_applied_  
_sciences_  
_Article_  
Design of Mode-Locked Fibre Laser with Non-Linear Power  
and Spectrum Width Transfer Functions with a  
Power Threshold  
[PERSON] 1, [PERSON] 1,2, [PERSON] 3,* and [PERSON] 1,4  
Footnote 16: https:language translation, to advanced research and industry-oriented tasks, such as inverse imaging problems or high-speed communications. Recently, RNNs have been applied in optical communications for non-linear signal distortion compensation [4, 5, 6] and non-linearity mitigation for advanced high-order modulation formats [7, 8, 9]. While RNNs enable the solution of complex tasks, the complexity of RNNs themselves increases the amount of time required for training and processing. This presents a computational challenge and requires novel computational approaches. Thus, there is a demand for novel electronic and optical hardware platforms to address speed and energy requirements for successful implementation of RNNs. An optical neuron is a key component of such technologies, from the seminal Fitzhugh-Nagumo neuron [10, 11] with its electronic realisation in the 1960s and the silicon neuron designed in the 1990s [12], to a silicon analogue of the eye's retina [13] and the Neurogrid network simulating one million neurons [14]. Among different underlying platforms, there are also the advanced silica-wafer-based SpiNNaker and BrainScaleS computing platforms of the Human Brain Project [15], while a 294-fold acceleration against a conventional computer was demonstrated in [16] using silicon photonic weight banks. At the same time, there have been developments in semiconductor RNNs with optical delayed feedback [17] and fibre-based RNNs with gigabyte-per-second speed [18]. Such architectures boast a large number of neurons and intrinsic layers. An echo state network (ESN) [19] is a type of reservoir computing (RC) [20] approach to the design of RNNs, which reduces training complexity as only output weights are required to be trained. The system can be realized by randomly connected non-linear nodes, which mimic neurons. Moreover, it has been demonstrated that only one non-linear node is required to realise RNNs as time- [21] or frequency-multiplexing [22] can be used to create virtual neurons. Such architectures have already been demonstrated to have practical applications, including in high-speed signal processing, in particular, in optical communications [23, 24, 25, 26, 27]. The all-optical implementation of ESN enables high-speed signal processing and can provide the basis for a new generation of high-speed computing hardware [17, 28] and address ever-growing demands on signal-processing speed. Optical neurons are often based on lasers, as it is a mature technology with a broad range of applications and enables control and flexibility in the design. Recently, semiconductor vertical-cavity surface-emitting laser (VCSEL) [29, 30] and graphene-based [31] laser neurons have been developed. Optical demonstration of ESN with GHz bandwidth was based on a semiconductor laser [17] for applications in optical networks and communications for amplitude-modulated-signal processing [24]. Higher bandwidth enables higher signal processing speed; moreover, current developments in optical communications and 5/6G employ ultra-wideband signals [32]. However, a fundamental limitation on optical RNN speed is imposed by the underlying material and is related to the relaxation time of the media. Fibre as a processing medium offers a tens-of-THz-order bandwidth window. Fibre-based ESN analogue (FESNA) [25] was the first demonstration of THz bandwidth neuromorphic signal processing. Moreover, fibre, in particular, multi-mode fibre, unlocks multiple degrees of freedom enabling multi-dimensional signal processing [25, 33], which is crucial for the solution of complex problems where multiple dimensions need to be processed simultaneously.