## Abstract

Gain dynamics and refractive index changes in fiber amplifiers are important in many areas. For example, the knowledge of the frequency responses for seed and pump power modulation are required to actively stabilize low noise fiber amplifiers. Slow and fast light via coherent population oscillations rely on the change of group index to delay or advance pulses, and refractive index changes in fiber amplifiers are a possible explanation for mode fluctuations in high power fiber amplifiers. Here, we analyze the frequency dependent influence of seed and pump power modulation on the fiber amplifier output power and the refractive index. We explain the observed power and refractive index modulation with an analytic model originally developed for telecom amplifiers and discuss a further simplification of the model.

© 2012 Optical Society of America

## 1. Introduction

Fiber amplifiers are a promising way to realize the high power single mode single frequency laser sources required for 3^{rd} generation gravitational wave detectors (GWD). To achieve the required power and phase noise levels for GWD, active stabilization will be necessary. For instance, the output power can be stabilized with an active feedback loop to the pump or seed power. For this active stabilization, a good understanding of the temporal dynamics in the fiber amplifier is required.

Power fluctuations also change the refractive index of the fiber due to temperature effects and the wavelength dependent gain, which couples via Kramers-Kronig-Relations (KKR) [1]. These refractive index changes are a possible explanation for mode fluctuations in high power amplifiers [2, 3] and can be exploited by using fiber amplifiers as phase shifters [1, 4, 5].

The influence of seed and pump power modulation on the output power has been investigated experimentally and theoretically [6, 7]. In a different context, this effect is also called slow and fast light via coherent population oscillations [8]. In this context, the idea is to change the group refractive index ${n}_{g}=n+\omega \frac{dn}{d\omega}$ by controlling the gain to delay or advance pulses. Although the nomenclature is different, the effects are essentially the same.

On the other hand, the refractive index change Δ*n* and corresponding shifts of the carrier phase have, to the best of our knowledge, only been investigated in time domain. From these time domain measurements [1, 4, 5], it is known that the optical phase shift in most rare earth doped active fibers is a result of the KKR and temperature contributions. A large contribution to the refractive index change is caused by highly absorbing transitions in the ultraviolet. This contribution to the refractive index is still significant in the infrared and results in an almost wavelength independent shift. This can be utilized using for example an ytterbium doped fiber as a phase shifter at 1550 nm or an erbium doped fiber as a phase shifter at 1064 nm [5, 9]. Also, the refractive index change can be investigated without amplifier gain and group velocity effects.

We will show that all of these processes can be accurately modeled with rate equations using erbium and ytterbium doped fiber amplifiers as representative systems. Furthermore, we will show how to estimate the relevant properties from experimentally accessible parameters and discuss the underlying physical processes and possible pitfalls for which the simple approximations are no longer valid. We will begin with the effect of pump and seed power modulation on output power and then continue with their effect on unabsorbed pump power and the optical phase of the signal beam.

## 2. Influence of pump and seed power modulation on output power

In fiber amplifiers a change of seed and/or pump power eventually changes the population of the upper laser level *N*_{2}, which in turn modifies the output power. The magnitude and the delay is determined by the rate equations. Plotting gain as a function of pump or seed power modulation reveals a transfer function with a low pass behavior for pump modulation and a damped high pass for seed modulation, as shown in Fig. 1 [6, 7]. Only, if the modulation frequency is lower than the corner frequency *ω*_{eff}, the population of the upper laser level can adapt: therefore pump fluctuations couple to output power below the corner frequency but not above. On the other hand increased seed power means less amplifier gain if the population adapts to the new situation, which is why relative seed fluctuations are suppressed at low frequencies. Because the underlying physical process – the population changes of the upper laser level – is identical for seed and pump modulation, the respective corner frequencies are identical. Therefore the two important properties are the corner frequency and for seed modulation the modulation suppression factor at low frequencies.

In principle, both time domain transients and frequency responses can be simulated numerically. Such numerical simulations can yield accurate results, if the required parameters are known. However, they are unsuitable to analyze the influence of specific system parameters which is required for design guidelines. For this, an analytic model as the one developed for telecom amplifiers by Novak and Moesle [6] is much better suited. Although this model does not include effects like amplified spontaneous emission (ASE) or stimulated Brillouin scattering (SBS), it is a very useful tool to understand the influence of the critical parameters. Since we are interested in a practical model, we rely on this analytical approach and will further simplify it for our specific cases whenever possible.

