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Case Study: Erbium-doped Fiber Amplifier for Rectangular Nanosecond Pulses

Design Goal

We want to design an erbium-doped fiber amplifier which can amplify a weak 1 nJ input pulse (3 ns, Gaussian spectrum with 5 nm bandwidth) at a wavelength of 1550 nm. We need an output energy of at least 10 μJ, and the signal output pulse should have a close to rectangular temporal shape.

We can anticipate that for this level of output pulse energy, there will be significant gain saturation during the pulse amplification, which leads to a distortion of the temporal pulse shape. We will thus consider different methods for coping with that challenge.

Initial Attempt

We first simply try with rectangularly shaped signal input pulses. In the Power Form “Fiber amplifier for continuous-wave signals”, we easily enter the signal input parameters and set some details of the amplifier:

Power Form inputs for an Er-doped fiber amplifier for pulses.
Figure 1: Part of the Power Form, where we specify details of the input signals and the amplifier stages. The input pulse is defined in both the time and frequency domain.

We initially choose from our long list of fiber data the commercial fiber “Liekki Er80-4-125” which has a core diameter of 4 μm. We assume that before pulse amplification the amplifier is pumped for a long enough time with as much as 1 W at 980 nm. In 1.3 m of the fiber, we can absorb most of the pump power. But that doesn't work for our signal: we get less than 9 μJ out, and that pulse is strongly distorted, far from the intended rectangular output:

Rectangular pulse amplified in a fiber with a 4 μm core diameter.
Figure 2: The time-dependent signal output power with a rectangular temporal shape. The pulse is amplified in an erbium-doped fiber with a core diameter of 4 μm. Gain saturation is related to the drop of the average Er excitation level (gray curve).

You see that the leading edge of the pulse (occurring at −1.5 ns) experiences a substantially higher gain than the trailing edge (at +1.5 ns), because the erbium excitation level drops substantially during the pulse due to energy extraction by the pulse. (One may be surprised that it does not drop more strongly, but note that the curve shows the erbium excitation averaged over the whole fiber length, and strong saturation occurs only near the output end, as the pulse energy is low before the pulse gets there.) Another problem is that we are producing substantial amplified spontaneous emission (ASE), which is wasting much of the used pump power.

So we see that for this substantial output pulse energy, we should use a fiber with larger mode area, resulting in a higher saturation energy. We can simply pick another fiber having 8 μm core diameter, which (for the same glass composition) has a 4 times higher saturation energy. Indeed, that already works much better; with the same fiber length and pump power, we get 14.5 μJ (more than needed). We can now actually reduce the pump power to 500 mW and still get 10.9 μJ; the pulse distortion is also significantly reduced:

Rectangular pulse amplified in a fiber with a 8 μm core diameter.
Figure 3: Same as before for a fiber with two times higher core diameter.

Still, this is far from the desired rectangular pulse. We will need to work on this.

In principle, we might try with a fiber having a far larger mode area, minimizing the effect of gain saturation. However, that approach would have its own serious limitations, such as requiring more pump power and making it more difficult to get single-mode operation for best beam quality. (You would hardly find such a fiber on the market.) Therefore, we decide for a different solution:

Pre-compensated Temporal Pulse Shape

The solution is to manipulate the input pulse shape in order to compensate for the shape distortion caused by the gain saturation. The idea is to compensate the lower amplifier gain of the trailing part by increasing the input power. We first try this with the simplest approach: a linear ramp of the signal input power. That is easy to implement with the Power Form:

  • Enable a second input pulse (and disable the original pulse), so that we can easily switch between the two pulses to compare.
  • Select the user-defined temporal pulse shape and enter a time-dependent expression defining the pulse shape (here, a linear ramp between −1.5 ns and +1.5 ns, Figure 4).
Specifying a user-defined linear ramp pulse shape along with a plot of pulse powers vs time.
Figure 4: It is easy to define a linear ramp input pulse in the Power Form. We have optimized the slope for best performance.
  • Activate the “input signal pulse” diagram to verify that the intended shape is obtained:
Specifying a user-defined linear ramp pulse shape along with a plot of pulse powers vs time.
Figure 5: The desired input pulse.

