I have a couple questions. I apologize beforehand if these seem obvious, I'm a beginner: 1) Why does the attenuation of the wave depend only on the z-coordinate? Is it because we assume the device to be uniform in the x and y directions and thus the only non-trivial derivative is in the z direction? or is there another explanation? 2) If the attenuation depends only in z, how is the extra absorption due to a slander path through the device being longer than a straight path accounted for? Is it accounted for by varying the magnitude of z-component of the k-vector in any way? Or is it in fact wrong that a slander path would attenuate more? 3) Does snell's law (i.e, the parallel component of the wave vector should be continuous through the interface) apply only to the real part of the k-vector or of both real and imaginary parts? Thank you, any help would be appreciated. I find this topic really interesting and your lectures invaluable. I would just like to grasp the concepts as firmly as possible.
My main motivation for creating these materials is to help beginners. Please feel comfortable asking questions! First, let me point you to the official course website where you can see the entire course organized and the links take you to the latest versions of the notes and videos, and other learning resources. This video is old and the sequence for TMM has been improved. Here is the link: empossible.net/emp5337/ 1) This is a very deep question. In TMM, waves cannot decay in the x and y directions. The applied wave is coming from a lossless medium and is a plane wave. Therefore it has a constant amplitude across the entire xy interface. Boundary conditions requires that the field on the other side of the interface to also have a constant amplitude over the entire xy interface. If that medium is lossy or the wave is cutoff, it can only decay in the z direction even though it may have an angle of incidence. Waves where the planes of equal phase are different than the planes of equal amplitude are called inhomogeneous waves and they have a complex angle. Here is a link to a short video I made about complex angles. ua-cam.com/video/aM2g1J1J8To/v-deo.html 2) I think I answered this question above. Decay is only ever in the z direction in TMM. 3) Both real and imaginary parts of the tangential component of k are continuous across the interface. In TMM with a lossless reflection region, the x and y components are only ever real. We are currently working on a generalized TMM, but we are not finished that work so I cannot comment completely how how to make the reflection region and transmission region lossy. I can say it is not as easy as just making the material properties complex. If you are enjoying this topic and just starting out, checkout our new book and other courses on EMPossible.net. empossible.net/fdfdbook/ empossible.net/ Hope this helps!!!
@@empossible1577 It helped a lot, thank you sooooooo much!!! Never imagined your response would be so fast and so complete, thank you! I think I have a better understanding now and I will check out all the material you just shared. Just one final question (just to recapitulate): since both real and imaginary parts of the tangential component of k are continuous across the interface and the incoming wave is coming from a lossless medium, could that be used as justification as to why attenuation only happens in the z-direction?: "Since tangential components of k-vector have to be continuous and the initial tangential components are only real, the subsequent tangential components will only ever be real and thus, attenuation may only happen in the z-direction since it's the only component of the k-vector that can be complex ". Would that be ok? or is it too simplistic?
@@andresyesidmorenovilla7888 I am not always this fast. Your question happened to come in when I was checking. I always try to respond within 24 hours. What you are saying is correct and not too simplistic.
Thank you for this lecture. I want to ask about something, if I want to add loss or gain to one or bunch of layers all I have to do is add imaginary value to the permittivity, right? if this is right, according to theory of parity-time symmetry, the transmission from left and right are equal but not the reflection. how I can extract the right and left reflection in this case from the code? thank you.
I may need to give this more thought, but I thought both T and R are the same. I "think" things will only differ when you introduce anisotropy or perhaps nonlinearities. Can you point to any literature? Oh yes...loss and gain are incorporated by making permittivity (or permeability) complex. The sign of the imaginary component indicates loss or gain. Which sign is loss or gain depends on the sign convention used.
@@empossible1577 Thank you. What i have tried after adding the imaginary part is to extract each term from the final global matrix and use the S11 and S22 for reflection left and right respectively, and after introducing the loss and the gain the transmission is the same on both sides. But still the final figure is not like the literature! In all parity-time literature they use the transfer matrix instead of scattering matrix but as you pointed out in the video it doesn’t matter as long as the code works in the first place, that’s why i wanted to discuss with you what modifications could be done to the code that could work in this situation.
@@omarelsheikh1060 I have never seen TMM fail when loss or gain is incorporated. That said, our current TMM formulation is not setup to have loss or gain in the external mediums. In fact, that is a long story, but the short version is that we are working on it. We want to generalize all our codes so that the external mediums can be complex. It turns out that is not easy!
The scattering matrix formulation starts of with using field equations derived in the PQ method (22:57). So is this formulation restricted to only isotropic material?
+_jp88 Yes. You can derive the PQ form for diagonally anisotropic materials. If you want to rotate the tensor at all, producing off diagonal terms, you have to implement TMM like in the previous lecture. It involves the 4x4 matrix, sorting modes, etc. However, when done it works quite well. We use our anisotropic TMM quite a lot.
Is it true to say that reflectance and transmittance are direction-agnostic quantities? So they are defined as just the total incident power (from whatever direction, even if that direction is not normal to the device) divided by the total reflected power (even if that power is in some wonky direction like very close to parallel to the device).
It is a little more than just power reflected divided by power transmitted. It is only the normal component of power that should go into this calculation. This may seem weird, but think of if this way. A wave travelling near parallel to the interface carries power to and from that device much less effectively than a wave at normal incidence. Another requirement is that reflectance and transmittance are relative to a single incident wave. I have some information about this in Lectures 3a and 3b here: empossible.net/academics/emp5337/
@@empossible1577 That is pretty counter-intuitive, but it does make sense now. Thanks! I think I have come up with a thought experiment that gets to the point. In the limit where we have an incident wave purely parallel to our device (kz = 0 in our incident media, free space, and say we are in 2D so kx = k0), then we can use the dispersion relation in the second media along with phase matching at the interface to find that kz in the second media is +/-sqrt(k0^2(*n2-1)). So phase matching seems to imply that for a plane wave parallel to the interface, there is a solution to Maxwell's equations where the plane wave is not moving at all towards the device in free space, but is moving towards/away in a second media. Thinking about it backward, this is total internal reflection, and we would expect no power to be transferred from the device side to the other side
Thanks for the great lecture! I would have a question though: I don't quite understand why Psi (the fields) is written as it is for each of the mediums (at minute 23:00) when formulating the boundary condition at each interface of Layer i . I get that the exponentials are 1 at the first interface inside Layer i. But I dont understand what happens with these terms for the medium (i.e. why is the matrix of exponentials missing from the left side of the boundary condition equation for the first interface and the right side of the second boundary eq.). Or am I just missing a simple point of how one can formulate the fields in a medium?
Great question. I am doing something rather strange and different here. Medium 1 and Medium two are not layers in the device. They are infinite half spaces. I have defined the psi terms in these external regions to perfectly define the fields immediate at the interface so no exponential terms are needed. Inside layer 1, I have defined the psi term to perfectly define the fields at the left interface so exponential terms are needed there. However, exponential terms are needed at the second interface since the Layer 1 psi term is propagated through the layer. The little dots at the end of the arrows indicate where the psi terms are defined.
Hi dr. Rumpf. I appreciate all the great content you are producing, thanks! Please help me get some clearity on the following: Elsewhere I have learned that from simply satisfying the boundary conditions the field in first region can be related to the field in the transmission region like so : [E0+, E0-] = [M11, M12; M21, M22] [Et+,0]. Reflection (r) and transmission (t) coefficients are then simply given by r = M21/M11 and t = 1/M11. What's the added value of taking this, what seems to me a much more complicated approach?
If TMM were handling a single interface, the approach you describe would probably be best. The main reason for the scattering matrices is to allow TMM to combine multiple layers together to simulate more complicated devices. This allows you to do advanced things like cascading and doubling to efficiently simulate devices with millions of layers (like fiber optic Bragg gratings).
No. Propagation happens due to a mutual coupling between the electric and magnetic fields so they both contribute. Since they are coupled, however, it becomes possible to calculate the power from just the electric field or from just the magnetic field.
