## Abstract

We investigate the performance of digital filter back-propagation (DFBP) using coarse parameter estimation for mitigating SOA nonlinearity in coherent communication systems. We introduce a simple, low overhead method for parameter estimation for DFBP based on error vector magnitude (EVM) as a figure of merit. The bit error rate (BER) penalty achieved with this method has negligible penalty as compared to DFBP with fine parameter estimation. We examine different bias currents for two commercial SOAs used as booster amplifiers in our experiments to find optimum operating points and experimentally validate our method. The coarse parameter DFBP efficiently compensates SOA-induced nonlinearity for both SOA types in 80 km propagation of 16-QAM signal at 22 Gbaud.

© 2013 OSA

## 1. Introduction

Next generation optical communication systems are predicated on systems with integrable components [1]. The nonlinear behavior of semiconductor optical amplifiers (SOAs) is being exploited in recent optical signal processing applications, e.g., wavelength conversion [2], intensity noise suppression [3] and regeneration [4]. Even when used simply for amplification, where nonlinear response is detrimental, SOA integrability along with its wide gain spectrum and cost effectiveness can make them more attractive than erbium-doped fiber amplifiers (EDFAs) [5–9]. The use of SOAs is imperative to compensate insertion loss in integrated transmitters, as shown in [10,11] where a Mach-Zehnder modulator is monolithically integrated with a SOA. The SOA nonlinearity can induce amplitude and phase distortions and deteriorate signal quality when used for M-ary quadrature amplitude modulation (M-QAM) signaling in coherent detection systems [12].

Various post- and pre-compensation schemes have been proposed to overcome the nonlinear effect of SOAs [13–18]. Both approaches rely on creating an inverse SOA to compensate for the SOA nonlinearity. In pre-compensation, the inverse SOA is applied at the transmitter before the signal is distorted by the SOA, whereas in post-compensation the inverse SOA is implemented at the receiver to deal with SOA distortion. While pre-compensation can be attractive when SOA parameters are known, post-compensation has the ability to blindly adapt itself to unknown SOA parameters.

A computationally efficient digital filter back-propagation (DFBP) technique is employed to post-compensate nonlinear effects of SOAs in [13,14]. At practical sampling rates, DFBP was shown in simulations and in experiments to outperform the more complex Runge-Kutta based method. The simplicity of DFBP stems from a linearized SOA dynamic gain equation used in its derivation. The complexity of DFBP can be further reduced by using one inverse gain block instead of multiple blocks; this was shown to provide good compensation [14]. In this paper we extend previous research by examining how to determine DFBP parameters without a characterization of the SOA. Indeed, we find parameters that are more effective in removing distortion than parameters based on knowledge of the transmitter SOA.

We propose a simple, low overhead method to optimize DFBP parameters. We used the error vector magnitude (EVM) as a figure of merit for the optimization (as opposed to bit error rate) to reduce complexity. We further propose a strategy to reduce the search space for optimal parameters, and show that coarse resolution can achieve compensation on a par with a computationally expensive search over a space with fine resolution. We demonstrate the efficiency of this technique in DFBP adaptation via both experiment and simulations. We examine SOAs designed for both linear and nonlinear performance. We determine the operating point of each SOA to obtain high gain while staying in a regime where post-compensation is effective. We target bit error rate (BER) below forward error correction (FEC) threshold with ~7% overhead. We investigate the propagation performance by including up to 80 km of fiber to the system.

The paper is structured as follows. In section 2, we discuss the relationship between parameters of the DFBP and parameters of the transmitting SOA. We also discuss our strategy for reducing the computational burden of DFBP parameter optimization. In section 3, the experimental setup is explained. Section 4 is devoted to experimental and numerical results. Finally, conclusions are drawn in section 5.

## 2. EVM based parameter estimation of DFBP

The efficiency of DFBP for post-compensating SOA nonlinearities in coherent communication systems has been investigated via both simulations and experiments [13, 14]. DFBP parameters were shown to play an important role in post-compensation performance. These parameters can be determined in one of two ways. In the first case the SOA can be characterized experimentally, e.g., the parameters can be inferred for the SOA gain and conversion efficiency (CE) curve [13,19]. The second method is to sweep the parameters, choosing those resulting in the best BER [14]. From a mathematical standpoint, and assuming only linear impairments and AWGN, performance should be similar.

