Gain dynamics in Er<sup>3+</sup>:Yb<sup>3+</sup> co-doped fiber amplifiers

M. Steinke; J. Neumann; D. Kracht; P. Wessels

doi:10.1364/OE.23.014946

1. Introduction

In order to achieve the required sensitivity, all current gravitational wave detectors (GWDs) are working with laser sources at 1064 nm that are externally frequency and power stabilized. The next generation of GWDs will most probably use different mirror substrates that can be cryogenically cooled in order to decrease the thermal noise and, thus, to increase the sensitivity of the interferometer. However, potential substrates require a change of the laser wavelength to 1.5 μm as they are not transparent anymore at 1.0 μm [1]. Since an output power of up to 1 kW (depending on the final design) at 1.5 μm, diffraction limited beam quality and single-frequency operation will be needed, fiber amplifiers are a good candidate for laser sources of the next generation of GWDs. In contrast to Yb³⁺ doped fiber amplifiers at 1.0 μm, the power scaling of single-frequency Er³⁺ doped fiber amplifiers at 1.5 μm is not that straight forward. If cladding pumped at 976 nm, purely Er³⁺ doped fibers suffer from low absorption and clustering effects at high doping concentrations, leading to long fiber lengths if high output power levels are required. Thus, without any mitigation strategies, nonlinear effects like stimulated Brillouin scattering (SBS) will limit the power scaling of such purely Er³⁺ doped single-frequency fiber amplifiers. Core pumping schemes might be an alternative, especially since it was shown that the output power of cascaded Raman fiber lasers at 1480 nm can exceed several hundred Watts [2]. However, such high power 1480 nm single mode sources are not commercially available, yet. Another promising method is in-band pumping at 1530 nm [3], but corresponding multimode diodes or fiber lasers suffer from low output power levels. The popular co-doping with Yb³⁺ ions increases pump light absorption at 976 nm, but also leads to unwanted parasitic processes at 1.0 μm at high pump powers. Several mitigation techniques of this effect have been reported, in particular the so called co-seeding method [4, 5], which was used lately to scale the output power of an Er³⁺:Yb³⁺ co-doped single-frequency fiber amplifier with excellent beam quality to 60 W [6]. Therefore, either a single Er³⁺:Yb³⁺ co-doped fiber amplifier or a coherent combination of the output power of multiple amplifiers could be used in the next generation of GWDs. In order to be able to stabilize the output power of a single Er³⁺:Yb³⁺ co-doped fiber amplifier or to combine several amplifiers coherently one needs to understand the gain dynamics of such amplifiers. To date, gain dynamics of purely Er³⁺ or Yb³⁺ doped fiber amplifiers have been studied in detail and a full analytical description of the corresponding transfer functions is possible [7, 8, 9]. In addition, gain dynamics of Er³⁺:Yb³⁺ co-doped glass lasers have been studied for CW [10] and pulsed [11] operation. Furthermore, the pulse-dynamics in high power Er³⁺:Yb³⁺ co-doped fiber amplifiers have been studied [12]. However, no analysis has been carried out for continuous-wave (CW) Er³⁺:Yb³⁺ co-doped fiber amplifiers, so far. Therefore, we analyzed to the best of our knowledge for the first time the gain dynamics of CW Er³⁺:Yb³⁺ co-doped fiber amplifiers theoretically and experimentally. In order to keep things simple we focused on Er³⁺:Yb³⁺ co-doped fiber amplifiers pumped at 976 nm and well below the threshold of parasitic processes at 1.0 μm, at which a mitigation scheme must be applied for further power scaling. In addition, we focused on the frequency range from 1 Hz to 100 kHz, as this domain is relevant for GWDs [1]. Because of the nonlinear term inside the rate equations, which describes the energy transfer process between the Yb³⁺ and Er³⁺ ions, it is, to the best of our knowledge, not possible to find a full analytical solution for the corresponding transfer functions. Therefore, we also used a numerical simulation to solve the system of partial differential equations describing an Er³⁺:Yb³⁺ co-doped fiber amplifier to gain deeper insight.

