Optical back propagation for fiber optic networks with hybrid EDFA Raman amplification

Xiaojun Liang; Shiva Kumar

doi:10.1364/OE.25.005031

1. Introduction

Signal propagation impairments in fiber optic links due to the interplay between fiber dispersion and nonlinear effects are one of the main limitations to increase system capacity. Signal-to-signal [1–6] and signal-to-noise [7–9] nonlinear interactions generate distortions that significantly degrade signal quality. Compensating nonlinear distortions is a challenging problem, especially in wavelength division multiplexing (WDM) network scenarios. Number of techniques have been proposed to mitigate nonlinear impairments both in digital [10–23] and optical domain [24-35]. Digital back propagation (DBP) is a widely investigated method for nonlinear compensation [12, 13]. Signal propagation in a fiber optic link is described by the nonlinear Schödinger equation (NLSE), which deterministically calculates nonlinear distortions. The NLSE is an invertible equation, indicating that the initial input signal field to a fiber can be recovered if the output signal is propagated backward along the same fiber. In DBP, this backward propagation is implemented in the digital domain to mitigate propagation impairments. DBP provides significant transmission performance improvement in single-channel point-to-point systems. However, DBP is usually limited to compensate for intra-channel nonlinear distortions only, due to the limitations on computational resources and the receiver bandwidth. Moreover, in practical fiber optic network scenarios, network nodes along the fiber link may use reconfigurable optical add-drop multiplexers (ROADMs) to drop some WDM channels and/or add new signal channels, and receiver-side DBP is not able to compensate for inter-channel nonlinear distortions due to the lack of propagation path information of the WDM signal modified by ROADMs before the receiver [34, 35].

Alternatively, nonlinear impairments may be compensated for in the optical domain. One method is to use midpoint optical phase conjugation (OPC), where an OPC is placed at the middle of a fiber optic link [24–27]. In fiber optic systems with lumped Erbium doped fiber amplifiers (EDFAs), midpoint OPC provides moderate nonlinear compensation performance, due to the fact that the signal power profile is not symmetric with respect to the OPC. In fiber optic systems with distributed Raman amplification, a much better symmetry of power profile with respect to the OPC can be obtained and midpoint OPC provides significant transmission performance improvement in point-to-point systems [27]. However, in fiber optic networks where network nodes have a mesh configuration, it is difficult to define the middle point of the link. One possible solution is to put an OPC at each network node. Even in this case, due to the add/drop of channels, the optical signal propagating after the OPC is not the same as before, which breaks down the signal-symmetry with respect to the location of OPC. As will be shown in this paper, this scheme has slightly better performance than single-channel DBP, since only a fraction of the inter-channel nonlinear distortions is compensated.

Another optical nonlinear compensation technique is optical back propagation (OBP). OBP compensates for chromatic and nonlinear effects by employing real photonic devices inline or at the receiver instead of virtual fibers of DBP. In [33], an OBP scheme is proposed to use OPC followed by dispersion decreasing fibers (DDFs) to compensate for propagation impairments and showed significant performance improvement for point-to-point WDM systems. In [35], the OBP module is applied after each transmission fiber (inline OBP) or at each network node (node OBP) in a practical network scenario with ROADMs. In this paper, we extend this idea to fiber optic systems with hybrid EDFA/Raman amplification. We derive a semi-analytical expression to calculate the dispersion profile of dispersion varying fiber (DVFs) in the OBP module. We use numerical simulations to show that the OBP brings significant transmission performance improvement as compared with the DBP in practical fiber optic network scenarios.

