1. Introduction

Optical amplifiers, such as erbium doped fiber amplifiers (EDFAs) or erbium ytterbium doped fibers (EYDFs), are critical components in optical communication systems [1–3]. Optimizing amplifier design can decrease required power margins and increase system throughput. Although physics-based numerical models are good predictors of optical amplifier gain and noise figure profiles in the C-band, their complexity leads to sparse examination of the large amplifier optimization space [4]. Physics-based models for extended L-band amplifiers have limited accuracy due to the difficulty of estimating the Giles-parameter. That estimation takes many experimental measurements, resulting in an accumulation of uncertainty. The use a in neural network (NN) model can avoid parametrization. We propose an experimental measurement campaign for L-band doped fibers that produces a training set for a neural network model. Our NN predictions of gain and noise figure are more accurate than physics-based models (variance of error reduced by 50%), and require much less computation (400 times faster simulations). We can greatly increase the search space in amplifier optimization.

Neural networks have been widely used to predict performance of deployed amplifiers with a fixed length of doped fiber [5,6]. Typically, the signal power and/or occupied spectrum is varied. In our research, we are the first to use a NN to capture the physics of a doped fiber at various lengths to be used in amplifier design. The length has a significant impact on inversion levels and is one of the most critical parameters for optimization. Our neural network includes length, signal power and pump power, direction, and pumping technique (cladding or core pumping). Our modeling of doped fibers could accelerate research into advanced amplifier design and its associated tools.

Simulating EDFAs with experimentally measured Giles parameters is a well-established technique in the C-band. These physics-based numerical models using measured parameters give accurate predictions of gain and noise figure. For the L-band, estimated parameters are more prone to error due to the presence of excited state absorption. The accuracy of the Giles-parameter in particular is limited by measurement dynamic range, fiber length precision, and the compounding of errors from the multiple independent measurements required [7] to form the estimate. Consequently, EDFAs designed with these simulators have performance that falls short of predictions [8]. Physics-based numerical models can be computationally expensive and time-consuming as they require solution of a system of differential equations via Runge–Kutta iterative methods. Exploring a wide range of design parameters for optimization can take significant time and computational resources. Realistically this leads to a smaller search space, or poor granularity in the parameter space.

We examine more accurate and less computationally expensive predictions of gain and noise figure by training a machine learning model with experimental training data. Machine learning techniques, such as neural networks, can learn complex, non-linear relationships between input and output variables [9], enabling more accurate predictions of amplifier behavior in across a range of input parameters. Most examinations of NNs for EDFAs focus on commercial amplifiers whose design is fixed (doped fiber design, fiber length, pump powers) [10,11]. The NN captures the changes in output gain or signal optical signal to noise ratio (OSNR) as the distribution of input power across wavelengths is varied. These models can exploit data captured from deployed systems for training set, and offer invaluable tools for optimizing the deployed EDFAs.

The problem we address is quite different from exploiting a deployed amplifier, as we focus on predicting the performance of an erbium doped fiber (EDF) for various fiber lengths and operating scenarios. We develop a tool for designing the amplifier. The authors in [12] used NN to build a model for bismuth-doped fiber amplifier (BDFA) in the E- and S-bands. They trained a NN to predict the gain and noise figure (NF) for a single fiber length with core pumping, but allowing variation of total signal power and forward/backward pump power. They examined several strategies for training sets, finding that random, uniformly distributed data points yield similar performance to a blanket grid of points. Our work adds the length of the doped fiber as a NN input. This is a critical parameter in amplifier design. We explore the performance of NN in both core pumping and cladding pumping scenarios. We contrast the accuracy of NN trained on experimental data with simulators using experimentally estimated model parameters. This extends our earlier work, where we had also considered a NN trained on synthetic data from a physics-based simulator [13].

The rest of paper is organized as follows. In section 2., we discuss the cost and accuracy of the predictors. We introduce neural networks as an alternative to traditional physics-based predictors. In section 3., we describe our experimental setup for core pumping and cladding pumping, as well as how we collect the data for training, validation, and testing. In section 4., we present error statistics showing the greater accuracy of the NN predictor. We analyze these statistics vis-a-vis designs with practical minimum gains. In section 5 we highlight the advantages in computational complexity of the NN solution. Finally, we conclude with some remarks.

2. Principle

Once a promising doping strategy has been found and a fiber has been produced, optimizing amplifier performance is essentially finding a good compromise between high gain, low NF and power conversion efficiency (PCE). For a given fiber design, we can select the optical amplifier length ($L$), as well as tweaking three operating parameters: signal power ($P_s$), forward pump power ($P_f$), and backward pump power ($P_b$). Signal power ($P_s$) is defined as the total signal power over the wavelengths examined, with each wavelength having equal power. The optimization of the amplifier operation involves examination of the four-dimensional parameter space.

An optimization across a four-dimensional input space, whether using exhaustive search or particle-swarm, must be completed with attainable computational effort. A less complex model can cover the optimization space more finely, and is more likely to find a globally optimum [$L$ $P_f$ $P_b$ $P_s$]. More complex models, such as physics-based numerical EDFA models, can only explore a coarser grid and may fall into local optima.