I am currently trying to implement this, and I am very confused. It looks like, for LHI media, if we multiply out PQ to get omega squared, then this omega squared matrix is just a multiple of the identity. (kx^2 + ky^2 - ur*er). So for *any* kx and ky (any direction we point the incident plane wave), the only thing that changes about omega is the eigenvalues, not the eigenvectors. Or, if the eigenvectors do change, we would have no way of finding them numerically (eigendecomposition of the identity matrix is ambiguous). So my questions are this: 1. Since the omega squared matrix is always a multiple of the identity for LHI media, how do we find the W matrix? The eigenvectors of the identity matrix are literally every vector, and not unique. The columns of W could be any set of orthonormal unit vectors in 2-D vector space (and I have found that with numerical noise they most definitely *will* be) 2. If we are interpreting the c+ as the forward propagating mode coefficients, and W * c+ as the forward-propagating x- and y-components of the electric field, then it seems like if we don't know the eigenvectors of W we can never figure out what the modes are in relation to the x and y components of the electric field. This seems to mean we cannot identify 'TE' and 'TM' modes from the matrix formalism proposed here. How do we find this relation? What information is missing from the matrix formalism?
I think I have figured out part of the answer. There is nothing missing from the matrix formalism, because it is the case for LHI media that any field polarized only in the x/y plane is a mode of this system. So we could actually choose any pair of orthonormal vectors as the columns of W, as long as we stuck with what we chose.
Hello Sir, Do you have any lecture / paper which you'd advise to investigate more about dispersion analysis? That is, how to find the effective parameters of a bulk unit cell (permeability and permittivity and eventually thickness) that behave the same way as a complex unit cell filled by whatever material and whatever geometry? Best,
Somehow I missed this question and I am four years too late! Anyway, take a look at Homogenization under Topic 5 here: empossible.net/academics/emp6303/
Thank you very much Sir for the lectures. I have three questions: 1. Is it because we have surrounded each layer with free space layers of thickness zero that we have to use Redheffer Star Product? or How will the Redheffer Star Product make the TMM stable? 2. How do you derive the vector k_inc (slide 38, 47:50) an the coefficient c_in (slide 39, 49:40) ? 3. In order to calculate reflectance, why can we just use the global Scattering Matrix: R= (Sglobal_11)^2?
1. Surrounding the layers with free space does not stabilize the method. This practice allows us to calculate scattering matrices without having to know anything about the surrounding layers. The Redheffer star produce would be needed no matter how you handle the scattering matrices. The TMM was stabilized by distinguishing forward and backward waves and treating them separately. 2. k_inc is not derived. It is an input argument to the method. It is the direction of the applied wave. The equation to calculate it is on Slide 38 of the video. 3. You can do this. In fact, it is the more efficient way. I take the longer approach to teach the flow of the matrices and information.
Sir I have a another question about the Transmittance, how should it be calculated for multilayer? is T= (Sglobal_21)^2? In the formula (slide 42, 54:10) , if the incident and transmitted waves are in the same medium, would it be T= |Etrn|^2/|Einc|^2 ?
Yes, the incident and transmission regions are same, the equation simplifies to what you have written. Reflectance looks similar, but is |Eref|^2/|Einc|^2
I have a question regarding homework 4. Should the inputs arguments SA & SB to the star() function be entered as block matrices ( with elements S11, S22,,, etc) or as struct data type ( with fields equal to S11, S22,...)?
No. It is extremely rare that the scattering matrix itself is ever used. Instead, the four elements of the scattering matrix S11, S12, S21, and S22 are stored and used separately. So, SA and SB are structures containing the four elements separately. So you will have SA.S11, SA.S12, SA.S21, SA.S22, SB.S11, SB.S12, SB.S21, and SB.S22.
First, let me point you to the official course website that has PDFs of the latest version of the course notes, links to the lecture videos, and more. empossible.net/academics/emp5337/ W0 and V0, I am now called Wg and Vg. These are the eigen-vectors of the gap medium. Pretty much anything can be used without much issue. However, some choices can lead to the case where kz=0 and the simulation will fail. On slide 30 of Lecture 2c (of the latest version of the notes) I show how to make a section that avoids this entirely.
Let me point you to the official course website. This entire section of notes on TMM has been considerably revised. See Topic 2. empossible.net/academics/emp5337/ As for the codes, I recommend trying to develop the codes on your own from just the theory slides. This will maximize your understanding of the method. If you are truly in a rush, stuck or have no idea where to even begin, there is a paid online course that will step you through every single line of code in MATLAB to implement the transfer matrix method. empossible.thinkific.com/courses/tmmmatlab Hope this helps!
Hi Sir, I am trying to apply this method for a metal/dielectric/metal structure (MIM). I need study the excitation of the surface plasmons propagating in this MIM structure. Does this formulation take care of the condition require to excite SPs? If not, Where would be the changes I need to make to find the eigen values and eigenfunction for SPs? Hope you would reply asap. Thanks
The TMM is a rigorous method so it is able to handle the physics of SPs. Turning the TMM into an eigen-mode solver is possible, but the process is not described in this lecture. It is a bit different and challenging. An alternative technique for you would be the finite-difference frequency-domain (FDFD) method described in later lectures. Also, FDFD will be able to simulate any type of excitation technique you may want to eventually study. The slab waveguide analysis described in 12 is exactly the code you want for predicting SPs. BTW, let me point you to the course website. It has links to all of these lectures but also contains the latest version of the notes. There have been some revisions, additions, and other improvements since the lectures were recorded. I think this may help. emlab.utep.edu/ee5390cem.htm
Hello sir, I could not understand why you have inserted the zero thick air film between the stacks of layers. I have three layers metal/dielectric/metal /substrate. I need to get the reflection from this system of layers. If wave would travel from one medium to another and it would change the phase on reflection from each layer. But if I would insert the air film in between it will distort the result what actually I am looking for. Please explain. Thanks.
There are several reasons this is done. For a complete answer, read the following journal article... R. C. Rumpf, "Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention," PIERS B, Vol. 35, 241-261, 2011. The short answer is this lets the scattering matrices not depend at all on the materials in adjacent layers. When this is the case, you can interchange the order of any scattering matrices you calculate. For your structure this is not very important, but for other more longitudinally periodic structures it is very useful. The materials outside of each layer are set to some homogeneous medium, not necessarily air. If you follow the formulation, when two scattering matrices are combined, the homogeneous layer between them will have zero thickness ans so it will not distort the amplitude or phase of your result at all. It is simply a numerical trick.
Fantastic video, I found it very helpful to create a working transfer matrix model. One question; is introducing loss into the model as simple as introducing a complex permeability?
Perhaps also the idea of frequency dependent permittivity, which comes up very frequently in physical applications. As a relative novice, I just iterate over the whole calculation for each different frequency, but I imagine there must be a more elegant way!
Dear sir, thank you for the great/extremely helpful video. I am currently working at implementing your method and extending to SP's in multilayer graphene system. You mentioned one could extract the dispersion relation, how would you achieve this with scattering matrix? I'm aware that in transfer-matrix method the pole of the reflection coefficient leads to the dispersion, and numerically one could also contour plot for different Omega values versus their corresponding k-vectors, which can be obtained from the eigenvalue of the transfer-matrix; but here does the eigenvalue of scattering matrix still give you omega-corresponding k-vectors? I would be grateful to hear your insights!
First, I want to thank you this great effort. Question: I can't undestand how Wh and Vh matrices of gap(or free space ) medium where calculated in Homework 3.1? any suggestions!
Wh and Vh are the eigen-vectors of the gap medium. See slide 26, Lecture 5b from the PDF notes on the official course website. emlab.utep.edu/ee5390cem.htm
Hello Sir, I want to plot my EM fields in a medium. I took three mediums of air with same thickness to be sure about what I'm supposed to have (that is, 100% transmission) From your Lecture TMM Extras, I extract the backward and forward coefficients in each medium. When illuminating at oblique incidence (theta = 30 degree), the norm of my backwark amplitude coefficients are not equal to 0 (just a bit larger than 0), while they should because of only air medium. Is there any numerical limitations to TMM, or should I reconsider the way I compute the forward / backward amplitudes coefficients? Thanks a lot for your work one more time! Best, Jules
The TMM as I presented is limited to lossless materials. Things are a bit more complicated when loss is incorporated. That is one limitation, but you are just simulating air (or probably vacuum) so I suspect this is not the problem. Otherwise, all should be okay. When you say "just a bit larger than zero," are saying like 10e-15 or something that small or more like 0.1? I would not worry about very small numbers (i.e. 10e-15), but I would worry about 0.1 scale numbers. One thing you can do is look at your intermediate scattering matrices. Other than phase, they should all essentially be identity matrices. Maybe one of them has an error? Also, for visualizing internal fields, have you seen the notes I created about this in the TMM extras lecture? I have not made a video of it, but you can get the notes from the official course website here: empossible.net/academics/emp5337/
@@empossible1577 Dear Sir, Thanks a lot for your answer. I have fixed my problem, I mixed things in the internal / external coefficients of a layer :). However, when looking at a slice of the field on a 2D plot (for example the Ex(z) field at an (x0,y0) coordinate ), I have a tiny but existing discontinuity at the interface between two layers. It's a really small discontinuity, maybe it comes from the way my program langage (Python) solve linear system? Who knows Thanks again for your lectures, it's really helpful Regards, Jules
+_jp88 The wave equation comes in different forms. You are referring to the time-domain wave equation. With a Fourier transform, you can also have a frequency-domain wave equation (Helmholtz equation) with no double time derivative. It is this that is cast into matrix form here.