2. Theory of optical back propagation

We consider signal propagation in a single optical fiber span as shown in Fig. 1(a). Signal propagation is described by the nonlinear Schödinger equation (NLSE) [1]:

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

where q(t, z) is the optical field envelope, α(z), β₂, and γ are the loss/gain, dispersion, and nonlinear coefficients of the transmission fiber, respectively. For fiber optic systems with lumped amplifiers, α(z) represents a constant loss coefficient of the transmission fiber. For fiber optic systems with distributed Raman amplification, α(z) represents the combined effect of fiber loss and distributed amplification gain, which is dependent on distance. Dispersion and nonlinear operators can be defined as:

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

Fig. 1 Schematics of back propagation based on (a) a virtual fiber and (b) a DVF. Tx: transmitter, Rx: receiver, TF: transmission fiber, OPC: optical phase conjugator, DVF: dispersion varying fiber.

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The output signal of the transmission fiber is

q (t, L) = M q (t, 0),

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

where L is the length of the transmission fiber and M is the propagation operator. To compensate for signal distortions due to fiber dispersion and nonlinear effects, a back propagation module is placed after the transmission fiber, consisting of an OPC and a virtual fiber, as shown in Fig. 1(a). If the back propagation module realizes an inverse transfer function of the fiber optic link, we recover the initial signal field envelope at the output if amplifier noise is ignored.

q_{o u t} (t) = M^{- 1} q (t, L) = q (t, 0) .

From Eq. (4), it follows that

M^{- 1} = \exp {- i \int_{0}^{L} [D (t) + N (t, L - z)] d z} .

Taking the complex conjugation of Eq. (5), we obtain

q_{o u t}^{*} (t) = M^{'} q^{*} (t, L) = q^{*} (t, 0),

where

M^{'} = \exp {i \int_{0}^{L} [D (t) + N^{*} (t, L - z)] d z} .

Comparing the transfer function of a transmission fiber in Eq. (4) and that of a virtual fiber in Eq. (8), we find that the desired virtual fiber parameters are as follows.

α^{'} (z) = - α (L - z), β_{2}^{'} = β_{2}, γ^{'} = γ .

The first equation in Eq. (9) indicates the requirement of power profile symmetry with respect to the location of the OPC in order to obtain exact dispersion and nonlinear distortion compensation. It is challenging to realize such a power profile symmetry in fiber systems, so we refer to it as a virtual fiber. Equation (7) is equivalent to solving the NLSE,

\frac{\partial q_{b}}{\partial z_{b}} + \frac{α^{'} (z_{b})}{2} q_{b} + i \frac{β_{2}^{'}}{2} \frac{\partial^{2} q_{b}}{\partial t^{2}} = i γ^{'} {| q_{b} |}^{2} q_{b},

with q_b (t, 0) = q^*(t, L), and z_b is the distance in the virtual fiber. Let

q_{b} (t, z_{b}) = \sqrt{P_{b}} e^{- W^{'} (z_{b}) / 2} u_{b},

and

d z_{b}^{'} = β_{2} d z_{b},

where P_b = P_ine⁻^W ⁽^L⁾ is the input power of the virtual fiber.

W (z) = \int_{0}^{z} α (z) d z

and

W^{'} (z) = \int_{0}^{z} α^{'} (z) d z

. Now, Eq. (10) is written as

\frac{\partial u_{b}}{\partial z_{b}^{'}} + \frac{i}{2} \frac{\partial^{2} u_{b}}{\partial t^{2}} = i \frac{γ P_{b} e^{- W^{'} (z b)}}{β_{2}} {| u_{b} |}^{2} u_{b},

Alternatively, the functionality of the back propagation model in Fig. 1(a) can be realized using an OBP module consisting of OPC, a dispersion varying fiber (DVF) and optical amplifiers, as shown in Fig. 1(b). In contrast to the virtual fiber, the DVF has a positive loss coefficient and a distance-dependent dispersion profile. The pre-amplifier with gain G′ may be employed to adjust the dispersion profile of DVF. The loss of DVF is compensated by another amplifier with gain G_d. Signal evolution in the DVF is given as

\frac{\partial q_{b}}{\partial z_{d}} + \frac{α_{d}}{2} q_{b} + i \frac{β_{2, d} (z_{d})}{2} \frac{\partial^{2} q_{b}}{\partial t^{2}} = i γ_{d} {| q_{b} |}^{2} q_{b},

where α_d, β_2,_d (z_d), and γ_d are the loss coefficient, dispersion profile, and nonlinear coefficient of the DVF, respectively. In the presence of a pre-amplifier, the input to the DVF is

q_{b} (t, 0) = \sqrt{G^{'}} q^{*} (t, L) .

