Optical Waveguides: Numerical Modeling

Optical Fiber

The simplest optical fiber is a dielectric cylindrical waveguide consisting of a central core with a higher refractive index that is surrounded by a cladding with lower refractive index. The cladding is usually enclosed in a protective jacket. We start with a short review of typical parameters of standard telecom fibers and discuss the guiding mechanism responsible for light propagation in these fibers.

Interestingly, modern technology often imitates—knowingly or accidentally—some structures possessed by biological organisms. Indeed, many years after the first telecommunication fiber had been invented, it was discovered that the spicules of the deep-sea “glass” sponge Euplectella have extraordinary fiber-optical properties, which are surprisingly similar to those of commercial modern telecom fibers. In fact, they even have significant advantages over the manmade fibers: the spicules are formed under normal ambient conditions contrasting the high-temperature manufacturing of telecom fibers, and are more robust, owing to an advanced crack-arresting mechanism. Figure 3 shows a schematic of a standard optical fiber (without a protective jacket) (a) and a "glass" sponge (b)-(d).

Figure 3. (a) A schematic of a standard optical fiber, (b) Mechanically cleaved spicules, (c) Interferogram (top) and corresponding refractive-index profile (bottom) of a spicule, (d) Wave guiding by individual spicules: Spicules acting as single-mode or few-mode waveguides (left); Spicules acting as multi-mode waveguides (right). Scale bar, 10 μm [Sundar et al., Nature 2003].

Manmade fibers are usually composed of silica glass or polymer. In glass fibers, the refractive index difference between the core and the cladding is achieved by adding appropriate dopands either to the core (in order to increase its refractive index) or to the cladding (to reduce its refractive index).

As other waveguides, optical fibers can be single-mode and multi-mode. The important parameter that defines a number of modes is the V parameter, given by (2)

where a is the radius of the core, l is a free space wavelength of light, is the refractive index of the core,  is the refractive index of the cladding, and D is the relative difference of the refractive indices. The fiber supports only one mode if V<2.405. In standard telecom fibers, the difference between  and  is typically small, so that the relative difference, given by << 1(3)

is much less than one. Optical fibers satisfying this condition are called weakly guiding. The typical value for a single-mode fiber is  

Total internal reflection (TIR) is the most common guiding mechanism in optical fibers, although other mechanisms mentioned in the introduction are becoming more and more widespread with the emergence of microstructured optical fibers (MOFs), also called photonic crystal fibers (PCFs). The TIR is an optical phenomenon that occurs when a ray of light strikes a medium boundary at an angle larger than the critical angle with respect to the normal to the surface. The critical angle is given by (4)

If the refractive index is lower on the other side of the boundary, then no light can pass through, so effectively all of the light is reflected.

Finally, other important fiber/waveguide parameters are the numerical aperture defined as


and the acceptance angle given by


that determine the cone of external rays that can be guided by the fiber and, therefore, is an essential parameter for coupling light in and out of the fiber. Light rays that impinge fiber at angles greater than the acceptance angle are refracted into the fiber, but are not guided for a long distance, as they are not totally reflected at the core/cladding interface, but rather are partially refracted into the cladding.

Figure 4. Major physical phenomena affecting light propagation in optical fibers.

Figure 4 illustrates several major phenomena that have an effect on light propagation in optical fibers. These include (a) dispersion, (b) nonlinear interactions, (c) the effect of loss, (d) the effects gain and noise. A brief review of the effects of dispersion and nonlinearity will be given in the next sections of this tutorial, while detailed discussions of other linear and nonlinear phenomena in optical fibers can be found in many textbooks listed in the Bibliography section. In particular, there are two excellent references on linear and nonlinear effects in optical fibers and on general nonlinear optics: Nonlinear Fiber Optics by G. P. Agrawal and Nonlinear Optics by R. W. Boyd.

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