Lecture 5 (EM21) -- Coupled-mode theory
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- Опубликовано: 16 июн 2024
- This lecture introduces the student to coupled-mode theory and its various forms. It is not intended to be a rigorous treatment of the math. Instead, it is intended to give the student a picture of coupled-mode theory and to identify the various forms the coupled-mode theory appears.
Prerequisite Topics: Electromagnetics and Differential Equations
As a complementary material for this amazing video, a good reference for this topic is the book: Fundamentals of Optical Waveguides by Katsunari Okamoto
Hi, Your lectures are really nice and helpful. Do you have any lecture, which shows how the coupling exactly happens between the wave guides ( how the power gets splits from one waveguide to the other)? Also Do you have any lectures on Ring resonators?
This lecture is all I have on the coupled mode theory. You could look into what is called the "overlap" integral. It essentially calculates how much of one mode matches a second mode. The more they have in common, the stronger the coupling. This may be the most insight you can have.
I do not currently have anything on ring resonators other than maybe on slide in the slow wave lecture. Sorry!
CEM Lectures Thank you very much for the reply. Could you please tell me in which lecture number could I find this "overlap" integral?
dipti mohanta Sorry. I don't think I have this in any of my slides. If I do add it as some point, it will be in this lecture.
Dear Dr. Raymond, thank you for this lecture. I feel it a bit strange for the last part of the video. Is it correct to say that in the mode-matching scheme, it is the eigenmode of the waveguide that is used as the basis for mode decomposition, they do not couple in the waveguide because they are orthogonal within the guide, whereas in the coupled-wave scheme, it is expanded using plane waves, as the plane-waves are not the eigenmode of the waveguide, they can be coupled? I think the word "mode" in these two different scenarios mean differently.
Exactly! The plane waves are coupled because they are not the eigen-modes. The word "mode" is confusing because it can mean so many things.
Thanks you very much for your great lectures. That would nice if you could include some references at the end.
Behrooz Semnani It is difficult to find good references for detailed derivation coupled-mode theory, but I will try.
super interesting
Thank you very much for this video. Is there a link where I can download your slides? Because they are really good.
Yes, absolutely. I have put most of my courses here:
emlab.utep.edu/academics.htm
Each of the graduate courses listed contains PDFs of the notes, links to recorded lectures, homeworks, and other resources. Click on "21st Century Electromagnetics" to get to this specific lecture.
Thx!
CEM Lectures That's great. Thank you very much. This video was very helpful to me
Hello, thank you very much for this helpful video. A paper that specifically discusses the derivation of the General Coupled Mode equation, as well as butt coupling, propagation coupling and little delta, is:
Marcatili, E. Improved Coupled-Mode Equations for Dielectric Waveguides (1986). IEEE Journal of Quantum Electronics, Vol. QE-22, No.6
Thank you very much! I will take a look at this paper and provide as a reference in the notes if I like it.
@@empossible1577 Thank you as well for your reply. I have a follow up question that you may be able to help with. I am trying to determine a useful/working definition on what constitutes weak coupling vs strong coupling. So far, what i have observed is qualitative definitions, which are also not applicable for my problem. Do you have a definition and a reference that would be uesful?
@@ND-im1wn If the fields where the modes overlap look similar, they will have stronger coupling. So you can weaken coupling by either moving the waveguides away from each other or have modes that are very orthogonal to each other (i.e. look very different).
@@empossible1577 Thank you again for your reply, Dr. Rumpf. To avoid an awkward discussion here, I have sent you an email with a more careful explanation of my problem. I hope that is alright.
Thank you so much for these amazing lectures. I think there is a sign mistake in the slide 34- dB/dz equation. According to the reprocity of kappa for this case, the sign to the right of the dB/dz equation would be positive
Thank you! I will check it the possible error!
Hello. I am looking to understand two things, but I am struggling. First, the fundamental question of why do these optical modes couple? And second, how can I predict the frequencies of these coupled modes prior to any simulation? I want to develop this theoretical intuition without just brute-forcing my simulations until I get the desired results. Any guidance in this area would be greatly appreciated.
