@ Adriaan

You got my point. The speed of light never changes in any medium. The observed reduction in speed is due to the delay introduced along the propagation path of a photon due to the repeated absorption and emission.

However during this transition of the constituent atoms/molecules from their ground to excited state and then back to the ground state is not a very efficient process. Some photon energy is dissipated in the medium in the form of heat and radiation. So there is some "net" absorption of energy for sure.

This relationship between the phase velocity and attenuation of wave is given by the dispersion relation,

Angular frequency (w) =  Phase velocity (c) x Spatial frequency (or, wave number, (k)) + Losses, 

which connects oscillations in space with oscillations in time.

As it can be seen that, lossless phase velocity occurs only if loss terms = 0. So, whenever there is some amplitude loss, it will affect its phase velocity too.

Attenuation and phase velocity are better related to each other using the Kramers-Kronig relationship (KKR) which is due to causality and the linearity principle. Please see wikipedia or relevant source for KKR.

I am looking for an analysis of the variation of the velocity such that the velocity changes continuously rather than discontinuously.

I try to find something in mechanics but I couldn't find an analysis showing how the velocity is decreased continuously before reaching a constant smaller velocity.

For example in the document bellow, as you suggested the masses play a role in the change of the velocity, but maybe there is not yet an analytical formula showing how the velocity changes continuously before having the new value. 

http://www.instructables.com/answers/What-is-the-actual-impact-velocities-of-two-coll/

"Assuming the initial speeds of m1 and m2 were v1 and v2 respectively, then conservation of momentum requires

m1*v1 + m2*v2 = (m1+m2)*v'

And as a result:

v' = (m1*v1 + m2*v2)/(m1+m2)"

That is a very good question.

When a radiation hits a given medium, then even if the medium is electrically neutral, it is disturbed by the incoming radiation (electric) field. The constituent atoms of the medium get momentarily polarized (pico to atto-second scale). Here polarization by radiation means, the negative and positive charge distribution within the atom are separated by a finite distance, giving rise to a finite dipole moment.

Note: the center of masses of negative and positive charge distribution coincides in the case of an electrically neutral atom and so the dipole moment is zero, but due to incident radiation field they get separated by a finite distance!

Once the individual atom is momentarily polarized (possesses a finite dipole moment), it is read to "talk" to the radiation. It absorbs the incoming radiation and transits to its excited state (following the dipole transition rules), but since the excited state is unstable and very short-lived, the atom has to emit the absorbed photon. The whole medium then acts just like a phased array of dipole antenna. These processes of absorption and emission, lead to a "delay" in the phase of the wave and so the phase velocity of the wave suffers in the medium.

See it as a relay race, where assume athletes run at a constant speed, but giving and taking of the baton (photon) leads to a delay!

Hope that clarifies.

the refractive index of a material is determined by its response to an electric and magnetic fields. For a dielectric material this is its permittivity that depends on the polarizability of the molecules, atoms and electrons of the material. Then this is the electromagnetic field of the light and its interaction with the components of the material that explains the change of its velocity. An important parameter is the frequency. At low frequency the molecules are polarized by the electric field of the light and interact with this light. At high frequencies this is the atoms and electrons that react to the field. And the refractive index is different in each case.

Zhongtao Hu remark is true only for thin laers with thikness near to wave length. For thick layers the c/n component is dominating.

I have a question about the explanations. Vikash answer looks correct and easy to understand. A "delay" is introduced by absorption and emission of the photon as it travels through the medium. In between these moments the photon continues to travel at speed c, because it cannot travel slower than that. There is no gradual slow down inside the medium, at its first interaction the photon changes its "effective" speed. This is in contradiction with the explanation that the photon can be slowed without absorption. Can someone clarify?

@ Adriaan

You got my point. The speed of light never changes in any medium. The observed reduction in speed is due to the delay introduced along the propagation path of a photon due to the repeated absorption and emission.

However during this transition of the constituent atoms/molecules from their ground to excited state and then back to the ground state is not a very efficient process. Some photon energy is dissipated in the medium in the form of heat and radiation. So there is some "net" absorption of energy for sure.

This relationship between the phase velocity and attenuation of wave is given by the dispersion relation,

Angular frequency (w) =  Phase velocity (c) x Spatial frequency (or, wave number, (k)) + Losses, 

which connects oscillations in space with oscillations in time.

As it can be seen that, lossless phase velocity occurs only if loss terms = 0. So, whenever there is some amplitude loss, it will affect its phase velocity too.

