A thin film on a substrate can act as an etalon, creating an interference pattern superimposed on the surface reflectivity when viewed in reflection. The spacing of the sinusoidal peaks, when combined with the index of refraction of the material, can be used to calculate the thickness of the material.
Destructive interference (i.e., minima) will occur when the following relation is satisfied:
M represents the minimum number. Just don't make the mistake of thinking that this is the number of oscillations you have! We can perform a number of mathematical steps to rearrange this in the form of a linear equation of the form y = mx + b.
This is now in the form of y = mx + b
If we then plot the frequency of the minima (n) versus the minimum number (M), we should see a linear graph where the slope, m, can be used to calculate the thickness given a known index of refraction. Reminder: n= c/l
Minima and maxima must be plotted separately. Both should have the same slope, although their intercepts will be different by M= 1/2 (since a maximum is always 1/2 period away from a minimum). Looking at the difference in the thickness obtained from the two slopes is one indicator of the accuracy of the measurement. To calculate the thickness from the slope:
You can also use a known thickness to determine the average refractive index over that wavelength range:
These calculations make the assumption that the index of refraction is constant. In reality, index of refraction varies with wavelength. How it varies depends on the material, so it is not possible to predict for all materials in a work of this limited scope. Knowing the functional form of the variation of the refractive index as a function of wavelength, one can modify the first equation (and subsequent derivations):
If you don't know what is meant by n(n), it is time to buy a software package to do the calculation for you!
Another assumption that is made is that the thickness of the sample is uniform. It is recommended that you repeat this measurement at several locations in order to get an idea of both the variation in thickness and error in the measurement.
As a first estimate of error in this measurement, plot and calculate the thickness using the maxima and minima separately. The difference between the two thicknesses obtained from the same data is the minimum error in your measurement.