I have set up an experiment such that microwave is converged and beamed to a polymer slab and the reflected wave is measured. I have recorded angle of incidences from 15 to 90 degree and I found out that at 80 degrees, the output is the highest, while at 45 and 85 degrees, the output is the lowest. The thickness of the polymer slab, t is measured to be 5.2cm. Assuming the refractive index of air is 1.00, determine the order of interference m and the refractive index of the polymer slab, n.
Angle of Incidence = [itex]\theta[/itex]
Optical Path Difference = OPD
Wavelength of Microwave = [itex]\lambda[/itex]
Destructive Interference: [itex]OPD = m\lambda =2t\sqrt(n^2 - (sin\theta)^2)[/itex]
Constructive Interference: [itex]OPD = (m+1/2)\lambda =2t\sqrt(n^2 - (sin\theta)^2)[/itex]
EDIT: I come across another equation and is not sure if it's correct: [itex]2t = \lambda/m[/itex]
The Attempt at a Solution
I know that a 180 degrees phase shift happened here and assume that at angle of incidence = 80, the 2 beams which result from reflection and refraction must be exactly 180 degrees difference in phase hence OPD must be odd multiples of half of wavelength [itex]\lambda[/itex] to form a constructive interference.
While at either 45 or 85 degrees, the OPD must be integer multiples of wavelength to result in a destructive interference.
Hence I tried putting all the values in with 45 degrees in the destructive equation and 80 degrees in the constructive equation and form a pair of simultaneous equations.
However without the wavelength of the microwave, it can't be solved. I tried to assume the wavelength to be 0.01m but the resulting refractive index result to be lesser than air which is incorrect.
Am I missing something or is there something that I did not measure?
EDIT: With the new equation [itex]2t = \lambda/m[/itex], I don't have to estimate the wavelength and after calculating 3 variables simultaneous equations:
I found n=2.008, m=1.371 and [itex]\lambda[/itex]=14.26cm if I take [itex]\theta=45[/itex] for the destructive interference.
And n=1.0003, m=0.3004 and [itex]\lambda[/itex]=3.124cm if i take [itex]\theta=85[/itex].
I'm not sure about m but I think both n and [itex]\lambda[/itex] are within acceptable values for [itex]\theta=45[/itex]. Can anybody confirm? Which [itex]\theta[/itex] am I supposed to use anyway? Both 45 and 85 shows a drop in output. I have included a graph to show my measurements.