I don't have a good reference handy but any graduate or late undergraduate physics text that deals with Xrays should tell you that just about anything has a refractive index less than 1 (and greater than 0) for Xrays. This has been known for a long time. Maybe the Braggs found it early 20th Century. It is about that old. This is the phase velocity of course, which is what determines refraction. group velocity (approximately) determines information flow, and group velocity remains <= C. Geronimo Jones
Geronimo is right. Phase velocity can be larger than c, but what we understand as "velocity of light" is the group velocity. In vacuum phase velocity is equal to the group velocity, but no so in other media. I didn't remember that refraction was determined by phase velocity, but if this is so, then it all makes sense. I don't know what recent experiments is Axel refering that shows group velocity larger than c, you should check your reference, and if it says so, share it. Most likely is an article by a journalist that doesn't know what he/she is talking about.
The article at http://physicsweb.org/article/world/13/9/3 claims that group velocities can be anything, even negative. I always understood "velocity of light" to mean the phase velocity (i.e. the velocity with which the wave peaks of monochromatic light move). Is that not correct?
Here's a Google threat about this topic: [1]
Also, I don't think it makes sense to speak of group velocity of monochromatic light, so the group velocity cannot be used for the definition of the refractive index, which requires light of one wavelength only. --AxelBoldt
Combine special relativity and a belief in causality and you very likely get a belief that information can't travel faster than C. I suppose you could take a position that relativity itself doesn't talk about this. Geronimo Jones
Article follows:
n = c/v (1)
where v is the velocity of light in the medium. Note that n can never be less than 1 because light can not go faster in a medium than in a vacuum.
Light travels slower in a medium because light is comprised of electromagnetic waves, and matter is made up of charged particles. Most of the time, the atoms that make up the matter that we deal with from day to day are neutrally charged, and thus produce no electric field. When these atoms are subjected to an electric field, however, the negatively charged electrons and the positively charged protons get pulled in different directions, creating a dipole. The field of the dipole runs counter to the original field, weakening it. When the field intensity is changing over time, the creation of the dipole effects how quickly the changes can take place, and thus the wavelength changes and the wave slows down. More precisely, the wavelength is scaled by:
λmedium = λvacuum/n (2)
This makes sense because it is required for mathematical consistency with he equation:
v = fλ (3)
When the velocity of the wave changes, the frequency and/or the wavelength of the wave must change. In this case the frequency remains constant, and the wavelength changes to account for the change in velocity.
sin(θexit) = noldsin(θapproach)/nnew
Recall than neither n can be less than one, and that the range of sine for any angle between 0o and 90o is between 0 and 1. In fact, the sine of any anlgle can never be greater than one. If nnew is less than nold, though, the right side of the equation can be greater than one if the angle of approach is great enough. When that happens, the light simply does not leave the medium and undergoes total internal reflection. Fiber optic cables are based upon this principle. Total internal reflection also explains why diamonds, with a whopping index of refraction of 4, sparkle so much.
The rainbow patterns come about because sometimes the reflected light interferes with itself constructively, sometimes destructively. Whenever light is incident on an interface between to mediums that have a different index of refraction, part of the light is reflected and part crosses the interface. To determine the type of the interference, the relative phases of the light reflected off the top of the film and off the bottom determine the type of interference. If they are in opposite phase, the light cancels out; same phase and the light looks brighter than other reflected wavelengths. Recall that:
sin(θ ± k180o) = i * sin(θ)
K is an interger. If k is even, then i = 1. If k is odd, then i = -1.
I don't know the details beyond how to figure out whether the the light waves are perfectly in phase or perfectly out of phase. I do have enough information to determine more, but I don't have the time right now to work it all out.
All phases are described in relation to the incident light wave.
First, determine the phase of the light reflected off the top of the film. If the index of refraction of the film is greater than that of the medium the light is in, then the light is 180o out of phase. Otherwise, the light is the same phase.
Second, determine whether reflecting off the bottom of the film will cause a 180o phase shift as above.
Third, determine how the thickness of the film will effect the phase. Specifically, λ/2 = 180o.
if 2t = k λfilm/2
k is an interger. If k is even, then there is no phase change. If k is odd, then there is a 180o phase change.
t is the thickness of the film. It is multiplied by two because the light must cross the flim twice (once going down, and again going up).
&lambdafilm is the wavelength of the light in the film, and:
λfilm = λvacuum/n
Fourth, count up the phase changes, and remember that two 180o phase shifts add up to no phase shift. If the net result is that the two reflected lights are opposite phase, then they cancel out. If they are the same phase, then they add up, and the light will be brighter than other wavelengths.
The film has to be thin, not more than a few dozen wavelengths of the incident light, in order for the reflection from the bottom of the film to get back in time to interfere with the reflection from the top.
This is what gives oily puddles a rainbow, as noted above, but also bubbles. Thin film reflection is also the underlying prinicple used by stealth fighters to obscure radar.
Could someone please provide a more mathematically rigorous definition for thin film reflection?
Ok, so here's the full story. Some people define the refractive index with respect to air, others with respect to vacuum. For most wavelengths, it doesn't matter, but for x-rays the difference is crucial. The more common and more logical definition is the one adopted by EB and by refractive index: use the vacuum. With the vacuum definition, the refractive index cannot be less than one, and that's the mistake in EB. Your reference above apparently uses the air definition; with that definition, refractive index of x-rays is indeed less than one.
Now for your other links: these are the experiments that show that the group velocity of a wave pattern can be larger than c. But group velocity of a wave pattern is different from the speed of the individual waves, which is commonly called "speed of light" and which is used in the definition of n. It can never exceed c. Now if you redefine n to use the group velocity instead of the light's speed, then you can indeed get a number between 0 and 1. EB doesn't do that, nor does our article. --AxelBoldt
![[Home]](wikipedia.png)