The nature of post-compensation is, however, that the source of the nonlinearity is remote from the digital signal processing. The optical and electrical systems in the link will introduce various impairments, as will the digital signal processing algorithms used for filtering, dispersion compensation and retiming before post-compensation can be applied. The accumulation of these effects will change the nature of the SOA-induced distortion, making the use of parameters gleaned from the SOA characterization less likely to be effective. Sweeping the parameter set to minimize BER yields better results. The parameter set essentially characterizes the overall channel, including but not limited to the SOA.

The structure of the DFBP was derived from SOA dynamics and is effective in combating the nonlinear transfer of intensity noise into phase noise, even though parameters used for the inverse SOA may differ from parameters for the physical SOA at the transmitter. Therefore we retain the parameter set of the physical SOA model, and propose a new methodology for fixing values in that set. In section 2.1 we describe that parameter set for the SOA and relate them to the DFBP implementation. In section 2.2 we define our strategy for searching for the optimal parameter set. We propose to reduce the search space based on SOA characteristics and previous experimental observations, and motivate the use of error vector magnitude as a figure of merit for optimization. In section 2.3 we present parameters obtained by characterizing SOAs in our lab that are used in simulations. These parameters are also applied in back-propagation to compare the performance with our proposed method based on coarse estimation of parameters. In section 3 these techniques will be applied in an experimental demonstration.

#### 2.1 DFBP parameters

The DFBP method is predicated on inversing the transmitter SOA behavior. The SOA model and the parameters capturing SOA behavior in forward propagation are critical for DFBP. The relationship between SOA input and output fields is given by [20]

where*E*(

_{in}*t*) and

*E*(

_{out}*t*) are respectively the SOA input and output fields,

*α*is the linewidth enhancement factor and

*h*(

*t*) represents the gain exponent or integrated material gain. The following SOA dynamic gain equation can be solved to find

*h*(

*t*):where

*h*

_{0},

*P*and

_{sat}*τ*are the unsaturated gain exponent, the saturation power and the carrier lifetime, respectively. Therefore, knowledge of

_{c}*h*

_{0},

*α*,

*P*and

_{sat}*τ*enables us to model the relationship between the SOA input and output fields. These four values are the parameter set describing SOA behavior and those needed to implement the DFBP. The input power |

_{c}*E*|

_{in}^{2}can be easily measured.

The DFBP is developed from a linearized version of the gain Eq. (2) to reduce complexity [13,15]. In this approach, *h*(*t*) is assumed to be equal to the sum of the average gain exponent and zero-average fluctuations, i.e., $h(t)=\overline{h}+\delta h(t).$ Using the following static gain equation, $\overline{h}$ can be found

*h*

_{0}is introduced to implement an “inverse SOA”. Let Δ

*t*be the sampling period, which is equal to symbol time since the DFBP input signal is one sample per symbol in our implementation (oversampling factor = 1). As shown in the DFBP block diagram in Fig. 1, we find the zero-average fluctuations,

*δh*(

*t*), using a digital filter which is derived from the linearized SOA model by taking z-transform [13]. For convenience we define two parameters,

*c*

_{1}and

*c*

_{2}, from the SOA parameter set and the easily measured input power |

*E*|

_{in}^{2}via

*h*(

*t*), the compensated output is found by multiplying the input by ${e}^{\frac{(1-j\alpha )}{2}h(t)}.$

#### 2.2 DFBP parameter estimation

The previous section presented the equations used in the DFBP exploiting knowledge of *h*_{0}, *α*, *P _{sat}* and

*τ*Previous examination has shown that DFBP performance is almost independent of

_{c}.*h*

_{0}[14]. Therefore we set

*h*

_{0}at a typical value of 4.6. In our experimental examination of both linear and nonlinear SOAs there is little variation in

*α*and we fix it at

*α*= 4.2. With two of the parameters fixed, the DFBP performance now mainly depends on

*P*and

_{sat}*τ*.

_{c}Our objective is to select the remaining two parameters to minimize the BER. Simulation of BER can be computationally costly and require use of training sequence (i.e., knowledge of bits transmitted). The EVM is known to correlate well with BER at reasonable SNR levels and is given by [21]

*E*,

_{i}*E*and

_{0,i}*N*are received symbol, ideal constellation point for received symbol and number of randomly transmitted data. The EVM figure of merit does not require knowledge of transmitted bits thus enabling blind adaptation. We propose the use of EVM for optimization of

*P*and

_{sat}*τ*in a computationally simple manner and show via simulation and experiment that such an optimization leads to significant BER improvement.