1. Is there an advantage to propagating the modes and their coefficients, rather than the fields? Conceptually it seems a little backward, but will this be useful in the future? 2. Also, is it necessary to retain both the E- and H-fields everywhere when doing calculations? It seems like we could just work with the E-fields the entire time, solve the uncoupled second-order differential equation, and then at the end once we have everything we want, calculate the H-fields. This seems like it would make the derivations simpler and we would have to store smaller matrices in memory, is there a computational reason we don't do this, or is it something more fundamental? Does this break when we have anisotropy?
I think I may have figured out the answer to my first question. It seems like we are at an impasse as to whether to deal directly with fields or with modes, and either would work. I think fundamentally this has to do with boundary conditions being easier to formulate in terms of fields, and propagation being easier to formulate in terms of modes. On the one hand, eigenmodes are very easy to propagate. You just multiply by exp() of a diagonal matrix whose elements are just the square root of the computed eigenvalues of the modes for that layer. Fields, on the other hand, propagate a little less cleanly - you have to exponentiate the entire matrix of the second-order differential equation, which you called capital omega. But after you propagate a mode through a layer, you want to convert it 'back' into a field so that you can match the boundary conditions. It still seems conceptually easier to work with fields by themselves.
1. For TMM there is little advantage. For other methods working with the mode coefficients requires fewer calculations so it is more efficient and I think it better unifies the various methods that can use scattering matrices. It is absolutely possible to formulate the scattering matrices to work directly with the fields. 2. For the most part, the TMM was done in terms of just electric fields. However, in order to solve the problem at the boundary the magnetic field terms are also needed because we would run out of information otherwise. Electric fields alone would be enough if waves only propagated forward, but since there are backward waves we need more information to solve the boundaries and this comes from the magnetic fields. However, we put the magnetic fields in terms of the electric fields so in a sense it is entirely done in terms of electric fields.
@@empossible1577 Thanks for the reply! It's not clear to me why there is a loss of information if we only formulate things in terms of the electric fields. We know a priori that the boundary conditions equate the tangential components of both the electric and magnetic fields (not just the electric field, so a total of 4 equations). If the H-field is fully determined by the E-field, then aren't two of these equations redundant? It seems like we basically have two equation pairs that say x1 = x2 and f(x1) = f(x2). The second equation is redundant. (where x1 is for example the x-component of the E-field in layer 1 and f is the H-field as a function of the E-field). Is there an advantage to retaining all four field components in the psi vector, or is it a necessity and I am just missing something?
@@JordanEdmundsEECS No. The difference is minimal. If you go through all of my lectures, you will see that I put a lot of work into making the methods look the same. Formulating based on mode coefficients is an artifact. There is absolutely nothing wrong with formulating as you have in mind based on the fields. It will be interesting for you to try and it and quantify the difference in terms of implementation and simulation time.
CEM Lectures Ahhh got it thanks. I realized why the boundary conditions are not redundant- they are vectorial, and the polarizations of E and H are not the same for waves propagating in different directions. My mistake XD
First, I want to point you to the course website that has all the latest version of the notes. There are a considerable amount of corrections and improvements since the videos were recorded. emlab.utep.edu/ee5390cem.htm Second, I would suggest using a different numerical technique for doing this, but you can certainly still do it with TMM and scattering matrices. See the first section in Lecture 6 that talks about dispersion analysis. Essentially you need both frequency (w) and wave number (k) and group velocity is calculated according to vg = dw/dk. You can get this information from the dispersion analysis technique. For a more detailed discussion of group velocity, see the first section in Lecture 22 in the EM21 course. Here is the website for that course: emlab.utep.edu/ee5390em21.htm Next, I would look through Lecture 7, also part of the EM21 course which covers the general theory for electromagnetic waves in periodic structures. I "think" all the theory you need is here, although I do not specifically address DoS. Last, take a look at the plane wave expansion method (PWEM) in the CEM course. If your device is 2D and dielectric, I suspect this will be your method of choice.
I just mean for a 1-D dielectric stack. I think I've got a working program, in that it spits out reasonable reflectance versus transmissivity. And I know that for some layer m, k_m = omega * n_m / c. Also, I know from literature that the DOS can be obtained from the complex transmission coefficient (t = x + iy) as 1/L * [(y' x - x' y)/(x^2 + y^2)] where L is the length of the crystal and the primes denote differentiation with respect to omega. So I guess my main question is how to obtain a complex transmission coefficient. Is this just obtained from the global scattering matrix element S_21?
Hi Professor. I am following your lectures closely. However I got stuck in two important points and I would like to ask for your help in order to clear the things up. I am interested in calculating the electric/magnetic field distribution along the stack. Whe solving the matrix differential equations for E and H, we get two vector coefficients named c^+ and c^-. It is clear for me how to get c^+, from the source parameter calculation slides, but I do not get to calculate the vector c^-. I drew the conclusion that it has to be calculated using the scattering matrix method in order to compute the backward coefficient of incident field and the transmitted forward-travelling coefficient. Then I got stuck in calculating the scattering matrices. I understand the fact that when computing its respective S-matrix, every layer has to be sandwiched in vacuum, so we can exchange the layers as we wish to compute the overall S-matrix. However it is not clear for me, where to include the zero-thickness rule of vacuum into the equations, in order to make it a numerical trick and that the computer knows that every surrounding vacuum layer has zero thickness, i.e. in which equation this thickness is included so we can say L -> 0 for media outside the layer. I thank you for your reply in advance. Yours sincerely, Juan
This question comes up often enough that you have motivated me to put together a few slides about this. I created a quick version so these do not represent my best work, but should be sufficient to learn from. Go to the course website and look at the PDF of the electronic notes for Lecture 6 TMM Extras. emlab.utep.edu/ee5390cem.htm It involves saving W, V, Lambda, and the intermediate global scattering matrix for each layer as you work from left to right through your layers. Then you work backward calculating first the internal mode coefficients and then the internal fields from the mode coefficients and the other saved parameters. If your device has a lot of layers, it may take quite a bit of memory in your computer to do this. Hope this helps!
Professor, thank you very much. Indeed the algorithm is much more complex as I was thinking. The slides are very clear. I just have a little question more regarding the calculation of single S-matrices for every layer. When saying that the sided medium is vacuum with zero-thickness, it is not clear for me where is this zero-thickness condition added that the computer knows it is a numerical trick. Or just calculate the S-matrices and putting them together with the Redheffer star product is enough to satisfy this condition? Thanks a lot
Before we published [1], the computational EM community was implementing the scattering matrices in a very crazy manner. The paper in Ref. [1] is worth the read and will explain everything. [1] R. C. Rumpf, "Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention," PIERS B, Vol. 35, 241-261, 2011. The scattering of a layer depends on the layers next to it. For this reason, a scattering matrix cannot be computed without knowledge of the layers next to it. Even more, once a scattering matrix is calculated, it can only ever be combined with the layers it was calculated with. This severely limits the usefulness of the scattering matrix. I wanted the scattering matrices to depend only on themselves and I wanted to combine them with other scattering matrices in any way I wished. This is the origin of the zero thickness concept. Each layer in a simulation is actually separated by homogeneous gaps filled with the same material in all gaps. This makes the layer scattering matrix independent of other layers and allows the scattering matrices to be combined in any order since all layers are now surrounded by the same medium. However, we do not want the homogeneous gaps to affect the structure electromagnetically. To do this, we make those gaps have zero thickness. There is not a place where there is a dimension set to zero. It is automatic based on how the scattering matrices are defined and formulated.
