What are the general parameters for the in-medium light group speed?

[greswd]

An electromagnetic wave has a phase speed and a group speed. Or velocities, for that matter.

In a medium, the phase speed of a wave is generally determined by the medium's permeability μ and permittivity ε.

What are the general parameters that determine the group speed of a wave in a medium?

 

[mfb]

The group velocity depends on how these things change as function of frequency. $\displaystyle v_g \ = \frac{\partial \omega}{\partial k}$ where $\omega$ is the angular frequency of the wave and $\displaystyle k = \frac{\omega}{v_p}$ with the phase velocity $v_p$. In general the phase velocity depends on $\omega$. If it doesn't change in some range then the group velocity is simply equal to the phase velocity.

 

[greswd]

The group velocity depends on how these things change as function of frequency. $\displaystyle v_g \ = \frac{\partial \omega}{\partial k}$

I understand, but I'm wondering what parameters of a medium in general affect this.

 

[mfb]

The same parameters as for the phase velocity.

 

[greswd]

The same parameters as for the phase velocity.

so, using both permeability and permittivity, I suppose it'll result in the same speed as the phase speed?

what about in media where both speeds are different?

 

[greswd]

The same parameters as for the phase velocity.

so, using both permeability and permittivity, I suppose it'll result in the same speed as the phase speed?

what about in media where both speeds are different?

I think the interesting situations are when the speeds differ, and I'd like to learn more about it.
(for a coherent, monochromatic beam)

 

[Lord Jestocost]

The group and phase velocity are the same in case the index of refraction of the considered material doesn’t depend on the light frequency. Generally, the permittivity and permeability and thus the index of refraction are frequency dependent in materials (dispersion), giving rise to different group and phase velocities.
Electromagnetism - Lecture 13 Waves in Insulators

 

[greswd]

The group and phase velocity are the same in case the index of refraction of the considered material doesn’t depend on the light frequency. Generally, the permittivity and permeability and thus the index of refraction are frequency dependent in materials (dispersion), giving rise to different group and phase velocities.

thanks, and sorry, I should've clarified, I'm interested in the case of a coherent, monochromatic beam, like that of a laser.

 

[greswd]

The group and phase velocity are the same in case the index of refraction of the considered material doesn’t depend on the light frequency. Generally, the permittivity and permeability and thus the index of refraction are frequency dependent in materials (dispersion), giving rise to different group and phase velocities.

thanks, and sorry, I should've clarified, I'm interested in the case of a coherent, monochromatic beam, like that of a laser.

Is it safe to say that for monochromatic waves in a medium, both speeds are always the same?

 

[mfb]

Is it safe to say that for monochromatic waves in a medium, both speeds are always the same?

No.

so, using both permeability and permittivity, I suppose it'll result in the same speed as the phase speed?

No. Check post 2. The group velocity depends on how the phase velocity changes with frequency. This is true even for an essentially monochromatic beam. An idealized monochromatic beam cannot form groups.

 

[greswd]

An idealized monochromatic beam cannot form groups.

so, for a real-world laser beam, it has a frequency spread, and therefore forms groups as mentioned in #7?

 

[Lord Jestocost]

Is it safe to say that for monochromatic waves in a medium, both speeds are always the same?

“What the index [of refraction, LJ] tells us is the speed at which the nodes (or crests) of the wave travel. The node of a wave is not a signal by itself. In a perfect wave, which has no modulations of any kind, i.e., which is a steady oscillation, you cannot really say when it “starts,” so you cannot use it for a timing signal. In order to send a signal you have to change the wave somehow, make a notch in it, make it a little bit fatter or thinner. That means that you have to have more than one frequency in the wave, and it can be shown that the speed at which signals travel is not dependent upon the index alone, but upon the way that the index changes with the frequency. This subject we must also delay (until Chapter 48).”

from chapter 31 “The Origin of the Refractive Index” in “The Feynman Lectures on Physics, Volume I”: http://www.feynmanlectures.caltech.edu/I_31.html

see also chapter 48 “Beats” in “The Feynman Lectures on Physics, Volume I”: http://www.feynmanlectures.caltech.edu/I_48.html

 

[mfb]

so, for a real-world laser beam, it has a frequency spread, and therefore forms groups as mentioned in #7?

And therefore can form groups.

 

[greswd]

“What the index [of refraction, LJ] tells us is the speed at which the nodes (or crests) of the wave travel. The node of a wave is not a signal by itself. In a perfect wave, which has no modulations of any kind, i.e., which is a steady oscillation, you cannot really say when it “starts,” so you cannot use it for a timing signal. In order to send a signal you have to change the wave somehow, make a notch in it, make it a little bit fatter or thinner. That means that you have to have more than one frequency in the wave, and it can be shown that the speed at which signals travel is not dependent upon the index alone, but upon the way that the index changes with the frequency. This subject we must also delay (until Chapter 48).”

from chapter 31 “The Origin of the Refractive Index” in “The Feynman Lectures on Physics, Volume I”: http://www.feynmanlectures.caltech.edu/I_31.html

see also chapter 48 “Beats” in “The Feynman Lectures on Physics, Volume I”: http://www.feynmanlectures.caltech.edu/I_48.html

Feynman’s lectures are good, however, in explaining the origin of the refractive index, he didn’t show how it relates to permeability and permittivity. Do you have a link which describes the origins in terms of them? Thanks

 

[Lord Jestocost]

http://www.feynmanlectures.caltech.edu/II_32.html
[PDF]SOLID STATE PHYSICS PART II Optical Properties of Solids - MIT