Nonlinear phase noise in coherent optical OFDM transmission systems

Xianming Zhu; Shiva Kumar

doi:10.1364/OE.18.007347

1. Introduction

Coherent optical orthogonal frequency division multiplexing (OFDM) has drawn significant attention in optical communications due to its high spectral efficiency using hundreds of subcarriers with higher-order modulation formats and its robustness to fiber chromatic dispersion and polarization mode dispersion [1–5]. However, due to the large number of subcarriers, OFDM is believed to suffer from high peak-to-average power ratio, which makes it less suitable for legacy optical communication systems with periodic inline chromatic dispersion compensation fibers [6]. In Ref [7], a simple formula for estimating the deterministic distortions caused by four-wave mixing (FWM) is developed, and it is found that the nonlinear limit in OFDM systems is independent on the number of OFDM subcarriers in the absence of dispersion. Ref [8]. analytically studied the combined effect of dispersion and FWM in OFDM multi-span systems and concluded that dispersion could significantly reduce the amount of FWM. Recently, significant research effort has been put in nonlinear compensation for coherent OFDM systems [9–16]. Of particular interest is the digital backward propagation [14–16], a technique in which the signal is propagated backwards in distance using digital signal processing (DSP) so that the deterministic linear and nonlinear impairments can be compensated. However, the nonlinear phase noise caused by the interaction between amplified spontaneous emissions (ASE) noise and fiber Kerr nonlinearity, also known as Gordon-Mollenauer effects [17], cannot be compensated using digital backward propagation. Nonlinear phase noise has been studied extensively for single-carrier systems [17–32]; however, to the best of our knowledge, nonlinear phase noise effects have not been investigated for OFDM systems.

In this paper, we derive an analytical formula to calculate the nonlinear phase noise induced by the interaction of ASE with SPM, XPM and FWM in coherent OFDM transmission systems. The analytical model is verified with numerical simulation results, enabling the study of the nonlinear phase noise in coherent OFDM systems without lengthy simulations. With the analytical model, we quantitatively compare the nonlinear phase noise induced by SPM, XPM and FWM, separately, and find that the nonlinear phase noise induced by FWM is dominant compared to that induced by SPM and XPM. This is in contrast to the results of Ref [27]. for WDM systems, in which it is found that ASE-FWM interaction is negligible in quasi-linear systems. This difference is likely due to the sub-carriers of OFDM systems interacting coherently, since they are derived from the same laser source. We also study the effects of fiber chromatic dispersion and the bit rate on the total phase noise in coherent OFDM systems and find that, the total phase noise decreases as fiber chromatic dispersion increases, achieving the limit of linear phase noise. The total phase noise scales up as bit rate increases.

The remainder of the paper is organized as follows. Section 2 describes the mathematical analysis of the nonlinear phase noise. In section 3 we show the validation of the mathematical model with numerical simulation of coherent OFDM systems, and study the impact of fiber chromatic dispersion, number of subcarriers, and bit rate on the variance of the total phase noise. Section 4 gives the conclusion.

2. Mathematical analysis for the nonlinear phase noise in coherent OFDM systems

The nonlinear Schrödinger equation governing light propagation in optical fiber is [33]

j \frac{\partial u (t, z)}{\partial z} - \frac{β_{2} (z)}{2} \frac{\partial^{2} u (t, z)}{\partial t^{2}} + γ \exp [- w (z)] {| u (t, z) |}^{2} u (t, z) = 0,

where

β_{2} (z)

is the dispersion profile, γ is the nonlinear coefficient,

w (z) = \int_{0}^{z} α (s) d s

, and

α (z)

is the fiber loss/amplifier gain profile.

There are large numbers of subcarriers in OFDM systems, making each subcarrier a quasi-cw wave due to low bit rate. The OFDM signal can be described as [8]

u (t, z) = \sum_{l = - N / 2}^{N / 2 - 1} u_{l} (t, z) \exp (j ω_{l} t),

where N is the total number of subcarriers,

u_{l} (t, z)

is the slowly varying field envelope, and

ω_{l} = 2 π l / T_{b l o c k}

is the frequency offset from a reference, with

T_{b l o c k}

as the OFDM symbol time. First we derive the analytical formula for the variance of nonlinear phase noise including the interaction of ASE noise with SPM and XPM. Second, we include the nonlinear phase noise variance induced by FWM.

