In this section we briefly review the research that has been performed in the PhD thesis of Kristof Vandoorne [1]. The main focus of his dissertation was to demonstrate the use of SOAs in the context of reservoir computing.
However, it is not immediately obvious why such a network of SOAs makes a good reservoir. The steady-state input-output curve, although similar to the upper half of the hyperbolic tangent function which is classically used, lacks the symmetric lower branch because optical power cannot be negative. Also, the topology is restricted, because the chip is planar, and too many crossings should be avoided due to cross-talk and losses (this was discussed previously in 4.6). The increased complexity of combiners and splitters with a high fan-in resp. fan-out further increase the topology constraints.
On the other hand, advantages are that the SOA has richer internal dynam- ics as opposed to the static neurons which are used in software. As we alluded to in the introduction chapter, the fact that an optical signal has a phase and amplitude, will play an important role in the performance of the reservoir.
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(b)Figure 5.2: After taking the average over time of the output classifiers, we ap-
ply the winner-take-all principle. The winning sample is the one with the highest positive output. In (a), the highest output corresponds to spoken digit ’7’, which is correctly recognized. In (b), digit ’5’ is chosen as wrong answer.
We begin by introducing the SOA. We then explain which topology was used, and highlight the most important results that were achieved.
5.2.1 Semiconductor Optical Amplifiers
Semiconductor Optical Amplifiers (SOAs) can provide a high gain over short dis- tances and can be electrically pumped. The operation of SOAs is similar to that of semiconductor lasers. Electrons from the valence band are excited to the conduction band, either through electrical or optical pumping. An incoming photon can interact with the excited electron, forcing the electron to release its energy and to return to the valence band. During the process, a new photon is emitted with exactly the same frequency, phase and direction. This process is called stimulated emission. More detail about the process can be found for example in [13]. SOAs are usually made from III-V compound semiconductors such as InP/InGaAsP and GaAs/AlGaAs.
5.2.2 Topology
Imagine a series of SOAs that are connected to each other on a single line. In- formation is always inserted from the left of each SOA, which means that infor- mation only flows from left to right. Because the SOA has no reflection, there is no communication from right to left. This is also true for a waterfall topology, as shown in Figure 4.11: SOA 2 does not influence SOA 1 if it is after SOA 2 (which means either right or down from SOA 1).
Because a unidirectional topology of non-reflecting components does not contain network memory1, a swirl topology is used, as shown in Figure 4.7(a). In this topology, there are a large number of feedback cycles, which improve the connectivity between the individual SOAs. Because the SOAs amplify the input signals, an additional attenuation term is used in order to avoid lasing2, or in other words, to avoid creating a loop somewhere in the circuit with a gain larger than one. Alternatively, the input current can be reduced to decrease the gain in the SOA.
1For the PhCC, however, we can afford to use the waterfall topology, because there each cavity
has a certain amount of reflection, which causes it to communicate to cavities on the left of, or above this cavity.
2With the current framework, laser operation, resulting from resonating cavities (where the cavity
is modeled as an explicit building block), cannot be simulated correctly. This is mainly because for laser operation, we need multiple modes that operate at different frequencies, i.e. equation 4.8. This is not yet supported in the current version. This is different from the case where we see the laser as a single black box component with rate equations: this can be modeled in the current framework because the rate equations are ordinary differential equations.
5.2.3 Summary of previous results
The metric that is used to evaluate the performance of the reservoir is the Word Error Rate (WER), which defines the ratio between wrongly predicted digits ver- sus the total number of available digits.
The main conclusion is that the SOA reservoir (i.e., with coherent simula- tions) outperforms the standard (incoherent) hyperbolic tangent reservoir (of the same size) for the isolated digit recognition task, for optimal parameters.
First, the influence of the delay between the neurons was investigated for both a classical ESN and the SOA network. As can be seen from Figure 5.3, the performance is better for an optimal delay that equals approximately half the duration of a spoken digit (in this case, this is a delay of 30 times the SOA delay). Also, the effect of using amplitude and phase (called a coherent simulation) compared to using only the magnitude of the signal (called an incoherent sim- ulation), is shown in Figure 5.3. The improved performance for a coherent sim- ulation was explained as follows: the state space of a complex-valued reservoir contains twice as many variables as that of a real-valued reservoir. Although the number of observed variables remains unchanged (only the magnitudes are used for the readout layer), there is internal interaction between magnitudes and phases (through complex addition of signals). As a result, the additional richness of transformations in state space is also present in the magnitudes. Because the internal state-space is richer, the observed signals are more diverse and therefore more useful to approximate the desired output.
The attenuation between the connections is also relevant: the SOAs amplify the signal, and it has to be attenuated again in order to prevent lasing and/or chaotic behavior of the network. This behavior will most likely render the reser- voir useless. This is equivalent to using a classical hyperbolic tangent discrete- time reservoir with a spectral radius that is very large (and larger than one).
5.2.4 Simulation framework
All research performed in the PhD of K. Vandoorne was done using the Matlab toolbox developed at the Department of Electronics and Information Systems (ELIS)3. Since then, a new framework was developed, called the OrGanic En- vironment for Reservoir computing (OGER) [14]. OGER, based on the free pro- gramming language Python, is an open-source machine learning toolbox based on the Modular toolkit for Data Processing (MDP) [15]. This framework allows one to easily run many simulations in parallel, along with a more modular ap- proach which makes it easier to set up a simulation with fewer lines of code and better reproducibility.
delay [ps] 4 6 8 10 12 14 0 50 100 150 200 250 300 350 400
coherent
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WER [%] SOA tanh incoherent coherent (Φ controlled)Figure 5.3: WER for the isolated digit recognition task, for a network of SOA
and a classical hyperbolic tangent network. The SOA network, when simulated in a coherent regime (i.e., using complex-valued signals), performs better than a classical, real-valued (incoherent) hyperbolic tangent network. Clearly, there is an optimal value for the intercon- nection delay, which turns out to be approximately half of the word length (picture courtesy of K. Vandoorne).