discuss the previous and the proposed methods and then explain why the proposed method can outperform previous methods in fiber nonlinearity tolerance.

Shown in Fig. 1(a) is the traditional method, denoted as TRA, for a PAS system. The shaped symbols from an amplitude shaper (AS) output block are temporally broken up and mixed with symbols from other AS blocks by a symbol inter-leaver that is typically present to reduce the post-FEC bit error rate (BER) in the presence of burst error. The sign bits are ignored in this section since they are irrelevant to the discussed methods. TRA fails to maintain the “temporal property” from the AS [11] so that it would not enjoy any nonlinear benefit. Shown in Fig. 1(b) is the finite-blocklength method with inter-AS pairing (referred as inter-DM pairing in [11]), denoted as FBLwItr. In this method, each AS block is directly transmitted with one tributary and the four blocks across the four tributaries are temporally aligned. To achieve this, the permutation caused by the bit/symbol inter-leaver has to be undone [10]. The effort to undo this permutation is considered as the extra complexity of this method. Since the temporal property from the AS is maintained at the transmitter output, FBLwItr has been demonstrated to have better fiber nonlinearity tolerance than TRA [9–11]. Shown in Fig. 1(c) is the finite-blocklength method with intra-AS pairing (referred as intra-DM pairing in [11]), denoted as FBLwIra. In this method, each AS block is equally-divided into 2 subblocks which are transmitted in parallel with both tributaries (Xi and Xq or Yi and Yq) of one polarization. The subblocks across the four tributaries are temporally aligned. Similar to FBLwItr, the permutation caused by the bit/symbol inter-leaver has to be undone. This method has shown a higher effective signal-to-noise ratio (SNR) than FBLwItr after nonlinear fiber transmission [11]. Finally, shown in Fig. 1(d) is the proposed super-symbol method, abbreviated as SUP. In this method, each AS block is equally-divided into four subblocks which is transmitted with the parallel use of all four tributaries (Xi, Xq, Yi and Yq). Again, similar to both FBLs, the permutation by the inter-leavers has to be undone. One super symbol is defined as a 4D block comprising all the shaped symbols from one and only one AS block and therefore its period (duration) will be ¼ that of an AS block. Although this method can be considered as an extension of FBLs that make full use of all tributaries of an optical carrier, in this paper we prefer to term it “super symbol” for consistency due to a previous work filing for patent application. [12]. It is worth mentioning that there is no restriction on how the shaped symbols of one AS block should be filled into a super symbol. As we will explain later, what matters to the nonlinear benefit is the constant energy of super symbols, rather than how the shaped symbols should be filled-in or their relative sequence in a super symbol.

Fig. 1. Four transmission methods for probabilistic amplitude shaping (PAS): (a) Traditional method (TRA): the shaped symbols from an amplitude shaper (AS) block are broken up due to the bit/symbol inter-leaver. (b) Previous finite-blocklength method with inter-AS paring (FBLwItr): each AS block is carried by one tributary and the four blocks over all tributaries are temporally-aligned. (c) Previous finite-blocklength method with intra-AS paring (FBLwIra): each AS block is divided into 2 subblocks carried by both tributaries of one polarization and the four subblocks across all tributaries are temporally-aligned. (c) Proposed super-symbol method (SUP): each AS block is divided into 4 subblocks carried by all the 4 tributaries and the four subblocks across all tributaries are temporally-aligned. A super symbol comprises all the shaped symbols from only a certain AS block. Note that for FBLwItr, FBLwIra, and SUP, the permutation caused by the bit/symbol inter-lever is assumed undone. A practical solution to undo this permutation is discussed in the end of Section 3.

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Now we will offer a simple model to explain the improved fiber nonlinearity benefit of SUP. In the presence of moderate amount of chromatic dispersion (CD), the self-phase modulation (SPM) can be estimated via Eqn. (3) in [13], which by ignoring the polarization cross term can be re-written as in Eq, (1) here:

(1)$$E_x^{\prime}(t )\approx \; {E_x}(t )\ast \textrm{exp}\{{j\kappa [{{{|{{E_x}(t )} |}^2} + {P_{xy}}(t )} ]\otimes {h_{spm}}(t )} \}$$

where ${E_x}^{\prime}(t )$ and ${E_x}(t )$ are the electric fields along the x polarization with and without the SPM distortions, ${P_{xy}}(t )= \; \; {|{{E_x}(t )} |^2} + {|{{E_y}(t )} |^2}$ is the “instantaneous power as a function of time (referred to as intensity waveform hereafter)” of both polarizations with ${E_y}(t )$ being the electrical field along the y polarization, $\otimes $ stands for the convolution operator, ${h_{spm}}(t )$ represents a linear filter which can be derived from the perturbation theory, and $\kappa $ is the nonlinear coefficient determined by the fiber parameters. Likewise, ${E_y}^{\prime}(t )$ can be expressed as in Eq. (2):

(2)$$E_y^{\prime}(t )\approx \; {E_y}(t )\ast \textrm{exp}\{{j\kappa [{{{|{{E_y}(t )} |}^2} + {P_{xy}}(t )} ]\otimes {h_{spm}}(t )} \}$$

From Eqs. (1) and (2) we see that the SPM power will scale with the variances of $[{{{|{{E_x}(t )} |}^2} + {P_{xy}}(t )} ]\otimes {h_{spm}}(t )$ and $[{{{|{{E_y}(t )} |}^2} + {P_{xy}}(t )} ]\otimes {h_{spm}}(t )$ for each polarization. If by any means we can reduce the variance of their common term ${P_{xy}}(t )\otimes {h_{spm}}(t ),$ we should be able to mitigate the SPM power for both polarizations. Since ${h_{spm}}$ in most practical scenarios has a low-pass filtering profile [13], we may reduce the SPM power effectively by suppressing the low-frequency components of ${P_{xy}}(t )$. In fact, both FBLs (abbreviated for both FBLwItr and FBLwIra) and SUP are found to suppress the low-frequency components of ${P_{xy}}(t )$ to different extents due to their temporal structures. To elaborate on this, we define the running digital sum (RDS) for the offset intensity waveform $[{{P_{xy}}(n )- {P_{avg}}} ]$ as $\mathrm{\gamma }(M )= \; \mathop \sum \nolimits_{n \le M} [{{P_{xy}}(n )- {P_{avg}}} ]$ where $M$ is a positive integer, n is the “symbol index”, $P_{a v g}=\left\langle\left|E_{x}(n)\right|^{2}+\left|E_{y}(n)\right|^{2}\right\rangle$ represents the channel average power with ${\langle\cdot\rangle} $ being the mean operator. From [14] it is expected that $[{{P_{xy}}(n )- {P_{avg}}} ]$ will create a spectral dip (or null) in the region of the direct-current (DC) frequency when ${\mathrm{\gamma }^2}$ is bounded by a finite value $\mathrm{\epsilon }$. In general, a smaller $\mathrm{\epsilon }$ would make this spectral dip wider which can suppress more low-frequency components; on the contrary, a larger or infinite $\mathrm{\epsilon }$ would result in negligible dip which will maintain all the low-frequency components. Therefore, a shaping goal to mitigate the SPM power would be the minimization of $\mathrm{\epsilon }$ for RDS. In TRA where we have no restriction on ${P_{xy}}(n )$, $\mathrm{\epsilon }$ could be very large which results in no spectral dip at all. In FBLwItr, the accumulation of $[{{P_{xy}}(n )- {P_{avg}}} ]$ will cancel every period of an AS block since $\mathop \sum \nolimits_{n \in \textrm{A}} [{{P_{xy}}(n )- {P_{avg}}} ]= 0$ where A is the index set of a complete shaping block, i.e. $A = \{{1,2, \ldots ,\textrm{N}} \}$ with $\textrm{N}$ being the size of an AS block. Here we assume that the CCDM is used as the AS so that $\mathop \sum \nolimits_{n \in \textrm{A}} [{{P_{xy}}(n )- {P_{avg}}} ]= 0$ will hold. Thus, ${\mathrm{\gamma }^2}$ of FBLwItr will be bounded by ${\epsilon _1} = {\left|{\mathop \sum \nolimits_{n \in B} ({{P_{xy}}(n )- {P_{avg}}} )} \right|^2}$ where $B \subset \{{1,\; 2,\; \ldots ,\; \textrm{N}} \}$ is the index set with all elements satisfying $[{{P_{xy}}(n )- {P_{avg}}} ]> 0$. In FBLwIra, the accumulation of $[{{P_{xy}}(n )- {P_{avg}}} ]$ will cancel every “half period” of an AS block due to the parallel use of two tributaries and therefore ${\mathrm{\gamma }^2}$ will be bounded by ${\epsilon _2} = {\left|{\mathop \sum \nolimits_{n \in C} ({{P_{xy}}(n )- {P_{avg}}} )} \right|^2}$ where $C \subset \{{1,\; 2,\; \ldots ,\; \textrm{N}/2} \}$ is the index set with all elements making $[{{P_{xy}}(n )- {P_{avg}}} ]> 0$ hold. Similarly, for SUP, the accumulation of $[{{P_{xy}}(n )- {P_{avg}}} ]$ cancels every “quarter period” of an AS block due to the parallel use of four tributaries and ${\mathrm{\gamma }^2}$ will be bounded by ${\epsilon _3} = {\left|{\mathop \sum \nolimits_{n \in D} ({{P_{xy}}(n )- {P_{avg}}} )} \right|^2}$ where $D \subset \{{1,\; 2,\; \ldots ,\; \textrm{N}/4} \}$ is the index set with every element subject to $[{{P_{xy}}(n )- {P_{avg}}} ]> 0$. By inspection we can find that the cardinality of B is larger than C and the cardinality of C is larger than D (i.e. $|B |$ > $|C |$ > $|D |$), which suggests the RDS bound of FBLwItr is larger than FBLwIra, and the RDS bound of FBLwIra is larger than SUP, i.e. ${\epsilon _1} > {\epsilon _2} > {\epsilon _3}$. This indicates that SUP has the widest spectral dip at ∼DC, followed by FBLwItra, and then FBLwItr. TRA has no dip at all. The wider dip of SUP, when interacting with the low-pass SPM filter, will lead to smaller SPM power, as illustrated in the principal diagram of Fig. 2; while TRA without any dip will result in the largest SPM power.