2.1 Complexity of modeling and simulation

We use the physics-based simulator for an L-band doped fiber for comparison with our NN model. The simulator solves the system of differential equations for propagation and population inversion iteratively (see appendix A). A fine segmentation of the fiber length is needed for convergence, which is computationally costly, particularly at long lengths. Resolution reduction and dimension reduction can bring simulation time to reasonable levels, but compromises accuracy.

The NN EDFA model has negligible computation time compared to the physics-based simulator. The computational effort to train the NN is done only once. Collecting the training set for the NN is where we must expend the greatest effort. We make a reasonable laboratory effort over several days to gather data for training and testing. We note that the physics-based modeling also requires significant experimental effort for parameter estimation. Unlike Giles-parameter estimation, NN data collection can be greatly automated. Each day we cut one length, $L$, of fiber. We collect gain and NF data for [$P_f$ $P_b$ $P_s$] unattended for the remaining 24-hour interval.

2.2 Accuracy challenges in physics-based EDFA modeling

The physics-based EDFA model relies on precise estimates of Giles parameters [8]. The error in the estimate of each parameter will contribute to the overall error of the model. In the C-band, the estimated parameters and the resulting numerical simulations have good accuracy and result in reliable EDFA performance prediction.

However, in the L-band, EDFs operate in the tail of the erbium emission band characterized by absorption and emission coefficients that are considerably lower, typically more than 20 times less than in the C-band [14]. In an effort to achieve accurate measurements and good dynamic range, longer fiber lengths are used. Longer fibers, however, result in heightened susceptibility to factors such as bending loss, non-uniformity in fiber fabrication, errors in EDF length estimates, and greater, non-negligible amplified spontaneous emission (ASE) [15]. Hence, characterization of the Giles parameters for modeling extended L-band EDFAs is challenging [7]. Inaccuracy in the Giles parameters can result in large discrepancies between the experimental and simulated performance of extended L-band EDFAs.

Further uncertainty is introduced when using cladding-pumping for which the pump power is distributed across the whole fiber cross-section. For accurate predictions with the physics-based model, we would require the pump propagation loss and absorption when injected in the cladding with the given pump power distribution.

2.3 Neural network EDFA modeling

By combining linear and nonlinear operations, feed forward neural network (FFNN) can capture the physical processes influencing EDFA behavior without directly modeling the physics behind the behavior. We find the FFNN hyperparameters that can accurately predict the minimum gain and noise figure of the systems studied.

The FFNN can be trained from synthetic data generated from a physics-based simulator or from experimental data. While we do not present these results here, we examined synthetic data in [13]. We observed that the FFNN can predict outputs from the physics-based simulator very well, but it also inherits the inaccuracies of that simulator. By training from experimental data we can capture behavior directly without knowledge of the physical parameters that are so challenging to estimate.

Figure 1 illustrates the NN structure we use for both core and cladding pumping approaches [16]. The input is the design vector [$L$ $P_f$ $P_b$ $P_s$] for core pumping and the input is [$L$ $P_f$ $P_s$] for cladding pumping as we consider only forward pumping. We examine seven wavelengths during data collection.

 

Fig. 1. Neural networks structure and the hyper-parameters in training process.

Download Full Size | PDF

We fixed the loss function to mean square error and the optimizer to Adam. We varied the number of hidden layers (from 2 to 7), the number of neurons per layer (from 10 to 300) and learning rate (from 0.01 to 0.0001) in preliminary trials with the collected data set. We evaluated several activation functions and found ReLU to provide the best results. We examined performance metrics including convergence, overfitting, and accuracy. We used only data collected for training and validation, and did not use the data collected for testing.

We found that a configuration with four hidden layers, each with the same number of neurons (100 for gain and 50 for NF) worked well on both training and validation stages. Larger neural network architectures were also examined, but the performance improvements observed were marginal. We found that hyperparameters summarized in the table on the right of Fig. 1 strike a favorable balance between effectiveness and computational complexity. These hyperparameters are used for results presented throughout the paper.

3. Data collection and metrics

3.1 Experimental setup

The amplifier gain, $G(\lambda )$, at wavelength $\lambda$ is expressed in dB as

The spontaneous emission process produces ASE noise, leading to a decrease in the signal-to-noise ratio at the amplifier output. The noise figure in dB is given by

Figure 2 shows the setup to gather experimental data sets for training, validation, and testing. We collect data from a core-pumped EDF and a cladding-pumped EYDF designed for the L-band and produced in our lab. Please refer to [17] for details on the characterization of these fibers.

 

Fig. 2. Experimental setup for gain and NF measurements with 7-channel input signal covering the extended L-band (OSA: optical spectrum analyzer).

Download Full Size | PDF

Given equipment availability in our laboratory, we use seven discrete lasers that are tuned to cover the extended L-band. Six Cobrite lasers are tuned to wavelengths are equally spaced in the range of 1575 to 1618 nm with < 0.1 nm linewidth. A seventh EXFO tunable laser (T100S-HP) laser is tuned to 1626 nm < 0.1 nm linewidth. We use a power meter to ensure that each wavelength has power of -4.95 dBm. We excite the EDF (or EYDF) using a total signal power varied via an attenuator.

For core-pumping, we independently set the forward pump power $P_f$ and backward pump power $P_b$. Both pumps are centered at $\lambda _p=$980 nm. For the cladding-pumped setup, we employ a fusionless side-pump combiner [18] to launch pump power in the cladding. As we have only one high-power pump source available in our lab, cladding pumping is strictly in the forward direction; the pump is at $\lambda _p\!=\!$915 nm so that only ytterbium ions absorb energy. We remove pump power from the output of the EYDF and use a 3-dB coupler, all to protect the optical spectrum analyzer (OSA) from large input powers.

