Optical back propagation for compensating nonlinear impairments in fiber optic links with ROADMs

Xiaojun Liang; Shiva Kumar

doi:10.1364/OE.24.022682

1. Introduction

One of the main limitations to increase the capacity and/or transmission reach of the fiber optic system is caused by the interplay between the chromatic dispersion (CD) and nonlinearity in optical fibers [1, 2]. Number of techniques have been proposed to mitigate these nonlinear impairments in optical [3–11] and digital domains [12–22]. Midpoint optical phase conjugation (OPC) provides effective mitigation of dispersion and nonlinear impairments only if the loss, nonlinearity and dispersion profiles are symmetric with respect to the location of OPC, which is hard to realize in practice. This is mainly because in practical systems, amplifier spacing is random ranging from 50 km to 125 km.

Digital back propagation (DBP) has been found to be effective in mitigating intra-channel nonlinear impairments [12,13]. Typically the step size of DBP is of the order of amplifier spacing or larger [12–15]. In principle, DBP could compensate for inter-channel nonlinear impairments of a point-to-point WDM system if the step size is less than 3 km [14], which significantly increases the computational cost. Assuming that future digital signal processing (DSP) could handle this kind of computational complexity, DBP could be useful for point-to-point WDM systems. However, it cannot be used for compensating the nonlinear impairments in a fiber optic communication system with reconfigurable optical add-drop multiplexers (ROADMs). This fact can be illustrated by the following example. Suppose channel 1 and channel 2 of a WDM system are co-propagating between node 1 and node 2 of a fiber optic network, as shown in Fig. 1. Channel 2 interacts with channel 1 and induces nonlinear noise on channel 1. At node 2, channel 2 is dropped. The receiver for channel 1 at node 3 has no access to channel 2 and hence, it would be impossible for the DBP at the node 3 to compensate for the nonlinear impairment caused by channel 2 on channel 1.

Fig. 1 Schematic of a fiber optic network with ROADMs.

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Optical back propagation (OBP) compensates for chromatic dispersion and nonlinearity by introducing real photonic devices inline or at the receiver instead of virtual fibers of DBP [6,7,9,11]. In [11], it was proposed to use OPC followed by dispersion-decreasing fiber (DDF) to compensate for dispersion and nonlinearity and such an OBP scheme showed significant performance improvement for the point-to-point WDM system. However, the DDF and OPC were considered to be ideal. In this paper, we extend this idea to include fiber optic networks and investigate the impact of non-idealities of DDF and OPC.

Midpoint OPC relies on the fiber optic link following the OPC to compensate for the dispersive and nonlinear impairments due to the fiber optic link preceding OPC. In contrast, in our scheme, we introduce dedicated photonic components to compensate for the CD and nonlinearity of the transmission fiber. The idea is similar to the insertion of dispersion-compensating fiber (DCF) in dispersion-managed systems which compensates for CD. However, in our case, DDF compensates for both CD and nonlinearity of the fiber simultaneously.

2. Theory of optical back propagation

Consider a simple case with a single fiber span as shown in Fig. 2(a). Signal propagation in the transmission fiber is described by the nonlinear Schödinger equation (NLSE):

\frac{\partial q}{\partial z} = i [D (t) + N (t, z)] q (t, z),

D (t) = - \frac{β_{2}}{2} \frac{\partial^{2}}{\partial t^{2}}, N (t, z) = γ {| q (t, z) |}^{2} + i \frac{α}{2},

where q is the optical field envelope, β₂, γ and α are the dispersion, nonlinear and loss coefficients of the transmission fiber, respectively. The output signal is

q (t, L_{a}) = \exp {i \int_{0}^{L a} [D (t) + N (t, z)] d z} q (t, 0),

where L_a is the length of the transmission fiber. After an optical phase conjugator (OPC), the signal becomes

q^{*} (t, L_{a}) = \exp {- i \int_{0}^{L a} [D (t) + N^{*} (t, z)] d z} q^{*} (t, 0) .

Fig. 2 Schematics of back propagation based on (a) a virtual fiber and (b) a DDF. Tx: transmitter, Rx: receiver, TF: transmission fiber, OPC: optical phase conjugator, DDF: dispersion-decreasing fiber.

