###### Abstract  
Recently, atmospheric-turbulence-induced fading in free-space optical (FSO) communication with pointing error impairment was modeled and studied using the Fisher-Snedecor \(\mathcal{F}\) distribution with a good fit to experimental data. In this letter, we investigate the end-to-end performance of dual-hop FSO fixed-gain relaying systems operating over \(\mathcal{F}\) turbulence channels. More specifically, we present closed-form expressions for the cumulative distribution function and the probability density function of the end-to-end signal-to-noise (SNR) ratio of the proposed system. Consequently, the outage probability, ergodic capacity, and average bit error rate performance are derived with tight asymptotic results in high-SNR regimes to gain more insight into the impacts of system parameters and channel turbulence conditions. Finally, Monte Carlo simulations are provided to validate the analytical results, revealing a significant performance gain compared to a single FSO link in the medium- to high-SNR range by using dual-hop FSO relaying. free-space optical (FSO) communication; \(\mathcal{F}\) turbulence; cooperative system +
Footnote †: journal: _Photonics_  
On the other hand, FSO provides high data rates only for short-range communication. In long-range transmission, the performance is significantly degraded by the effects of atmospheric-turbulence-induced fading, fog attenuation, and pointing errors, which lead to frequent link outages. To address various turbulence issues with other limitations of long-range communication, relaying techniques have been employed as an efficient alternative. Various relaying-assisted systems have proposed integrating FSO into outdoor and indoor broadcasting systems. A mixed FSO-radio-frequency relaying system was considered in [3] with the GG turbulence channel and Rayleigh fading RF channel. In [9], the authors presented a comprehensive outage performance analysis of a mixed RF-FSO communication system, where the RF link was distributed over the Nakagami-\(m\) channel, while the FSO link followed GG turbulence. Additionally, the performance of a cooperative power-line communication-FSO communication system was analyzed in terms of the outage probability and average bit error rate (BER) in [10]. However, dual-hop FSO systems have attracted little attention in the literature when considering the impact of the \(\mathcal{F}\) distribution, which is the primary purpose of this letter. In this letter, with the great modeling performance and low computational cost of the \(\mathcal{F}\) distribution, we extend previous studies on cooperative communication by considering the \(\mathcal{F}\) statistical model for the FSO link and presenting an exact closed-form analysis of the end-to-end performance of dual-hop FSO systems. Although the proposed system enables optical wireless connectivity to reach remote areas via the open nature of optical wireless communication, the use of amplify-and-forward (AF) relays complicates the expression of the end-to-end signal-to-noise ratio (SNR) and requires the manipulation of special mathematical functions. More precisely, we present an analytical framework that takes into account the impacts of both pointing errors and various types of turbulence to analyze the performance of the proposed systems with both heterodyne detection (HD) and intensity modulation/direct detection (IM/DD) techniques. More specifically, the probability density function (PDF) and cumulative distribution function (CDF) of the proposed system's end-to-end SNR are derived. As a consequence, the statistical characterization of the SNR is then used to develop tractable closed-form expressions for the outage probability (OP), ergodic capacity (EC), and average bit error rate (BER). Additionally, we provide asymptotic expressions for the achievable diversity gain to obtain important insights into the impacts of system parameters.