Each length of fiber constitutes one measurement campaign. An optical attenuator was controlled by computer. The $P_{s-in}(\lambda )$ was calculated from the initial power less the attenuation. An OSA (model ANDO AQ6317B) trace was captured for each [$P_s$, $P_b$, $P_f$]. The $P_{s-out}(\lambda )$ was found for each wavelength from the OSA trace. The ASE power $P_{ASE}$ was estimated from the OSA trace using a resolution bandwidth of $B_o$=0.1 nm.

3.2 Training and testing

Experimentally, we can sweep [$P_f$ $P_b$ $P_s$] with fine granularity via automatically controlled equipment (pump sources and attenuators). The preparation of various fiber lengths [$L$] is manual and cumbersome. To complete data collection in a reasonable time, we limited our training set to 4-6 fiber lengths. We install one fiber length $L$, and sweep the signal and pump powers via computer control. We record the OSA spectrum at 0.1 nm resolution to calculate gain and NF. Each 24-hours we change the fiber length and again sweep pump and signal. Training data is collected at fiber lengths that are multiples of 10 m. Test data is collected at lengths not found in the training set; to test the limitations of predictions, we capture test data at the midpoint of fiber lengths in the training set.

The range of swept parameters is given in Table 1. Core pumping and cladding pumping ranges are different due to the much lower power conversion efficiency of cladding pumping. For the core-pumping case, we test both forward and backward pumping using standard pump sources. The test set fiber lengths are (35, 45, and 55 m) for core-pumping and (25, 35, 45, and 55 m) for cladding pumping. Technical difficulties force us to use a grid for fiber lengths, but the fiber lengths in the test set are distinct from those in the training set. Unlike the grid-based training dataset, we use randomly selected triplets ($P_s$, $P_f$, and $P_b$) for the test set. Each element of the triplet is chosen from a uniform distribution across the range of parameters in Table 1 with a very fine spacing (0.1 dB for $P_s$, and 1 mW for $P_f$ and $P_b$). This strategy allowed us to veer away from the grid-based approach used in the training set, and thus, none of the test set vectors appear in the training set. The measurement time is given in Appendix B, Table 3.

Table 1. Swept parameters (core and cladding pumping) for measurement training set; size of search space given in brackets.

View Table | View all tables in this article

We use the mean square error (MSE) as the objective function for optimization of the NN weights. The gain and NF vary with wavelength, so we choose to average the MSE across the $L$-band for our objective function. For $M$ wavelengths measured in the $L$-band at $N$ points [$L$ $P_f$ $P_b$ $P_s$] in the training set, the MSE for an epoch is found by

The flowchart on the left side of Fig. 3 shows the training process. We collect on-grid data, with a larger training set size for cladding pumping. We randomly divided the grid points to form the training and validation subsets: training 90%, validation 10%. We train two separate NN, one for gain and one for noise figure. That is, two sets of weights are trained, one minimizing the MSE in 4 with $Y$=$G_{min}$ and the other with $Y$=$NF$. On the right side of Fig. 3, we collect the test set from off-grid $L$ values and [$P_f$ $P_b$ $P_s$] chosen randomly from a fine grid. Again, the cladding-pumped test set is larger than the core-pumped. We collect statistics on the accuracy of both classic simulation and NN for gain and for noise figure.

 

Fig. 3. Flowchart of the training process for experimental NNs and testing of prediction accuracy. The on-grid training set is captured at fiber lengths that are multiples of 10 m, while the off-grid test set is captured at other lengths.

Download Full Size | PDF

Typical convergence behavior for the MSE vs. epoch can be seen in Fig. 4(a). The training error decreases sharply for the first 200 epochs. The MSE continues to decrease, and we run 7200 epochs in total. As seen in Fig. 4(b), we achieve MSE of 0.01 dB$^2$. We see no evidence of overfitting from the validation curve, i.e. the validation curve follows the same behavior as the training.

 

Fig. 4. a) MSE vs the number of epochs in the training process of core-pumped gain, and b) training/validation loss for the last epochs.

Download Full Size | PDF

Given the high impact of length on inversion level, and of inversion level on gain, coverage of fiber lengths of interest is critical. We are confident the NN has been effectively trained and can generalize to unseen data within the range of inputs included in the training set. Our outlook is murkier outside the range covered. Therefore, in our experiments we were careful to select a range of lengths that covered all useful scenarios, as determined during preliminary investigation of the doped fiber under test. Physics-based models (provided they had perfect knowledge of parameters) can cover arbitrary fiber lengths; it remains unclear to what extent the neural network can extrapolate gain dynamics outside the range of lengths used in training.