2.1 SPM and XPM induced nonlinear phase noise

Inserting Eq. (2) into (1) and considering the effects of SPM and XPM only, we have

\begin{array}{l} j (\frac{\partial u_{l}}{\partial z} - β_{2} ω_{l} \frac{\partial u_{l}}{\partial t}) - \frac{β_{2}}{2} ω_{l} \frac{\partial^{2} u_{l}}{\partial t^{2}} + \frac{β_{2}}{2} ω_{l}^{2} u_{l} = \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} - γ \exp [- w (z)] ({| u_{l} |}^{2} + 2 \sum_{m \neq l} {| u_{m} |}^{2}) u_{l} . \end{array}

Within each OFDM block, $u_{l}$ is constant; therefore, the first and second order derivative of $u_{l}$ with respect of time, appearing in Eq. (3), can be ignored. Now the exact solution of Eq. (3) can be written as

u_{l} (t, z) = u_{l} (t, 0) \exp [j \frac{β_{2}}{2} ω_{l}^{2} z + j γ L_{e f f} (z) ({| u_{l} |}^{2} + 2 \sum_{m \neq l} {| u_{m} |}^{2})],

where

L_{e f f} (z) = \frac{1 - \exp (- α z)}{α} .

We define $L_{s}$ as the fiber span length, and the signal is periodically amplified by amplifiers located at $κ L_{s}$ , κ = 1, 2, …, M, where M is the total number of fiber spans. Consider the noise added by the amplifier located at $κ L_{s}$ . Let us expand the noise field as a discrete Fourier transform

n (t) = \sum_{l = - N / 2}^{N / 2 - 1} n_{l} \exp (j ω_{l} t) .

Strictly speaking, the noise field should be expressed as a Fourier transform instead of a discrete Fourier transform. In other words, we have approximated the noise as a field with 2N degrees of freedom (DOF) instead of infinite degrees of freedom. For a linear system that employs matched filter at the receiver, 2N DOFs accurately describe the noise process [32]. In Ref [17]. it is argued that 2 DOFs per carrier (or total 2N DOFs) is sufficient to describe the noise field even in a nonlinear system. The total field immediately after the amplifier located at $κ L_{s}$ is

u^{+} (t, κ L_{s}) = \sum_{l = - N / 2}^{N / 2 - 1} (u_{l} (t, κ L_{s}) + n_{l}) \exp (j ω_{l} t) .

Let

\begin{array}{l} u_{l}^{+} (t, κ L_{s}) = u_{l} (t, κ L_{s}) + n_{l} \\ \begin{matrix}  \end{matrix} = [u_{l} (t, 0) + n_{l}'] \cdot \exp [j β_{2} ω_{l}^{2} κ L_{s} / 2 + j γ L_{e f f} (κ L_{s}) ({| u_{l} |}^{2} + 2 \sum_{m \neq l} {| u_{m} |}^{2})] . \end{array}

We assume that the ASE noise is a white noise process with power spectral density $ρ_{A S E}$ , from which it follows that,

\begin{array}{l} 〈 n_{l}' \cdot n_{k}'^{*} 〉 = 〈 n_{l} \cdot n_{k}^{*} 〉 = \frac{ρ_{A S E}}{T_{b l o c k}} δ_{l, k}, \\ 〈 n_{l}' \cdot n_{k}' 〉 = 0, \end{array}

where

δ_{l, k}

is the Kronecker delta function,

δ_{l, k} = {\begin{cases} 1 \begin{matrix} if \begin{matrix} l = k \end{matrix} \end{matrix} \\ 0 \begin{matrix} otherwise
. \end{matrix} \end{cases} .

Now treating $u_{l}^{+} (t, κ L_{s})$ as the initial field, Eq. (3) is solved to obtain the field at the end of the optical system, located at $z = M L_{s}$ , as

u_{l}^{+} (t, M L_{s}) = [u_{l} + n_{l}'] \exp {j Φ_{D} + j γ \cdot (M - κ) L_{e f f} (L_{s}) \cdot (u_{l} n_{l}'^{*} + u_{l}^{*} n_{l}')},

where

Φ_{D}

is the deterministic phase shift caused by dispersion, SPM and XPM, which has no impact on the nonlinear phase noise, and is expressed as

Φ_{D} = β_{2} ω_{l}^{2} M L_{s} / 2 + γ M L_{e f f} (L_{s}) ({| u_{l} |}^{2} + 2 \sum_{m \neq l} {| u_{m} |}^{2}) .

