A method which does work with a gradient index of refraction is
the Fresnel-Snell method. This method deals with measuring the reflected
light from the tanks inner surface, and comparing it to that of the
initial beam.
Here is a diagram of the setup I used for this experiment :
Here are some links for Fresnel's equations and Snell's Law:
Wikipedia
: Fresnel's Equations
Hyper
Physics : Fresnel's Equations
Wikipedia : Snell's
Law
Hyper
Physics : Snell's Law
I stared at Fresnel's equations for quite some time trying to figure
out how to get the unknown index separated from the equation. I could not
do it. I needed a way to relate one of the variables I did
not know to the other. This is where Snell's law comes into
play. Snell's
Law allowed me to related the angle of transmitted light to the unknown
index of refraction of the medium. It all seemed to be very easy from
this point until I subbed in the equations for theta-t. I ended up with a
formula in the form of one unknown, but I could not solve for that
unknown. Marty Cohen brought in a book that has a whole mess of math
proofs and I used that to taylor expand my formula and figure out the
relationship that unlocked the door of this whole mystery.
A question that arose recently about my method of measuring the
GRIN, and it is important so I think I should address it. This is the
question, " How do you know that no light is being lost from the
initial reflection on the outer part of the glass?" In other words, the
air-glass reflection coefficient. The answer is Brewster's angle. I used
Brewster's angle for air and glass to minimize the amount of reflected
light on the outside reflection. That tells me that almost all of the
light is being transmitted into the glass. Knowing this I used a simple
method of measuring Reflection Coefficients for reflected parallel
polarized light; and in turn figured out the varying indices of refraction
at different depths.
Here are some links for Brewster's Angle :
Science
World : Brewster's Angle
Hyper
Physics : Brewster's Angle
Once the two mediums started to mix and I shined the laser beam into the solution I noticed an odd phenomenon. When the beam curved from the water and crossed into the corn syrup, there was an odd aberration that made it look like the laser doubled back. Here is a picture of that phenomena:
I further investigated this aberration and
realized that it is nothing more than an optical illusion. It is the same
effect of sticking your finger in water and seeing it slightly
displaced. The only difference is that the displacement is gradual over a
small distance so the beam appears to double back. If there was no
gradient then this illusion would not take place
Recording the reflection Coefficients at different depths in
the GRIN tank allowed me to measure the indices of refraction at those
different depths. My graph for this data is a little choppy due to the
varying intensities of the laser beam. Here is the graph of my data:
The red line represents the Reflection
Coefficient, (Primary
Y-Axis), and the blue line represents Index of Refraction, (Secondary
Y-Axis), both with respect to depth on the X-Axis. The general behavior
of the curve for index of refraction is interesting, but expected. a
gradual change in index, then a very steep change where the medium changes
to corn syrup, then a return to that gradual change.
Using a perfectly distributed GRIN I modeled what the path of
light should look like if a beam was introduced at an incident angle of 89
degrees. Here is a graph of the beams path in that situation:
The black dashes , which is the data, matches up with a parabola graphed in teal. This is evidence that the light follows a parabolic curve when it passes through a gradient index.