How are the parameters P_TE and P_TM calculated? It seems to me like they should be an additional parameter in the model, as we have said nothing about the polarisation of the EM source.
Those parameters are defined in the "dashboard" of your code. That is the piece of code at the beginning that defines everything about your problem. It is all the hard coded numbers. P_TE is the complex amplitude of the TE polarization and P_TM is the complex amplitude of the TM polarization. There is also THETA and PHI parameters the describe the angle of incidence of the source.This information is conveyed in to different places on the course website:emlab.utep.edu/ee5390cem.htmFirst, the dashboard for a typical TMM simulation is given in Homework #3. Second, you can get some example numbers from the benchmarking document. In this document, you will find all the dashboard parameters along with all the intermediate parameters as they are calculated in a typical TMM simulation. This way you can troubleshoot your own code and better understand the order of the calculations.Hope this helps!
It is possible that order of eigen-modes can be different between your results and mine. There may also be a mistake in the benchmarking documents, but they have survived students in several courses so that is less likely. Just keep at it!
+Vladimir Vargas Calderon You can find discussion of scattering matrices in most circuit theory and microwave engineering texts under the general title "Network Theory." In fact, the scattering matrices used in CEM are N-port networks. Even for just TMM we have a four-port system, not a two-port system, because there are two possible polarizations on each side giving a total of 4 ports. Take a look at the final scattering matrices and you will see that they are 4x4, not 2x2. Now in terms of our improved version of scattering matrices, the only place I know of that being discussed is in our publication on the concept. www.jpier.org/pierb/pier.php?paper=11083107 The paper is open-source so you should be able to download it freely. Hope this helps!
A big thank you prof Ray for this resources. I am very much curious about the mathematical derivations as I get confident when I understand them. So on slide 21, you combine the inverse of [Wi Wi ; Vi -Vi] with [Wj Wj ; Vj -Vj] to get 1/2[Aij Bij; Bij Aij]. I cant seem to find any matrix algebra or assumptions that led to this result. On expanding, what I got was [Wi^-1Wj + Wi^-1Vj Wi^-1Wj - Wi^-1Vj ; Vi^-1Wj - Vi^-1Vj Vi^-1Wj + Vi^-1Vj]. Can you please provide some hint that led to this result? Your always prompt response is very much appreciated. Thank you.
I realized that, by finding the general Schur complement of an inverse 2x2 block matrix and substituting the Wi,j and (-)Vi,j elements with some few basic algebra steps leads to that result. Thanks.
Thanks for posting this online and it helped me a lot understanding the concept o TMM. There is one thing I think I should point out. You used exp(jwt-jkz) most of the time but you used exp(jkz-jwt) for TMM here. However, you ended up with the same set of the differential equations you developed previoiusly. Thus I suppose you used different normalization for H here. You used H_normalized= -j*sqrt(u0/e0)*H. But here did you use H_normalized= j*sqrt(u0/e0)*H? Thanks again for posting.
Victor Wang It is great to see you analyzing this in such detail. The manner in which I normalize H is always the same regardless of the sign convention. The biggest impact of the sign convention is that you have to also consider the sign of the imaginary parts of mu and eps if you are incorporating loss. Otherwise, it is only subtle sign changes in the formulation, if any changes.
Victor Wang Also, there is one thing I don't quite understand. You used exp(ikz) convention. However, you expressed Ex(z) as Ex(z)= (a_+)*exp(-omega*z)+(a_-)*exp(omega*z). Is that a conflict with the sign convention?
Victor Wang I appears I may have goofed my notation. Thanks for pointing it out. I will look this over in more detail and fix what may be wrong as soon as I can.
Victor Wang So I looked at this in a bit more detail and I now remember what my thoughts were at the time I did this sign convention change. First, I tend to like the exp(+jkz) convention so that the imaginary parts of permittivity and refractive index are positive for lossy materials. This is more consistent with what is published in materials databases. However, the convention exp(-jkz) is used more often in electrical engineering. Both work and both are used. The TMM lecture here is intended to get the students ready for learning rigorous coupled-wave analysis in a later lecture. In that formulation I wanted to stick with what was in the literature which is the exp(-jkz) convention. That is why I made the switch in the TMM lecture to this sign convention. I guess now the question is, why did I ever show the exp(+jkz) convention? I wanted students to see both! Also, my TMM and FDFD codes all now use the exp(+jkz) convention. Yes, I even confuse myself many times when I cannot remember which convention I used. If you would like to see a revised set of slides that stick with the exp(+jkz) convention, send me an e-mail. I have them. Also, incase you are not aware, you can access the electronic notes, recorded lectures, and other resources through the course website here: emlab.utep.edu/ee5390cem.htm
Hi Raymond, the code works fine and I compared it with benchmark results. Next step, on slide 40 for Wavelength or Frequency Parameter Sweep, inside the for loop you introduced "LAMBDA". What is this LAMBDA? lam0 is a free space wavelength which is in this case have a range of different values. Could you please guide a bit about this parameter sweeping part?
In a sense, frequency and free space wavelength are synonymous. Optics people like to characterize waves in terms of wavelength, while radio frequency people use frequency. I present both in my lectures. In this case LAMBDA is an array of free space wavelengths over which to perform simulations. You can easily convert this to an array of frequencies FREQ if you wish by using FREQ = c0./LAMBDA; Once either FREQ or LAMBDA is defined, you will setup a for loop to iterate through that array and perform a separate TMM simulation for each frequency or wavelength in that array. In the end, you will plot transmission and reflection as a function of frequency or wavelength.
Thanks for the lectures, they help me a lot! By the way, I find it bizarre since people always wrote in papers that the global matrix is the product of individual layer matrices, does it mean matrix multiplication or star product? How are they different in the result?
If you use the pure transfer matrix method, it is standard matrix multiplication. However, the pure transfer matrix method is not stable as you saw in Lecture 4. It is possible to fix TMM so that it works, but it is no longer ordinary matrix multiplication. Most of the world has adopted scattering matrices, although other approaches are faster. Combining two scattering matrices is definitely not a matter of just multiplying the two scattering matrices. Unfortunately, it is more complicated than that and is called the Redheffer star product. When you implement scattering matrices, you almost never store it as a single matrix. You store the elements S11, S12, S21, and S22 as separate matrices. Combining two scattering matrices involves maybe a dozen matrix operations. See Redheffer star product equations. If you tried standard multiplication to combine S matrices, I don't know what would happen, but the results would be garbage and differ significantly from using the Redheffer star product. I am not sure what the meaning would be of multiplying two scattering matrices. Somebody may know differently, but I don't think there is ever a time that is done.
Thank you very much, Sir, for the lectures . In homework 3, everything is going fine except that when I calculate X = exp(omega*k0*L), it gives incorrect value, compared to the benchmark. I reviewed my code several times but i can't identify what went wrong. Did anybody had similar issue ?
@@empossible1577 Thanks, sir, it worked. The only remaining issue now is that the real and imaginary parts of the final global scattering matrices ( after connecting to the external regions ) are different compared to their counterparts in the benchmark, making the conservation equal to 0.9869. What do you think the issue might be ?
@@OMohamedSamehO So your reflection side and transmission side scattering matrices match the benchmarking document? If so, check the star product equations.
I have a couple questions. I apologize beforehand if these seem obvious, I'm a beginner:
1) Why does the attenuation of the wave depend only on the z-coordinate? Is it because we assume the device to be uniform in the x and y directions and thus the only non-trivial derivative is in the z direction? or is there another explanation?
2) If the attenuation depends only in z, how is the extra absorption due to a slander path through the device being longer than a straight path accounted for? Is it accounted for by varying the magnitude of z-component of the k-vector in any way? Or is it in fact wrong that a slander path would attenuate more?
3) Does snell's law (i.e, the parallel component of the wave vector should be continuous through the interface) apply only to the real part of the k-vector or of both real and imaginary parts?
Thank you, any help would be appreciated. I find this topic really interesting and your lectures invaluable. I would just like to grasp the concepts as firmly as possible.
My main motivation for creating these materials is to help beginners. Please feel comfortable asking questions!