3.3 NN prediction figures of merit

We evaluate the accuracy of the converged NN model using several figures of merit. Each test case involves $M$ wavelengths, $N_t$ test cases in total. First, we consider the distribution of the absolute error between prediction $\hat {Y}_j$ and true value $Y_j$ (gain or NF). The distribution of this error encompassing all $M\times N_t$ measured wavelengths is

The minimum gain will vary across test cases, shifting the gain profile higher and lower. The absolute error may be higher when gains are higher, but still be a reasonable percentage of the gain. Therefore we also consider the relative error in a prediction, referenced to the true gain. The prediction error is calculated directly in dB without conversion to a linear scale. The distribution of the worst-case prediction across wavelengths in percentage is given by

To illustrate error metrics, we present in Fig. 5 a typical measured gain profile (black, square) with the prediction from neural network (red, star) and physics-based simulator (blue, diamond). The simulator profile shows large excursions from the true measured profile, while the NN is much more accurate. The worst-case absolute error occurs at $\lambda _1$ for NN and at $\lambda _4$ for the simulator. The worst-case relative error occurs at $\lambda _7$ for both NN and simulator. The minimum gain, $G_{min}$, of 12 dB shows that even the worst case NN error at $\lambda _1$ is relatively small.

 

Fig. 5. A typical gain profile using measurements, NN, and physics-based simulation where $L=45~$m, $P_f=790~$mW, $P_b=470~$mW, and $P_s=0.9~$dBm.

Download Full Size | PDF

4. Results

4.1 Core-pumping, gain and noise figure

We train the NN using the 1600 spectra collected for core-pumping, as illustrated on the left of Fig. 3. An additional 250 spectra are collected for the testing phase, illustrated on the right of Fig. 3, are used to assess prediction accuracy. Using the definition in (5), we collect statistics on the accuracy of the NN and physics-based simulator in terms of the absolute error in dB for all data at all wavelengths. Histograms are presented in Fig. 6(a) for gain and 6(b) for noise figure. The mean, $\mu$, and standard deviation, $\sigma$, are included as annotations, as well as an estimate of the underlying continuous probability density function. Note that the physics-based simulator used Giles-parameter measured per [7] and the set of differential equations given in the appendix A.

 

Fig. 6. Core pumping: histograms from NN and the physics-based simulator for absolute error (5) in dB for a) gain, and b) noise figure; red curves are an estimate of the underlying probability density function.

Download Full Size | PDF

The impact of slight inaccuracies in the Giles-parameters leads to very broad histograms for the physics-based simulator in the L-band. The variance is high, with outliers having error as high as 2.7 dB for gain, and 2 dB for noise figure. To appreciate the source of these inaccuracies, consider the absorption and emission values, which we measured at 0.325 dB/m and 1.155 dB/m at 1626 nm, respectively. With these values, the physics-based simulator predicts a 40 m fiber length achieving a minimum gain of 11 dB at the optimum inversion level (e.g., $N2$ = 0.44). An error of as little as 0.05 dB/m in absorption and emission can result in a 2 dB difference in predicted gain.

The mean error for the NN, and the histogram is much narrower and concentrated around the mean, with outliers at 0.9 dB for gain and 0.6 dB for noise figure. The mean error for the NN is significantly lower than for the physics-based simulator - around a factor of 6 smaller. While errors for noise figure are generally lower than for gain, trends are similar. For the sake of brevity, we confine further results to gain prediction accuracy.

4.2 Core-pumping vs. cladding-pumping

In Fig. 7 we present histograms of the worst-case error for a given vector [$L$ $P_f$ $P_b$ $P_s$], i.e., the error on the wavelength experiencing the least accurate prediction per (6). Comparing histograms in Fig. 6(a) with Fig. 7(a), the mean is now higher as only one (the highest) of the seven errors at each data point is including in the histogram. The variance of the physics-based simulator errors goes down, but the histogram remains much broader and farther to right (higher error) than the NN case.

 

Fig. 7. Histograms of absolute error (dB) on worst-case wavelength of gain prediction from NN and the physics-based simulator for a) core pumping and b) cladding pumping.

Download Full Size | PDF

We present cladding-pumped results in Fig. 7(b). Contrasting the physics-based simulator (red histograms) for core and cladding pumped, we see much greater variance in the cladding pumped case. The mean errors in physics-based simulation are almost identical for the cladding and core pumping. The variance of 0.63 vs. 0.9 is, however, quite significant. To have confidence that an amplifier design achieves its target gain requires a low standard deviation in the tool used for the design. The outliers can reach as high as 4 dB of error for cladding pumping.

The challenges in correctly estimating the physical parameters for the simulator are greater in cladding pumping, and the effect on accuracy is severe. The lowest value of the worst error from simulation for core-pumping exceeds 1 dB, while for cladding it is 0.6 dB. This is most likely due to the larger sample size (600 vs. 250 test cases). The mean, variance, and shape of the core-pumping and cladding-pumping histograms for the NN are similar. As measurement with cladding-pumping is more challenging, the mean error is larger.

The error can be put into context by examining the relative error (8) for the wavelength experiencing the worst prediction error in a given gain profile. If large errors occur when gain is small, the usefulness of the model would be compromised. Figure 8 shows the histogram of relative error in both core-pumping and cladding-pumping scenarios. For core-pumping in Fig. 8(a) we again see the entire histogram for the physics-based simulator has higher error than the NN histogram. However, the weakness of the physics-based simulator is accentuated. The shapes of the histograms are more extreme than for absolute error for worst case wavelength shown in Fig. 7. The NN histogram has narrowed, and the physics-based simulator has widened.

 

Fig. 8. Histograms of relative error (%) on worst-case wavelength of gain prediction from NN and physics-based simulator for a) core pumping and b) cladding pumping.