The linear phase noise is embedded in the term $u_{l} + n_{l}'$ , and the nonlinear phase noise caused by SPM and XPM by the amplifier located at $z = κ L_{s}$ is:

δ Φ_{S P M + X P M, κ} = γ \cdot (M - κ) L_{e f f} (L_{s}) \cdot (u_{l} n_{l}'^{*} + u_{l}^{*} n_{l}') .

Squaring Eq. (13) and making use of Eq. (9), we obtain the variance of the nonlinear phase noise caused by SPM and XPM

〈 δ Φ_{S P M + X P M, κ}^{2} 〉 = 2 γ^{2} \cdot {(M - κ)}^{2} L_{e f f}^{2} (L_{s}) ({| u_{l} |}^{2} + 2 \sum_{m \neq l} {| u_{m} |}^{2}) \frac{ρ_{A S E}}{T_{b l o c k}} .

Assuming that the number of subcarriers carrying data is $N_{e}$ (equivalently the oversampling factor is $N / N_{e}$ ) and each subcarrier has equal power, summing Eq. (14) over all amplifiers, we obtain the nonlinear phase noise variance caused by SPM and XPM

〈 δ Φ_{S P M + X P M}^{2} 〉 = \frac{1}{3} M (M - 1) (2 M - 1) \cdot γ^{2} L_{e f f}^{2} (L_{s}) (2 N_{e} - 1) P_{s c} \frac{ρ_{A S E}}{T_{b l o c k}},

where

P_{s c}

is the power per subcarrier. Equation (15) is our final equation for the nonlinear phase noise variance taking into account the interaction of ASE with SPM and XPM. The analytical model will be validated in the next section.

2.2 FWM induced nonlinear phase noise

Substituting Eq. (2) in Eq. (1), and considering only the FWM effect, we obtain the following equation with the quasi-cw assumption

\frac{\partial u_{l}}{\partial z} - j \frac{β_{2}}{2} ω_{l}^{2} u_{l} = j γ \exp [- w (z)] \cdot \sum_{\begin{array}{l} p + q - r = l \\ p \neq l, q \neq r \end{array}} u_{p} u_{q} u_{r}^{*} \exp {- j (ω_{p}^{2} + ω_{q}^{2} - ω_{r}^{2}) \frac{β_{2}}{2} z} .

The solution to Eq. (16) for the field $u_{l}$ is as follows

\begin{array}{l} u_{l} (M L_{s}) = u'_{l, z_{0}} + j \sum_{p + q - r = l} \int_{z_{0}}^{M L_{s}} γ e^{- α z'} u_{p, z_{0}} u_{p, z_{0}} u_{r, z_{0}}^{*} \exp [- j Δ β_{p, q, r, l} (z')] d z' \\ \begin{matrix}  \end{matrix} = u'_{l, z_{0}} + j \sum_{p + q - r = l} u_{p, z_{0}} u_{q, z_{0}} u_{r, z_{0}}^{*} Y_{p, q, r, l} (z_{0}, M L_{s}), \end{array}

where

u'_{l, z_{0}} = u_{l, z_{0}} \exp (- j \frac{β_{2}}{2} ω_{l}^{2} z_{0}),

with

u_{l, z_{0}} = u_{l} (z_{0})

.

Δ β_{p, q, r, l} (z)

is the phase-mismatch factor given by

Δ β_{p, q, r, l} (z) = (ω_{p}^{2} + ω_{q}^{2} - ω_{r}^{2} - ω_{l}^{2}) \frac{β_{2} z}{2} .

and

Y_{p, q, r, l} (z_{0}, M L_{s}) = \int_{z_{0}}^{M L_{s}} γ e^{- α z'} \exp [- j Δ β_{p, q, r, l} (z')] d z' .

To obtain Eq. (17), we have ignored the depletion of FWM pumps appearing on the right hand side (RHS) of Eq. (16), which is known as the un-depleted pump approximation [34]. We have also assumed that the chromatic dispersion has been completely compensated using digital backward propagation [14–16] at the end of the system.

