Birefringence in in uniaxial optical media and Snell's law

[SchroedingersLion]

Hey guys,

is anyone here familiar with birefringence in uniaxial optical media?
In such a medium, there are only two types of polarizations allowed for a wave to propagate.
A wave with any other polarization will split into two waves with the allowed polarizations (ordinary + extraordinary wave with two different refractive indices).

I was asking myself how Snell's law comes into play when the light enters such a crystal.
I found a nice tutorial on Youtube on how to figure out in which direction the new waves will propagate inside of the crystal:


I am the guy from the comment section, so I repeat my question from there:
"Now, when a wave initially has a "right" polarization and it falls orthogonal on a crystal surface, there will be no refraction and there will be no birefringence (as no splitting of the wave is necessary, since it already has one of the two allowed polarizations).
However, what happens if my wave has one of the two polarizations, but enters the crystal with an angle? Due to Snell's law, there should be refraction. But as soon as the beam refracts, it won't have the correct polarization anymore. How do I know what will happen in such a case?"

And to add the reverse case: What happens if my wave has a not allowed polarization, but hits with such an angle that Snell's law will refract it classically so that it now has an allowed polarization? Birefringence shouldn't occur there, should it?

Regards,
SchroedingersLion

 

[blue_leaf77]

But as soon as the beam refracts, it won't have the correct polarization anymore.

If my memory serves me right, the directions of the refracted rays are correlated with the special directions in the crystal. So, the refraction will take place in such a way that the polarizations of the refracted rays are along the two special directions. I will need to have a review of my old book in order to give a more thoughtful answer though.

What happens if my wave has a not allowed polarization, but hits with such an angle that Snell's law will refract it classically so that it now has an allowed polarization? Birefringence shouldn't occur there, should it?

Since polarization is a vector, you can always decompose it to the components in any given directions.

 

[SchroedingersLion]

If my memory serves me right, the directions of the refracted rays are correlated with the special directions in the crystal. So, the refraction will take place in such a way that the polarizations of the refracted rays are along the two special directions. I will need to have a review of my old book in order to give a more thoughtful answer though.

Since polarization is a vector, you can always decompose it to the components in any given directions.

Yes, it is as you said, the ordinary and extra ordinary waves will have polarizations in the two "special directions". That much is clear and not part of my question.
In the video, it is explained how to figure out refraction direction in such a case. But ONLY under the assumption, that I will really have birefringence, that I really get a splitting of the waves.
Now, if my initial polarization was already in one of the two allowed directions, I could be fooled to say: There will be no birefringence, since I already have an allowed polarization. Then I could just use Snell's law from 1st semester optics, nothing special.
However, after the refraction, I will not have the allowed polarization anymore, so birefringence SHOULD occur. Then I should have used the method from the video.
It's a bit contradicting and confusing...

 

[blue_leaf77]

However, after the refraction, I will not have the allowed polarization anymore, so birefringence SHOULD occur.

The special polarization directions inside the medium are determined by the incoming ray angle with respect to the interface, it's not a predefined property of the anisotropic medium. What is certain is that, the special polarization directions will always be the one perpendicular to the normal plane (the plane formed by the incoming ray and the normal vector on the surface) and that which is parallel to this plane. For instance, if you shine your laser with incoming polarization parallel to the normal plane then you will only get the extraordinary ray after entering the birefringence material with such polarization that is determined by the index ellipsoid.
I suggest that you read optics and photonics textbooks, most of them cover this subject.

 

[SchroedingersLion]

The special polarization directions inside the medium are determined by the incoming ray angle with respect to the interface, it's not a predefined property of the anisotropic medium. What is certain is that, the special polarization directions will always be the one perpendicular to the normal plane (the plane formed by the incoming ray and the normal vector on the surface) and that which is parallel to this plane. For instance, if you shine your laser with incoming polarization parallel to the normal plane then you will only get the extraordinary ray after entering the birefringence material with such polarization that is determined by the index ellipsoid.
I suggest that you read optics and photonics textbooks, most of them cover this subject.

That's probably what I was missing. So it makes no sense to talk about allowed directions before the beam entered the medium? So the method from the video is always applicable.
What is in the case of orthogonal incidence, the case where there shouldn't be classical refraction. In this case I could be inclined to think:
Ok, there won't be classical refraction. So if my beam now has one of the allowed polarization directions, there will be only an ordinary wave?

 

[blue_leaf77]

Upon some reflection on my past lectures, I could be mistaken when I wrote

What is certain is that, the special polarization directions will always be the one perpendicular to the normal plane (the plane formed by the incoming ray and the normal vector on the surface) and that which is parallel to this plane.

The normal plane should most likely be replaced with incidence plane, which is the plane formed by the incident ray and the optic axis (not normal vector as in my previous post). The special directions of polarization are those parallel and perpendicular to the plane of incidence. Nevertheless it does not change the fact that the special directions depend on the incident ray angle.

What is in the case of orthogonal incidence, the case where there shouldn't be classical refraction.

Depends on which direction the incident polarization is, if it's in the plane of incidence then you will have only e-ray, if perpendicular to this plane then you get only o-wave.

 

[SchroedingersLion]

Alright, thank you :)