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Suppose the output of OPC passes through a virtual fiber which is identical to the transmission fiber except that the loss coefficient is −α, i.e. dispersion and nonlinearity operators for the virtual fibers are

D_{v} (t) = - β_{2} \frac{\partial^{2}}{\partial t}, N_{v} (t) = γ {| q (t, z) |}^{2} - i \frac{α}{2},

respectively. The output of the virtual fiber is

q_{out} (t) = \exp {i \int_{0}^{L a} [D_{v} (t) + N_{v} (t, z)] d z} q^{*} (t, L_{a}) .

Since D_v (t) = D(t) and N_v (t, z) = N^∗(t, z), using Eq. (4) in Eq. (6), we find

q_{out} (t) = q^{*} (t, 0) .

Equation (7) shows that the input field can be recovered using an OPC followed by a virtual fiber, except for a conjugation operation.

A negative loss coefficient of the virtual fiber means a gain coefficient, which is difficult to realize in practice. In digital back propagation (DBP), a virtual fiber is implemented in the digital domain by converting the optical signal into an electrical signal. To realize back propagation in the optical domain, a dispersion-decreasing fiber (DDF) can function as the virtual fiber as shown in Fig. 2(b) [11]. In contrast to the virtual fiber, the DDF has a positive loss coefficient and a dispersion profile that decreases with distance:

β_{2, d} (z) = \frac{e^{- α_{d} z}}{\frac{γ e^{- α L_{a}}}{γ_{d} G} + α (\frac{1 - e^{- α_{d} z}}{α_{d}})} β_{2},

where α_d and γ_d are the loss and nonlinear coefficients of the DDF, respectively. G denotes the gain of a pre-amplifier. The desired DDF length for exact compensation of nonlinear effects in the transmission fiber is [11]

L_{d} = - \frac{1}{α_{d}} l n [1 - \frac{α_{d} γ e^{- α L_{a}}}{γ_{d} G α} (e^{α L_{a}} - 1)] .

As shown in Fig. 2(b), an OBP module that fully compensates the propagation impairments of a transmission fiber consists of an OPC, a pre-amplifier (G is usually small), a DDF and an amplifier to compensate for the DDF loss. In a fiber optic network, this OBP module can be placed after each transmission fiber (referred to as inline OBP) or at each optical network node (referred to as node OBP). Both configurations are applicable to compensating for propagation impairments in WDM optical communication systems where ROADMs present, since the propagation impairments are compensated in the optical domain before a ROADM modifies a WDM signal. However, for WDM systems with ROADMs, conventional receiver-side compensation techniques, such as DBP, cannot mitigate inter-channel nonlinear impairments due to the lack of full propagation path information.

The DDF may be interpreted as a matched nonlinear filter that provides the exact compensation of CD and nonlinear effects. DDFs have been used for soliton compression [23] and reduction of timing jitter due to soliton collision [24]. DDFs can be made by tapering the fiber cross-section during the drawing process, which changes the waveguide dispersion [25]. Changing the cross-section of the fiber alters the effective core area of the fiber, which changes the nonlinear coefficient. However, the dependence of the effective core area on the dimensions of the cross-section is much weaker than that of group velocity dispersion [25]. Fabrication of a fiber with a predetermined continuous dispersion profile requires special facilities and exacting manufacturing conditions. An alternative approach is to replace a continuous dispersion profile by a step-like dispersion profile, which relieves fabrication difficulties [26].

With the proper choice of pre-amplifier gain G, the length of the DDF can be made sufficiently short so that the loss introduced by the DDF is small. The initial dispersion and length of the DDF is chosen to be roughly the same as that of the DCF used in dispersion managed systems. Only difference is that for DCF, the dispersion is constant whereas for the DDF it decreases with distance.

3. Modeling of optical phase conjugation

Optical phase conjugation (OPC) has been intensively investigated for optical signal processing [3–11]. A good review of OPC techniques and applications can be found in [27]. OPC can be realized by four-wave mixing (FWM), stimulated Brillouin scattering (SBS) or stimulated Raman scattering (SRS). Fig. 3 shows a schematic diagram of an OPC based on the FWM effects in a highly nonlinear fiber (HNLF). One or two laser fields can be used as pump inputs to the HNLF.

Fig. 3 (a) OPC based on a HNLF, (b) OPC with one laser pump, (c) OPC with two laser pumps, (d) OBP module for multi-channel systems. MUX: multiplexer, HNLF: highly nonlinear fiber, BPF: band pass filter, DeMUX: demultiplexer, OPC: optical phase conjugator, DDF: dispersion-decreasing fiber. ω₀ is the carrier frequency of the optical signal.

