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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1If memory serves, there's a comment in Zangwill about this very subject. I'll try to remember to look it up when I'm in my office on Monday. – Michael Seifert Mar 09 '19 at 22:52
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2Zangwill's discussion is in Section 16.8. I may write a summary at some point, but it'll require some thought before I can do so. In the meantime, check out Transverse wave vs the hairy-ball theorem, How do you make a spherical radio wave?, and How do coherent isotropic radiators evade the hairy-ball theorem? – Michael Seifert Mar 11 '19 at 21:22
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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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