Download Full Size | PDF

The uncertainty of predictions in cladding-pumping in Fig. 8(b) is again greater than that of core pumping. This can be attributed to uncertainty in both the coupling efficiency for the pump and its spatial distribution. While both histograms show greater error, the NN is less effected by the experimental challenges. The physics-based model must estimate a small number of parameters very well, while NN model can rely on a large data set for training to average out any experimental errors. The low relative errors in NN predictions suggest that the model is reliable and can produce accurate results.

4.3 Analysis of errors

We present histograms of the wavelength experiencing the worst-case prediction error for a given gain spectra produced by the NN in terms of absolute error (7) in Fig. 9(a). The gain at the shortest wavelength, 1575 nm, varies much more widely with amplifier design than other wavelengths. Small changes in the inversion level can cause significant changes in gain at this wavelength. This makes modeling more challenging, and both core and cladding pumping have the most worst-case absolute errors at this wavelength.

 

Fig. 9. For core and cladding pumping, histogram of wavelength with maximum a) absolute, and b) relative error; inset in (b) is minimum gain distribution.

Download Full Size | PDF

For relative error (9), we see different behavior in Fig. 9(b). There is a much greater concentration of worst-case error at the highest wavelength, 1626 nm. For cladding pumping we see most of the occurrences at either 1575 nm or 1626 nm. To explain the difference we include an inset where we indicate the showing wavelengths at which the minimum gain occurs. For core pumping the minimum gain almost always occurs at 1626 nm. The relative error is highest here since the denominator in (9) is small. For cladding pumping the minimum gain sometimes occurs at 1575 nm, so the relative error does also. The distribution of absolute error (see Fig. 7) has low variance, with values clustered at the mean. Hence, the relative error is greatly influenced by the gain which explains why the relative error distribution resembles the minimum gain distribution.

We use the minimum gain to put error statistics into context. In Fig. 10 the top two histograms give the distribution of minimum gain for our test set used to produce the error histograms. As expected, the core pumping has higher minimum gain. The cladding pumping has a much wider distribution of minimum gain.

 

Fig. 10. For NN predictions of core and cladding pumping, the top two figures give histograms of the minimum gain (same x-axis for all figures); the bottom figure gives average worst-case error in bins with similar minimum gain, where left y-axis is relative error (%) and right y-axis is absolute error (dB).

Download Full Size | PDF

In Fig. 10 the lower plot gives the error statistics from Fig. 7 and Fig. 8, but now grouped according to the test set measurements falling in the same bin of minimum gain. The figure highlights the accuracy of the model in those cases of gain of practical interest.

In the region where the minimum gain has the highest probability, over 10.5 dB for the core-pumping and over 7.5 dB for the cladding-pumping, both the core and cladding scenarios have a similar level of error. In the regions with the highest probability for the minimum gain, the error is flat. That is, here the error exhibits no dependence on the level of gain, with a relative error of 3 % and an absolute error of less than 0.5 dB. Large excursions of relative error (up to 10 %) only occur in cladding-pumping and only for minimum gains with both very low probability and very little practical interest.

5. Exploiting fast computation time

Having established the accuracy of the NN predictions for gain and noise figure, we next consider exploitation of the NN model for amplifier design. In particular, we will exploit of the reduction in computation time for the NN model compared to the physics-based simulator. We fix the signal power to 0 dBm and execute an exhaustive search of the remaining 3-dimensional parameter space for core-pumping. We use a very fine mesh, with increments of one meter for fiber length and increments of 10 mW for pumps, yielding 156,271 vectors. The NN takes one hour to predict the gain and NF profiles, while the physics-based simulator needs 20 days. Roughly speaking, we can generate predictions 400 times faster and thus cover a vastly larger search space.

We find the minimum gain and maximum noise figure for each point in the search space. For visualization, we group together results for all points that yield the same total pump power. Among the set of equal pump power, we retain the point yielding the best performance in the group.

The color bars in Fig. 11(a) and (b) give minimum gain and maximum NF over the seven wavelengths as a function of fiber length and total pump power. We only plot the subset (30,000 points) that achieves minimum gain greater than 8 dB. The color maps illustrate that larger fiber length and pump power improves minimum gain, but degrade the NF. The fast NN-based simulator quantifies this trade-off.

 

Fig. 11. Results for core pumping when sweeping NN at $P_s=0$ dBm showing a) minimum gain, b) maximum NF, and c) the backward/forward breakout of pumping.

Download Full Size | PDF

We can use the data set to examine the forward and backward pumping strategies. In Fig. 11(c) the color bar indicates the ratio $P_f/P_b$ where the solutions with minimum gain over 8 dB. While much of the plot is for fairly equal division of pump power, we see two clear regions where power is more concentrated in one direction. In Fig. 11(c), we see that more backward pump than forward pump is only optimal at short fiber and low total pump power. In Fig. 11(a), we see this case is of limited interest as it yields overall low minimum gain. More forward pump is optimal at long fiber length and moderate total pump power, but this choice can give high NF. Keeping shorter lengths helps NF. With our NN-based simulator we can examine all mixes of backward and forward to make better trade-offs.

