Without you I wouldn't have been able to get through my masters degree, your lectures were one of the biggest resources that allowed me to finish my thesis. Thank you very much. Hope others get to benefit from your incredible insights as well.
Thank you for your lecture! In 2D photonic crystal structures, I think there are no consideration of slab thickness (bottom boundary condition), also in simulation tool. I'm wondering that solving 2D PhC eigenmodes includes assumption of infinite thickness.
If you have a photonic crystal slab, boundaries at the top and bottom must both be considered in order to get an accurate simulation at all frequencies. Sometimes it is still instructive to look at scattering from a single interface, a bit like the meaning of the Fresnel equations. As formulated, photonic bands are typically calculated for a lattice that is infinite in all directions. This GREATLY simplifies the analysis the conclusions are typically very consistent with the properties of a finite lattice. It is only when the lattices are very thin or perhaps other situations I am not thinking of where the properties deviate significantly from the properties of an infinite lattice. BTW, if you want to learn more about the simulation tools, I have creates some resources for that. First, below is a book in finite-difference frequency-domain that I think is the best first method to learn. The book is intended for the complete beginner. empossible.net/fdfdbook/ Another good method is finite-difference time-domain. For this, I created an online course also intended for the complete beginner. Below is a link to a video showcasing the course as well as a link directly to the course. ua-cam.com/video/uBiprIN8gfY/v-deo.htmlsi=STJlTYSny4FMh6DY empossible.thinkific.com/collections/FDTD-in-MATLAB If you want to learn some more advanced methods like the plane wave expansion method or rigorous coupled-wave analysis, checkout this Computational Electromagnetics course: empossible.net/academics/emp5337/ Hope this helps!
I don't understand why you're explaining that propagating electromagnetic waves is forbid in bandgap materials. Isn't it possible to say that electromagnetic waves pass through? thanks for the good lecture
It gets into a bit of semantics. Waves that are "forbidden" are more accurately described as cutoff. This means they can still propagate through the photonic crystal, but they decay in amplitude as they propagate. When calculating a photonic band diagram, the lattice is assumed to be infinitely periodic. Even waves that decay very slowly will decay to zero in an infinite lattice. We tend to ignore the fact that the waves can actually propagate some short distance, but you are correct that they actually do. The same can be said for modes in a waveguide that are cutoff.
Isn't that weird? e^(j*b.r), shouldn't it be e^(j*arccos(b.r))? When I see vectors being fed into e like that, there seems to be a higher dimensional thing going on. Like, I should be looking at contour maps, domain coloring, or doing square matrix transformations in ℂ^2 or ℂ^3.
There is a dot product happening...beta dot r. The dot product of two vectors is a single scalar quantity. Maybe the symbols got messed up in the video render. I will have to check.
regarding the parameter of PhC structure, it's have any equation related to lattice constant, air holes radius and slab tickness? I cannot justify my proposal thesis for my result
Typically some kind of a simulation is done on a photonic crystal to calculate the band diagram, isofrequency contours, or something else. Usually it is desired to design a photonic crystal for a specific wavelength so the lattice constant is calculated from the simulation in order to have the PhC operate at the desired wavelength. As for geometric parameters like hole radius, there may be equations for this for some canonical lattices, but the physics is complicated enough that this information usually must come from a simulation and not an analytical equation. To justify something for your proposal, you will either need to point to some literature on your topic that justifies your ideas or run some simulations to justify your ideas. I suspect you are just starting out and simulations may be a new area for you. You have two options. First, you can use commercial software or open-source software. Second, you can develop your own codes and capabilities. I recommend the second approach because the commercial software lacks some capabilities that you will likely need if you are doing something new or different. I recently wrote a book specifically intended for people to get started in computational electromagnetics/photonics (CEM). It not only teaches a powerful and versatile method, it teaches the art of CEM. The art includes benchmarking, troubleshooting, convergence, setting up codes to simulate different things, sources, boundary conditions, etc. This is all done using very plain language. Here is a link to the book website: empossible.net/fdfdbook/ Another good place to get started is an online course on finite-difference time-domain. Here is a link to a video showing what is in this course: ua-cam.com/video/uBiprIN8gfY/v-deo.html Here is a link to the courses themselves: empossible.thinkific.com/collections/FDTD-in-MATLAB I hope something here helps!!
Do you have a lecture about bragg law and its relation with BZ? I am looking for a concept about superprism and angle-dependant bandgap. I saw in some papers that both superprism effect and the partial gap ( dependet on the angle of incident light) stem from the modes on surface plane of BZ not bulk modes (inside the BZ).
