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Admin
AIUI, a negative refraction index is a sort of reflection.
Admin
Remember this guy?
http://www.rappers.org/wp-content/uploads/2009/06/flavaflav-111.jpg
Admin
Fortunately, no. I shall now endeavor to forget him again.
Admin
A nice example of what happens when your "High Availability" solution actually helps you lose your service entirely.
Admin
Believe it or not, I've never heard that one. Works.
Admin
They bend light “the wrong way“ allowing for non-standard optics. Funky stuff. No idea what they mean in relation to light velocities though. IIRC, a lot of the negative RIs only happen at selected wavelengths; they're only observed in fancy inductor-based metamaterials, and the feature size and spacing of the inductors is critical.
Admin
Phase, Group and Signal Velocity http://en.wikipedia.org/wiki/Phase_velocity http://en.wikipedia.org/wiki/Group_velocity http://www.colorado.edu/physics/EducationIssues/baily/courses/BailySP12_GroupVelocity.pdf http://www.phy.duke.edu/~qelectron/pubs/SFLProgressInOptics.pdf
Even the TL;DR is a bit long: No information can be conveyed by a pure sine wave, which is infinite in time and space. To convey information, the wave must be modulated is some way, which introduces a wave (or waves) of a modulating frequency (or frequencies). This produces a "beating" effect such as you may have heard when a musical instrument is slightly out of tune, or may have seen a diagram of an AM radio signal: http://www.mathpages.com/home/kmath210/kmath210_files/image006.gif
The phase velocity, vp, is the rate at which any given component of the wave mixture propagates. However, since in all (?) transparent media (i.e., not in vacuum), the speed of light (radio, x-rays and other EM waves) is a function of frequency, the modulation envelope propagates at a velocity that depends on the difference between the velocities of the frequencies. This velocity is called the group velocity, vg.
Index of refraction is c/vp (where c = speed of light in vacuum, and vp is the phase velocity of a particular frequency component of the EM wave). In vacuum, the speed of light is constant and independent of frequency, vp = vg = c, and n = 1.000000...
In most ordinary materials at optical frequencies, vp of higher frequencies < vp of lower frequencies, vg < vp < c, and information propagates at vg. However, there are some materials and frequencies (commonly x-rays) for which lower frequencies travel slower than higher frequencies. In this case, the sign of the difference is reversed, resulting in a negative group velocity. There are also conditions under which either vp or vg (but not both) may be > c. For example, vp > c at optical frequencies in very thin films of gold, resulting in 0 < n < 1.
These anomalous behaviors seem to be associated with atomic resonances in the materials through which the waves are propagating. The last paper linked above seems (to the extent I understood it) to do a pretty good job of explaining anomalous vg, but I didn't find a good explanation of anomalous vp.
These anomalies do not violate relativity (v > c) or causality (v < 0), because the anomalous v doesn't actually transmit any information. The information is only transmitted at min(vp, vg). Or at the speed the photons are moving, which is always c, independent of n. Or something like that.
Yeah, I have no idea either.Admin
And anything from working at all upwards is of course better than nonfunctional.
Admin
We should add that the equations of motion are relativistically invariant under change to an inertial system that moves along the wave conductor. It follows that the product of phase velocity and group velocity is always the square of the speed of light in vacuum.
v_p * v_g = c^2
For common microwaves, the phase velocity exceeds c (and hence the group velocity is lower).
(Btw, that relationship between the relativistic invariance of the equations of motion and the above equation works component wise. That is, if the equations are invariant under movement into x direction, vpx * vgx = c^2, and if they are invariant under movement into y direction, vpy * vgy = c^2, and so on. Btw^2, only one of the speed vectors behaves like a decent vector, the other behaves rather strangely, but the vector of the inverses of the speed components behaves like a usual vector again. Btw^3, that is why B. Philbert calls that kind of vector of reciprocal speed components a "time gradient", which I btw^4 thought superfluous at first, but because it is a decent vector, now think to be the more adequate notion.)
Considering the (relative) time dilatation and therefore a frequency shift that is dependent on the relative speed of the inertial systems, we find that it is possible to find an inertial system where the group velocity is zero and therefore the group velocity is infinite, that is the system moving together with the groups. (This is the case of the standing wave at cutoff frequency.)
(If the other inertial system moves at even higher speeds, the phases travel backwards. (As the phases don't carry information or energy, that doesn't violate the special theory of relativity.) The group velocity changes sign, too.)
That v_p > c means that the index of refraction is between 0 and 1 for every frequency whose waves can travel though the wave guide, and 0 for the cutoff frequency. ( n(f) = c / v_p(f) )
It's just a pity that below the cutoff frequency the index of refraction becomes imaginary instead of negative which leads to a standing wave that attenuates exponentially with the way into the tube. Otherwise we'd have a simple (to physicists' standard) example of a negative index of refraction.
I remember having heard about group velocity and phase velocity having different directions, which would mean a negative index of refraction, but I don't remember what it was anymore.