Polarization and intensity of a Gaussian beam

[Sciencestd]

The equation above (from Wikipedia), assumed that the Gaussain beam has polarization in x-direction, as I know that the polarization means that the oscillation direction of the electric field and so the intenisty... so how we get circular intensity in every direction which means in x and y, above mentioned it with "r"... I can't connect between the two things..
Thank you.

 

[Gordianus]

The versor along the x direction is arbitrary. Any direction, perpendicular to z, is valid.

 

[Gordianus]

In addition, intensity is a scalar. We don't have "linear" or "circular" intensity.

 

[Sciencestd]

In addition, intensity is a scalar. We don't have "linear" or "circular" intensity.

Thank you so much for the answers. So what decide the intensity shape?
And the second question can I say that the polarization vector of the electric field is shorter and shorter in the sides?

 

[Gordianus]

When you move away from the center the electric field gets smaller but it points in the same direction.