The Mie series solution for the scattering of an electromagnetic plane wave by a homogeneous sphere is introduced. Though ‘exact’, ‘a mathematical difficulty develops which quite generally is a drawback of this “method of series development”: for fairly large particles … the series converge so slowly that they become practically useless’ (Sommerfeld 1954). What is still worse, numerical studies reveal the occurrence in Mie cross sections of very rapid and complicated fluctuations, known as the ‘ripple’, that are extremely sensitive to small changes in the input parameters. They are related to orbiting and resonances, which may also be detected through their contribution to nonlinear optical effects, such as lasing, that are observed in liquid droplets.

The Mie solution

We consider a monochromatic linearly polarized plane electromagnetic wave with wave number k incident on a homogeneous sphere of radius a and a complex refractive index (relative to the surrounding medium) N = n + iκ, where n is the real refractive index and κ is the extinction coefficient (Jackson 1975). The time factor exp(–ωt) where ω = ck is the circular frequency, is omitted.

Taking the origin at the center of the sphere, the z axis along the direction of propagation of the incident wave and the x axis along its direction of polarization, the incident electric field is E0 = exp(ikz) ◯. In spherical coordinates, the components of the scattered electric field at large distances are of the form (Bohren & Huffman 1983)

CAM theory provided for the first time a detailed physical explanation of the meteorological glory. It confirmed van de Hulst's conjecture (Sec. 4.3) about the importance of surface waves of the type illustrated in fig. 4.3, but it showed that higher-order Debye contributions, both from surface waves and from the shadow of higher-order rainbows, are very important. What is very remarkable is that all these contributions arise from complex critical points, so that the glory is a macroscopic tunneling effect. Several other physical effects must be taken into account, rendering the glory one of the most complicated of all known scattering phenomena.

The meteorological glory is observed in backscattering. CAM theory predicts the existence as well of a forward optical glory, that will be discussed in Chapter 13.

Observational and numerical glory features

Observations of natural glories, together with laboratory observations, have shown some conspicuous differences between glories (sometimes called anti-coronae) and coronae (van de Hulst 1957).

1. (i) Variability. The appearance of the glory rings varies considerably from one observation to another, and sometimes even in the course of a single observation. Indeed, as will be illustrated below, the near-backward intensities are rapidly varying functions of β, θ and N.

2. (ii) Angular distribution. The angular distribution falls off away from the central region much more slowly than in the coronae, so that the outer rings are more pronounced: the observation of as many as five sets of rings has been reported.

3. […]

Introduction

The theoretical studies of long-distance soliton transmission in optical fibres with periodic Raman gain were outlined in Chapter 2 ‘Solitons in Optical Fibres: An Experimental Account’ by L. F. Mollenauer. From this work, it was anticipated that the overall information rates made possible by the all-optical nature of the system were potentially orders of magnitude greater than in conventional communication systems. Rate–length products were predicted to be as high as approximately 30000 GHz · km for a single wavelength channel. In the following sections we will describe the first set of experiments designed to explore the fundamentals of such long-distance repeaterless soliton transmissions.

Experimental investigation of long-distance soliton transmission

In the following experiments, a train of soliton pulses was injected into a closed loop of fibre, the length of which corresponded to one amplification period in a real communication system. The pulses propagated around the loop many times until the net required fibre pathlength had been reached (Mollenauer and Smith, 1988). The experimental configuration is shown schematically in Figure 3.1. A 41.7 km length (L) of low-loss (0.22 dB/km at 1.6 μm) standard telecommunications single-mode fibre was closed on itself with a wavelength selective directional coupler (an all-fibre Mach–Zehnder (FMZ) interferometer). The interferometer allowed efficient coupling of the bidirectional pump waves at approximately 1.5 μm (provided by the 3 dB coupler at the pump input) into the loop. At the same time, the signals at approximately 1.6 μm (50 ps sech2 intensity profile pulses at a 100 MHz repetition rate from a synchronously mode-locked NaCl:OH– colour centre laser) were efficiently recirculated (<5% coupled out).

