Achievable information rates calculation for optical OFDM few-mode fiber long-haul transmission systems

Changyu Lin; Ivan B. Djordjevic; Ding Zou

doi:10.1364/OE.23.016846

1. Introduction

The demand for high-speed transmission and long-haul connectivity is putting enormous pressure on the optical networks. Spatial multiplexing by using few-mode fibers (FMFs) is a promising pathway to achieve higher bit rates and represents a hot research topic [1–4]. Due to the fiber nonlinearities, the increase of the power does not lead to the increase of spectral efficiency after certain critical launch power, and this effect is commonly referred to as the nonlinear Shannon limit. Linear and quasi-linear channel models for Shannon limit calculation of FMF-based systems are well studied by many researchers [5–12]; however, the nonlinearity cannot be ignored in long-haul spatial multiplexing transmission. Moreover, the channel memory effects due to dispersion and nonlinearity interaction as well as signal and amplified spontaneous emission (ASE) noise interaction, cannot be underestimated since it might limit the data rate in transmission.

A theoretic analysis of capacity limits of optical fiber network has been presented in [13]. This paper discusses basic digital signaling problems, including the pulse shaping, spectrum analyzing, and finally estimates the lower bound of capacity in optical fiber links in particular for SMF applications. However, the practical method to estimate the achievable information rates (AIRs) for a given MIMO-SDM channel, in particular for FMF with channel memory is still an open problem.

Another great work dealing with calculation of channel capacity for FMF links is that due to [5,6]. These papers focus on the fully loaded OFDM signal and study the nonlinear effects such as four-wave mixing (FWM), which leads to the reduction of achievable rate. The paper shows that even in the presence of high spatial overlapping in FMF transmission, the overall channel capacity is still linear up to the certain number of spans. Our AIR results show similar trend even for FMF channels with memory.

In this paper, we study the achievable information rates for a given multi-dimensional coherent optical (CO) OFDM scheme in both standard single mode fiber (SSMF) and few-mode fiber for long-haul transmission. We propose an algorithm for achievable information rate calculation, suitable for optical channels with memory [14] including SMF and various fibers for SDM such as FMF, few-core fiber, and few-mode-few-core fiber.

The paper is organized as follows. In Section 2, we describe the FMF channel model and coherent optical OFDM (CO-OFDM) system setup. We introduce a modified nonlinear Schrödinger equation used for modeling of signal propagation in few-mode fibers, which includes nonlinear interaction among the spatial modes and signal-ASE noise nonlinear interaction. In the same section, the channel estimation and compensation of mode-coupling and dispersion effects is described as well. In section 3, we describe the proposed algorithm for AIR calculation. Both theoretic and practical analyses of the AIR calculation for SMF and FMF links are provided in Section 4. Some important concluding remarks are presented in Section 5. We consider both SMF and FMF links with/without memory for QPSK/16QAM transmission over CO-OFDM system.

2. System diagram and FMF channel model

The OFDM is used to deal with dispersion effects and to reduce the memory of the channel. The CO-OFDM system under study is shown in Figs. 1(a) and 1(b).

Fig. 1 OFDM system: (a) overall system configuration, (b) modulation and demodulation; (c) and compensation scheme.

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To achieve higher spectral efficiency with larger number of sub-channels, we deployed the OFDM system with 2048 subcarriers in 50GHz band. The OFDM modulator accepts the independent parallel data streams and maps them to QPSK/16QAM points for each spatial channel. After up-sample the modulated signal, an ideal low pass filter is used and various spatial channels get multiplexed together before transmission takes place. Received symbols are processed by a typical OFDM demodulator and passed to channel compensation block. To minimize the channel imperfections, we used training symbol and pilot symbol in the OFDM scheme for estimation [15,16]. The training symbols are part of data frames that are known at the receiver side, and the pilot symbols represent the redundancy added at each data frame for efficient channel estimation. To compensate for the mode crosstalk, the pilot symbols are properly inserted for least-square (LS) and least minimum mean square error (LMMSE)- based compensation algorithms, as shown in Fig. 1(c). It is important to mention that the procedure discussed above is performed for each spatial mode. In the pure linear channel model, the ASE noise is the only limiting factor in addition to the mode-coupling. To obtain the SNR estimate for LMMSE, we perform a symbol-based SNR estimation after the LS compensation. The LMMSE technique improves the OFDM symbol estimate based on LS compensated symbols. Notice that we perform the LS and LMMSE estimation based on each OFDM symbol, which is not optimal for full channel inversion as described in [13]. The chromatic dispersion compensation is applied before the compensation of the mode-coupling. Our assumption is that the chromatic dispersion affects all spatial modes in a similar fashion if the refractive index profile is parabolic-like and properly designed.

