Effects of coherence length on optical interference filter

[Tom.G]

Following may, or may not, be relevant (out of my field ); see if it helps.

For Optical Coherence Tomography (OTC), perhaps take a look at pg. 4 in:

https://www.edmundoptics.com/ViewDocument/OCT_Whitepaper_18.pdf

 

[Charles Link]

@Tom.G Interesting, but it doesn't cover the Fabry-Perot interference. It was good to get some feedback from someone in any case. Right now it seems we must have a very limited number of viewers who are reading through the posts in any detail.

Edit: I do think students should find this thread as a very good way to learn all about the Fabry-Perot effect. I'm really surprised that we don't seem to be hearing from any students.

 

[Charles Link]

I want to take a couple of minutes one more time to show IMO what the basis of the Fabry-Perot effect is all about. This is very relevant to any calculations you would do with it involving thin films.

If you take a single interface with air on one side and a material of index $n$ on the other side such that you get a 50-50 split in energy for any electromagnetic wave incident on it at normal incidence where one half the energy is transmitted and one half reflected, you really get an interesting case when you consider two waves incident on this interface from opposite directions. They interfere with each other and you no longer get a 50-50 split. You need the Fresnel coefficients to calculate the result because they remain as good numbers even though the energy reflection coefficient $R$ is no longer a good number when two waves are present.

It just takes a little algebra to show that if the two waves are in phase with each other at the interface, all of the emerging energy will go to the right, because there is a $\pi$ phase change for the reflected E field that is incident on the higher index material, causing destructive interference with the transmitted wave that comes from the right.

If there is a $\pi$ phase difference at the interface between the two incident waves, then all the emerging energy goes to the left.

I think this may not be the way this is usually presented in the textbooks, but it really isn't at all in the category of personal theory. I'm just applying the well-established Fresnel coefficients to the simple case of two sources incident from opposite directions onto a single interface. I urge you to try it for yourselves. So far, I've received almost zero feedback on this concept, but I think trying the calculation for yourselves may well be worth the while. To me, it really explains what is going on with the Fabry-Perot effect.

See https://www.physicsforums.com/insights/fabry-perot-michelson-interferometry-fundamental-approach/

 

[renormalize]

I'm just applying the well-established Fresnel coefficients to the simple case of two sources incident from opposite directions onto a single interface. I urge you to try it for yourselves. So far, I've received almost zero feedback on this concept, but I think trying the calculation for yourselves may well be worth the while. To me, it really explains what is going on with the Fabry-Perot effect.

@Charles Link , I've refrained from commenting on your Insight article because I don't see the novelty of your approach. Analyzing two oppositely-directed waves at single dielectric interfaces is the "bog standard" transfer-matrix method that's used to design stack filters. Here's a random example from online:
Temperature dependence of refractive indices...
An excerpt from that article:

 

[Charles Link]

@renormalize I learned the above method many years ago in graduate school. It was only in the last several years though that I recognized it works even for a single interface.

If you look at it with two or more interfaces, it looks like a system of multiple reflections with phase differences, etc. Perhaps it is obvious to many that it works as well with a single interface. It wasn't obvious to me. For a single interface, you need to introduce a second source, and I really was surprised to find the energy is indeed conserved with the result that depends on the relative phases of the two sources.

Edit: Looking it over more carefully, in graduate school I did not learn the matrix method, which when I computed a couple terms just now, it takes a little extra algebra. (We treated the multi-layer system though with a left-going wave and a right going wave on each side of every interface). One can write e.g. $E_{2r}=\rho_{12}E_{2l}+\tau_{12}E_{1r}$, where with the matrix method I found some Fresnel coefficients wind up in the denominator for the expressions. The matrix method is a very mechanical algebraic process, and it really doesn't provide for much insight into the underlying physics.

 

[Charles Link]

@renormalize I looked over the write-up above somewhat carefully from post 64 and I believe (4) is in error.
In nomenclature you should be able to follow, I get the following:

$E_{1r}=E_{2r}/\tau_{12}-\rho_{12} E_{2l}/\tau_{12}$ and

$E_{1l}=\rho_{21}E_{2r}/\tau_{12}+(\rho_{21}^2/\tau_{12}+\tau_{12})E_{2l}$.

This gives $(M_{21}/M_{11})^2=\rho_{21}^2$, but that is not the energy reflection coefficient of the system. The energy reflection coefficient of the system is $R=(E_{1l}/E_{1r})^2$.

The expression $R=\rho_{21}^2$ only holds for a single source incident on an isolated interface. The person who wrote this up seems to have blundered with their formula (4) above. I'm not infallible, but on this one I'm pretty sure I am right, and the "book" is wrong.[Edit: See below=the book did get it right]. Even though the Fresnel coefficient $\rho_{21}$, (which has $R=\rho_{21}^2$ when there is a single source incident on the isolated interface), remains a good number for multiple sources and multiple layers, this $R$ is no longer a good number, but instead needs to be determined by what the whole system is doing.

Edit: I looked it over some more and wondered if perhaps I made a mistake, because their $M$ is for the whole system. and I think I must retract the above, because I see how they get (4). Since the final interface has no left going wave on the right side, they set the right going wave to some arbitrary number, and the left-going wave to zero. They then have $R=(E_{1l}/E_{1r})^2=(M_{21}/M_{11})^2$. My mistake.

 

[Charles Link]

Even though it's been a couple months, I'd like to comment on @renormalize post 64. It may be obvious to some that you can and do get interference from two sinusoidal sources incident on a single interface, I was surprised myself when I discovered about fifteen years ago that that is the case. Yes the transfer matrix works for that, and the case at normal incidence is solved the same way as the Michelson interferometer case with angles of incidence at 45 degrees, but I think it may be worthwhile for instructors to point this out in teaching the subject. (It is really even simpler than using the transfer matrix. You simply employ the Fresnel coefficients).

It was 40+ years ago that I was a teaching assistant for an upper level undergraduate optics course, and we learned all about the mathematical method for multi-layer systems with the Fabry-Perot effect, but it wasn't until about 25 years later that I recognized you do get an interference that conserves energy from two sinusoidal sources incident on a single interface from opposite directions, with the result dependent upon the relative phase of the two sources.