Optical waveguides are dielectric structures that transmit electromagnetic waves in the direction parallel to their axis at visible or infrared frequencies. They are fundamental building blocks of many optical systems, including fiber-optic communications links (Fig. 1a), fiber lasers and amplifiers for high-power applications, as well as all-optical photonic integrated circuits (Fig. 1b).

Generally, optical waveguides can be analyzed by solving Maxwell’s equations, (1)

where E and H are the electric and magnetic fields, B and D are the electric and magnetic flux densities, r and J are electric charge and electric current densities, or their reduced form, the electromagnetic wave equation, with appropriate boundary conditions determined by the properties of the waveguide and cladding materials.

From design and modeling viewpoints, optical guiding structures can be divided into four categories:

The first class of structures includes waveguides that are invariant along their lengths. The solution of the wave equation for single frequency propagation involves calculation of the eigenfunctions, or modes of the waveguide, at a fixed frequency and the eigenvalues that correspond to the axial propagation constant of the wave in the waveguide. In several cases, modal solutions can be found analytically. However, most practical cases rely on numerical solution of the wave equation.
The second class of structures includes waveguides that are non-uniform in the direction of wave propagation. Examples of such structures include waveguide tapers, gratings and photonic crystal waveguides, Y-splitters, S-bends and helical waveguides. The accurate description of light propagation in these structures relies on efficient numerical methods.
The third class of problems includes studies of optical pulse propagation in nonlinear waveguides. While linear pulse propagation can be relatively easily solved in the frequency domain, its nonlinear counterpart requires more advanced numerical approaches.
Finally, a novel class of optical waveguides includes structures composed of metamaterials. In this case, boundary conditions should be modified to take into account the magnetic properties of the metamaterials that are not present in conventional optical waveguides. In addition, the frequency dependence of the material parameters, such as the dielectric permittivity, the magnetic permeability, and the refractive index has to be included.
Also, optical waveguides can be classified based on

their geometry (as shown in Fig. 2): lens-and mirror-waveguides (a)-(b),slab and strip waveguides (c)-(d), standard and microstructured fibers (e)-(f), and photonic crystal waveguides (g),
guiding mechanism: total internal reflection, antiguiding, photonic bandgap, and antiresonant guiding, as well as some more exotic mechanisms such as total external reflection in metamaterials-based waveguides),
mode structure: single-mode or multi-mode,
waveguide material:glass, polymer, semiconductor, metal, artificially created materials.

Several approaches to modeling light transmission in various types of optical waveguides have been described in the literature. These include Beam Propagation Methods (BPM), Finite Element Method (FEM), Finite Difference Time Domain (FDTD) Method, Transfer Matrix Method (TMM), Multipole Method, and so forth. Strategically, there are two basic approaches to the numerical solution of a particular research problem. The first approach includes development of a homemade numerical code. The second approach relies on adaptation of commercial or freely available Computer Added Design (CAD) tools. Each of these approaches has advantages and disadvantages. The advantages of developing homemade codes include full access and control over the source code and therefore, flexibility with regard to code modification and improvement. On the other hand, reliable ready-to-use CAD tools are applicable to a variety of different physical problems and could be optimized for speed, memory usage, and so forth. Therefore, these CAD tools are being widely utilized in industrial product development as well as in academic research.

Numerical methods and their implementation in various free and commercial software packages are addressed in the Numerical Methods section of this website. In this tutorial, we discuss some important properties of light propagation in optical waveguides, taking as an example optical fiber.