Laguerre Mathematics in Optical Communications

Mario Martinelli and Paolo Martelli

The mathematical studies done by Edmond Nicolas Laguerre in the 19th century laid the foundation for contemporary optical communications.



Laguerre’s name comes up in the world of optical communications as many times as Fourier’s does in diffraction optics. Perhaps one of the first times his name was invoked was in 1965, when the laser was in its infancy. At an important Congress of that pioneer period, Roy J. Glauber noticed that the photocount distribution of a monomodal field, which results from the superposition of a coherent excitation and a chaotic one, presented a generating function with “the same form as the generating function for the Laguerre polynomials.” (The conference, titled “The Physics of Quantum Electronics,” took place in San Juan, Puerto Rico, on June 28-30, 1965.) As a consequence, he derived a photocount distribution expressed in terms of the Laguerre polynomials Ln(x), which are defined by:

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