1.6 FIELD EXPANSIONS

In the analysis of communication systems, it is often convenient to use orthogonal expansions of signals to formulate a more general procedure, and sometimes to gain further insight into results [10]. Orthogonal expansions produce mathematically valid waveform representations that permit signals to be treated in pieces (multiple dimensions) rather than as a single entity. Likewise, in optical communications it is convenient to represent optical fields in terms of similar orthogonal expansions as an alternative to field description.

Consider again a general complex optical field defined over an area A as

images

where we have denoted a(t, r) as the complex envelope and ωo as the optical source frequency. Let us now consider a spatial orthogonal expansion of this complex envelope as

images

where {Φi(r)} are a set of complex two-dimensional orthogonal space functions defined over A. That is,

images

with the asterisk denoting complex conjugates. The corresponding expansions functions are given by

images

That is, each ai(t) is the complex time function corresponding to the spatial function Φi

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