Rotating Polarization with Mirrors

Light and its behaviour and properties
Dinesh

Rotating Polarization with Mirrors

Post by Dinesh »

Nice pictures of a cube beamsplitter. They've even got the descritption right! Nothing there about a mirror above the cube beamsplitter.
kaveh1000 wrote:The portion of the beam reflected by the prism is s-polarized with respect to the prism face, but p with respect to the mirror.
True. The polarisation is determined by the plane of incidence. The plane of incidence from the cube (prism) goes up and into the paper (my y-z plane) and so the beam is s, as you say. The plane of incidence of the reflected light from the upper mirror goes along the paper and so is p, as you say. But, that still doesn't explain why the output light is p and not s, which it can be. If it were s, then the beam would be horizontally polarised. My question: why isn't it s, since the input beam to the system is s?
Johnfp

Rotating Polarization with Mirrors

Post by Johnfp »

I don't know Dinesh. I see it, saw it as soon as Kaveh turned me on to it, implemented it on the table and was able to rotate my polarization easily when going from side illumination to overhead illumination in H1 set ups. Maybe I don't know what your asking.

Think of the mirror as having a ring on it of some diameter. The mirror is at a 45 to the ground and the ring is glued at normal to the mirror. Now take a rope and feed it through the ring. The end that runs parallel to the ground you are holding. The other end hangs down. Now take the end of the rope and wiggle it up and down such that the waves are not bigger then the diameter of the ring. The waves move up and down until they get to the ring then they move near to far on the rope end that is hanging down.

Now take my original post and arrange the mirrors as such and put two rings in the system. You will find the polarization has been rotated by 90 degrees (Of couse there will be gross losses in a mechanical system like this.) But it should get the process across.
kaveh1000
Posts: 56
Joined: Wed Jan 14, 2015 5:04 pm

Rotating Polarization with Mirrors

Post by kaveh1000 »

Dinesh, I can only think you are trying to analyse the problem too deeply. I don't know how to answer your question.
djm
Posts: 39
Joined: Mon Jan 19, 2015 6:11 pm

Rotating Polarization with Mirrors

Post by djm »

Dinesh wrote:The beam going along the positive y axis ("up") has the y-z plane as the plane of incidence while the polarisation vector is along the x axis, and so the beam as shown is s polarised.
Yes, the beam going up towards the mirror is s-polarized in relation to the cube. But it is p-polarized in relation to the mirror. The transmitted beam through the cube is p-polarized in relation to the cube and is vertically polarized if the beam entering the cube is horizontally aligned.
Dinesh wrote:When the beam hits the mirror atop the cube, the beam changes direction to go along the positive x direction. Now the plane of incidence is the x-y plane and the polarisation vector is along the y axis, which means it's p polarised.
Yes, after the mirror the beam is still p-polarized in relation to the mirror. Which means it is vertically polarized if the original beam to the cube is horizontally aligned.
Dinesh wrote:But, that still doesn't explain why the output light is p and not s, which it can be. If it were s, then the beam would be horizontally polarised. My question: why isn't it s, since the input beam to the system is s?
The reason is because the light going up from the polarizing beamsplitter cube can only be s-polarized with respect to the cube and then consequently only p-polarized with respect to the mirror (in this arrangement).

This effect, that a polarizing beamsplitter cube is reflecting s-polarized light and transmitting p-polarized light, is what we utilize together with a waveplate to get a variable beamsplitter for our polarized laser beams.
djm
Posts: 39
Joined: Mon Jan 19, 2015 6:11 pm

Rotating Polarization with Mirrors

Post by djm »