The corner frequency derived by Novak [6] is

Here,*τ*is the fluorescence lifetime and ${P}_{s}^{0}(L)$ and ${P}_{p}^{0}(L)$ are the average signal and pump power at the end of the fiber given in number of photons per second. The coupling factors

*B*= Γ

_{s}*(*

_{s}*σ*

_{12}+

*σ*

_{21})/

*A*and

*B*= Γ

_{p}

_{p}σ_{13}/

*A*depend on the overlap with the doped regions Γ, the signal cross sections for absorption and emission

*σ*

_{12},

*σ*

_{21}, the pump light absorption cross section

*σ*

_{13}, and the mode area

*A*. For in-band pumping, it is necessary to include the reemission of pump light

*σ*

_{31}in

*B*.

_{p}Since power scaling of single frequency amplifiers is usually achieved by scaling the core size to increase the SBS threshold, the corner frequency will be more or less independent of the design output power.

In most continuous wave fiber amplifiers the term
${P}_{s}^{0}(L){B}_{s}$ is much larger than the others. For a typical erbium doped fiber amplifier (EDFA), one can estimate the corner frequency *f*_{eff} = *ω*_{eff}/2*π* to be in the range of 1–10 kHz. Compared to a fluorescence lifetime in the range of 10 ms, *f*_{eff} = 1/(2*πτ*) ≈ 16 Hz, this is a very large value. So, unless the ion lifetime is very small, which is not the case for erbium, or the fiber does not absorb a significant amount of pump light (which can be necessary in some cases, for example to increase the SBS threshold),
${\omega}_{\text{eff}}\approx {P}_{s}^{0}(L){B}_{s}$ is a good estimation of the corner frequency.

#### 2.1. Signal power modulation

To completely model the seed modulation, knowledge of the low frequency suppression is required. In an ideal amplifier with a seed power significantly larger than the saturation power, one would expect fast modulations to be amplified in the same way as the average power, because the inversion cannot follow in time. On the other hand low frequencies should not be amplified, as the inversion adapts to the changing power levels and decreases the amplification factor for higher seed power levels and vice versa.

To investigate whether this intuitive picture is justified, we start with the expressions for signal modulation as derived by Novak et al. [6]

*m*is the signal modulation index,

_{s}*θ*the phase shift and

_{s}*ω*the modulation frequency defined via ${P}_{s}={P}_{s}^{0}(1+{m}_{s}\cdot \text{sin}(\omega t))$. For high frequencies

*m*′

*=*

_{s}*m*, which is true for all amplifier parameters. At

_{s}*ω*= 0 and using Eq. (1) this can be rewritten to

As long as the seed power is large compared to the saturation power and the unabsorbed pump power (second and third term in Eq. (5)) the suppression is indeed the inverse gain. It is worth noting that fiber parameters such as *B _{p}*,

*B*and

_{s}*τ*are only included in the second and third term and therefore not required for this first estimate. The second term is the low gain approximation of amplifier saturation power. Therefore, ${m}_{s}^{\prime}\approx {m}_{s}{P}_{s}^{0}(0)/{P}_{s}^{0}(L)$ is only a good approximation, as long as the seed power is larger than the saturation power. More seed power decreases the amount of ions decaying spontaneously. It also reduces the amount of ASE, which decreases the suppression just like spontaneous emission [10].

According to the third term in Eq. (5), unabsorbed pump power has to be considered as well. Increasing the seed power cannot increase the amount of absorbed pump light, if the pump light is already completely absorbed. On the other hand, if there is a large amount of unabsorbed pump light, a small change in seed power at low frequencies changes how much pump light is absorbed. Therefore, more seed power causes more pump power to be absorbed and the gain for low frequencies increases, resulting in a lower suppression.

The effect is the same for erbium and ytterbium. However, in an ytterbium doped fiber the cross sections are about one order of magnitude larger at 976 nm than at 1064 nm and therefore *B _{p}* ≫

*B*. Thus, the pump contribution in Eq. (5), which is weighted by the cross sections, becomes significant at lower unabsorbed pump power levels. The same is also true for the saturation power, as the lifetime of ytterbium ions is one order of magnitude smaller than the lifetime of erbium ions. Therefore, similar power levels can lead to very different behavior.

_{s}High average seed power compared to saturation power of the amplifier and to unabsorbed pump power reduces the impact of both the second and third term. Therefore, increasing seed power and operating in the saturated regime is a suitable method to reduce the impact of the second and third term, which makes the transfer function simple.

