1. Introduction

Although the current interferometric gravitational wave detectors (GWDs) operate with laser sources emitting at a wavelength of 1064 nm, the next generation will most probably utilize cryogenically cooled silicon mirrors to reduce the thermal noise. That change would demand laser sources at 1.5 μm [1, 2], since silicon is not transparent at 1 μm but at wavelength above 1.4 μm. Moreover, there is a clear tendency to increase the optical power in order to reduce the quantum noise and increase the sensitivity at high frequencies. This makes Er³⁺-doped fiber amplifiers (EDFAs) promising candidates for the next generation of GWDs. In this context, a robust knowledge of temporal and gain dynamics of the amplifier is required to design proper frequency and power stabilization systems which are able to work in a relatively wide range of frequencies. Pumping Er³⁺ at 1480 nm permits modeling the amplifier as a 2-level system, where the excited energy level decays to the ground energy level producing optical emission at 1.5 μm. Such systems have been widely described [3, 4]. However, because of the low power of commercially available single-mode lasers at 1480 nm, it is common to utilize pump lasers at 976 nm [5]. This not only reduces the quantum efficiency due to a higher quantum defect, but also implies the fast relaxation transition from the upper state level to the metastable intermediate level, which modifies the amplifier’s temporal dynamics. Nevertheless, these systems are usually modeled as 2-level systems as well [6], neglecting this fast relaxation process and leading to discrepancies and behaviors in the temporal dynamics that a 2-level model cannot predict. On the other hand, an extensively used configuration are Er³⁺:Yb³⁺co-doped fibers to take advantage of the higher absorption of Yb³⁺around 980 nm. In this kind of amplifiers, the upper energy level of Yb³⁺is excited, from which the energy is then transferred to the upper state level of Er³⁺ [7]. This solution also involves fast relaxation processes which must be modeled and, in any case, it is not possible to find a full analytical solution for the corresponding transfer function [8]. In purely Er³⁺-doped amplifiers, the pump power modulation couples to the amplified output by the gain, since the modulation of the population density of the intermediate energy level leads to modulation of emission at 1.5 μm. The pump power modulation is coupled to the output phase via temperature produced due to the high quantum defect of Er³⁺ pumped around 980 nm. Additionally, the pump power modulation can couple to the output phase by Kramer-Kronig-Relation (KKR) which can become noticeable when pump power is too low to produce a relevant population inversion [9,10].

In this paper we report an analytical model to describe the temporal dynamics of a 3-level system by means of its pump to amplified output transfer functions. We carried out two different experiments. In the first experiment we measured the pump to signal transfer function of an EDFA up to 150 kHz as higher frequencies are not relevant for GWDs. We compare these transfer functions with the model’s prediction showing a good agreement with both the magnitude and phase of the amplified signal. In the second experiment we simulated and measured the transfer function of the pump power to the phase of the output signal, which suggests a low-pass behavior of the upper energy level, responsible for the fiber heating by multiphonon emission when the transition to the intermediate energy level occurs, i.e. the quantum defect.

2. Analytical model

The relevant energy levels and transitions of Er³⁺ doped silica when it is pumped at 976 nm are shown in Fig. 1. At first, the energy level ⁴I_11/2 is excited, increasing its population density. This level then decays to the level ⁴I_13/2 in a non-radiative process which releases phonons, i.e. heat. After that, a radiative transition towards the ground energy level ⁴I_15/2 takes place emitting photons around 1.5 μm.

 

Fig. 1 Energy level diagram of Er³⁺doped in silica. n_i represents the population densities of the energy levels, W_ji are the rates between energy levels, and τ_mp and τ are the spontaneous lifetimes of the ⁴I_11/2 and ⁴I_13/2 levels respectively.

