Hello to Egypt! There are quite a few people from Egypt watching these lectures. That is great to see! I really hope to travel there some day to see the ancient sites.
Thanks for the videos! What are some advantages of using 3D simulations to calculate the modes propagating in a waveguide compared to a 2D geometry simulation?
Some things just cannot be simulated in 2D, like helix antennas or some devices with anisotropy. The accuracy and trustworthiness of a 3D simulation is also better for 3D structures. However, if it is ever possible to simulate a 3D structure in either 1D or 2D, I think it is always good practice to do so. This will get you 95% to the final design without the need of a big 3D simulation. Almost all of our 3D simulations are done mostly in 1D and 2D. Often, our 3D simulation is only done to confirm what our 2D simulations predict.
I have a long lasting question in this topic Which method is faster in 2D scalar paraxial wave equation: 1 - FD-BPM (Crank Nicolson Scheme, which mostly costs a linear system solution) 2 - FFT-BPM (Costs 1 fft, 1 ifft and 1 multiplication) ? I've simulated both of these methods in MATLAB and, to me, FFT-BPM seems faster, but I've come across two articles comparing both methods and they've concluded the opposite. On the Crank Nicolson scheme I'm using the backslash operator in MATLAB to solve the sparse linear system and though my tests it seems as fast as the Thomas Algorithm. And I'm also using sparse matrices. Thank you for your time and your amazing content. I've been watching your videos since I've started my research in optics as an undergraduate and the content that you put out there is really valuable to a lot of people.
Short answer -- I don't know! However, I will make a guess that the FFT-BPM is the faster method when the transverse direction is large. At what point it becomes faster I am not sure. Also, the hardware really affects which is faster. I might be that on a GPU the FFT approach is faster while on CPU the FD approach is faster. When comparing the methods, don't just compare for one simulation. Instead, compare the speed and memory as problem size grows (or grid resolution gets finer). I think that will be more telling.
@@empossible1577 I love that you answer so quickly hahaha. I have a follow-up question. Is there any method for Transparent Boundary Conditions for FFT-BPM? I've searched quite a lot and found nothing. It is quite common to find TBC for FD methods, but I've not seen a TBC for FFT-BPM yet.
@@carl00s01 Hmmm...I don't think there is a TBC, but you have at least two options. First, you can make the transverse dimension large enough that the boundary conditions do not interfere with your answer. Second, you may be able to incorporate a PML or at least graded loss at the boundaries to absorb waves.
@@empossible1577 Thanks for your response. I've been doing the first way for quite a while. Do you have any reference for the second one? I've never seen a PML for FFT-BPM
@@carl00s01 I recommend a uniaxial PML so that the PML is independent of the implementation. It is only a matter of incorporating the loss terms in the permittivity and permeability. What are you simulating? I might be able to suggest something better.
Dear CEM team, first of all I would like to thank you for providing this great online resource, I enjoyed your lecture series a lot! I implement a Pade [1,1] 2D (x,z) BPM method following your instructions and tried to model the propagation of a Gaussian beam (3um beam radius, 370nm wavelength) in an anti guiding step index fiber (the fiber had a cladding refractive index of 1 and the 6um core had a refractive index of 0.98) with it. According to my BPM model the beam was still somewhat confined in the core after propagation one Rayleigh length, which surprised me. I would have expected that the beam would diverge significantly stronger than a beam that is propagating in a homogeneous structure. I was wondering if this BPM method is capable of describing such anti-guide problems correctly or if have to implement higher Pade[2,2] with multi steps. Cheers!
I am curious about your choice of refractive indices of 1 and 0.98. Wouldn't 1.5 and 1.499 be more realistic? Also, the refractive index of the core should be slightly higher, unless you have something special in mind.
My choice of refractive indices is in fact a little unusual but exactly what I would like to model. My question is if you see any fundamental reason for this method to fail when using refractive indices smaller than 1
@@stefangaus8316 I am not coming up with any reasons that should not work. Have you tried simulating something with a known answer that has refractive indices around 1.5, just to benchmark your code?
Hi, actually I think the bidirectional BPM at least scales better than RCWA, the propagation operator here is just a finite difference matrix and we don't really have to solve any painful eigenvalue problems, isn't it?