#### 2.2. Pump power modulation

As the next step we will discuss pump power modulation, which is a low pass and can be expressed as [6]:

Novak et al. compared the result at *ω* = 0 to the seed power modulation to show that *m*′* _{p}* reaches its maximum at approximately the seed power, where

*m*′

*has a minimum, as long as spontaneous emission is small. At*

_{s}*ω*= 0 and because of energy conservation

*m*′

*≈*

_{p}*m*. At the same time the suppression for seed modulation is large unless the amplifier is unsaturated. On the other hand, if $\frac{{P}_{s}^{0}(0)}{{P}_{s}^{0}(L)}\approx 1$, pump modulation is suppressed, while seed modulation is not. More generally speaking, if seed modulation is suppressed, pump modulation will increase and vice versa.

_{p}#### 2.3. Unabsorbed pump power

Finally, we will discuss the transfer functions of signal and pump power modulation on the unabsorbed pump light, which corresponds to the slow light case. Although these equations were not discussed by Novak et al. [6], we will just state the results here, because the derivation is analogous to the signal transfer functions (2, 3 and 6, 7)

*m*″

*is the modulation index for seed power modulation and*

_{s}*m*″

*is the modulation index for pump power modulation and*

_{p}*θ*″

*is the corresponding phase. Therefore, unabsorbed pump light behaves the same way signal light does, but with the roles of seed and pump power modulation exchanged. However, due to the absorption of the pump power $\left({B}_{p}^{0}(0)>{B}_{p}^{0}(L)\right)$, the transfer function from pump light to unabsorbed pump light is not a damped high-, but a damped low pass.*

_{s/p}## 3. Experimental results – power modulation

To validate our approximations experimentally we built two all–fiber fiber amplifiers as depicted in Fig. 2. We will discuss the results achieved with an erbium doped fiber amplifier (EDFA) co–pumped at 1480 nm first. The EDFA was seeded by a 1550 nm distributed feedback (DFB) diode with 46 mW average power. This is well above the 0.2 mW saturation power of this fiber. For simplicity, we chose to directly modulate the seed power via current modulation. Since the corner frequency rarely surpasses 10 kHz, this technique is sufficient. During the measurement the modulation index was kept as small as possible to avoid nonlinearities.

The transfer functions for varying pump power levels are shown in Fig. 3. The characteristics are as we expected, i.e. the low frequency modulations were not amplified regardless of the pump power level, while the high frequency modulation was amplified and its amplification increased with pump power level. One can also see that the corner frequency shifted to higher frequencies as we increased the pump power as predicted by Eq. (1). The phase shift (Fig. 3, bottom) is directly related to the slope of the amplification because of causality. The largest phase shift (or fractional pulse advancement when thinking of it as fast light) is possible for the largest slope. This also corresponds to the highest suppression of low frequency modulations. Since an undamped high pass causes a 90° phase shift, this is the principal limit for the achievable phase shift with a single amplifier. Since the approximation
${m}_{s}^{\prime}\approx {m}_{s}{P}_{s}^{0}(0)/{P}_{s}^{0}(L)$ is valid for this amplifier, the accuracy of *ω*_{eff} will mostly depend on the accuracy of the parameter *B _{s}*. The parameters we used are shown in Table 1.

To investigate the influence of the saturation power we used an ytterbium amplifier with comparable parameters. In principle, the setup was the same as in Fig. 2, but now using a 976 nm pump diode, a 1064 nm seed operating at 10 mW and 30 mW average power and an ytterbium doped fiber. Due to the one order of magnitude larger saturation power of ytterbium, it is better suited to discuss the regime besides the simple saturated amplifier. In fact, the seed power is in the range of the saturation power (Table 1). The resulting transfer functions shown in Fig. 4 reveal why the different saturation power levels have to be considered. The low frequency modulations are amplified as well and the amplification increases with pump power.

According to Eq. (5) this increase could only be caused by the third term, because the second term is a constant. However, this was not the case, as there was no significant amount of unabsorbed pump power at low pump power. Thus, one has to keep the limitations of the approximation in mind, which relies on ${P}_{s}^{0}(L)$ being significantly larger than the saturation power.

However, if ${P}_{s}^{0}(0)$ is at the level of the saturation power and the gain is small as well, the approximation does not hold and overestimates the saturation contribution. Fortunately, this will only happen for very low field intensities in the fiber, which are very unlikely in any realistic cw amplifier design. On the other hand, the increase at higher pump power levels was caused by unabsorbed pump light, i.e. term 3 in Eq. (5).

