Lecture -- Slab waveguides

This video introduces the concepts of a slab waveguide. The video is intended to explain the waveguide with as little electromagnetic theory as possible, yet give the student enough information to understand how to analyze a slab waveguide and understand the results.
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[@user-eb3jv4jv2v]

Why would the speed travel in different mode be different ? It seem different to my seoundary school physic that the speed is determine by the medium😢

[@empossible1577]

To get a more intuitive feel for a guided mode, you can think of a ray bouncing back and forth between the upper and lower interfaces of the waveguide. It turns out only discrete angles are allowed and these become the distinct guided modes. Each mode propagates at a different angle so they propagate at a different speed.

[@localfuture1484]

Hello, I have several questions about Waveguides for quite a long time. Based on the formula (transverse resonance condition/interferes with itself), it demands several (fixed) combinations of values of wavenumber(k), incident angle (theta), and mode number m for a given d (separation/height). Q.1 When sending EM signals into Waveguide, how can I adjust the incident angle? Q.2 If incident angle isn't correct, where does those energy go? Is there a certain percentage of energy simply lost as Heat? If incident angle or wavenumber is not that accurate, is the waveguide sensitive to that inaccuracy and stop working immediately? Q.3 If the incident EM signal is full of all angles (emitting in all directions), can we conclude that most of signals would get lost and only a very tiny amount of signals will get transferred through waveguide?

[@empossible1577]

Q.1 Exciting guided modes is actually a bit more complicated than it seems. The main problem is that you have to match a free space wave to a guided mode and these do not usually just match. To match them, you need to make the external wave look like the guided mode. A guided more looks a bit like a Gaussian beam so these tend to excite guided modes most effectively. A good model of a guided mode is a wave bouncing back and forth between the interfaces via total-internal-reflection. Only certain angles correspond to guided modes because the other angles would lead to a case where the wave bouncing back and forth would interfere with itself and leak out of the waveguide. The angle of incidence you are asking about corresponds to the allowed angle. Different modes will have different angles. I think a better model is thinking of the guided mode as a single packet of energy. It also has a propagation constant which describes how quickly the guided mode accumulates phase in the longitudinal direction. When you launch a wave at some angle of incidence, you are matching the component of its waveguide vector in the direction of the mode to the phase constant of the guided mode. This whole store gets more complicated when you want to excite higher-order modes, but I will not address that here. The short answer goes back to making the external wave look like the guided mode. In practice, at optical frequencies this would entail precisely aligning a beam using micropositioners to the angle and focusing down to a small spot at the entrance of the waveguide. There are other mechanisms such as grating couplers that can excite guided modes as well. Q.2 If your exciting is not correct, you will see the power lead from the waveguide. At visible frequencies, optical fibers will glow red and dim with distance. It is not usually an exact thing and there will be some tolerance. If you are aligning a beam to a waveguide, take your time and be patient. An alignment could take a while. I know people who spend 30 minutes or more getting an alignment and even then it is not perfect. It is not just about getting the angle correct, but also making the external wave look like the guided mode. If that is not perfect, power will eventually leak out of the waveguide. Q.3 Correct. This is not generally an efficient way to excite a guided mode. You will see people do things to collimate the wave before entering the waveguide in order to prevent what you are talking about. Hope this helps!!

[@localfuture1484]

@@empossible1577 Thanks a lot!! Very Detailed and helpful!

[@vahehgolestanian3390]

Hello, why the electric field is just function of x,y, and z. why it is not a function of time

[@empossible1577]

This is a frequency-domain analysis. It is assumed the fields are oscillating harmonically in their steady-state. If you wanted to add the time element, you would simply multiply your solution by exp(j*w*t) and then take the real part of the overall solution. You may be curious about what happens before the steady-state. This requires a time-domain analysis and it resolves the transient response. The finite-difference time-domain (FDTD) is a great numerical method for doing this. See kzbin.info/www/bejne/q3PMoaV_g52anLs

[@byquip]

Hello, I have a question about "quality" of solution in terms of datatypes. I wrote program for 1D solution and for windows version of python the best complex datatype is float 64 bit for real and imaginary part of number (128 bit). And if I use "numpy.finfo(numpy.complex128).eps" I will get something like 2^-52 (~2.2e-16). This is machine limits of float64. Which lowest epsilon for matlab or another prog. language? There is also numpy.complex256 with eps equal to 2^-63 (1.08e-19) BUT scipy.sparse library that have eigensolver does not support this datatype and this datatype available only for python on linux systems. Thank you for yours videos.

[@empossible1577]

Hmmm...I think the accuracy of the finite-difference method itself is way less than the numbers you are giving. I would not worry at all about the numerical precision for small things like slab waveguides. The numerical precision becomes a problem for extremely large problems. As for which language might be best...I actually do not know. I would guess that might be Fortran.

[@byquip]

@@empossible1577 Thank you for the answer. Also for 2D problem I have another question. PQ matrix is 2x2 shape, each element of it is shape n^2 x n^2. So PQ is 2 x 2 x nx^2 x ny^2 shape. Where nx, ny number of cells for media represented as refractive index. Is there a way to calculate eigenvalues and vectors by using elements of PQ matrix? (I can use only square matrices but PQ is 4D matrix for 2D waveguide problem).

[@empossible1577]

@@byquip Matrices are only ever 2D arrays no matter how many dimensions are being simulated. If there are Nx by Ny points on the grid, there is a total of M=Nx*Ny points on the grid. The P and Q matrices will each be MxM. The produce P*Q will also be of size MxM.

[@byquip]

@@empossible1577 I thought that it was 2x2x(Nx*Ny)x(Nx*Ny). For example in code for P matrix: Pxx = - A_x * inv_eps_z * D_y # 00 Pxy = A_x * inv_eps_z * D_x + I_m # 01 Pyx = - A_y * inv_eps_z * D_y - I_m # 10 Pyy = A_y * inv_eps_z * D_x # 11 All the matrices are MxM shape. But how is possible to be P matrix MxM shape as well? Could you please explain? It's the last key.

[@empossible1577]

@@byquip If Pxx, Pxy, Pyx and Pyy are all MxM, then P would be 2Mx2M. Sorry if I mispoke.

[@lutfunnaharasha5636]

You have written wrong about critical angle..totally opposite idea😓

[@empossible1577]

I would like to correct anything I have presented incorrectly and your help would be greatly appreciated. You can be more specific about where I presented something incorrectly? Thank you!

[@whysoserious0609]

Hey I also want to know why is incorrect, I feel it's same as my textbook