To evaluate the achievable information rate of the FMF system, we follow the model in [15,17–19]. The electric field in a group of $N / 2$ degenerate spatial modes is represented by $N$ -dimensional complex valued vector $\vec{E}$ . For example, if we consider two modes fiber (LP01 and LP11), then the $\vec{E}$ can be expanded as,

\vec{E} = {[\begin{matrix} E_{1} & E_{2} \end{matrix}]}^{T} = {[\begin{matrix} E_{1x} & E_{1y} & E_{2x} & E_{2y} \end{matrix}]}^{T},

which is constructed by stacking the Jones vectors of N spatial modes one on top of each other. In Eq. (1),

\vec{E_{i}}

denotes the i-th spatial mode. Besides, the original Manakov equation can be obtained from [20]. We separate the linear and nonlinear parts as follow:

\frac{d \vec{E}}{d z} = (\hat{D} + \hat{N}) \vec{E},

where the

\hat{D}

is the linear part and the

\hat{N}

is the nonlinear part.

The linear part vector form NLSE can be written as,

\hat{D} = - \frac{1}{2} \vec{α} - \vec{β_{1}} \frac{d y}{d x} - j \vec{β_{2}} \frac{\partial^{2}}{\partial t^{2}} + \frac{1}{3!} \vec{β_{3}} \frac{\partial^{3}}{\partial t^{3}},

where in the ideal case,

\vec{α}

is a zero matrix and

\vec{β_{i}}

is proportional to the identity. In general cases, the

\vec{α} = α \vec{I}

(

\vec{I}

is identity matrix), assuming that all modes are attenuated to the same level.

\vec{β_{1}}

can be calculated as

\vec{β_{1}} = U \cdot Α \cdot U^{†},

where

U

and

Α

are random unitary Gaussian matrix [21] and different modes delays matrix, respectively. The matrices

U

and

Α

are of size 2N-by-2N and

U^{†}

is the conjugated transpose of

U

.

Α

can be further represented as,

Α = (\begin{matrix} \vec{τ_{1}} + \frac{τ_{P M D}_{_{1}}}{2} \vec{σ_{1}} & \dots & \vec{0_{2 \times 2}} \\ ⋮ & ⋱ & ⋮ \\ \vec{0_{2 \times 2}} & \dots & \vec{τ_{N}} + \frac{τ_{P M D}_{_{N}}}{2} \vec{σ_{1}} \end{matrix}),

where

τ_{PMD}_{_{i}}

is a two-by-two diagonal matrix with elements equal to differential mode group delay (DMGD):

τ_{PMD}_{_{i}} = \frac{D_{P M D}_{_{1}}}{2 \sqrt{2 \cdot L_{correlation}}},

where

L_{correlation}

is the correlation length of the FMF. Similarly,

\vec{α}

,

\vec{β_{2}}

and

\vec{β_{3}}

can be extended to matrix form [16,17]. The nonlinear operator

{\hat{N}}_{i}

can be defined as,

{\hat{N}}_{i} = j γ [f_{s e l f} \frac{8}{9} {| \vec{E_{i}} |}^{2} \vec{I} + \sum_{k \neq i}^{N} f_{c r o s s} \frac{4}{3} {| \vec{E_{k}} |}^{2} \vec{I} - \frac{1}{3} ({\vec{E_{i}}}^{†} \vec{σ_{3}} \vec{E_{i}}) \vec{σ_{3}}] i = 1, 2, \dots, N,

where

\vec{E_{i}}

denotes the i-th propagation mode, which includes both x and y polarization modes. The terms

f_{s e l f}

and

f_{c r o s s}

are general self- and cross-mode coupling coefficients [17]. The i-th spatial mode power

{| \vec{E_{i}} |}^{2}

can be calculated as

{| \vec{E_{i}} |}^{2} = {| \vec{E_{i x}} |}^{2} + {| \vec{E_{i y}} |}^{2}

and

N

is the number of propagation modes. The nonlinearity factor

γ

in Eq. (7) can be obtained by [18],

γ = \frac{2 {πn}_{2} f_{ref}}{{cA}_{eff}},

where

n_{2}

is the nonlinear index,

f_{ref}

is the reference frequency and

A_{eff}

is the core area.

