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Now consider a common picture illustrating Snell's law.
The two planes are no longer arbitrary, they are: (a) plane
of incidence (defined by the incoming beam and the normal)
and (b) plane perpendicular to it. Note that the plane (b) is
also uniquely defined by the beam direction.
Similar situations exist in crystal optics. Thinking that
planes of polarization are arbitrary, rather than imposed
by atomic structures, may be an obstacle to understanding.