# Case Study: Dispersion Engineering for Telecom Fibers

## Key questions:

- How can we design fibers to achieve low chromatic dispersion in the 1.5-μm spectral region? Which types of refractive index profiles may be suitable for that?
- What side effects can that optimization have, for example concerning propagation losses?
- Can we also achieve spectrally flat chromatic dispersion?

The performance of a fiber-based telecom system is often substantially influenced by effects of the chromatic dispersion of the used transmission fibers. Therefore, dispersion engineering is often applied to optimize performance. In this case study, we explore some typical methods to find designs for dispersion-shifted fibers and dispersion-flattened fibers.

In this case study, we consider the usual germanosilicate fibers, which are silica fibers where the fiber core contains some amount of germania (GeO_{2}). Further, we assume that the 1.5-μm spectral region is of interest, which is widely used in telecom because (a) the propagation losses of silica fibers are minimal there and (b) we have well working erbium-doped fiber amplifiers (EDFAs) as essential building blocks for recovering the signal strength as needed. Finally, here we limit ourselves to single-mode fibers as used for larger transmission distances, where chromatic dispersion is important.

For the modeling, we use the software RP Fiber Power, which offers a Power Form titled “Modes of a germanosilicate fiber”. Here, we can define an arbitrary germania doping profile, from which the software calculates the refractive index profile, and from that all the properties of the fiber modes, including the group velocity dispersion (GVD).

## Step-index Fibers

We begin with the simplest kind of refractive index profile. Step-index fibers have a constant refractive index within the fiber core, somewhat higher than that of the fiber cladding. So we have only two parameters to play with: the core diameter and the germania concentration. The latter determines the refractive index contrast, which in turn determines the numerical aperture (NA).

Several properties can be important for a telecom fiber, and should thus be considered:

- We want sufficiently robust guidance of light. For that, the numerical aperture should be at least around 0.1.
- We want low propagation losses. That is more easily achieved with a not too high NA, as the losses through Rayleigh scattering at fluctuations of the core–cladding boundary are sensitive to the NA.
- We should operate the fiber in the single-mode regime, and such that we get sufficiently robust guiding. In particular, we should have the mode field well confined to the core, if at all possible. Another indication is that the cut-off wavelength of the next higher mode (LP
_{11}) should not be too far away. - A reasonable value of the effective mode area is also often desired – for example, a value around 80 μm
^{2}to 100 μm^{2}is typical for telecom single-mode fibers, and ideally a dispersion-shifted fiber would have a similar value for easy low-loss splicing. Note that efficient transfer of light from one single-mode fiber to another requires a good match of the two mode fields, which usually given if the mode areas are similar. Also, smaller mode areas lead to stronger nonlinear effects, which are usually undesirable. - For this study, we mainly want to optimize the dispersion characteristics. Note that different magnitudes of group velocity dispersion (GVD) may be required in practice, depending on the system architecture.

### Standard Step-index Fiber

We first analyze Corning's SMF-28, which is a standard single-mode fiber which is widely used in telecommunications. In the mentioned Power Form, we enter the core diameter of 8.2 μm and a germania concentration of 4.5%, as this is needed to obtain the known NA of 0.14. This, however, leads to a somewhat small effective mode area of 64.4 μm^{2}; Corning specified the effective mode field diameter as 10.5 ± 0.5 μm at 1550 nm, suggesting a mode area around 87 μm^{2}. Therefore, let us assume only 3.5% GeO_{2}, leading to an NA of 0.123 and a mode area of 76 μm^{2} – still a bit low, but note that conversion from mode field diameter to mode area also depends on the exact mode shape, which we do not know.

Here is the part of the Power Form showing the inputs and the calculated mode parameters:

The resulting GVD is just slightly weaker with the lower NA: −21034 fs^{2}/m instead of −22300 fs^{2}/m. The LP_{11} cut-off wavelength (not seen in the table if not choosing a wavelength out of the single-mode regime) is 1314 nm, which fits well with Corning's specification.

