Rotating birefringent calcite crystals: precession of image?

[PathEnthusiast]

I have been doing some reading about birefringence in order to understand colors observed in different birefringent crystals when I came across the following page in connection with calcite crystal birefringence.

https://www.microscopyu.com/tutorials/birefringence-in-calcite-crystals

I think I understand why there are two images produced and why the polarizers can eliminate light from each image: the refractive index the crystal imposes for light polarized perpendicular to the optical axis of the crystal is close enough to air that the light path is not obviously distorted for light polarized in that direction, whereas light polarized parallel to the optical axis experiences a refractive index far enough from air that the light path is significantly distorted, producing an image offset from what one would expect for an ordinary transparent object. Because the refractive indices and therefore the light paths are polarization-dependent, an appropriately oriented polarizer can eliminate light corresponding to each of the images. So far so good. What I'm having trouble understanding is why rotating the crystal causes precession. If I imagine replacing the crystal with a material having a single refractive index, it seems to me I should be able to recreate the scenario for each of the rays of light. But when I do that exercise I don't see any reason the image should rotate. If I rotated an ordinary glass of water, I don't expect to see the refracted image shift as I rotate the glass. So why is the crystal causing rotation? Is there a decent mathematical treatment of this phenomenon somewhere that I could review to better understand the rotational aspect?

Thanks in advance!

 

[Andy Resnick]

I think I understand why there are two images produced and why the polarizers can eliminate light from each image: the refractive index the crystal imposes for light polarized perpendicular to the optical axis of the crystal is close enough to air that the light path is not obviously distorted for light polarized in that direction, whereas light polarized parallel to the optical axis experiences a refractive index far enough from air that the light path is significantly distorted, producing an image offset from what one would expect for an ordinary transparent object.

Not exactly- but in order to really understand that applet you need to be comfortable with 'polarization states'. The polarizer decomposes the electromagnetic field into two orthogonal linear states and blocks one of them, and when a calcite crystal axis is oriented correctly, the two preferred orthogonal linear polarization states are 'ordinary' and 'extraordinary'.

The relative orientation of polarizer and crystal means that the output polarization state of the polarizer maps to a linear combination of O and E- polarization states input to the crystal.

O- and E states propagate through the crystal differently as they experience different refractive indices- this results in angular displacement of the two states as they propagate, again as long as the crystal axis is oriented properly. If the crystal is thick enough, the final image consists of two visibly laterally sheared images, each in one polarization state. Rotating the polarizer shows how the mapping varies with angle.

Does that help?

 

[PathEnthusiast]

Not exactly- but in order to really understand that applet you need to be comfortable with 'polarization states'. The polarizer decomposes the electromagnetic field into two orthogonal linear states and blocks one of them, and when a calcite crystal axis is oriented correctly, the two preferred orthogonal linear polarization states are 'ordinary' and 'extraordinary'.

The relative orientation of polarizer and crystal means that the output polarization state of the polarizer maps to a linear combination of O and E- polarization states input to the crystal.

O- and E states propagate through the crystal differently as they experience different refractive indices- this results in angular displacement of the two states as they propagate, again as long as the crystal axis is oriented properly. If the crystal is thick enough, the final image consists of two visibly laterally sheared images, each in one polarization state. Rotating the polarizer shows how the mapping varies with angle.

Does that help?

Thanks for the reply! What you've said is in line with my understanding of the ordinary/extraordinary terminology and that the two refractive indices of the crystal are polarization-dependent. But I still don't understand the rotational behavior of the images. The way I am conceptualizing things, for light polarized in a given direction, one component of the polarization experiences one refractive index, let's call it n1, and the other experiences another refractive index, let's call it n2. In the applet, one of the images does not change position with rotation. I assume light corresponding to this image is experiencing a fixed refractive index--say n1. That would seem to mean the other light is experiencing the refractive index n2. But if that's correct, I should be able to replace the fancy birefringent crystal with an ordinary material of refractive index n2, rotate it, and recreate the rotational behavior of the image. But I don't see why there should be rotation of the image through an ordinary refractive material--for example, that is not how I would expect a cup of water to behave if I rotated the cup. So what accounts for the rotation of the image? I must have an incorrect assumption somewhere, but I'm having trouble finding my mistake.

 

[Andy Resnick]

I must have an incorrect assumption somewhere, but I'm having trouble finding my mistake.

