# No double refraction (birefrigence)?

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## Main Question or Discussion Point

Consider reflection and refraction of a plane light wave on a vacuum-uniaxial crystal interface.

As it is well-known, when the optical axis of the uniaxial crystal is parallel or perpendicular to the normal vector of the interface, there is no double refraction for a natural (unpolarized) light at normal incidence.

For the optical axis neither parallel nor perpendicular to the normal vector, it seems to me, also there is a no-double-refraction case, where the incident angle is set so that the refractive waves (both o-ray and e-ray) propagate along the optical axis. However I failed to find any textbooks which presents such a case. Did I miss something?

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Andy Resnick
What I mean is that setting the incident angle so that ke//OA (Optical Axis) holds, then we have Se//ke and ko//ke, no double refraction, because e-wave and o-wave have the same refractive index in such a case.
I think I understand what you mean- there's some special geometry such that both rays are refracted 'into' the optical axis? I'll have to check Born &Wolf (and a few other sources) first.

Andy Resnick, I think that Maxwell EM theory and Huygens principle give the same conclusion, but Huygens principle is more intuitive. Do you think I am wrong? If not, why do textbooks not tell this?

Andy Resnick
Andy Resnick, I think that Maxwell EM theory and Huygens principle give the same conclusion, but Huygens principle is more intuitive. Do you think I am wrong? If not, why do textbooks not tell this?
Textbooks do have this information, you just have to know where to look. I found a nice description in "Introduction to the Methods of Optical Crystallography" (Bloss).

I think the answer is 'yes'. There is only one direction relative to the optical axis that the indicatrix is rotationally symmetric, and so off-axis angles of incidence will only refract into that direction for not only a specific angle of incidence but also specific incident polarizations, this appears to be the method used to determine the orientation of the optical axis by conoscopic observations of the location of isogyres.

The situation is more complex for biaxial crystals, but again, conoscopic observation of the isogyres seems to be a method used to locate the optical axes.

... for not only a specific angle of incidence but also specific incident polarizations,...
Many thinks to you. Unfortunately, I don't have access to this book.
I think
"... for not only a specific angle of incidence but also specific incident polarizations,..."
should be
"... for not only a specific angle of incidence but also specific incident plane,..."
Namely the plane on which the normal vector of the interface and the optical axis lie. At this special incidence of a natural (unpolarized) light beam, there is no double refraction.

I got quite a few pieces of CaCO3 (uniaxial) crystal, but I did not find such a phenomenon. Maybe not easy to see. Are there any journal papers which present such experimental observations? Thanks again.

blue_leaf77
Homework Helper
Looking at the picture below, it seems possible to realize an arrangement such that ke//OA, Se//ke, and ko//ke. Namely, rotate the OA clockwise till it coincides the vector line of $k_o$.

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Andy Resnick
Many thinks to you. Unfortunately, I don't have access to this book.
I think
"... for not only a specific angle of incidence but also specific incident polarizations,..."
should be
"... for not only a specific angle of incidence but also specific incident plane,..."
Namely the plane on which the normal vector of the interface and the optical axis lie. At this special incidence of a natural (unpolarized) light beam, there is no double refraction.
That's entirely possible- I am easily confused by crystal optics.... too many planes/axes to consider.

I got quite a few pieces of CaCO3 (uniaxial) crystal, but I did not find such a phenomenon. Maybe not easy to see. Are there any journal papers which present such experimental observations? Thanks again.
I'm sure there are. I found these:

http://www.uwgb.edu/dutchs/petrology/intfig1.htm
http://www.minsocam.org/ammin/am43/am43_1029.pdf
http://edafologia.ugr.es/optmine/xplconos/futallw.htm[/PLAIN] [Broken]
http://www.geo.arizona.edu/geo3xx/geo306_mdbarton/classonly/306%20Web%20Materials/306_Lecture041027.htm[/URL] [Broken]

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... I'm sure there are. I found these:
http://www.uwgb.edu/dutchs/petrology/intfig1.htm
The web page you cited presents how to explore the optical properties of minerals (crystals) by creating an interference figure, nothing to do with what we are discussing. Sorry.

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Looking at the picture below, it seems possible to realize an arrangement such that ke//OA, Se//ke, and ko//ke. Namely, rotate the OA clockwise till it coincides the vector line of $k_o$.
I think you are right. In fact, by directly drawing a common tangent line of the circle and the ellipse, cutting the crystal so that the cutting line intersects with the common tangent line, and drawing the normal of the cutting plane, we can get the refractive angle --- Huygens principle.

