# How can I estimate the elasto-optic coefficients ($p_{11}$ and $p_{12}$) of a material?

I am attempting to estimate the elasto-optic coefficients ($p_{11}$ and $p_{12}$) of $\mathrm{TiO}_2$ and $\mathrm{ZrO}_2$, where $p_{11}$ and $p_{12}$ refer to the elements of a strain-optic tensor for a homogeneous material as given in Hocker (Fiber-optic sensing of pressure and temperature, 1979).

I have found a document which specifies that the longitudinal elasto-optic coefficient ($p_{12}$) can be estimated using the Lorentz-Lorenz relation that it gives as

$$p_{12} = \frac{(n^2 - 1)(n^2 + 2)}{3n^4}$$

however no reference is given, and other sources give the Lorentz-Lorenz relation as something rather different. For example Wikipedia says that the equation relates the refractive index of a substance to its polarizability and gives it as

$$\frac{n^2 - 1}{n^2 + 2} = \frac{4\pi}{3}N\alpha$$

which bares only a vague relation to the earlier equation.

Does anyone know of any other ways in which to estimate the elasto-optic coefficients of a material?

-
Excellent question! Unfortunately I don't know much of anything about optics but I'll see if I can draw some attention to this. – David Z Aug 22 '12 at 21:03
Thanks - hopefully some insight will be forthcoming soon :) – user714852 Aug 24 '12 at 17:25
Did you ever make any progress on this? I'm wondering the same thing. Thanks. – user42528 Mar 14 '14 at 16:45
Ugh, I tried looking through various books (like Korpel's Acousto-Optics") but it seems like it's not going to be straightforward to find out. Maybe you could try emailing Carl Mungan at usna.edu/Users/physics/mungan/index.php and ask him where he got the formula from? – DumpsterDoofus Mar 20 '14 at 0:13

Elasto-optic properties are complex tensorial properties, and I don't think there is any good way to estimate them short of:

• measuring them experimentally
• calculating them through quantum chemistry methods (CRYSTAL14 is one code with such features)
• finding them in the literature

Luckily for you, a simple search reveals that values have been measured, at least for TiO2:

Source here; it's Google's first hit for a search of “elasto-optic tensor TiO2”.

-
Nice :-) I'm not sure if the OP is still around to accept the answer but this does seem to cover it! – David Z Mar 21 '14 at 0:32

Boyd is a useful reference. See for example:

E.L. Buckland and R.W. Boyd, "Electrostrictive contribution to the intensity-dependent refractive index of optical fibers," Opt. Lett. 21:1117 (1 Aug. 1996)

where the dielectric constant epsilon is the square of the refractive index n.

-
Only mentioning a reference might not help people who do not have that book, since this is an equivalent or maybe even worse as a link only answer. Could you summarize the relevant section of your mentioned reference? – fibonatic Dec 29 '14 at 16:47