Introduction

In a communication system, it is desirable to launch the pulses close to each other so as to increase the information-carrying capacity of the fibre. But the overlap of the closely spaced solitons can lead to mutual interactions and therefore to serious performance degradation of the soliton transmission system. This has been pointed out independently by three groups (Chu and Desem, 1983a; Blow and Doran, 1983; Gordon, 1983).

Karpman and Solov'ev (1981) first considered the two-soliton interaction in their study of the non-linear Schrödinger equation (NLS) by means of single-soliton perturbation theory. Although they did not have optical fibre transmission in mind their results are applicable to fibres since the soliton propagation in optical fibres are described by the same equation. However, their method is restricted to large soliton separation only.

Our numerical investigations (Chu and Desem, 1983a) show that soliton interaction can lead to a significant reduction in the transmission rate by as much as ten times. At about the same time, Blow and Doran (1983) showed that the inclusion of fibre loss also leads to dramatic increase of soliton interactions. Gordon (1983) derived the exact solution of two counter-propagating solitons (of nearly equal amplitudes and velocities) and analysed the interaction by obtaining the approximate equations of motion corroborating the results of Karpman and Solov'ev (1981).

A considerable amount of research effort has been spent on the reduction of soliton interactions.

Introduction

Although soliton phenomena arise in many distinct areas of physics, the single-mode optical fiber has been found to be an especially convenient medium for their study. As described in the previous chapters of this book, soliton propagation of bright optical pulses has been verified in a number of elegant experiments performed in the negative group velocity dispersion (GVD) region of the spectrum; most recently, transmission of 55-ps optical pulses through 4000 km of fiber was achieved, by use of a combination of non-linear soliton propagation to avoid pulse spreading and Raman amplification to avoid losses (Mollenauer and Smith, 1988). For positive dispersion (λ < 1.3 μm in standard single-mode fibers), bright pulses cannot propagate as solitons and the interaction of the non-linear index with GVD leads to spectral and temporal broadening of the propagating pulses. These effects form the basis for the fiber-and-grating pulse compressor (Nakatsuka et al., 1981; Grischkowsky and Balant, 1982; Tomlinson et al., 1984), which was utilised to produce the shortest optical pulses (6 fs) ever reported (Fork et al., 1987). For both signs of GVD, the experimental results are in quantitative agreement with the predictions of the non-linear Schrödinger equation (NLS).

Although bright solitons are allowed only for negative GVD, the NLS admits other soliton solutions for positive GVD (Hasegawa and Tappert, 1973; Zakharov and Shabat, 1973). These solutions are ‘dark pulse solitons’, consisting of a rapid dip in the intensity of a broad pulse of a c.w. background.

In this chapter, soliton amplification and transmission using erbium-doped fiber amplifiers are presented. First, the general features of the erbium-doped fiber amplifier are described. A new method called dynamic soliton communication is presented, in which optical solitons can be successfully amplified and transmitted over an ultralong dispersion-shifted fiber by using the dynamic range of N = 1–2 solitons. Multi-wavelength optical solitons at wavelengths of 1.535 and 1.552 μm have been amplified and transmitted simultaneously over 30 km with an erbium-doped fiber repeater. It is shown that there is saturation-induced cross talk between multi-channel solitons, and the cross talk (the gain decrease) is determined by the average input power in high bit-rate transmission systems.

The amplification of pico, subpico and femtosecond solitons and 6–24 GHz soliton pulse generation with erbium-doped fiber are also described, which indicates that erbium fibers are very advantageous for short pulse soliton communication. Some soliton amplification characteristics in an ultralong distributed erbium fiber amplifier are also presented. Finally, we describe 5–10 Gbit/s transmission over 400 km in a soliton communication system using the erbium amplifiers.

