Analytical modeling of XPM in dispersion-managed coherent fiber-optic systems

Xiaojun Liang; Shiva Kumar

doi:10.1364/OE.22.010579

1. Introduction

The modeling of cross-phase modulation (XPM) distortion in fiber-optic systems with direct detection [1–8] and with coherent detection [9–19] has drawn significant interest. Poggiolini et al. [9] and Carena et al. [10] have modeled the nonlinear interference noise (NLIN) in dispersion-uncompensated (DU) transmission systems as excess additive Gaussian noise and its variance is calculated using a perturbation technique. Dar et al. [15] have analyzed the properties of NLIN in a DU fiber-optic system with large dispersion and found that the NLIN is not additive Gaussian, but rather it depends strongly on the data transmitted in the channel. Shahi et al. [17] have developed an analytical expression for the XPM variance of a probe Gaussian pulse due to its interaction with other channels of a wavelength division multiplexing (WDM) system and validated the model numerically. In this paper, we extend the analysis of [17] to arbitrary pulse shapes and study the scaling laws of the XPM variance with residual dispersion per span, the number of WDM channels and transmission distance.

For a non-Gaussian pulse, the XPM variance cannot be calculated analytically without additional approximation. In DU systems, the stationary phase approximation can be used to circumvent the difficulty in explicitly evaluating integrals and analytical expressions are still available for the variance of nonlinear distortions [15,20]. However, the stationary phase approximation technique requires a large accumulated dispersion, which becomes inaccurate for dispersion-managed (DM) systems. The modeling of XPM variance in DM systems has become a challenging problem and in this paper, we focus on the development of a simple analytical expression for the XPM variance in DM systems with arbitrary pulse shapes. We use the summation of time-shifted Gaussian pulses to fit a non-Gaussian pulse, which not only provides a good approximation of the non-Gaussian pulse but also allows analytical derivation of the XPM variance. We note that a similar method has been successfully applied in quantum chemistry, where a summation of Gaussian functions ( $\sum_{k} \exp (- a_{k} x^{2})$ ) is used to approximate an exponential function ( $\exp (- b x)$ ) [21]. The summation of Gaussian functions is used in modeling electronic orbitals, named as Gaussian type orbitals (GTOs) [22]. Despite the fact that Gaussian functions are not mutually orthogonal, GTOs form a complete basis set and have gained dominance in calculating electronic wave functions due to its simplicity in explicitly evaluating integrals. In this paper, we use a similar technique and approximate a non-Gaussian pulse by a summation of time-shifted Gaussian pulses. The fitting parameters are calculated using the least squares method (LSM).

The analytical expressions developed in this paper are used to study the XPM impairments in DM systems based on quadrature phase shift keying (QPSK) and 16-quadrature amplitude modulation (16-QAM). Scaling laws of the XPM variance with launch power, residual dispersion per span, number of channels and number of spans are studied. We found that the XPM variance depends on the power levels of the probe channel for 16-QAM, i.e., the variance is higher for constellation points close to the edge than those close to the center. This is the consequence of strong phase noise due to the XPM [14] and the manifestation of the non-additive nature of XPM noise [15]. The frequency-domain approach of [9–11] treats the XPM variance as white noise whose variance is independent of the signal power levels. Although such an approach is acceptable for systems in which amplified spontaneous emission (ASE) noise is the dominant penalty, it could lead to inaccurate results when XPM variance is comparable to ASE variance. The XPM variance is highest for the resonant dispersion map with zero accumulated dispersion per span (Ψ_res = 0) and it decreases monotonically with |Ψ_res|. The XPM variance increases almost quadratically with the number of spans when Ψ_res = 0 due to the coherent addition of XPM distortions. However, when |Ψ_res| is large, the XPM variance scales almost linearly with the number of spans since the correlation between XPM distortions occurring in different spans becomes close to zero.

2. Analytical XPM model

2.1 Gaussian pulse

The evolution of the optical field envelope in a fiber-optic link is described by the nonlinear Schrödinger equation (NLSE). Ignoring the higher-order dispersion and higher-order/delayed nonlinear effects, the NLSE can be written as

j \frac{\partial q}{\partial z} - \frac{β_{2} (z)}{2} \frac{\partial^{2} q}{\partial T^{2}} + γ_{0} {| q |}^{2} q = - j \frac{α (z)}{2} q,

where q is the optical field envelope; β₂(z), γ₀ and α(z) are the dispersion, nonlinear and loss/gain profiles, respectively. Using the transformation

q (z, T) = \exp [- w (z) / 2] u (z, T),

where

w (z) = \int_{0}^{z} α (s) d s

, we obtain the lossless form of the NLSE as

j \frac{\partial u}{\partial z} - \frac{β_{2} (z)}{2} \frac{\partial^{2} u}{\partial T^{2}} + γ (z) {| u |}^{2} u = 0,

where

γ (z) = γ_{0} \exp [- w (z)]

