# Can two semi-infinite plane waves undergo perfectly constructive interference?

This is building off of a question I asked here.

When discussing the linked problem with some friends, the consensus seemed to be that the reason two identical semi-infinite plane waves cannot undergo perfect constructive interference is because their non-infinite spatial extent means they necessarily cannot be monochromatic, and so they cannot perfectly constructively interfere everywhere.

While this seems logical, I feel there is an assumption here that shouldn't be taken for granted: this assumes that the spatial distribution of frequencies cannot be the same for both waves. Is this the case? Or is it possible to generate two coherent semi-infinite beams whose spatial frequency distributions are perfectly in phase?

This question is, I believe, equivalent to asking if two identical semi-infinite plane waves can perfectly constructively interfere. In the linked question you can see my energy-based argument as to why I expect it is not possible. I'm looking for a more rigorous workthrough.

• Is a semi infinite plane wave even a valid solution of Maxwell’s equations? – Dale Oct 16 '18 at 22:15
• @Dale it is valid. It's easy to see this: just consider an initial state where a superposition of infinite plane waves forms a semi-infinite plane wave (all the wave vectors in the superposition point the same direction). Then, since vacuum is dispersionless, this sem-infinite wave will move with $c$ as a rigid structure. – Ruslan Mar 23 at 9:05