Colors Observed in Photoelastic Media

[ishkutay]

When a photoelastic material experiences stress, its internal structure changes such that a position-dependent birefringence occurs. This means that the material effectively becomes a wave-plate, changing the polarization of incident light. It has long been known that when placed between two crossed polarizers (polarizers at 90° with respect to each other), isochromatic lines appear that indicate areas experiencing the same stress.

From my understanding, summarized above, I know these isochromatic lines have the same, rotated polarization... but why are the colors from isochrome to isochrome different?! Color, of course, is a frequency-dependent phenomenon, and I don't see how varying polarizations lead to varying colors.

 

[sophiecentaur]

I don't know the details but the colours that are seen will the result of interference between multiple paths through the material. The path, in wavelengths will be different at different frequencies so the interference effects will depend on the wavelength and produce colours. The simple phenomenon of thin oil film colours is explained by the path length beingdependent on the viewing angle so a similar argument could explain the different colours.

 

[ishkutay]

I understand why you would suggest a connection to oil films; the two do certainly look quite similar. However, the color effects in an oil film originates from the fact that light at every point on an oil film is the superposition of two reflections- one at the film surface and the other at the interface of the oil and water. It is the path difference associated with these different reflection points, and the spatially varying oil film thickness, that is responsible for the perceived colors patterns. In this effect, however, that path difference doesn't seem to be present, so I can't think of what two beams would be interfering to cause this color effect.

 

[sophiecentaur]

If the material is birefringent, the ordinary and extraordinary ray can have different path lengths. That can cause interference.

 

[ishkutay]

That is very true! So a color-dependent phase difference is introduced to the extraordinary or ordinary polarization, and an analyzing filter (looking at either the extraordinary or ordinary polarization in isolation) will show a color corresponding to which colors were able to interfere constructively. Since birefringence $\propto$ stress, color observed from interference effects $\propto$ stress.

 

[sophiecentaur]

I think that looks right. The colours your picture show are typical of a Subtractive filtering - i.e -R, -G and -B are the Cyan, Magenta and Yellow in the picture. The blue would be -(R+G) when the null covers both those, I guess. Very pretty, saturated colours. Where did the set square come from?

 

[ishkutay]

Haha the square was just the first result that popped up on google images when I searched for "photoelasticity" XD Thanks for helping me out!

 

[pixel]

I don't know the details but the colours that are seen will the result of interference between multiple paths through the material.

Just wondering - the object shown in post #1 is not "thin," i.e. a small number of wavelengths, as is usually the requirement for interference to occur.

 

[ishkutay]

I'm not so sure that width has much to do with the occurrence of interference as a phase difference of $n\cdot(2\pi) + \theta$ ,where $n$ is an integer, is physically equivalent to $\theta$ when considering interference effects... correct? i.e. even if the reflecting object has greater width, the phase difference, $2nk_0L$, where L is the width of the object, may have a value $2n\pi + \theta$, but only $\theta$ is needed for interference analysis.

 

[sophiecentaur]

Just wondering - the object shown in post #1 is not "thin," i.e. a small number of wavelengths, as is usually the requirement for interference to occur.

Interference occurs when there is a path difference of a few half wavelengths. If the Ordinary and Extraordinary waves have speeds that only differ by a very small amount then the actual path length can be enormous before the phase differences are a half wavelength. That has been my reasoning. It is possible that the stress effects are greater at the surface, which would also favour my argument.

 

[ishkutay]

Also, isn't it true that if you had a path difference of $n\lambda + \Delta \lambda$, the interference effects would be the same as if there was a path difference of $\Delta\lambda$?

 

[Andy Resnick]

When a photoelastic material experiences stress, its internal structure changes such that a position-dependent birefringence occurs. This means that the material effectively becomes a wave-plate, changing the polarization of incident light. It has long been known that when placed between two crossed polarizers (polarizers at 90° with respect to each other), isochromatic lines appear that indicate areas experiencing the same stress.

From my understanding, summarized above, I know these isochromatic lines have the same, rotated polarization... but why are the colors from isochrome to isochrome different?! Color, of course, is a frequency-dependent phenomenon, and I don't see how varying polarizations lead to varying colors.

Photoelasticity is highly nontrivial. In general, the optical phenomenon is "most easily" characterized as a variation in the electric impermeability tensor η=ε₀ ε^-1 by the strain tensor σ, which in linearized form η(σ)=η(0)+P⋅σ where P is the 4th-rank strain-optic tensor. Typically, in photoelastic studies, the full strain tensor is reduced to in-plane strain, slightly simplifying the above.

The origin of colors in photorefractive materials is (to me) somewhat unclear, but in essence the strained birefringent material acts like a de Senarmont compensator plate:

https://www.olympus-lifescience.com.../techniques/polarized/desenarmontcompensator/

These are "bad" quarter wave plates, where the retardation varies strongly with wavelength, allowing for a good mapping between phase retardation and color. Similarly, the strained material becomes birefringent, introducing a spatially-varying retardance that "somehow" is converted to color.

That said, I don't understand the details- for example, higher-order fringes don't appear to desaturate as they do for The Michel-Lévy Interference Color Chart (https://www.mccrone.com/mm/the-michel-levy-interference-color-chart-microscopys-magical-color-key/). This thread is motivating me to get a decent reference text, most likely Cloud's "Optical Methods of Engineering Analysis"

https://www.cambridge.org/core/book...ing-analysis/C59B725E48AE6B75FDD30A303E5E6A5C

 

[ishkutay]

Oh goodness, okay. I myself am not at all comfortable working with tensors... certainly not the electric impermeability tensor/ strain-optic tensor. And yes, I'm still completely lost on the mechanism responsible for converting spatially -varying polarization (via spatially-varying retardance) to color. The book you mentioned looks highly applicable.

 

[sophiecentaur]

Also, isn't it true that if you had a path difference of $n\lambda + \Delta \lambda$, the interference effects would be the same as if there was a path difference of $\Delta\lambda$?

You can only rely on the effect working for low values of n because of the lack of coherence in normal ambient light.

 

[pixel]

Interference occurs when there is a path difference of a few half wavelengths. If the Ordinary and Extraordinary waves have speeds that only differ by a very small amount then the actual path length can be enormous before the phase differences are a half wavelength.

Okay, and I guess you mean to say the "optical path difference," in this case (n_o-n_e)d.

 

[Andy Resnick]

Oh goodness, okay. I myself am not at all comfortable working with tensors... certainly not the electric impermeability tensor/ strain-optic tensor. And yes, I'm still completely lost on the mechanism responsible for converting spatially -varying polarization (via spatially-varying retardance) to color. The book you mentioned looks highly applicable.

I found this online:

http://edaet.usuarios.rdc.puc-rio.br/AET8-Photoelasticity-EOLSS.pdf

And it does seem to show that higher-order fringes obey a Michel-Lévy type behavior (see Figure 14).

 

[ishkutay]

Thank you for finding/posting the article! It looks like the question has been answered : )