Now consider the noise added by the amplifier located at $κ L_{s}$ . The optical field immediately after the amplifier is shown in Eq. (7). Equation (17) is solved using the initial condition of Eq. (7). Replacing $u_{l, z_{0}}$ in Eq. (17) with $u_{l}^{+} (κ L_{s})$ , we obtain the optical field at the end of the fiber span as

\begin{array}{l} u_{l}^{+} (M L_{s}, κ) = u_{l}^{+} (κ L_{s}) \exp (- j \frac{β_{2}}{2} ω_{l}^{2} κ L_{s}) + j \sum_{p + q - r = l} u_{p, κ}^{+} u_{q, κ}^{+} u_{r, κ}^{+ *} Y_{p, q, r, l} (κ L_{s}, M L_{s}) \\ \begin{matrix}  \end{matrix} = (u_{l, κ} + n_{l}) \exp (- j \frac{β_{2}}{2} ω_{l}^{2} κ L_{s}) + \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} j \sum_{p + q - r = l} (u_{p, κ}^{} + n_{p}) (u_{q, κ}^{} + n_{q}) (u_{r, κ}^{*} + n_{r}^{*}) Y_{p, q, r, l} (κ L_{s}, M L_{s}) . \end{array}

Ignoring the higher order term of $n_{l}$ , we have

\begin{array}{l} u_{l}^{+} (M L_{s}, κ) \approx (u_{l, κ} + n_{l}) \exp (- j \frac{β_{2}}{2} ω_{l}^{2} κ L_{s}) + \\ \begin{matrix}  \end{matrix} j \sum_{p + q - r = l} (u_{p, κ} u_{q, κ} u_{r, κ}^{*} + n_{p} u_{q, κ} u_{r, κ}^{*} + n_{q} u_{p, κ} u_{r, κ}^{*} + n_{r}^{*} u_{p, κ} u_{q, κ}) Y_{p, q, r, l} (κ L_{s}, M L_{s}) . \end{array}

From Eq. (22), we have

u_{l}^{+} (M L_{s}, κ) = u_{l, κ} \exp (- j \frac{β_{2}}{2} ω_{l}^{2} κ L_{s}) + u_{F W M, κ} + δ u_{l} (M L_{s}, κ),

where

u_{F W M, κ}

is the deterministic distortion caused by FWM, expressed as

u_{F W M, κ} = j \sum_{p + q - r = l} u_{p, κ} u_{q, κ} u_{r, κ}^{*} Y_{p, q, r, l} (κ L_{s}, M L_{s}) .

This distortion can be compensated with the digital backward propagation, and thus it has no impact on the nonlinear phase noise. The third term on the RHS of Eq. (23) $δ u_{l} (M, κ)$ describes the ASE-FWM interaction and can be expanded as

δ u_{l} (M L_{s}, κ) = n_{l} \exp (- j \frac{β_{2}}{2} ω_{l}^{2} κ L_{s}) + j \sum_{q = 1}^{N} (n_{q} A_{q} + n_{q}^{*} B_{q}),

where

A_{q} = 2 \sum_{p = - N / 2}^{N / 2 - 1} u_{p + l - q, κ} u_{p}^{*} Y_{q, p + l - q, p, l} (κ L_{s}, M L_{s}), \begin{matrix} p \neq q, l \neq p + l - q \end{matrix}

B_{q} = \sum_{p = - N / 2}^{N / 2 - 1} u_{q + l - p, κ} u_{p} Y_{q + l - p, p, q, l} (κ L_{s}, M L_{s}), \begin{matrix} p \neq q, l \neq q + l - p \end{matrix}

From Eq. (25), we have

〈 {| δ u_{l} |}^{2} 〉 = 〈 {| n_{l} |}^{2} 〉 + \sum_{q = 1}^{N} 〈 {| n_{q} |}^{2} 〉 ({| A_{q} |}^{2} + {| B_{q} |}^{2}),

〈 δ u_{l}^{2} 〉 = j 〈 {| n_{l} |}^{2} 〉 2 B_{l} \exp (- j \frac{β_{2}}{2} ω_{l}^{2} κ L_{s}) - \sum_{q = - N / 2}^{N / 2 - 1} 〈 {| n_{q} |}^{2} 〉 2 A_{q} B_{q} .

After the digital backward propagation removes the deterministic distortions, the phase noise of the received field is

δ Φ_{l, κ}^{} \approx \frac{Im (δ u_{l})}{| u_{l} |} = \frac{δ u_{l} - δ u_{l}^{*}}{2 j | u_{l} |},

where

Im (\cdot)

denotes the imaginary part. As

〈 δ Φ_{l, κ}^{} 〉 = 0

, we can calculate the variance of the phase noise as

〈 δ Φ_{l, κ}^{2} 〉 \approx 〈 - \frac{{(δ u_{l} - δ u_{l}^{*})}^{2}}{4 {| u_{l} |}^{2}} 〉 = \frac{2 〈 {| δ u_{l} |}^{2} 〉 - (〈 δ u_{l}^{2} 〉 + 〈 δ u_{l}^{* 2} 〉)}{4 {| u_{l} |}^{2}} .

