Sinusoidal shape and continuity of the electromagnetic wave

I hope my question will be understandable. I wonder about the sinusoidal shape of an electromagnetic wave. I have the impression that something fundamental is behind this shape. To simplify things, let's put ourselves in the case of a single electromagnetic wave. Is the curve of an electromagnetic wave always sinusoidal and continuous? Is it possible that this is a periodic staircase curve, for example at the Planck scale? Especially in a discrete space-time like that of the LQG (Loop Quantum Gravity)? Isn't the sinusoidal shape and continuity just a matter of scale and/or measuring device?

• My answer here also applies to the first part of this question. – The Photon Aug 10 '20 at 21:45

But it comes to the Earth as a plane wave, where $$\mathbf B$$ and $$\mathbf E$$ are expressed as:
$$\mathbf E = \mathbf E(u)$$, and $$\mathbf B = \mathbf B(u)$$ where $$u = \mathbf {k.x} - \omega t$$ and $$\omega = |\mathbf k|c$$.