photonspolarizationquantum mechanicsquantum-optics
The photons are completely polarized, i.e their polarization states can be expressed as $a|R\rangle+b|L\rangle$, where $|R\rangle$ and $|L\rangle$ are two helicity eigenstates of the photon. For example, the $|R\rangle$ photon is right circularly polarized and the $|H\rangle=\frac{1}{\sqrt{2}}(|R\rangle+|L\rangle)$ is horizontally linearly polarized photon.
Is the polarized light a pure state and the unpolarized light a statistical mixture of photons with different polarizations?
linearly - then there's exactly the same number of both spins.
This is incorrect. If you have a collection of photons in which half are left hand circularly polarized ($L$) and half are right ($R$), then you have unpolarized light (not linearly polarized). If you have linear polarized light, then each photon is in a (quantum) superposition of R and L at the same time. It is equally true to say that circularly polarized light is in a superposition of horizontally ($H$) and vertically ($V$) polarized light.
What this means is if you take circular light and shine it on a linear polarizing beamsplitter, half will go in each path (i.e. each photon's wavefunction will collapse to either $H$ or $V$ polarized with 50% probability). However if you tried to measure or seperate the light based on $L/R$, all the light would be measured in one path.
how can be linearly polarized light different, than not polarized at all?
If the light is unpolarized then if you try to measure the polarization in the $L/R$ basis, half the light will be found in each path (just like linear polarization), however if you try to measure $H/V$, half the light will also be measured in each path in this experiment as well (unlike linearly polarized light).
I'm basically repeating Peter Diehr's comment here but as an answer.
H and V polarization usually refer to some arbitrary reference frame. This reference frame is usually defined by the direction of gravity (call this $\hat{z}$ direction). If light is travelling "horizontally" (meaning perpendicular to the direction of gravity, for example in the $\hat{y}$ direction) then we can talk about horizontally and vertically polarized light. For vertically polarized light the electric field is parallel to the direction of gravity so it is in the $\hat{z}$ direction. For horizontally polarized light the electric field is perpendicular to the direction of gravity, that is in the $\hat{x}$ direction. Note that in both cases the electric field is perpendicular to the direction of light propagation.
This definition of horizontal and vertical polarization also makes sense for light travelling at some reasonable angular deviations from the horizontal plane. We can talk about polarizations which mostly don't have components in the $\hat{z}$ direction, horizontal polarizations, and polarizations which mostly do have components in the $\hat{z}$, vertical polarizations.
However, imagine a beam travelling directly upwards from the ground in the $\hat{z}$ direction. In this case the polarization lies in the $\hat{x}$, $\hat{y}$ plane. It doesn't make sense to me to call either of these vertical polarization. Rather, we would be better served calling them "East-West" and "North-South" polarized, for example. The key point here is the geometric and coordinate dependent nature of how we call polarizations.
s and p polarizations are different from horizontal and vertical polarizations. As described above, horizontal and vertical usually refer to the direction of the electric field with respect to gravity. s and p polarizations are instead defined with respect to a reference frame which is related to the surface with which the light is interacting. The surface can be oriented in an arbitrary direction with respect to gravity so we see that s and p are unrelated to horizontal and vertical polarization.
s and p are defined with respect to the plane of incidence as described in the Wikipedia article. The plane of incidence is defined by two vectors. One vector is the normal vector to the surface (the vector which points out of the surface). The other vector is the direction of propagation for the light under consideration. Often these two vectors are not parallel and thus they describe a plane*. This plane is called the plane of incidence.
p-polarized light is light where the electric field vector lies in this plane of incidence. s-polarized light is light where the electric field vector is perpendicular to this plane of incidence. If you think about this in terms of the material surface you can see that the electric field vector either pokes in or out of the material surface for p-polarized light while it "lies flat" in the plane of the material surface.
If the electric field is directly incident on the surface the propagation direction and surface normal are parallel so we cannot define a plane of incidence. In this case all polarizations "lie flat" in the plane of the material surface so it is sort of like all polarizations are "s-polarized". This is comparable to the case when light was travelling directly upwards against gravity and all polarizations looked like horizontal polarizations so it didn't make sense to talk about horizontal and vertically polarized light anymore. In this case we must introduce a new different reference frame if we want to talk about the polarization of the light field.
Finally to unite all of these concepts. We see that horizontal and vertical polarization refer to the direction of polarization compared to the direction of gravity and s and p-polarization refer to the direction of polarization compared to the plane of incidence. If the surface normal is pointing either in the direction of gravity (as implied by the figure in the OP) or perpendicular to the direction of gravity (a surface which is vertical like a mirror in the restroom) then there is a relationship between s and p-polarized and horizontal and vertically (h and v) polarized light.
For a surface with a surface normal pointing in the direction of gravity we can see that:
p-polarized ~ v-polarized
s-polarized = h-polarized
I put the tilde on the p to v comparison because for glancing angles (where the light is travelling almost horizontal, large angle of incidence) it makes sense to talk about vertical polarized light but if the light is travelling more and more perpendicular to the surface (smaller angles of incidence) it makes less and less sense to talk about vertically polarized light.
For a vertically mounted surface (like a mirror in the restroom) with light travelling perpedicular to gravity we have:
p-polarized = h-polarized
s-polarized = v-polarized
For a vertically mounted surface with light travelling with a component mostly in the direction of gravity we have:
p-polarized ~ v-polarized
s-polarized = h-polarized
Note regarding glare and sunglasses and the Brewster angle: You have pointed out that unpolarized light incidence at (or near) the brewster angle will be filtered and (mostly) s-polarized upon reflection. If the surface under consideration is horizontal (asphalt on the road, or standing water, for example) then we know that s-polarized light is h-polarized light. Thus if we want to filter this out using sunglasses for example we should make the sunglasses eliminate h-polarized light. Polarizing sunglasses are designed to eliminate h-polarized light when the wearer is standing upright and holding their head in the usual way. If they tilt their head, however, then their sunglasses will now let in more h-polarized light and allow in more v-polarized light. The angle of the sunglasses defines a new coordinate system. Note also that if the surface is vertically oriented (such as the rear window of the car in front of you) then since sunlight is typically coming from above, the s-polarized reflected light will again be mostly horizontally polarized. This means that generally polarizing sunglasses are good and reducing glare from all sorts of objects assuming the sunlight is coming from above. However, during sunrise and sunset the sun is lower in the sky so the reflected s-polarized light may now have large components in the vertical direction so the sunglasses will not work as effectively.
*However for the case of light which is shining directly incident onto a material surface the two vectors are parallel and no plane is defined. I'll discuss this case shortly.
Best Answer
Yes, a photon in a polarized light is found in a pure state such as $|H\rangle$, $|V\rangle$, $|L\rangle$, $|R\rangle$, or any complex linear combination of them. A photon in (completely) unpolarized light is described by the density matrix $$ \rho = \frac{1}{2} \left( |L\rangle \langle L| + |R\rangle \langle R| \right) = \frac{1}{2} \left( |H\rangle \langle H| + |V\rangle \langle V| \right)$$ Note that you omitted the relationship for the vertically polarized state, $|V\rangle = i(|R\rangle - |L\rangle)/\sqrt{2}$, up to an overall sign which is a convention (well, the whole phase including $i$ is physically inconsequential, so it doesn't matter at all but one must be self-consistent with the conventions).