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The polarization of a plane wave traveling in free space is well defined and traverse to the direction of propagation from classical electromagnetic theory. Spherical waves are another type of frequently used solutions. However, I am finding it difficult to visualize how are the polarization vectors arranged over the spherical wavefront, referring to the hairy ball theorem.

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A realistic antenna is a dipole. The dipole radiation has no field in the directions you would expect singularities from the hairy ball theorem.

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