CIP? Do you mean "Particle in Cell (PIC)?" If so, I would probably only mention that in passing in this lecture series. I teach another class in finite-difference time-domain. The next time I teach it, I will record the lectures. It is here that I may include a lecture on PIC. I don't have any immediate plans to make on lecture on the FVTD, although it is probably worthy of mentioning. You may want to see the lecture notes on the FDTD class here, which is a similar method: emlab.utep.edu/ee5390fdtd.htm
In the lecture, you assume the the device is uniform along y direction and get the equation for ey (the vector for the y component of electric field). But what if this assumption is not satisfied? Can we still use this method to get the equation for ey? I was trying to do this but I only got equation with both ey and ex(or ez) in it. I think I need use the divergence Maxwell equation. But because of the difference between DE(the derivative for E) and DH(the derivative matrix for H), ex and ez can't be cancelled at the same time. Could you answer my question? Thank you
Yes, this is absolutely possible. You will have to retain and solve for more field components. For isotropic materials, this will be Ex and Ey (or Hx and Hy). Take a look at RCWA in this lecture series. It does just this, but is a rigorous method. I may at some point post a slide or two on the 3D BPM. You can also refer to the literature. Key words include “full vector,” “3D,” etc. I added a section to the electronic notes on the formulation of 3D FD-BPM for you. I did it quickly so beware that there may be a mistake somewhere. If you find one, please let me know! You can get them here at Lecture 16: emlab.utep.edu/ee5390cem.htm As you will see, Ex and Ey are no longer separate and are calculated at the same time. It is a much larger numerical problem (as 3D always is). If you have a polarized beam, say in the y direction, then your source would simply just set Ex to zero, but it will still need to be there. As long as you adopt the Yee grid for your finite-differences, there is no need to include divergence equations.
Really, thanks a lot for your effort. It is really appreciated effort. I am following you from Egypt. Thanks Again.
Hello to Egypt! There are quite a few people from Egypt watching these lectures. That is great to see! I really hope to travel there some day to see the ancient sites.
@@empossible1577 Sorry for late reply. It is our pleasure to visit Egypt and see you. I just sent you an email accordingly. Please check it.
Thanks for the videos! What are some advantages of using 3D simulations to calculate the modes propagating in a waveguide compared to a 2D geometry simulation?
Some things just cannot be simulated in 2D, like helix antennas or some devices with anisotropy. The accuracy and trustworthiness of a 3D simulation is also better for 3D structures. However, if it is ever possible to simulate a 3D structure in either 1D or 2D, I think it is always good practice to do so. This will get you 95% to the final design without the need of a big 3D simulation. Almost all of our 3D simulations are done mostly in 1D and 2D. Often, our 3D simulation is only done to confirm what our 2D simulations predict.
CEM Lectures thanks a lot for the help! Would love to add you to my network! This channel is amazing
I have a long lasting question in this topic
Which method is faster in 2D scalar paraxial wave equation:
1 - FD-BPM (Crank Nicolson Scheme, which mostly costs a linear system solution)
2 - FFT-BPM (Costs 1 fft, 1 ifft and 1 multiplication)
?
I've simulated both of these methods in MATLAB and, to me, FFT-BPM seems faster, but I've come across two articles comparing both methods and they've concluded the opposite. On the Crank Nicolson scheme I'm using the backslash operator in MATLAB to solve the sparse linear system and though my tests it seems as fast as the Thomas Algorithm. And I'm also using sparse matrices.
Thank you for your time and your amazing content. I've been watching your videos since I've started my research in optics as an undergraduate and the content that you put out there is really valuable to a lot of people.
Short answer -- I don't know!
However, I will make a guess that the FFT-BPM is the faster method when the transverse direction is large. At what point it becomes faster I am not sure. Also, the hardware really affects which is faster. I might be that on a GPU the FFT approach is faster while on CPU the FD approach is faster. When comparing the methods, don't just compare for one simulation. Instead, compare the speed and memory as problem size grows (or grid resolution gets finer). I think that will be more telling.
@@empossible1577 I love that you answer so quickly hahaha. I have a follow-up question. Is there any method for Transparent Boundary Conditions for FFT-BPM? I've searched quite a lot and found nothing. It is quite common to find TBC for FD methods, but I've not seen a TBC for FFT-BPM yet.
