Photonics and Lasers

Chapter 4 - Cylindrical Waveguides

Chapter 4

 

Cylindrical Waveguides

The previous chapter treated propagation of light in a planar waveguide, in which the E field varies in only one transverse dimension (1-D), for example the x direction. We now extend this to two dimensions (2-D), in which the field varies in both the x and y transverse directions. An important special case is that of the optical fiber, which has (usually) cylindrical symmetry about the fiber axis (z axis). A full treatment of the 2-D waveguide modes is beyond the scope of this book. However, many of the features of light propagation in an optical fiber can be understood, at an intuitive and semiquantitative level, by simple extensions of the 1-D treatment to 2-D.

4-1. ACCEPTANCE ANGLE AND NUMERICAL APERTURE

 

Consider the cylindrical dielectric waveguide shown in Fig. 4-1, with a solid cylindrical core of refractive index n1surrounded by a concentric cladding shell of refractive index n2. The medium outside the fiber will be taken to have index n0. A ray of light that enters the fiber end at an angle from the fiber axis will be refracted upon entering, striking the core-cladding boundary at an angle of incidence . Total internal reflection will occur at the core-cladding boundary if n2 < n1provided that the internal waveguide angle is greater than the critical angle c = sin-1( n2/n1). As with the planar waveguide, the angle with the boundary is preserved upon reflection, and the ray will continue to propagate without reflection loss. The waveguide modes corresponding to such rays are termed guided modes or propagating modes, since they are guided in a near-lossless propagation down the fiber. Losses other than reflection, such as absorption and scattering, will be considered in Chapter 5.

As the entrance angle is increased from the value shown in Fig. 4-1, the angle of incidence on the core-cladding boundary will decrease, until at some maxthe critical angle is reached, = c. Rays having > maxwill still enter the fiber, but the reflection at the core-cladding boundary will only be partial, as shown in Fig. 4-2. After a short distance down the fiber, the light will mostly have been lost from the core, and the modes corresponding to such rays are termed unguided modes or nonpropagating modes. The fiber will therefore accept light into the guided modes only for entrance angles within the range 0 < < max. In three dimensions for a cylindrical fiber, this corresponds to an acceptance angle cone of half-angle max, as shown in Fig. 4-3. Light incident on the fiber core within this range of angles is accepted into guided modes, whereas light incident outside of this range goes into unguided modes. This same cone angle applies for light leaving the end of the fiber.

 

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