I am just curious, why such a simple but important interesting case is not presented in popular textbooks.

Andy Resnick
The web page you cited presents how to explore the optical properties of minerals (crystals) by creating an interference figure, nothing to do with what we are discussing. Sorry.
I thought we were discussing crystal optics? The interference figure is a way to image the indicatrix.

Whatever, I tried to answer you as best I could.

I thought we were discussing crystal optics? The interference figure is a way to image the indicatrix.
Whatever, I tried to answer you as best I could.
Many thanks to you, although no answer to my question could be provided. Maybe someone in this Physics Forum can, and I am waiting.

blue_leaf77
Homework Helper

Let me repeat. Consider reflection and refraction of a plane light wave on a vacuum-uniaxial crystal interface.

As it is well-known, when the optical axis of the uniaxial crystal is parallel or perpendicular to the normal vector of the interface, there is no double refraction for a natural (unpolarized) light at normal incidence.

For the optical axis neither parallel nor perpendicular to the normal vector, it seems to me, also there is a no-double-refraction case, where the incident angle is set so that the refractive waves (both o-ray and e-ray) propagate along the optical axis. If I am right,
(1) Are there any textbooks which presents such a case?
(2) Are there any journal papers which present such experimental observations?

blue_leaf77
Homework Helper
I thought that picture I posted answers your question without resorting to papers. I don't know if there are any paper on this subject though.

I thought that picture I posted answers your question without resorting to papers. I don't know if there are any paper on this subject though.
(1) Your answer does not seem sufficiently convincing because there are no supporting references.
(2) As mentioned before, I got quite a few pieces of CaCO3 (uniaxial) crystal, but I did not find such a phenomenon ---- not convincing myself.

Many thanks to you.

blue_leaf77
Homework Helper
(1) Your answer does not seem sufficiently convincing because there are no supporting references.
You can find the picture in "Fundamental of Photonics" by Saleh and Teich.
but I did not find such a phenomenon
So, you have tried it yourself experimentally, do you know the direction of the OA in those crystals?

You can find the picture in "Fundamental of Photonics" by Saleh and Teich.

So, you have tried it yourself experimentally, do you know the direction of the OA in those crystals?
(1) I did not find the presentation about "no-double refraction case" (we are talking about) in "Fundamental of Photonics" by Saleh and Teich, except for a picture (Figure 6.3-13), similar to that you gave.
(2) The pieces of CaCO3 (uniaxial) crystal I got are for education, and I can know the principal section with the help of my notebook (liquid crystal screen), which gives polarized light. Then try different directions and always there is a double refraction for a natural light.

blue_leaf77
Homework Helper
I did not find the presentation about "no-double refraction case"
Why do the authors have to make specialized sections for each possible outcome of a certain phenomena the chapter is discussing about? How thick do you think the book will be if such idea was implemented? Sometime, the reader is expected to deduce their own conclusion upon certain phenomena which is not explicitly mentioned in a literature but the basics of which are already elaborated. This is the case with that picture. In the book, the picture comes with the explanation of how to interpret the geometric meaning of the indicatrix at the interface of different media, which eventually leads to that picture. If you know how to read that diagram, you should be able to infer that it's indeed possible to have no-double refraction even if the ray is not perpendicularly incident on the interface and the OA is neither parallel nor perpendicular to the interface. Now, let's define $\alpha$ to be the angle between the OA and the interface. That picture suggests that for $\alpha < 90^o-\theta_c$, where $\theta_c = \arcsin \frac{n_{air}}{n_o}$ is the critical angle between the two media, the situation of no-double refraction cannot occur regardless of the incident angle. This might be the case in your experimental attempt. It might be that the angle $\alpha$ is such that it doesn't allow you to have observe no-double refraction.

...If you know how to read that diagram, you should be able to infer that it's indeed possible to have no-double refraction even if the ray is not perpendicularly incident on the interface and the OA is neither parallel nor perpendicular to the interface. ...
Many thanks for your answer to my question. I think I can say:
(1) No references have clearly claimed that there is a case of no-double refraction for the Optical Axis neither parallel nor perpendicular to the vacuum-uniaxial crystal interface.
(2) No references have presented such experimental observations.

Andy Resnick