Attenuation and phase velocity are better related to each other using the Kramers-Kronig relationship (KKR) which is due to causality and the linearity principle. Please see wikipedia or relevant source for KKR.

Vikash, thank you. It's perfectly clear.

Thank you all for your enlightening responses and explanations.

I will read more about Kramers-Kronig relationship as proposed by Vikash and Hugo. The difference for high and low frequencies as emphasised by Andre (molecules/atoms and electrons) is also very important.

I wonder if there are in the literature analytical formulas for the delay (due to repeated absorption/emission) as a function of the diameter of the molecules or of the atoms?

Also, can we say that the light medium (vacuum/ether) introduces also a delay (due to its unknown constitutive "atoms") which is directly related to c?

Best regards

First, it does not matter on the size of the atoms.

Second, for all practically purposes, vacuum means nothing, though there are pair production phenomenon associated with it. Vacuum does not introduce any delay!

Light speed is derived from electric and magnetic properties of the media. There should be no expectation that different media would have the same properties. Energy of a photon does not decrease unless it is scattered.

Energy may be described as [ E= hf ] where h is not expected to change much if any.

Light speed is also described by frequency and wave length. Since frequency is near constant to conserve energy, wave length should change in conventional thought.

From SR and also GR several metrics are suggesting frequency and wave length change by the same fraction in an altered medium, but locally energy is still conserved and h as a variable is not well accepted.

So the questions is a good one although it might seem a bit old fashioned. The possible answers enter into modified PV theory.

Halim Boutayeb -"Also, can we say that the light medium (vacuum/ether) introduces also a delay (due to its unknown constitutive "atoms") which is directly related to c?"

Will take look on Maxwell's mechanical model of electromagnetism and nature of ether. This mechanical model transformed to wellknown Maxwell's equations.

Our disscussion is reiteretion of very old classical discussion with known winners ;-)

https://en.wikipedia.org/wiki/James_Clerk_Maxwell

http://nemo-ieee.org/about/about-the-logo

Thank you Vladimir Kubov, I didn't know that NEMO logo comes from Maxwell's attempting model of the aether. This is interesting that this model seems to anticipate the future discovery of internal atomic structure.

James C. Rautio's papers are also interesting to read.

I think part of science is continuously questioning usual theories and it is possible that a satisfactory theory of the aether will be developed in the future.

The effect can be demonstrated with VHF, UHF or microwaves using artificial dielectrics.  Artificial dielectrics were used in or just after the second world war to make lenses that didn't weight several tons (metamaterials have been around for 70 years or more!).  They were made out of an array of evenly spaced metal spheres held in place on strings.  The electromagnetic wave causes currents on the spheres.  These currents re-radiate, but the re-radiation from each sphere lags the initial field by 90 degrees (I think).  The size and packing density of the spheres determines the proportion of re-radiation to existing wave, and so the phase lag and the effective dielectric constant.  It will take several periods of the sphere lattice before the bulk slow-down factor is reached, so it is not simultaneous at the boundary.  This is the sort of thing that can be calculated if you want to try that, and it could be measured at VHF if you had the time, equipment and space.

Interesting that when it is atoms that are re-radiating the photons seem to retain their identity (coherence), as I believe quantum communication has been demonstrated through a dielecric (air).

See this about an artificial dielectric made from an array of metal discs.

http://www.dtic.mil/dtic/tr/fulltext/u2/a005687.pdf

@ Malcom White: "These currents re-radiate, but the re-radiation from each sphere lags the initial field by 90 degrees (I think)...  Interesting that when it is atoms that are re-radiating the photons seem to retain their identity (coherence).."

This can maybe be explained by the fact that for dielectric, the displacement current in Maxwell-Ampere equation is much greater than the conduction current (sigma/(2pi f epsilon) is small for dielectric). Whereas for a conductor, the conduction current is much greater than the displacement current. Maxwell-Ampere equation shows that displacement current and conduction current are out of phase of 90 degrees for an harmonic field.

If is interesting to note that a same medium can be seen as a conductor at low frequency and as a dielectric at higher frequency (sigma/(2pi f epsilon) decreases with frequency f).

I am presently reading texts concerning Lorentz's dispersion theory for gaz since it seems to be a very usual model. From my readings, the dispersion in solids and liquids seems to start with a similar technique except that the interactions between molecules are considered, which is not the case for gaz in Lorentz's theory.

Dear all,

I have found Abraham-Minkowski dilemma intersting in relation to this thread, because it deals with light momentum. Particularly this paper:

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.104.070401

I would be grateful if someone can share his knowledge on this subject.

Best regards