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The capacity and geographical coverage of the global communications network continue to expand. One consequence of this expansion is a steady growth in the overall energy consumption of the network. This is the first of two papers that explore the fundamental limits on energy consumption in optical communication systems and networks. The objective of these papers is to provide a framework for understanding how this growth in energy consumption can be managed. This paper (Part I) focuses on the energy consumption in optically amplified transport systems. The accompanying paper (Part II) focuses on energy consumption in networks. A key focus of both papers is an analysis of the lower bound on energy consumption. This lower bound gives an indication of the best possible energy efficiency that could ever be achieved. The lower bound on energy in transport systems is limited by the energy consumption in optical amplifiers, and in optical transmitters and receivers. The performance of an optical transport system is ultimately set by the Shannon bound on receiver sensitivity, and depends on factors such as the modulation format, fiber losses, system length, and the spontaneous noise in optical amplifiers. Collectively, these set a lower bound on the number of amplifiers required, and hence, the amplifier energy consumption. It is possible to minimize the total energy consumption of an optically amplified system by locating repeaters strategically. The lower bound on energy consumption in optical transmitters and receivers is limited by device and circuit factors. In commercial optical transport systems, the energy consumption is at least two orders of magnitude larger than the ideal lower bounds described here. The difference between the ideal lower bounds and the actual energy consumption in commercial systems is due to inefficiencies and energy overheads. A key strategy in reducing the energy consumption of optical transport systems will be to reduce these inefficiencies and overheads.