The different contributions for low frequency gain at 10 mW and 30 mW seed power are shown in Fig. 5. The measured low frequency gain (squares) and the high frequency amplifier gain (bullets) are shown in dependence of output power. The lines are calculated values according to Eq. (4) and the approximation according to Eq. (5) is also shown. Naturally compared to 10 mW the amplification factor is smaller for the 30 mW seed and the achievable output power levels are higher. At the same time the achievable low frequency suppression is comparable. For the 10 mW seed the low frequency gain increases up to an output power of 75 mW. At this point the low frequency gain starts to saturate, but only to increase dramatically at an output power beyond 150 mW. The regime up to 150 mW is dominated by the saturation power of the ytterbium doped fiber, beyond that unabsorbed pump power increases and becomes the dominating effect. As discussed before, Eq. (5) does not properly reflect the increase of the saturation contribution with increasing gain, which can be clearly seen in the parts where the approximated low frequency gain is larger than the measured amplifier gain. As one can see, the approximation fits well as long as the amount of unabsorbed pump power is small and the output power is large compared to the saturation power.

Finally, we measured the unabsorbed pump light’s frequency response (Fig. 6, ytterbium doped fiber, 10 mW average seed power, 155 mW average pump power). The transfer functions clearly show the difference between pump and seed power modulation. Between signal and pump power modulation the phase is flipped by 180°, which corresponds to *m*″* _{s}* in Eq. (10) being negative. Increasing the seed power decreases the amount of unabsorbed pump power, while increasing pump power also increases the amount of unabsorbed pump power.

## 4. Design guidelines

To conclude this part we will give some general design guidelines. The corner frequency *ω*_{eff} depends on the ion lifetime, which cannot be changed except by different dopants and thus operating wavelengths. On the other hand, the fiber output intensities can be changed. For slow light applications, generally a large *ω*_{eff} is desirable to shift shorter pulses. When operating the fiber amplifier at its maximum output power this value is also close to its maximum. Deliberately increasing the amount of unabsorbed pump power can further increase this value, but is not always desirable.

If on the other hand a low *ω*_{eff} is required, this can be achieved with a long and/or highly doped fiber and very low power and is only limited by the fluorescence lifetime and practical detection levels.

In first order, increasing the low frequency suppression for seed power modulation is as simple as increasing the gain. However, this works much better, if the required saturation power is low (e.g. erbium). Also most of the pump light must be absorbed, especially, if the respective cross sections are larger than the seed cross sections. If seed suppression is high, this also results in a steeper slope and a larger phase shift. Therefore, a larger pulse delay/advance for slow light applications can be achieved. On the other hand, decreasing this suppression is beneficial for low noise amplifiers as long as the seed source exhibits less noise than the pump source. In this case it is preferable to reduce the influence of the pump source instead. This can be achieved with a high level of unabsorbed pump power again. There is a practical limit, however, as too much unabsorbed pump power is usually undesirable.

## 5. Modulation of the optical phase

The refractive index contribution from KKR should be proportional to the number of ions in the excited state. Therefore, it should be possible to use the model by Novak et al. [6] to model the dynamics of the phase shift via the population. The equations for pump and seed modulation are

*N*

_{2}modulation fraction

*δ*. All the other quantities are the same as before. Compared to output power modulation, the refractive index change is more difficult to measure. The main obstacle is the separation of the gain and the group velocity from the actual refractive index- and corresponding optical phase change. For this reason, we used a 1550 nm probe laser in a Mach-Zehnder configuration to detect the refractive index change induced in an ytterbium doped fiber (Fig. 7). This wavelength is not amplified in the ytterbium fiber and therefore not affected by the gain or group velocity effects. One could argue that the phase shift does not have the same magnitude at 1064 nm – which is true – but we only expect a frequency independent conversion factor. Due to large absorption bands in the ultraviolet, a change of the

*N*

_{2}population causes a phase shift at 1064 nm as well as 1550 nm [1], therefore the dynamics should be the same for both wavelengths.

Because of additional noise contributions, the Mach-Zehnder interferometer needs to be actively stabilized. We used unequal interferometer arm lengths, to allow active control of the interferometer operating point via frequency tuning of the 1550 nm probe light. For this we utilized a mid fringe feedback loop to control the current of the 1550 nm DFB diode. However, this changes not only the frequency but also the output power. As the power modulation of the probe signal is undesired, we used an acousto optic power modulator to stabilize the probe power. Then we modulated the 976 nm pump or the 1064 nm seed power and monitored the control signal and the error signal in the probe light’s frequency control loop.