Just to clarify.
Dinesh wrote:But, that still doesn't explain why the output light is p and not s, which it can be.
No, in this optical setup the beam exiting the mirror can never be s-polarized with respect to the mirror.
Dinesh wrote:If it were s, then the beam would be horizontally polarised.
Yes, but the beam is not s-polarized with respect to the mirror, but p-polarized, and then consequently vertically polarized.
Dinesh wrote:My question: why isn't it s, since the input beam to the system is s?
In the illustrated setup the input beam to the system is both s- and p-polarized with respect to the cube's polarizing beamsplitter surface. The beam reflected from the cube going up towards the mirror is s-polarized with respect to the cube. It will allways be s-polarized (with an ideal cube). This also means that with respect to the mirror, that is reflecting the beam in 90 degrees to the original beam entering the cube, the beam will then always be p-polarized. However, if the mirror instead had an orientation with the beam exiting the mirror parallel to the original beam, then the reflected beam would be s-polarized.
Dinesh

Rotating Polarization with Mirrors

Post by Dinesh »

My understanding of polarisation states is that the polarisation state is dependant on the plane of incidence, not on the optical component that it impinges on. The optical component merely defines a plane of incidence. So, a beam may be "vertically" or "horizontally" polarised, but these polarisation states depend on the gravitational field in which they exist. Where there is no gravitational field, "vertical" and "horizontal" have no meaning. The beam may also be defined wrt the plane defined by two vectors: the surface normal of the reflecting surface, n, and the direction of the beam,k. These two vectors form the plane of incidence and the polarisation - the E vector - is perpendicular to k. If it's also perpendicular to n it's then s polarised, and if it perpendicular to k and parallel to n it's p polarised. It's the n and k vectors that define the state of polarisation, "up" and "down" are superfluous accidents of the earth's gravitational field. Which kind of makes you wonder what the polarisation states of light inside a black hole would be? But I digress

In the case given here, the light exiting the cube is s polarised because its k vector is along the y axis (in the coordinate system I described earlier) and the surface normal, n, of the reflecting surface (inside the cube) is in the y-z plane. The E vector is along x-y and so the E vector is both perpendicular to the n vector and the k vector. If the mirror above the cube had never been there, it would be an s polarised beam. I don't think that merely placing a mirror in this set-up now creates the same beam with two polarisation states for the same E vector since this would imply two mutually perpendicular k vectors - a violation of conservation of energy. Once the mirror is placed, the surface normal to the reflecting mirror,n', is in the x-y plane and the new propagation vector, k' is in the direction of x. Hence there are two solutions for the E vector: Either perpendicular to both the n' and the k' and so in the z direction and s polarised, or parallel to n' and perpendicular to k' and so in the y direction and thus p polarised. Again, it cannot be both s and p since this would imply two k vectors for the same beam.

However, on thinking further, the beam from the cube, E_up is:

E_up = E_x(y)*exp(ikz) (where k is the unit vector in the z direction)

Thus, the vector component of dipole moment on the mirror is

E_up*sin(theta) = E_x(y)*sin(theta) where theta is the tilt of the mirror wrt the horizontal.

This means there is no component of the dipole moment in the z direction. Thus, the beam reflecting from the mirror also cannot have an E vector in the z direction. This would imply that the diagram is correct. However, if the mirror were tilted azimuthally, then there would be a cos(phi) component to the dipole moment and so there would be a "horizontal" component to the polarisation.

I think the problem here is really the difference between theorists and experimentalists. To the experimentalist, mathematics follows reality, while to the theorist, reality follows mathematics!
Dinesh

Rotating Polarization with Mirrors

Post by Dinesh »

Sorry, I made a mistake. The light going up from the cube is not

E_up = E_x(y)*exp(ikz) (where k is the unit vector in the z direction)

It's

E_x(y) = E_0*exp(ikjy) where j is the unit vector in the y direction. The dipole moment is then q*E_x(y)*sin(theta) which is in the x-y plane and so the E vector of the reflected beam is in the y direction
kaveh1000
Posts: 56
Joined: Wed Jan 14, 2015 5:04 pm

Rotating Polarization with Mirrors

Post by kaveh1000 »

Dinesh, you are clearly a genius with maths, but you have to ask yourself why all that theoretical stuff is getting in the way of understanding a very simple principle that seems to be obvious to everyone else in this forum.
Dinesh

Rotating Polarization with Mirrors

Post by Dinesh »

kaveh1000 wrote:Dinesh, you are clearly a genius with maths, but you have to ask yourself why all that theoretical stuff is getting in the way of understanding a very simple principle that seems to be obvious to everyone else in this forum.
Genius, huh? Flattery will get you everywhere....well, at least it'll get you a beer at the MIT bar (they do have one, don't they?)

