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.

In the earlier days of nonlinear optics, the materials used for experiments and devices were mainly inorganic dielectric crystals, vapours, liquids and bulk semiconductors. The search for ‘good’ nonlinear-optics media was made amongst the known materials. However in more recent years, with the growing interest in optical devices and applications, attention has focused increasingly on new artificial solid-state materials which may offer higher nonlinearity; in particular, those that will allow nonlinearoptical devices to operate efficiently at relatively low power levels, such as the outputs from semiconductor-diode lasers. Organic materials offer great scope since modern methods of synthesis allow considerable flexibility in the design of materials at the molecular level. As mentioned in §4.4.3, the macroscopic nonlinear-optical properties of many organic crystals are given by the tensor sum of the properties of the constituent molecules, with due regard to local-field factors and molecular orientation. It is this feature of organic materials that allows a ‘molecular-engineering’ approach to the optimisation of macroscopic properties. Several materials with large second-order nonlinearity have been successfully fabricated (Chemla and Zyss, 1, 1987). Some of these newer organic optical materials also exhibit other desirable properties, such as a greater resistance to optical damage. These have applications in devices such as compact optical-frequency doublers and parametric- amplifiers and -oscillators. However, materials with large third-order nonlinearity are perhaps of greater interest currently, since the nonlinear refractive-index effect can be exploited for switching, optical bistability, phase conjugation and other types of signal processing (Gibbs, 1985).

Semiconductors contain free carriers. That is the characteristic feature which makes them different from the other systems which we have considered hitherto. The optical nonlinearities discussed in previous chapters arose from bound charges. Similar effects arise from bound charges in semiconductors but we do not consider them here. The optical nonlinearities which arise from free carriers in semiconductors are particularly important for applications because of the high degree of control which we have over the free-carrier densities and therefore on the performance of devices which make use of them.

When the free-carrier densities are changed by optical excitation we are concerned with real transitions (see §6.6). The resulting nonlinear processes proceed via a real exchange of energy from the optical field to the medium, and are often referred to as ‘dynamic nonlinearities’ (Miller et al, 1981a; Oudar, 1985); this is the nomenclature used here.

Another feature of semiconductors which has become of particular significance for applications in recent years is the ability to fabricate multiple ‘quantum well’ structures, in which the carriers are confined in one direction in repeated layers of the order of 5 nm wide. Within the layers the carrier motion is two-dimensional, which drastically affects their behaviour. This topic belongs more naturally to the next chapter. In this chapter, we confine our attention to the nonlinear-optical properties of bulk semiconductors.

In §§8.1 and 8.2 we outline the one-electron band structure and the behaviour of phonons in Group IV and III – V semiconductors.

In recent years there has been a rapid expansion of activity in the field of nonlinear optics. Judging by the proliferation of published papers, conferences, international collaborations and enterprises, more people than ever before are now involved in research and applications of nonlinear optics. This intense activity has been stimulated largely by the increasing interest in applying optics and laser technology in tele-communications and information processing, and has been propelled by significant advances in nonlinear-optical materials.

The origins of these recent developments can be traced through three decades of work since the invention of the laser and the first observations of nonlinear-optical phenomena by Franken et al (1961). From the earliest days it was recognised that such phenomena can have useful practical applications; for example, effects such as optical-frequency doubling allow the generation of coherent radiation at wavelengths different from those of the available lasers. In the 1960s many of the most fundamental discoveries and investigations were made. Work was then mainly concerned with the interaction of high-power lasers with inorganic dielectric crystals, gases and liquids. Effects such as parametric wave-mixing, stimulated Raman scattering and self-focusing of laser beams were investigated intensively. The invention of the wavelength-tunable dye laser paved the way for the development during the 1970s of many tunable sources utilising nonlinear effects, such as harmonic generation, sum- and difference-frequency mixing, and stimulated Raman scattering. In this way tunable coherent radiation became available in a wide spectral range from the far infrared to vacuum ultraviolet.