You are almost there- the missing piece is the orientation of the crystal axis- the c-axis, which homogeneous materials do not posses. As you can see in the URL below, the c-axis is not aligned to the optical axis of the various elements but exists at some angle to it:

https://lh3.googleusercontent.com/proxy/e-BOnpN3eR76DdPo_Emdt-CWQtivXH9Wa6bGqnqmp5SMFIf14dLONZxDfdo1BSE719BBUR2g9K8jdA8GQvuqbj_Nrxik8Nk

So, as the crystal rotates about the optical axis, the c-axis rotates as well. In the figure, the o-rays are parallel to the optical axis and so are not displaced during rotation.

Does that help?

 

[PathEnthusiast]

You are almost there- the missing piece is the orientation of the crystal axis- the c-axis, which homogeneous materials do not posses. As you can see in the URL below, the c-axis is not aligned to the optical axis of the various elements but exists at some angle to it:

https://lh3.googleusercontent.com/proxy/e-BOnpN3eR76DdPo_Emdt-CWQtivXH9Wa6bGqnqmp5SMFIf14dLONZxDfdo1BSE719BBUR2g9K8jdA8GQvuqbj_Nrxik8Nk

So, as the crystal rotates about the optical axis, the c-axis rotates as well. In the figure, the o-rays are parallel to the optical axis and so are not displaced during rotation.

Does that help?

That link is giving me a permissions-error, unfortunately, so I can't see it. In general terms, a solution along those lines would make sense, but I'm having trouble visualizing the way shifting the axis would affect the position of the image, so a diagram would certainly be helpful.

 

[Andy Resnick]

That link is giving me a permissions-error, unfortunately, so I can't see it. In general terms, a solution along those lines would make sense, but I'm having trouble visualizing the way shifting the axis would affect the position of the image, so a diagram would certainly be helpful.

Hmm... that's odd. How about this link?

http://www.nordskip.com/viking/calcitelight.jpg

 

[PathEnthusiast]

Hmm... that's odd. How about this link?

http://www.nordskip.com/viking/calcitelight.jpg

That one works! I'm definitely getting closer. I think I can formulate my question a little more clearly with the help of that diagram. So the unpolarized light, in my imagining of the rotating crystal scenario, is entering at some "fixed" point around which the crystal is rotated. The ordinary ray proceeds o-ray proceeds in a straight line from that point, unperturbed by the rotation. As the crystal rotates, the e-ray also rotates. That describes the behavior--but *why* does the e-ray rotate? My understanding is that the path of the e-ray is governed by refraction, because it experiences a particular refractive index, n(e) that is different from that experienced by the o-ray, n(o). But if the e-ray experiences the *same* n(e) at each point of rotation, I would expect it to refract in the same way--that is, its path wouldn't change with rotation. So, my assumption would be that the refractive index, n(e), is somehow shifting in such a way that it leads to rotation of the light path--but that's a) hard to visualize and b) seems at odds with my (quite primitive) understanding of birefrigent crystals. Am I getting any closer to the right model or just drifting further away?

 

[Andy Resnick]

That one works! I'm definitely getting closer. I think I can formulate my question a little more clearly with the help of that diagram. So the unpolarized light, in my imagining of the rotating crystal scenario, is entering at some "fixed" point around which the crystal is rotated. The ordinary ray proceeds o-ray proceeds in a straight line from that point, unperturbed by the rotation. As the crystal rotates, the e-ray also rotates. That describes the behavior--but *why* does the e-ray rotate? My understanding is that the path of the e-ray is governed by refraction, because it experiences a particular refractive index, n(e) that is different from that experienced by the o-ray, n(o). But if the e-ray experiences the *same* n(e) at each point of rotation, I would expect it to refract in the same way--that is, its path wouldn't change with rotation. So, my assumption would be that the refractive index, n(e), is somehow shifting in such a way that it leads to rotation of the light path--but that's a) hard to visualize and b) seems at odds with my (quite primitive) understanding of birefrigent crystals. Am I getting any closer to the right model or just drifting further away?

You are definitely getting there- just remember that in this example of a rotating uniaxial crystal, the c-axis and e-ray are always at the same relative angle by design- not accidental. Cleaving/slicing the crystal in other planes such that the c-axis is misaligned to the optical axis will generate different optical effects, most of them uninteresting.

Biaxial crystal produce additional effects: conical refraction is the most well-known:

https://www.photonics.com/Articles/Conical_Refraction_The_Forgotten_Phenomenon/a34819