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The linear polarized plane wave solution for the electromagnetic field equations in vacuum has many planes where the electric field $\mathbf{E}=0$, and these planes travel at the speed of light.

Is there a field solution in which $\mathbf{E}=0$ at isolated point(s) instead of planes? Kind of like a 'kink' in the field which travels at the speed of light?

Or if that is impossible, what about some field configuration where the electromagnetic field becomes zero at some points periodically, allowing one to trace out a path (even if only at discrete points) at the speed of light?

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It is possible to make the electric field zero on lines in 3D. Ever heard of optical vortices? If you add three or more plane waves with slightly different directions of propagations (not lying in the same plane) then the electric field will become zero on lines that propagate along the with the beam. So, on a plane perpendicular to the direction of propagation, the electric field will be zero at isolated points, called optical vortices. Around these points the phase increases (or decreases) by $2\pi$.

One can also produce special fields where the electric field becomes zero at an isolated point in 3D, but this tends to be very contrived. It is not a generic feature of electromagnetic fields (optical fields) to become zero like that.

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