_{c}As indicated in Fig. 1, we capture a block of received samples (~4000 symbols) to be used for finding the optimal parameter set. We apply noise filtering, dispersion compensation (if necessary) and retiming on the captured data. To implement the DFBP, we use fixed values of *h*_{0}, *α* and search over 45 possibilities for (*P _{sat}*,

*τ*), i.e., 9 for

_{c}*τ*and 5 for

_{c}*P*as shown in Fig. 1. Our spread of values (

_{sat}*P*,

_{sat}*τ*), covers a wide gamut of possibilities for linear and nonlinear SOAs. One might limit them having knowledge about SOA parameters. For each pair, the DFBP block compensates for SOA nonlinearity. We then perform phase recovery and minimum mean square error (MMSE) equalization for those 4000 symbols and calculate EVM. At the end, we choose the parameter pair corresponding to minimum EVM.

_{c}While we searched all 45 parameter pairs, optimization methods, e.g. gradient descent, can be used to find the pairs giving minimum BER. When SOA parameters are fairly constant the computational burden of a complete search is acceptable. For applications where DFBP parameters need to be tuned regularly optimization methods could reduce latency. In this work, however, we focus on simplifying parameter estimation for DFBP to compensate SOA nonlinearity efficiently and examine the size of the search space.

#### 2.3 SOA characterization

In the previous section we proposed a method to find the optimized parameters for DFBP. Although this estimation technique does not require information on the SOA at the transmitter, SOA characterization allows us to have more precise simulations which can be used to verify the experimental results. In addition, we also apply parameters from characterization of the SOA directly in the DFBP to contrast performance testing the two parameter extraction methods.

We use the simulator presented in [22] and consider a SOA waveguide as cascade of *K* small sections for greater accuracy. The SOA distributed loss and amplified spontaneous emission (ASE) is included in this model. The propagation equations are solved using Runge-Kutta fourth order algorithm when *K* = 80 as “ground truth”. For back-propagation, we set *K* = 1 (one inverse gain block) to reduce complexity, as it has been shown this causes minimal penalty [14].

Table 1 indicates the parameters extracted by characterization of the two SOAs used in our experiments. We examined two commercial SOAs with differing design objectives; the first stresses gain and linearity (linear Covega BOA-2679 or L-SOA with noise figure of 7.3 dB), while the second enhances the nonlinear response for optical signal processing applications (nonlinear CIP-NL-OEC-1550 or NL-SOA with noise figure of 8.4 dB). We varied the three parameters *h*_{0}, *P _{sat}* and

*L*(SOA loss) and fitted them to the average gain versus SOA input power curve. To extract the remaining two parameters,

*α*and

*τ*, we measured CE for four-wave mixing in SOAs as a function of frequency detuning. We compared the theoretical curve of CE versus frequency detuning with experimental results, and varied

_{c}*α*and

*τ*to find a good fit between the two curves [19].

_{c}## 3. Experimental setup

In this section we take the techniques laid out in section 2 and validate them experimentally. Our investigation includes back-to-back (B2B) measurements as well as propagation for up to 80 km. Please note that while our transmitter is constructed from discrete components for experimental convenience, DFBP for post-compensation targets integrated transmitter sources with a SOA booster stage, or other subsystems (e.g., wavelength converters) where SOAs may be operated in saturation.

Figure 2 shows the experimental setup. An external cavity laser (ECL) source with less than 100 kHz 3-dB linewidth is used at 1550 nm. Each of two 22-Gb/s 4-level electrical signals is obtained by combining two 2^{20}-1 pseudo-binary random sequences (PRBSs). These signals form the 16-QAM signal set that drives the I/Q (in-phase/quadrature) Mach-Zehnder external modulator (SHF 46213D). The SOA is used in a booster configuration at the transmitter to increase launched power. The 1-nm optical band-pass filter after the SOA has 3.8-dB loss, and limits ASE.

We performed our experiment in two different configurations. In configuration (A) we examine back-to-back performance with the SOA present. In configuration (B) the signal is launched through 80 (or 60) km of standard single mode fiber (SSMF) with 0.19 dB/km loss. The received optical signal-to-noise ratio (OSNR) is adjusted using a variable optical attenuator (VOA) and an EDFA. After a polarization controller (PC), a VOA adjusts the received power to −7 dBm for configuration (A) and −10 dBm for configuration (B). The local oscillator (LO) was a narrow linewidth ECL with 14 dBm output power. We used these values for LO and receiver input power to operate the coherent receiver at its optimum working regime.

