Lecture 3c -- Wave Dispersion & Polarization

[@localfuture1484]

Hi, @4:55, I found there may be something wrong with the orientation of "Left-Hand Circular Polarization". I think at the fixed point when viewed from behind, the rotation should be counterclockwise (you meantion it in another PDF). The drawing here is not correct. Could you help check it? In other words, on the right figure, the spiral part is "Left-Hand Circular", however, for the circle on the "Plane", it's "Right-hand Circular".

[@empossible1577]

You are right. This section of the notes has been considerably revised and you are watching an older and obsolete video. I don't delete them because there are good comments and discussion with the video. I recommend accessing the information through the course website. You can see the information organized, download the notes, get links to the videos, and see all the other learning resources. You will also have the latest version of everything. The discussion related to polarization is now Lecture 6d. Here is a link the course website: empossible.net/academics/emp3302/

[@1729malli]

Thank you for very clear explanation. I have a question related to basic understanding. In Slide 11, Polarisation vector is mentioned as Pa a^ + Pb b^. a^,b^ are unit vectors. Each Pa, Pb are complex numbers which are reprensented by euler notation (magnitude * e^(j*phi)). I can understand how to represent vectors with i,j,k. "How to represent them in real plane (x,y,z (i,j,k unit directions)) if components of i,j,k are complex", If components are real it is clear. For example, In slide 15, P = x^ + y^ j to represent Circular polarization, Here y^ direction itself phase of 90 from x^. Not sure how to understand them. In one of the application notes i read they mention circular polarization as cos(wt) x^ + sin(wt) y^, Here both components are real, as "wt" changes this combined vector traces a circle, when it is varied in z direction we get Circular polarised wave. I am having confusion on HOW TO VISUALISE VECTORS WITH COMPLEX COMPONENTS IN REAL PLANE.

[@empossible1577]

That is a great question and I am not making any attempt to do that here, but should. The short answer is that each direction can have a unique amplitude and phase. A video will be worth a 1000 words here and I think I will make a short video about this and post it soon. BTW…you are watching an old video. This portion of my electromagnetics course has been considerably improved and revised. I recommend accessing these videos through the course websites as your main portal. You can download notes, get links to the latest version of everything, and other learning resources. This video is in Topic 6 at the following link. empossible.net/academics/emp3302/ Hope this helps!

[@empossible1577]

In the case of polarization, it is a bit misleading to look at the vector with x, y, and z components. The polarization must be perpendicular to the direction of the wave. This means the polarization vector really only has an a^ and b^ direction. It may be easier to think of this as x^ and y^ where the wave propagates along z^. These two directions are both a linearly polarized wave. They are completely separate waves. They just happen to be occupying the same space. Each wave has its own amplitude, phase, and electric field direction.

[@1729malli]

@@empossible1577Thanks for the replies. These are really helpful for my understanding. I have read few application notes on complex vectors and time harmonic fields. From that, what i understood was every complex vector has transformation to time harmonic representation some are Physically intuitive but some are not. In terms of notation, if a complex vector a = are + j aim, here are, aim are real vectors, like are = arex * x^ + arey * y^ where a'rex' is x^ component. Similarly for aim also. This vector gets transformed to time harmonic representation by Re{a * exp(j*w*t)} which is "are * cos(wt) - aim * sin(wt)"--> Now in vector form [arex * cos(wt) - aimx * sin(wt)] x^ + [arey * cos(wt) - aimy * sin(wt)] y^. from this, i can transform any complex vector to time harmonic real vector whom i can represent in real plane as (wt) varies. PLEASE CORRECT ME IF I AM WRONG. For example on Slide 15, x^ + j y^, are =1, aim =1, So, time harmonic representation is cos(wt)-sin(wt) which if plot in x,y plane as (wt) varies i see a circle. from this i can say it is CP. In general "time harmonic vector "F1* cos(wt) + F2* sin(wt)" where F1, F2 are real vectors traces an ellipse" I am not sure whether we can call this time harmonic vector representation which came with Re{complex vector * exp(j*wt)} is same as Frequency domain reprentation. I think time harmonic representation is to visualise the complex vectors in real plane. "To complete i am adding this information, for a complex vector a(as above), a is LP if a x a* = 0 => arex/arey = aimx/aimy = k(constant) a is CP if a.a = 0 for that |are| = |aim| and are . aim = 0"

[@1729malli]

@@empossible1577 Thanks for sharing the course. My original intention was to learn Computational electromagntics. I think from initial videos i see i lack basic principles required for computational Electromagnetics course. I will check above(EMP3302) course first.