. To the first order accuracy, XPM only involves the interaction between two channels, so we consider only two channels of a WDM system. The total field envelope can be written as

u = u_{1} + u_{2},

where u_k is the field envelope of the kth channel, k = 1, 2. Substituting Eq. (4) into Eq. (3) and ignoring the four wave mixing terms, we get

j \frac{\partial u_{k}}{\partial z} - \frac{β_{2} (z)}{2} \frac{\partial^{2} u_{k}}{\partial T^{2}} = - γ (z) [{| u_{k} |}^{2} + 2 {| u_{l} |}^{2}] u_{k}, k = 1, 2 a n d l = 3 - k .

Without loss of generality, we consider the interaction between a pulse of channel 1 (probe) in symbol slot 0 and multiple pulses in channel 2 (pump) modulated with random data. We assume that the leading order solution of Eq. (5) is linear and treat the nonlinear terms on the right-hand side as perturbations. The input fields to the optical fiber can be written as

u_{1} (0, T) = \sqrt{P} a_{0} g (0, T),

u_{2} (0, T) = \sqrt{P} \sum_{n = - N}^{N} b_{n} g (0, T - n T_{s}) \exp (- j Ω T),

g (0, T) = \exp (- \frac{T^{2}}{2 T_{0}^{2}}),

where P is the power, (2N + 1) is the total number of symbols, T_s is the symbol interval, Ω is the channel separation in radians, T₀ is the half-width at 1/e-intensity point of the Gaussian pulse. For QAM format, the data a₀ and b_n are given by

a_{0} o r b_{n} = \frac{x_{n} + j y_{n}}{\sqrt{2}},

where x_n = ± 1, ± 3, ± 5, …, ± (X-1), y_n = ± 1, ± 3, ± 5, …, ± (Y-1). X and Y are the number of amplitude levels of the in-phase and quadrature components, respectively. Using the perturbation technique, we take γ₀ as a small parameter and expand the field in channel k into a series

u_{k} = u_{k}^{(0)} + γ_{0} u_{k}^{(1)} + γ_{0}^{2} u_{k}^{(2)} + ..., k = 1, 2,

where

u_{k}^{(m)}

denotes the mth-order solution. The linear solution satisfies Eq. (5) with γ₀ = 0,

j \frac{\partial u_{k}^{(0)}}{\partial z} - \frac{β_{2}}{2} \frac{\partial^{2} u_{k}^{(0)}}{\partial T^{2}} = 0.

Solving Eq. (11), we find the linear solution as [17]

u_{1}^{(0)} (z, T) = \frac{\sqrt{P} T_{0}}{T_{1}} a_{0} \exp (- \frac{T^{2}}{2 T_{1}^{2}}),

u_{2}^{(0)} (z, T) = \frac{\sqrt{P} T_{0}}{T_{1}} \sum_{n} b_{n} \exp (- \frac{{(T - τ_{n})}^{2}}{2 T_{1}^{2}} - j Ω T + j θ (z)),

where

T_{1} = \sqrt{T_{0}^{2} - j S (z)},

τ_{n} = n T_{s} + S (z) Ω,

θ (z) = S (z) Ω^{2} / 2,

S (z) = \int_{0}^{z} β_{2} (s) d s .

To find the first-order solution, we substitute Eq. (10) into Eq. (5) and collect all the terms that are proportional to γ₀. We find the governing equation for the first-order solution as

j \frac{\partial u_{k}^{(1)}}{\partial z} - \frac{β_{2} (z)}{2} \frac{\partial^{2} u_{k}^{(1)}}{\partial T^{2}} = - e^{- w (z)} [{| u_{k}^{(0)} |}^{2} + 2 {| u_{l}^{(0)} |}^{2}] u_{k}^{(0)}, k = 1, 2 a n d l = 3 - k .

Before solving Eq. (18), consider the following differential equation

j \frac{\partial f}{\partial z} - \frac{β_{2} (z)}{2} \frac{\partial^{2} f}{\partial T^{2}} = F (z, T),

where the forcing function F(z, T) is in the form

F (z, T) = η (z) \exp {- \sum_{k = 1}^{3} {[T - C_{k} (z)]}^{2} R_{k} (z)} .