We can also use the NN-based simulator to find a reasonable balance between gain and NF in an amplified link. We take the example of a link that compensates for 100 dB attenuation using $N_a$ amplifiers. We use the development in [19] that assumes for each segment $GL_s$ = 1 ($L_s$ represents losses of section) and that all wavelengths experience the same gain. To respect these assumptions, we follow our amplifier designs with an appropriate gain flattening filter so that all wavelengths see the minimum gain $G_{min}$ of the amplifier. An amplifier is considered only if the minimum gain is sufficient compensate the loss of a single span (100 dB split across $N_a$ spans). Each span is considered a single-stage amplifier followed by a gain flattening filter. The gain flattening filter will affect the overall noise figure, but as it located after amplification, the affect at a single stage will be negligible. We approximate the total noise figure by the amplifier NF.

Using the development in [19], the gain and NF determine the OSNR per wavelength in dB as

The worst case OSNR across wavelengths is presented in Fig. 12 as a function of the total pump power and the amplifier fiber length. The number of amplifiers in the link, $N_a$, varies with [$L$ $P_f$ $P_b$]; the regions of fixed $N_a$ at its value are shown in white in Fig. 12. The high gain region is where the fewest (8) amplifiers are required, but this region has low OSNR. We zoom in on the region with highest OSNR, i.e., at the short amplifier fiber lengths. We highlight three points with different optimization strategies. Point A, with the lowest pump power, has the best OSNR of 23 dB but needs 12 optical amplifiers. Point B requires twice the pump power, but reduces the number to 10 optical amplifiers with only a 0.5 dB reduction in OSNR compared to point A. At point C, three amplifiers can be saved, but triple the pump power of point A is required, and the OSNR is reduced by 1 dB. The fast computation allows us to quantify the trade-offs available.

 

Fig. 12. (a) OSNR after compensating 100 dB of attenuation in an amplified link (inset numbers are required number of amplifiers), and (b) zoom on good OSNR area and three potential optimizations.

Download Full Size | PDF

6. Discussion

The exploitation of an amplifier follows many steps. First there is the design of one or several doped fibers. Secondly, these fibers are incorporated into a design that may include multiple stages and multiple fibers. Finally, in responding to specific system needs, deployed amplifiers may be adjusted via manipulation of pump powers, pumping direction, and the use of dummy signal channels, etc. The work in this paper relates to the second step, working with results produced in the first step. Our neural networks are for single stage amplifiers, but they set the stage for models of mid-stage amplifiers as well.

There has been considerable interest in the use of neural networks in the final step of exploitation of fiber amplifiers. Data can be collected in large quantity during live deployment. Data has been collected for deployed chains of amplifiers as the network sees variations in signal power over multiple wavelength channels [6]. The data is used to train NN tools permitting informed adjustment of operating parameters for the network (dummy signal channels, pumps, add/drop decisions, etc.) [5]. Thus, the NN is an operational tool rather than a design tool. Clearly decisions have already been made on what doped fiber to use, the fibers have already been cut, the pumps deployed, etc.

Our work falls squarely in the role of design tool. The signal inputs to the model can be used to target a specific power input specification, or to explore designs that are robust to signal power variations. The pump inputs to the model can be used to meet gain targets or to find power efficient solutions. The length input to the model is the most critical and unique in our solution. The length has a significant impact on inversion levels and thus optimization. Our examination of a single stage with two pumping directions and two pumping modalities (core and cladding) is a proof of concept for the use of neural networks to replace physics-based simulators. Our NN models are a step towards developing NN models for more complex amplifiers with, for example, hybrid fiber designs [20] or less know amplifier materials such as bismuth [21].

Future work should expand the neural network modeling to meet requirements for multistage amplifier designs. Multiple stages are the path to high performance, and they require very expert judgement on choice of doping profile, intermediate gain targets, etc. The NN model can improve that decision process. Our tool may also be of interest for deployed networks. A model used in design could be streamlined to model only the final product with the length fixed. The model could be used to adjust operating parameters to meet client-specific performance targets, both static and dynamic.

7. Conclusion

In summary, we were able to collect data for training and testing our NN model experimentally. Compared to the parameter-based simulation method, our trained NN provided a faster and more accurate prediction of gain and NF for core pumping and cladding pumping scenarios. The results revealed that the accuracy of prediction using NN remains unaffected by both the fiber type and the pumping method employed. Moreover, we showed that the average of absolute error and relative error remains consistent and does not rely on the gain values, exhibiting a stable and constant trend. Our approach of combining the NN with fine resolution sweeping enables efficient exploration of the design space and provides insights into the trade-offs between gain and noise figure for different amplifier configurations. We could find the best amplifier design for a given link, with trade-offs that can be customized.

A. Physics-based simulator

The model is based on the standard set of power propagation and population rate equations of Er/Yb-co-doped fibers which neglects back-transfer and considers only the Er$^{3+}$ I$_{15/2}$ and I$_{13/2}$ energy levels. Since our doped fiber amplifiers are designed as single-mode amplifiers, we only consider LP$_{01}$ for the signal in simulations. For cladding pumping, we assume the pump power distribution is uniform. Table 2 lists the parameters and constants in the model. The following propagation equations for the pump power, $P_p$, the signal, $P_s$, and the ASE, $P_{ASE}$, are solved using 4$^{th}$ order Runge-Kutta. Note that the emission and absorption cross-sections were estimated by measurements on our fabricated fibers.

