Derivation
The Gaussian beam formalism begins with the wave equation for an electromagnetic field in free space or in a homogeneous dielectric medium:
where may stand for any one of the six field components, or . The Gaussian beam formalism proceeds by writing down a solution of the form
where it is assumed that the beam is sufficiently collimated along the axis that may be neglected. Substituting this solution into the wave equation above yields the paraxial approximation to the wave equation:
Solving this differential equation yields an infinite set of functions, of which the Gaussian beam is the lowest-order solution or mode.
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