The solution of Eq. (19) is given by [17]

f (z, T) = - j \int_{0}^{z} \frac{η (s)}{\sqrt{δ (z, s) R (s)}} \exp [- \sum_{k = 1}^{3} C_{k}^{2} R_{k} + \frac{C^{2}}{R}] \exp [\frac{{(D + j T)}^{2}}{δ (z, s)}] d s,

where

R = R_{1} + R_{2} + R_{3},

C = C_{1} R_{1} + C_{2} R_{2} + C_{3} R_{3},

D = j C / R,

δ (z, s) = [1 - j R A (z, s)] / R,

A (z, s) = 2 [S (z) - S (s)] .

For the first order correction for

u_{1}

due to the XPM term

2 {| u_{2}^{(0)} |}^{2} u_{1}^{(0)}

in Eq. (18), the corresponding forcing function F(z, T) is

\begin{array}{l} F (z, T) = - 2 e^{- w (z)} {| u_{2}^{(0)} |}^{2} u_{1}^{(0)} \\ = 2 P^{3 / 2} a_{0} η (z) \sum_{m} \sum_{n} b_{m} b_{n}^{*} \exp {- \sum_{k = 1}^{3} {[T - C_{k} (z)]}^{2} R_{k} (z)}, \end{array}

where

η (z) = \frac{- T_{0}^{3} \exp [- w (z)]}{T_{1} (z) {| T_{1} (z) |}^{2}},

C_{1} (z) = τ_{m} (z), C_{2} (z) = τ_{n} (z), C_{3} (z) = 0,

R_{1} = R_{3} = \frac{1}{2 T_{1}^{2}}, R_{2} = \frac{1}{2 {(T_{1}^{*})}^{2}},

Using Eq. (21), the first order correction for

u_{1}

due to XPM is obtained as

u_{1}^{(1), X P M} (L_{t o t}, T) = j 2 P^{3 / 2} a_{0} \sum_{m} \sum_{n} b_{m} b_{n}^{*} X_{m n} (L_{t o t}, T),

where

X_{m n} (L_{t o t}, T) = \int_{- \infty}^{\infty} U_{m n} (L_{t o t}, T^{'}) h_{R X} (T - T^{'}) d T^{'},

U_{m n} (L_{t o t}, T) = \int_{0}^{L_{t o t}} \frac{- η^{'} (s)}{\sqrt{δ (L_{t o t}, s) R (s)}} \exp [\frac{{(D + j T)}^{2}}{δ (L_{t o t}, s)}] d s,

η^{'} (s) = η (s) \exp (- \sum_{k = 1}^{3} C_{k}^{2} R_{k} + \frac{C^{2}}{R}) .

Here, L_tot is the total transmission distance and h_RX(T) is the impulse response of the receiver filter.

From Eq. (10), we see that the first order XPM distortion is

δ u_{1} = γ_{0} u_{1}^{(1), X P M} .

The XPM variance can be calculated by

V a r {δ u_{1}} = E {{| δ u_{1} |}^{2}} - {| E {δ u_{1}} |}^{2} .

For QAM signals, we define

K_{1} = E {{| b_{n} |}^{2}}, K_{2} = E {{| b_{n} |}^{4}} .

For QPSK, K₁ = K₂ = 1, while for 16-QAM, K₁ = 5, K₂ = 33. Using Eqs. (31), (35) and (36), the XPM variance is found as [17]

σ_{X P M}^{2} = 4 γ_{0}^{2} P^{3} {| a_{0} |}^{2} ((K_{2} - K_{1}^{2}) \sum_{m} {| X_{m m} |}^{2} + K_{1}^{2} \underset{m \neq n}{\sum_{m} \sum_{n}} {| X_{m n} |}^{2}) .

2.2 Non-Gaussian pulse

A non-Gaussian pulse h(T) is approximated by a summation of time-shifted Gaussian pulses as

h^{'} (T) = \sum_{k = 1}^{K} ξ_{k} \exp [- \frac{{(T - μ_{k} T_{s})}^{2}}{2 {(θ_{k} T_{s})}^{2}}],

where ξ_k, μ_k and θ_k are fitting parameters; K is the number of time-shifted Gaussian functions. We use the least squares method (LSM) to optimize the fitting parameters. The detailed derivation of the LSM is given in the Appendix A. As an example, Fig. 1 shows the Gaussian fitting of the two most commonly used pulse shapes in optical communications: the Nyquist pulse x₁(t) and the raised cosine pulse x₂(t), which are respectively defined as:

x_{1} (t) = \sin c (\frac{t}{T_{s}}) \frac{\cos (a π t / T_{s})}{1 - {(2 a t / T_{s})}^{2}},

x_{2} (t) = {\begin{matrix} 1, | t | < \frac{1 - a}{2 T_{s}} \\ \frac{1}{2} [1 - \sin (\frac{π}{a T_{s}} | t | - \frac{π}{2 a})], \frac{1 - a}{2 T_{s}} \leq | t | \leq \frac{1 + a}{2 T_{s}} \\ 0, | t | > \frac{1 + a}{2 T_{s}} \end{matrix},

where a is the roll-off factor. The fitting parameters optimized by LSM are given in Tables 1 and 2. This example shows that only a few (K = 6) time-shifted Gaussian pulses are enough to achieve good fitting of the Nyquist pulse and the raised cosine pulse. We found that the required number of Gaussian pulses increases when the roll-off factor decreases. For example, the required numbers are 10 and 16 when the roll-off factor is 0.2 and 0.1, respectively. When the roll-off factor is greater than 0.6, six Gaussian pulses are sufficient.

Fig. 1 Fitting a Nyquist pulse x₁(t) and a raised cosine pulse x₂(t) using a summation of time-shifted Gaussian functions. (roll-off factor = 0.6, number of Gaussian functions = 6).

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Table 1. Fitting Parameters Optimized by LSM (Nyquist Pulse, x₁(t))

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Table 2. Fitting Parameters Optimized by LSM (Raised Cosine Pulse, x₂(t))

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Using Eq. (39), the XPM variance for systems with non-Gaussian pulses can be derived following the similar procedure as in Section 2.1. The XPM distortion for the case of non-Gaussian pulse is given by (see the Appendix B)

u_{1}^{(1), X P M, N G} (L_{t o t}, T) = j 2 P^{3 / 2} a_{0} \sum_{m} \sum_{n} b_{m} b_{n}^{*} Y_{m n} (L_{t o t}, T),

where

Y_{m n} (L_{t o t}, T) = \int_{- \infty}^{\infty} V_{m n} (L_{t o t}, T^{'}) h_{R X} (T - T^{'}) d T^{'},

V_{m n} = \int_{0}^{L_{t o t}} \sum_{k_{1} = 1}^{K} \sum_{k_{2} = 1}^{K} \sum_{k_{3} = 1}^{K} \frac{- η^{'} (s)}{\sqrt{δ (L_{t o t}, s) R (s)}} \exp [\frac{{(D + j T)}^{2}}{δ (L_{t o t}, s)}] d s,

where the parameters are the same as those defined in Section 2.1, except for the following ones:

T_{1, k} = \sqrt{{(θ_{k} T_{s})}^{2} - j S (z)},

η (z) = - e^{- w (z)} ξ_{k_{1}} ξ_{k_{2}} ξ_{k_{3}} \frac{θ_{k_{1}} θ_{k_{2}} θ_{k_{3}} T_{s}^{3}}{T_{1, k_{1}} T_{1, k_{2}}^{*} T_{1, k_{3}}},

C_{1} (z) = τ_{m, k_{1}}, C_{2} (z) = τ_{n, k_{2}}, C_{3} (z) = μ_{k_{3}} T_{s},

R_{1} (z) = \frac{1}{2 T_{1, k_{1}}^{2}}, R_{2} (z) = \frac{1}{2 {(T_{1, k_{2}}^{*})}^{2}}, R_{3} (z) = \frac{1}{2 T_{1, k_{3}}^{2}},

τ_{n, k} = (n + μ_{k}) T_{s} + S (z) Ω .

The XPM variance for systems with non-Gaussian pulses is

σ_{X P M}^{2} (T) = {| a_{0} |}^{2} G (T),

where

G (T) = 4 γ_{0}^{2} P^{3} ((K_{2} - K_{1}^{2}) \sum_{m} {| Y_{m m} |}^{2} + K_{1}^{2} \underset{m \neq n}{\sum_{m} \sum_{n}} {| Y_{m n} |}^{2}) .

