Consider an unpolarized laser beam propagating
perpendicularly to a page. We often say that it can be
represented by two uncorrelated components polarized
in two mutually perpendicular planes, for example,
horizontal and vertical. The implication is that any two
mutually perpendicular planes can be chosen to define
components.
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. In other words,
lines and dots used to show how the incoming light is
polarized are no longer arbitrary. They must be in planes (a)
and (b), otherwise Fresnel's laws of refraction (Chapter 25
in "Fundamental of Optics" by Jenkins and White, for
example) have no meaning.
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.
Somehow I was not aware of this.
Ludwik Kowalski