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Considering this energy level diagram, the corresponding rate equations for the population densities of the energy levels in an EDFA are

Substituting Eq. (8) in Eq. (3) and Eq. (9) in Eq. (2) and applying the variation of the Bononi-Rusch model reported by Novak and Moesle [12] by integrating along the fiber leads to

Substituting Eqs. (12) and (13) in Eqs. (10) and (11) yields

Assuming now a small modulation (m_p≪1) of the input pump power

Since δ₂ and δ₃ are small, we can approximate $e^{{\delta }_{2\cdots }}\approx 1+{\delta }_2\dots$ and $e^{{\delta }_{3\cdots }}\approx 1+{\delta }_3\dots$. Furthermore, we can neglect higher order terms (e.g. ${\delta }_2^2$ or δ₂δ₃). Using Eq. (12) finally gives

Using Eq. (20) and multiplying Eq. (22) with $\frac{e^{-i\omega t}}{m_{\mathrm{p}}}$ yields

In general, the term $\frac{N_k^0{\delta }_k}{m_{\mathrm{p}}}{e}^{i{\varphi }_k}$ represents the complex transfer function of the population density in the level k since $N_k^0{\delta }_k$ is the magnitude of the population density modulation around its steady state, m_p is the input modulation and $e^{i{\varphi }_k}$ is the phase factor. Therefore, as can be seen in Eq. (23), the transfer function $\frac{N_3^0{\delta }_3}{m_{\mathrm{p}}}{e}^{i{\varphi }_3}$ of the upper level ⁴I_11/2 depends on the transfer function $\frac{N_2^0{\delta }_2}{m_{\mathrm{p}}}{e}^{i{\varphi }_2}$ of the meta-stable level ⁴I_13/2. If we now assume full pump power absorption, as desired in practical fiber amplifiers, we can use $P_{\mathrm{p}}^0(z=L)\approx 0$, and the transfer function $\frac{N_3^0{\delta }_3}{m_{\mathrm{p}}}{e}^{i{\varphi }_3}$ of the upper level becomes

As mentioned, under the assumption of a complete pump light absorption, $\frac{N_3^0{\delta }_3}{m_{\mathrm{p}}}{e}^{i{\varphi }_3}$ is a single low-pass with cutoff frequency $\frac{1}{{\tau }_{\text{mp}}}$ and thus, the transfer function $\frac{N_2^0{\delta }_2}{m_{\mathrm{p}}}{e}^{i{\varphi }_2}$ of the meta-stable level is a double low-pass with cutoff frequencies at $\frac{1}{{\tau }_{\text{mp}}}$ and $B_s{P}_s^0(z=L)+\frac{1}{\tau }$.

In order to obtain the transfer function of the amplified signal, we use Eq. (13) and assume again that the population densities N₂(t) and N₃(t) and the seed power P_p(z = L) are modulated around their steady state solutions such that

Substitution of Eqs. (26)–(28) in Eq. (13), using again the approximation e^δ… ≈ 1 + δ … (δ ≪1) and neglecting higher order terms (e.g. ${\delta }_2^2$ or δ₂δ₃) yields

Multiplication with $\frac{e^{-i\omega t}}{m_{\mathrm{p}}}$ leads to

Thus, the transfer function of the amplified seed is a low-pass $\frac{1}{{\omega }_2+i\omega }$ multiplied with the term $\frac{{\omega }_3+i\omega }{{\omega }_1+i\omega }$ that is either a damped low-pass if ω₃ > ω₁ or a damped high-pass if ω₁ > ω₃. Comparing Eqs. (35) and (37), it can be established a threshold of the amplified output power level, in watts, to define this feature as

Where h is the Planck constant and λ_s is the seed wavelength. For example, if we consider a fiber with core diameter of 8.5 μm, ${\sigma }_{\mathrm{s}}^{\text{abs}}=1.18\times {10}^{-25}{\mathrm{m}}^2$, τ_mp = 9 μs [13] and Γ_s≈1, then ω₃ > ω₁ is satisfied while $P_{\mathrm{s}}^0(z=L)<6.75\mathrm{W}$, otherwise ω₁ > ω₃.