So far we assumed that there is only one symbol (at the symbol slot 0) in the probe channel and Eq. (50) provides the variance as a function of time T for this case. Here, we consider the modification of Eq. (50) for the case of a known symbol a₀ in the symbol slot 0 of the probe channel and random symbols in the neighboring slots of the probe channel. Since the Nyquist pulse x₁(T) has a pulse width larger than the symbol period, the XPM variance given by Eq. (50) has non-zero values even when T is larger than the duration of the symbol slot 0, which means that the interaction of a pulse in symbol slot 0 of the probe channel with the symbol sequence of the pump channel (given by Eq. (50)) introduces distortion on the pulse in the neighboring symbol slot k, k∈[-M, M], k ≠ 0, and M is the number of neighboring symbol slots up to which the XPM distortion is significant. In the case that multiple random symbols present in the probe channel, the calculation of the overall XPM variance at the center of any given symbol slot (e.g. symbol slot 0) needs to take into account both contributions from the interaction of the current symbol with the pump as well as those from the interaction of the neighboring symbols with the pump. As the data on the neighboring symbol slots of the probe channel is random, the variance contribution of the neighboring symbols needs to be averaged over the random data

{| a_{n} |}^{2}

. The overall XPM variance at the center of the symbol slot 0 can be calculated as

σ_{X P M_o v e r a l l}^{2} = {| a_{0} |}^{2} G (0) + E {{| a_{n} |}^{2}} \sum_{\begin{array}{l} k = - M \\ k \neq 0 \end{array}}^{M} G (k T_{s}) .

The first term in Eq. (52) represents the XPM variance of the pulse in the symbol slot 0 of the probe channel with the pump and the second term represents that of the pulses in the neighboring symbol slots of the probe channel with the pump. For QPSK,

E {{| a_{n} |}^{2}} = 1

, while for 16-QAM,

E {{| a_{n} |}^{2}} = 5

. The first term in Eq. (52) is proportional to

{| a_{0} |}^{2}

, which implies that the XPM variance could have multiple levels as an M-QAM constellation has multiple power levels. For example, the 16-QAM system has three XPM variance levels, i.e.,

{| a_{0} |}^{2}

can take any one of the values [1, 5, 9]. The capability of estimating multiple XPM variance levels rather than only one averaged level will permit more accurate bit error rate (BER) estimation in practical fiber-optic communication systems.

In summary, the analytical XPM model can be applied in both DU and DM systems, and can deal with arbitrary pulse shapes. Also, the model is valid for M-QAM modulation formats and can be used to estimate the multiple levels of the XPM variance.

3. Results and discussion

To validate the analytical model, we carried out Monte-Carlo simulations of DM coherent fiber-optic systems to find numerical XPM variances, which were then compared with the analytical results calculated using Eq. (52). The WDM fiber-optic system considered in this paper is shown in Fig. 2. Unless otherwise specified, the system configuration is as follows: symbol rate per channel = 28 Gbaud, modulation = QPSK or 16-QAM, pulse shape = Nyquist pulse (x₁(t)) with a roll-off factor of 0.6, number of WDM channels = 5, channel spacing = 50 GHz, amplifier spacing = 80 km, number of fiber spans = 10, number of symbols simulated = 32768 per channel. The dispersion, loss, and nonlinear coefficients of the transmission fiber are D_TF = 16.5 ps/nm/km, α_TF = 0.2 dB/km, and γ_TF = 1.1 W⁻¹km⁻¹, respectively. For the dispersion compensating fiber (DCF), D_DCF = −117.7 ps/nm/km, α_DCF = 0.5 dB/km, and γ_DCF = 4.4 W⁻¹km⁻¹. Gain of the first and second stages of the amplifiers are G₁ = 16 dB and G₂ = 5.2 dB, respectively. The residual dispersion per span (defined as: Ψ_res = D_TFL_TF + D_DCFL_DCF) is 100 ps/nm. Half of the accumulated residual dispersion is compensated in the digital-domain at the transmitter, and the other half at the receiver. We ignore the ASE noise, laser phase noise and polarization effects, since the primary focus is to validate the analytical model for XPM variance. The central channel is demultiplexed using a 50 GHz optical filter. The received signal passes through the coherent receiver front end and then through the digital signal processing (DSP) unit with four samples per symbol. The intradyne receiver selects the channel of interest (middle channel in our case). In the DSP unit, digital back propagation (DBP) is followed by a post- dispersion compensation filter and then by a noise limiting filter. The intra-channel distortion is removed by DBP at the receiver [23]. The step size of DBP is 0.1 km. DBP compensates only for dispersion and nonlinear effects of the fiber-optic channel. To compensate for pre-dispersion compensation, post-dispersion compensation filter is used. A second order Gaussian filter with a 3-dB bandwidth of 28 GHz is used as a noise limiting filter. Without loss of generality, we consider only the XPM variance of the central channel. The numerical variances are calculated for various power levels of QAM.

Fig. 2 Schematic of a dispersion-managed WDM fiber-optic transmission system. Tx: transmitter; Rx: receiver; MUX: multiplexer; DMUX: demultiplexer; DCF: dispersion compensating fiber.

