The field of photonic crystals (aka periodic photonic structures) is experiencing an unprecedented growth due to the dramatic ways in which such structures can control, modify, and harvest the flow of light.

The idea of writing this book came to M. Skorobogatiy when he was developing an introductory course on photonic crystals at the Ecole Polytechnique de Montréal/ University of Montréal. The field of photonic crystals, being heavily dependent on numerical simulations, is somewhat challenging to introduce without sacrificing the qualitative understanding of the underlying physics. On the other hand, exactly solvable models, where the relation between physics and quantitative results is most transparent, only exist for photonic crystals of trivial geometries. The challenge, therefore, was to develop a presentational approach that would maximally use intuitive analytical and semi-analytical models, while applying them to complex and practically relevant photonic crystal structures.

We would like to note that the main purpose of this book is not to present the latest advancements in the field of photonic crystals, but rather to give a systematic, logical, and pedagogical introduction to this vibrant field. The text is largely aimed at students and researchers who want to acquire a rigorous, while intuitive, mathematical introduction into the subject of guided modes in photonic crystals and photonic crystal waveguides. The text, therefore, favors analysis of analytically or semi-analytically solvable problems over pure numerical modeling. We believe that this is a more didactical approach when trying to introduce a novice into a new field.

In static networks essentially all functionality resides in the network access stations (NASs). The performance of the network is therefore determined by how the NASs provide logical connectivity and throughput to satisfy the network's traffic requirements. This chapter explores the performance issues in static networks, viewing them all as special cases of shared media, as described in Section 5.1. Existing and potential uses of shared media abound, the most important of these being to provide efficient local access for end users to a larger optical network. The multiplexing and multiple-access techniques required to achieve multipoint logical connectivity in these networks are treated in Section 5.2. Sections 5.3 through 5.6 deal with capacity allocation and control to serve prescribed traffic requirements. We first point out some general flow conservation constraints that must be satisfied in any shared-channel system. Then the problems of traffic scheduling and control are discussed in settings with increasing degrees of complexity: dedicated connections (Section 5.4), demand-assigned connections (Section 5.5), and packet switching in the optical layer (Section 5.6). Section 5.7 discusses network access applications of static multipoint architectures. These include broadcast star-based and wavelength-router-based passive optical networks (PONs) that provide the foundation of fiber to the home/premises. In these applications the static network is the link between the end user and an optical core or metropolitan area network.

Shared Media: The Broadcast Star

The simplest form of a transparent optical network, the static network, was defined in Chapter 3 as a collection of fixed (passive) splitting/combining nodes without wavelength selectivity, interconnected by fibers that provide full or partial connectivity among a set of NASs.

Given a source s and destination d for a point-to-point connection on a selected waveband in an LLN, the Min-Int algorithm presented here attempts to find a minimum-interference optical path p = 〈s, d〉 for that connection on the given waveband. The exact sense in which interference is minimized requires some explanation and is defined in Section E.3.

The Image Network

The approach used to find a path that minimizes interference is based on shortest path calculations, where the path “length” takes into account weights or “lengths” representing currently active interfering signals. These weights are associated with nodes rather than links. A useful way of visualizing the node-weighting procedure is shown in the image network of Figure E.1. In the network shown in the figure, each node of the original network is “blown up” to create additional intranodal links between each input/output port pair. This is nothing more than a representation of the internal structure of the LDC on the chosen waveband (see Figure 2.19[b]). The image network of Figure E.1 corresponds to the state of activity in the network of Figure 6.55. Two optical connections, (1, 1*) and (2, 2*), are active, with signal S1 transmitted from station 1 to 1* and signal S2 transmitted from station 2 to 2*.

We shall denote an internodal link from node i to node j by (i, j) and assign it a positive weight d(i, j).

The concept of the limiting cut, introduced in Section 6.3.1.2, stems from the Min Cut–Max Flow relation in multicommodity flow problems. We first give a brief summary of this problem and then present a heuristic for finding limiting cuts.

