Derivation from simultaneous equations (Fabry-Perot etalon)

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Problem Statement
I am trying to understand how the three equations below were solved simultaneously to get to the equation shown below. This equation was presented in a paper and it relates the input and output fields of a Fabry-Perot etalon.
Relevant Equations
This is the equation I am trying to derive:
$$\frac{E_{2}}{E_{1}}=\exp\left[i\left(\pi+\varphi\right)\right]\frac{\tau-r\exp\left(-i\varphi\right)}{1-r\tau\exp\left(i\varphi\right)}$$
Here are three equations:
$$E_{2}=rE_{1}+itE_{3} \tag{1}$$
$$E_{4}=rE_{3}+itE_{1} \tag{2}$$
$$E_{2}=\tau\exp\left(i\varphi\right)E_{4} \tag{3}$$

I started by substituting Eqn. 3 into Eqn. 2,

$$\frac{E_{2}}{\tau}\exp\left(-i\varphi\right)=rE_{3}+itE_{1}, \ \therefore E_{3} = \frac{1}{r}\left[\frac{E_{2}}{\tau}\exp\left(-i\varphi\right)-itE_{1}\right],$$

and then I substituted this result into Eqn. 1 to get

$$E_{2}=rE_{1}+it\frac{1}{r}\left[\frac{E_{2}}{\tau}\exp\left(-i\varphi\right)-itE_{1}\right]$$
$$\left[1-\frac{it}{\tau r}\exp\left(-i\varphi\right)\right]E_{2}=\left[r+\frac{t^{2}}{r}\right]E_{1}\Rightarrow\frac{E_{2}}{E_{1}}=\frac{\left[r+\frac{t^{2}}{r}\right]}{\left[1-\frac{it}{\tau r}\exp\left(-i\varphi\right)\right]}.$$

After applying the complex conjugate I got

$$\frac{r+\frac{t^{2}}{r}+i\left[\frac{t^{3}}{\tau r^{2}}+\frac{t}{\tau}\right]\exp\left(-i\varphi\right)}{1+\left(\frac{t}{\tau r}\right)^{2}\exp\left(-i2\varphi\right)}.$$

I couldn't manipulate this further to get to the desired equation. Am I on the right track? :oldconfused: :

Also, how did the authors introduce the ##\pi## into the argument of the exponential?

Any help is greatly appreciated.
 

Charles Link

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I think ## t ## is the same as ## \tau ##. Try multiplying numerator and denominator by ## ire^{i \phi} ## in the last step. I don't think their algebra is flawless, but see if that doesn't get you closer to their result.## \\ ## Meanwhile ## i=e^{i \frac{\pi}{2}} ## and not ## e^{i \pi} ##.
 
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Hi @Charles Link

Are you sure that they took ##t## to be the same as ##\tau##? They are different variables, ##t## is the amplitude-transmittance of the beam-splitter (coupler), while ##\tau## represents the attenuation.

Following your advice, here is what I got:

$$\frac{\left[r+\frac{t^{2}}{r}+i\left[\frac{t^{3}}{\tau r^{2}}+\frac{t}{\tau}\right]\exp\left(-i\varphi\right)\right]ir\exp\left(i\varphi\right)}{\left[1+\left(\frac{t}{\tau r}\right)^{2}\exp\left(-i2\varphi\right)\right]ir\exp\left(i\varphi\right)}$$

$$= \frac{i\left[r^{2}+t^{2}\right]\exp\left(i\varphi\right)-\left[\frac{t^{3}}{\tau r}+\frac{tr}{\tau}\right]}{ir\exp\left(i\varphi\right)-\frac{t^{2}}{\tau^{2}r}\exp\left(-i\varphi\right)}$$

If we set ##t=\tau##, this becomes:

$$\frac{i\left[r^{2}+\tau^{2}\right]\exp\left(i\varphi\right)-\left[\frac{\tau^{2}}{r}+r\right]}{ir\exp\left(i\varphi\right)-\frac{1}{r}\exp\left(-i\varphi\right)}$$

Do you think this is getting closer to the result?

Here is the paper. I wouldn't be completely surprised if there was a mistake with their algebra (they also misidentified the conditions for the 3 coupling regimes).
 

