Lecture 7 (EM21) -- Theory of periodic structures

[@pablanche]

Thank you so much for this comprehensive explanation of band diagrams. Invaluable.

[@empossible1577]

Thank you!

[@elliottlovell763]

I have never commented on anything on youtube before but I thought this was worth it. This lecture series is so clear and so well put together, thanks heaps.

[@empossible1577]

+Elliott Lovell Thank you!

[@joohigarg8324]

Excellent

[@jingxuanwei2173]

Awesome, very informative! Thanks so much!

[@leoneldavidquinterospalomi1536]

Thank you! great video and very useful

[@pendalink]

fantastic. thank you!

[@empossible1577]

BTW, you are watching an old video. I have considerably improved and revised this topic and I think you will really like the new material. See Topic 4 on the official course website. empossible.net/academics/emp6303/ I recommend using the course websites as your main portal to the courses. You can download the notes, get links to the latest versions of the videos, and get access to other learning resources. Hope this helps!!

[@jumjim88]

A great video ! Thanks a lot!

[@kaiwang5546]

This is very useful and clear!

[@haoqiu4364]

Thank you so much for your lectures! It's really helpful! I have some questions about the slide 42&43 at around 45:00 in this video. If the phase velocity Vp calculated as Slide 42 (The slope of the line connecting T to the point on the band corresponds to phase velocity.), the Vp at the point of 2nd (or higher) band just right above the point T will be close to infinite. Am I right? I am really confused. Looking forward to your reply.

[@empossible1577]

I don't see any T's so I am not sure I can answer your question. Maybe you mean the gamma point and you are asking about the second band directly above the gamma point. This is not infinite phase velocity because that band has been folded. Think of the second band as a continuation of the first band as it exits the band diagram on the right. So the exact equation of w/k for phase velocity only really applies to the first band if the folding is not considered.

[@haoqiu4364]

Yes, I mean the gamma point. I got it. Thanks!

[@aquilabobber556]

Hello, thanks a lot for your lecture videos. I learned FDTD from your youtube series. Do you also have a lecture on the eigenvalue problem on slide 39?

[@empossible1577]

First, let me point you to the official course website. This is an old video and the topic has been considerably revised. If you use the course website as your main portal, you can see the latest versions of the notes and videos as well as have other resources provided on the website. This specific subject is now taught in Topic 4. empossible.net/academics/emp6303/ Solving the eigen-value problem is taught using the plane wave expansion method (PWEM) in Topic 6 of the Computational Electromagnetics course. empossible.net/academics/emp5337/ I show how to solve the problem in a very indirect manner using finite-difference time-domain (FDTD) in Lecture 3i here: empossible.net/academics/emp5304/ It is also possible to solve it using the finite-difference frequency-domain method (FDFD). I have some very crude notes in Lecture 4i here: empossible.net/academics/emp5337/ I am releasing a new book on the FDFD method that I hope is available before the end of this year. I have an entire chapter devoted to doing this. Hope this helps!

[@aquilabobber556]

@@empossible1577 Thanks for your quick response. Have you read/used the book Photonic Crystals molding the flow of light by John Joannopoulos to develop the eigenvalue problem in any of the lectures/courses. I am reading this book and looking for an example where the eignenvalue problem was solved and the method stated in the book discussed.

[@empossible1577]

@@aquilabobber556 That is a great book! They use the plane wave expansion method (PWEM) for calculating photonic bands.

[@aquilabobber556]

@@empossible1577 Have you done a lecture on using the PWEM method for the eigenvalue problem discussed in the book? Solving for H instead of E. Your explanations and visual presentation is best I have ever seen in a lecture.

[@empossible1577]

@@aquilabobber556 You can find the lectures for the plane wave expansion method in Topic 6 here: empossible.net/emp5337/ I describe both E and H. You get the same answer for both and there is no significant speed difference that I can tell.

[@somayehnodehi3657]

I am a little bit confused by the symmetry points of IBZ for a fcc structure. For example at MPB software, they introduce the points like X=(0,0.5,0.5), U=(0,0.625,0.375), L=(0 , 0.5, 0), Gamma=(0,0,0), W=(0.25, 0.75,0.5), K=(0.375, 0.75, 0.375) but in some texes they are defined like X=(0,0.5,0.5), U=(0.25,0.625,0.625), L=(0.5 , 0.5, 0.5), Gamma=(0,0,0), W=(0.25, 0.75,0.5), K=(0.375, 0.75, 0.375). Is there any difference between these to defiened set? if I change the componente of symmetery poins at MPB, do I get the same result? I need to sweep some results at angle of zero respect to the normal direction of a fcc lattice. In some texts, the direction of Gamma-L is introduced as the perpendicular direction of the unit cell. Now because the components of L are diffrent at different text and softwares , I am doubtfull about the direction.

[@empossible1577]

Visualization is the key to computation. I suggest generating a 3D plot showing the BZ and these key points of symmetry. It is hard to conclude anything just from looking at the numbers, but there are different ways to define the key points. Both sets of numbers could be correct here.

[@somayehnodehi3657]

You are right!!!!! both sets are correct!! :D

[@medboukhicha6484]

Thank you for the clear explanation but you didn't mention the difference between electronic and photonic gap as it might be confusing !

[@empossible1577]

Great point!

[@RounakSIngh17]

First thanks for the videos. It is really helpful. however, I would like to point out that the mathematical form of reciprocal lattice vector is wrong. The denominator is same for all formulas.

[@empossible1577]

Very sorry for any errors. Can you be more specific about what is wrong and on what slide? I definitely want to fix any errors.

[@RounakSIngh17]

The slide 19 which consists of the formulas for reciprocal lattice vector. The denominators of all three vectors are same for all reciprocal lattice vectors. Shouldn't they be different ?

[@empossible1577]

This is actually correct. You might recognize the vector triple product in the denominator as the volume of the unit cell.

[@RounakSIngh17]

Thanks professor :) . I got it now.