Let

q_{b} (t, z_{b}) = \sqrt{P_{d}} e^{- α_{d} z_{d} / 2} u_{b},

and

d z_{d}^{'} = β_{2, d} (z_{d}) d z_{d},

where P_d = G′e⁻^W ⁽^L⁾P_in is the input power of the DVF. Equation (14) may be written as

\frac{\partial u_{b}}{\partial z_{d}^{'}} + \frac{i}{2} \frac{\partial^{2} u_{b}}{\partial t^{2}} = i \frac{γ_{d} P_{d} e^{- α_{d} z_{d}}}{β_{2, d} (z_{d})} {| u_{b} |}^{2} u_{b} .

Comparing Eq. (13) with Eq. (18), the following conditions are required for the OBP module to act as the virtual fiber of Fig. 1(a), i.e. to fully compensate for the propagation distortions in the transmission fiber,

\frac{γ e^{- W^{'} (z_{b})}}{β_{2}} = \frac{G^{'} γ_{d} e^{- α_{d} z_{d}}}{β_{2, d}},

and

d z_{b}^{'} = d z_{d}^{'} .

Using Eqs. (12) and (17) in Eq. (20), we find

\frac{β_{2}}{β_{2, d} (z_{d})} = \frac{d z_{d}}{d z_{b}} .

Using Eq. (21) in Eq. (19) and integrating, we obtain

\int_{0}^{z_{b}} e^{- W^{'} (z_{b})} d z_{b} = \frac{G^{'} γ_{d} (1 - e^{- α_{d} z_{d}})}{γ α_{d}} .

Let

A (x) = \int_{0}^{x} e^{- W^{'} (x)} d x .

Now, Eq. (22) may be written as

A (z_{b}) = \frac{G^{'} γ_{d} (1 - e^{- α_{d} z_{d}})}{γ α_{d}} .

Equation (19) may be written as

β_{2, d} (z_{d}) = \frac{G^{'} β_{2} γ_{d}}{γ} e^{W^{'} (z_{b}) - α_{d} z_{d}},

where

z_{b} = A^{- 1} [\frac{G^{'} γ_{d} (1 - e^{- α_{d} z_{d}})}{γ α_{d}}] .

The length of DVF is obtained by the dispersion compensation condition $β_{2} L = \int_{0}^{L_{d}} β_{2, d} (z_{d}) d z_{d}$ , and is given as

L_{d} = - \frac{1}{α_{d}} l n [1 - \frac{α_{d} γ A (L)}{G^{'} γ_{d}}] .

Equations (25)–(27) determines a DVF that makes the OBP module to exactly compensate for the signal-to-signal nonlinear interactions in the transmission fiber. The analysis of [33] assumed that the fiber loss coefficient is constant and analytical expressions were derived for the dispersion profile and length of DVF. The OBP theory developed in this paper is a generalization of the theory in [33], in order to account for the distance-dependent loss/gain profiles. Due to this change, an explicit analytical expression for the dispersion profile of the DVF cannot be found. Equations (23)–(26) are solved numerically to obtain the dispersion profile. The integration of Eq. (23) is calculated using a Matlab function, integral(). And the inversion function in Eq. (26) is calculated using a looking up table.

