# Case Study: Erbium-doped Fiber Amplifier for Multiple Signals

## Key questions:

- What influence does the fiber length have for the variation of amplifier gain over some wavelength range? What role does signal reabsorption play here?
- How can we apply spectral filtering at the input or output end, and how to those methods compare in terms of effectiveness and efficiency?

## Design Goal

We want to design an erbium-doped fiber amplifier which can amplify ten weak (5 μW) continuous-wave input signals with evenly spaced wavelengths from 1530 nm to 1548 nm. We want a total output signal power of 50 mW in each signal (500 mW total), implying 40 dB gain). Importantly, the signal outputs should all have about the same optical power – we care about gain equalization. This is relevant, for example, in the context of wavelength division multiplexing.

With the given range of wavelengths, we have to expect significant variations of amplifier gain. To equalize the signal output powers, we will need to consider some special optimization. We will investigate the resulting noise figures for different equalization methods.

## Initial Attempt

We first try with a single amplifier stage.

For the simulations, we use the RP Fiber Power software. In the Power Form “Fiber amplifier for continuous-wave signals”, we easily enter the parameters of signal input and amplifier:

Using the tab structure for input signals, we activate all 10 available signals, specifying a wavelength and an input power (5 μW) for each. In principle, we could also define a single signal with 10 equidistant spectral lines, but that way we could get the noise figure only for the central line, and here we want to know all of them.

We choose the commercial fiber “Er-IXF-EDF-FGL” (as one of many for which we have data) and have to decide on a fiber length and the pump power at 980 nm. Making use of one of the convenient variation diagrams (see Figure 2), we determine that a fiber length of 3 m with 1 W of pump power (applied in forward direction) gives a total signal output power of 535 mW (total gain of 40.3 dB), meeting our requirements:

When we use too long a fiber, the efficiency is degraded due to reabsorption of the signal power by non-excited erbium ions.

Unfortunately, when we examine the powers of the individual signals (see Figure 3), we see that the longer wavelength signals are amplified substantially less than those at the shorter wavelengths:

However, we can improve this simply by using a somewhat longer fiber. This is because the above-mentioned signal reabsorption mostly occurs at the shorter wavelengths. Note that the reabsorbed signal power will not be entirely lost, but rather contributes to the signal amplification at longer wavelengths. Therefore, we won't lose much in terms of efficiency.

To investigate the optimum fiber length, we can again use the variation diagram for the fiber length, but now showing the signal powers individually (see Figure 4, selecting the `show individual signals`

option). Here, we can see that a fiber length of 4.5 m leads to a substantially lower variation of signal output powers:

While the short-wavelength signals are amplified a lot more than the others at the beginning, their powers soon reach a peak and then drop in a region where the longer-wavelength signals are still amplified substantially: they experience much less reabsorption.

As a result, we indeed see that although the total power is optimal at around 3 m (see Figure 2), a fiber length of 4.5 m gives a better output power distribution over the signals (see Figure 4), while still achieving a total output power of 514 mW:

## Filtering of signals

To achieve even better power uniformity, we can now apply filtering by spectrally dependent attenuation – either at the output or at the input side.

### Output Filtering

We first consider filtering at the amplifier output, i.e., applying a wavelength-dependent loss there. That can be achieved with various types of optical filters – for example, involving spatial separations of wavelength components through a pair of diffraction gratings and a spatially dependent loss obtained from a liquid crystal modulator.

In the form, we could just manually enter the required output loss values for each signal. However, it is more convenient to use a little script code to get the loss for each signal calculated automatically from the output power without loss. This is easy, not requiring any iterations, since the output losses have no back-action on the amplifier.

For achieving the required output power of 50 mW in each signal, we now need to substantially increase the pump power to 1.82 W. While the weakest signal at 1538 nm needs no attenuation, some attenuation is needed for shorter and longer wavelengths:

As expected, we get the perfectly even signal power distribution, but we need substantially more pump power than before. Substantial power is lost by first amplifying some of the signals to high powers and then discarding the excess power.

The noise figures (between 3.34 dB and 3.57 dB) are nearly unaffected by the attenuation at the output: attenuation of already amplified signals has little impact on the resulting amplifier noise.

For obtaining a higher amplifier efficiency, we can use a modified method of power equalization:

### Input Filtering

The second approach is to attenuate the power of the signals at the input of the amplifier. This is more economical in terms of power: attenuating weak input signals means a much lower power loss, since the attenuated signals extract less power from the amplifier. It turns out that 1.01 W of pump power are just sufficient in this case.

With this technique, it is more difficult to find the right parameter set, as modifying one signal input attenuation affects not only the output power of that signal, but also that of all other signals because of the modified gain saturation. Therefore, we need to iteratively calculate the required attenuation. This could be done with manual tweaking in several rounds, but we can also automate that with a few lines of script code entered into the form. In each iteration, it simply readjusts the input attenuation values for all signals according to the last obtained output powers. It turns out that we need more than 10 iterations to get accurate results, as after each one the gain of the fiber rises somewhat. However, we can e.g. do 20 iterations in a fraction of a second.

The results:

So with input filtering, we got about the same result as before, but spending only 1.01 W instead of 1.82 W pump power. On the other hand, the noise figure values are now significantly higher than before: between 3.67 dB and 6.11 dB, while before we had between 3.34 dB and 3.57 dB. Particularly, the shortest and longest wavelength signals, which need most attenuation, take a strong hit on their noise figures.

The reason why attenuation affects the noise figure can be understood with a simple picture, considering a light beam as a sequence of photons. Attenuation means that some of those photons are randomly removed; that randomness implies the addition of quantum noise. That effect is far stronger for attenuation at the input, since there the photon flux is much weaker, so that the statistical effect of removing some of the photons is larger. Further, the influence of subsequent amplification is then stronger.

So we see that there is a trade-off between power efficiency and signal noise. If pump power and efficiency are important, input filtering is clearly the best choice, but that is quite detrimental concerning the noise performance.

## Dual-stage Amplifier

A trade-off between efficiency and noise performance can be achieved by using a dual-stage amplifier system, where the attenuation is done between the stages. With our Power Form, this is again simple to try out: we just use output losses of the first stage and configure a second amplifier stage without further attenuation. We use only 1.5 m of fiber for the first stage, pumped with 500 mW, and 3 m of fiber for the second stage, pumped with 730 mW, and achieve about the same signal power outputs as before, where the noise figures are between 5.01 dB and 7.38 dB.

We have chosen the same total fiber length as before in order to get the right balance of gain between short and long wavelengths. It turns out that this leads to quite incomplete pump absorption in both stages, and as a result we need 500 mW + 730 mW = 1.23 W pump power total – a bit more than for one amplifier stage with input attenuation. However, after the first stage we have 0.3 W pump power left, and if we manage to feed that into the second stage, rather than wasting it, we need only 0.93 W total. So indeed we get a better power efficiency combined with noise powers which are in between those for the other two solutions. Of course, the higher complexity and component cost would be a significant disadvantage.

One could try to further optimize the noise performance, e.g. by applying more pump power to the first amplifier stage.

In principle, an experienced person could have at least qualitatively anticipated all relevant aspects. However, only a full system simulation gives certainty on how things work out, and that nothing important has been overlooked.

## 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 study:

- Signals with substantially different wavelength will generally have substantially different amplifier gains. A rough equalization may be achieved by optimizing the fiber length.
- Fine tuning can be done with wavelength-dependent attenuation – either at the input or at the output. Both approaches have their strengths and disadvantages.
- With a two-stage amplifier design, one may get a higher performance.

With a suitable simulation software, you can easily analyze such aspects and optimize your amplifier design.

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