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3.1. One-pump OPC

HNLF is a dispersion-shifted fiber with zero dispersion at 1550 nm. Around this wavelength, we can ignore fiber dispersion. The total input field to the HNLF can be written as

q = q_{s} + q_{p} e^{i Ω_{p} t},

where q_s and q_p are the envelopes of the signal field and laser pump, respectively. Here, we take the optical signal carrier frequency ω₀ as reference and Ω_p denotes the relative frequency of the pump with respect to signal. Consider the nonlinear term in NLSE,

i γ {| q |}^{2} q = i γ {{| q_{s} |}^{2} q_{s} + 2 {| q_{p} |}^{2} q_{s} + q_{s}^{2} q_{p}^{*} e^{- i Ω_{p} t} + 2 {| q_{s} |}^{2} q_{p} e^{i Ω_{p} t} + {| q_{p} |}^{2} q_{p} e^{i Ω_{p} t} + q_{p}^{2} q_{s}^{*} e^{i 2 Ω_{p} t}},

which generates a conjugated signal

q_{s}^{*}

at the frequency of 2Ω_p. As shown in Fig. 3(a), a band pass filter (BPF) is used to obtain the conjugated signal at the specific frequency band. Assuming the HNLF dispersion to be zero (β₂ = 0), the NLSE simplifies as

\frac{\partial q}{\partial z} + \frac{α}{2} q = i γ {| q |}^{2} q .

The generated conjugation field (q_c) at the frequency of 2Ω_p evolves as

\frac{\partial q_{c}}{\partial z} + \frac{α}{2} q_{c} = i γ q_{p}^{2} q_{s}^{*} .

Let

q_{p} = A_{p} e^{- α z / 2}, A_{p} = \sqrt{P_{p}},

where P_p is the pump power. Also let

q_{s} (t, z) = A_{s} (t, z) e^{- α z / 2} .

Substituting Eqs. (14) and (15) into Eq. (13) and solving for q_c with the initial condition of q_c (t, 0) = 0, we find the conjugated signal at the output of the HNLF as

q_{c} (t, L) = i γ P_{p} A_{s}^{*} L_{e f f} e^{- α L / 2},

where L is the HNLF length,

L_{e f f} = \int_{0}^{L} e^{- α z / 2} d z

is the effective length. The output power of the conjugated signal is

P_{out} \equiv {| q_{c} (t, L) |}^{2} = {(γ P_{p} L_{e f f})}^{2} e^{- α L} P_{in},

where

P_{in} = {| A_{s} |}^{2}

is the input signal power. The power loss is defined as

{loss}_{1} = \frac{P_{out}}{P_{in}} = {(γ P_{p} L_{e f f})}^{2} e^{- α L} .

Here, the subscript “1” indicates that there is a single pump.

The central frequency of the conjugated signal is shifted by 2Ω_p, which is undesirable for fiber optic network application since the conjugated signal may interfere with the other channels of a WDM system. In [28], two OPC systems are placed in cascade so that the central frequency of the conjugated signal is the same as that of the input signal. This technique, although effective, lowers the output power of the conjugated signal and hence, optical signal-to-noise ratio (OSNR) of the transmission system becomes smaller as compared to the system with single stage OPC. In this paper, we investigated the possibility of two OPC systems in parallel, as shown in Fig. 3(d). For example, suppose the WDM spectrum ranges from 1540 nm to 1560 nm and the OPC pump wavelength is 1550 nm. The WDM spectrum is demultiplexed into two bands [see Fig. 3(d)] – left band 1540 nm to 1550 nm and the right band 1550 nm to 1560 nm. The left (right) band passes through the upper (lower) branch and the conjugated signals swap the wavelength bands, i.e. the conjugated signal in the upper branch extends from 1550 nm to 1560 nm. The band pass filter (BPF) removes the pump and signal bands, and the conjugated signal in the upper and lower branches are multiplexed together to get a full conjugated signal. Although the channel numbers may be shifted by a constant, the network software can easily keep track of the channel number.

3.2. Two-pump OPC

For the two-pump case shown in Fig. 3(c), the total input field to the HNLF is

q = q_{s} + q_{p 1} e^{i Ω_{1} t} + q_{p 2} e^{i Ω_{2} t} .