Fig. 2. Principle diagram of the resultant SPM power for (a) traditional method, (b) previous finite blocklength method with inter-AS pairing, (c) previous finite blocklength method with intra-AS pairing, (d) proposed super-symbol method. PSD: power spectral density, ${P_{xy}}$: intensity waveform of both polarizations, ${P_{avg}}:$ the average power of both polarizations, ${h_{spm}}$: the SPM filter, typically a low-pass filter. Note this principle diagram is also applicable to XPM by replacing ${P_{xy}}$, ${P_{avg}}$ and ${h_{xpm}}$ with ${P_{xy,u}}$, ${P_{avg,u}}$ and ${h_{xpm,u}}$, respectively, with u being the channel index of interfering channel.

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We stress that the above model can apply also for the cross-phase modulation (XPM) which can be similarly estimated as Eqs. (3) and (4) below with the polarization cross terms ignored [15]:

(3)$$E_x^{\prime\prime}(t )\approx \; {E_x}(t )\ast \textrm{exp}\{{j\kappa [{{{|{{E_{x,u}}(t )} |}^2} + {P_{xy,u}}(t )} ]\otimes {h_{xpm,u}}(t )} \}$$

(4)$$E_y^{\prime\prime}(t )\approx \; {E_y}(t )\ast \textrm{exp}\{{j\kappa [{{{|{{E_{y,u}}(t )} |}^2} + {P_{xy,u}}(t )} ]\otimes {h_{xpm,u}}(t )} \}$$

where $E_x^{\prime\prime}(t )$ and $E_y^{\prime\prime}(t )$ are the electric fields along the x and y polarizations with XPM distortions, u is the channel index of interfering channels, ${E_{x,u}}(t )$ and ${E_{y,u}}(t )$ are electric fields along the x and y polarizations of the $u$-th channel, ${P_{xy,u}}(t )$ is equal to ${|{{E_{x,u}}(t )} |^2} + {|{{E_{y,u}}(t )} |^2}$, and ${h_{xpm,u}}$ is the linear XPM filter of the $u$-th channel. Similar to ${h_{spm}}$, the XPM filter ${h_{xpm,u}}$ also exhibits a low-pass filtering profile. In general, the bandwidth of ${h_{xpm,u}}$ is a function of spectral spacing between the probe and interfering channels, and is even smaller than that of ${h_{spm}}$ for the interfering channels farther away, as we will see in Section 3. Therefore, SUP in principle will enjoy not only the lower SPM penalty but also the lower XPM penalty in a homogeneous SUP-based transmission system.