(11)$$\begin{cases} R_{12,Er,k}(z)=\frac{\sigma_{p,Er}\Gamma_{p,k}P_p(z)}{h\nu_p} \\ W_{12,k}(z)=\sum_{\lambda,s}\frac{\sigma_{a,Er,\lambda}\Gamma_{\lambda,k}P_{s,\lambda}(z)}{h\nu_\lambda}+\sum_{\lambda,ASE}\frac{\sigma_{a,Er,\lambda}\Gamma_{\lambda,k}P_{ASE,\lambda}(z)}{h\nu_\lambda} \\ W_{21,k}(z)=\sum_{\lambda,s}\frac{\sigma_{e,Er,\lambda}\Gamma_{\lambda,k}P_{s,\lambda}(z)}{h\nu_\lambda}+\sum_{\lambda,ASE}\frac{\sigma_{e,Er,\lambda}\Gamma_{\lambda,k}P_{ASE,\lambda}(z)}{h\nu_\lambda} \\ R_{12,Yb,k}(z)=\frac{\sigma_{p,Yb}\Gamma_{p,k}P_p(z)}{h\nu_p} \end{cases}$$

(12)$$\begin{cases} n_{2,Er,k}(z)=\frac{\left(R_{12,Er,k}(z)+W_{12,k}(z)+A_kK_{tr}n_{2,Yb,k}(z)\right)\rho_{Er,k}(z)}{R_{12,Er,k}+W_{12,k}(z)+W_{21,k}(z)+A_kK_{tr}n_{2,Yb,k}(z)+\frac{A_k}{\tau_{Er}}} \\ n_{2,Yb,k}(z)=\frac{R_{12,Yb,k}(z)\rho_{Yb,k}(z)}{R_{12,Yb,k}(z)+A_kK_{tr}n_{2,Yb,k}(z)+\frac{A_k}{\tau_{Yb}}} \\ \rho_{Er,k}(z)=n_{1,Er,k}(z)+n_{2,Er,k}(z) \\ \rho_{Yb,k}(z)=n_{1,Yb,k}(z)+n_{2,Yb,k}(z) \end{cases}$$

(13)$$\begin{cases} \text{core pumping: } \frac{dP_p(z)}{dz}={-}\sigma_{p,Er}\bigg{(}\sum_{k=1}^{K}n_{1,Er,k}(z)\Gamma_{p,k}\bigg{)}P_p(z) \\ \text{cladding pumping: } \frac{dP_p(z)}{dz}={-}\sigma_{p,Yb}\bigg{(}\sum_{k=1}^{K}n_{1,Yb,k}(z)\Gamma_{p,k}\bigg{)}P_p(z) \\ \begin{aligned}\text{forward-propagation: }\frac{dP_s(z)}{dz}=&\bigg{(}\sigma_{e,Er,\lambda}\bigg{(}\sum_{k=1}^{K}n_{2,Er,k}(z)\Gamma_{\lambda,k}\bigg{)}\\ &-\sigma_{a,Er,\lambda}\bigg{(}\sum_{k=1}^{K}n_{1,Er,k}(z)\Gamma_{\lambda,k}\bigg{)}-\alpha_s\bigg{)}P_{s,\lambda}(z)\end{aligned} \\ \begin{aligned}\text{backward-propagation: }\frac{dP_s(z)}{dz}=&\bigg{(}-\sigma_{e,Er,\lambda}\bigg{(}\sum_{k=1}^{K}n_{2,Er,k}(z)\Gamma_{\lambda,k}\bigg{)} \\&+\sigma_{a,Er,\lambda}\bigg{(}\sum_{k=1}^{K}n_{1,Er,k}(z)\Gamma_{\lambda,k}\bigg{)}-\alpha_s\bigg{)}P_{s,\lambda}(z) \end{aligned}\\ \begin{aligned}\frac{dP_{ASE}(z)}{dz}=&\bigg{(}\sigma_{e,Er,\lambda}\bigg{(}\sum_{k=1}^{K}n_{2,Er,k}(z)\Gamma_{\lambda,k}\bigg{)}-\sigma_{a,Er,\lambda}\bigg{(}\sum_{k=1}^{K}n_{1,Er,k}(z)\Gamma_{\lambda,k}\bigg{)}-\alpha_s\bigg{)}P_{ASE,\lambda}(z)\\ &+\sigma_{e,Er,\lambda}\bigg{(}\sum_{k=1}^{K}n_{2,Er,k}(z)\Gamma_{\lambda,k}\bigg{)}2h\nu_\lambda\Delta\nu_\lambda \end{aligned}\end{cases}$$

Table 2. Main parameters in Er/Yb-co-doped fiber simulation.

View Table | View all tables in this article

B. Measurement time

Once the setup is in place, the measurement time is dominated by the OSA capture to file. During the core pumping measurements our script took approximately 135 seconds for each measurement. We updated our methodology (capturing entire sweeps rather than measurements on individual wavelengths) for the cladding pumping measurements, reducing the time to 83 seconds. Large data set collection (training sets) was conducted overnight, requiring approximately 15 hours for both scenarios. The smaller data sets for test (at midpoint fiber lengths) were collected during working hours. Hence, within a 24-hour period, we gathered a large training set and a short test set. We summarized these details in an appendix.

