The first expression, which is correct, is written using formula (A4) of the linked document:
$$\frac{J'_{\nu} (u)}{u J_{\nu}(u)} = \frac{J_{\nu - 1} (u)}{u J_{\nu}(u)} - \frac{\nu}{u^2} = \xi_1(u) - \frac{\nu}{u^2}$$
Formula (A6), used for the second expression, is wrong. It should be...
Temporarily putting aside the ##\frac{k_1^2 + k_2^2}{k_1^2}## terms signs, consider the part which should be ##0##. The first 4 terms come from the espansion of the LHS (which involves ##\xi_1##, ##\xi_2##) of the original characteristic equation. The last 3 terms directly come from the RHS of...
Hello!
In Optical fibers, let ##k_1## and ##k_2## be respectively the propagation constants in core and cladding, ##\beta## the propagation costant of a mode along the direction ##z##, ##a## the radius of the fiber. Using the normalized quantities ##u=a \sqrt{k_1^2 − \beta^2}## and ##w=a...
Yes, it is not a good starting point. I have read about the dielectric slab, for example, where modes are much simpler and immediate. My problem is not about modes themselves, but about some unclear information on what modes actually propagate in optical fibers (refer to the quote I just posted...
Ok! My doubt arouse because of sentences like:
from this document, page 6. However, if I correctly understood what you state, TE and TM modes are, at least conceptually, valid and existing modes, and they are able to propagate by their own.
Yes, of course. This depends on the frequency of...
Yes, I get it.
Sorry, I don't know them.
If I correctly understood, yes, modes somehow represent the spectrum of the optical fiber. Any real field propagating in this structure can be described as a composition of modes.
But my post was about a slightly different scope: are all these modes...
In a step-index optical fiber, considering Bessel functions of order ##\nu = 0## and no ##\phi## dependence, it is possible to obtain two separate sets of components, which generate respectively TE and TM modes. In the former case, only ##E_{\phi}##, ##H_r##, ##H_z## are involved; in the latter...
Hello!
For dielectric slab waveguides, starting from the field expressions, it is possible to draw the Electric field corresponding to a specific mode, showing its (possible) zeros inside the core and its exponential decay in the cladding:
A Google search can provide plenty of images like...
If you are interested in modes, in this page the mode field expressions are obtained for a dielectric slab waveguide. It starts from Maxwell's equations and then uses boundary conditions (after introducing refraction and some basic concepts). Modes in optical fibers are obtained through a...
There is of course a purely mathematical answer to this question (the Wikipedia pages can contain it), but it's not the only one.
As regards (metallic, dielectric, ...) waveguides: a mode is a field configuration which
is a solution of Maxwell's equations and satisfies the boundary conditions...
Ok, it could make sense. So we are stating that a wave which can't be always itself along the guide, can't propagate there. Why are we so sure about it? I mean: a wave can change along the guide, without destroying itself.
I don't think this is a difficult question, just I can't figure out the problem. The question can be asked in a simpler way.
A plane-wave is propagating with a certain angle of incidence into a parallel-plate waveguide: it will bounce off the upper plate, then the lower plate, and so on. Why can...
Thank you for your post. I found that relation, but still can't understand how it is justified.
Let's look at this image: in Figure (a), a wave coming from left propagates with a certain \mathbf{k_u}. It then bounces a first time (in the upper plate), as in Figure (b), and a second time (in the...
Hello!
In this previous post, most replies point out that it is not possible to predict the angle of refraction (and so the frequency) at a certain interface, given the wavelength of the original signal.
In particular,
But when dealing with optical waveguides, it seems to be different. I am...
Don't worry, I am late this time more than you. Thank you instead for all your detailed observations.
I followed all your steps and I agree with them. But maybe there is another solution and I would like to ask you for your opinion.
Consider your equation:
A e^{-j k_1 y \sin \theta_i} + B e^{-j...
First of all thank you for your answer.
In the link you suggested, the othogonality seems to be just for sines of the form \sin (mx) and \sin (nx) with n \neq m and so this situation is quite different; anyway, the sine functions k_1 y \sin \theta_i and k_1 y \sin \theta_r are actually equal...
Hello!
This post is strictly related to my previous one. Let's consider the same context and the same image. Regarding the oblique incidence of a wave upon an interface between two dielectric, all the texts and all the lectures write an equation like the following:
e^{-j k_1 y \sin \theta_i} +...
That's right. I implicitly considered the geometry in the attached image just a few posts above, but the paper does not specify anything.
Yes, but just now :s.
And if my computation is exact, this agrees with Snell's law too.
Thank you for all your help,
Emily
Yes, I know and you're right. But my doubt was simply regarding the math.
I completely agree.
This could be the case, if I consider the wave vector as a phasor.
Ok! But what about that way? Is it the direction orthogonal to the boundary or the direction parallel to the boundary?
In the...
Yes, but fields are not considered in Snell's law. It deals only with rays (incident and refracted) and the \theta_r should be forced to the value \pi / 2 and this should be evident without considering the fields. How?
Hello everybody!
I read that if one of the two materials involved in Snell's law is a conductor, the refraction angle \theta_r is about \pi / 2 and is independent of the incident angle \theta_i (I think \theta_r will be precisely \pi / 2 if the conductor is ideal). My question is: why...