2.3: LINEAR LIGHTWAVE PROPAGATION IN AN OPTICAL FIBER

2.3.1 Electromagnetic Preliminaries

Any treatment of light guiding in a fiber must begin with the Maxwell equations and describe their solution to some degree of mathematical detail. Many excellent treatments of dielectric waveguides exist [14 19], and the reader would benefit by consulting one or more of these. We draw heavily on Buck s treatment [19].

The Maxwell equations in MKS units can be written as

where D= E and B= H, where ? and are the permittivity and permeability, respectfully, of the medium. In a source-less medium, J=0 and ? _free =0. Using standard manipulations, the wave equations for propagating E and H fields can be derived from the Maxwell equations as

The formulas in Eq. (2.2) are each three-wave equations, one for each vector component of E and H. Assuming time harmonic fields, we may generally write (for a wave propagating in the z direction)

where ?= ?/ ?= ?n/c is the propagation constant, or the phase shift per length, of a sinusoidal wave measured along the z axis, and is the wave velocity.

The explicit form of the time dependence can be used to simplify the form of the Maxwell equations [17] to

Defining , we can derive the wave equation in phasor form [19] as the vector Helmholtz equations:

The wave number k has...