Introduction

Recent progress on erbium-doped fiber amplifiers (EDFA) has been very rapid since they show great potential for opening a new field in high-speed optical communication (Mears et al., 1985; Poole et al., 1985; Desurvire et al., 1987; Snitzer et al., 1988; Kimura et al., 1989; Suzuki et al., 1989; Nakazawa et al., 1989).

Introduction

In optical fibers, solitons are non-dispersive light pulses based on non-linearity of the fiber's refractive index. Such fiber solitons have already found exciting use in the precisely controlled generation of ultrashort pulses, and they promise to revolutionise telecommunications. In this chapter, I shall describe those developments, and the experimental studies they have stimulated or have helped to make possible. Thus, besides the first experimental observation of fiber solitons, I shall describe the invention of the soliton laser, the discovery of a steady down-shift in the optical frequency of the soliton, or the ‘soliton self-frequency shift’, and the experimental study of interaction forces between solitons.

As early as 1973, Hasegawa and Tappert (1973) pointed out that ‘single-mode’ fibers – fibers admitting only one transverse variation in the light fields – should be able to support stable solitons. Such fibers eliminate the problems of transverse instability and multiple group velocities from the outset, and their non-linear and dispersive characteristics are stable and well-defined. The first experiments (Mollenauer et al., 1980), however, had to wait a while, for two key developments of the late 1970s. The first was fibers having low loss in the wavelength region where solitons are possible, and the second was a suitable source of picosecond pulses, the mode-locked color center laser.

But the first experiments led almost immediately to further developments.

Introduction

The earliest experimental schemes for generating optical solitons, see for example the chapter by Mollenauer in this book or the original paper by Mollenauer et al. (1980), relied on launching pulses with power and transform limited spectral characteristics which matched the soliton requirements for the particular optical fibre used. However, it was shown by Hasegawa and Kodama (1981), that a pulse with any reasonable shape could evolve into a soliton. In such a case, the energy not required to establish the soliton appears as a dispersive wave in the system.

An alternative mechanism for soliton generation was proposed by Vysloukh and Serkin (1983), based on stimulated Raman scattering in fibres, which was later verified by Dianov et al. (1985), through compression in multisoliton Raman generation from a pulsed laser source. Since then, there has been a considerable number of experimental reports of soliton generation through stimulated Raman scattering in various configurations and using several different pump sources, Islam et al. (1986), Zysset et al. (1986), Kafka and Baer, (1987), Gouveia-Neto et al. (1987), Vodopyanov et al. (1987), Nakazawa et al. (1988) and Islam et al. (1989). More generally, it has been shown theoretically by Blow et al. (1988a), that soliton formation is possible in the case where there is coupling between waves leading to energy transfer, specifically via a gain term in the non-linear Schrödinger (NLS) equation description of the system.

Introduction

The study of non-linear waves has always been associated with numerical analysis since the discovery of recurrence in non-linear systems (Fermi et al., 1955) and elastic soliton–soliton scattering (Zabusky and Kruskal, 1965). Since then the mathematics of non-linear wave equations has grown into the industry of inverse scattering and numerical analysis has developed a number of techniques for studying non-linear systems. Inverse scattering theory has given us much insight into integrable non-linear systems and has supplied many useful exact solutions of non-linear partial differential equations. Numerical analysis has mostly been used in the complementary field of non-integrable systems. Since most non-integrable systems of interest, in physics, are ‘close’ to integrable ones the combination of perturbation theory and numerical analysis provides a powerful tool for investigating such systems. In this chapter we will show through illustrative examples how simple concepts and enhanced understanding can be derived for complex non-linear problems through insight gained from numerical simulation. A good example of this is described in Section 4.6 where the concept of a ‘soliton phase’ emerges from numerical simulation; this is a particularly simple and useful concept enabling the design of a number of soliton switching systems.

Before we begin let us clarify the use of the term soliton. When used by mathematicians the term soliton has a precise meaning in the context of inverse scattering theory and carries with it the associated properties of a localised non-linear wave, elastic scattering amongst solitons, stability and being part of an integrable system.