The Multicommodity Flow Problem and Limiting Cuts

In the most common version of the multicommodity flow problem, a set of demands are prescribed between source-destination node pairs in a network with a given topology and link capacities. (Each source-destination demand is known as a commodity, and the network can be anything—gas pipelines, airline routes, highways, and so on.) The basic issue is whether the prescribed demands can be satisfied within the capacity constraints; that is, whether all commodities can be routed through the network (in a bifurcated manner if necessary) so that the total flow of all commodities on each link does not exceed its capacity. If so, the demands are said to be feasible.

In wavelength-routed networks (WRNs), the commodities (demands) are LCs, each requiring one λ-channel, so the capacity of a cut Ci is FiW, where Fi is the number of fiber pairs in the cut and W is the number of available wavelengths. Because a channel in a WRN is a single point-to-point entity, bifurcated routing is not permitted in a WRN. (An exception would be a case in which several λ-channels are required to carry the flow on one LC.)

The relations between cut capacities and feasible demands were stated in Section A.1.8 for the single-commodity case. In the multicommodity case, which is of interest here, the relations are considerably more complex.

SONET, the acronym for synchronous optical network, is currently the prevailing standard for high-speed digital transmission in North America. Introduced in the 1980s, it replaced an earlier standard, the plesiochronous digital hierarchy (PDH), which had been in place for more than two decades prior to the introduction of the SONET standard [Ballart+89]. The most frequently used lower levels of the PDH system are the DS1 (1.544 Mbps, designed to carry 24 64-Kbps digitized voice signals plus synchronizing overhead) and DS3, running at 44.736 Mbps. An architecture similar to SONET, the synchronous digital hierarchy (SDH), is currently used in Europe and Japan, replacing an earlier European hierarchy similar to the PDH system. SONET can carry PDH bit streams as well as many other types of digital traffic (e.g., ATM cells) as part of its payload. One of the most important features of SONET is its highly organized protection capability [Wu92].

The basic building block (i.e., the first level) of the SONET signal hierarchy is called the synchronous transport signal-level 1 (STS-1). STS-1 has a bit rate of 51.84 Mbps and is divided into two portions, transport overhead and information payload, and the transport overhead is divided further into line and section overheads. (A line is composed of one or more sections in series, separated by electronic regenerators.) The line overhead is terminated at SONET terminals and add/drop multiplexers (ADMs), and the section overhead is terminated at regenerators.

Survivability against failures, including failure recovery, is important in any telecommunications network but is highly critical for high-bandwidth optical networks. As more traffic is concentrated on fewer routes, the number of customers that can be potentially affected by a failure is increased. An analysis of failures in the Public Switched Telephone Network over a two-year period in the 1990s showed that human error, acts of nature, and overloads were the major sources of failure. The impact of the failures was measured in terms of how many times a particular failure occurred, duration of the outage, and number of customers and number of customer minutes affected during that outage. During that period, the average number of customers affected due to cable cuts or cable component failures was 216,690, costing 2,643 million in customer minutes. Similarly, the average number of customers affected by each equipment failure was 1,836,910, costing 3,544.3 million in customer minutes . Cable cuts and hardware/equipment failures account for approximately half of the failures encountered in the network during that period.

Fiber cuts are considered one of the most common failures in fiber-optic networks. Furthermore, the use of WDM over these fibers produces an extremely high volume of traffic on a cable. Commercially available fiber-optic transmission systems can run at 10 Gbps or more per channel with 80 or more channels (wavelengths) per fiber. This translates to more than 800 Gbps per fiber.

Throughout this book the approaches taken to system design and performance evaluation are based on the constraints of the enabling technology. Available fiber capacity is assumed to be limited by the constraints and imperfections of optical transceivers, amplifiers, and cross-connects. These constraints affect maximum available spectrum, wavelength spacing, and maximum bit rates per channel. Optical connections are assumed to have limited reach, both geographically and in terms of the number of optical cross-connects they may traverse. Sizes of switches as well as their speed, complexity, and functionality are also assumed to be limited by cost and performance constraints, ultimately going back to the limits of the underlying technologies. Trade-offs between optical and electronic methods of implementing connectivity and routing are suggested, in which the optimal design point depends again on relative cost and performance of the enabling technologies.