Charles Link

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Try again: I get ## \frac{(r^2+t^2)i e^{i \phi}}{1+ire^{i \phi} }## assuming your second to the last expression in post 1 is correct. ## \\ ## I can not read the journal article because I am not a member. And it is ok for us to make algebra mistakes, but that's really unacceptable for them to have any in a published article.. ## \\ ## I suggest you start over with a book that sets up the diagram carefully and derives it correctly. I don't know that the equations you started with are correct. Try Hecht and Zajac for this. These authors are really wasting our time... These equations are really rather routine. At each interface on each side, you have a wave going to the left and one to the right except for the last one on the right. And a couple of them are related by a simple phase factor. This is really very routine. I can't see the article to tell what they did, but it appears they did a sloppy job of it... ## \\ ## For something that I think gets it right, try reading this: http://web.mit.edu/2.710/Fall06/2.710-wk8-a-ho.pdf And yes, ## t_{12} \neq t_{21} ##.
 
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Charles Link

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An alternative derivation takes the incident amplitude after it crosses the first interface, and then part of it crosses the second interface, and portions undergo two more reflections, and then subsequently cross the second interface, with portions getting reflected, etc, in an infinite geometric series, with part always getting out after 2 more reflections. That derivation is perhaps easier to compute than the algebraic one above.## \\ ##See also the above post.
 
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Hi @Charles Link and @BvU

I agree with you. I have access to that through my university but the paper is also freely available on researchgate. And here is a screenshot of the relevant part. Eqn. 4 is the expression I'm trying to derive/verify.

The reason I'm using this paper is that it takes into account the intrinsic resonator losses (##\tau = 1-\text{loss}##). Your link and also Hecht (chapter §9) do not take this loss into consideration. I am trying to simulate fiber ring resonators where this loss is very important (unlike the simple air-spaced Fabry-Perot etalon).

In this paper, they get the following as the transmission:

$$\frac{\tau^{2}-2r\tau\cos\varphi+r^{2}}{1+r^{2}\tau^{2}-2r\tau\cos\varphi}.$$

The standard equation in most textbooks (no losses) is:

$$\frac{2r-2r\cos\left(\varphi\right)}{1+r^{2}-2r\cos\left(\varphi\right)} = \frac{F\sin^{2}}{1+F\sin^{2}},$$

where ##F=\frac{4r}{\left(1-r\right)^{2}}## is the coefficient of Finesse (Hecht erroneously defines ##F=\left(\frac{2r}{1-r^{2}}\right)^{2}##).

Try again: I get ## \frac{(r^2+t^2)i e^{i \phi}}{1+ire^{i \phi} }## assuming your second to the last expression in post 1 is correct.
Did you use ##t=\tau## in the simplification?

I'm a bit confused because the required expression (in my first post or Eqn. 4 of the paper) needs to have ##r## and ##\tau## but not ##t##.
 

Charles Link

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See page 4 of the "link" I gave you from MIT. You are mixing up ## \frac{E_t}{E_i} ## (ratio of E-field amplitudes) with ## (\frac{E_t}{E_i})^2 ## (energy transmission coefficient), which you went to in your last post. Try to be more specific as it is hard to follow, when you started post 1 with ## \frac{E_t}{E_i} ##.
Meanwhile, the paper you are using for a source do a poor job of things IMO.
One thing that needs to be determined in a lossy system is the equations that govern the loss. I did find a treatment of this in the book "Lasers " by Siegman p.415, but he doesn't do the algebra that you are looking for.
I will try to work through the algebra of the screenshot you supplied when I get a chance.
 

Charles Link

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And a follow-on: This author seems to introduce a ## \frac{\pi}{2} ## phase shift (a factor of ## i ## ) in his beam-splitter equations. See https://www.physicsforums.com/threads/interference-puzzle-where-does-the-energy-go.942715/#post-5963655 for the origin of this. In general, such a phase shift does not occur in a dielectric Fabry-Perot etalon. The author of your paper does a very sloppy job with his diagrams=so much so, that it is really a waste of time to try to figure out what he is doing. (There are basically 7 different E-field amplitudes to account for with a Fabry-Perot etalon=he's cutting the corners by just showing 4. A couple of them are related by a simple factor, but you should really work with all 7 for a complete derivation. This is what I did, without losses, and I got the same results that the MIT paper did.) Meanwhile, though, his introduction of the factor of ## i ##, for one of the processes in the equations that he works with, would likely make his results different from what you are needing to compute. ## \\ ## I recommend you try to find a better presentation than the one that this author gives. Siegman does treat the lossy system, but doesn't do the algebra for you. If you do the algebra carefully, and write out the complete equations with the various E-field amplitudes for incident and reflected waves at each of the interfaces, you should be able to get a correct result.
## \\ ## You might also find the following Insights article that I authored as good reading: https://www.physicsforums.com/insights/fabry-perot-michelson-interferometry-fundamental-approach/
 
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