In previous sections we have introduced a notation for describing the on-axis properties of electric fields.
Specifically, we have described the electric fields along an optical axis as functions of frequency (or time)
and the location z. Models of optical systems may often use this approach for a basic analysis even though
the respective experiments will always include fields with distinct off-axis beam shapes. A more detailed
description of such optical systems needs to take the geometrical shape of the light field into account. One
method of treating the transverse beam geometry is to describe the spatial properties as a sum of
‘spatial components’ or ‘spatial modes’ so that the electric field can be written as a sum of the
different frequency components and of the different spatial modes. Of course, the concept of
modes is directly related to the use of a sort of oscillator, in this case the optical cavity. Most of
the work presented here is based on the research on laser resonators reviewed originally by
Kogelnik and Li [35]. Siegman has written a very interesting historic review of the development of
Gaussian optics [52, 51] and we use whenever possible the same notation as used in his textbook
‘Lasers’ [50
].
This section introduces the use of Gaussian modes for describing the spatial properties along the transverse orthogonal x and y directions of an optical beam. We can write
with
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