It must be noticed that Eq. (32) appears as a double low-pass if ω₃ is sufficiently large and ω₃ can also be expressed as function of the absorption and emission cross sections by using the definition of B_s shown in Eq. (15)

3. Pump-to-output signal transfer function

To validate the model, in particular Eq. (32), we compared it to the measured pump power to output power transfer function of an Er³⁺-doped fiber amplifier pumped at 976 nm up to 150 kHz. In this section we present the setup, results and fit of the analytical model.

3.1. Setup

The experimental setup is shown in the Fig. 2. It consisted of a commercial 14-meters long single-mode Er³⁺-doped silica fiber with core diameter of 8.5 μm and cladding diameter of 125 μm manufactured by CorActive. The nominal absorption at 976 nm was 1 dB per meter. The active fiber was pumped by a FBG-stabilized single-mode laser at 976 nm with a maximum output power of 600 mW and seeded at 1572 nm by a DFB diode with ~MHz linewidth. The current of the pump laser was modulated with the sweep signal from a commercial network analyzer, producing the subsequent optical modulation. The modulation depth was always kept at 10% of the mean pump power and the input seed power was constant at 65 mW. A 976/1572 nm WDM was spliced to the output of the active fiber to eliminate any possible residual pump power. Additionally, a band-pass filter centered at 1570 nm with a 12 nm bandwidth was used. The signal was detected by a photodiode with a bandwidth of 1.2 GHz. A sample of the pump power was taken as reference signal from the 1572/976 nm WDM just before the active fiber and detected by another photodiode with a 1.2 GHz bandwidth. At the maximum pump power, the output signal was 250 mW.

 

Fig. 2 Setup used to measure the pump to signal transfer functions of an EDFA. WDM: Wavelength division multiplexer, PD: Photodiode, BPF: Band-pass filter.

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3.2. Results

The measured pump power to output power transfer functions are shown in Fig. 3. The magnitude shows a low-pass behavior with its roll-off frequency increasing with the output power. Furthermore, all curves reach a slope of −40 dB/dec at frequencies above ~10 kHz. Besides this, the phase presents a maximum delay of approximately −150°. In Fig. 4, the model developed in Section 2, i.e. Eq. (32), is fitted to the transfer function measured at a pump power of 482 mW and 250 mW output power. It shows a good agreement with the magnitude and the phase in the whole frequency range. Note that the frequency axis was extended in order to show how the model predicts the magnitude and phase at higher frequencies. The limiting factor for measuring higher frequencies, and eventually observe the predicted damped low-pass in the magnitude, was the sensitivity of the network analyzer.

 

Fig. 3 Pump to amplified signal transfer functions for different pump and output power levels. Top: magnitude. Bottom: phase. The legend shows the pump power followed by the output signal power. The magnitude has been calculated as $20{\log }_{10}\left(\frac{V_{\text{in}}}{V_{\text{ref}}}\right)$, where V_in and V_ref are the voltage signal delivered by the output photodiode, and the voltage signal delivered by the reference photodiode, respectively (see Fig. 2).

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Fig. 4 Pump power to output power transfer function for a pump power of 482 mW and an output power of 250 mW together with the fitted model. The frequency axis is deliberately extended to show the model prediction up to 1 MHz.

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As stated in Section 2 (see Eqs. (32)–(37)), the relations between the cutoff frequencies and the spontaneous lifetimes are given by

Table 1. Effective lifetimes 1/ω₁ and 1/ω₂ for each measured output signal power level, corresponding to the energy levels ⁴I_13/2 and ⁴I_11/2 respectively.

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Fig. 5 Evolution of ω₁ with output power and a corresponding linear fit. The inset shows the extension to an output power of zero such that ω₁ = 1/τ.

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4. Pump-to-output phase-shift transfer function

An effective technique to increase the optical power of laser sources that can be potentially used in the future GWDs is the coherent beam combination (CBC) of multiple fiber amplifiers. It has been demonstrated that Er³⁺-doped and Yb³⁺-doped fibers can be used as phase actuators by pump power modulation [15–17], avoiding the disadvantages of piezo-electrical or electro-optical actuators. In such configuration it becomes critical to stabilize the phase of the fiber amplifiers, for which a good understanding of the pump-power to optical phase-shift transfer function is required.

