Ideal optical backpropagation of scalar NLSE using dispersion-decreasing fibers for WDM transmission

Xiaojun Liang; Shiva Kumar; Jing Shao

doi:10.1364/OE.21.028668

1. Introduction

The maximum reach of a long haul fiber optic system with advanced modulation formats is mainly limited by fiber nonlinear impairments. The backpropagation techniques can be used to compensate for dispersion and nonlinear effects of the transmission fiber (TF). The compensation schemes can be divided into three types: digital [1–11], optical [12–19], and the combination of both [20]. The optical backpropagation (OBP) has the following advantages/disadvantages over digital backpropagation (DBP). (i) A very large bandwidth (~4 THz) is available for OBP while the bandwidth of the DBP is limited by the bandwidth of the coherent receiver. (ii) DBP requires significant computational resources, especially for wavelength division multiplexing (WDM) system and hence it is currently limited to off-line signal processing. In contrast, OBP provides compensation in real time. (iii) Number of samples per symbol available for DBP is limited by the sampling rate of the analog-to-digital converter (ADC). Although it is possible to do upsampling on the digital signal processor (DSP), it leads to additional computational complexity. However, for OBP, the signal processing is done on the analog optical waveform. (iv) OBP requires a real fiber which has loss. So, amplifiers are needed to compensate for fiber loss in the OBP section which enhances the noise in the system.

In [16,18], an OBP scheme consisting of optical phase conjugation (OPC), dispersion compensation fiber (DCF)/fiber Bragg grating (FBG), and highly nonlinear fiber (HNLF) is investigated. DCF/FBG is used to compensate for dispersion, and HNLF is used to compensate for nonlinearity. The dispersion and nonlinear effects are compensated in a split-step fashion analogous to split-step Fourier scheme (SSFS) used to solve the nonlinear Schrödinger equation (NLSE). Although this technique is quite effective for a single channel, for a WDM system, small step size is required and hence the insertion losses due to DCF/FBG and HNLF increase which limit the transmission performance. In this paper, we investigate the possibility of introducing a single optical device which can exactly compensate for dispersion and nonlinearity. A dispersion-decreasing fiber (DDF) with a specific dispersion profile is found to meet our requirements.

In the proposed scheme, an OPC is placed at the end of the transmission link which is followed by N spans of DDFs where N is the number of TF spans. The DDFs introduce a small amount of losses which are compensated by amplifiers placed in the OBP section. Numerical simulation results show that the OBP with DDF outperforms DBP and midpoint-OPC schemes. The transmission reach of a WDM system can be significantly enhanced using the proposed scheme as compared to linear compensation in the receiver or DBP. We found that the DBP is limited mainly by the sampling rate of the ADC. In this paper, the study is limited to scalar NLSE and it does not fully address the general polarization-multiplexed WDM transmission system.

2. OBP theory

The evolution of the optical field envelope in a fiber optic link is described by the 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 β₂, γ and α are dispersion, nonlinear and loss coefficients of TF, respectively. The formal solution of Eq. (1) for a single span of TF is

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

Here L_a is the fiber length. Let the output signal field of the fiber pass through an OPC, as shown in Fig. 1(a). The output of the OPC is

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

Let the output of OPC propagate through an ideal optical backpropagation fiber (OBPF) that is identical to the TF except that the sign of the loss coefficient of OBPF is inverted. In this case, from Eq. (2), we see that the nonlinear operator corresponding to OBPF is N*(t,z). Using Eq. (4), the output of the OBPF is

Fig. 1 A single-span fiber optic system with (a) OBP using an ideal optical backpropagation fiber with negative loss coefficient; (b) OBP using a DDF and amplifiers. Tx: transmitter; TF: transmission fiber; OPC: optical phase conjugator; OBPF: optical backpropagation fiber; DDF: dispersion-decreasing fiber; Rx: receiver.