Table 3. Measurement time

View Table | View all tables in this article

Estimating the Giles parameters presents challenges very different from training and test set collection. The physics-based simulator requires estimation of several parameters and multiple pieces of equipment and measurement techniques. For instance, we must measure absorption for pump power, estimate background loss, excited-state absorption (ESA), pair induced quenching (PIQ), erbium concentration, fiber core size, among other factors, contributes to the complexity of the estimation.

The data collection methodology is different, but of similar effort in terms of human intervention for both NN and Giles parameter estimation. Much of data collection for NN can be automated, so longer collection times (to increase the training set size to improve accuracy) do not translate to more researcher hours. The automated approach would not aide Giles parameter estimation, as the uncertainties and inaccuracies stem from inherent challenges in the L-band (the low emission cross-section and the presence of excited state absorption) [7].

Funding

Natural Sciences and Engineering Research Council of Canada (CRDPJ 538379-18).

Disclosures

The authors declare that they have no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. P. J. Winzer, D. T. Neilson, and A. R. Chraplyvy, “Fiber-optic transmission and networking: the previous 20 and the next 20 years,” Opt. Express 26(18), 24190–24239 (2018). [CrossRef]  

2. G. P. Agrawal, Fiber-optic communication systems (John Wiley & Sons, 2012).

3. F. Q. Kareem, S. R. Zeebaree, H. I. Dino, et al., “A survey of optical fiber communications: challenges and processing time influences,” Asian J. Res. Comput. Sci. 7, 48–58 (2021). [CrossRef]  

4. J. Yu, S. Zhu, C. L. Gutterman, et al., “Machine-learning-based EDFA gain estimation,” J. Opt. Commun. Netw. 13(4), B83–B91 (2021). [CrossRef]  

5. F. Musumeci, C. Rottondi, A. Nag, et al., “An overview on application of machine learning techniques in optical networks,” IEEE Commun. Surv. Tutorials 21(2), 1383–1408 (2019). [CrossRef]  

6. A. D’Amico, S. Straullu, A. Nespola, et al., “Using machine learning in an open optical line system controller,” J. Opt. Commun. Netw. 12(6), C1–C11 (2020). [CrossRef]  

7. H. Feng, S. Jalilpiran, F. Maes, et al., “Characterization of Giles parameters for extended L-band erbium-doped fibers,” J. Opt. Soc. Am. B 39(7), 1783–1791 (2022). [CrossRef]  

8. M. Sharma, F. Maes, L. Wang, et al., “Data-driven optimization of giles parameters of super L-band erbium doped fibers,” in European Conference and Exhibition on Optical Communication, (Optica Publishing Group, 2022), pp. We5–1.

9. K. Choudhary, B. DeCost, C. Chen, et al., “Recent advances and applications of deep learning methods in materials science,” npj Comput. Mater. 8(1), 59 (2022). [CrossRef]  

10. S. Zhu, C. L. Gutterman, W. Mo, et al., “Machine learning based prediction of erbium-doped fiber WDM line amplifier gain spectra,” in 2018 European Conference on Optical Communication (ECOC), (IEEE, 2018), pp. 1–3.

11. Y. You, Z. Jiang, and C. Janz, “OSNR prediction using machine learning-based EDFA models,” in 45th European Conference on Optical Communication (ECOC 2019), (IET, 2019), pp. 1–3.

12. A. Donodin, U. C. De Moura, A. M. R. Brusin, et al., “Neural network modeling of bismuth-doped fiber amplifier,” J. Eur. Opt. Society-Rapid Publ. 19(1), 4 (2023). [CrossRef]  

13. H. Rabbani, S. Jalilpiran, S. LaRochelle, et al., “Neural networks for fiber amplifier design optimization using experimental training sets,” in CLEO: Science and Innovations, (Optica Publishing Group, 2023), pp. SF3H–2.

14. S. Jalilpiran, V. Fuertes, J. Lefebvre, et al., “Baria-silica erbium-doped fibers for extended l-band amplification,” Journal of Lightwave Technology (2023).

15. M. Bolshtyansky, I. Mandelbaum, and F. Pan, “Signal excited-state absorption in the L-band EDFA: Simulation and measurements,” J. Lightwave Technol. 23(9), 2796–2799 (2005). [CrossRef]  

16. “How Feedforward neural networks Work,” https://builtin.com/data-science/feedforward-neural-network-intro.

17. L. Wang, M. Sharma, F. Maes, et al., “Low cost solution for super L-band fiber amplifier based on single-mode and multi-mode hybrid pumping scheme,” in Optical Fiber Communication Conference, (Optica Publishing Group, 2022), pp. W3J–4.

18. C. Matte-breton, S. LaRochelle, S. Duval, et al., “Method of coupling optical fibers, and optical coupler,” (2022). US Patent App. 17/636,397.

19. G. Keiser, Optical fiber communications, vol. 2 (McGraw-Hill New York, 2000).

20. C. Matte-Breton, L. Wang, F. Maes, et al., “Concentric layers with heterogeneous doping for cladding-pumped l-band fiber amplifiers,” in Next-Generation Optical Communication: Components, Sub-Systems, and Systems XI, vol. 12028 (SPIE, 2022), p. 1202802.

21. Y. Wang, S. Wang, A. Halder, et al., “Bi-doped optical fibers and fiber amplifiers,” Opt. Materials: X 17, 100219 (2023). [CrossRef]