How do I work out what m value (0, 1/2, 1 etc) to put in the thin film interference equations like 2nt = (m + 1/2)*lambda? Does it depend if it's constructive or destructive? Could someone help explain, thanks!
No, it's always constructive. The question is what path difference causes constructive interference.
Forget about reflections and thin films for a minute. Suppose I had two sources that were emitting waves exactly in phase. Different places in space will be closer to one point than the other, so there's a path difference. Where will I get constructive interference?
Answer: When the path difference is an integer number of wavelengths.
Now, what if one of my sources was 180 degrees out of phase with the other one? If they travel an equal distance, they cancel out. So where do I have constructive interference?
Answer: When the path difference is an integer + 1/2 number of wavelengths. If the path difference is 1/2 wavelength, then that adds a 180 degree phase shift, which added to the original 180 degree phase difference puts them in phase.
And that's the key to thin films. Look at the two interfaces. You get a 180 degree phase shift when the interface is going from a lower to higher index of refraction, for instance air (n = 1.0) to water (n = 1.33) or water to oil with n = 1.50.
If you get a phase shift from one surface but not the other, then it's going to take an path difference that is an integer + 1/2 number of wavelengths to get them back in phase.
If there's no phase shift at either interface, or there's a phase shift at both interfaces, then to get the two waves in phase means the path difference is an integer number of wavelengths.
Wait, this is not right. There are destructive interference in thin-film interference. That's the whole point of anti-reflective coating!
To the OP: Here's a page out of my class lecture notes that may help:
Here, "t" is the film thickness, and "n" is the index of refraction of the film itself. The rest should be self-explanatory.
And to answer your question, the value of "m" that you should use depends on the question being asked. Often, you will be asked to find the thinnest film that will cause such-and-such. In that case, you want the smallest "t", meaning that you choose m=0 or 1. Any other value of m will produce larger t. For the top equation, using m=0 makes no sense, because it means that there is no film at all.
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