After coherent detection with a 22-GHz 3-dB bandwidth integrated receiver, the signal is digitized using two channels of a commercial 80-GS/s real-time oscilloscope with 30 GHz bandwidth. Signal processing is performed offline on 2 million captured samples. In the digital signal processing (DSP), we apply a Gaussian low-pass filter and do dispersion compensation, if needed. We then perform resampling and timing recovery. Afterwards, we numerically boost the signal power to its inferred value at the SOA output and utilize the DFBP method to post-compensate the SOA-induced nonlinearity. We remove the frequency offset between the LO and the received signal using the estimator suggested in [23]. We apply a MMSE filter to mitigate the effect of limited receiver bandwidth. We then employ a decision-aided maximum likelihood algorithm to estimate the carrier phase [24]. Finally, we choose the closest symbol to the received I/Q coordinates from 16-QAM constellation and carry out symbol to bit mapping. We synchronize to the transmitted PRBS, count errors and estimate bit error rate (BER). The signal constellations for two SOAs are shown as an inset in Fig. 2. The amplitude and phase distortions deteriorate signal performance, nevertheless, application of coarse parameter DFBP leads to recovery of the 16-QAM signal.

## 4. Experimental and numerical results

In our first set of results, we show the effect of the induced nonlinearity on BER when varying SOA input power and applying different bias currents to specify a suitable working point for each SOA. We devote the second subsection to comparing the performance of the proposed DFBP with parameters optimized over the discrete parameter set (Fig. 1) to a DFBP whose parameters were optimized over the same range of values, but with much greater granularity. Having the appropriate working conditions, we concentrate on the results from configuration (B) to find the OSNR penalty due to application of SOA as compared to the B2B case for which we have no SOA and fiber. By transmitting the distorted signal from SOA over a certain length of fiber, it is expected to experience degradation of performance comparing to results from first subsection due to interaction between SOA nonlinearity and effects originated from transmission, e.g., dispersion. Therefore, in choosing suitable operating points for the SOAs, sufficient distance of BER from the FEC limit was targeted.

#### 4.1 Appropriate SOA operating condition

We change the input power to the SOA in configuration (A) of the experimental setup and examine system performance for different SOA saturation levels. The BER versus launched power, *P _{launched}*, for NL-SOA and L-SOA are presented in Figs. 3(a) and 3(b), respectively. We observe significant improvement (~6 dB) in launched power to the fiber when applying DFBP to mitigate nonlinearity for both SOAs. As expected, with higher bias currents, the SOA gives more gain which in turn leads to higher launched powers. The performance is, however, limited due to severe nonlinearity.

In Fig. 3(a), the launched power is 1 dB less for *I _{bias}* = 160 mA as compared to launched powers for

*I*= 250 mA and 300 mA. Although BER less than a FEC limit of 3.8e-3 is achievable for all three examined bias currents, we choose

_{bias}*I*= 160 mA as a good working point for two reasons. First, the BER distance from the FEC limit is more reliable for this case and second, the SOA power consumption is less for lower bias currents. We show the gain versus bias current of the CIP nonlinear SOA as an inset to demonstrate that the obtained gain decreases quickly for currents below around 160 mA. Therefore,

_{bias}*I*= 160 mA is a good compromise between gain and performance. As shown in Fig. 3(b), the difference between launched power for

_{bias}*I*= 400 mA and 600 mA is 1 dB for the linear Covega SOA, as well. Therefore, we select

_{bias}*I*= 400 mA as operating point for the Covega SOA considering above the previously mentioned reasons, and especially to lower power consumption. Simulation results for the selected bias currents are shown with diamond makers for both SOAs in Fig. 3 verifying the experimental results for coarse parameter DFBP. As mentioned in section 2.3, forward propagation parameters for SOA in simulations are adjusted using the information from SOA characterizations.

_{bias}#### 4.2 Coarse vs. fine parameter estimation

We have already mentioned that parameters *h*_{0} and *α* either vary little among SOAs or have little impact on the DFBP improvement. The parameters *P _{sat}* and

*τ*, however, vary over wide ranges and DFBP performance is sensitive to these values. For instance, we observed that for our SOAs,

_{c}*P*can vary from 6 to 14 dBm, while

_{sat}*τ*can vary from 45 to 285 ps. This large search area can be examined in a brute force manner with fine resolution (0.2 dB steps for

_{c}*P*and 5 ps steps for

_{sat}*τ*), or a reduced search area can be adopted to decrease the delay and computational overhead for parameter estimation. We examine experimentally the impact of using a coarse resolution (5 values for

_{c}*P*and 9 values for

_{sat}*τ*) for estimation.

