# Varying index of refraction Experiment ( )

## Main Question or Discussion Point

Varying index of refraction Experiment (URGENT)

I have a experiment that measures the index of refraction. (see references for a better explanation)

Basically is like this: You have a cell made of glass with water. You put unmixed sugar (or other thing) on it, creating a gradient of index of refraction. With time, it will "mix" (diffusion).

You pass a laser througth it and measure the deflection. With it you can calculate the index of refraction. If you do this in different times you can measure too the diffusion using the index of refraction.

The bad thing about using sugar is that the diffusion is slow. Others solutions are too slow too. And testing different kinds of solutions to see what is the fastest is not worth it.

My idea was to use a heat source. The diffusion is faster if we heat it. If we heat it slowly with a stead heat, we can accelerate the diffusion without interfering too much on the process.

The trouble is, How can we correlate the diffusion equation, that is measured in terms of concentration, with the heat equation, that is a temperature?

And my way to measure the refractive index will also be different. I will take a snapshot of the path of the laser. $\rigth(x(t),y(t)\left)$ and use fermat principle with $n(y(t))$ (the index of refraction varies only in the direction normal to the floor).
With the snapshot and a scale, I can find $n$

Leting the path $x=f(y)$ and noting that this isn't a proper function in this case. We find that: $\frac{n(y)^2}{n_0^2}=\frac{1}{f'(y)^2}+1=y'(x)^2+1$

Another question. In 1 it says we can use the refractive index to measure the diffusion, but it don't show how. This is a different question than the other I mead, since the other uses heat equation too. But this is much "simpler".
Or the only way to make the correlation between them is using Abbe spectrometer and empirically determining the index of refraction in function of the concentration? And putting all results in the diffusion equation and finding the diffusion coefficient?

Who the heat source will affect the diffusion? It's plausible that we can consider the temperature in the solution "almost" equal? since we are slowly heating it? The refractive index will depend on $x,z$ too with the addition of the heat source? How much greater the temperature will need to be made for the diffusion to occur, in, say, 1 hour experiment or less?.

In all articles there is no data, only formulaes and 1 picture. That is pretty much useless for a physicist. Since we can't make sure the assumptions were correct. So the reson of my questions.
If someone want's to help me, it really isn't urgent, but I want the Math part to be donne, so I can work in the apparatus. Change the solutions. Withouth the propoer math, it will be a waste of time to make different experiments to find what it's better.

Thanks

References
1 Spatially varying index of refraction: An open ended undergraduate topic (American Journal of Physics -- March 1980 -- Volume 48, Issue 3, pp. 183-188)
2 Measurement of refractive index gradients by deflection of a laser beam (American Journal of Physics -- July 1975 -- Volume 43, Issue 7, pp. 573-574)
3 Bouncing Light Beam (American Journal of Physics -- June 1972 -- Volume 40, Issue 6, pp. 913-914)

I don't know If I can put the articles here. Probably not, so I will at least give the references.

The same post in Art Of problem Solving]