In the following, we always consider the GVD in units of ps / (nm km), as is the usual way in the telecom context. (The encyclopedia article on chromatic dispersion explains the conversion.) The dispersion profile shows that the GVD of that kind rises towards longer wavelengths, getting substantially anomalous in the 1.5-μm region:

### Step-index Fiber with Small Core

Now we try what happens with a modified step-index design, having a smaller core diameter and also an increased numerical aperture such as to conserve the V number at 2.05. Specifically, we now reduce the core diameter from 8.2 μm to 4 μm and find that we need 14.7% GeO_{2} in the core; this scales with the square of the core diameter, or just linearly with the core area. That already shifts the zero dispersion wavelength <$\lambda_0$> to 1565 nm:

So this works as expected. However, there are some caveats:

- The effective mode area has been strongly reduced to 18.2 μm
^{2}. So that fiber would no longer be compatible with a standard single-mode fiber, e.g. concerning splicing. Even if you splice two such fibers together, you will probably find it difficult to achieve low splice losses with such a small core, as the position tolerance becomes rather tight. Also, the tendency for nonlinear effects would be strongly increased. - The propagation losses will be much higher than for the standard fiber. This is due to the high refractive index contrast (causing strong scattering at any irregularities) and also because of mechanical stress at the core–cladding interface.
- A substantial dispersion slope remains, which is detrimental for some applications.

The main problem is the small mode area. We can try to mitigate that by now reducing the NA and thus the V number. If we keep our core diameter of 4 μm and reduce the GeO_{2} concentration to 10% (<$V$> = 1.69), the zero dispersion wavelength becomes as large as 1668 nm. So we can actually increase the core diameter again, say to 4.28 μm, shifting <$\lambda_0$> back to 1564 nm. Now, the mode area is 25.6 μm^{2} – still rather small. The V number is 1.81, still reasonable.

Already knowing that the step-index approach does not work that well, we turn to modified profiles.

## Triangular Profile Fiber

A triangular profile, having a linear decrease in refractive index with increasing radial position within the core, is a common choice. (Strictly speaking, the literature often presents profiles where <$n^2$> rather than <$n$> is triangular, but that is a small difference, given the typically rather small index contrast.)

In our Power Form, we can conveniently enter an expression for the GeO_{2} concentration vs. radial coordinate `r`

. We just enter `c0 * (1 - (r / r_co))`

, and separately define `c0`

, the GeO_{2} concentration at the center.

We again just have two parameters to play with: `d_co`

and `c0`

. With manual tweaking, we easily find a combination which seems to work reasonably well: 5 μm core diameter and 8% peak concentration lead to <$\lambda_0$> = 1559 nm – similar as in the last example –, although the dispersion slope is increased, and the effective mode area gets rather large at 106 μm^{2} – larger than ideal.

The tweaking of the available parameters is currently cumbersome, since each time we need to check three parameters, not finding it easy to determine whether a design change overall improved or deteriorated the results. To make this substantially more convenient, we can define a figure of merit (FOM), also called a loss function, which reflects our design goal. Let us define the goal to be the following:

- zero dispersion wavelength: <$\lambda_{0\rm d}$> = 1550 nm
- effective mode area: <$A_{\rm eff,d}$> = 80 μm
^{2}

If we could reach those goals simultaneously, the FOM should get 0. Any deviations should be penalized with positive contributions to the FOM – with proper relative weights. A reasonable definition of FOM is then:

$${\rm FOM} = \sqrt{(\frac{\lambda_0 - \lambda_{0\rm d}}{\text{ 10 nm}})^2 + (\frac{A_{\rm eff} - A_{\rm eff,d}}{10 \: \mu {\rm m}^2}^2)^2}$$where we give the same penalty to a 10-nm deviation of zero dispersion wavelength as for a 10-μm^{2} deviation from the wanted mode area. In the Power Form, we define that function and use it for displaying the FOM value in the Output area. That way, we can see at a glance whether any design modification made it overall better or worse: we simply want the FOM to be reduced.

