Engineers who design transistors, lasers, and other semiconductor components want to understand and control the cause of resistance to current flow so that they may better optimize device performance. A detailed microscopic understanding of electron motion from one part of a semiconductor to another requires the explicit calculation of electron scattering probability. One would like to know how to predict electron scattering from one state to another – something quantum mechanics can do.

In classical mechanics, a particle of mass $m$ subject to a restoring force linear in displacement, $x$, from a potential minimum such that $F(x)=-{\kappa }_{0}x$, where ${\kappa }_0$ is the force constant, results in one-dimensional simple harmonic motion with an oscillation frequency $\omega =\sqrt{{\kappa }_0/m}$.

Often there are situations in which the solutions to the time-independent Schrödinger equation are known for a particular potential but not for a similar but different potential. Time-independent perturbation theory provides a means of finding approximate solutions using an expansion in the known eigenfunctions.

The history of the laser dates back to at least 1951 and an idea of Townes. He wanted to use ammonia molecules to amplify microwave radiation. Townes and two students completed a prototype device in late 1953 and gave it the name maser or microwave amplification by stimulated emission of radiation.

This chapter focuses on the propagation of vortex beams inside a GRIN medium. After an overview in Section 8.1 of polarization-related topics such as the Stokes vector and the Poincare sphere, the concept of a phase singularity is discussed in Section 8.2. This concept is used to form specific combinations of the modes that act as vortices with different state of polarizations. In Section 8.3, we discuss the techniques used for generating different types of vortex beams. Section 8.4 shows that a vortex beam also exhibits the self-imaging property during its propagation inside a GRIN medium. The impact of random mode coupling is also discussed in this section. Vortex-based applications of GRIN fibers are covered in Section 8.5.

This chapter provides an introduction to the subject known as gradient-index optics. In Section 1.1, we present a historical perspective on this subject before introducing the essential concepts needed in later chapters. Section 1.2 is devoted to various types of refractive-index profiles employed for making gradient index devices, with particular emphasis to the parabolic index profile because of its practical importance. In Section 1.3, we discuss the relevant properties of such devices such as optical losses, chromatic dispersion, and intensity dependence of the refractive index occurring at high power levels. The focus of Section 1.4 is on the materials and the techniques used for fabricating gradient-index devices in the form of a rod or a thin fiber

This chapter focuses on the effects of loss or gain in a graded-index (GRIN) medium. In Section 6.1, we discuss the impact of losses on the modes of such a medium. Section 6.2 considers the mechanisms used for providing optical gain inside a GRIN medium. Section 6.3 is devoted to Raman amplifiers and Raman lasers, built with GRIN fibers and pumped suitably to provide optical gain. Parametric amplifiers are discussed in Section 6.4, together with the phase matching required for four-wave mixing to occur. The focus of Section 6.5 is on amplifiers and lasers made by doping a GRIN fiber with rare-earth ions. Section 6.6 includes the nonlinear effects and describes the formation of spatial solitons and similaritons inside an active GRIN medium.

This chapter is devoted to the study of dispersive effects that affect short pulses inside a graded-index fiber. An equation governing the evolution of optical pulses inside a GRIN medium is found in Section 4.1. The dispersion parameters appearing in this equation change, depending on which mode is being considered. Section 4.2 focuses on the distortion of optical pulses resulting from differential group delay and group velocity dispersion. Section 4.3 deals with the effects of linear coupling among the modes, occurring because of random variations in the core’s shape and size along a fiber’s length. A non-modal approach is developed in Section 4.4 for the propagation of short optical pulses inside a GRIN medium. The focus of Section 4.5 is on the applications where optical pulses are sent through a GRIN rod or fiber