Dear Professor, I have noticed here you mentioned both mode expansion method and FDTD should end up with the same result of bandstructure ( for one structure). I have simulated a structure at mode expansion which seems correct. But when I simulate it using FDTD, there are big differences (some extra points and some missing bands) between its result and the one from mode expansion. at FDTD I have used a supercell ( 4 primitive cells). In any cell, I have used four dipoles (16 at supercell). Also, the meshing step is about 0.04 micrometer. The time for simulation is 1000 seconds. I was wondering if the number of dipoles is low that's why I am missing some bands or I should go for the smaller step of meshing. I appreciate your opinion on this problem. Thanks
Why are you arraying four unit cells? I suspect this is the problem. Just use one. Another thing you can do is find a similar structure in the literature and calculate the bands with both PWEM and FDTD. See which matches the correct results. Maybe your PWEM is the one getting the wrong result.
The structure is an FCC, that's why I have made a supercell from four primitive cell. Indeed the PWEM result is correct because it is similar to a literature but the FDTD results are different. I have changed the meshing and increase the number of dipoles to 24, but no changes. what is the main factor that cause some noise or missing points in FDTD method?
I don't think you are implementing the supercell technique incorrectly. There should not be a need to include anything more than one unit cell in memory. Doing this can cause strange things like mystery bands, missing bands, and incorrect bands. Just use one unit cell in FDTD and see if that fixes your problem.
Thank you. Just to let you know, this section of the course has been considerably revised. First, let me point you to the official course website below. I recommend using this as your main portal to the course because you see all the lectures organized, download the notes, and get links to the latest version of everything. empossible.net/academics/emp6303/ For a great tutorial on photonic crystals, work through "Solid State Electromagnetics" under Topic 4 and then "Photonic Crystals" in Topic 5. Hope this helps!!
Thanks a lot for the channel you prepared! I do love the lectures and follow them. I simulated some 3D pc and derived the DOS. In different meshing that people provided in order to have a precise DOS, I saw they avoiding Gamma point. So all the meshing were in a way that skip the center (0,0,0). Indeed they are right, because when I change the meshing step somehow that includes the Gamma point , DOS result showed a wierd trending at zero to higher frequencies ( like descending).What is the point here?
I am not sure what calculating methods those others are using, but the gamma point has the Bloch wave vector magnitude of zero. This can cause some numerical methods to fail or yield crazy answers. I think this is the only reason they are avoiding the gamma = 0 point.
Ok, so I went ahead and created a diagram to go with what I was trying to explain in my previous reply. Take a look at slide 12 in the latest version of Lecture 19 here: emlab.utep.edu/ee5390cem.htm
Thanks a lot for this lecture, I am following your lectures about photonic crystals, I have simulated some photonic band structure and provided the related DOS as well. In some structure, I saw DOS at zero frequency is equal to DOS at some highr frequencies, all the time ( I mean for all the structures), we don't get the exponential graph. Should I trust the DOS in this situation? because I can see people mention the same exponential graph of DOS for their structure at their papers.
At plane wave mode expansion method, I use MPB to derive the bandstructure, also they have provided a code that counting Gaussian smoothing function around any frequency point. This one shows zero frequency has the same value of DOS like the higher frequencies of the first band. When the frequencies increase also the number of band increase, then DOS shows exponential trending ( It happens after the first band). Also, at FDTD, I use Lumerical. I derive the data from bandstructure graph to matlab and count the number of peaks at any frequencies to calculate DOS. Sometimes, I see the value of DOS at zero is more than that of the other frequencies. Even it is possible to guess it from bandstructure graph, because there are a lot of point at frequency=0 line at the graph. Is it possible, it happens because of the tolerance, I mean the counted peak is not a real answer because the value of the peak is quite low compare to the others.
I know ligh line is w=ck|| . The definition of light line you expressed here was something new to me. What is the relation of the first definition and the one you mentioned? Indeed, How can I calculate a related light line of a dispersion curve? Thanks
I am being lazy in my language. What I call a light line is not entirely a light line. Instead, it is the band for a homogenized unit cell. Those bands starting at the gamma point will be the light line until that line reaches the edge of the Brillouin zone and then it folds. A true light line does not fold. The point is to show where the photonic crystal band deviates from the homogeneous band, which is the light line at this point. Did this help or did I make it more confusing?
Let me point you to the course website: emlab.utep.edu/ee5390em21.htm The site contains links to the videos, latest version of the notes, and other resources. Some of the notes have been considerably improved ad revised so you will notice some differences.
Without you I wouldn't have been able to get through my masters degree, your lectures were one of the biggest resources that allowed me to finish my thesis. Thank you very much. Hope others get to benefit from your incredible insights as well.
This is awesome to hear!! You made my day!! Thank you!!!
Thank you for your lecture!
In 2D photonic crystal structures, I think there are no consideration of slab thickness (bottom boundary condition), also in simulation tool.