Several notes are worth mentioning before we end this section. First, based on our explanation, the nonlinear gain of both FBLs and SUP comes from the DC spectral dip of ${P_{xy}}(n )$, which is resulted from the constant energy of each shaping block. The sequence or temporal correlation of the shaped symbols within an AS block has no direct relation to this spectral dip and thus may not have significant influence on the nonlinear benefit. This says, as long as all the symbols from an AS block are packed into a super symbol, no matter how they are filled in, similar nonlinear benefit will always be observed. Second, although in the explanation we assume CCDM is used as the AS which leads to finite RDS due to the constant energy of each AS block, we’ve found that sphere shaping, such as enumerative sphere shaping (ESS) [9] and shell mapping [16], which delivers shaped blocks with various energy, will similarly form a noticeable spectral dip at ∼DC frequency as well. This is because the majority of the shaped blocks from sphere shaping will have similar energy due to the fact that the higher-energy blocks can accommodate more information bits. This is certainly a good news since sphere shaping is known for better efficiency in providing linear shaping gain [9]. Third, we have used RDS and its finite-value constraint to explain the nonlinear gain origin of SUP. Conversely, we may use finite-valued $\mathrm{\gamma }$ as a nonlinear shaping constraint for mitigating fiber nonlinearities. For instance, the cost of a shaped amplitude sequence now can be modified to a weighted average of the sequence energy and the square of sequence RDS: $\mathrm{\alpha }\mathop \sum \nolimits_{\forall n} {P_{xy}}(n )$ + $({1 - \mathrm{\alpha }} ){\mathrm{\gamma }^2}$, with $\mathrm{\alpha }$ taking the balance between the linear and nonlinear shaping gain. We will leave this as an open subject for readers to explore.

3. Simulation results

For the results we will be more focusing on the benefit of the SUP method, rather than its optimum blocklength exploration. We will adopt shell mapping [16] to demonstrate the proposed method in the simulations. However, other shaping algorithms should in principle result in similar conclusions as in this paper.

In simulations, the WDM transmission performance of 25-GHz-spaced, 26 × 22.5-GBaud dual-polarized PAS-64QAM signals with TRA, both FBLs, and SUP are evaluated. Assuming a FEC code rate at 0.9 (11% overhead) and an AS rate (previously DM rate [17]) at 0.863, each 22.5-GBaud channel carries a net data rate of ∼218 Gb/s (net spectral efficiency = 4.85 bits/symbol per polarization) with a pre-FEC BER threshold at ∼2.1e-2. For the AS we use shell mapping and by which we will study three shaping blocklengths with equal capacity: ($\textrm{K}$, $\textrm{N}$) = (69, 40), (207, 120) and (345, 200) where $\textrm{K}$ is the number of input information bits {0, 1} for an AS block and N is the number of the output real-valued symbols {1, 3, 5, 7} of an AS block. These block lengths are considered feasible with manufacturing technologies today. The level probability of each blocklength is given in Table 1, based on which the block length at 200 is expected to provide ∼0.04 and ∼0.2 dB benefit over block lengths at 120 and 40, respectively, in terms of linear shaping gain. Note that for a fixed blocklength all the TRA, FBLs, and SUP should have the same level probability since these methods simply change the sequential order of the produced shaped symbols.

Table 1. Level distribution of shell mapping with three block lengths at 40, 120, and 200.

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Same shaped symbols are produced and used for the four methods. For TRA we use a permutation function (emulating the symbol inter-leaver) to randomize the sequence of the shaped symbols for each tributary. For FBLwItr, every shaped block is directly transmitted with one tributary and the blocks over the four tributaries are temporally-aligned. For FBLwItra, every shaped block is divided into two equal-sized subblocks which are transmitted parallelly with both tributaries of one polarization. The subblocks over the four tributaries, two from one AS block and the remaining two from another AS block, are temporally-aligned. Regarding the symbol-to-tributary distribution, we use a modulo-2 symbol index to determine which tributary the $n$-th symbol is assigned to, for example: $\textrm{mod}({n,\; 2} )= 0$ and $1$ will assign the $n$-th symbol respectively to the in-phase and quadrature-phase tributary. For SUP, every shaped block is divided into 4 equal-sized subblocks which are parallelly transmitted with all four tributaries. The four subblocks from the same AS block are temporally-aligned over the four tributaries. A modulo-4 symbol index, $\textrm{mod}({n,\; 4} )$, is used to determine which tributary the $n$-th shaped symbol is assigned to, i.e. $\textrm{mod}({n,\; 4} )= \{{0,\; 1,\; 2,\; 3} \}$ will assign the $n$-th symbol to the {Xi, Xq, Yi, Yq} tributary, respectively. However, other symbol-to-tributary distribution should result in similar transmission performance, as we have explained in the previous section. Afterwards, for all the four methods, an uniformly-distributed binary bit, serving the sign bit of PAS, is added to each real-valued shaped symbol, leading to the desired dual-polarized PAS-64QAM format.