In this section we present both simulations and experimental measurements of the pumppower to output phase-shift transfer function in an EDFA. The results are in agreement with the expected phase shift due to temperature, which is the dominant mechanism, and also show consistency with the analytical model in Eq. (32) and measurements presented beforehand in Section 3.2.

4.1. Setup

In Fig. 6 the setup designed to measure the pump power to output phase-shift transfer function of the amplifier is presented. It consisted of a Mach-Zender interferometer with the EDFA described in Section 3.1 in one of its arms. The sweep signal from the network analyzer was used to modulate the pump laser power, thus producing a modulation of the population density in the energy level ⁴I_11/2 of the Er ⁺ ions. This leads to a modulation of the phonon emission due to the quantum defect. Therefore, the temperature generated in the fiber is modulated as well, producing slight changes in the refractive index of the core via the thermo-optical effect that eventually change the optical path difference between the interferometer arms. A probe signal passing through both arms of the interferometer can be used to determine the phase-shift by measuring the power of the resulting interference at the output. In the setup shown in Fig. 6 the pump and seed lasers are the same ones used in Section 3.1. To measure the relative phase-shift, a laser diode at 1310 nm was added to provide the probe signal which was introduced to both arms of the interferometer by a 50:50 coupler. It was a DFB laser with a ∼MHz linewidth delivering an optical power of 5 mW. In the passive arm, a 40 m-long fiber stretcher was used as an actuator to stabilize the interferometer and a polarization controller was installed to improve the contrast of the interference. In order to reduce excessive coupling of the frequency noise of the 1310 nm laser to the power at the output of the interferometer, a piece of approximately 26 m of standard single-mode passive fiber (SMF-28) was attached after the EDFA. The interferometer was closed by another 50:50 coupler and the resulting interference signal was filtered by a bandpass filter centered at 1310 nm with a 12 nm bandwidth. Again, two broadband photodiodes with 1.2 GHz bandwidth were used to sense the interference and reference signals from the interferometer output and the EDFA input pump power respectively. A commercial feedback controller Digilock-110 from Toptica Photonics was used to stabilize the interferometer output power by compensating the optical phase acting with a control signal on the stretcher. In the range of frequencies in which the stabilization loop had gain to lock the interferometer, the pump power to phase-shift transfer function was measured by means of measuring the control signal acting on the stretcher. This signal is provided together with the reference signal to the network analyzer to obtain the pump power to phase-shift transfer function. At frequencies at which the stabilization loop did not have enough gain to lock the interferometer, the error signal measured by the photodiode at the output of the interferometer was used instead of the control signal.

 

Fig. 6 Setup used to measure the pump power to phase-shift transfer function. FPGA: Digilock field programmable gate array. WDM: Wavelength division multiplexer. SMF-28: Passive single-mode fiber. BPF: Bandpass filter.

