Preface

Nonlinear optics is one of the most important and rapidly developing fields of modern physics related to the nonlinear interaction between light and matter. Generally, all media are optically nonlinear. However, these nonlinearities are very weak. For this reason, nonlinear optical phenomena had been first observed experimentally in the early 1960s after the invention of the laser as a source of coherent and sufficiently strong optical radiation. Typical nonlinear optical effects are sum and difference frequency harmonics generation, higher harmonic generation, self-focusing of light beams, self-phase modulation of optical pulses, soliton formation and propagation, stimulated light scattering, four-wave mixing, nonlinear dynamics of lasers and optical amplifiers, etc. The theoretical analysis of nonlinear optical effects is based on the simultaneous solution of Maxwell's equations and the equations of motion of a medium excited by optical waves. Such an approach results in a system of nonlinear differential equations, which is typically solved by using the slowly varying envelope approximation (SVEA). Nonlinear optical effects are widely used in modern optical communications and optical signal processing. These applications require novel theoretical and experimental investigations in nonlinear optics.

The objective of this book is to discuss novel results concerning both theoretical analysis and experimental observation of optical pulse generation and stimulated light scattering in optical fibers and nanostructures.

The book consists of eight chapters divided into four sections. Section 1 is an introduction. In Chapter 1, the basic equations and theoretical approach to the analysis of nonlinear optical phenomena are summarized. Essential nonlinear optical effects are briefly reviewed. The contents of Chapter 1 should facilitate an understanding of the following sections.

Section 2 consists of three chapters. In this section, novel results in mathematical methods of nonlinear optical effects analysis are presented. In Chapter 2, novel methods of the nonlinear Schrödinger equation (NLSE) solution for optical pulse propagation in optical fibers are presented. Fiber losses, higher-order dispersion coefficients, noise, and different modulation formats are taken into account. In Chapter 3, three novel solutions of NLSE are introduced. They represent the nonlinear superposition of real and complex exponential and trigonometric functions. In Chapter 4, a new theory of Vavilov–Cherenkov radiation (VCR) is presented.

Section 3 consists of two chapters. In this section, nonlinear effects related to optical pulse generation are discussed. In Chapter 5, nonlinear effects that accompany nanosecond pulse generation in optical fibers are investigated theoretically and experimentally. In Chapter 6, methods of nonlinear optical generation efficiency enhancement are demonstrated experimentally.

**II**

**Section 4**

Nonlinear Optical Processes in Micro- and Nanostructures **107**

**Chapter 7 109**

**Chapter 8 127**

Widely Tunable Quantum-Well Laser: OPO Diode Around 2 μm

*by Alice Bernard, Jean-Michel Gérard, Ivan Favero and Giuseppe Leo*

Stimulated Raman Scattering in Micro- and Nanophotonics

Based on a Coupled Waveguide Heterostructure

*by Maria Antonietta Ferrara and Luigi Sirleto*

Section 4 consists of two chapters where the peculiarities of nonlinear optical phenomena in micro- and nanostructures are studied. In Chapter 7, the design of the tunable quantum well (QW) laser based on waveguide heterostructure is proposed. In Chapter 8, experimental results for stimulated Raman scattering (SRS) in microand nanophotonics are reviewed.

> **Dr. Boris I. Lembrikov** Holon Institute of Technology (HIT), Holon, Israel

> > Section 1

Introduction

1

Section 1 Introduction

Chapter 1

Boris I. Lembrikov

1. Introduction

waves A exp i k!

given by [1–3]

3

� r ! � � � <sup>ω</sup><sup>t</sup> h i where <sup>k</sup>

> E !

envelope (SVE) such that [1–3]

ð Þ¼ z; t

∂2 A ∂z<sup>2</sup> � � � �

� � � � <sup>≪</sup> <sup>k</sup> <sup>∂</sup><sup>A</sup> ∂z

� � � �

1 2

Introductory Chapter: Nonlinear

The number of publications concerning different aspects of nonlinear optics is enormous and hardly observable. We briefly discuss in this chapter the fundamental nonlinear optical phenomena and methods of their analysis. Nonlinear optics is related to the analysis of the nonlinear interaction between light and matter when the light-induced changes of the medium optical properties occur [1, 2]. The nonlinear optical effects are weak, and their observation became possible only after the invention of lasers which provide a highly coherent and intense radiation [2]. A typical nonlinear optical process consists of two stages. First, the intense coherent light induces a nonlinear response of the medium, and then the modified medium influences the optical radiation in a nonlinear way [1]. The nonlinear medium is described by a system of the dynamic equations including the optical field. The optical field itself is described by Maxwell's equations including the nonlinear polarization of the medium [1, 2]. All media are essentially nonlinear; however, the nonlinear coupling coefficients are usually very small and can be enhanced by the sufficiently strong optical radiation [1, 2]. For this reason, to a first approximation, light and matter can be considered as a system of uncoupled oscillators, and the nonlinear terms are some orders of magnitude smaller than the linear ones [2]. Nevertheless, the nonlinear effects can be important in the long-time and longdistance limits [2]. Generally, the light can be considered as a superposition of plane

> ! , ω, r

quency, radius vector in the space, and time, respectively [1, 2]. The medium oscillators can be electronic transitions, molecular vibrations and rotations, and acoustic waves [2]. Typically, only a small number of linear and nonlinear oscillator modes are important that satisfy the resonance conditions [1–3]. In such a case, the

optical fields can be represented by a finite sum of discrete wave packets E

where c:c: stands for the complex conjugate and A zð Þ ; t is the slowly varying

� � � � ; ∂2 A ∂t2 � � � �

Here we for the sake of definiteness consider the one-dimensional case. The evolution of the waves (1) is described by the system of the coupled equations in the

� � � � <sup>≪</sup> <sup>ω</sup>∂<sup>A</sup> ∂t

� � � �

!, t are the wave vector, angular fre-

½ � A zð Þ ; t exp i kz ð Þþ � ωt c:c: (1)

� � � � ! ð Þ z; t

(2)

Optical Phenomena