For systems with distributed Raman amplifiers using backward pump, the loss/gain profile may be written as [1]

α (z) = α_{s} - K e^{α_{p} z},

where

K = \frac{g_{R} P_{p} e^{- α_{p} L}}{A_{p}},

and α_s is the fiber loss at signal frequency, α_p is the fiber loss at Raman pump frequency, g_R is the Raman pump gain coefficient, A_p is the effective area, and P_p is the pump power. The DVF dispersion profile for this case can be calculated using Eqs. (25)–(27). For systems with lumped EDFA only, the loss/gain profile is simplified as α(z) = α_s. In this case, the semi-analytical expressions (25) and (27) reduce to analytical expressions [33],

β_{2, d} (z_{d}) = \frac{e^{- α_{d} z_{d}}}{\frac{γ}{γ_{d} G^{'}} + α_{s} (\frac{1 - e^{- α_{d} z_{d}}}{α_{d}})} β_{2},

and

L_{d} = - \frac{1}{α_{d}} l n [1 - \frac{α_{d} γ}{G^{'} α_{s} γ_{d}} (e^{- α_{s} L} - 1)],

respectively.

3. Dispersion profiles

Figure 2 shows the dispersion profiles of the DVF for full EDFA systems. We see that the pre-amplifier gain can be used to adjust the dispersion profile. From Fig. 2, it can be seen that the dispersion decreases monotonically with distance and the required length of DVF can be reduced by increasing the pre-amplifier gain G′. Figure 3 shows the DVF dispersion profiles for different pre-amplifier gains for the scheme of hybrid EDFA/Raman amplification. In this case, the dispersion increases initially and then decreases, which can be qualitatively understood as follows. In the presence of distributed Raman amplification, the signal power profile along the transmission fiber decreases and then increases, so does the local nonlinear phase shift. To compensate for nonlinear distortion in such a system, the required nonlinear phase shift in the virtual fiber [see Fig. 1(a)] should decrease initially and then increase. From Eq. (18), we see that the local nonlinear phase shift in inversely proportional to the dispersion of DVF. In the OBP module, the DVF dispersion profile is expected to increase initially and then decrease for systems with distributed Raman amplification so that the local nonlinear phase shift would decrease initially and then increase. Figure 4 shows the DVF dispersion profiles for the scheme of full Raman amplification. Comparing the profiles for full EDFA [Fig. 2] and full Raman [Fig. 4] systems, we see that the required length of DVF and also the gain G′ for the full Raman system is much less than that for the full EDFA system and hence, the loss due to OBP is lower for the full Raman system.

Fig. 2 Dispersion profiles of optical back propagation fibers for the case of full EDFA amplification. G′ is the gain of the pre-amplifier.

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Fig. 3 Dispersion profiles of optical back propagation fibers for the case of hybrid amplification (EDFA gain = 4.8 dB, Raman gain = 7.2 dB). G′ is the gain of the pre-amplifier.

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Fig. 4 Dispersion profiles of optical back propagation fibers for the case of full Raman amplification. G′ is the gain of the pre-amplifier.

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To study the impact of DVF dispersion profile fluctuation, which may happen due to non-ideal effects during fabrication, we introduce a perturbed dispersion 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. (25)x(z) are zero-mean Gaussian random variables with a standard deviation of σ_DVF. In this paper, we choose σ_DVF = 0.02 in Monte Carlo simulations. Figure 5 shows the perturbed dispersion profiles for the three schemes, with σ_DVF = 0.02.

Fig. 5 DVF dispersion profile fluctuations. The dispersion profile is modeled as: ${\hat{β}}_{2, d} (z) = [1 + x (z)] β_{2, d} (z)$ , where β_2,_d (z) is a desired dispersion profile (ideal case), x(z) are zero-mean Gaussian random variables with a standard deviation of 0.02.

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A distance-dependent dispersion profile can be achieved by controlling the drawing speed during fiber fabrication, where the change in fiber core diameter modifies the dispersion coefficient. It is also feasible to control dispersion slope during fabrication. In terms of dispersion and dispersion slope, the DVF belongs to the family of dispersion compensating fibers and dispersion-slope compensating fibers for which dispersion slope could be zero or negative [36]. The change in fiber core diameter will vary the effective area and nonlinear coefficients. We note that the change in nonlinear coefficients could be quite small. Therefore, we assumed a constant nonlinear coefficient in the derivations. It is straightforward to modify the derivation and include the distance dependence of nonlinear coefficient.