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Figure 3 shows the overall XPM variance as a function of average launch power per channel, P_ave. P_ave is related to P (the power defined in Eq. (6)) by the pulse shape and the modulation format. For the Nyquist pulse (x₁(t)) with a roll-off factor of 0.6, P_ave(dB) = P(dB) - 0.71 dB for QPSK, and P_ave(dB) = P(dB) + 6.27 dB for 16-QAM. The XPM variance is proportional to P³, as shown in Eq. (51). For 16-QAM, the overall XPM scales linearly with |a₀|², as shown in Eq. (52). Figure 3(c) shows the mean XPM variance for 16-QAM, by taking an average over |a₀|². The analytical results agree quite well with the numerical results for both QPSK and 16-QAM systems. In the analytical calculation, the XPM variance contributions of six neighboring symbols are included, namely M = 3 in Eq. (52). In the systems considered here, symbols further than M = 3 have negligible impact on the overall XPM variance. The discrepancies are within 7% and 8% for QPSK and 16-QAM systems, respectively. The discrepancy is defined as

Fig. 3 XPM variance vs. average launch power per channel.

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d i s c r e p a n c y = \frac{| σ_{X P M}^{2} (n u m e r i c a l) - σ_{X P M}^{2} (a n a l y t i c a l) |}{σ_{X P M}^{2} (n u m e r i c a l)} \times 100 % .

Figure 4 shows the constellation of the received signal, after compensating for the average nonlinear phase rotation. As can be seen, the XPM noise on the edge constellation points of 16-QAM is higher than those close to the center since larger power leads to larger XPM variance (see the first term of Eq. (52)). The phase noise component in the case of 16-QAM is significantly higher than for QPSK, in accordance with the findings of [14] and [15]. From Fig. 3(b), we see that the XPM variance depends on the power levels of the probe pulse.

Fig. 4 Constellation of received signal (P_ave = 4 dBm, number of fiber span = 1, Ψ_res = 0).

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Figure 5 shows the overall XPM variance as a function of residual dispersion per span. The XPM variance reaches a peak value in the case of resonant dispersion map (i.e., Ψ_res = 0), since the XPM fields of each fiber span add up constructively in this case. The XPM variance decreases monotonically with |Ψ_res|. The discrepancies are within 7% and 8% for QPSK and 16-QAM systems, respectively.

Fig. 5 XPM variance vs. residual dispersion per span (P_ave = 0 dBm).

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Figure 6 shows the overall XPM variance as a function of the number of WDM channels. As the number of channels increases beyond 5, the variance increase sublinearly, similar to the results obtained for the direct detection systems [1]. The discrepancies are within 5% and 8% for QPSK and 16-QAM systems, respectively.

Fig. 6 XPM variance vs. number of WDM channels (P_ave = 0 dBm).

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Figure 7 shows the overall XPM variance as a function of the number of fiber spans $N_{s}$ . For 16-QAM, the results for the case of ${| a_{0} |}^{2} = 5$ are plotted. Using a curve fitting to the simulation results, we find that the XPM variance for the QPSK system scales as $N_{s}^{1.99}$ , $N_{s}^{1.71}$ and $N_{s}^{1.56}$ when the residual dispersion per span is 0, 50 and 100 ps/nm, respectively. For the 16-QAM system, the XPM variance scales as $N_{s}^{1.96}$ , $N_{s}^{1.82}$ and $N_{s}^{1.72}$ when the residual dispersion per span is 0, 50 and 100 ps/nm, respectively. For a resonant dispersion map (i.e., Ψ_res = 0), the XPM field of every fiber span is identical and will coherently add up so that the XPM noise scales linearly with distance and hence, the variance increases quadratically with distance. As the residual dispersion increases, the degree of correlation between XPM noise occurring in different spans decreases and hence the total XPM variance scales linearly with distance for large |Ψ_res|. Typically, the XPM variance scales as $N_{s}^{r}$ where r∈ [1, 2] for different values of residual dispersion, similar to the intra-channel four wave mixing (IFWM) variance [20]. The discrepancies are within 8% and 10% for QPSK and 16-QAM systems, respectively.

Fig. 7 XPM variance vs. transmission distance (P_ave = 0 dBm).