The problem is, Kaveh, that the the obvious is sometimes not so obvious. Many times, the obvious is wrong. I've seen too many situations where the "obvious" was assumed and proved to be not quite as obvious. There is a very well known holograpy company led by a very competent optical scientist who thought it was "obvious" that woodgrain was due to spurious reflections off the walls of the lab and would not believe it was a polarisation effect. We wasted an entire day covvering the walls of the lab with black velvet and still the woodgrain occurred! The very first dot matrix machines had some of the dots brighter than other dots. Everyone thought it was "obvious" that the laser was mode-hopping. I managed to prove mathematically that the cube in the dot matrix was twisted. Even though the mathematics was pretty compelling proof, it took quite a while for the company to act on it because the person in charge thought that it was "obvious" and "simple" that the laser was mode-hopping.

In this case, for instance, it seems "obvious" and "simple" that the beam is apparently both p and s polarised at the same time relative to both cube and mirror. This is simple, right? However, let's throw in a bit of complexity. Let's have a beamsplitter instead of a mirror. Let's then place a sheet of glass above the beamsplitter at an angle of 60 degrees odd. We're agreed that the beam is clearly s polarised when it exits the cube at the top. So, according to Brewster's law, the glass atop the beamsplitter will reflect some of the light (as you know, only p polarised at Brewster's angle does not reflect). Now, vary the beamsplitter's reflectivity. At some point, due only to the varying of the splitter, the beam now takes on the characteristics of both p and s. So, apparently, the reflectivity of the piece of glass atop the cube will now drop because of the p component, while simultaneously not drop because of the s component.

The problem is, people seem to accept truth based on a belief system and, when the belief system does not solve the problem, people add all sorts of spurious ad hoc assumptions to the belief system to force fit the result into the belief system. When I was in school and in university, back in the bad ol' 60's, the educational system tried to teach us rational thinking and analytical skills. I was actually told by Dr Bondi (one of the proposers of the Steady State theory of cosmology and a lecturer at Kings) that the whole point of going to university was to teach you to think and analyse. If you simply wanted to learn physics, you could do a correspondance course! My apparent skill in mathematical manipulation is not that great; these are after all, simple algebraic equations. I'd like to believ my skill is in the analysis of a speculation or problem, the mathematisation is a fairly simple and straightforward process coming out from the analysis. But these days,I find that any analysis of any problem is frowned upon and deemed to be socially unacceptible. It seems that the socially correct thing to do, when presented with facts and/or speculation is to assume a "group think", "politically correct" stance such that analysis has a faint whiff of elitism. When did educational system get to the stage that the results of education in the analysis of a speculation become a shameful thing done in the dark. When did we deny the ability to analyse, and put down those who did analyse, and simply accept a "group hug" kind of mentality to problem solving?
BobH
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Location: Mesa, AZ

Rotating Polarization with Mirrors

Post by BobH »

Dinesh wrote: I think the s polarisation state of the input beam will be preserved and the reflected beam from the mirror above the cube will have a polarisation vector along the z axis, ie horizontally.

I strongly disagree with this. Dinesh my friend, sometimes a mathematical analysis can go very wrong and this is one case if it led you to that conclusion. There is no way the polarization of the beam coming off the top mirror in the drawing referred to above is "horizontal" if the mirror itself is in the "vertical" direction with respect to the plane of incidence defined by the beamsplitter cube. No way. A page of argument won't change that, and only confuses anyone reading this topic.
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