Insert Eqs. (28) and (29) into (31) and use Eq. (9), we have:

〈 δ Φ_{l, κ}^{2} 〉 = \frac{ρ_{A S E}}{2 P_{s c} T_{b l o c k}} + \frac{ρ_{A S E}}{4 P_{s c} T_{b l o c k}} {2 \sum_{q = 1}^{N} {| A_{q} + B_{q} |}^{2} + 4 Im (B_{l} \exp (- j \frac{β_{2}}{2} ω_{l}^{2} κ L_{s}))} .

The first term on the RHS of Eq. (32) is the variance of the linear phase noise, and the second term on the RHS of Eq. (32) is the variance of the nonlinear phase noise caused by FWM. Summing Eq. (32) over all amplifiers in the fiber system, we obtain the phase noise variance caused by linear phase noise and FWM as follows:

〈 δ Φ_{l i n e a r, l}^{2} 〉 = \sum_{κ = 1}^{M} 〈 δ Φ_{l i n e a r, l, κ}^{2} 〉 = M \frac{ρ_{A S E}}{2 P_{s c} T_{b l o c k}} .

\begin{array}{l} 〈 δ Φ_{F W M, l}^{2} 〉 = \sum_{κ = 1}^{M} 〈 δ Φ_{F W M, l, κ}^{2} 〉 = \frac{ρ_{A S E}}{4 P_{s c} T_{b l o c k}} {2 \sum_{κ = 1}^{M} \sum_{q = 1}^{N} {| A_{q} + B_{q} |}^{2} \\ \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} + \sum_{κ = 1}^{M} 4 Im (B_{l} \exp (- j \frac{β_{2}}{2} ω_{l}^{2} κ L_{s}))} . \end{array}

The first term on the RHS of Eq. (34) is the nonlinear phase noise induced by FWM, and the second term on the RHS of Eq. (34) is the interaction between the linear and nonlinear phase noise.

2.3 Total phase noise

In summary, the total phase noise for the l ^th subcarrier in an OFDM system including the linear phase noise and nonlinear phase noise (induced by interaction between ASE and SPM, XPM and FWM) is as follows:

〈 δ Φ_{l}^{2} 〉 = 〈 δ Φ_{l i n e a r, l}^{2} 〉 + 〈 δ Φ_{S P M + X P M}^{2} 〉 + 〈 δ Φ_{F W M, l}^{2} 〉,

where the first, second, and third terms on the RHS of Eq. (35) are given by Eqs. (33), (15), and (34), respectively.

3. Results and discussions

In this section, we validate our analytical model for the variance of the total phase noise in OFDM systems given by Eq. (35) with numerical simulation. The following parameters are used throughout the paper unless otherwise specified: the bit rate is 10 Gb/s, the amplifier spacing is 100 km, and the noise figure (NF) is 6 dB. A single type of fiber is used between amplifiers. To separate the deterministic (although bit pattern dependent) distortions due to nonlinear effects from the ASE-induced nonlinear noise effects, we use digital backward propagation [14–16]. Since digital backward propagation compensates for both dispersion and deterministic nonlinear effects, we do not use the cyclic prefix. 2048 OFDM frames are used to get a good Monte Carlo statistics. Each OFDM subcarrier is modulated with binary-phase-shift-keying (BPSK) data. Figure 1 shows the coherent OFDM system structure in our simulation.

Fig. 1 Structure of coherent OFDM transmission systems

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For Figs. 2 -3 , we choose a fiber dispersion D of 1 ps/nm/km and a total launch power of 0 dBm. Here we use only one subcarrier (N_e = 1) to carry data while the total number of subcarriers is 8 (8th-folder oversampling), so that the nonlinear phase noise model that includes SPM effects alone can be validated. The subcarrier carrying data is located at the central of the OFDM spectrum, corresponding to the first subcarrier U ₀ in Fig. 1 due to FFT operation. The signal spectrum before entering into the fiber span is shown in Fig. 2. And in Fig. 3, the solid lines show the analytical linear phase noise and nonlinear phase noise variance induced by SPM only, the dashed line with circles show the numerical simulation results for the variance of linear phase noise and SPM induced nonlinear phase noise, as a function of fiber propagation distance. As can be seen, the agreement is quite good.