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4. Conclusions

We have developed an analytical model for the XPM variance in dispersion-managed coherent fiber-optic systems. A first order perturbation technique is used to calculate the XPM distortion and then statistical analysis is carried out to calculate XPM variance for systems based on QAM. For arbitrary pulse shapes, the XPM variance cannot be calculated analytically, due to the difficulty in evaluating integrals explicitly. We have introduced a method of using the summation of time-shifted Gaussian pulses to fit arbitrary pulse shapes. Using the LSM to optimize the fitting parameters, a good approximation of the Nyquist pulse and the raised cosine pulse is achieved by using only a few time-shifted Gaussian pulses. As a result, analytical derivation of the XPM variance becomes possible for arbitrary pulse shapes. Moreover, the analytical model can be used to estimate the multiple levels of XPM variance in an M-QAM system rather than only one averaged variance level, which will permit more accurate BER estimation. The analytical XPM model has been validated by extensive numerical simulations of dispersion-managed coherent WDM fiber-optic systems. The discrepancies between the analytical XPM variances and the numerical results are within 10% in all the simulations.

Appendix A

The LSM and steepest descent algorithm are used for curve fitting. For a target function h(T), we fit it by a summation of time-shifted Gaussian functions

h^{'} (T) = \sum_{k = 1}^{K} ξ_{k} \exp [- \frac{{(T - μ_{k} T_{s})}^{2}}{2 {(θ_{k} T_{s})}^{2}}] .

The error function is

e (T) = h (T) - h^{'} (T) .

We discretize the time axis by

T = m Δ T,

where ΔT is the step size in the T axis; m is an integer. In order to find a good fitting, we need to minimize the cost function χ,

χ = \sum_{m = 1}^{M} e_{m}^{2} = \sum_{m = 1}^{M} {[h_{m} - {h^{'}}_{m}]}^{2},

where

e_{m} = e (m Δ T), h_{m} = h (m Δ T), {h^{'}}_{m} = h^{'} (m Δ T)

.

Substituting Eq. (54) into Eq. (57) and taking the derivatives of χ with respect to ξ_k, μ_k and θ_k respectively, we find

\frac{\partial χ}{\partial ξ_{k}} = - 2 \sum_{m} (h_{m} - {h^{'}}_{m}) \exp [- \frac{{(m Δ T - μ_{k} T_{s})}^{2}}{2 {(θ_{k} T_{s})}^{2}}],

\frac{\partial χ}{\partial μ_{k}} = - 2 \sum_{m} (h_{m} - {h^{'}}_{m}) \frac{ξ_{k} (m Δ T - μ_{k} T_{s})}{θ_{k}^{2} T_{s}} \exp [- \frac{{(m Δ T - μ_{k} T_{s})}^{2}}{2 {(θ_{k} T_{s})}^{2}}],

\frac{\partial χ}{\partial θ_{k}} = - 2 \sum_{m} (h_{m} - {h^{'}}_{m}) \frac{ξ_{k} {(m Δ T - μ_{k} T_{s})}^{2}}{θ_{k}^{3} T_{s}^{3}} \exp [- \frac{{(m Δ T - μ_{k} T_{s})}^{2}}{2 {(θ_{k} T_{s})}^{2}}] .

The steepest descent algorithm is used to optimize ξ_k, μ_k and θ_k. The parameters at the (i + 1)th iteration are updated from the ith iteration by

ξ_{k}^{(i + 1)} = ξ_{k}^{(i)} - \frac{\partial χ}{\partial ξ_{k}} Δ ξ, μ_{k}^{(i + 1)} = μ_{k}^{(i)} - \frac{\partial χ}{\partial μ_{k}} Δ u, θ_{k}^{(i + 1)} = θ_{k}^{(i)} - \frac{\partial χ}{\partial θ_{k}} Δ θ,

where Δξ, Δμ and Δθ are the step sizes. At the 0th iteration,

ξ_{k}^{(0)}

,

μ_{k}^{(0)}

and

θ_{k}^{(0)}

are chosen to be random numbers within the interval (0 1).

Appendix B

Consider two channels of a WDM system using a non-Gaussian pulse h(T) which can be approximated by h'(T) in Eq. (39). The input fields to the optical fiber are

u_{1} (0, T) = \sqrt{P} a_{0} h^{'} (T),

u_{2} (0, T) = \sqrt{P} \sum_{n = - N}^{N} b_{n} h^{'} (T - n T_{s}) \exp (- j Ω T),

The linear solution can be obtained by solving Eq. (11) in the frequency domain. Using the identity