The transmission link consists of 12 spans of 80-km SSMF with fiber loss = 0.2 dB/km (span loss = 16 dB), CD parameter = 16.8 ps/(km.nm), and nonlinearity coefficient = 1.31 (1/W/km). No optical dispersion management is applied. The loss of each span is compensated by an EDFA with noise figure = 5 dB. Ideal transmitters and receivers (infinite bandwidth, linear transfer function of all components, and zero laser linewidth) are assumed to focus on the transmission impairments which include the amplified spontaneous emission (ASE) noise, CD, and fiber nonlinearities. At the transmitter, a root-raised cosine (RRC) filter is applied with a roll-off factor = 0.1. At the receiver, another RRC filter with a roll-off factor = 0.1 (i.e. the matched filter) as well as a perfect CD compensator is applied first. Then, an adaptive T/2-spaced, 21-tap 2 × 2 time-domain equalizer based on least mean square (LMS) algorithm, coupled with a decision-directed phase locked loop (DD-PLL), is employed decoupling the X and Y polarization signals as well as mitigating residual linear distortions. Afterwards, bit error rate (BER) as a function of OSNR is derived by direct error counting and the OSNR margin to the BER threshold at 2.1e-2 is evaluated. Each BER is obtained with 3 runs of different noise seeds and each run involves 100K symbols per tributary.

In Fig. 3(a), the power spectral density (PSD) of $[{{P_{xy}}(t )- {P_{avg}}} ]$, denoted as $\rho (f )$, of the four methods are depicted with a block length at 40. The resolution is fixed at 10 MHz. It can be observed that all the four methods result in very similar $\rho (f )$ except for the low frequency components near DC. A spectral dip near DC is found for both FBLs and SUP with different width while no dip is found for TRA. In Fig. 3(b), the spectral profiles of the SPM filter and the XPM filters with the 1^st to 4^th neighboring channels (respectively labelled as 25 GHz, 50 GHz, 75 GHz and 100 GHz) for single span are illustrated. To obtain the SPM filter profile, the discrete-time impulse response of the SPM filter is first derived based on Eqn. (1) in [18] with $m$ = 0:

(5)$${h_{spm}}(n )= \mathop \smallint \nolimits_0^L dzf(z )\smallint \; dt{|{U({z,t} )} |^2}\ast {|{U({z,t - n{T_s}} )} |^2}$$

where n is the symbol index, $L$ ( = 80 km) is the total distance, $f(z )$ is the power profile, $U({z,t} )$ is the pulse waveform at z distance, and ${T_s}$ is the symbol duration. Here n runs from −32 to 31 which can well capture the entire response of ${h_{spm}}(n )$. After 2x interpolation and inverse fast Fourier transform (IFFT), the SPM filter profile is obtained and presented in the figure. The XPM filter profile is directly obtained with Eqn. (8c) in [15]. The associated parameters for deriving these filters are equal to those used in the transmission simulation, except that only one span is considered here. We’ve found that the SPM filter and all the XPM filters indeed exhibit low-pass filtering profiles which highlights the importance of spectral dips of FBLs and SUP in suppressing the SPM and XPM power. We’ve also found that the XPM filter bandwidth is even smaller than that of the SPM filter, especially for the interfering channels farther away from the probe. This may imply that, suppressing the SPM power requires a wider dip than that suppressing the XPM power. The bandwidth reduction of the XPM filter with larger spacing is due to the larger walk-off caused by CD.

Fig. 3. (a) The measured power spectral density of the offset intensity waveform $[{{P_{xy}}(\textrm{t} )- {P_{avg}}} ]$ of the four methods with a block length at 40. (b) The passband power profiles of the SPM filter and the XPM filters with the 1^st to 4^th neighboring (interfering) channels. The labels on the XPM filters indicate the spacing between the probe and interfering channel. The spectral resolution is fixed at 10 MHz.