Although emphasizing that these technological constraints are paramount, we purposely keep as much of a separation as possible between the architectures discussed in the book and the limitations of any specific technology. The reason is obvious: Today's technology is likely to be obsolete tomorrow. After more than a decade of gestation in the laboratory, photonic and optoelectronic technology has matured to the point where a wide range of technological choices are available for implementing each function in a network, so that cost-effectiveness and viability in the field are the primary issues now rather than proof of concept, which was the issue in the network testbeds just a few years ago.

The first edition of this book was published when optical networks were just emerging from the laboratory, mostly in the form of government-sponsored testbeds. Since then there have been fundamental changes in many aspects of optical networking, driven by the move from the laboratory to commercial deployment and by the twists and turns of the world economy. The investment climate in which optical networks have developed has had two major swings as of this writing. During the technology bubble that began at the end of the 20th century, investment in research, product development, and network deployment increased enormously. The activities during this time of euphoria produced advances in the technology base that would not have been possible without the extraordinary momentum of that period. At the same time, commercial network deployment provided a reality check. Some ideas that were pursued in the late 1990s dropped by the wayside because they did not meet the test of commercial viability, and new ones came along to take their place. When the bubble burst after less than a decade of “irrational exuberance,” the pendulum swung the other way. Investors and executives who a short time earlier thought the sky was the limit now wondered if demand would ever materialize for all of the fiber capacity in the ground. At this writing a more reasoned approach has taken hold; that seemingly elusive demand has materialized and, hopefully, a more rational and sustainable growth period will ensue.

Ultimately, the performance of a network is limited by the quantity and functionality of its physical resources. In this chapter we examine the various functions performed in a multiwavelength network, emphasizing the role of the optical resources (located in the physical layer of Figure 1.3) in providing connectivity and throughput. For the most part we use the terms transparent optical, purely optical, and just optical interchangeably to refer to entities in the physical layer. The implication is that there is a clean break between the underlying technology and functionality in the physical layer and that in the logical layer. The physical layer contains optical components executing linear (transparent) operations on optical signals, whereas the logical layers contain electronic components executing nonlinear operations on electrical signals. In reality, as mentioned in Chapter 1, the picture in real networks is more nuanced. For example, some simple signal processing (either electronic or optical) may be present in the physical layers of today's networks, making them “opaque” to a greater or lesser degree. Conversely, as optical technology for signal processing matures, it is beginning to make its way into the logical layers. Nevertheless, the somewhat simplified view of a transparent (linear) optical layer underlying an electronic (nonlinear) logical layer is very helpful in providing a generic model for most multiwavelength networks. It will be used throughout this book, with exceptions duly noted as they appear. To provide a proper framework for the discussion that follows, we start in Section 2.1 with a description of layers and sublayers of the multiwavelength network architecture.

At various points in the book, we use stochastic traffic and queueing models to represent the behavior of a network under conditions of random demand. These are based on Markov processes as well as some more general queueing models, which are summarized in this appendix. A readable and comprehensive treatment of these models may be found in [Kleinrock75].

Random Processes

Random processes, such as connection requests, contents of packet queues, and so forth, can be described as sequences of random variables, often called the states of the process, with state transitions occurring at successive (isolated) time points. (Between state transitions, the state remains constant.) In discrete state processes, the states take on discrete (typically integer) values, whereas in continuous state processes the states take on a continuum of values. For example, a discrete state process might be the length of a packet queue, whereas a continuous state process might be the random noise generated in an electrical circuit. In discrete time processes, the transitions are spaced regularly in time so that a complete description of the process is given by the state sequence alone. In continuous time processes, the transitions may occur randomly, at any point in time.

