# On the Capacity of Optical Backbone Networks
## 1 Introduction  
In recent years, the volume of network traffic has been experiencing almost exponential growth. This phenomenon can be attributed to various factors, such as the widespread adoption of cloud services, the expansion of video distribution platforms and social media, and the emergence of new technologies like 5G and beyond (5G++) [1]. To address this situation, optical networks have emerged as a crucial asset. Optical networks are high-capacity communication infrastructures that utilize light for transmission, processing, and routing of information These networks vary in terms of distance and capacity, falling into several tiers: (1) long-haul networks, such as submarine cables and backbone networks, span extensive geographic distances and offer huge capacities (in the order of dozens of Tbit/s); (2) medium-haul networks, like metro networks, cover cities or metropolitan areas, handling data transmission in the range of hundreds of Gbit/s; (3) short-haul networks, such as access networks, also known as \"last-mile networks\", encompass small areas, connecting end-users to the network providers and delivering data rates in the order of a few Gbit/s. In particular, the advent of the fifth-generation fixed network (F5G) allows for supporting access data rates per user larger than 1 Gb/s, with future enhancements promising rates per user up to 10 Gb/s [2]. The most remarkable feature of optical networks is the fact that they use optical fibers as a transmission medium. An optical fiber is a very thin glass filament with a diameter the size of a human hair that has the tremendous advantage of incurring very small losses as the light propagates through it. The first proposal to use optical fibers to carry information was made in 1966 and came from [PERSON] [3]. Since that proposal, optical fiber communications have witnessed enormous progress over the last six decades. Several breakthroughs have contributed to that progress; among them, one can refer to the emergence, during the 1990s, of erbium-doped fiber amplifiers (EDFA) andwavelength division multiplexing (WDM), as well as optical coherent detection in the early 20 th century [4; 5]. Optical amplifiers like EDFAs make it possible to compensate for the fiber losses in the optical domain, avoiding the use of expensive repeaters. Optical coherent detection allows us to increase the number of degrees of freedom to carry information in comparison with traditional direct detection, since it makes use of both the amplitude and phase of the optical signals, while direction detection relies only on intensity. WDM is a fundamental technology in the optical networking field as it enables the transmission of large amounts of data across long distances. It works by simultaneously transmitting multiple optical signals, often referred to as optical channels, through a single optical fiber, with each channel utilizing its own wavelength. An optical backbone network is structured using interconnected nodes and WDM links in a suitable configuration. Typically, the nodes are based on reconfigurable optical add-drop multiplexers (ROADMs) (for their functionality, see Section 2.2). These ROADMs enable the establishment of paths that traverse the network in optical form, known as lightpaths, which, in turn, correspond to optical channels. The channel capacity is an important parameter of optical channels. The concept of channel capacity was introduced by [PERSON] in 1948 [6]. It was defined as the maximum data rate at which information can be reliably transmitted through a noisy channel without errors. The fundamental assumptions behind this definition are that the noise is additive, white, and Gaussian (AWGN), and that the channel is linear, i.e., the capacity always increases with increasing signal power. However, this last assumption does not hold for optical fiber channels, which are non-linear by nature.