4. Results and discussions

We investigated a wavelength division multiplexing (WDM) fiber optic system with EDFA and/or Raman amplifications, as shown in Fig. 6(a). The transmitter generates a 7-channel WDM signal with a channel spacing of 50 GHz. The symbol rate per channel is 28 Gbaud and the modulation format is 64-QAM. A root raised cosine (RRC) with a roll-off factor of 0.6 is used. 32,768 symbols per channel are used in the Monte Carlo simulations. Signal propagation in fibers is simulated using the split-step Fourier scheme (SSFS). The entire fiber optic link includes 12 network nodes, including transmitter and receiver nodes. The distance between each two adjacent network nodes is 4 or 5 spans of 60 km long transmission fibers. The actual distance between the nodes is randomly picked out of these values. The total number of fiber spans between the transmitter and the receiver is 50. The parameters of the transmission fiber are as follows: α_TF = 0.2dB/km, β_2,_TF, = 5ps²/km, β_3,_TF = 0.05ps³/km, γ_TF = 2.2W⁻¹km⁻¹. The parameters of the DVFs are α_DVF = 0.4dB/km, β_3,_DVF = −0.07ps³/km, γ_DVF = 4.86W⁻¹km⁻¹. We investigated three different amplification schemes: (i) full EDFA amplification, (ii) hybrid EDFA/Raman amplification (EDFA gain = 4.8 dB, Raman gain = 7.2 dB), and (iii) full Raman amplification. The DVF dispersion profiles in Fig. 5 are used in simulations. The pre-amplifier gains for the three different amplification schemes are 13.0 dB, 10.8 dB and 7.8 dB, respectively. The distributed Raman amplifier uses backward pump configuration. The Raman pump powers are 20.1 dBm and 22.3 dBm for scheme (ii) and (iii), respectively. An OBP module is placed at each network node prior to a ROADM. The OBP has two OPCs so that the conjugated signal does not interfere with other WDM channels, as shown in Fig. 6(b) [35]. The OPC is realized using highly nonlinear fibers (HNLF) with two laser pumps. The details about OPC modeling can be found in [35], in which non-ideal effects of OPC are considered, including OPC loss, phase noise and relative intensity noise (RIN) of pump lasers. The OPC loss is calculated as $L o s s = 4 P_{p 1} P_{p 2} {(γ_{H N L F} L_{e f f})}^{2} e^{- α_{H N L F} L}$ , where P_p₁ and P_p₂ are the power of two pump lasers, α_HNLF and γ_HNLF are the loss and nonlinear coefficients, respectively, $L_{e f f} = \int_{0}^{L} e^{- α_{H N L F} z / 2} d z$ is the effective length, and L is the length of HNLF fiber [35]. The OPC loss is compensated for by an optical amplifier with a noise figure of 4.8 dB. For simplicity, the nonlinear distortions of OPC as discussed in [37] are ignored in this paper since it can be partially mitigated. 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. In the OBP module, the loss of DVF fiber is compensated for by an optical amplifier with a noise figure of 4.8 dB. We take the central WDM channel as the channel of interest. After OBP, a ROADM at each network node will randomly drop three out of the six neighbor channels and then add new signals to the same frequency of the dropped channels, while the signal on the central channel is not modified by a ROADM. At the receiver, a matched filter is used to remove out of band noise, and the accumulated phase noises are compensated using a feedforward method in the digital domain [38]. 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 $Q = \sqrt{2} e r f c^{- 1} (2 \times B E R)$ . We investigated the performance of DBP for comparison. Due to the loss of propagation path information at ROADMs, only single-channel DBP is feasible at the receiver. In DBP, the received signal is downsampled to 2 samples/symbol and the back propagation step size is 3 km.

