Does an electromagnetic wave have any spatial extent transverse to its direction of propagation?

Question

Electromagnetic waves (classical, non-quantum conception) seem to be typically depicted as mutually orthogonal plane waves with amplitudes varying orthogonal (transverse) to the direction of propagation of the waves (not sure if that's the most accurate verbal description of graphical depictions, but I think it's pretty close).

Although such a depiction suggests a transverse spatial extent of the wave, isn't this only because the electric and magnetic field magnitudes are depicted by lines of lengths corresponding to vector magnitudes?

Rather, is it the case that an electromagnetic wave has no transverse spatial extent?

(One can imagine that if the varying electric and magnetic field vectors were visually depicted by some other means -- e.g. gray-scale shading -- there would be no suggestion of transverse spatial extent).

I noticed other posted questions that were close in concept, but did not seem to address this specific aspect. Please advise if I missed one that is essentially the same question.

Also, I understand that there may be no meaningful non-quantum answer to this, but thought I'd ask anyway.

Answer 1

Yes, it has a transverse spatial extent. If it didn't, then it would occupy zero volume, and when you integrated its energy density you would get zero.

Another way to see that it has to have some transverse size is that if it didn't, the fields would be discontinuous functions of the transverse coordinates, and then you wouldn't be able to define the partial derivatives appearing in Maxwell's equations.