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

This chapter focuses on the impact of partial coherence on the propagation of optical beams inside a GRIN medium. Section 11.1 introduces the basic coherence-related concepts needed to understand the later material. Section 11.2 uses the evolution of cross-spectral density to study whether periodic self-imaging, an intrinsic property of a GRIN medium, is affected by partial coherence of an incoming beam. Section 11.3 employs the Gaussian-Schell model to discuss how the optical spectrum, the spectral intensity, and the degree of coherence associated with a Gaussian beam change with the beam’s propagation inside a GRIN medium. The focus of Section 11.4 is on Gaussian beams that are only partially polarized. The concept of the polarization matrix is used to study how the degree of polarization evolves when such a partially coherent Gaussian beam is transmitted through a GRIN medium

This chapter focuses on photonic analog of the spin-orbit coupling of electrons occurring inside a graded index medium. Section 9.1 describes two physical mechanisms that can produce changes in the state of polarization of an optical beam. The vectorial form of the wave equation is solved in Section 9.2 to introduce a path-dependent geometrical phase. The photonic analog of the spin-orbit coupling and its implications are also discussed in this section. Section 9.3 considers how the scalar LPlm modes change when the coupling term is taken into account. We treat this term first as a perturbation and then obtain the exact vector modes of a GRIN medium. A quantum approach is used in Section 9.4 to discuss various polarization-dependent effects.

Propagation of electromagnetic waves inside a GRIN medium is studied in this chapter. Section 2.1 starts with Maxwell’s equations and uses them to derive a wave equation in the frequency domain. A mode based technique is used in Section 2.2 for solving the wave equation for a GRIN device fabricated with a parabolic index profile. The properties of both the Hermite’Gauss and the Laguerre-Gauss modes are discussed. Section 2.3 is devoted to other power-law index profiles and employs the Wentzel-Kramers Brillouin method to discuss the properties of modes supported by them. We discuss in Section 2.4 the relative efficiency with which different modes are excited by an optical beam incident on a GRIN medium. The intermodal dispersive effects that become important for pulsed beams are also covered. Section 2.5 describes several non-modal techniques that can be used for studying wave propagation in GRIN media.

The focus of this chapter is on longitudinal variations of the refractive index and how such variations affect the propagation of light inside a GRIN medium. Section 7.1 describes the ray-optics and wave-optics techniques that can be used for this purpose. Section 7.2 focuses on tapered GRIN fibers and describes the impact of tapering on the periodic self-imaging for a few different tapering profiles. The analogy between a GRIN medium and a harmonic oscillator is exploited in Section 7.3 by employing several quantum-physics techniques for solving the GRIN problem. Section 7.4 is devoted to the case of periodic variations in the refractive index that are induced by changing the core’s radius of a GRIN fiber along its length in a periodic fashion.

The focus of this chapter is on focusing and self-imaging of optical beams occurring in a graded-index rod. Section 3.1 provides a geometrical-optics perspective and shows why optical rays follow a curved path inside a GRIN medium. The modes of such a medium are used in Section 3.2 to find a propagation kernel and use it discuss the phenomenon of self-imaging. Section 3.3 is devoted to studying how a GRIN rod can be used as a flat lens to focus an incoming optical beam. Imaging characteristics of such a lens are also considered in this section. Several important applications of GRIN devices are discussed in Section 3.4.

Considerable effort has been directed toward developing new types of artificial materials known now as photonic crystals and metamaterials. Even though the initial focus was not on creating a spatially varying refractive index, it was soon realized that such materials can be fabricated with an index gradient in one or more dimensions. In this chapter, we focus on the novel GRIN devices whose design is based on photonic crystals and metamaterials. Section 10.1 introduces the basic concepts needed to understand the physics behind these two types of materials. Section 10.2 is devoted to GRIN structures based on the concept of photonic crystals. Metamaterials designed with an index gradient are discussed in Section 10.3. The focus of Section 10.4 is on a subgroup of metamaterials, known as metasurfaces, which contain nanoscale objects made with dielectric or metallic materials and are thinner than the wavelength of radiation they are intended for.

The focus in this chapter is on intensity-dependent changes in the refractive index of a GRIN medium, responsible for the Kerr effect. In Section 5.1, we consider self-focusing of an optical beam inside a GRIN medium. Pulsed beams are considered in Section 5.2, where we derive a nonlinear propagation equation and discuss the phenomena of self- and cross-phase modulations. Section 5.3 is devoted to modulation instability and the formation of multimode solitons. Intermodal nonlinear effects are considered in Section 5.4 with emphasis on four-wave mixing and stimulated Raman scattering. Nonlinear applications discussed in Section 5.5 include supercontinuuum generation, spatial beam cleanup, and second harmonic generation.