Fig. 6 (a) Fiber optic system with an OBP module at each network node. (b) OBP module for multi-channel systems. Tx: transmitter, Rx: receiver, TF: transmission fiber, OBP: optical back propagation, ROADM: reconfigurable optical add-drop multiplexer, DeMUX: demultiplexer, BPF: band pass filter, MUX: multiplexer, OPC: optical phase conjugator, DVF: dispersion varying fiber, Att.: attenuator.

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In Fig. 7, we compare the performance of OBP with DBP and linear compensation. For the case of ideal OBP, ideal dispersion profile with no fluctuations and ideal OPC operation without loss or noise are assumed, while for the case of realistic OBP, we include dispersion profile fluctuation of DVFs, loss due to OPC, and phase noise and RIN noise of pump lasers in OPCs. As compared with linear compensation, single-channel DBP brings only 0.8 dB, 1.5 dB, and 1.5 dB Q-factor gains for full EDFA, hybrid EDFA/Raman, and full Raman systems, respectively. These small performance gains are due to the fact that only intra-channel nonlinear impairments are compensated. In contrast, OBP compensates for both intra-channel and inter-channel nonlinear distortions. The Q-factor gains of realistic OBP are 5.8 dB, 5.9 dB, and 6.1 dB as compared with linear compensation, for full EDFA amplification, hybrid EDFA/Raman amplification, and full Raman amplification schemes, respectively. The scheme with distributed Raman amplification has better performance than the scheme with EDFA amplification, due to the better noise performance of distributed Raman amplifiers and shorter length of DVF. In the linear regime where the launch power is small, the OBP has slightly worse performance than the linear compensation scheme which is due to the extra noises added by the amplifiers for OPC, OPC noise and the amplifiers for compensating DVF loss. The OBP fully compensates for signal-to-signal nonlinear interactions. The main limiting factors to transmission performance of fiber optic networks with OBP are signal-to-noise nonlinear interactions [7] and nonlinear polarization mode dispersion (PMD) [39]. As compared to the DBP, the signal-to-noise nonlinear interactions are considerably reduced for the OBP due to frequent optical phase conjugation [40]. In the case of optical networks, it is not possible to have multi-channel DBP since it requires access to all the channels that have caused nonlinear distortion on a given channel (say the central channel) on its entire path. The DBP at a given node does not have access to channels that are dropped in previous nodes. In other words, the DBP can effectively compensates for single channel nonlinear impairments and the main impairment that limits the DBP is cross-phase modulation due to other channels. In contrast, the OBP can compensate for both single channel and inter-channel nonlinear impairments in optical networks.

Fig. 7 Q-factor vs. launch power per WDM channel. Transmission distance = 3000 km, modulation = 64 QAM.

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It is interesting to compare the OBP with an OPC-only scheme that puts an OPC at each network node. For the case of distributed Raman amplification, signal power firstly decreases with distance due to fiber loss and then increases due to Raman amplification. This Raman amplification scheme provides power profile symmetry between two adjacent fiber spans, to some extent. Consider a simple case with two fiber spans, transmission fiber 1 (TF1) and transmission fiber 2 (TF2), and with an OPC placed between the two fiber spans. The power profiles in TF1 and TF2 are shown in Fig. 8. In order for TF2 to exactly compensate for the propagation impairments in TF1, a power profile in TF2 should be symmetric to the power profile in TF1. The mismatch between the green curve and the red curve shows the inaccuracy in symmetry. Although the power profiles are not exactly symmetric, OPC based nonlinear compensation has shown excellent performance in point-to-point systems [27]. However, there is another fundamental limit for OPC-only compensation schemes in fiber optic networks where the WDM channels may be modified by a ROADM at a network node and the number of fiber spans between two nodes is variable. The OPC-only systems require not only power symmetry, but also signal symmetry with respect to the location of OPC. Due to the add/drop of channels at a node, the signal symmetry with respect to OPC is not maintained, and hence it provides only the partial compensation of inter-channel nonlinear impairments. In our simulations, four WDM channels carry the same signals before and after an OPC, while the other three WDM channels have different signals before and after an OPC. In the OPC-only system, if the signals before and after the OPC are different, it cannot compensate for the cross phase modulation (XPM) distortions on the central channel caused by the neighboring three channels that have undergone changes. Figure 9 compares the performance of OBP and the OPC-only scheme. The ideal OPC-only scheme ignores the loss and noise in OPC and shows 0.3 dB Q-factor gain over single-channel DBP, which is due to the fact that the OPC-only scheme partially compensates for inter-channel nonlinear distortions in a network scenario. It also shows that the OBP outperforms the OPC-only scheme by 4.6 dB.