Fig. 2 OFDM signal spectrum before entering into fiber spans. Total number of subcarriers is 8, with one subcarrier carrying data.

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Fig. 3 Variance of the total phase noise as a function of propagation distance for SPM effect only. Total number of subcarrier is 8 with only one subcarrier carrying data.

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In order to validate the nonlinear phase noise model including the ASE interaction with SPM, XPM, and FWM effects in Eq. (35), we use 8 subcarriers to carry data for an OFDM system with 64 subcarriers. The subcarrier carrying data is located at the central of the OFDM spectrum, corresponding to the subcarriers U₀~U₃ and U₆₀~U₆₃ in Fig. 1. Figure 4 shows the OFDM signal spectrum, and Fig. 5 shows the variance of the linear phase noise and nonlinear phase noise for numerical simulation (dashed line with circles) and analytical calculation (solid line), respectively. We see that very good agreement is achieved, which validates our model for the nonlinear phase noise considering SPM, XPM, and FWM effects.In Ref [7], the authors showed that the nonlinear degradation due to FWM effects in OFDM systems is nearly independent of the number of ODFM subcarriers used in the system in the absence of chromatic dispersion, while in Ref [8]. the authors studied the chromatic dispersion effects on the FWM and showed that chromatic dispersion could decrease the FWM effects significantly. However, both of these analyses focused on the deterministic nonlinear effects. In this section, we will study the dependence of the nonlinear phase noise effects on fiber dispersion and bit rate in an OFDM system with digital backward propagation.

Fig. 4 OFDM signal spectrum before entering into fiber spans. Total number of subcarriers is 64, with 8 subcarriers carrying data.

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Fig. 5 Variance of the total phase noise as a function of propagation distance considering the ASE interaction with SPM, XPM and FWM effects. Total number of subcarriers is 64 with 8 subcarriers carrying data.

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In Figs. 6 -8 , we fix the transmission distance to be 1000 km, the total number of subcarriers is 128 with 64 subcarriers carrying data (two-fold oversampling). Figure 6 shows the OFDM signal spectrum. Figure 7 shows the variance of the total phase noise (linear + nonlinear phase noise) as a function of the launch power, for D = 17 ps/nm/km and D = 0 ps/nm/km. Solid lines and circles show the analytical results and the numerical simulation results, respectively. As can be seen from Fig. 7, the nonlinear tolerance increases significantly as the fiber chromatic dispersion parameter increases. It is also shown in Fig. 7 that the total variance of the phase noise initially decreases with launch power since the linear phase noise is dominant at low launch powers. However, as the launch power increases beyond −2 dBm, the variance increases with D = 0 ps/nm/km since nonlinear phase noise becomes dominant at higher powers.

Fig. 6 OFDM signal spectrum before entering into fiber spans. Total number of subcarriers is 128 with 2-folder oversampling (64 subcarriers carrying data).

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Fig. 8 Variance of the total phase noise as a function of bit rate in Gb/s. The total number of subcarriers is 128 with two-fold oversampling, total channel power is −3 dBm, and transmission distance is 1000 km.

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Fig. 7 Variance of the total phase noise as a function of channel power. The total number of subcarriers is 128 with two-fold oversampling, bit rate is 10 Gb/s, and transmission distance is 1000 km.

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In Fig. 8, we show the impact of the bit rate on the total phase noise for a transmission fiber with D = 17 ps/nm/km and D = 0 ps/nm/km. The total launch power is −3 dBm. Solid lines show the analytical results, while filled circles show the numerical simulation results. From Fig. 8, we see that the variance of the total phase noise scales linearly with the bit rate. This could be explained by the fact that, with the increase of the bit rate, the OFDM symbol time $T_{b l o c k}$ decreases, which leads to the increase of the total phase noise as described in Eqs. (15), (33) and (34). The qualitative explanation for the increase in phase noise when the bit rate increases is as follows: as the bit rate increases, OSNR requirement for a given BER increases. This is because the receiver filter bandwidth scales with bit rate, which leads to the increase of the total noise within the receiver bandwidth. The same thing happens with phase noise: the total phase noise within the receiver bandwidth increases as the bit rate increases.