First, let me point you to the official course website where you can see the entire course organized and the links take you to the latest versions of the notes and videos, and other learning resources. This video is old and the sequence for TMM has been improved. Here is the link:
empossible.net/emp5337/
1) This is a very deep question. In TMM, waves cannot decay in the x and y directions. The applied wave is coming from a lossless medium and is a plane wave. Therefore it has a constant amplitude across the entire xy interface. Boundary conditions requires that the field on the other side of the interface to also have a constant amplitude over the entire xy interface. If that medium is lossy or the wave is cutoff, it can only decay in the z direction even though it may have an angle of incidence. Waves where the planes of equal phase are different than the planes of equal amplitude are called inhomogeneous waves and they have a complex angle. Here is a link to a short video I made about complex angles.
ua-cam.com/video/aM2g1J1J8To/v-deo.html
2) I think I answered this question above. Decay is only ever in the z direction in TMM.
3) Both real and imaginary parts of the tangential component of k are continuous across the interface. In TMM with a lossless reflection region, the x and y components are only ever real. We are currently working on a generalized TMM, but we are not finished that work so I cannot comment completely how how to make the reflection region and transmission region lossy. I can say it is not as easy as just making the material properties complex.
If you are enjoying this topic and just starting out, checkout our new book and other courses on EMPossible.net.
empossible.net/fdfdbook/
empossible.net/
Hope this helps!!!
@@empossible1577 It helped a lot, thank you sooooooo much!!! Never imagined your response would be so fast and so complete, thank you! I think I have a better understanding now and I will check out all the material you just shared. Just one final question (just to recapitulate): since both real and imaginary parts of the tangential component of k are continuous across the interface and the incoming wave is coming from a lossless medium, could that be used as justification as to why attenuation only happens in the z-direction?: "Since tangential components of k-vector have to be continuous and the initial tangential components are only real, the subsequent tangential components will only ever be real and thus, attenuation may only happen in the z-direction since it's the only component of the k-vector that can be complex ". Would that be ok? or is it too simplistic?
@@andresyesidmorenovilla7888 I am not always this fast. Your question happened to come in when I was checking. I always try to respond within 24 hours.
What you are saying is correct and not too simplistic.
Thank you for this lecture.
I want to ask about something, if I want to add loss or gain to one or bunch of layers all I have to do is add imaginary value to the permittivity, right?
if this is right, according to theory of parity-time symmetry, the transmission from left and right are equal but not the reflection. how I can extract the right and left reflection in this case from the code?
thank you.
I may need to give this more thought, but I thought both T and R are the same. I "think" things will only differ when you introduce anisotropy or perhaps nonlinearities. Can you point to any literature?
Oh yes...loss and gain are incorporated by making permittivity (or permeability) complex. The sign of the imaginary component indicates loss or gain. Which sign is loss or gain depends on the sign convention used.
@@empossible1577 Thank you.
What i have tried after adding the imaginary part is to extract each term from the final global matrix and use the S11 and S22 for reflection left and right respectively, and after introducing the loss and the gain the transmission is the same on both sides.
But still the final figure is not like the literature!
In all parity-time literature they use the transfer matrix instead of scattering matrix but as you pointed out in the video it doesn’t matter as long as the code works in the first place, that’s why i wanted to discuss with you what modifications could be done to the code that could work in this situation.
@@omarelsheikh1060 I have never seen TMM fail when loss or gain is incorporated.
That said, our current TMM formulation is not setup to have loss or gain in the external mediums. In fact, that is a long story, but the short version is that we are working on it. We want to generalize all our codes so that the external mediums can be complex. It turns out that is not easy!
@@empossible1577 Ah i see, that’s why people still stick to TMM in their publications even in 2023!
Okay thank you for your replay Dr. Raymond.
The scattering matrix formulation starts of with using field equations derived in the PQ method (22:57). So is this formulation restricted to only isotropic material?
+_jp88 Yes. You can derive the PQ form for diagonally anisotropic materials. If you want to rotate the tensor at all, producing off diagonal terms, you have to implement TMM like in the previous lecture. It involves the 4x4 matrix, sorting modes, etc. However, when done it works quite well. We use our anisotropic TMM quite a lot.
+CEM Lectures Right. Thank you again.
Is it true to say that reflectance and transmittance are direction-agnostic quantities? So they are defined as just the total incident power (from whatever direction, even if that direction is not normal to the device) divided by the total reflected power (even if that power is in some wonky direction like very close to parallel to the device).
It is a little more than just power reflected divided by power transmitted. It is only the normal component of power that should go into this calculation. This may seem weird, but think of if this way. A wave travelling near parallel to the interface carries power to and from that device much less effectively than a wave at normal incidence. Another requirement is that reflectance and transmittance are relative to a single incident wave.
I have some information about this in Lectures 3a and 3b here:
empossible.net/academics/emp5337/
@@empossible1577 That is pretty counter-intuitive, but it does make sense now. Thanks! I think I have come up with a thought experiment that gets to the point. In the limit where we have an incident wave purely parallel to our device (kz = 0 in our incident media, free space, and say we are in 2D so kx = k0), then we can use the dispersion relation in the second media along with phase matching at the interface to find that kz in the second media is +/-sqrt(k0^2(*n2-1)). So phase matching seems to imply that for a plane wave parallel to the interface, there is a solution to Maxwell's equations where the plane wave is not moving at all towards the device in free space, but is moving towards/away in a second media. Thinking about it backward, this is total internal reflection, and we would expect no power to be transferred from the device side to the other side
Thanks for the great lecture!
I would have a question though:
I don't quite understand why Psi (the fields) is written as it is for each of the mediums (at minute 23:00) when formulating the boundary condition at each interface of Layer i . I get that the exponentials are 1 at the first interface inside Layer i. But I dont understand what happens with these terms for the medium (i.e. why is the matrix of exponentials missing from the left side of the boundary condition equation for the first interface and the right side of the second boundary eq.). Or am I just missing a simple point of how one can formulate the fields in a medium?
Great question. I am doing something rather strange and different here. Medium 1 and Medium two are not layers in the device. They are infinite half spaces. I have defined the psi terms in these external regions to perfectly define the fields immediate at the interface so no exponential terms are needed. Inside layer 1, I have defined the psi term to perfectly define the fields at the left interface so exponential terms are needed there. However, exponential terms are needed at the second interface since the Layer 1 psi term is propagated through the layer. The little dots at the end of the arrows indicate where the psi terms are defined.
thanks fo the fantastic video.
Hi dr. Rumpf. I appreciate all the great content you are producing, thanks!
Please help me get some clearity on the following:
Elsewhere I have learned that from simply satisfying the boundary conditions the field in first region can be related to the field in the transmission region like so : [E0+, E0-] = [M11, M12; M21, M22] [Et+,0]. Reflection (r) and transmission (t) coefficients are then simply given by r = M21/M11 and t = 1/M11. What's the added value of taking this, what seems to me a much more complicated approach?
If TMM were handling a single interface, the approach you describe would probably be best. The main reason for the scattering matrices is to allow TMM to combine multiple layers together to simulate more complicated devices. This allows you to do advanced things like cascading and doubling to efficiently simulate devices with millions of layers (like fiber optic Bragg gratings).
At circa 54:00 - Does this mean that all the power resides in the electric component of the wave?
No. Propagation happens due to a mutual coupling between the electric and magnetic fields so they both contribute. Since they are coupled, however, it becomes possible to calculate the power from just the electric field or from just the magnetic field.
I am currently trying to implement this, and I am very confused. It looks like, for LHI media, if we multiply out PQ to get omega squared, then this omega squared matrix is just a multiple of the identity. (kx^2 + ky^2 - ur*er). So for *any* kx and ky (any direction we point the incident plane wave), the only thing that changes about omega is the eigenvalues, not the eigenvectors. Or, if the eigenvectors do change, we would have no way of finding them numerically (eigendecomposition of the identity matrix is ambiguous). So my questions are this:
1. Since the omega squared matrix is always a multiple of the identity for LHI media, how do we find the W matrix? The eigenvectors of the identity matrix are literally every vector, and not unique. The columns of W could be any set of orthonormal unit vectors in 2-D vector space (and I have found that with numerical noise they most definitely *will* be)
2. If we are interpreting the c+ as the forward propagating mode coefficients, and W * c+ as the forward-propagating x- and y-components of the electric field, then it seems like if we don't know the eigenvectors of W we can never figure out what the modes are in relation to the x and y components of the electric field. This seems to mean we cannot identify 'TE' and 'TM' modes from the matrix formalism proposed here. How do we find this relation? What information is missing from the matrix formalism?