3. Description of algorithm for AIR calculation for SMF/SDM fiber models with memory

The first step in proposed algorithm for AIR calculation is to group the multichannel received signal into matrix. The number of spatial modes is denoted by N, and the sequence length is denoted by n, so that the received symbol array is an N-by-n matrix with complex elements. The second step is to form a received array with channel memory consideration. As an illustration, in Fig. 2, we assume that the current symbols get affected by previous and incoming symbols, so that the channel memory related to the current symbols is m = 2. Consequently, the new array size will be $(m + 1) \cdot N$ -by- $(n - m)$ , and each column of the array contains the current and neighboring symbols. In third step, each column of the array is grouped into a new multidimensional state, and the corresponding multi-dimensional conditional probability density functions (PDFs) are estimated by evaluating the corresponding histograms after accumulating enough received symbols. Finally, the achievable information rate algorithm is executed based on the conditional PDFs and received symbols.

Fig. 2 Data block arrangement for AIR calculation in multidimensional systems.

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The transmitted symbol array can be denoted as $\vec{x}$ , whose size is N-by-n, for simplicity, we write it as

\vec{x} = (x_{1} x_{2}, ..., x_{n}),

where

x_{i} = (\begin{matrix} x_{i, 1} \\ x_{i, 2} \\ ⋮ \\ x_{i, N} \end{matrix}) \in ℵ = {0, 1, ..., M - 1}^{N},

wherein

x_{i}

is an element of

ℵ

, which is an

N

dimensional vector space over transmitted symbols (constellation points).

\vec{x}

is a super-symbol vector (more precisely matrix) containing a sequence of

x_{i}

and

M

is the number of points in the corresponding

M

-ary signal constellation (e.g.,

M = 4

for QPSK). Every symbol carries

l = \log_{2} M

bits information. Similarly,

\vec{y}

is received super-symbol array represented as a sequence of

y_{i}

as follows

y_{i} = (\begin{matrix} y_{i, 1} \\ y_{i, 2} \\ ⋮ \\ y_{i, N} \end{matrix}) \in ℂ^{N},

with

ℂ

denoting the complex field. The information rate, expressed in bits/channel use, is determined by

I (Y; X) = H (Y) - H (Y | X),

where the

X

and

Y

represent input and output super-symbol random variables and the

H (\cdot)

denotes the entropy of a random variable which can be calculated as [22]

H (Y) = - E [\log_{2} \Pr (Y)] = - \lim_{n \to \infty} \frac{1}{n} \log_{2} \Pr (\vec{y [1, n]}),

where the

E [\cdot]

denotes the mathematical expectation operator. By using the logarithmic-chain rule, part of right side of Eq. (11) can be written as

\log_{2} \Pr (\vec{y [1, n]}) = \sum_{i = 1}^{n} \log_{2} \Pr (y_{i} | \vec{y [1, i - 1]}) .

So the entropy of random variable Y (Eq. (11)) can be now obtained by,

H (Y) = - \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^{n} \log_{2} \Pr (y_{i} | \vec{y [1, i - 1]}) .

