What is TEM00 and TEM01... etc?

TEM = Transvers Electromagnetic Mode. 

A transverse mode of a beam of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular (i.e., transverse) to the propagation direction of the beam. Transverse modes occur in radio waves and microwaves confined to a waveguide, and also in light waves in an optical fiber and in a laser's optical resonator.

See below for some different modes and their corresponding dots/shapes.

 

 

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