I think I have figured out part of the answer. There is nothing missing from the matrix formalism, because it is the case for LHI media that any field polarized only in the x/y plane is a mode of this system. So we could actually choose any pair of orthonormal vectors as the columns of W, as long as we stuck with what we chose.
Hello Sir,
Do you have any lecture / paper which you'd advise to investigate more about dispersion analysis? That is, how to find the effective parameters of a bulk unit cell (permeability and permittivity and eventually thickness) that behave the same way as a complex unit cell filled by whatever material and whatever geometry?
Best,
Somehow I missed this question and I am four years too late! Anyway, take a look at Homogenization under Topic 5 here:
empossible.net/academics/emp6303/
Thank you very much Sir for the lectures. I have three questions:
1. Is it because we have surrounded each layer with free space layers of thickness zero that we have to use Redheffer Star Product? or How will the Redheffer Star Product make the TMM stable?
2. How do you derive the vector k_inc (slide 38, 47:50) an the coefficient c_in
(slide 39, 49:40) ?
3. In order to calculate reflectance, why can we just use the global Scattering Matrix: R= (Sglobal_11)^2?
1. Surrounding the layers with free space does not stabilize the method. This practice allows us to calculate scattering matrices without having to know anything about the surrounding layers. The Redheffer star produce would be needed no matter how you handle the scattering matrices. The TMM was stabilized by distinguishing forward and backward waves and treating them separately.
2. k_inc is not derived. It is an input argument to the method. It is the direction of the applied wave. The equation to calculate it is on Slide 38 of the video.
3. You can do this. In fact, it is the more efficient way. I take the longer approach to teach the flow of the matrices and information.
Thank you
Sir I have a another question about the Transmittance, how should it be calculated for multilayer? is T= (Sglobal_21)^2? In the formula (slide 42, 54:10) , if the incident and transmitted waves are in the same medium, would it be T= |Etrn|^2/|Einc|^2 ?
Yes, the incident and transmission regions are same, the equation simplifies to what you have written. Reflectance looks similar, but is |Eref|^2/|Einc|^2
I have a question regarding homework 4. Should the inputs arguments SA & SB to the star() function be entered as block matrices ( with elements S11, S22,,, etc) or as struct data type ( with fields equal to S11, S22,...)?
No. It is extremely rare that the scattering matrix itself is ever used. Instead, the four elements of the scattering matrix S11, S12, S21, and S22 are stored and used separately. So, SA and SB are structures containing the four elements separately. So you will have SA.S11, SA.S12, SA.S21, SA.S22, SB.S11, SB.S12, SB.S21, and SB.S22.
On slide 28, what to use for W0 and V0? it must be obvious, but i don't see it. Thanks!
First, let me point you to the official course website that has PDFs of the latest version of the course notes, links to the lecture videos, and more.
empossible.net/academics/emp5337/
W0 and V0, I am now called Wg and Vg. These are the eigen-vectors of the gap medium. Pretty much anything can be used without much issue. However, some choices can lead to the case where kz=0 and the simulation will fail. On slide 30 of Lecture 2c (of the latest version of the notes) I show how to make a section that avoids this entirely.
@@empossible1577 Excellent. Thanks for the response and these lectures are quite informative.
Great video!
How to write a MATLAB code for this? Please someone help me.
Let me point you to the official course website. This entire section of notes on TMM has been considerably revised. See Topic 2.
empossible.net/academics/emp5337/
As for the codes, I recommend trying to develop the codes on your own from just the theory slides. This will maximize your understanding of the method. If you are truly in a rush, stuck or have no idea where to even begin, there is a paid online course that will step you through every single line of code in MATLAB to implement the transfer matrix method.
empossible.thinkific.com/courses/tmmmatlab
Hope this helps!
@@empossible1577 thanks a lot!
Hi Sir,
I am trying to apply this method for a metal/dielectric/metal structure (MIM). I need study the excitation of the surface plasmons propagating in this MIM structure. Does this formulation take care of the condition require to excite SPs? If not, Where would be the changes I need to make to find the eigen values and eigenfunction for SPs? Hope you would reply asap. Thanks
The TMM is a rigorous method so it is able to handle the physics of SPs. Turning the TMM into an eigen-mode solver is possible, but the process is not described in this lecture. It is a bit different and challenging. An alternative technique for you would be the finite-difference frequency-domain (FDFD) method described in later lectures. Also, FDFD will be able to simulate any type of excitation technique you may want to eventually study. The slab waveguide analysis described in 12 is exactly the code you want for predicting SPs.
BTW, let me point you to the course website. It has links to all of these lectures but also contains the latest version of the notes. There have been some revisions, additions, and other improvements since the lectures were recorded. I think this may help.
emlab.utep.edu/ee5390cem.htm
Hello sir,
I could not understand why you have inserted the zero thick air film between the stacks of layers.
I have three layers metal/dielectric/metal /substrate. I need to get the reflection from this system of layers. If wave would travel from one medium to another and it would change the phase on reflection from each layer. But if I would insert the air film in between it will distort the result what actually I am looking for. Please explain. Thanks.
There are several reasons this is done. For a complete answer, read the following journal article...
R. C. Rumpf, "Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention," PIERS B, Vol. 35, 241-261, 2011.
The short answer is this lets the scattering matrices not depend at all on the materials in adjacent layers. When this is the case, you can interchange the order of any scattering matrices you calculate. For your structure this is not very important, but for other more longitudinally periodic structures it is very useful.
The materials outside of each layer are set to some homogeneous medium, not necessarily air. If you follow the formulation, when two scattering matrices are combined, the homogeneous layer between them will have zero thickness ans so it will not distort the amplitude or phase of your result at all. It is simply a numerical trick.
Fantastic video, I found it very helpful to create a working transfer matrix model. One question; is introducing loss into the model as simple as introducing a complex permeability?
That is it! I really should work that concept into the lecture and homework.
Perhaps also the idea of frequency dependent permittivity, which comes up very frequently in physical applications.
As a relative novice, I just iterate over the whole calculation for each different frequency, but I imagine there must be a more elegant way!
No that is it. Just calculate a different permittivity for each frequency. You can get values from tables or from materials models.
Dear sir, thank you for the great/extremely helpful video. I am currently working at implementing your method and extending to SP's in multilayer graphene system. You mentioned one could extract the dispersion relation, how would you achieve this with scattering matrix? I'm aware that in transfer-matrix method the pole of the reflection coefficient leads to the dispersion, and numerically one could also contour plot for different Omega values versus their corresponding k-vectors, which can be obtained from the eigenvalue of the transfer-matrix; but here does the eigenvalue of scattering matrix still give you omega-corresponding k-vectors? I would be grateful to hear your insights!
First, I want to thank you this great effort.
Question: I can't undestand how Wh and Vh matrices of gap(or free space ) medium where calculated in Homework 3.1? any suggestions!
Wh and Vh are the eigen-vectors of the gap medium. See slide 26, Lecture 5b from the PDF notes on the official course website.
emlab.utep.edu/ee5390cem.htm
Hello Sir,
I want to plot my EM fields in a medium. I took three mediums of air with same thickness to be sure about what I'm supposed to have (that is, 100% transmission)
From your Lecture TMM Extras, I extract the backward and forward coefficients in each medium. When illuminating at oblique incidence (theta = 30 degree), the norm of my backwark amplitude coefficients are not equal to 0 (just a bit larger than 0), while they should because of only air medium.
Is there any numerical limitations to TMM, or should I reconsider the way I compute the forward / backward amplitudes coefficients?
Thanks a lot for your work one more time!
Best,
Jules
The TMM as I presented is limited to lossless materials. Things are a bit more complicated when loss is incorporated. That is one limitation, but you are just simulating air (or probably vacuum) so I suspect this is not the problem.
Otherwise, all should be okay. When you say "just a bit larger than zero," are saying like 10e-15 or something that small or more like 0.1? I would not worry about very small numbers (i.e. 10e-15), but I would worry about 0.1 scale numbers. One thing you can do is look at your intermediate scattering matrices. Other than phase, they should all essentially be identity matrices. Maybe one of them has an error?