Similarly, the second half of the Eq. (10) can be calculated by,

\begin{array}{l} H (Y | X) = - E [\log_{2} \Pr (Y | X)] = - \lim_{n \to \infty} \frac{1}{n} \log_{2} \Pr (\vec{y [1, n]} | \vec{x [1, n]}) \\ = - \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^{n} \log_{2} \Pr (y_{i} | \vec{y [1, i - 1]}, \vec{x [1, n]}) . \end{array}

At this point, we can see that the mutual information between

X

and

Y

can be obtained by averaging of individual increments of mutual information

I (Y; X)

as follows

\begin{array}{l} I (Y; X) = \lim_{n \to \infty} \frac{1}{n} \log_{2} \Pr (y_{i} | \vec{y [1, i - 1]}, \vec{x [1, n]}) \\ - \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^{n} \log_{2} \Pr (\vec{y [1, n]}) = \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^{n} d I_{i}, \end{array}

where the

d I_{i}

can be obtained by

d I_{i} = \log_{2} \Pr (y_{i} | \vec{y [1, i - 1]}, \vec{x [1, n]}) - \log_{2} \Pr (y_{i} | \vec{y [1, i - 1]}) .

To calculate these two conditional probabilities, we use the forward recursion of the multilevel/multidimensional BCJR algorithm. The states in BCJR are defined as the super symbols with memory

s_{i} = [x_{i - a}, ..., x_{i - 1}, x_{i}, x_{i + 1}, ..., x_{i + b}],

with

s_{i} \in ℑ = {0, 1, ..., D},

where the

D

is the size of sample space of the state, which depends on the memory length and transmission format (constellation size), and m is the memory length as

m = a + b ..

In Eq. (19), ‘

a

’ and ‘

b

’ are memory of the observed symbol on the left and right, respectively. So the sample space size

D

is calculated by

D = M^{\dim} = M^{(m + 1) \cdot N} .

Figure 3 describes one-in-one-out efficient moving window type algorithm which can significantly reduce the computer memory requirements in calculation. Note that if we consider larger constellation with longer memory, the number of states

D

will be extremely large. For example, consider 16-QAM with single mode fiber with two polarizations even without memory, the space size D is equal

16^{1 \times 4} = 65536

.

Fig. 3 Finite state machine description for AIR calculation.

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For each state defined above, we accumulate the samples to form the approximation of the density function. Let us write the multidimensional PDF as

\Pr (\vec{y [i - a, i + b]} | s_{i}) .

Note the

\vec{y [i - a, i + b]}

denotes the samples containing ‘

a

’ previous, current, and ‘

b

’ incoming multidimensional samples, in the similar form as

s_{i}

in Eq. (17). Due to the nonlinear coupling effects, the use of Gaussian distribution is not accurate; instead we estimate PDFs by evaluation of histograms. We estimate the PDF of each state separately for I and Q channels in one polarization by histograms collection.

When the distance is short and for high SNR, most of the samples collected will be concentrated around the center bins when we use the uniform bins’ distribution. This will lead to some empty edge bins and coarse PDF estimation in the center. To better estimate the PDFs, we use a non-uniform quantization of bin separation for each dimension. Let $z$ denote the original uniformly distributed bin separation and let the $f (z)$ denote the non-uniform bin separation, which can be written as

f (z) = sgn (z) (\frac{1}{μ}) ({(1 + μ)}^{| z |} - 1) | z | \leq V .

By properly choosing the parameters

μ

and upper limit

V

, we can avoid the existence of empty bins for more accurate estimation of PDFs.

Finally, we use the forward recursion of BCJR algorithm to calculate the probabilities in Eq. (13). Considering all possible state transitions, the number of possible edges, denoted as $N_{e d g e s}$ , for each state is given by

N_{e d g e s} = M^{N} .

Also, by assuming uniformly distributed source

\Pr (x) = \frac{1}{M^{N}} .

Given one channel output super-symbol and collected histogram for estimating the PDF of ach state, we are able to obtain the first half of Eq. (16). Note that

(\vec{y [1, i - 1]}, \vec{x [1, n]})

uniquely defines a state at

y_{i}

from Eq. (16). This calculation only requires the forward metric and branch metric from BCJR algorithm, but not the backward metric to perform the second half calculation of second term in Eq. (16).

In the forward recursion of BCJR algorithm, two important metrics, the forward and branch metrics, are defined as follows

Forward metric:

α_{i} (s) = \log [\Pr (s_{i} = s, \vec{y [1, i]})];

Branch metric:

γ_{i} (s', s) = \log [\Pr (y_{i} | s) \Pr (x_{i})] .

