In order to realize environmentally friendly networks, energy-efficient technologies are essential. In Fiber Wireless (FiWi) networks that consists of optical and wireless networks, there are two main power saving mechanisms, namely, Power Saving Mode (PSM) and Optical Network Unit (ONU) sleep. PSM works in wireless networks, and ONU sleep works in optical networks. These two mechanisms work independently in FiWi networks. Since both PSM and ONU sleep turn off the node's transmission device to save energy, the throughput of the network decreases. Thus, a trade-off relationship between throughput and energy consumption exists. Therefore, taking account of this relationship is essential when discussing energy efficiency in these networks. In this paper, we focus on the effect of PSM and ONU sleep on throughput and energy consumption. We analyze both the energy consumption and the throughput in FiWi networks. Through analysis, we point out energy and throughput inefficiencies in FiWi networks and propose a novel method that determines the optimal ONU sleep period and behavior in order to increase throughput and decrease the energy consumption. Furthermore, we validate our proposed method through numerical analysis, and confirm that it improves both throughput and energy consumption.