[@empossible1577]

@@1729malli Not all complex vectors are interpreted the same way in electromagnetics. For example, there is a polarization vector that directly affects amplitude. Then there is the wave vector that directly affects phase. Both can be complex but their interpretation and effects on a wave are different. I am not sure there is a single unified way to interpret a complex vector that would help in electromagnetics.

[@전기전자-j9f]

Thank you very much, this is the perfect contents i think.

[@timuralmabetov2213]

what LHI stands for?

[@empossible1577]

Linear, homogeneous and isotropic (LHI) Basically, these are assumptions made about the material properties to make the math the simplest. Linear - Some materials actually change their properties depending on the strength of the electric field. This is a nonlinear material. Saying a material is linear is just saying the properties do no change with field strength. Homogeneous - This means the material is smooth and continuous. It contains no bumps, holes, edges, corners, or any other type of discontinuity. You can think of this as being in the middle of the ocean, or space, or air, or something like that. Isotropic - A material is isotropic when the field sees the same property regardless of the direction of the field. There do exist materials that exhibit different properties when the direction of the field is changed. These are called anisotropic. Hope this helps!

[@timuralmabetov2213]

@@empossible1577 thank you for your explanation 🙏

[@empossible1577]

@@timuralmabetov2213 BTW...this portion of materials have been considerably revised and improved. Checkout the course website to get links to the latest version of everything: empossible.net/academics/emp3302/

[@mishuk2008]

Best em learning materials for undergrads on the internet.

[@empossible1577]

Thank you!

[@한두혁]

Thank you for the great lecture. I think you are using fourier transformed wave eq..where can I find the lecture for this? I am not sure about the derivation and why we are using it. Thank you.

[@empossible1577]

Below is the link to the official course website. I recommend using that as your main portal to the course because it organizes the information, you can download the notes, get links to the latest versions of the notes and videos, and get other learning resources. empossible.net/emp3302/ The video you are watching has been moved to Topic 6 where you can watch the videos that come before it. I have reorganized the information, but it is all still there.

[@geez6666]

These are really good set of lectures you have. Thank you. I hope you're doing well wherever you are. Is there a reason you omitted 180 deg from the table under linear polarization?

[@empossible1577]

Good point. I should have said something like m*180, meaning any integer multiple of 180.

[@geez6666]

@@empossible1577 okay thanks

[@AbuSayed-er9vs]

Mind blowing as usual!!!

[@youssefelhajji530]

Thank you ,great lecture

[@empossible1577]

Thank you!

[@vahidj3196]

The Best

[@empossible1577]

Thank you!

[@meghjitmajumder3468]

But sir in the corresponding pdf, named as "Dispersion Relation"... First, there are only 9 slides Second, there's nothing about Polarization. (Only same till the slide no. 6)

[@empossible1577]

The notes have been improved and revised since they were recorded. You are seeing some of the differences. I need to go back and rerecord some of the lectures for greater consistency. Very sorry! Here is a link to the official course website. empossible.net/academics/emp3302/

[@GabrielSantos-th7ww]

The first equation in the slide number one is the Helmholtz's equation, not the wave equation.

[@empossible1577]

In your opinion, what is the difference between the frequency-domain wave equation and the Helmholtz equation?

[@GabrielSantos-th7ww]

@@empossible1577 As far as I know, if I take the Fourier transform of the wave equation, I end up with the Helmholtz equation. So yeah, the Helmholtz equation is the frequency-domain wave equation. EDIT: My bad, it's not taking the Fourier transform of the equation, but assuming that the solution of the wave equation can be represented as a Fourier integral. That's the way I learned how to derive the Helmholtz equation.

[@empossible1577]

@@GabrielSantos-th7ww That is my understanding as well. I think it is still proper to call the frequency-domain wave equation the wave equation.

[@GabrielSantos-th7ww]

​@@empossible1577 Well, yes. But I still think it's necessary to make those clarifications in the video given that it's only presented as the wave equation.

[@Hertzultra]

If one assumes this to be understood as is in electrodynamic propagation of waves and paralleled with wave mechanics then WHY do we NOT HAVE a GRAND UNIFIED THEORY??