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).
Figure 1. (a) Fiber-optic communication system, (b) Photonic integrated circuit (courtesy of Professor Ben Eggleton, CUDOS, University of Sydney).
Generally, optical waveguides can be analyzed by solving Maxwell's
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
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
- mode structure: single-mode or multi-mode,
- waveguide material:glass, polymer, semiconductor,
metal, artificially created materials.
Figure 2. (a) Lens waveguide, (b) Mirror waveguide, (c) Slab waveguide, (d) Strip waveguide, (e) Standard optical fibers, (f) Microstructured fiber, (d) Photonic crystal waveguide.
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.
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