# Light in fibre optic cables

Imagine a fibre optic cable stretching from London to New York (5571 km) carrying a data signal

This data signal is split into two parts, red and blue light

Both signals start travelling down the wire at the same time. Would they reach the other end at the same time? If not, what would be the distance and time delay between the two signals at the other end? (what is the equation to work this out?)

This may seem an simple question, but a truly correct answer is actually a bit more complicated.

This is because there are different speeds and the question is which one is relevant. There is

• phase velocity (this is the "ordinary" speed of light, and the one that is described with the refractive index). BUT this is not the speed at which information travels, or even a pulse.
• group velocity. This is the speed a pulse (more specific its envelope) travels. This is much closer to the speed of information
• front velocity. This is the true speed at which information is traveling (and this cannot be higher than speed of light in vaccuum). This is the speed you are asking for, and in the case of an optical fiber it is very close to the group velocity (which is an easier to work with term than the front velocity).

So the answer, with the refractive index is wrong. It is the group index which is relevant here.

The group index of silica can be found here (http://www.rp-photonics.com/group_index.html).

At 400nm (blue) it is 1.515 and at red 750nm) 1.475. So the time a pulse travels is given by $$L/c_0 \times n_g$$

which is for L=5571km, 28.15 ms for blue vs. 27.41 for red. So red beats blue.

EDIT: I forgot something. Since a fiber is a waveguide, the group velocity is probably still be affected by the special waveguide structure. Graphically this can be understood by imagining, that the light bounces back and forth in the fiber and hence travels a longer path, than given from the physical length of the fiber. This effect depends on the diameter of the fiber core, and is also different for different wavelengths. So the true answer is specific to a special fiber. The above answer is just neglecting this waveguide effects.

• thanks Andreas, a beautiful answer. This might mean a bank would prefer a red wavelength transmission, to a blue wavelength transmission. I wonder if they get to choose. Maybe this is why important telephones are red ;) – pluke Oct 2 '13 at 19:37
• Interesting point ;). But I am quite sure that long distance telecommunication is exclusively done at 1.55um wavelength, because there the attenuation is lowest. So there is no choice for banks ... – Andreas H. Oct 2 '13 at 21:56
• Moreover the fibre refractive index profile is chosen to shift the zero dispersion point (i.e. $\mathrm{d}_\lambda v_g = 0$, where $v_g$ is the groupspeed) from around 1320nm (where it is naturally) to 1550nm. So the waveguide structure is very important here. The refractive index profiles are called "depressed cladding fibres". – WetSavannaAnimal Oct 3 '13 at 13:23

light of different frequencies travel at different speeds in matter. this leads to the light reaching the destination at different times. you need to obtain the refractive indices of red and blue light in fibre.

velocity of your light is $v = c/n$, where c is the velocity of light in vacuum, and n is your specific refractive index.

• would blue beat red? or vice versa? EDIT: it appears that Red light would win, now to work out by how much! answers.yahoo.com/question/index?qid=20070927031643AAvXyG8 – pluke Oct 2 '13 at 16:44
• This is not correct, the group index is relevant here. the OP asks for the speed of information, which is the front velocity (that is close to group velocity) – Andreas H. Oct 2 '13 at 16:55
• Well as a practical matter, neither a blue wavelength, nor a red wavelength have any chance at all of travelling from London to New York on a fiber optic cable, without repeater stations. Practical fiber optic long distance cables operate at infra-red wavelengths, often around 1,500 nm wavelength. , so you might want to rethink your question; perhaps over some much shorter route, like a few meters at most. – user26165 Oct 3 '13 at 18:34
• I'm aware of repeaters and the question was purely hypothetical – pluke Oct 4 '13 at 9:17