Also, for visualizing internal fields, have you seen the notes I created about this in the TMM extras lecture? I have not made a video of it, but you can get the notes from the official course website here:
empossible.net/academics/emp5337/
@@empossible1577 Dear Sir,
Thanks a lot for your answer. I have fixed my problem, I mixed things in the internal / external coefficients of a layer :).
However, when looking at a slice of the field on a 2D plot (for example the Ex(z) field at an (x0,y0) coordinate ), I have a tiny but existing discontinuity at the interface between two layers. It's a really small discontinuity, maybe it comes from the way my program langage (Python) solve linear system? Who knows
Thanks again for your lectures, it's really helpful
Regards,
Jules
5:34 Why is it called matrix wave equation? There is no double differential with respect to time.
+_jp88 The wave equation comes in different forms. You are referring to the time-domain wave equation. With a Fourier transform, you can also have a frequency-domain wave equation (Helmholtz equation) with no double time derivative. It is this that is cast into matrix form here.
1. Is there an advantage to propagating the modes and their coefficients, rather than the fields? Conceptually it seems a little backward, but will this be useful in the future?
2. Also, is it necessary to retain both the E- and H-fields everywhere when doing calculations? It seems like we could just work with the E-fields the entire time, solve the uncoupled second-order differential equation, and then at the end once we have everything we want, calculate the H-fields. This seems like it would make the derivations simpler and we would have to store smaller matrices in memory, is there a computational reason we don't do this, or is it something more fundamental? Does this break when we have anisotropy?
I think I may have figured out the answer to my first question. It seems like we are at an impasse as to whether to deal directly with fields or with modes, and either would work. I think fundamentally this has to do with boundary conditions being easier to formulate in terms of fields, and propagation being easier to formulate in terms of modes. On the one hand, eigenmodes are very easy to propagate. You just multiply by exp() of a diagonal matrix whose elements are just the square root of the computed eigenvalues of the modes for that layer. Fields, on the other hand, propagate a little less cleanly - you have to exponentiate the entire matrix of the second-order differential equation, which you called capital omega. But after you propagate a mode through a layer, you want to convert it 'back' into a field so that you can match the boundary conditions. It still seems conceptually easier to work with fields by themselves.
1. For TMM there is little advantage. For other methods working with the mode coefficients requires fewer calculations so it is more efficient and I think it better unifies the various methods that can use scattering matrices. It is absolutely possible to formulate the scattering matrices to work directly with the fields.
2. For the most part, the TMM was done in terms of just electric fields. However, in order to solve the problem at the boundary the magnetic field terms are also needed because we would run out of information otherwise. Electric fields alone would be enough if waves only propagated forward, but since there are backward waves we need more information to solve the boundaries and this comes from the magnetic fields. However, we put the magnetic fields in terms of the electric fields so in a sense it is entirely done in terms of electric fields.
@@empossible1577 Thanks for the reply! It's not clear to me why there is a loss of information if we only formulate things in terms of the electric fields. We know a priori that the boundary conditions equate the tangential components of both the electric and magnetic fields (not just the electric field, so a total of 4 equations). If the H-field is fully determined by the E-field, then aren't two of these equations redundant? It seems like we basically have two equation pairs that say x1 = x2 and f(x1) = f(x2). The second equation is redundant. (where x1 is for example the x-component of the E-field in layer 1 and f is the H-field as a function of the E-field). Is there an advantage to retaining all four field components in the psi vector, or is it a necessity and I am just missing something?
@@JordanEdmundsEECS No. The difference is minimal. If you go through all of my lectures, you will see that I put a lot of work into making the methods look the same. Formulating based on mode coefficients is an artifact. There is absolutely nothing wrong with formulating as you have in mind based on the fields. It will be interesting for you to try and it and quantify the difference in terms of implementation and simulation time.
CEM Lectures Ahhh got it thanks. I realized why the boundary conditions are not redundant- they are vectorial, and the polarizations of E and H are not the same for waves propagating in different directions. My mistake XD
How would one extend this to also calculate the DOS and group velocity?
First, I want to point you to the course website that has all the latest version of the notes. There are a considerable amount of corrections and improvements since the videos were recorded.
emlab.utep.edu/ee5390cem.htm
Second, I would suggest using a different numerical technique for doing this, but you can certainly still do it with TMM and scattering matrices. See the first section in Lecture 6 that talks about dispersion analysis. Essentially you need both frequency (w) and wave number (k) and group velocity is calculated according to vg = dw/dk. You can get this information from the dispersion analysis technique.
For a more detailed discussion of group velocity, see the first section in Lecture 22 in the EM21 course. Here is the website for that course:
emlab.utep.edu/ee5390em21.htm
Next, I would look through Lecture 7, also part of the EM21 course which covers the general theory for electromagnetic waves in periodic structures. I "think" all the theory you need is here, although I do not specifically address DoS.
Last, take a look at the plane wave expansion method (PWEM) in the CEM course. If your device is 2D and dielectric, I suspect this will be your method of choice.
I just mean for a 1-D dielectric stack. I think I've got a working program, in that it spits out reasonable reflectance versus transmissivity. And I know that for some layer m, k_m = omega * n_m / c. Also, I know from literature that the DOS can be obtained from the complex transmission coefficient (t = x + iy) as
1/L * [(y' x - x' y)/(x^2 + y^2)]
where L is the length of the crystal and the primes denote differentiation with respect to omega. So I guess my main question is how to obtain a complex transmission coefficient. Is this just obtained from the global scattering matrix element S_21?
You actually already are obtaining a complex transmission coefficient. It is the S21 coefficient in your global scattering matrix.
Hi Professor. I am following your lectures closely. However I got stuck in two important points and I would like to ask for your help in order to clear the things up. I am interested in calculating the electric/magnetic field distribution along the stack. Whe solving the matrix differential equations for E and H, we get two vector coefficients named c^+ and c^-. It is clear for me how to get c^+, from the source parameter calculation slides, but I do not get to calculate the vector c^-. I drew the conclusion that it has to be calculated using the scattering matrix method in order to compute the backward coefficient of incident field and the transmitted forward-travelling coefficient. Then I got stuck in calculating the scattering matrices. I understand the fact that when computing its respective S-matrix, every layer has to be sandwiched in vacuum, so we can exchange the layers as we wish to compute the overall S-matrix. However it is not clear for me, where to include the zero-thickness rule of vacuum into the equations, in order to make it a numerical trick and that the computer knows that every surrounding vacuum layer has zero thickness, i.e. in which equation this thickness is included so we can say L -> 0 for media outside the layer.
I thank you for your reply in advance.
Yours sincerely,
Juan
This question comes up often enough that you have motivated me to put together a few slides about this. I created a quick version so these do not represent my best work, but should be sufficient to learn from. Go to the course website and look at the PDF of the electronic notes for Lecture 6 TMM Extras.
emlab.utep.edu/ee5390cem.htm
It involves saving W, V, Lambda, and the intermediate global scattering matrix for each layer as you work from left to right through your layers. Then you work backward calculating first the internal mode coefficients and then the internal fields from the mode coefficients and the other saved parameters. If your device has a lot of layers, it may take quite a bit of memory in your computer to do this.
Hope this helps!
Professor, thank you very much. Indeed the algorithm is much more complex as I was thinking. The slides are very clear. I just have a little question more regarding the calculation of single S-matrices for every layer. When saying that the sided medium is vacuum with zero-thickness, it is not clear for me where is this zero-thickness condition added that the computer knows it is a numerical trick. Or just calculate the S-matrices and putting them together with the Redheffer star product is enough to satisfy this condition?
Thanks a lot
Before we published [1], the computational EM community was implementing the scattering matrices in a very crazy manner. The paper in Ref. [1] is worth the read and will explain everything.
[1] R. C. Rumpf, "Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention," PIERS B, Vol. 35, 241-261, 2011.
