Talk:Dispersion (optics)

Group velocity formula

I recently added what I thought was the more general formula for the group velocity:

$v_{g}=v-\lambda {\frac {\partial v}{\partial \lambda }}$

which user:Stevenj has removed saying it's not generally true, only in the case of low dispersion. I though this came from the definition of the phase and group velocities:

$v=\omega /k$

and

$v_{g}={\frac {\partial \omega }{\partial k}}$

hence:

$v_{g}=v+k{\frac {\partial v}{\partial k}}$

and from $k=2\pi /\lambda$, giving:

$v_{g}=v-\lambda {\frac {\partial v}{\partial \lambda }}$.

So, what am I missing? Are my velocity definitions wrong? (I accept that none of the equations are valid for inhomogeneous media.) -- Bob Mellish 17:58, 5 October 2005 (UTC)

Ah, now I see what you meant. My problem is with the $k=2\pi /\lambda$. Ordinarily, in equations like this (or in most of optics for that matter) λ means the vacuum wavelength, i.e. $\lambda =2\pi c/\omega$. Using the wavelength in the material is problematic — not only for inhomogeneous media, but also for anisotropic homogeneous media — but mainly I think people don't use the medium-dependent λ because it is just more convenient to talk in terms of conserved quantities. (I wish this article would start with the general definitions and then give the specific formulas for special cases like homogeneous media, but I don't have time to do major re-working myself right now.) —Steven G. Johnson 04:59, 6 October 2005 (UTC)

It looks like essentially every usage of λ in the article means vacuum wavelength, but it never says that explicitly. Sigh. —Steven G. Johnson 05:02, 6 October 2005 (UTC)

• Hrrm, I didn't think of k being wrong. Yes, vacuum wavelength makes more sense to use. Unfortunately I've forgotten most of the derviation of this stuff, if you've got a good modern reference I could look it up and try to improve the article, if you're too busy. Personally I'd prefer it to start with the simple cases and move to the most complex ones, but then I've never worked with photonic crystals or the like. -- Bob Mellish 16:35, 7 October 2005 (UTC)

The classic reference on all of this stuff is:

Léon Brillouin, Wave Propagation and Group Velocity (Academic: New York, 1960).

You can also find some discussion in Jackson (Classical Electrodynamics) etc. However, even so most of this discussion is limited to homogeneous media. In general, the group velocity is defined as dω/dk, where k is the Bloch wavevector in a periodic system (of which a homogeneous system is a special case). In a periodic system, on the other hand, there is no perfectly satisfactory definition of phase velocity, since k is only defined up to a reciprocal lattice vector. A proof that dω/dk is equal to the energy velocity for dispersionless, lossless systems (including periodic systems and also including waveguides) can be found in Sakoda, Optical Properties of Photonic Crystals (Springer: Berlin, 2001). This can be extended to systems with material dispersion (using the proper definition of energy density in a dispersive medium, as can be found in Jackson), but I'm not aware of any single reference that contains the complete derivation for all cases (arbitrary inhomogeneity and dispersion). Once you include loss, of course, or when you are looking at evanescent modes (complex k), then the group velocity is no longer the energy velocity; this is described in the Brillouin reference, which also describes the front velocity and other concepts. A good description (without too much math) of the phenomena and consequences of dispersion (both material and waveguide, but not in general periodic media) in communications systems can be found in Ramaswami and Sivarajan, Optical Networks: A Practical Perspective (Academic Press, 1998); this is an excellent introductory textbook overall (although a little specialized to fibers, in which things simplify because the inhomogeneity is weak). In particular, Ramaswami also gives a more general definition of the dispersion parameter D, which is especially pertinent to this article. —Steven G. Johnson 18:49, 7 October 2005 (UTC)

• Heh, I actually have the Brillouin book right here; I grabbed that equation from page 3 (eqn. 9b). I've been reading it with aim to improving the front velocity and signal velocity articles. I'll try seeing if the library has the other works you mention. -- Bob Mellish 18:59, 7 October 2005 (UTC)

waveguide dispersion vs. modal dispersion

what's the difference? Pfalstad 15:47, 4 January 2006 (UTC)

Modal dispersion comes because the waveguide supports multiple modes at the same frequency that travel at different speeds. Waveguide dispersion refers to the fact that for a single mode the speed depends on the relative size of the wavelength and the waveguide geometry, which causes the solution's field pattern to change. —Steven G. Johnson 17:37, 4 January 2006 (UTC)