This results in a substantially improved output pulse shape, with an output energy of 10.8 μJ:

Output after an linearly ramping pulse is amplified in a erbium-doped fiber amplifier.
Figure 6: The output pulse obtained with a linear ramp of the input pulse.

However, the pulse shape is still not perfect, as we still see some bow shape. In order to get an even better output pulse shape, we try an exponential ramp of the input signal power. For that, we make a third input pulse, where the time-dependent power is given as if abs(t) < 1.5 ns then exp(t/ns * 0.347):

Specifying a user-defined exponential ramp pulse shape along with a plot of pulse powers vs time.
Figure 7: Input pulse with an exponential ramp.

The result now indeed very good:

Output after an exponentially ramping pulse is amplified in a erbium-doped fiber amplifier.
Figure 8: The output pulse obtained with an exponential ramp of the input pulse.

The exponential ramp nearly perfectly compensates for the effect of gain saturation (Figure 7), resulting in a nearly rectangular temporal pulse shape with an output energy of 10.8 μJ, achieving our goal.

Why actually does an exponential ramp work so much better than a linear one? This is because for a constant output power during the pulse, we get an approximately linearly decay of the erbium excitation and the exponential gain coefficient, which translates into an exponential decay of the power amplification factor.

Note that such temporal power ramps can be easily realized with a suitable electronic driver for a laser diode. The output power of a laser diode would closely follow the provided current ramp.

Repetitive Operation

Previously, we only considered amplification of a single pulse, assuming steady-state conditions of the amplifier before the pulse comes. But we can also investigate what happens in repetitive operation – say, with a pulse repetition rate of 5 kHz, corresponding to a pulse period of 200 μs. As that is well below the upper-state lifetime of erbium (here: 9 ms), one will probably expect that the amplifier can by far not recover its fully pumped state between two consecutive pulses. On the other hand, an average output power of 5 kHz · 10.8 μJ = 54 mW is far below the applied pump power of 500 mW. So from that viewpoint, it could work. But does it really?

Let's test it. Simply check “Repetitive operation” in the form, with 8 pumping/amplification cycles, and activate the diagram for the pumping dynamics over all cycles. Further, we now assume that the amplifier is initially unpumped. The result:

pumping dynamics over 8 amplification cycles
Figure 9: The pumping dynamics over the first eight amplification cycles, starting with an unpumped amplifier.

In the first few cycles, we hardly get any pulse energy out because the amplifier gain is still very small. However, once the steady state has been reached, the recovery of gain after a pulse is rather quick – it largely happens during only 100 μs, i.e., in a time far shorter than the upper-state lifetime. That is possible essentially because we are pumping at a power level far above the pump saturation energy. In this regime, which is often encountered for fiber lasers and amplifiers, certain common rules (e.g., relaxation towards the steady state with a time constant equal to the upper-state lifetime) do no longer hold.

Conclusions

You can learn various things from this demonstration:

  • With such simulations, you will often discover that the system won't work just as you expected. Also, it may work fine in a regime where you didn't expect it to work. The simulations make it easy for you to test your expectations and to correct them if necessary; that way, you develop a solid understanding of such devices, as you could never do solely by reading textbooks and articles. By the way, we will also be happy to help you explaining what you found!
  • With such a model, you can quickly and efficiently find a working device design – often, within very few hours. Ordering the parts, building the amplifier, testing it and trying to analyze and solve the issues there (possibly after ordering further parts) would be far more tedious, costly and time-consuming!
  • The modeling is quite straightforward with our software: simply enter the envisaged system parameters and inspect the resulting numerical and graphical outputs. You can rapidly explore parameters and configurations. The Power Form gives you plenty of freedom, e.g. for using user-defined inputs, multiple stages, multiple signals and pump sources, etc., besides offering many different diagrams. And if you encounter any difficulties, we will help you within our diligent support!