\int_{- \infty}^{\infty} \exp (- a x^{2} - b x) d x = \sqrt{\frac{π}{a}} \exp (\frac{b^{2}}{4 a}),

we find the linear solution as

u_{1}^{(0)} (z, T) = \sqrt{P} a_{0} \sum_{k = 1}^{K} ξ_{k} \frac{θ_{k} T_{s}}{T_{1, k}} \exp [- \frac{{(T - μ_{k} T_{s})}^{2}}{2 T_{1, k}^{2}}],

u_{2}^{(0)} (z, T) = \sqrt{P} \sum_{n = - N}^{N} b_{n} \sum_{k = 1}^{K} ξ_{k} \frac{θ_{k} T_{s}}{T_{1, k}} \exp [- \frac{{(T - τ_{n, k})}^{2}}{2 T_{1, k}^{2}} - j Ω T + j θ (z)],

where

T_{1, k} = \sqrt{{(θ_{k} T_{s})}^{2} - j S (z)},

τ_{n, k} = (n + μ_{k}) T_{s} + S (z) Ω .

For first-order correction of u₁, the forcing function of Eq. (19) is

F (z, T) = 2 P^{3 / 2} a_{0} \sum_{m} \sum_{n} b_{m} b_{n}^{*} \sum_{k_{1} = 1}^{K} \sum_{k_{2} = 1}^{K} \sum_{k_{3} = 1}^{K} η (z) \exp [- \sum_{k^{'} = 1}^{3} {[T - C_{k^{'}} (z)]}^{2} R_{k^{'}} (z)],

where

η (z) = - e^{- w (z)} ξ_{k_{1}} ξ_{k_{2}} ξ_{k_{3}} \frac{θ_{k_{1}} θ_{k_{2}} θ_{k_{3}} T_{s}^{3}}{T_{1, k_{1}} T_{1, k_{2}}^{*} T_{1, k_{3}}},

C_{1} (z) = τ_{m, k_{1}}, C_{2} (z) = τ_{n, k_{2}}, C_{3} (z) = μ_{k_{3}} T_{s},

R_{1} (z) = \frac{1}{2 T_{1, k_{1}}^{2}}, R_{2} (z) = \frac{1}{2 {(T_{1, k_{2}}^{*})}^{2}}, R_{3} (z) = \frac{1}{2 T_{1, k_{3}}^{2}} .

Using Eq. (21), the first order correction for u₁ due to XPM is found as

u_{1}^{(1), X P M, N G} (L_{t o t}, T) = j 2 P^{3 / 2} a_{0} \sum_{m} \sum_{n} b_{m} b_{n}^{*} Y_{m n} (L_{t o t}, T),

where

Y_{m n} (L_{t o t}, T) = \int_{- \infty}^{\infty} V_{m n} (L_{t o t}, T^{'}) h_{R X} (T - T^{'}) d T^{'},

V_{m n} = \int_{0}^{L_{t o t}} \sum_{k_{1} = 1}^{K} \sum_{k_{2} = 1}^{K} \sum_{k_{3} = 1}^{K} \frac{- η^{'} (s)}{\sqrt{δ (L_{t o t}, s) R (s)}} \exp [\frac{{(D + j T)}^{2}}{δ (L_{t o t}, s)}] d s .

References and links

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k	1	2	3	4	5	6
ξ_k	0.5342	0.5342	−0.2588	−0.2588	0.0289	0.0289
μ_k	0.0042	−0.0042	0.9884	−0.9884	1.8314	−1.8314
θ_k	0.5909	0.5909	0.4932	0.4932	0.3781	0.3781

k	1	2	3	4	5	6
ξ_k	0.0350	0.0350	−0.0632	−0.0632	0.9106	0.9106
μ_k	0.0482	−0.0482	0.7283	−0.7283	0.2542	−0.2542
θ_k	0.0470	0.0470	0.0931	0.0931	0.2239	0.2239

k	1	2	3	4	5	6
ξ_k	0.5342	0.5342	−0.2588	−0.2588	0.0289	0.0289
μ_k	0.0042	−0.0042	0.9884	−0.9884	1.8314	−1.8314
θ_k	0.5909	0.5909	0.4932	0.4932	0.3781	0.3781

k	1	2	3	4	5	6
ξ_k	0.0350	0.0350	−0.0632	−0.0632	0.9106	0.9106
μ_k	0.0482	−0.0482	0.7283	−0.7283	0.2542	−0.2542
θ_k	0.0470	0.0470	0.0931	0.0931	0.2239	0.2239

Analytical modeling of XPM in dispersion-managed coherent fiber-optic systems

Abstract

1. Introduction

2. Analytical XPM model

2.1 Gaussian pulse

2.2 Non-Gaussian pulse

3. Results and discussion

4. Conclusions

Appendix A

Appendix B

References and links

Cited By

Figures (7)

Tables (2)

Equations (75)

Optics Express