  We analyze a secure two-hop mixed radio frequency (RF) and underwater
wireless optical communication (UWOC) system using a fixed-gain
amplify-and-forward (AF) relay. The UWOC channel is modeled using a unified
mixture exponential-generalized Gamma distribution to consider the combined
effects of air bubbles and temperature gradients on transmission
characteristics. Both legitimate and eavesdropping RF channels are modeled
using flexible $\alpha-\mu$ distributions. Specifically, we first derive both
the probability density function (PDF) and cumulative distribution function
(CDF) of the received signal-to-noise ratio (SNR) of the mixed RF and UWOC
system. Based on the PDF and CDF expressions, we derive the closed-form
expressions for the tight lower bound of the secrecy outage probability (SOP)
and the probability of non-zero secrecy capacity (PNZ), which are both
expressed in terms bivariate Fox's $H$-function. To utilize these analytical
expressions, we derive asymptotic expressions of SOP and PNZ using only
elementary functions. Also, we use asymptotic expressions to determine the
optimal transmitting power to maximize energy efficiency. Further, we
thoroughly investigate the effect of levels of air bubbles and temperature
gradients in the UWOC channel, and study nonlinear characteristics of the
transmission medium and the number of multipath clusters of the RF channel on
the secrecy performance. Finally, all analyses are validated using Monte Carlo
simulation.