Fig. 8 (a) Signal power profile in transmission fiber 1 (TF1) before an OPC (blue line). (b) Signal power profile in transmission fiber 2 (TF2) after an OPC (green line), and the desired power profile in TF2 (red line) that is symmetric to the power profile in TF1.

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Fig. 9 Comparison of OBP with OPC only case. The OPC only case has an OPC at each node along the transmission link. Transmission distance = 3000 km, modulation = 64 QAM.

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We have also investigated the nonlinear compensation of point-to-point systems, where no ROADM presents and the WDM channels are not added/dropped throughout the entire fiber optic link. Figure 10 compares the performance of various nonlinear compensation schemes. The multi-channel DBP is based on the coupled NLSE [14] and is implemented with a step size of 3 km using 2 samples per symbol. The multi-channel DBP brings 3.4 dB gain as compared with linear compensation, since it compensates for both intra-channel and inter-channel nonlinear distortions. The OBP brings 4.5 dB Q-factor gain over the OPC-only scheme which puts an OPC at every network node. In the simulations, the distance between two network nodes are randomly chosen between 4 or 5 fiber spans, which affects the performance of the OPC-only scheme. The performance of OPC-only system could be improved to some extent by placing the OPC at the optimal location [41]. Even in point-to-point systems without ROADMs, the OBP outperforms the multi-channel DBP by 2.4 dB mainly for the following two reasons. Firstly, the multi-channel DBP is implemented using 2 samples per symbol, while the OBP performs analog signal processing and is not restricted by the sampling rate. Secondly, the multi-channel DBP compensates for the XPM effect with a step size around a few kilometers. However, in order to compensates for the four wave mixing (FWM) effect, a much smaller step size in the order of 100 meters is required [14]. In contrast, the OBP compensates for nonlinear distortions in a continuous manner so that it compensates for both XPM and FWM effects in real time.

Fig. 10 Comparisons of different nonlinear compensation schemes for a point-to-point system. The WDM channels are not added/dropped throughout the fiber optic link. Transmission distance = 3000 km, modulation = 64 QAM.

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5. Conclusions

We have investigated an optical back propagation (OBP) technique to compensate for propagation impairments in fiber optic systems with hybrid EDFA/Raman amplification. An OBP module consisting of an optical phase conjugator (OPC), optical amplifiers, and dispersion varying fibers (DVFs) is placed at each network node before a ROADM modifies a WDM signal, therefore both intra-channel and inter-channel nonlinear distortions can be compensated. We have obtained a semi-analytical expression for the novel dispersion profiles of DVFs of the OBP module for fiber optic systems with EDFA and/or Raman amplification. Simulation results show that the OBP method has significant performance improvements as compared with digital back propagation (DBP) or OPC-only systems. It is shown that the OBP module placed at each node could fully compensate for nonlinear impairments while DBP or OPC-only systems provide only partial compensation of nonlinear impairments.

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Optical back propagation for fiber optic networks with hybrid EDFA Raman amplification

Abstract

1. Introduction

2. Theory of optical back propagation

3. Dispersion profiles

4. Results and discussions

5. Conclusions

References and links

Cited By

Figures (10)

Equations (32)

Optics Express