@@carl00s01 Hmmm...I don't think there is a TBC, but you have at least two options. First, you can make the transverse dimension large enough that the boundary conditions do not interfere with your answer. Second, you may be able to incorporate a PML or at least graded loss at the boundaries to absorb waves.
@@empossible1577 Thanks for your response. I've been doing the first way for quite a while. Do you have any reference for the second one? I've never seen a PML for FFT-BPM
@@carl00s01 I recommend a uniaxial PML so that the PML is independent of the implementation. It is only a matter of incorporating the loss terms in the permittivity and permeability.
What are you simulating? I might be able to suggest something better.
Can we use effective index method for optical fiber? Will it be as accurate as the slab waveguide?
Yes. I am sure there are papers that use BPM to study optical fibers.
Dear CEM team,
first of all I would like to thank you for providing this great online resource, I enjoyed your lecture series a lot! I implement a Pade [1,1] 2D (x,z) BPM method following your instructions and tried to model the propagation of a Gaussian beam (3um beam radius, 370nm wavelength) in an anti guiding step index fiber (the fiber had a cladding refractive index of 1 and the 6um core had a refractive index of 0.98) with it. According to my BPM model the beam was still somewhat confined in the core after propagation one Rayleigh length, which surprised me. I would have expected that the beam would diverge significantly stronger than a beam that is propagating in a homogeneous structure. I was wondering if this BPM method is capable of describing such anti-guide problems correctly or if have to implement higher Pade[2,2] with multi steps.
Cheers!
I am curious about your choice of refractive indices of 1 and 0.98. Wouldn't 1.5 and 1.499 be more realistic? Also, the refractive index of the core should be slightly higher, unless you have something special in mind.
My choice of refractive indices is in fact a little unusual but exactly what I would like to model. My question is if you see any fundamental reason for this method to fail when using refractive indices smaller than 1
@@stefangaus8316 I am not coming up with any reasons that should not work. Have you tried simulating something with a known answer that has refractive indices around 1.5, just to benchmark your code?
Hi, actually I think the bidirectional BPM at least scales better than RCWA, the propagation operator here is just a finite difference matrix and we don't really have to solve any painful eigenvalue problems, isn't it?
Sir is it possible to solve the time independent non linear Schrödinger equation (2+1) D through beam propogation method
I would think so, but I have never done it. I cannot think of why that would not be possible.
thank you for your lectures. I'm studying from your channel.
I was wondering if you would upload lecture about CIP and FVTD methods.
CIP? Do you mean "Particle in Cell (PIC)?" If so, I would probably only mention that in passing in this lecture series. I teach another class in finite-difference time-domain. The next time I teach it, I will record the lectures. It is here that I may include a lecture on PIC.
I don't have any immediate plans to make on lecture on the FVTD, although it is probably worthy of mentioning. You may want to see the lecture notes on the FDTD class here, which is a similar method:
emlab.utep.edu/ee5390fdtd.htm
In the lecture, you assume the the device is uniform along y direction and get the equation for ey (the vector for the y component of electric field). But what if this assumption is not satisfied? Can we still use this method to get the equation for ey? I was trying to do this but I only got equation with both ey and ex(or ez) in it. I think I need use the divergence Maxwell equation. But because of the difference between DE(the derivative for E) and DH(the derivative matrix for H), ex and ez can't be cancelled at the same time. Could you answer my question? Thank you
Yes, this is absolutely possible. You will have to retain and solve for more field components. For isotropic materials, this will be Ex and Ey (or Hx and Hy). Take a look at RCWA in this lecture series. It does just this, but is a rigorous method. I may at some point post a slide or two on the 3D BPM. You can also refer to the literature. Key words include “full vector,” “3D,” etc.
I added a section to the electronic notes on the formulation of 3D FD-BPM for you. I did it quickly so beware that there may be a mistake somewhere. If you find one, please let me know! You can get them here at Lecture 16:
emlab.utep.edu/ee5390cem.htm
As you will see, Ex and Ey are no longer separate and are calculated at the same time. It is a much larger numerical problem (as 3D always is). If you have a polarized beam, say in the y direction, then your source would simply just set Ex to zero, but it will still need to be there.
As long as you adopt the Yee grid for your finite-differences, there is no need to include divergence equations.
Can I get the electronic notes for FD BPM 3D? I search the em lab page but couldn't find the same.