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q_{O B P F, o u t} (t) = e^{i \int_{0}^{L_{a}} [D (t) + N^{*} (t, z)] d z} q^{*} (t, L_{a})

\begin{array}{l} = e^{i \int_{0}^{L_{a}} [D (t) + N^{*} (t, z)] d z} e^{- i \int_{0}^{L_{a}} [D (t) + N^{*} (t, z)] d z} q^{*} (t, 0) \\ = q^{*} (t, 0) . \end{array}

Thus, the input field envelope can be recovered by performing a phase conjugation in the electrical domain at the receiver. Equation (5) is equivalent to

\frac{\partial q_{b}}{\partial z_{b}} = i [D (t) + N^{*} (t, z_{b})] q_{b} (t, z_{b}),

with

q_{b} (t, 0) = q^{*} (t, L_{a}),

and

z_{b}

is the distance in OBPF. Using

q_{b} = \sqrt{P_{i n}} e^{- α (L_{a} - z_{b}) / 2} u_{b},

and

d {z^{'}}_{b} = β_{2} d z_{b},

Eq. (7) can be rewritten as

i \frac{\partial u_{b}}{\partial {z^{'}}_{b}} - \frac{1}{2} \frac{\partial^{2} u_{b}}{\partial t^{2}} + \frac{γ P_{i n}}{β_{2}} e^{- α (L_{a} - z_{b})} | u_{b} |^{2} u_{b} = 0,

where P_in is the power launched to the TF. Equation (10) describes the field propagation in an ideal fiber with a constant β₂ and a negative loss coefficient (or equivalently the power increasing with distance) that exactly compensates for dispersion and nonlinearity of the TF. However, it is hard to realize such a fiber in practice. For an ideal OBP, we like to have a short length of a fiber (so that its insertion loss is small) which provides the same response as that of the ideal OBPF given by Eq. (10). Here, we derive an equivalent way of realizing Eq. (10) by using amplifiers and a DDF with positive loss coefficient α_d and a dispersion profile β_2,d(z_d) [see Fig. 1(b)]. The optical field envelope in the DDF is described by

i \frac{\partial q_{b}}{\partial z_{d}} - \frac{β_{2, d} (z_{d})}{2} \frac{\partial^{2} q_{b}}{\partial t^{2}} + γ_{d} | q_{b} |^{2} q_{b} + i \frac{α_{d}}{2} q_{b} = 0,

where

α_{d}

and

γ_{d}

are the loss and nonlinear coefficients of DDF, respectively,

z_{d}

is the distance in the DDF,

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

and G' is the gain of the amplifier preceding DDF. Using transformations

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

and

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

Equation (11) can be rewritten as

i \frac{\partial u_{b}}{\partial {z^{'}}_{d}} - \frac{1}{2} \frac{\partial^{2} u_{b}}{\partial t^{2}} + \frac{γ_{d} P_{d} e^{- α_{d} z_{d}}}{β_{2, d} (z_{d})} | u_{b} |^{2} u_{b} = 0,

where

P_{d} = G P_{i n} = G^{'} e^{- α L_{a}} P_{i n}

is the input power of the DDF. Equations (10) and (14) are identical only if

d {z^{'}}_{b} = d {z^{'}}_{d},

and

\frac{γ P_{i n}}{β_{2}} e^{- α (L_{a} - z_{b})} = \frac{γ_{d} P_{d} e^{- α_{d} z_{d}}}{β_{2, d} (z_{d})} .

Substituting Eqs. (9) and (13) in Eq. (15), we find

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

w \equiv β_{2} z_{b} = \int_{0}^{z_{d}} β_{2, d} (z_{d}) d z_{d},

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

Substituting Eqs. (18) and (19) in Eq. (16), we obtain

\frac{d w}{d z_{d}} e^{α w / β_{2}} = (\frac{γ_{d} P_{d} β_{2}}{γ P_{i n}}) e^{α L_{a}} e^{- α_{d} z_{d}} .

Integrating Eq. (20), we find

\frac{β_{2}}{α} (e^{\frac{α}{β_{2}} w (z_{d})} - 1) = (\frac{γ_{d} P_{d}}{γ P_{i n}}) e^{α L_{a}} \frac{1 - e^{- α_{d} z_{d}}}{α_{d}} .

Simplifying Eq. (21), we obtain

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

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

The length of DDF L_d is found as follows. Total accumulated dispersion of the ideal OBPF [Fig. 1(a)] should be the same as that of the DDF, i.e.,

β_{2} L_{a} = w (L_{d}) = \int_{0}^{L_{d}} β_{2, d} (z_{d}) d z_{d},

or

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)} .

