# Case Study: Telecom Fiber with Parabolic Index Profile

## Key questions:

- To which extent does a parabolic index profile minimize intermodal dispersion? Does that work for all guided modes?
- How precisely does the parabolic shape need to be realized? What is the effect of small deviations?

We consider a telecom fiber of multimode type with a parabolic refractive index profile. It is known that this profile is suitable for reaching rather weak intermodal dispersion, i.e., weak differences in group velocity between the guided modes. This governs the differences in transit times between different modes, which typically limit the possible bit rate. With such a fiber design, one can exploit advantages of multimode fiber systems (e.g., cheaper transmitters) while still achieving a reasonably high bit rate.

With some numerical simulations, we will explore how strong the achieved reduction in intermodal dispersion is and how sensitive it is to slight changes in the doping profile.

We consider a germanosilicate fiber, having a fiber core with a mixture of GeO_{2} (germanate) and SiO_{2} (silica), while the fiber cladding consists of pure SiO_{2}. Instead of directly assuming a refractive index profile, we start with the GeO_{2} concentration profile, from which the index profile is then calculated. (That way, we have the index profile with its full frequency dependence.)
For the simulation, we use the software RP Fiber Power, which offers the Power Form “Mode Properties of a Germanosilicate Fiber”.

## Analysis of the Nominal Fiber Design

We assume a core diameter of 50 μm and start with a parabolic concentration profile of the fiber core with a peak GeO_{2} concentration of 7%. The cladding is undoped, so that we got a constant refractive index there. The operation wavelength is 1550 nm (telecom C band). We enter all this into the form:

We see that the maximum doping concentration corresponds to a reasonable value of the numerical aperture of 0.174. (That quantity is really meaningful only for step-index fibers, but can still be used to get a feeling for the index contrast.)

We get 22 guided modes, or 39 when separately counting different mode orientations.

The table already shows that the group indices (`n_g`

) are mostly quite similar, but for a better overview, we create a diagram:

Most modes have quite similar values of the group index, but just a few of them have far lower values. Using the table in the form, we can easily identify them: the lowest two values are for LP_{05} and LP_{24} (at the bottom of the table, visible only after scrolling). The diagram also shows that these modes are relatively close to their cut-off wavelengths: they would cease to exist for slightly longer wavelengths. So this behavior is not surprising: such modes tend to have intensity profiles which reach substantially into the cladding, where of course we depart from the parabolic shape of the profile.

How problematic is that aspect for telecom applications? Potentially, it could really spoil the performance if at least a few of the modes have transit times differing a lot from those of all the others. However, one may launch the light such that highest-order modes do not have significant optical powers so that they become irrelevant.

We quickly compare this result with that for a step-index fiber with the same core diameter and a constant GeO_{2} concentration of 3.35% in the core, which leads to about the same number of guided modes. As expected, here we get a much larger variation of group indices:

Here, not even the lowest-order modes have similar group indices. So we would definitely get far stronger intermodal dispersion. Let's quickly estimate how much that is in practice. For example, a difference in group index of <$\delta n_{\rm g}$> = 0.003 causes a change in transit time of <$\delta n_{\rm g} \: L / c$> = 10 ns through 1 km of fiber. That would already be fairly disturbing when trying to reach a bit rate of 1 Gbit/s, and is far worse than what group velocity dispersion in a single-mode fiber would do, e.g. for 20 nm difference in wavelength. Therefore, the improvement achieved with the parabolic doping profile is really important.

## Modification of the Doping Profile

We now investigate the effects of some deviation from the parabolic doping profile – namely, adding a 4^{th} order term. The form allows us to conveniently enter some additional definitions; we use the following:

```
c4 := 0.1
c2 := 1 - c4
c(r) := 7 * (1 - c2 * (r / r_co)^2 - c4 * (r / r_co)^4)
```

Then we can simply reference `c(r)`

in the field for defining the doping profile. Our function has been defined such that for <$r = r_{\rm co}$> we get zero concentration, i.e., a continuous transition to the undoped cladding, and for <$r = 0$> (edge of the core) the same peak concentration as before. The parameter `c4`

can be used for modifying the strength of deviation from the parabolic profile. We first look at the index profile and mode functions as before:

We see that the index profile got slightly modified. (Compare, for example, the refractive index for <$r$> = 10 μm with the situation of Figure 2.) Also, we got a few more guided modes, since the radial decay of refractive index is now initially a bit slower.

Now let's check the resulting group indices:

The variation of group indices is now substantially larger than for the parabolic profile, but still far smaller than for the step-index profile.

## Conclusions

The RP Fiber Power software is an invaluable tool for such work – very powerful and at the same time pretty easy to use!

- As expected, the fiber with a parabolic index profile exhibits much reduced intermodal dispersion (comparing with a step-index fiber), although for the highest-order modes this is not true.
- We can easily check how strong the impact of any deviations from the parabolic profile would be.

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