# Speed of light in a medium other than vacuum

#### fluidistic

Gold Member
I'm wondering if the speed of light in a medium other than vacuum is well defined. I explain myself: Say I am underwater and I create a laser pulse. I know that at any given time, the speed of the photons constituting the light is always c. However I also know that photons will get absorbed by the atoms of water molecules and then be re emitted in the same direction as they were. So that light will take a longer time to go through a distance than if it was traveling in vacuum.
Say I have a photon detector situated at a distance of 300000 km from the laser, still under water (I know it's not possible on Earth but anyway). Some (maybe very few) photos will never meet a single atom between leaving the laser and being absorbed by the photon detector. Some other photons will be absorbed and re emitted say n times, other m times, etc. So what started as a pulse, isn't a pulse anymore when it goes to the detector. Some photons will reach the detector well before the others. Then why do we even talk about a speed of light in water? What does this really mean? Is it an approximation for small distances (in contrast with my example of 300000 km)? Is it an average of the "apparent speed" of all the photons? Or I'm all wrong and indeed, the speed of light in any medium is well defined?

Edit: I just thought that maybe such a huge distance would make that almost all photons were absorbed and re emitted a very close number of times. So if it is the case, consider a very few photons going through a medium distance (say 40000 km) in a gas at a very low pressure.

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#### Dr Lots-o'watts

I'm wondering if the speed of light in a medium other than vacuum is well defined. I explain myself: Say I am underwater and I create a laser pulse. I know that at any given time, the speed of the photons constituting the light is always c.
If it is a pulse, then it contains many wavelengths, and the truth is that each wavelength has a different speed when propagating in a medium other than air i.e. the pulse will spread.

Some (maybe very few) photos will never meet a single atom between leaving the laser and being absorbed by the photon detector.
I think it is safe to says that they will all meet atoms, even for the smallest distances. Water molecules and photons have a cross-section, and water molecules practically touch each other. If you change this to a low density gas, then the index of refraction gets closer to 1 (n(air) is almost equal to n(vacuum) = 1).

Is it an average of the "apparent speed" of all the photons?
If you want to treat the photons individually, yes, but the average speed is well-defined.

I'm not sure if I'm answering your general question, but I hope it helps a bit.

#### Integral

Staff Emeritus
Gold Member
The speed of light in any material is given by

$$c = \sqrt { \frac 1 {\epsilon \mu } }$$

Where $\epsilon$ and $\mu$ are the permeability and permittivity of the material.

So the answer to your question is yes the speed of light in materials is well defined.

Dr L. W. beat me to it, $\epsilon$ and $\mu$ can be frequency dependent of course.

#### fluidistic

Gold Member
Ok thanks guys. Sorry, I used wrongly the word "pulse". I meant like a flash. In other words, I turn on and then off the (monochromatic) laser as fast as possible.
To integral: Is this formula valid for quantum electrodynamics? I'm pretty sure it holds in Classical Electrodynamics but I'm just curious if it still holds when you consider light as photons instead of plane waves (as in a laser).
If your formula still holds, it means I'll always see that all the photons hit the receptor in the same instant. If so, how can that be possible?

In a case where I send 2 parallel monochromatic photons through a very short (around 1 meter long) gas at very, very low pressure (number of atoms/molecules is around say 10 in 1 m^3). Imagine one photon reach the receptor without having hit any atom while the other photon hit say 1 atom and then is re emitted. It will reach the receptor a very bit later compared to the other photon, yet both were released at the same time. Is my reasoning wrong?

#### Dr Lots-o'watts

Ok thanks guys. Sorry, I used wrongly the word "pulse". I meant like a flash. In other words, I turn on and then off the (monochromatic) laser as fast as possible.
This can still be decomposed in a Fourier series. The most perfect laser is still not perfectly monochromatic (Heisenberg's principle, heat of the emitting crystal etc.).

In a case where I send 2 parallel monochromatic photons through a very short (around 1 meter long) gas at very, very low pressure (number of atoms/molecules is around say 10 in 1 m^3). Imagine one photon reach the receptor without having hit any atom while the other photon hit say 1 atom and then is re emitted. It will reach the receptor a very bit later compared to the other photon, yet both were released at the same time. Is my reasoning wrong?
I agree with your reasoning. In this case, the index of refraction can still be defined, and it is only slightly above vacuum's. And as I said, the speed of light of electromagnetism (as written by Integral) is an average. A perfectly defined average.

#### fluidistic

Gold Member
Ok thank you very much. Question totally solved.

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