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 can, please, give a quantitative description with some examples, like the cosinusoidal functions in my post. Also the introduction of the exact name for each quantity (transferred energy, available power, reactive power, etc.) surely would be another help to understand.
Dear ZapperZ, what you say can intuitively make sense, but it surprises me as well as Chandra Prayaga, maybe because we are lacking some mechanical waves concepts and using a different perspective.
When for example Electro-magnetic waves are introduced in a classical approach, the Electric field...
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!
However, as regards the first derivative, d\omega / dk = v_g and dk/d\omega = 1/v_g, so they are exactly reciprocal. If you take the unit measures, they are reciprocal too. So, here is still my doubt.
Considering the simplest case, the one regarding plane waves,
k = \omega / v
with v constant.
d\omega/dk = v = v_g is the group velocity and dk/d\omega = 1/v = 1/v_g is the reciprocal of the group velocity.
d^2 \omega/dk^2 = \alpha = 0 is the group velocity dispersion; so, the reciprocal of...
Hi!
Dealing about wave propagation in a medium and dispersion, wavenumber k can be considered as a function of \omega (as done in Optics) or vice-versa (as maybe done more often in Quantum Mechanics). In the first case,
k (\omega) \simeq k(\omega_0) + (\omega - \omega_0) \displaystyle \left...
Yes, of course it is (as explained in the document): this is straightforward. Anyway I can't get how "the evolution of the amplitude A(x,t) of the harmonic wave e^{i (k_0 x - \omega (k_0) t)} in (8.7) is governed by the" Schrödinger equation (page 5).
Yes, of course, in fact in my post I wrote the 2nd derivative has to be considered.
Ok, I understand.
Thank you. Did you follow the whole procedure? I can't get how he obtains a Schrödinger equation describing the amplitude of the (8.7) (page 5).
Thanks for this one too.
Emily
Hello!
Starting from a gaussian waveform propagating in a dispersive medium, is it possible to obtain an expression for the waveform at a generic time t, when the dispersion is not negligible?
I know that a generic gaussian pulse (considered as an envelope of a carrier at frequency k_c) can be...
Maybe my question was simpler than it seems. In fiber optics the light source must generate a signal whose angle of incidence is not greater than the acceptance angle, in order for the signal to be guided.
As regards dielectric slab guides, instead: is the condition about the angles more...
Hello!
Let's consider a plane wave represented by a ray, propagating in a 2D dielectric slab. It has a medium with refractive index n_1 as its core and a medium with refractive index n_2, n_2 < n_1, as its cladding. In order for this ray to represent a mode, it must satisfy two conditions:
-...
I know that, in Quantum Mechanics, talking about "actual" quantities is inappropriate.
Anyway, it is common to compute the number of electrons in conduction band, for example, in a semiconductor (e.g. here, penultimate formula). During the computation, Fermi distribution and density of states...
TeethWhitener, I agree with you, the refractive index is at the same power as c and in different media photons have different velocities.
Henryk, it was difficult because the book spoke about "density" without specifying anything else; so I thought it was per unit frequency.
Thank you both!
Hello!
Let's consider again a system of atoms with only two permitted energy levels E_1 and E_2 > E_1. When electrons decay from E_2 level to E_1, they generate a photon of energy E_{21} = E_2 - E_1 = h \nu. The number of photons (per unit frequency, per unit volume) emitted by such a system in...
:wink:
Yes, exactly!
I understand what you are meaning and yes, it is correct. Obviously, very often you are not interested in counting just the number of states between two available modulus values k_1 and k_2: instead, you will be interested in counting the total number of states, so...
Did you deduce this from my post or by your own?
The first post was maybe not clear about this. Don't consider my problem as real and don't take its numbers as absolute: it was just an example to show how the Fermi distribution is used to count the electrons in a band of energies.
The strange...
Hello!
Let E_1, E_2, \ldots, E_n be n allowed energy levels for a system of electrons. This system can be described by the Fermi-Dirac distribution f(E).
Each of those levels can be occupied by two electrons if they have opposite spins.
Suppose that E_1, E_2, \ldots, E_n are such that...
1) If you read again my answer, I didn't say that dk has not a physical meaning. I said instead that dk is not related to the spacing between the states
Be careful: the \mathbf{k}-space is a vector space. Your variable \mathbf{k} = k_x \mathbf{x} + k_y \mathbf{y} + k_z \mathbf{z}, is in fact a...
You're welcome :).