I'm wondering that solving 2D PhC eigenmodes includes assumption of infinite thickness.
If you have a photonic crystal slab, boundaries at the top and bottom must both be considered in order to get an accurate simulation at all frequencies. Sometimes it is still instructive to look at scattering from a single interface, a bit like the meaning of the Fresnel equations.
As formulated, photonic bands are typically calculated for a lattice that is infinite in all directions. This GREATLY simplifies the analysis the conclusions are typically very consistent with the properties of a finite lattice. It is only when the lattices are very thin or perhaps other situations I am not thinking of where the properties deviate significantly from the properties of an infinite lattice.
BTW, if you want to learn more about the simulation tools, I have creates some resources for that. First, below is a book in finite-difference frequency-domain that I think is the best first method to learn. The book is intended for the complete beginner.
empossible.net/fdfdbook/
Another good method is finite-difference time-domain. For this, I created an online course also intended for the complete beginner. Below is a link to a video showcasing the course as well as a link directly to the course.
ua-cam.com/video/uBiprIN8gfY/v-deo.htmlsi=STJlTYSny4FMh6DY
empossible.thinkific.com/collections/FDTD-in-MATLAB
If you want to learn some more advanced methods like the plane wave expansion method or rigorous coupled-wave analysis, checkout this Computational Electromagnetics course:
empossible.net/academics/emp5337/
Hope this helps!
I don't understand why you're explaining that propagating electromagnetic waves is forbid in bandgap materials. Isn't it possible to say that electromagnetic waves pass through? thanks for the good lecture
It gets into a bit of semantics. Waves that are "forbidden" are more accurately described as cutoff. This means they can still propagate through the photonic crystal, but they decay in amplitude as they propagate. When calculating a photonic band diagram, the lattice is assumed to be infinitely periodic. Even waves that decay very slowly will decay to zero in an infinite lattice. We tend to ignore the fact that the waves can actually propagate some short distance, but you are correct that they actually do. The same can be said for modes in a waveguide that are cutoff.
Isn't that weird? e^(j*b.r), shouldn't it be e^(j*arccos(b.r))? When I see vectors being fed into e like that, there seems to be a higher dimensional thing going on. Like, I should be looking at contour maps, domain coloring, or doing square matrix transformations in ℂ^2 or ℂ^3.
There is a dot product happening...beta dot r. The dot product of two vectors is a single scalar quantity. Maybe the symbols got messed up in the video render. I will have to check.
Awesome lecture!
Thank you!
Here is a link to the course website if you want to see the the videos that come before or after this one:
empossible.net/academics/21cem/
Incredible lectures! learned from you a lot. Thanks buddy.
Great to hear!! Thank you!!
regarding the parameter of PhC structure, it's have any equation related to lattice constant, air holes radius and slab tickness? I cannot justify my proposal thesis for my result
Typically some kind of a simulation is done on a photonic crystal to calculate the band diagram, isofrequency contours, or something else. Usually it is desired to design a photonic crystal for a specific wavelength so the lattice constant is calculated from the simulation in order to have the PhC operate at the desired wavelength.
As for geometric parameters like hole radius, there may be equations for this for some canonical lattices, but the physics is complicated enough that this information usually must come from a simulation and not an analytical equation.
To justify something for your proposal, you will either need to point to some literature on your topic that justifies your ideas or run some simulations to justify your ideas. I suspect you are just starting out and simulations may be a new area for you. You have two options. First, you can use commercial software or open-source software. Second, you can develop your own codes and capabilities. I recommend the second approach because the commercial software lacks some capabilities that you will likely need if you are doing something new or different. I recently wrote a book specifically intended for people to get started in computational electromagnetics/photonics (CEM). It not only teaches a powerful and versatile method, it teaches the art of CEM. The art includes benchmarking, troubleshooting, convergence, setting up codes to simulate different things, sources, boundary conditions, etc. This is all done using very plain language. Here is a link to the book website:
empossible.net/fdfdbook/
Another good place to get started is an online course on finite-difference time-domain. Here is a link to a video showing what is in this course:
ua-cam.com/video/uBiprIN8gfY/v-deo.html
Here is a link to the courses themselves:
empossible.thinkific.com/collections/FDTD-in-MATLAB
I hope something here helps!!
Do you have a lecture about bragg law and its relation with BZ? I am looking for a concept about superprism and angle-dependant bandgap. I saw in some papers that both superprism effect and the partial gap ( dependet on the angle of incident light) stem from the modes on surface plane of BZ not bulk modes (inside the BZ).
Not really. I talk a little bit about it in the coupled mode lecture, but I do not think I have what you are looking for. Very sorry!