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Now we compare their spectral dip widths with different blocklengths. Shown in Fig. 4(a), (b) and (c) are $\rho (f )$ with a blocklength at 40, 120, and 200, respectively, over a frequency range of −2 GHz to + 2 GHz. Notably Fig. 4(a) is the zoomed plot of Fig. 3(a) for the details of dip. To define the dip width we draw a horizontal line which is about −3 dB relative to the average power at ± 2GHz. This line will have two intersections with the PSD of each method of which the spacing in Hertz is defined as the dip width. First we can see that TRA consistently shows no dip regardless of the blocklength. With a blocklength at 40, the dip width of FBLwItr, FBLwIra, and SUP is ∼400 MHz, 800 MHz, and 1.6 GHz, respectively, which is found to be proportional to the number of tributaries used for transmitting each AS block, i.e. the degree of parallelism for each AS block. With a blocklength at 120 and 200, the dip width of SUP is reduced to ∼540 MHz and ∼400 MHz, respectively. Similar width reduction can be found as well for both FBLs. These results prove that, a short blocklength can result in a wide spectral dip for $\rho (f )$ which can be further widened (up to x4 times) through the use of SUP.

Fig. 4. PSD at ∼DC frequencies with different blocklengths. (a) blocklength = 40, (b) 120, and (c) 200. Resolution bandwidth is 10 MHz.

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In Fig. 5, the transmission results of OSNR margin to the BER threshold at 2.1e-2, as a function of launch power per channel, is presented for all methods with different blocklengths. The optical noise bandwidth is 0.1 nm. We will first compare their tolerances against SPM and then tolerances against XPM. Shown in Fig. 5(a) is the single-channel transmission result with blocklength at 40. In the low-power regime, all methods tend to converge because of their similar linear shaping gain from the same blocklength; in the high-power regime, different methods are found to diverge: SUP performs best, FBLwIra better, FBLwItr well, and TRA worst. To quantify the nonlinear benefit we define the gain as the difference of the maximum margin between two methods. Then SUP is found to provide < 0.1-dB, 0.2-dB, and 0.63-dB gain over FBLwItr, FBLwIra, and TRA, respectively. The result shows that the spectral dip can help mitigate the SPM power and a wider dip width can indeed mitigate more, which directly verifies our model in Section 2. For reference purpose, the inset depicts the respective constellation diagrams at 2-dBm launch power without ASE loading.

Fig. 5. The optical -signal-to-noise ratio (OSNR) margin to achieve BER = 2.1e-2 after transmission over 12 × 80 km SSMF, (a) single channel with a block length at 40. The inset depicts the respective constellation diagrams at ∼2 dBm without ASE loading. (b) 26 channels with a block length at 40, (c) 26 channels with a block length at 120, and (d) 26 channels with a block length at 200. The insets of (b)-(c) illustrate the respective constellation diagrams at −0.5 dBm without ASE loading.

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Shown in Fig. 5(b) is the 26-channel transmission result with a blocklength at 40. For 26-channel result only the probe (14^th from the lower frequency) channel is evaluated. In the high-power regime where we are more interested in this paper, SUP shows a comparable performance as FBLwIra, and offers ∼0.18-dB and ∼0.6-dB gain over FBLwItr and TRA, respectively. Interestingly, the SUP-over-FBLwItra and SUP-over-FBLwItr gain are found reduced, though slightly, compared with the 1-channel transmission result. The reason of this SUP gain reduction is below: the wider dip width of SUP can suppress more SPM power than both FBLs, therefore leading to an observable performance difference between SUP and FBLs in the scenario of 1-channel transmission. However, for XPM, the wider dip width of SUP may suppress only a little bit more power than FBLs since the XPM filters have narrower bandwidth (compared with the SPM filter) which relaxes the requirement for a wider dip. Therefore, the resultant XPM power between FBLs and SUP would be comparable which closes their performance gap in the scenario of 26-channel transmission. However, this would only occur when the dip width of FBLs is still wide to cover the bandwidth of XPM. With a longer yet feasible blocklength, where the dip width of FBL becomes insufficient for XPM, the gap between SUP and FBL may enhance.