Yes, it is. If \mathbf{k} is a vector with components k_x, k_y, k_z, the minimum spacing between two adjacent values of k_x (or k_y, or k_z) is \pi / L. So, each of the acceptable values of \mathbf{k} result to be "alone" inside a volume (\pi / L)^3 (and then you obtain the...
Ok, and so first of all you can say now that the density of the allowed states in the K-space is
\displaystyle \frac{1}{\left( \displaystyle \frac{\pi}{L} \right)^3}
that is (volume of k-space of one octant of a spherical shell)*(density of allowed states in K-space.)
No, absolutely, for...
Yes, a factor of 2 is included in the density of states. My question is: if an energy level E_1 is such that (for example) f(E_1) = 0.1 and there are two possible states with opposite spins for the electrons at E = E_1: will both the states have the same 0.1 probability of being occupied? Or...
Hello!
In order to obtain the number of actual electrons in the conduction band or in a range of energies, two functions are needed:
1) the density of states for electrons in conduction band, that is the function g_c(E);
2) the Fermi probability distribution f(E) for the material at its...
To DrClaude: ok, now it is more clear, thank you.
To marcusl: I think the "particle in a box" is the simplest representation of an electron into a lattice; it is good as a first approximation. The effective mass includes the effects of lattice and so the electrons can be treated as free...
Remember that we are counting states in the Fermi sphere in order to derive the state density. States are acceptable solutions of the Schrödinger equation for this problem. So, should they also be linearly independent?
In the Wikipedia page you linked, it is stated that negative values of p, q, r are neglected because "they give wavefunctions identical to the positive" p, q, r "solutions except for a physically unimportant sign change". It is what thephystudent mentioned: \sin(x) =? \sin(-x). So I would like...
To marcusl: no, it is general, but referred to semiconductor materials.
To navrit: I think it is related to the wavefunctions more than the symmetry of the Fermi sphere.
To thephystudent: no, \sin(x) = - \sin(-x) and so they seem to be not equivalent.
I think the approach to be followed is the...
Hello!
When computing the density of states of electrons in a lattice, a material with dimensions L_x, L_y, L_z can be considered. The allowed \mathbf{k} vectors will have components
k_x = \displaystyle \frac{\pi}{L_x}p
k_y = \displaystyle \frac{\pi}{L_y}q
k_z = \displaystyle \frac{\pi}{L_z}r...
Hi!
When dealing with a pn homojunction, it is easy to see the features it has at equilibrium, and also the features it has with forward/reverse bias. Plots show the constant Fermi level at equilibrium and the different Fermi levels for a forward bias; moreover, examples show how much the bands...
Ok, thank you.
By "vacuum level" I mean the energy of a free electron outside the crystal, as stated http://ecee.colorado.edu/~bart/book/book/chapter2/ch2_3.htm, par. 2.3.3.2. Anyway, maybe the vacuum level is bent like the energy bands across the junction due to the electric field.
Hello!
The (potential) energy of an electron in a solid structure is always negative; also the E_c and E_v levels (conduction band and valence band limits) are negative, in the band diagram of a pn junction.
When the junction is built and thermal equilibrium is reached, the depletion region...
Ok and thank you! As far as you know, which could be a typical height choosen for barriers in heterostructures? (For example, in the case of AlGaAs - GaAs - AlGaAs)
For example, let's refer to this document, page 7, figure (a). If electrons migrated from the right (n-AlGaAs) region to the central (GaAs) region, overstepping that high barrier potential, how can we be sure that they won't also overstep the barrier between the central and the left (p-AlGaAs)...
Hello!
Double heterostructures are used in LEDs and lasers to provide both the confinement of the charge carriers and the confinement of the generated light.
This image is a comparison between a homojunction and a heterojunction.
As regards the unbiased junctions, when the n region and the p...
Thank you for your clarifications. But in particular during the transient just after the contact between the n region and the p region, how can electrons increase their energy and move to the conduction band of the p side?
It is not for thermal energy. May the charge concentration gradient be...
Hello!
Some of the processes caused by a pn junction are not clear for me. Just after the contact between the p and the n region, a migration of charges happens in a semiconductor junction in order to reach an equilibrium condition. A valence band and a conduction band are present in both...
I copied it from some photocopies. Moreover, I have compared the equation with another guy's notes and it is the same. So I am sure that the equation is the same as in the original, and yes, it is about the matter I needed.
I have it by way of notes. I just suppose that these notes are based on an Italian textbook, which I have not found in Amazon. It deals with Quantum Mechanics in order to deal with laser and optic signals transmission.