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By Gaussian, I mean that for a beam propagating in the z direction, a cross-section of the beam in the x-yplane has

|E| = k exp( -(x^2+y^2)/w^2 )

see for instance

http://en.wikipedia.org/wiki/Gaussian_beam

I gather that real laser beams tend to have this sort of "Gaussian" profile. However, I haven't been able to find out anything with exact formulas for Ex, Ey, and Ez. [itex]\nabla \cdot E = 0[/itex] seems to imply that Ez is not zero.