Figures 5(a) and (b) have basically showed that the spectral dip can effectively enhance the tolerance against both SPM and XPM, which agrees well with [9] where ESS is adopted as the AS. However, from Fig. 5(b) we’ve observed that the proposed SUP cannot provide noticeable gain over FBLwIra. A following question is whether this is still true with longer blocklengths. Shown in Figs. 5(c) and (d) are the 26-channel transmission results with a blocklength at 120 and 200, respectively. From Figs. 5(b)-(d) we can obtain the following observations. First, the performance ranking, in terms of the optimum margin, of all methods is found consistent for different blocklengths: SUP ≥ FBLwIra > FBLwItr > TRA. This would be a solid evidence showing the relation between the spectral dip width and nonlinear performance. Second, the SUP-over-TRA gain reduces as the blocklength increases, which is directly resulted from the narrower dip width with longer blocklengths. Notably, the gain reduction from 40 to 120 is only ∼0.03 dB (negligible gain reduction) which implies the dip width of SUP with blocklength at 120 (∼530 MHz) is still sufficiently wide for efficient nonlinearity mitigation while the gain reduction from 120 to 200 is ∼0.13 dB which hints the dip width with blocklength at 200 (∼400 MHz) is somewhat narrower than sufficient. In the extreme case, the SUP-over-TRA gain will reduce to zero with an infinitely-long blocklength. Third, both the SUP-over-FBLwIra gain and SUP-over-FBLwItr gain is enlarged as the blocklength increases, which stems from a fact that the wider dip of SUP (≥ 2x of FBLs) can better sustain nonlinear benefit when a longer blocklength is used. In other words, both FBLs are losing more nonlinear benefit than SUP when considering a long yet feasible blocklength for pursuing a larger linear shaping gain. This then emphasizes the strength of SUP: it can better take care of both the linear and nonlinear shaping gains without additional complexity compared with both FBLs. For reference, the insets to the right of the main figure from Figs. 5(b) to (c) illustrate the respective constellation diagrams at −0.5-dBm launch power without ASE loading.

The complexity of SUP should be similar to FBLs. In Fig. 6 we illustrate one possible architecture to implement FBLs and SUP [12]. At the transmitter, an extra symbol-level “pre-deinterleaver” is inserted before the FEC encoder to “pre-cancel” the permutation caused by the symbol inter-leaver, that is typically placed after the FEC encoder, so that the shaped symbols from the same AS block can be packed together for FBLs or even packed as a super symbol for SUP. At the receiver, an inverse operator, “post-interleaver”, is added after the FEC decoder to restore the sequence of transmitted symbols. Thus, the additional complexity for FBLs and SUP, is the pair of pre-deinterleaver and the post-interleaver, which should have similar and affordable complexity as the original pair of inter-leaver and de-interleaver.

Fig. 6. One possible PAS architecture to accommodate the proposed SUP. This may applicable to both FBLs.

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At last we stress that, although in this paper the shell mapping is used as the AS for demonstration, it is expected that, other block-wise shaping algorithms, such as ESS, CCDM, and etc, can also provide nonlinear benefit with SUP, as long as the majority of shaped blocks exhibits similar/comparable energy.

4. Summary

We have proposed a super-symbol transmission method for PAS that improves fiber nonlinearity tolerance. To explain the nonlinear benefit of SUP, we’ve introduced a new figure of merit, RDS, to indicate the nonlinear performance of a sequence. We’ve found that a tightly-bounded RDS, such as SUP, can create a wide spectral dip at DC which can suppress both the SPM and XPM power. After 12 × 80-km dispersion-uncompensated SSMF transmission, the proposed method (SUP) shows an ∼0.15-dB gain over the previous finite blocklength method with intra-DM pairing (FBLwIra), an ∼0.26-dB gain over the previous finite blocklength method with inter-DM pairing (FBLwItr), and an ∼0.44-dB gain over the traditional method (TRA), all with a feasible blocklength at 200. The enhanced nonlinearity tolerance, as well as its feasible implementation, makes SUP a promising transmission scheme for the next-generation optical transport network.

Disclosures

The authors declare no conflicts of interest.

References

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Block length	40	120	200
Prob(1)	0.4270	0.4388	0.4419
Prob(3)	0.3218	0.3221	0.3222
Prob(5)	0.1800	0.1727	0.1708
Prob(7)	0.0712	0.0664	0.0651

Transmission method of improved fiber nonlinearity tolerance for probabilistic amplitude shaping

Abstract

1. Introduction

2. Working principles

3. Simulation results

4. Summary

Disclosures

References

Cited By

Figures (6)

Tables (1)

Equations (5)

Optics Express