_{c}We captured over 1 million symbols per SOA input power level to compare the performance of EVM-optimized coarse parameter estimation vs. BER optimized, fine parameter estimation. In the first case, we examine only 4000 symbols and find *P _{sat}* and

*τ*(examining 45 pairs) that minimizes the EVM for those symbols. We then take the EVM optimized

_{c}*P*and

_{sat}*τ*and find the BER over the entire captured data set. In the second case, we take all captured data and minimize the BER by examining in turn a total of ~300 pairs of values for

_{c}*P*and

_{sat}*τ*. The same captured data is used in both cases. We repeated the procedure for each SOA type examined.

_{c}The BER versus input power of SOA is reported in Fig. 4. Results are given for the nonlinear NL-SOA (160 mA bias current) and for the L-SOA (400 mA bias current). BER for the DFBP found with EVM optimization using coarse resolution is given with circle markers, while results for the DFBP found with BER optimization using fine resolution are given with triangle markers. In addition, the results for parameters attained by SOA characterization are shown with square markers. For both SOA types, we see negligible BER degradation when using the simpler, less computationally expensive EVM based optimization. Our demonstration establishes that post-compensation can be applied without use of a training sequence, and with minimal delay and computation. However, when using the parameters obtained by SOA characterization, we observe significant performance degradation especially for L-SOA. These results suggest that, in contrast to our proposed method, parameters given by measurement do not guarantee efficient compensation. The degradation stems from two effects: first, the SOA operating point for characterization and for coherent detection are in most cases different and second, the accumulation of various effects during transmission, e.g. dispersion and filtering effect, are not captured during characterization of SOA parameters.

In Fig. 5 we report the sensitivity of BER to variations of parameters *P _{sat}* and

*τ*. We sweep these parameters with fine resolution (0.2 dB for

_{c}*P*and 5ps for

_{sat}*τ*), calculating the BER for each parameter pair. The search space is chosen to have BER contours fall in the region of the FEC limit. Figure 5(a) corresponds to the nonlinear NL-SOA, while Fig. 5(b) gives results for the linear L-SOA. The plots are reported for

_{c}*P*= −19 and −13 dBm for NL-SOA and L-SOA, respectively. The dark central section of the contour represents BER performance well below the FEC threshold. The points highlighted correspond to the EVM optimized parameters found.

_{in}Consider the EVM based search space: *P _{sat}* between 6 and 14 dBm, and

*τ*between 45 and 285 ps. We can see that this parameter space clearly covers the BER range of interest– that above and below the FEC limit. Despite having covered a wide range with limited resolution, our approach fell safely in the below-FEC level. Even with no prior information on the SOA used (either SOA characterization or even linear/non-linear category), we can blindly find a DFBP solution that moves us below the FEC level. Figure 5 suggests that DFBP performance is more sensitive to

_{c}*P*than

_{sat}*τ*

_{c}. The width of the low-BER region for

*P*is around 1 dBm for both plots which validates our choice for resolution of

_{sat}*P*in coarse estimation.

_{sat}#### 4.3 Propagation performance – two SOA types

In this section we examine the efficiency of DFPB in the presence of fiber propagation for two types of SOA. Figure 6 shows BER versus received OSNR measured in 0.1 nm resolution bandwidth; the BER curve with square markers corresponds to B2B without SOA (and hence no nonlinearity or DFBP), i.e., no SOA and no fiber. Two types of SOA, linear and nonlinear, are examined using a DFBP with parameters found per the method described in section 2. The SOA currents were fixed at the values determined in section 4.1. The SOA input power was set to *P _{in}* = −21 and −17 dBm (or equivalently,

*P*≈1 dBm) for NL-SOA and L-SOA, respectively. We launched the 16-QAM signal into 80 and 60 km of SSMF fiber for each SOA. The OSNR penalty is less than 4 dB for all cases when we apply the DFBP algorithm. While not shown in Fig. 6, we also observed that with 100 km of fiber, the BER is above the FEC limit within the achievable OSNR range.

_{launched}## 5. Conclusion

We presented a simple method based on EVM to coarsely estimate parameters of DFBP. The proposed scheme determines DFBP parameters by processing a small portion of data (~4000 symbols). We demonstrated, via experiment and simulations, the efficiency of a DFBP using an EVM optimized, coarse parameter estimation for mitigating SOA-induced phase and amplitude distortions in 16-QAM with coherent detection at 22 Gbaud. The penalty due to coarse estimation of DFBP parameters is negligible compared to DFBP with very fine parameter estimation. We examined the OSNR penalty induced by application of SOA as booster at transmitter in transmission of signal over 80 km of SSMF. The experimental results show less than 4 dB OSNR penalty at FEC threshold (BER = 3.8e-3) for both nonlinear and linear SOAs employed in our experiment.

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