A realization of a random process is a specific sequence. In the case of discrete time processes, a realization is completely specified as a sequence of states. In continuous time processes, the transition times must also be specified.

In Chapter 5 we discussed shared-channel networks, and the emphasis was on satisfying traffic requirements on a static, multipoint physical topology (a broadcast star or its equivalent). The traffic requirements were expressed in terms of flows on logical connections (LCs), and satisfaction of these requirements involved multiplexing and multiple access to share the available channels efficiently. When combined time and wavelength division techniques were employed, the optical connections supporting the LCs were set up and time shared by rapidly tuning the transceivers over a given set of wavelengths. Because all optical connections shared a common broadcast medium in a static configuration, all optical paths supporting these connections were permanently in place. We now move on to optical connection routing and wavelength/waveband assignment – issues that were absent in the static case. We treat both point-to-point and point-to-multipoint (multicast) logical connections.

Introduction

In this chapter we focus on the optical layer of the architecture shown in Figure 2.1(a); that is, we treat purely optical (transparent) networks with reconfigurable optical paths, in which reconfiguration is achieved by space switching together with wavelength and/or waveband routing. Unless otherwise stated, we assume that there is no wavelength conversion in these networks, so the constraint of wavelength continuity is in force. The earliest proposals for wavelength-routed networks (WRNs) appeared in [Brain+88] and [Hill88].

In much of the subsequent work on these networks, a recurring issue has been to determine the number of wavelengths required to achieve a desired degree of connectivity as a function of network size and functionality of network nodes (e.g., static wavelength routers, static wavelength interchangers, or WSXCs).

Since the beginning of the 21st century there has been a burgeoning demand for communications services. From the ubiquitous mobile phone, providing voice, images, messaging, and more, to the Internet and the World Wide Web, offering bandwidth-hungry applications such as interactive games, music, and video file sharing, the public's appetite for information continues to grow at an ever-increasing pace. Underneath all of this, essentially unseen by the users, is the optical fiber-based global communications infrastructure – the foundation of the information superhighway. That infrastructure contains the multiwavelength optical networks that are the theme of this book.

Our purpose is to present a general framework for understanding, analyzing, and designing these networks. It is applicable to current network architectures as they have evolved since the mid-1990s, but more importantly it is a planning and design tool for the future. Our approach is to use a generic methodology that will retain its relevance as networks, applications, and technology continue to evolve.

Why Optical Networks?

Since the fabrication of the first low-loss optical fiber by Corning Glass in 1970, a vision of a ubiquitous and universal all-optical communication network has intrigued researchers, service providers, and the general public. Beginning in the last decades of the 20th century enormous quantities of optical fiber were deployed throughout the world. Initially, fiber was used in point-to-point transmission links as a direct substitute for copper, with the fibers terminating on electronic equipment.

Optical networks as described in previous chapters of this book have progressed steadily since the mid-1980s from point-to-point transmission systems to broadcast stars to ring networks to fully reconfigurable multiwavelength mesh networks utilizing a wide range of optical layer equipment: reconfigurable add/drop multiplexers, optical cross-connects, optical amplifiers, and optical access subnets. The next frontier in optical networking is the optical packet-switched network.

The present generation of multiwavelength optical networks are circuit-switched in their core, meaning that they are connection oriented. In these networks, regardless of the specific scheme used to set up an optical connection, a significant delay (typically of the order of milliseconds or more) is always incurred during the setup period, during which the intermediate switches between source and destination are configured to support data transport. This means that circuit switching is efficient only when the average duration of the connections is much longer than the setup time; i.e., seconds or more.

In current applications, exemplified by Internet browsing, a typical source will transmit data in short bursts (on microsecond timescales), possibly changing destinations with each burst. This is completely incompatible with the circuit-switched approach, where a source-destination pair holds a dedicated connection for an extended period of time. Of course, the currently accepted solution to this problem is to maintain the circuit-switched optical infrastructure and provide a packet-switched logical layer (e.g., a network of electronic IP routers) over the optical layer to deal with the bursty traffic.