Dear Professor, I have noticed here you mentioned both mode expansion method and FDTD should end up with the same result of bandstructure ( for one structure). I have simulated a structure at mode expansion which seems correct. But when I simulate it using FDTD, there are big differences (some extra points and some missing bands) between its result and the one from mode expansion.
at FDTD I have used a supercell ( 4 primitive cells). In any cell, I have used four dipoles (16 at supercell). Also, the meshing step is about 0.04 micrometer. The time for simulation is 1000 seconds.
I was wondering if the number of dipoles is low that's why I am missing some bands or I should go for the smaller step of meshing. I appreciate your opinion on this problem.
Thanks
Why are you arraying four unit cells? I suspect this is the problem. Just use one.
Another thing you can do is find a similar structure in the literature and calculate the bands with both PWEM and FDTD. See which matches the correct results. Maybe your PWEM is the one getting the wrong result.
The structure is an FCC, that's why I have made a supercell from four primitive cell. Indeed the PWEM result is correct because it is similar to a literature but the FDTD results are different. I have changed the meshing and increase the number of dipoles to 24, but no changes.
what is the main factor that cause some noise or missing points in FDTD method?
I don't think you are implementing the supercell technique incorrectly. There should not be a need to include anything more than one unit cell in memory. Doing this can cause strange things like mystery bands, missing bands, and incorrect bands. Just use one unit cell in FDTD and see if that fixes your problem.
Do you still study about PCFs?
so cool I enjoyed it alot
Thank you. Just to let you know, this section of the course has been considerably revised. First, let me point you to the official course website below. I recommend using this as your main portal to the course because you see all the lectures organized, download the notes, and get links to the latest version of everything.
empossible.net/academics/emp6303/
For a great tutorial on photonic crystals, work through "Solid State Electromagnetics" under Topic 4 and then "Photonic Crystals" in Topic 5.
Hope this helps!!
Thanks a lot for the channel you prepared! I do love the lectures and follow them.
I simulated some 3D pc and derived the DOS. In different meshing that people provided in order to have a precise DOS, I saw they avoiding Gamma point. So all the meshing were in a way that skip the center (0,0,0). Indeed they are right, because when I change the meshing step somehow that includes the Gamma point , DOS result showed a wierd trending at zero to higher frequencies ( like descending).What is the point here?
I am not sure what calculating methods those others are using, but the gamma point has the Bloch wave vector magnitude of zero. This can cause some numerical methods to fail or yield crazy answers. I think this is the only reason they are avoiding the gamma = 0 point.
Ok, so I went ahead and created a diagram to go with what I was trying to explain in my previous reply. Take a look at slide 12 in the latest version of Lecture 19 here:
emlab.utep.edu/ee5390cem.htm
Thanks a lot for this lecture, I am following your lectures about photonic crystals, I have simulated some photonic band structure and provided the related DOS as well. In some structure, I saw DOS at zero frequency is equal to DOS at some highr frequencies, all the time ( I mean for all the structures), we don't get the exponential graph. Should I trust the DOS in this situation? because I can see people mention the same exponential graph of DOS for their structure at their papers.
Not sure. How are you calculating the DOS?
At plane wave mode expansion method, I use MPB to derive the bandstructure, also they have provided a code that counting Gaussian smoothing function around any frequency point. This one shows zero frequency has the same value of DOS like the higher frequencies of the first band. When the frequencies increase also the number of band increase, then DOS shows exponential trending ( It happens after the first band).
Also, at FDTD, I use Lumerical. I derive the data from bandstructure graph to matlab and count the number of peaks at any frequencies to calculate DOS. Sometimes, I see the value of DOS at zero is more than that of the other frequencies. Even it is possible to guess it from bandstructure graph, because there are a lot of point at frequency=0 line at the graph.
Is it possible, it happens because of the tolerance, I mean the counted peak is not a real answer because the value of the peak is quite low compare to the others.
I know ligh line is w=ck|| . The definition of light line you expressed here was something new to me. What is the relation of the first definition and the one you mentioned?
Indeed, How can I calculate a related light line of a dispersion curve?
Thanks
I am being lazy in my language. What I call a light line is not entirely a light line. Instead, it is the band for a homogenized unit cell. Those bands starting at the gamma point will be the light line until that line reaches the edge of the Brillouin zone and then it folds. A true light line does not fold. The point is to show where the photonic crystal band deviates from the homogeneous band, which is the light line at this point.
Did this help or did I make it more confusing?
Yes, I get the point.
Great!
Very wonderful lesson! Could you please provide us the PPT in this lecture?
Let me point you to the course website:
emlab.utep.edu/ee5390em21.htm
The site contains links to the videos, latest version of the notes, and other resources. Some of the notes have been considerably improved ad revised so you will notice some differences.
Tanks!
very nice
We also think so! It's amazing
thanks prof
Yes, thank you, it is great!
𝐧𝐢𝐜𝐞!
Agreed!!