Equations (23) and (25) are the main results of this paper. If the dispersion profile of the DDF is tailored to satisfy Eq. (23), the combination of the amplifiers and DDF provides the ideal response described by Eq. (10), and hence, signal-signal nonlinear interactions can be exactly compensated. The amplifier with gain

G_{d} = e^{α_{d} L_{d}}

is introduced after the DDF [see Fig. 1(b)] to compensate for the loss of DDF. Figure 2 shows the dispersion profiles of DDF that satisfy Eq. (23). As can be seen, relatively shorter length of DDF can compensate for the dispersion and nonlinear effects of the TF.

Fig. 2 Dispersion profiles of DDF. TF parameters: α = 0.2 dB/km, β₂ = 5 ps²/km, γ = 2.2 W⁻¹km⁻¹, L_a = 60 km. DDF parameters: α_d = 0.4 dB/km, γ_d = 4.86 W⁻¹km⁻¹. (a) G = 1.0: β_2,d(0) = 175.1 ps²/km, L_d = 20.5 km; (b) G = 1.26: β_2,d(0) = 220.6 ps²/km, L_d = 12.1 km; (c) G = 1.5: β_2,d(0) = 262.6 ps²/km, L_d = 9.0 km.

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So far we considered the compensation of dispersion and nonlinearity of a single-span fiber optic link. For a multiple-span transmission system, Fig. 3 shows the schematic of a WDM fiber optic transmission system consisting of M transmitters, N spans of TFs, the OBP module, and M coherent receivers. The OBP is applied at the end of the transmission link. A pre-amplifier with gain G is introduced so that the required dispersion profile and length of the DDF can be adjusted according to Eqs. (23) and (25), respectively. A band pass filter (BPF) is introduced to remove the out of band amplified spontaneous emission (ASE) noise. During backpropagation, amplifiers with gain G_d are used to fully compensate for the loss of each span of DDF.

Fig. 3 Schematic diagram of a WDM fiber optic transmission system with OBP. MUX: multiplexer; BPF: band pass filter; DMUX: demultiplexer.

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In DBP, the compensation of fiber dispersion and nonlinearity is implemented in a step-wise manner and the performance is usually limited by the step size which has to be traded off against computational cost or system complexity. In WDM systems, the required computational load may prevent DBP from real time implementation. In the OBP with DDF, the compensation of dispersion and nonlinearity is realized by a gradually decreasing dispersion profile, which inherently has a very small step size. The DDF with exponentially dispersion decreasing fibers have been fabricated before [21,22]. The step size of the order of a few meters in DDF can be realized and hence, nearly ideal OBP can be realized using DDF. The DDF can be fabricated by tapering the fiber during drawing process which alters the waveguide contribution to the dispersion [21]. The maximum dispersion required for OBP fiber is of the same order as the commercially available dispersion compensation fiber and of the same sign.

3. Simulation results

We simulate a WDM fiber optic transmission system with OBP at the receiver with the following parameters: number of WDM channels = 5, channel spacing = 100 GHz, symbol rate per channel = 25 Gsymbols/s, modulation = 32 quadrature amplitude modulation (QAM), number of symbols simulated = 32768 per channel. The linewidths of the transmitter and local oscillator lasers are 100 kHz each. The dispersion, loss, and nonlinear coefficients of the TF are β₂ = 5 ps²/km, α = 0.2 dB/km, and γ = 2.2 W⁻¹km⁻¹, respectively. This type of fiber has been fabricated before and it is known as negative dispersion fiber (NDF) [23,24]. The amplifier spacing is 60 km, and the spontaneous emission noise factor is n_sp = 1.5. The BPF shown in Fig. 3 is a second order Gaussian filter with full bandwidth of 450 GHz. For the DDF, α_d = 0.4 dB/km, γ_d = 4.86 W⁻¹km⁻¹, and L_d = 12.1 km [see Fig. 2(b)]. The corresponding amplifier gain for compensating the DDF loss is 4.84 dB. In all the simulations, 32 samples per symbol are used in the transmission link so as to obtain a frequency window covering all the WDM channels. In DBP simulations, 2 samples per symbol are used after the ADC unless otherwise specified, while in OBP simulations, backpropagation is in the optical domain and 32 samples per symbol are used. Using the method of [2], the coupled NLSE is used to compensate for the inter-channel nonlinear impairments ignoring four-wave mixing (FWM). However, the OBP scheme compensates for both cross-phase modulation (XPM) and FWM simultaneously. The central channel is demultiplexed using a second order Gaussian filter with full bandwidth of 50 GHz. In the coherent receiver, for OBP, two samples per symbol are used after the ADC and phase noise compensation is done using the approach of [25]. A low pass filter (LPF) of bandwidth 25 GHz is used prior to phase noise compensation. For the DBP scheme, coupled NLSE is solved in digital domain prior to phase noise compensation. The optical and electrical filter bandwidths are optimized in both OBP and DBP schemes.