To avoid the dependence on all previous received symbols

\vec{y [1, i]}

, we need to update the

α_{i} (s)

in an iterative fashion. The calculation of

α_{i} (s)

solely depends on previous value of

α_{i - 1} (s)

and current branch metric

γ_{i} (s', s)

.The

γ_{i} (s', s)

will be updated for every given received symbol

y_{i}

in current one-in-one-out window. The algorithm outputs the

d I_{i}

without saving the

γ_{i} (s', s)

for different time indices, which can significantly reduce the memory requirements of regular BCJR algorithm especially when the number of states is large. The i-th iteration updating rule can be written as

γ_{i} (s', s) = \log [\Pr (s_{i} = s, y_{i}, s_{i - 1} = s')];

α_{i} (s) = \max^{*} [α_{i - 1} (s') + γ_{i} (s', s)],

where the

\max^{*} (\cdot)

operation for two elements is the defined as

\max^{*} (c, d) = \ln (e^{c} + e^{d}) = \max (c, d) + \ln (1 + e^{- | y - x |}) .

For the

\max^{*} (\cdot)

calculation operating on more than two elements, we can perform calculation in an iterative fashion, by applying the max*-operator to two elements at a time. For three elements example,

\max^{*} (c, d, e) = \max^{*} (c, \max^{*} (d, e)) .

The initial values of $α_{i} (s)$ can be set as

{\begin{matrix} α_{0} (s) = - \infty \forall {s \in ℑ, s \neq 0} \\ α_{0} (s = 0) = 0 \\ α_{1} (s) = - \infty \forall {s \in ℑ} \end{matrix} .

Notice that arbitrary choice of the starting state of

α_{i} (s)

can result in some accuracy loss of the beginning AIR estimation output before all states can be possibly reached. To compensate this, we can simply skip some of the output

d I_{i}

and start the averaging of Eq. (20) when all states can be attained. The result should converge to a certain values when n is sufficient large regardless of the starting point.

Based on the estimated PDFs and the moving windows, we are able to calculate the $d I_{i}$ for each input and take the average. The calculation steps in this algorithm can be summarized as follows: 1) group the super-symbol with or without memory, 2) form the states and multi-dimensional histograms for multilevel BCJR, and 3) by using the forward and branch metrics calculate the $d I_{i}$ by using Eq. (16) and average it over long enough super-symbol group sequence.

4. Numerical results

To analyze the lower bound of AIR of the CO-OFDM over FMF system, we simulate the propagation of the mode-multiplexed OFDM signals over more than 200 spans, each of length 80 km. The fiber and system parameters are presented in the table below. The simulation is very time consuming. The most consuming parts in this study are the FMF nonlinear propagation and calculation of AIR with memory assumption. To speed-up simulation, the whole sequence has been split into K parts by multi-thread technology in C + + programs, where K is the number of threads being used (typical K≥24). We verified that the same performance is obtained for single thread, but with much faster simulation process. Besides, the matrix calculation has been speed-up by employing the Eigen library, which is known to be very handy and efficient. The parameters of the CO-OFDM AIR calculation system, as well as the fiber parameters we used in simulations, are summarized in Table 1.

Table 1. Fiber parameters and AIR calculation OFDM system parameters

View Table

Notice that the AIR calculation does not take the redundancy added to the OFDM frames into account since the only data carrying subcarriers are used in AIR calculation.

As an illustration of simulation results, Fig. 4 shows the receiver side constellation diagrams of the single OFDM symbol for different lengths, before and after channel impairments compensation, when QPSK/16QAM format is applied. Both SMF and FMF link with two spatial modes and two polarization states are observed.

Fig. 4 Received constellations: (a), (c) and (e) correspond to QPSK over SMF for number of spans equal to 50, 100, and 150, respectively. On the other hand, (b), (d) and (f) correspond to 16QAM over FMF (with two spatial and two polarization modes) for number of spans equal to 1, 50, and 100, respectively.

