Photonics and Lasers

Chapter 3 - Planar Wavelguides

Chapter 3

 

Planar Waveguides

This chapter treates the propagation of light between parallel planes having different indices of refraction. We start with the planar geometry because it is easier to treat mathematically than the cylindrical geometry of optical fibers. This material has direct application to integrated optic and semiconductor devices, and will also allow us to develop an intuitive feeling for optical modes and dispersion that can be carried over into the later chapters on optical fibers.

3-1. WAVEGUIDE MODES

 

Consider the planar dielectric waveguide shown in Fig. 3-1, with medium 1 of refractive index n1sandwiched between two semiinfinite media with index of refraction n2. If n2 < n1, a ray of light propagating in medium 1 will undergo total internal reflection at the upper boundary, provided that the angle of incidence 1 on the boundary is greater than the critical angle c = sin-1(n2/n1). The angle with the boundary is preserved upon reflection, so the ray will then strike the lower boundary at the same angle of incidence, assuming parallel surfaces. A ray with a well-defined initial direction will continue to propagate down the waveguide in this zigzag path without loss from the reflections. It is this perfect reflection of light energy by the waveguide boundaries that allows light to propagate such great distances down optical fibers.

The view of light as a narrow ray with a well-defined direction is called the geometric or ray optics picture, and is valid when the wavelength is much smaller than the width of the ray. In this limit, there is no restriction on the beam direction in the waveguide (other than c < 1< 90°), and the concept of a waveguide mode does not apply. However, when the wavelength is larger, we must use the wave optics picture of light, according to which an initially collimated beam of diameter D will eventually spread out due to diffraction (see Chapter 2). Because of this diffraction, any light beam of finite width inside a waveguide that starts out at a particular angle 1 will spread out into other angles, and the angular distribution will change as the light propagates down the waveguide. What we would like to find is a pattern of light distribution that remains constant along the waveguide. Such a pattern is referred to as a mode.

It is important to understand the concept of a mode, because we will refer to modes a lot in this book. An intuitive view of a mode can be obtained by picturing two people holding a rope that is stretched between them. If one person shakes the rope in just the right way, a stable pattern of oscillations will be seen, and this corresponds to a vibra-tional mode of the rope. If the rope is shaken the "wrong" way, then it still vibrates, but there is no stable pattern. The essential feature of a mode is that there is a pattern that is stable in time.

 

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