Even manual tweaking is now relatively comfortable, but automatic optimization is still much nicer. No problem! Although the Power Form does not explicitly offer optimization, we can enter the following code, to be executed before the final simulation:

```
vary d_co_ex in [2 um, 10 um], c0 in [0, 20],
for minimum of (CalcModes(); FOM())
```

Within a couple of seconds, the optimization drives the FOM down dramatically. We get quite precisely the wanted zero dispersion wavelength and mode area with a core diameter of 5.71 μm and a peak concentration of 7.6%. Here is the dispersion profile:

Such fibers have been found to work quite well [1] and are indeed widely used. Only, for this type of design, we have a mode field extending far into the cladding:

This will not give us perfectly robust guidance of LP_{01} in the 1.5-μm region – in particular, some sensitivity to micro-bend losses. There are designs fixing this problem as well, but in this case study, we don't pursue this further.

## Dispersion-flattened Fiber With W Profile

With dispersion-shifted fibers, we will still get a substantial dispersion slope. This is not good for applications requiring low dispersion over a substantial bandwidth. Therefore, *dispersion-flattened fibers* have been developed which achieve that. A prominent design type is based on a “W profile”: a high-index region in the center is surrounded by a ring with lower index, even lower than the cladding. This index depressed ring could be achieved with fluorine doping, but we can also use pure silica and use a cladding (at least some area around the ring) with some germania. We try the latter approach.

The parameters of that design are:

- <$r_1$>: radius of the high-index region
- <$c_1$>: GeO
_{2}concentration in the high-index region - <$r_{\rm co}$>: core radius (= outer radius of the ring)
- <$c_{\rm cl}$>: GeO
_{2}concentration in the cladding

In our model, we assume that the whole cladding is germania-doped, although in practice it is okay to have pure silica outside some radius which the light cannot reach.

Of course, we need a different type of figure of merit (FOM) here: no longer based on the zero dispersion wavelength, but rather a sum of squared dispersion values for a range of wavelengths. The code entered into the form:

```
FOM() := sqrt(
sum(l_t := 1500 nm to 1600 nm step 10 nm, (GVD_lm(0, 1, l_t) / (100 fs^2))^2)
+ ((A_eff_lm(0, 1, l0_d) - A_eff_d) / (10 um^2))^2)
```

The code for the optimization:

```
vary
d_co_ex in [2 um, 10 um],
r1 in [1 um, 8 um],
c1 in [0, 20],
c_GeO2_cl in [0, 5e-2],
for minimum of (SetCoreSize(); CalcModes(); FOM()),
ytol = 1e-3
```

That optimization results in rather good dispersion characteristics:

What is also nice: the mode size is reasonably well confined to the core, leading to robust guiding:

Only, the mode area is quite small (21.1 μm^{2}). Even if we give that a much stronger weight in the FOM, we cannot reach a substantially larger mode area with this design: the required dispersion is obtainable only with a small core.

There are plenty of other designs which have been tried out – for example, with a triangular structure at the center plus one or two rings around it. Such designs have more parameters to play with, but this also makes it more difficult to find a reasonable parameter set, even as a starting point of a numerical optimization. One could employ a Monte-Carlo technique with that, letting the computer try out thousands of randomly chosen starting points. Fabrication tolerances are also an important aspect which should be included in the optimization. But we don't go that far in this case study, even though that would definitely be possible with the RP Fiber Power software.

## Conclusions

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

You can learn various things from this case study:

- Simple step-index fiber designs with small core and high NA can provide low chromatic dispersion in the 1.5-μm region. However, such dispersion-shifted fibers exhibit significantly increased propagation losses and a substantial dispersion slope.
- Fibers with a triangular refractive index profile reach similar chromatic dispersion (still with a substantial dispersion slope), but with lower propagation losses.
- There are more sophisticated designs for dispersion-flattened fibers, having a much lower dispersion slope, thus offering low dispersion in a much wider wavelength range. These are more difficult to design and typically require tighter fabrication tolerances.

The case study also shows that numerical simulations are the only way to find out how such fiber designs really work. Without such tools, it is hard to estimate what dispersion characteristics will result from a specific fiber design, and even harder to find such a design. A highly flexible software such as RP Fiber Power is indispensable for such work.

## More to Learn

Encyclopedia articles:

### Bibliography

[1] | B. J. Ainsley and C. R. Day, “A review of single-mode fibers with modified dispersion characteristics”, J. Lightwave Technol. LT-4 (8), 967 (1984) |

### Video

Here, you can see how the simulations for this case study were done with our software RP Fiber Power:

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