For a BPSK system with coherent detection, from [35], one would obtain the bit error rate BER as a function of the phase noise variance $σ^{2}$ as

B E R = \frac{1}{2} e r f c (\frac{1}{\sqrt{2 σ^{2}}}),

where

e r f c (\cdot)

is the complementary error function [35]. From Eq. (36), we would get that, for a BER of

1 \times 10^{- 5}

, the required phase noise variance would be

5.5 \times 10^{- 2}

. However, this result would only be valid when linear phase noise is dominant. In the presence of nonlinear phase noise, it is hard to evaluate the BER without knowing the probability density function of the nonlinear phase noise, which would be a subject for future investigation.

In Fig. 9 , we show the impact of the number of subcarriers on the variance of total phase noise, obtained analytically using Eq. (35). Two-folder oversampling is used in the simulation. The total launch power is −3 dBm, the bit rate is 10 Gb/s. Figure 9 shows that, in the absence of dispersion, the variance of total phase noise scales linearly with the number of subcarriers, while with moderate levels of dispersion, the variance of total phase noise is almost constant because the linear phase noise is dominant for such systems.

Fig. 9 Variance of the total phase noise as a function of number of subcarriers, obtained analytically. Two-folder oversampling is used in the simulation. Bit rate is 10 Gb/s, total channel power is −3 dBm, and transmission distance is 1000 km.

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Finally Fig. 10 shows the variance of the nonlinear phase noise as a function of propagation distance for SPM induced nonlinear phase noise alone (dashed line), XPM induced nonlinear phase noise alone (dashed line with ‘x’), and FWM induced nonlinear phase noise alone for D = 0 ps/nm/km (solid line with circles), D = 10 ps/nm/km (solid line with triangles) and D = 17 ps/nm/km (solid line with diamonds), obtained analytically using Eqs. (15) and (34). From Fig. 10, we see that, for an OFDM system with large number of subcarriers, nonlinear phase noise induced by FWM is significantly larger than that induced by SPM and XPM. This is in contrast to the results of Ref [27]. for WDM systems, in which it is found that ASE-FWM interaction is negligible in quasi-linear systems. This difference is likely due to the fact that the subcarriers of OFDM system are derived from the same laser source and interact coherently. We also see that, with moderate levels of fiber chromatic dispersion, the nonlinear phase noise induced by FWM decreases since the phase matching becomes more difficult.

Fig. 10 Variance of the nonlinear phase noise due to separate effects of SPM, XPM, and FWM, as a function of propagation distance, obtained analytically. Total number of subcarriers is 128 with 2-folder oversampling. Bit rate is 10 Gb/s with −3 dBm launch power.

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4. Conclusions

In summary, we derive an analytical formula for the variance of the nonlinear phase noise in an OFDM system, taking into account the interaction of ASE noise with SPM, XPM and FWM effects. Our analytical results agree well with the numerical simulation. In addition, we quantitatively compared the nonlinear phase noise induced by SPM, XPM and FWM, and found that the nonlinear phase noise induced by FWM is the dominant nonlinear effect in OFDM systems with digital backward propagation. We also studied the effect of dispersion and bit rate on the nonlinear phase noise. Our results show that nonlinear phase noise can be suppressed with fiber chromatic dispersion, approaching the limit of linear phase noise for large dispersion values. Finally, we show that for OFDM systems, the variance of the total phase noise scales linearly with the channel bit rate.

References and links

1. W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–588 (2006). [CrossRef]

2. A. Lowery, L. Du, and J. Armstrong, “Performance of optical OFDM in ultra-long-haul WDM lightwave systems,” J. Lightwave Technol. 25(1), 131–138 (2007). [CrossRef]

3. J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009). [CrossRef]

4. A. Sano, E. Yamada, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K. Ishihara, and Y. Takatori, “No-guard-interval coherent optical ODFM for 100-Gb/s long-haul WDM transmission,” J. Lightwave Technol. 27(16), 3705–3713 (2009). [CrossRef]

5. S. Jasen, I. Morita, T. Schenk, and H. Tanaka, “121.9-Gb/s PDM-OFDM transmission with 2-b/s/Hz spectral efficiency over 1000 km of SSMF,” J. Lightwave Technol. 27(3), 177–188 (2009). [CrossRef]

6. Y. Yang, Y. Ma, and W. Shieh, “Performance impact of inline chromatic dispersion compensation for 107-Gb/s coherent optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1042–1044 (2009). [CrossRef]

