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# APPENDIX E

# Pulse Propagation in Optical Fiber

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In mathematical terms, chromatic dispersion arises because the propagation constant β is not proportional to the angular frequency ω, that is, dβ/dω ≠ constant (independent of ω). dβ/dω is denoted by β_{1}, and β_{1}
^{−1} is called the *group velocity*. As we will see, this is the velocity with which a pulse propagates through the fiber (in the absence of chromatic dispersion). Chromatic dispersion is also called *group velocity dispersion*.

If we were to launch a pure monochromatic wave at frequency ω into a length of optical fiber, the magnitude of the (real) electric field vector associated with the wave would be given by

(E.1)

Here the *z* coordinate is taken to be along the fiber axis, and *J* (*x, y*) is the distribution of the electric field along the fiber cross section and is determined by solving the wave equation. This equation can be derived as follows.