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In Fig. 5(a), we depict the AIRs of SMF vs. number of spans for both QPSK and 16QAM with and without memory. We group all symbols in both polarizations into super-symbol for more accurate calculation. In this case of insufficient memory, the slope of the AIR curves will decrease as distance increases. Since the signals of 16QAM in SMF and QPSK in FMF are extremely sensitive to nonlinear ISI [23], this requires to investigate memory effects in order to achieve high AIRs at long distance. As shown in Fig. With memory effects taken into consideration, both QPSK and 16QAM can attain the full rate over 125 spans (10,000 km), at the expense of increased complexity of corresponding nonlinear-MAP detector. When memory effects are taken into account we essentially group several symbols transmitted in successive time intervals into a super-symbol so that the nonlinear intra-channel interaction between symbols in a super-symbol is not relevant anymore. The overall AIRs get significantly improved on such a way, at the cost of significantly increased MAP detection complexity. Figure 5(b) shows the simulation results of lower bound AIR calculation for FMF link for two spatial modes and two polarization states.

Fig. 5 AIR results for CO-OFDM transmission over: (a) SMF and (b) FMF.

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The importance of this study is profound in two ways. Firstly, it clearly demonstrates the inaccuracy of models or systems that do not take the channel memory into account. The existing models just consider the nonlinear inter-symbol interference due to dispersion and nonlinearities as a random process, whereas, in fact, it has both deterministic and random components. Secondly, we show that by taking memory effects into account it is possible to improve the achievable information rates. This indicates that the deterministic component of nonlinear ISI can be, in principle, compensated for at the receiver side with more advanced detection scheme. Additionally, it is evident that the SDM system has the similar AIR performance as SMF up to the certain distance, after which the nonlinear interaction among spatial becomes important and AIR decays more rapidly than that in SMF.

5. Concluding remarks

An algorithm has been proposed to calculate the achievable information rates in SDM systems by taking the nonlinear coupling of modes into account. The lower bounds of AIRs for high-speed CO-OFDM optical transmission over FMF system have been calculated using the forward recursion of the multilevel BCJR algorithm. Both linear and nonlinear effects in FMF are taken in to account for more accurate channel model. We consider the inter-spatial-channel linear/nonlinear effects and employ the super-symbol AIR calculation algorithm over the entire spatial channels. It is important to notice that the proposed approach is applicable to any MIMO nonlinear system, including the optical recording, magnetic recording, and free-space optics.

We show that the SDM system has the similar AIR performance trend as SMF up to the certain distance, after which the nonlinear interaction among spatial modes becomes important and AIR decays more rapidly than that in SMF. We have demonstrated that the models of either channels or systems that do not take the channel memory into account underestimate the AIRs. By jointly considering all spatial modes/channels, we provide a more general AIR calculation solution for MIMO-SDM system.

Acknowledgments

This work was supported in part by the National Science Foundation (NSF) under Grant CCF-0952711 and NSF CIAN ERC.

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Description	Variable	Values	Description	Values
Length of step	$L$	1000 [m]	Number of modes	2/4
Length of span	$L_{s p a n}$	80 [km]	Modulation format	QPSK/ 16QAM
Correlation length	$L_{c o r r}$	80 [m]	FFT/IFFT length	4096
Attenuation	$α$	0.2 [dB/km]	Data subcarrier size	1919
Dispersion	$D_{λ}$	$16 \cdot 10^{- 6}$ [s/m²]	Pilot subcarrier size	129
Dispersion Slope	$S_{λ}$	$0.08 \cdot 10^{3}$ [s/m³]	Number of subcarrier	2048
Mode dispersion	$D_{M D}$	$\begin{array}{l} (2.0 \cdot 10^{- 12}) / 31.62 \\ [{s/m}^{2}] \end{array}$	Cyclic extension length	256
Nonlinear Index	$γ$	$2.6 \cdot 10^{- 20}$ [m²/w]	Symbol rate	50GS/s
Effective Core Area	$A_{eff}$	$126 \cdot 10^{- 12}$ [ m²]	Oversampling factor	8
EDFA noise figure	$N F$	6 [dB]	Histogram bins	64/128

Achievable information rates calculation for optical OFDM few-mode fiber long-haul transmission systems

Abstract

1. Introduction

2. System diagram and FMF channel model

3. Description of algorithm for AIR calculation for SMF/SDM fiber models with memory

4. Numerical results

5. Concluding remarks

Acknowledgments

References and links

Cited By

Figures (5)

Tables (1)

Equations (32)

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