7. A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15(20), 13282–13287 (2007). [CrossRef] [PubMed]

8. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

9. A. J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM,” Opt. Express 15(20), 12965–12970 (2007). [CrossRef] [PubMed]

10. L. B. Du and A. J. Lowery, “Improved nonlinearity precompensation for long-haul high-data-rate transmission using coherent optical OFDM,” Opt. Express 16(24), 19920–19925 (2008). [CrossRef] [PubMed]

11. X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express 16(26), 21944–21957 (2008). [CrossRef] [PubMed]

12. X. Liu, F. Buchali, and R. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. 27(16), 3632–3640 (2009). [CrossRef]

13. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008). [CrossRef] [PubMed]

14. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008). [CrossRef] [PubMed]

15. E. Ip and J. Kahn, “Compensation of dispersion and nonlinear effects using digital back-propagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]

16. E. Yamazaki, H. Masuda, A. Sano, T. Yoshimatsu, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K. Ishihara, M. Matsui, and Y. Takatori, “Multi-staged nonlinear compensation in coherent receiver for 16,340-km transmission of 111-Gb/s no-guard-interval co-OFDM,” ECOC2009, paper 9.4.6.

17. J. Gordon and L. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990). [CrossRef] [PubMed]

18. P. Winzer and R. Essiambre, “Advanced modulation formats for high capacity optical transport networks,” J. Lightwave Technol. 24(12), 4711–4728 (2006). [CrossRef]

19. A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol. 12(11), 1993–2000 (1994). [CrossRef]

20. K.-P. Ho, “Probability density of nonlinear phase noise,” J. Opt. Soc. Am. B 20(9), 1875–1879 (2003). [CrossRef]

21. K.-P. Ho, “Asymptotic probability density of nonlinear phase noise,” Opt. Lett. 28(15), 1350–1352 (2003). [CrossRef] [PubMed]

22. A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett. 29(7), 673–675 (2004). [CrossRef] [PubMed]

23. A. G. Green, P. P. Mitra, and L. G. Wegener, “Effect of chromatic dispersion on nonlinear phase noise,” Opt. Lett. 28(24), 2455–2457 (2003). [CrossRef] [PubMed]

24. S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett. 30(24), 3278–3280 (2005). [CrossRef]

25. C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. 27(21), 1887–1889 (2002). [CrossRef]

26. C. McKinstrie and C. Xie, “Phase jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. 8(3), 616–625 (2002). [CrossRef]

27. M. Hanna, D. Boivin, P. Lacourt, and J. Goedgebuer, “Calculation of optical phase jitter in dispersion-managed systems by the use of the moment method,” J. Opt. Soc. Am. B 21(1), 24–28 (2004). [CrossRef]

28. K.-P. Ho and H.-C. Wang, “Effect of dispersion on nonlinear phase noise,” Opt. Lett. 31(14), 2109–2111 (2006). [CrossRef] [PubMed]

29. K.-P. Ho, “Error probability of DPSK signals with cross-phase modulation induced nonlinear phase noise,” IEEE J. Sel. Top. Quantum Electron. 10(2), 421–427 (2004). [CrossRef]

30. X. Zhu, S. Kumar, and X. Li, “Analysis and comparison of impairments in differential phase-shift keying and on-off keying transmission systems based on the error probability,” Appl. Opt. 45(26), 6812–6822 (2006). [CrossRef] [PubMed]

31. A. Demir, “Nonlinear phase noise in optical fiber communication systems,” J. Lightwave Technol. 25(8), 2002–2032 (2007). [CrossRef]

32. S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. 27(21), 4722–4733 (2009). [CrossRef]

33. G. P. Agrawal, Nonlinear Fiber optics, New York: Academic, 3^rd Edition, 2001.

34. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. 17(11), 801–803 (1992). [CrossRef] [PubMed]

35. J. G. Proakis, Digital Communications, New York: McGraw-Hill, 4^th Edition, 2000, pp. 269–274.

Nonlinear phase noise in coherent optical OFDM transmission systems

Abstract

1. Introduction

2. Mathematical analysis for the nonlinear phase noise in coherent OFDM systems

2.1 SPM and XPM induced nonlinear phase noise

2.2 FWM induced nonlinear phase noise

2.3 Total phase noise

3. Results and discussions

4. Conclusions

References and links

Cited By

Figures (10)

Equations (36)

Optics Express