So the question I had was that if we have two mirrors as shown in the diagram, what exactly occurs to the light ray that is incident on the vertex of the mirror? If we would like to construct the reflected light ray, How do we do so when we can draw 2 normals to the given mirrors. How to measure angle of incidence and reflection?
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$\begingroup$ Related: Intuition for construction of a wave reflected from a general corner reflector (though this is for wave optics, but the point is that geometric optics can't describe this situation well) $\endgroup$– RuslanCommented May 21 at 21:17
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2 Answers
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Mathematically, it depends at which angle you hit the corner. You can show that the ray will be reflected symmetrically along the angle bisector.
In a physical mirror, the corner would not be a perfectly sharp angle since there is only a finite resolution to which physical objects can be constructed. At best, your resolution is bounded by individual atoms making up the mirrors. If you actually managed to align two mirrors to this level of accuracy (which might even be possible in principle), the interaction between the light and the atoms lying in the corner would be governed by scattering processes in quantum electrodynamics.
More realistically, the mirrors will only be smooth and aligned to several orders of magnitude above the size of individual atoms. This means whatever happens if the light ray hits the corner is determined by whatever the geometry of the mirrors looks like in the region you define as the "corner"
Now, to address the elephant in the room: So far I have made the unphysical assumption of a perfectly one-dimensional beam of light. In reality, the beam will have a nonvanishing area, again larger than whatever imperfections the setup may have. This means that whatever happens when the beam hits ghe "corner", what really matters is what happens in the near vicinity of the corner, where you should be able to use usual geometric optics to figure out how the beam behaves.
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In geometrical optics there is no such a thing as a single ray by itself, there are only rays of finite ray bundles (pencils). A ray is one of the infinitude of orthogonal trajectories of the equiphase surfaces, the tangent of the rays are the gradients of the surfaces, and as such there is always a finite bundle of rays associated with an arbitrarily small but finite size propagating (equiphase) surface.
With respect to your question, the reflective properties of that particular single ray hitting the corner point is undetermined because that point is a geometric singularity at which there is no tangential plane that would determine the reflective ray. All the rays of the bundle of which that particular one hits the corner will have well defined reflective properties per the Malus-Dupin law as long as the bundle is small enough but still finite.