Due to FWM, a conjugated signal will be generated at the frequency of Ω₁ + Ω₂. Following a similar procedure in Section 3.1, we find the conjugated field as

q_{c} (t, L) = 2 i γ A_{p 1} A_{p 2} A_{s}^{*} L_{e f f} e^{- α L / 2},

where

A_{p 1} = \sqrt{P_{p 1},} A_{p 2} = \sqrt{P_{p 2}}

,

P_{p 1}

and

P_{p 2}

are the pump powers. The power loss is found as

{loss}_{2} = 4 P_{p 1} P_{p 2} {(γ L_{e f f})}^{2} e^{- α L} .

Comparing Eqs. (18) and (21), we find that the output power is four times larger for the case of two pumps when $P_{p}^{2} = P_{p 1} P_{p 2}$ .

3.3. Pump noise

The non-ideal effects, such as phase noise and relative intensity noise (RIN) of a pump laser can be modeled as

A_{p} = (A_{p 0} + δ A_{p 0}) e^{i θ_{p}},

where A_p is the complex envelope of a pump field, A_p₀ is the mean field amplitude, δA_p₀ is a zero-mean Gaussian random variable that accounts for the RIN noise, θ_p is a phase noise. The pump field of Eq. (22) is plugged into Eqs. (16) and (20) to account for laser non-ideal effects.

4. Results and discussions

We investigate a WDM fiber optic system with ROADMs, as shown in Fig. 4. A 7-channel WDM signal is generated at the transmitter, with a channel spacing of 50 GHz. The WDM spectrum extends from 193.37 THz to 193.72 THz. The symbol rate per channel is 28 Gbaud and the modulation format is 32-QAM. A raised cosine pulse with a roll-off factor of 0.6 is used. The split-step Fourier scheme (SSFS) is used for the simulation of fiber propagation, with a simulation bandwidth of 600 GHz. In simulations, 32768 symbols per channel are transmitted. Between each two neighboring network nodes there are 4 spans of 60 km long transmission fiber. The parameters of the transmission fiber are as follows: α_TF = 0.2dB/km, β₂,_TF = 5ps²/km, β_3,TF = 0.05ps³/km, γ_TF = 2.2W⁻¹km⁻¹. The total transmission distance is 2400 km. A ROADM at each network node will randomly drop one WDM channel and add new signal to the same channel, except for the central channel. In this paper, we like to study the impact of inter-channel nonlinearity, such as cross phase modulation (XPM) and FWM on the central channel. At the receiver, the accumulated phase noises are compensated using a feedforward method in the digital domain [29]. Accumulated third order dispersion is also compensated in the digital domain. The bit error rate (BER) of the center channel is calculated at the receiver and Q-factor is obtained using $Q = \sqrt{2} e r f c^{- 1} (2 \times B E R)$ . For the DBP scheme, the received signal is downsampled to 2 samples/symbol and the back propagation step size is 6 km. In the inline OBP case as shown in Fig. 4(a), an OBP module is placed after each transmission fiber. In the node OBP case as shown in Fig. 4(b), an OBP module is placed at each network node. The parameters of DDF are α_d = 0.4dB/km, β₃,_d = −0.07ps³/km, γd = 4.86W⁻¹km⁻¹. The DDF dispersion profile β₂,_d(z) is calculated using Eqs. (8) and (9) with G = 1.26. The DDF length is 12.1 km. The HNLF in OPC has the following parameters: α_HNLF = 0.97dB/km, γ_HNLF = 11.5W⁻¹km⁻¹, L = 1km. The laser pumps in OPC have an average power of 15 dBm, a linewidth of 15 kHz, and a RIN of −155 dB/Hz.

Fig. 4 Fiber-optic mesh networks using OBP to compensate for signal propagation impairments. (a) Inline OBP, (b) Node OBP. Tx: transmitter, Rx: receiver, TF: transmission fiber, OBP: optical back propagation, ROADM: reconfigurable optical add-drop multiplexer.

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The DDF dispersion profile calculated by Eq. (8) is an ideal profile that will provide exact compensation of propagation impairments of a transmission fiber. However, non-ideal effects may occur during practical fabrications resulting in dispersion profile fluctuations. To study the impact of dispersion profile fluctuation, we introduce a perturbed profile as

{\hat{β}}_{2, d} (z) = [1 + x (z)] β_{2, d} (z),

where β_2,d (z) is the ideal profile of Eq. (8)x(z) are zero-mean Gaussian random variables with a standard deviation of σ_DDF. Fig. 5 shows the perturbed dispersion profiles with different values of σ_DDF. Fig. 6 shows the Q-factor of the central channel as a function of dispersion profile fluctuation. The Q-factor penalty is only 0.4 dB when σ_DDF increases from 0.01 to 0.6.