To be more specific, I have been simulating 1D photonic crystals with TMM. I first noticed that when there are more than one defects in the crystal that the the transmission frequencies are no longer at the frequency they were "designed" to be at (i.e. at their lambda/(2*n) frequency). I realize that this may be a naïve consideration, but this is what first got me expose to the ideas of coupled-mode theory and the idea that resonant cavities can interact with each other in a system like this. I know that there a lot of analogies in photonics to solid-state physics, so I immediately thought of the tight-binding model. Searching for this lead me to the idea of quasi-modes, Wannier Functions, and more. However, I have not been able to find any fundamental material in this area, or anything that I can understand. More simply, I do not know what I am looking for...
Let's talk about this in the context of two waveguides in close enough proximity that they are coupled and will exchange power. Thinking of having two independent modes that are coupled is a useful way of thinking about this, especially when it comes to applications, but it is not actually correct. The two modes are not coupled. Instead, this is just one "super mode" that encompasses both waveguides. That supermode looks very similar to the two independent modes would but still look a bit different. If you were to plot the amplitude of the super mode as it propagates, you would see the same picture, but it is really just the left and right halves of the super mode coming in and out of phase. The closer the waveguides, the quicker the power seems to jump back and forth.
Unfortunately, even I cannot look at a configuration and made an intuitive guess about what frequency is coupled. In fact, all frequencies are coupled it is just that the strength of the coupling changes with frequency, allowing us to make useful devices that are frequency sensitive. In the end, I have to run a simulation to figure out the coupling. I only know to size things just a bit smaller than the wavelength. I still need to simulation to figure out the specifics.
This approach is based on the orthogonality of the modes. Is there some analytical theory that regards coupling between nonorthogonal modes?
That depends what you mean by orthogonal. If the modes are orthogonal, they will not couple. There needs to be something common between the modes to couple them.
From the orthogonality conditions, I see that modes 1 are orthogonal to modes 2 so coupling is achieved if a mode of waveguide 2 is also a mode of waveguide 1. Is this correct?
@@andersonoliveira8991 Assuming that you have a clean waveguide 'mode' (at a specific z). This field can be written as a superposition of an 'infinite number' of modes on another basis - for example, a basis consisting of modes that are supported by the other waveguide. In this framework, it is easier to think about orthogonality and energy transfer.
Dear Prof. Raymond
What is the reference for more details of deriving the equations in this lecture?
Unfortunately, I do not know of any good references for this materials. The few things I have seen are very sparse. I suspect if anything good does exist, it is in an old and out-of-print textbook.
@@empossible1577 thanks for your lecture and this is true since most of available references do not discuss this topic completely.
@@Islam_K_Abu_Almohtade_ If you find something good, let me know!
Hello sir! What are the different modes that exist in coupled waveguides?
The approximation that is made is that the modes in coupled waveguides are the same as single waveguides. This is an approximation and not reality, but it surely simplifies the math. In reality, the modes change their shape and propagation characteristics a bit in a coupled waveguide. It is more accurate to think of the coupled waveguides as one large weird waveguide.
I have realized that these concepts, like rectangular dielectric waveguides or coupled-mode theory lack of a good, universal and broad literature. I’m trying to get a good understanding of this theory, but most of the notes are fragmentary, and pretty old, in books that are very difficult to follow. Many other books seem to make arbitrary postulations and fly from one idea to another, without good explanations. I’m wondering why there is no solid theory on these very interesting topics, like for example in classical microwave theory.
It is definitely an old theory. It takes a lot of effort to develop it and so I think researchers are willing to just regurgitate the results from others and use that extra time to do new things. That is my excuse anyway!
Dear Raymond, can you please point me to a reference detailing all the derivation steps?
Many thanks.
Unfortunately, I do not know of any good references for this material. The few things I have seen are very sparse. I suspect if anything good does exist, it is in an old and out-of-print textbook. What you see in this lecture is what I have collected over the years in my personal notes. If you find anything good, please let me know!
@@empossible1577 I found few good references by Hermann Haus and Amnon Yariv. However very hard to digest, I was hoping I could find a tutorial paper or a book chapter. Happy to share/email the papers. Thanks very much for making a lot of great materials available online
@@empossible1577 would it be accurate to consider mode matching more rigorous and practical than coupled mode theory?
@@outrospection4all I can get the papers. Can you just give me the citations? Thx!
@@outrospection4all I think both methods are equally rigorous and I am not sure one is better than the other. Mode matching is probably more intuitive and probably more efficient, especially if the modes have analytical solutions.