If our model is suitable, both optical phase transfer functions, i.e. seed and pump power modulation, should be a low pass with the same corner frequency as the output power transfer function for pump modulation. The measured transfer functions are shown in Fig. 8 for 105 mW average pump power and 10 mW average seed power. As predicted, all of them show a low pass behavior and almost the same corner frequency. Compared to the output power, the phase transfer functions show additional features in the low frequency range up to 3.5 Hz. We attribute these features to thermal effects, which are not accounted for in a model only based on KKR.

Since the gain modulation model explains the phase shift’s frequency response, we can also use it to explain time domain effects previously demonstrated. In our experiments with an erbium fiber as a phase shifter at 1064 nm [5], we found that in absence of seed power increasing pump power accelerates the phase response. In the high pumping limit
${P}_{p}^{0}(L)$ increases and therefore also *ω*_{eff} increases. Only in the very low pumping limit, where the pump light is completely absorbed, the response is dominated by the fluorescence lifetime.

Furthermore, we observed that increasing the average pump power decreased the magnitude of the phase change due to pump modulation. While this is to be expected since the pop ulation saturates, we have also shown that this effect can be reduced by inclusion of a seed. Rewriting ${m}_{p}={P}_{p}^{\text{mod}}/{P}_{p}^{0}(0)$ Eq. (14) becomes

*ω*

_{eff}. This is usually desired, but already decreases the actuator range. Additionally, the term $\left[1-\frac{{P}_{p}^{0}(L)}{{P}_{p}^{0}(0)}\right]$ approaches zero as the fraction of absorbed pump light becomes small. This second effect is prevented by an additional seed which reduces the amount of unabsorbed pump light and keeps $\left[1-\frac{{P}_{p}^{0}(L)}{{P}_{p}^{0}(0)}\right]$ close to one.

Compared to pump modulation, we were able to achieve a larger phase shift with seed modulation, when we operated in the amplified regime. Equation (16) is of the same type for seed power modulation with the pump power swapped for the corresponding seed power. However, for seed power modulation the absolute value of $\left[1-\frac{{P}_{s}^{0}(L)}{{P}_{s}^{0}(0)}\right]$ can become much larger, if ${P}_{s}^{0}(L)\gg {P}_{s}^{0}(0)$. The equation also predicts two regimes for seed modulation: depending on the pump power the seed can either be amplified or absorbed $\left({P}_{s}^{0}(L)<{P}_{s}^{0}(0)\right)$. Between both regimes there should be a 180° phase flip due to the changed sign.

To verify this, we measured the influence of seed modulation on the refractive index with and without the pump light as shown in Fig. 9 (black, red, 10 mW seed power, 105 mW pump power). As predicted, there is a phase flip of 180° between both regimes. When increasing the pump power the phase shift decreases at first (as the seed absorption decreases towards transparency), before it increases again.

Interestingly, between both regimes the low frequency temperature contribution also flips its phase. The sign of the phase change for seed power modulation without pump is the same as for pump power modulation (also including the seed) (Fig. 9, blue). Therefore, a seed power increase increases the fiber’s temperature, if the pump diode is switched off and decreases the temperature, if the pump diode is switched on. This is in contrast to the quantum defect heating model, which is usually employed when modeling fiber amplifiers. Similar effects have been reported in bulk amplifiers and lasers [11, 12]. In this field thermal management is even more critical than in high power fiber amplifiers.

## 6. Conclusion

We discussed the impact of dynamic gain in fiber amplifiers on fiber amplifier output power and optical phase. We applied the analytic model developed by Novak and Moesle [6] to explain dynamic effects in our ytterbium and erbium fiber amplifiers. An additional approximation further reduces the complexity of the equations. We have shown that this approximation is valid as long as the unabsorbed pump power and the (amplified) spontaneous emission is low. The model can also be used to predict the temporal behavior of the KKR contribution to the refractive index. Of course, the refractive index changes in active fibers are not limited to the refractive index change via KKR. For example, thermal effects play an important role in the low frequency range. However, the KKR related phase shift is dominating in the frequency range from 1–10 kHz.

## Acknowledgment

This research was made possible by the Cluster of Excellence Centre for Quantum Engineering and Space-Time Research (QUEST) funded by the German Research Foundation (DFG). H. Tünnermann acknowledges a Ph.D. grant from the Hannover School for Laser, Optics and Space-Time Research (HALOSTAR).

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