## 1 Introduction
However, PAM is not without challenges, mainly when deployed in high-speed communication scenarios, as it is susceptible to intersymbol interference (ISI) and system non-linearities [13]. Consequently, in numerous optical communication systems, PAM is often accompanied by an equaliser, such as a decision feedback equaliser (DFE) [13; 14]. This strategic pairing ensures the mitigation of potential issues, enhancing the reliability and performance of VCSEL-based OWC systems employing PAM. While DFEs effectively address ISI, non-linearity renders them inefficient, particularly for higher modulation levels. Therefore, non-linear DFEs like the Volterra-series equaliser and neural networks (NNs) are utilised to overcome ISI and non-linear effects. For instance, ref. [15] demonstrated that recurrent NNs outperform conventional DFEs in non-linear channels. Feedforward NNs, such as multilayer perceptrons (MLPs), radial basis functions (RBFs), and long short-term memory (LSTM), have been tested on optical channels using PAM-\(M\)[8; 16; 17]. Specifically, ref. [16] showed that MLP outperforms both conventional DFEs and Volterra-series equalisers. To the best of the authors' knowledge, few experimental studies have reported using 850 nm VCSELs with PAM-\(M\) for OWC transmissions, especially with launched optical powers below the AEL. The work outlined in [18] that uses 850 nm VCSELs reported a bit rate of 25 Gbps, but this was with a 7 \(\times\) 7 array with an optical power of \(\sim\)22 dBm. The study in [8] employed a deep LSTM NN for effective ISI compensation in VCSEL-based OWC, achieving a data rate of 13.5 Gbps. However, the experimental work was limited to PAM-2, and the NN used at least 100 neurons per layer. This paper presents the first experimental evaluations of directly modulated eye-safe single-mode VCSELs (SM-VCSELs) operating at 850 nm for high-speed OWC links. The evaluation incorporates PAM-\(M\) and employs an NN equaliser conducted over a 2.5 m OWC link. The SM-VCSEL is launched with a transmitted optical power of \(-1.47\) dBm, which is securely below the AEL of \(-1.10\) dBm, ensuring its adherence to the class-1 laser safety standards and thus complying with eye safety. Compared to multimode VCSELs (MM-VCSELs), SM-VCSELs offer advantages such as narrower spectral widths and lower-intensity noise [6]. We obtain a bit rate of about 29 Gbps at a link distance of 2.5 m with PAM-4 modulation and an NN equaliser. The main contributions of this work are summarised as follows:  
1. We show the first use of NN equalisers in an eye-safe 850 nm SM-VCSEL-based OWC that employs PAM-\(M\) schemes of up to 16 levels. The equalisers demonstrate superior performance to the conventional DFEs, particularly for high-level PAM [8; 16], which requires less bandwidth but suffers from higher device non-linearity. This is especially beneficial in OWC links using photodiodes with larger detection areas that offer higher power margins but lower bandwidths. While some studies have employed NN equalisers for OWC with PAM up to eight levels [8; 17; 19; 20], this paper is, to the best of the authors' knowledge, the first experimental work to achieve a multigigabit OWC link with PAM-16 and NN equalisers by leveraging the SM-VCSELs' high signal-to-noise ratio (SNR). 2. Previous work on NN equalisation for VCSEL-based OWC employed multiple hidden layers with over 100 neurons per layer. The NN equaliser for this study require less computational complexity, utilising only one hidden layer with six neurons. 3. We highlight the non-linearity compensation capability of the NN equaliser by examining two modulation conditions for the VCSELs, with the first condition exhibiting lower non-linearity than the second. Notably, NN equalisers perform better in the second scenario. Additionally, for low-level PAM [2; 4], NN equalisers provide superior bit rate performance compared to DFEs with MM-VCSELs, despite MM-VCSELs having higher non-linearity than SM-VCSELs.