Figure 4 shows the bit error ratio (BER) as a function of the launch power per WDM channel when the transmission distance is 1200 km. The solid curve represents the BER of OBP using DDFs, and the dashed and dotted curves represent the BER of DBP with 3 km and 10 km step sizes, respectively. The DBP step size of the simulated WDM system is limited by the walk-off length [2], which is 3.2 km. We found that there is no obvious performance improvement when a step size smaller than 3 km is chosen for DBP, consistent with the results of [2]. Also, Fig. 4 shows the simulation results of DBP with 4 samples/symbol ADC sampling rate and DBP with DSP upsampling [1] from 2 to 4 samples/symbol. The DBP performance can be improved by increasing ADC sampling rate or DSP upsampling, at the cost of increased system complexity or computational cost. The OBP outperforms DBP (2 samples/symbol, step size = 3 km) by 2.0 dBQ. The relatively poor performance of DBP as compared to OBP is mainly due to the down sampling penalty and the lack of FWM compensation. The performance of midpoint OPC is worse than DBP, because the power profile is unsymmetrical with respect to the location of OPC. The performance of OBP is worse than that of DBP (with step size = 3 km) when the launch power is less than −2 dBm which is due to the optical signal to noise ratio (OSNR) penalty resulting from OBP amplifiers. The OSNR penalty due to OBP amplifiers is found to be 0.56 dB. From Fig. 4, it can also be seen that the DBP with a step size of 10 km performs worse than the DBP with a step size of 3 km even at lower launch powers (−10 dBm to −6 dBm) due to residual nonlinearity. The curve with ‘ + ’ shows the case where no OBP (or DBP) is applied and fiber dispersion and laser phase noise are compensated in the receiver. As can be seen, the performance of this system is much worse than the system with DBP or OBP.

Fig. 4 BER versus launch power per WDM channel. Transmission distance = 1200 km.

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Figure 5 shows the minimum BER as a function of transmission distance. The BER_min is obtained by optimizing the launch power for each distance. At the BER of 2.1 × 10⁻³, the transmission reaches of linear compensation only and midpoint OPC are 300 km and 360 km, respectively. For DBP with a 10 km step size and 2 samples/symbol sampling rate, the reach is 760 km, which can be increased to 1600 km by using a 3 km step size at the cost of more than tripling the computational effort. The transmission reach of OBP with DDF is 2460 km. Although the OBP fully compensates for signal-signal nonlinear interactions, it neither compensates for signal-ASE nonlinear interactions [26,27] nor mitigates nonlinear polarization mode dispersion (PMD) [28], which are the limiting factors to enhance the reach in systems based on OBP.

Fig. 5 BER_min versus transmission distance.

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4. Conclusions

We have investigated the performance of an OBP scheme consisting of an OPC and N spans of DDFs followed by amplifiers to compensate for dispersion and nonlinear effects of an N-span fiber optic WDM system. We have identified the conditions under which the nonlinear effects (both intra- and inter-channel nonlinearities) can be fully compensated and obtained an analytical expression for the novel dispersion profile of the DDF which provides the exact compensation of intra- and inter-channel signal-signal nonlinear impairments. The performance of the proposed OBP scheme is compared with DBP and midpoint OPC and simulation results show that the transmission reach can be significantly enhanced using the OBP with DDF.

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Ideal optical backpropagation of scalar NLSE using dispersion-decreasing fibers for WDM transmission

Abstract

1. Introduction

2. OBP theory

3. Simulation results

4. Conclusions

References and links

Cited By

Figures (5)

Equations (25)

Optics Express