Although the concept of solitons has been around since Scott Russell's various reports on solitary waves between 1838 and 1844, it was not until 1964 that the word ‘soliton’ was first coined by Zabusky and Kruskal to describe the particle-like behaviour of the solitary wave solutions of the numerically treated Korteweg–deVries equation. At present, more than one hundred different non-linear partial differential equations exhibit soliton-like solutions.

However, the subject of this book is the optical soliton, which belongs to the class of envelope solitons and can be described by the non-linear Schrödinger (NLS) equation. In particular, only temporal optical solitons in fibres are considered, omitting the closely related work on spatial optical solitons.

Hasegawa and Tappert in 1973 were the first to show theoretically that, in an optical fibre, solitary waves were readily generated and that the NLS equation description of the combined effects of dispersion and the non-linearity self-phase modulation, gave rise to envelope solitons. It was seven years later, in 1980 before Mollenauer and co-workers first described the experimental realisation of the optical soliton, the delay primarily being due to the time required for technology to permit the development of low loss single-mode fibres.

Over the past ten years, there has been rapid developments in theoretical and experimental research on optical soliton properties, which hopefully is reflected by the contents of this book.

Introduction

The traditional methods for soliton generation in optical fibers use laser sources which generate stable transform-limited ultra-short light pulses, the pulse shape and spectrum of which coincide with those of soliton pulses in fibers. For a long time only color-center lasers satisfied these conditions, and these lasers were used in the majority of soliton experiments (see, for example, Mollenauer et al., 1980; Mollenauer and Smith, 1988). The soliton laser (Mollenauer and Stolen, 1984) is also based on a color-center laser. The successes in semiconductor laser technology has made it possible to use laser diodes in recent soliton experiments (see, for example, Iwatsuki et al., 1988).

In this chapter we shall discuss alternative methods for soliton generation in fibers. In these methods the laser radiation coupled into the fiber is not a fundamental soliton at the fiber input, but the fundamental solitons are formed from the radiation due to the non-linear and dispersive effects in the fiber. Methods for the generation of a single fundamental soliton as well as high-repetition rate (up to THz range) trains of fundamental solitons, which are practically non-interacting with each other, will be described. High-quality adiabatic fundamental soliton compression and the effect of stabilisation of the femtosecond soliton pulse width in fibers with a slowly decreasing second-order dispersion will also be discussed. We shall discuss the problem of adiabatic soliton compression up to a duration of less than 20 fs, so we shall also consider a theoretical approach for the description of ultrashort pulse (USP) propagation through the fiber.

Theoretical properties of light wave envelope propagation in optical fibers are presented. Generation of bright and dark optical solitons, excitation of modulational instabilities and their applications to optical transmission systems are discussed together with other non-linear effects such as the stimulated Raman process.

Introduction

The envelope of a light wave guided in an optical fiber is deformed by the dispersive (variation of the group velocity as a function of the wavelength) and non-linear (variation of the phase velocity as a function of the wave intensity) properties of the fiber. The dispersive property of the light wave envelope is decided by the group velocity dispersion (GVD) which may be described by the second derivative of the axial wavenumber k (= 2π/λ) with respect to the angular frequency ω of the light wave, ∂2k/∂ω2 (= k″). k″ is related to the coefficient the group velocity delay D in ps per deviation of wavelength in nm and per distance of propagation in km, through k″ = Dλ2/(2πc) where λ is the wavelength of the light and c is the speed of light. For a standard fiber, D has a value of approximately –10 ps/nm · km for the wavelength of approximately 1.5 μm. D becomes zero near λ = 1.3 μm for a standard fiber and near λ = 1.5 μm for a dispersion-shifted fiber.

The non-linear properties of the light wave envelope are determined by a combination of the Kerr effect (an effect of the increase in refractive index n in proportion to the light intensity) and stimulated Brillouin and Raman scatterings.