A very nice video! Super clear and insightful. Dear Dr Raymond and EMPossible, I have derived for Hy and the final formulation is exactly the same as the Ey(x) part but instead of Ey, it is Hy. Because of this, I am confused about which part is different in the code between Ey (shown in the video) and Hy. Thank you for the content!
The final equation should look the same except you should also have mu and eps swapped. If this is not the case, you have made a mistake somewhere. If you swap E for H and mu for eps, the two equations should be identical. This is due to duality of electromagnetics.
@@empossible1577 Thank you for the fast response! Yes, I have swapped the [eps/mu]^0.5 with [mu/eps]^0.5, and the result is identical: (n_eff^2 Hy = [(d^2/dx^2) + eps_r] Hy). Does this mean the n_eff value from Ey shown in the video and from Hy are exactly the same? (i.e., n_eff from TE and TM modes)
@@lbrahimarsy Yes. If you swap mu and eps, then the H mode will give you the exact same neff as the E mode did with the original mu and eps. That is because it is the same differential equation. In a practical setting, the two equations will give different neff because most devices are made of only eps and not mu. The waveguides support two different types of modes and if you want to know everything about the waveguide you will need to calculate both. BTW...the terminology "E mode" and "H mode" is just my terminology. The labels "TE" and "TM" are much more common. My issue with those labels is that TE and TM are defined differently in different cases and can be a bit ambiguous. For numerical analysis, "E mode" and "H mode" removes this ambiguity, but you will only hear that from me. If you like this kind of analysis, you might be interested in a book I recently published through Artech House. Here is a link to the book website: empossible.net/fdfdbook/
Thank you for such a great lecture. It helps me big time! I am trying to digest it and to compare it with the analytic method shown in books. there are some things that I still need to understand. 1. in analytical method, we use the boundary conditions at the core-cladding interfaces while here, we do not include that, do we? Could you explain how it is possible? 2. do you have the video where you explain the incoming angles calculated numerically for each mode?
1. Boundary conditions are handled by how the numerical method is implemented. For finite-differences, adopting the Yee grid scheme ensures that the physical boundary conditions are satisfied. It is as simple as this. 2. If you are calculating modes supported in a waveguide, it does not makes sense to think about any sort of source or angle of incidence. Calculating the modes tells you what modes are possible. It is an entirely different simulation to figure out which of those those modes get excited by a particular excitation.
Thanks for the nice lecture, but i find that i am still confused by solving the mode of a channel waveguide using the same method. Since I can't substitute by d/dy=0. I am wondering if a channel waveguide can be seen as the superposition of a horizontal slab waveguide and a vertical slab waveguide, such that it carries the modes of both slab waveguides.
You can certainly analyze a channel waveguide this way as an approximate method. Fortunately, it is quite easy to do it rigorously. Incase you do not know, this video is part of a series of videos from our Computational Methods course: empossible.net/academics/emp4301_5301/ Topics 6 and 7 teach a very powerful way to implement the finite-difference method. The slab waveguide is an example. With this background in the finite-difference method, you can learn how to analyze channel waveguides in Topic 4 in our Computational Electromagnetics course: empossible.net/academics/emp5337/ I will also point you to a book that I recently wrote that teaches computational electromagnetics to the complete beginner using the finite-difference frequency-domain method. The book covers everything you are doing here and MUCH more and in better detail. Here is a link to the book website to learn more about the contents: empossible.net/fdfdbook/ Hope this helps!!
The only thing that changes is: (1) which material tensors you build and retain, (2) calculation of your A and B matrices, and (3) calculation of other field components if you even do that.
At 45:05 : if my matrix is too large, of the order of 10^9, can I use the eigs command instead of eig ? Will it have a marked effect on the result? Thanks!
The function eig() operates on full matrices and will calculate all of the eigen values and eigen-vectors. The function eigs() operates on sparse matrices. Sparse matrices are used when matrix size is very large. Typically it is not practical to calculate all of the eigen-values or eigen-vectors so eigs() is meant only to calculate a subset. It gives you lots of ways to determine which of the solutions you calculate. That is the primary difference.
is there no Ex ? or Ex will also exist if we solve maxwells equations and use elimination or For TEz in slab waveguide in this orientation axes choosen only Ey will exist? please let me know why are we not talking about Ex
For slab waveguides, Maxwell's equations split into two independent sets of equations. This means there are two types of modes that the waveguide will support. One of those modes will have Ex and the other will not.
What do you mean by "ve?" I checked the math and all looks correct to me. Let me point your to the latest version of the notes. While this equation is the same, the formulation has been revised and may be a little simpler to follow. This is now Lecture 7b here: empossible.net/academics/emp4301_5301/
In this case, the notes themselves are the source. The information was created exclusively for the course. There are certainly other resources that cover similar topics. For slab waveguides, search for information on optical waveguides and/or integrated optics. Here are some... www.amazon.com/Fundamentals-Optical-Waveguides-Optics-Photonics/dp/0125250967/ref=sr_1_1?keywords=optical+waveguides&qid=1556715022&s=books&sr=1-1-spell www.amazon.com/Introduction-Optical-Waveguide-Analysis-Schrodinger/dp/0471406341/ref=sr_1_2?keywords=optical+waveguides&qid=1556715042&s=books&sr=1-2-spell
The find the dispersion equation, you need to perform a closed-form analysis of the waveguide. I did this for a standard parallel plate waveguide in lecture 5b here: emlab.utep.edu/ee4347appliedem.htm If you want to do it numerically, you will end up with a plot instead of an equation. You will simply plot the eigen-frequency of your mode as a function of beta, the phase constant.
You are right! I usually calculate this (and ya for 2D) and use it for graphics later in the code. In this case, I drew the graphics differently so xa is never used at all. You can certainly delete it if you wish.
Wow,Thx so much for response. I want to ask other question. We suppose that the refractive index in x direction slowly vary in this case,but if we consider it and add the term which is partial refractive index partial x into our differential equation,it will change the answer which is nearby boundary only?
@@user-we6yk1yt7r I don't think I understand you question. I think you are asking about graded-index waveguides? This is only a matter of modifying how you build the waveguide onto your grid. Nothing else in the code has to change because it was formulated and implemented to be rather generic like this. I would certainly expect the answer you get to change a bit since you are simulating a different waveguide.
Thank you very much! It's really very helpful,but there is one thing I can't understand , why do we need to sort our modes, later in the matlab code, and how it's done, I will be very grateful if you can explain it to me, thank you again.
The mode sorting is only needed if you want to identify the fundamental mode, second order mode, etc. Without it, all modes are still calculated correctly. It is just that they may be in the wrong order in the eigen-vector matrix.
Thank you very much sir, while rewatching this video I noticed one thing that bugged me, you defined beta as k0*n*sin(theta), when Katsunari Okamoto in Fundamentals of Optical Waveguides, defines it as k0*n*cos(theta) with same axis directions, can you clarify this please.
I suspect Okamoto is defining the angle differently. I am defining it off of the surface normal. Check to see if Okamoto is defining the angle off of the longitudinal axis. In my definition, when theta is 90 degrees, beta is in the exact direction of the waveguide and should be beta=k0*n. This happens correctly when it has sin(theta) because theta=90 degrees in this case.