The scattering of a layer depends on the layers next to it. For this reason, a scattering matrix cannot be computed without knowledge of the layers next to it. Even more, once a scattering matrix is calculated, it can only ever be combined with the layers it was calculated with. This severely limits the usefulness of the scattering matrix. I wanted the scattering matrices to depend only on themselves and I wanted to combine them with other scattering matrices in any way I wished. This is the origin of the zero thickness concept. Each layer in a simulation is actually separated by homogeneous gaps filled with the same material in all gaps. This makes the layer scattering matrix independent of other layers and allows the scattering matrices to be combined in any order since all layers are now surrounded by the same medium. However, we do not want the homogeneous gaps to affect the structure electromagnetically. To do this, we make those gaps have zero thickness. There is not a place where there is a dimension set to zero. It is automatic based on how the scattering matrices are defined and formulated.
CEM Lectures Professor once again, thank you very much for your soon reply and your clear explanations. They indeed helped me a lot.
Juan
How are the parameters P_TE and P_TM calculated? It seems to me like they should be an additional parameter in the model, as we have said nothing about the polarisation of the EM source.
Those parameters are defined in the "dashboard" of your code. That is the piece of code at the beginning that defines everything about your problem. It is all the hard coded numbers. P_TE is the complex amplitude of the TE polarization and P_TM is the complex amplitude of the TM polarization. There is also THETA and PHI parameters the describe the angle of incidence of the source.This information is conveyed in to different places on the course website:emlab.utep.edu/ee5390cem.htmFirst, the dashboard for a typical TMM simulation is given in Homework #3. Second, you can get some example numbers from the benchmarking document. In this document, you will find all the dashboard parameters along with all the intermediate parameters as they are calculated in a typical TMM simulation. This way you can troubleshoot your own code and better understand the order of the calculations.Hope this helps!
Thank you that is very helpful! Although I have to say, Im having some difficulty getting my code to return the benchmarked results!
It is possible that order of eigen-modes can be different between your results and mine. There may also be a mistake in the benchmarking documents, but they have survived students in several courses so that is less likely. Just keep at it!
Is the Qh and Vh mentioned in the benchmark what youve called Q0 and V0 in the video? I can't seem to decipher what they are
Same thing. Sorry for the inconsistent notation.
Is there any textbook that develops the ideas more deeply? Thanks a lot for the great content!
+Vladimir Vargas Calderon You can find discussion of scattering matrices in most circuit theory and microwave engineering texts under the general title "Network Theory." In fact, the scattering matrices used in CEM are N-port networks. Even for just TMM we have a four-port system, not a two-port system, because there are two possible polarizations on each side giving a total of 4 ports. Take a look at the final scattering matrices and you will see that they are 4x4, not 2x2. Now in terms of our improved version of scattering matrices, the only place I know of that being discussed is in our publication on the concept.
www.jpier.org/pierb/pier.php?paper=11083107
The paper is open-source so you should be able to download it freely. Hope this helps!
What is W0 on slide 29? How would we calculate W0?
Me also, I can't undestand how Wh and Vh matrices where calculated in Homework 3.1? any suggestions!
A big thank you prof Ray for this resources. I am very much curious about the mathematical derivations as I get confident when I understand them. So on slide 21, you combine the inverse of [Wi Wi ; Vi -Vi] with [Wj Wj ; Vj -Vj] to get 1/2[Aij Bij; Bij Aij]. I cant seem to find any matrix algebra or assumptions that led to this result. On expanding, what I got was [Wi^-1Wj + Wi^-1Vj Wi^-1Wj - Wi^-1Vj ; Vi^-1Wj - Vi^-1Vj Vi^-1Wj + Vi^-1Vj]. Can you please provide some hint that led to this result? Your always prompt response is very much appreciated. Thank you.
I realized that, by finding the general Schur complement of an inverse 2x2 block matrix and substituting the Wi,j and (-)Vi,j elements with some few basic algebra steps leads to that result. Thanks.
Thanks for posting this online and it helped me a lot understanding the concept o TMM. There is one thing I think I should point out. You used exp(jwt-jkz) most of the time but you used exp(jkz-jwt) for TMM here. However, you ended up with the same set of the differential equations you developed previoiusly. Thus I suppose you used different normalization for H here. You used H_normalized= -j*sqrt(u0/e0)*H. But here did you use H_normalized= j*sqrt(u0/e0)*H? Thanks again for posting.
Victor Wang It is great to see you analyzing this in such detail. The manner in which I normalize H is always the same regardless of the sign convention. The biggest impact of the sign convention is that you have to also consider the sign of the imaginary parts of mu and eps if you are incorporating loss. Otherwise, it is only subtle sign changes in the formulation, if any changes.
Victor Wang Also, there is one thing I don't quite understand. You used exp(ikz) convention. However, you expressed Ex(z) as Ex(z)= (a_+)*exp(-omega*z)+(a_-)*exp(omega*z). Is that a conflict with the sign convention?
Victor Wang I appears I may have goofed my notation. Thanks for pointing it out. I will look this over in more detail and fix what may be wrong as soon as I can.
Victor Wang So I looked at this in a bit more detail and I now remember what my thoughts were at the time I did this sign convention change. First, I tend to like the exp(+jkz) convention so that the imaginary parts of permittivity and refractive index are positive for lossy materials. This is more consistent with what is published in materials databases. However, the convention exp(-jkz) is used more often in electrical engineering. Both work and both are used. The TMM lecture here is intended to get the students ready for learning rigorous coupled-wave analysis in a later lecture. In that formulation I wanted to stick with what was in the literature which is the exp(-jkz) convention. That is why I made the switch in the TMM lecture to this sign convention. I guess now the question is, why did I ever show the exp(+jkz) convention? I wanted students to see both! Also, my TMM and FDFD codes all now use the exp(+jkz) convention. Yes, I even confuse myself many times when I cannot remember which convention I used.
If you would like to see a revised set of slides that stick with the exp(+jkz) convention, send me an e-mail. I have them. Also, incase you are not aware, you can access the electronic notes, recorded lectures, and other resources through the course website here:
emlab.utep.edu/ee5390cem.htm
Hi Raymond, the code works fine and I compared it with benchmark results. Next step, on slide 40 for Wavelength or Frequency Parameter Sweep, inside the for loop you introduced "LAMBDA". What is this LAMBDA? lam0 is a free space wavelength which is in this case have a range of different values. Could you please guide a bit about this parameter sweeping part?
In a sense, frequency and free space wavelength are synonymous. Optics people like to characterize waves in terms of wavelength, while radio frequency people use frequency. I present both in my lectures. In this case LAMBDA is an array of free space wavelengths over which to perform simulations. You can easily convert this to an array of frequencies FREQ if you wish by using
FREQ = c0./LAMBDA;
Once either FREQ or LAMBDA is defined, you will setup a for loop to iterate through that array and perform a separate TMM simulation for each frequency or wavelength in that array. In the end, you will plot transmission and reflection as a function of frequency or wavelength.
Thanks for the lectures, they help me a lot! By the way, I find it bizarre since people always wrote in papers that the global matrix is the product of individual layer matrices, does it mean matrix multiplication or star product? How are they different in the result?
If you use the pure transfer matrix method, it is standard matrix multiplication. However, the pure transfer matrix method is not stable as you saw in Lecture 4. It is possible to fix TMM so that it works, but it is no longer ordinary matrix multiplication. Most of the world has adopted scattering matrices, although other approaches are faster. Combining two scattering matrices is definitely not a matter of just multiplying the two scattering matrices. Unfortunately, it is more complicated than that and is called the Redheffer star product. When you implement scattering matrices, you almost never store it as a single matrix. You store the elements S11, S12, S21, and S22 as separate matrices. Combining two scattering matrices involves maybe a dozen matrix operations. See Redheffer star product equations. If you tried standard multiplication to combine S matrices, I don't know what would happen, but the results would be garbage and differ significantly from using the Redheffer star product. I am not sure what the meaning would be of multiplying two scattering matrices. Somebody may know differently, but I don't think there is ever a time that is done.
Thank you very much, Sir, for the lectures . In homework 3, everything is going fine except that when I calculate X = exp(omega*k0*L), it gives incorrect value, compared to the benchmark. I reviewed my code several times but i can't identify what went wrong. Did anybody had similar issue ?
Try expm(), not exp()
@@empossible1577 Thanks, sir, it worked. The only remaining issue now is that the real and imaginary parts of the final global scattering matrices ( after connecting to the external regions ) are different compared to their counterparts in the benchmark, making the conservation equal to 0.9869. What do you think the issue might be ?
@@OMohamedSamehO So your reflection side and transmission side scattering matrices match the benchmarking document? If so, check the star product equations.