This was beautiful to watch. So clear :')

The fact at 20:57 is just a simple consequence of the Jordan canonical form (you give only the case where algebraic and geometric multiplicities are equal so all Jordan blocks are 1x1). The matrix exponential is defined by its power series: e^A = I + A + A^2/2! + ..., so when you have the Jordan form A = USU^T, with U unitary, you get e^A = I + USU^T + USU^TUSU^T/2! + ... = I + USU^T + US^2U^T/2! + ... = Ue^SU^T.

For math stuff on why f(A) = W*f(lambda)*inv(W), see "similar matrices" and Jordan canonical form

Prof Raymond: I do not know if this has been stated before, or if I'm misunderstanding the words, but when a matrix is represented as PDP where D is a diagonal matrix, this particular process is known as diagonalization. The entire topic is close to similarity transformations, which is what you may have been referring to. (Re: ~20th minute)

Hello Professor Raymond, I think there's a subtle mistake in the sign convention. When you replaced the d/dt operator in Maxwell's equations, you assumed an exp(i*omega*t) dependence and hence replaced d/dt by i*omega. Once this is done, when your plane wave is assumed to have an exp(i*k*x) dependence, this automatically means it's traveling in the -x direction. If you wanted your exp(i*k*x) wave to travel in the + x direction, you should have changed the time dependence to exp(-i*omega*t). Now, a problem arises when you get to the matrix differential equation, let's call it d(psi)/dz = A * psi, where A is the matrix you call capital omega. Whether you assumed an exp(i*k*x) dependence or exp(-i*k*x) dependence, you're going to arrive at the same matrix A. In doing this, the equations are blind to the fact as to whether the wave is traveling in the +x or - x direction as it should because the solution will have forward and backward travelling waves anyway. (Also mathematically, kx and ky occur squared or as kx*ky and hence flipping both signs will not change the matrix). So, in essence what you arrive at in your slides is the matrix equation assuming that exp(-i*k*x) means propagation on the +x direction. A subtle thing occurs when you try to sort your modes using the Poynting vector in the example you provided. Because now the direction of propagation is reversed, you find that the Poynting vector of a mode points in +z while the eigenvalue says it should be -z. Here I think we're under the mercy of the eigensolver because if we multiply the eigenvector by a constant, it's still an eigenvector. Hence to fix the apparent contradiction, I find that if we multiply the eigenvectors all by complex i, the Poynting vectors and the eigenvalues match the desired direction of propagation. Probably what needs to be done is to calculate the Poynting vector and then use the eigenvalue to find out if it implies the same direction of propagation or not. If not, we need to multiply the Poynting vector by -1 and hence multiply E by i and H by i to fix this before proceeding. Let me know what you think. I also really appreciate the effort of recording the lectures and putting them here; please keep them coming :) as I use them to teach myself computational E and M. Cheers and thank you very much for making these lectures public.

@21:00 another way to go about this is... e^x = x + x^2/2! + x^3/3! + ... and set x to your matrix. You should get the same answer. Not that I've ever needed to take e to the power of a matrix, but yeah, that's how you could do it. I saw it in an MIT linear algebra class sometime last year. Another advantage of this approach is that you can use SVD on the matrices in the expansion to get an exact solution. Very cool shortcut in my view. Awesome lecture btw, I'll probably never need this stuff, but it's far better than soaking up endless hours of PUBG and Fortnite 😅. Yes, I'm a gamer 😂👽☮️

Hi Professor Raymond, I was trying to solve for anisotropic case using the TMM taught in this lecture, and after mode sorting, I ended up with a messy 4x4 eigenvector and diagonal eigenvalue matrices. However this is also means that the amplitude c will be a 4x1 column matrix (the first two values of the column matrix indicate forward propagation and the other two represents backward propagation), instead of 2x1 column matrix shown in the lecture video. My question is, does each of the four amplitude c has unique meaning? Thank you.

Dear Prof. Raymond. First of all, thank you for these amazing lecture series. I was wondering about the sign of the eignevalue exponentials in the fully ansiotropic case. After sorting the eigenvalues, you have also changed the sign of the lambda ^{minus} matrix. This is appearing also in the corrected notes. I was wondering if the change of the sign is necessary? If one does this he/she ends up in having four forward propagating modes instead of two forward and two backward waves. Or I miss something? Thanks again for the amazing work you are are doing.

Thanks very much for the video. I have some questions: how to calculate the absorption in each layer, and how to calculate the absorption in each position?

Lovely sir, Is there any Matrix equation containing simple Fresnel equation for air to die-electric medium ?

Following up on my last comment, we do not need to adjust the eigenvectors. It's just the definition you used for the Poynting vector is ExH instead of ExH* and this fixes things in the end. Prefer though that we start with the exp(-i*omega*t) convention from the start.

Hi Professor Raymond,i am looking for MMT code for Matlab simulation, but i have no idea how i can get this package or code, my question how can i get MMT implemented Matlab Mathworks, thank you for reading

Prof Raymond, I do not understand fully why the TMM method is unstable. Yes it treats everything as forward moving waves, however in the eigenvalue matrix, we get eigen-values with positive and negative signs indicating the direction of propagation of the wave. This should take care of the direction of propagation of the waves in the final solution after multiplication with the mode coefficient. In (47:01), when you make a new matrix with exp(+lambda*z) and exp(-lambda*z), are you not forcing the backward wave to become forward wave?

Hello Professor Raymond, I'm trying to figure out the Eigenvector W, and Eigen-value (Lambda). However, how can I reach to that answer as described in the 'Getting a Feel for the Numbers'? There is no equation or formula for that. I'm stuck in that points. :) Thanks,

At 15:20 you provide the format for anisotropic materials. Would it be incorrect to use the diagonal representation of the relative permeability and permittivity, in which case it is only slightly more complicated? Or is there some physics that gets lost when assuming a diagonal (but multi-valued) tensor dielectric function.

Awesome. Do you have a lesson on how to implement in matlab?

1. Why does the magnetic field have different eigenvectors but the same eigenvalues as the electric field? (Lec 2a, slide 54). The eigenvalues being the same make sense because the E- and H-fields propagate together, and the eigenvectors being different makes sense if P and Q do not commute (in the case of these matrices they do, but I suspect their generalizations do not). 2. How are we justified in assuming that the solution for the magnetic field has the same c+ and c- coefficients as for the electric field, not different ones? It seems like these should just be arbitrary coefficients in both cases, and I'm not sure how they should be related.

Hello, is it possible to implement the TMM in 2D and 3D cases ? Thank you for the lectures

SIr, I am an electrical engineering student. Sir how Transfer Matrix Method is used for solar pv module as it a multilayered structure.

Respected Sir May you live long!. why we need to use TMM instead of using Frensel's equation. i am apologies to say that sound quality is very pour and have distortion. It will be blessing for students if you use whiteboard in your future classes. so that its become students friendly for non native English speakers

Hi Professor Raymond,i am looking for TMM .P file for Matlab simulation, so that after implementation we can verify it with your result. my question how can i get sanctification for TMM implemented Matlab Mathworks is 100% correct, thank you for reading