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4.2. Simulation and results

Using a simulation tool previously developed in our group [18], we performed simulations of the temperature-induced phase-shift in the EDFA. The simulation assumes the generation of heat in the core and calculates its radial propagation in the fiber depending on its geometry. However, it does not take into account the physical origin of the heat generation. Furthermore, the simulated transfer function excludes any temperature-independent effect that may occur in the EDFA such as KKR. In Fig. 7, both the measured and simulated phase-shift magnitudes are shown in a frequency range from 100 mHz to 80 kHz. The maximum measurable frequency was limited by the sensing photodiode and network analyzer sensitivity. It can be seen how both phase-shift magnitudes rapidly fall at frequencies between 100 mHz and 3 Hz and then keep falling slower until 10 kHz. Within this range the simulation predicts fairly well this tendency. This shows that the phase-shift is dominated by temperature and filtered by the fiber geometry and thermal properties. Note that the simulations have demonstrated a strong dependency of the frequency response on the fiber geometry at certain frequency ranges, specially with the core and cladding diameters, thermal conductivity, heat transfer coefficient [18] and, similarly, their possible small variations along the fiber. Thus, the accurate knowledge of these parameters is particularly important to achieve an accurate reproduction of the phase-shift dynamics. Nonetheless, a discrepancy starting around 20 kHz between the measured and simulated transfer functions can be noted, which could not be reproduced in the simulations using realistic parameters within the margins guaranteed by the fiber specifications. To study this further, we performed new measurements and simulations with higher frequency resolution to calculate the equivalent transfer function of this discrepancy as shown in Fig. 8. Since this means that the temperature generation reacts to a lesser extent to pump power modulation than in the simulation, we can assume this effect is a consequence of the transfer function of the population density modulation of the energy level ⁴I_11/2, which governs the phonon emission. Therefore it should follow Eq. (24), which corresponds to a single low-pass function. By fitting a low-pass function to the corresponding discrepancy between the simulated and measured transfer functions, the resulting cutoff frequency is f ≈ 16.8 kHz, leading to an effective lifetime of ${\tau }_{\text{mp}}=\frac{1}{2\pi f}\approx 9.4\mu \mathrm{s}$, which is very close to the τ_mp obtained in the experiment in Section 3.2 and typical values on the literature for the lifetime of the upper state level ⁴I_11/2 [13]. Thus, we can state that the temperature-induced phase-shift of the EDFA is filtered by the transfer function of the energy level ⁴I_11/2.

 

Fig. 7 Pump power modulation induced phase-shift, i.e. the magnitude of the corresponding transfer function. The curves have been scaled in order to overlap and the absolute magnitudes are arbitrary.

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Fig. 8 Detail of the discrepancy between the phase-shift transfer functions.

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Nevertheless, it is worth to note that the fitted low-pass function and the mentioned discrepancy start differing slightly at approximately 80 kHz. This gap is a mere 5 dB/dec and there are several factors that could explain it. The most immediate is the inexact knowledge of the fiber properties and their slight variations along it. In this context, the small variations of the core size are specially influential [18]. Besides this, at high frequencies it is also possible that refractive index changes due to the Kramers-Kronig relations [19] and transitions between Stark splits in the energy levels ⁴I_11/2 and ⁴I_13/2 become relevant, since the temperature-induced phase-shift gets less dominant as the frequency increases, as stated before. None of these features are included in the simulations and further investigations are needed to fully understand their influence. Moreover, these additional mechanisms are difficult to characterize by means of the transfer function because fitting and simulation inaccuracies cannot be screened. However, the small gap shown in Fig. 8 between the calculated discrepancy and the single low-pass function can be an early evidence of their effects.

5. Conclusions

We have presented an analytical model of the temporal dynamics of EDFAs pumped at 976 nm and experimentally tested it by means of the pump power to output power and pump power to output phase-shift transfer functions. Even though we experimentally demonstrated the suitability of the developed model with an Er³⁺-doped fiber amplifier, it can also be applied to any 3-level system, e.g. thulium [20]. We found good agreements with the measurements in the range from 100 mHz to 150 kHz and with the typical values of the lifetimes of the energy levels used in the literature. At sufficiently high frequencies and assuming full pump power absorption, the pump power to output power transfer function is a low-pass multiplied by a damped low-pass. The phase of this transfer function does not reach −180°, a fact that simplifies the design of stabilization loops at such frequencies. The first cutoff frequency of the pump power to output power transfer function, and hence the effective lifetime τ is given by the transfer function of the ion population density of the energy level ⁴I_13/2 and depends on its spontaneous lifetime and the output power, whereas the second cutoff frequency, which remains constant, is caused by the transfer function of the population density of the energy level ⁴I_11/2. The modulation of the population density in the energy level ⁴I_11/2 produces a modulation of the phonon emission rate, leading to a heat generation modulation. The pump power then couples to the output phase by the thermo-optical effect. It has been shown that the transfer function of the population density in the energy level ⁴I_11/2, described in the analytical model, can also predict the filtering of the output phase-shift induced by temperature.