Fig. 5 DDF dispersion profile fluctuations. The dispersion profile is modeled as: ${\hat{β}}_{2, d} (z) = [1 + x (z)] β_{2, d} (z)$ , where β₂,_d (z) is a desired dispersion profile (ideal case), x(z) are zero-mean Gaussian random variables with a standard deviation of σ_DDF.

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Fig. 6 Q-factor vs. DDF dispersion profile fluctuation. Inline OBP using one laser pump, transmission distance = 2400 km, launch power per WDM channel = −2 dBm.

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In Fig. 7, we compare the performance of inline OBP and node OBP. In both cases, we also show the performance of ideal OPC and DDF case, which ignores dispersion profile fluctuations as well as OPC insertion loss, pump laser phase noise and RIN. The performances of DBP and linear compensation are also shown. As compared with linear compensation, DBP provides only 1.3 dB Q-factor gain, since only intra-channel nonlinear impairments are compensated. For ideal cases, inline OBP and node OBP have 8.7 dB and 8.1 dB Q-factor gains as compared to the case of linear compensation, respectively. When the non-idealities are ignored, inline OBP outperforms node OBP, which is due to the fact that inline OBP compensates for nonlinear phase noise arising from the interaction of fiber nonlinearity and amplified spontaneous emission (ASE) noise (so called Gordon-Mollenauer phase noise [30]) better than the node OBP. Practical cases include OPC and DDF non-ideal effects, with σ_DDF = 0.6. When the noise due to OPC and DDF fluctuations are included, node OBP performs slightly better than inline OBP. This is because the number of OPC for the case of inline OBP is four times larger than that of the node OBP in our example, which lowers the OSNR of the inline OBP. For inline OBP, OPCs with 1 pump and 2 pumps bring 3.5 dB and 4.9 dB Q-factor gains, respectively. For node OBP, OPCs with 1 pump and 2 pumps bring 4.3 dB and 5.6 dB Q-factor gains, respectively. Two-pump OPC case outperforms one-pump OPC case due to the smaller power loss (see Eqs. (18) and (21)) and lower laser phase noise. Also, it is interesting to notice that for practical cases, although node OBP provides higher maximum value of Q-factor than inline OBP does, the optimal launch power per channel of inline OBP is 2 dB higher than that of node OBP.

Fig. 7 Q-factor vs. launch power per WDM channel. transmission distance = 2400 km, σ_DDF = 0.6 for non-ideal DDF cases.

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5. Conclusions

In WDM systems where ROADMs present, signal channels may be added or dropped by a ROADM at an optical network node. As a result, a receiver node has no information of the signal channels that are dropped at previous network nodes. Due to the lack of full propagation path information, receiver-side compensation techniques, such as digital back propagation (DBP), are unable to mitigate inter-channel nonlinear impairments. In contrast, optical back propagation (OBP) can compensate for both intra- and inter-channel nonlinear impairments in the optical domain before a ROADM modifies a WDM signal. We have studied a WDM system with 2400 km transmission distance and 32-QAM format and showed that DBP only brings 1.3 dB Q-factor gain as compared with linear compensation. The OBP compensation module can be implemented after each transmission fiber (inline OBP case) or at each network node (node OBP case). We showed that inline OBP brings 3.5 dB and 4.9 dB Q-factor gains for the cases of one and two laser pumps in OBP module, respectively, as compared to the case of linear compensation. Node OBP brings 4.3 dB and 5.6 dB Q-factor gains for the cases of one and two laser pumps in OBP module, respectively. We have included non-ideal effects of the OBP module, such as pump laser phase noise and RIN. We have also showed that the performance of OBP has good tolerance to the fluctuation of DDF dispersion profile that may occur during practical fabrication process.

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Optical back propagation for compensating nonlinear impairments in fiber optic links with ROADMs

Abstract

1. Introduction

2. Theory of optical back propagation

3. Modeling of optical phase conjugation

3.1. One-pump OPC

3.2. Two-pump OPC

3.3. Pump noise

4. Results and discussions

5. Conclusions

References and links

Cited By

Figures (7)

Equations (23)

Optics Express