2. Analytical model

The Er³⁺:Yb³⁺ level system as well as the relevant energy transfer between the Yb³⁺ and Er³⁺ ions are presented in Fig. 1. The resonant energy transfer [13] corresponds to a de-excitation of an Yb³⁺ ion from its excited state ²F₅_/₂ to the ground state ²F₇_/₂ and an excitation of a neighboring Er³⁺ ion from its ground state ⁴I₁₅_/₂ to the upper state ⁴I₁₁_/₂. Due to the very short lifetime of the upper state a quick relaxation to the metastable Er³⁺ state ⁴I₁₃_/₂ takes place and, thus, any back-transfer of energy is suppressed. The seed signal is amplified by stimulated transitions between the Er³⁺ metastable state ⁴I₁₃_/₂ and the ground state ⁴I₁₅_/₂. The corresponding rate equations without any upconversion processes, energy back-transfer, and without absorption of pump light by the Er³⁺ ions can be expressed as [14]

\begin{array}{l} 0 = p_{Er} - n_{1} - n_{2} - n_{3} \\ \frac{\partial n_{2}}{\partial t} = W_{12} n_{1} - W_{21} n_{2} - \frac{n_{2}}{τ_{21}} + \frac{n_{3}}{τ_{32}} \\ \frac{\partial n_{3}}{\partial t} = - \frac{n_{3}}{τ_{32}} + R n_{6} n_{1} \\ 0 = p_{Yb} - n_{5} - n_{6} \\ \frac{\partial n_{6}}{\partial t} = W_{56} n_{5} - W_{65} n_{6} - \frac{n_{6}}{τ_{65}} - R n_{6} n_{1} \end{array}

where t represents the time, n_i is the population density of the i-th state and τ_ij is the lifetime of the transition from the i-th to the j-th state. R is the cross-relaxation coefficient, responsible for the energy transfer and p_Er and p_Yb are the Er³⁺ and Yb³⁺ doping concentrations. The W_ij terms represent the stimulated transfer rates between the individual Yb³⁺ and Er³⁺ states and can be expressed as

\begin{array}{l} W_{56} = \frac{Γ_{p} σ_{56} P_{p} (z, t)}{A_{c}} & W_{65} \end{array} = \frac{Γ_{p} σ_{65} P_{p} (z, t)}{A_{c}}

\begin{array}{l} W_{12} = \frac{Γ_{s} σ_{12} P_{s} (z, t)}{A_{c}} & W_{21} \end{array} = \frac{Γ_{s} σ_{21} P_{s} (z, t)}{A_{c}}

where A_c is the fiber core area. σ_ij represents either the absorption or emission cross section between the i-th and j-th state and P_p₍_z,t₎ and P_s₍_z,t₎ are the pump and seed power in units of photons per second. The overlap of the pump and seed light with the doped core can be expressed as

Γ_{p, s} = 1 - e^{\frac{- 2 r_{c}^{2}}{ω_{p, s}^{2}}}

where r_c is the core diameter and

ω_{p, s} = r_{c} (0.65 + \frac{1.619}{V_{p, s}^{1.5}} + \frac{2.879}{V_{p, s}^{6}})

is either the mode radius of the pump or seed radiation [15], which can be calculated with the so called V-Parameter

V_{p, s} = \frac{2 π}{λ_{p, s}} r_{c} N A_{c}

where NA_c is the NA of the core and λ_p and λ_s is either the wavelength of the pump or the seed light. In case of a cladding pumped amplifier, the overlap of the pump radiation with the core must be rescaled for example by a factor of

\frac{A_{c}}{A_{clad}}

where A_clad is the pump cladding area.

Fig. 1 Energy-level diagram of the co-doped Er³⁺:Yb³⁺ system. Not included are any upconversion processes, back-transfer of energy or absorption of pump light by the Er³⁺ ions.

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Any small modulation of the seed or pump power leads to a corresponding modulation of the population density of the co-doped Er³⁺:Yb³⁺ system around its steady state solution. The exact lifetime of the upper Er³⁺ state ⁴I₁₁_/₂ depends on the actual core composition of the fiber under use and typically it is kept as small as possible to avoid back-transfer of energy. Thus, in the literature, in particular for the evaluation of the steady state population densities, a vanishing lifetime is assumed commonly (see for example [16] and references within). As we will only consider modulation frequencies up to 100 kHz, we will assume here that the lifetime of the upper Er³⁺ state ⁴I₁₁_/₂ is not much larger than 10 μs and, thus, the upper Er³⁺ state will react to any change in pump or seed light instantaneously. Indeed, from our experimental results, as will be shown, there is no evidence that this approximation could be wrong. Under this approximation, any modulation of the corresponding population will transfer without any loss to the metastable state ⁴I₁₃_/₂ and its corresponding population. Therefore, in the given case the rate equations of the co-doped Er³⁺:Yb³⁺ system can be simplified by neglecting the upper Er³⁺ state ⁴I₁₁_/₂, which then leads to an energy transfer directly between the upper Yb³⁺ state ²F₅_/₂ and the metastable Er³⁺ state ⁴I₁₃_/₂ that we call from here on the Er³⁺ upper state for the sake of convenience. The simplified rate equations can now be written as

\begin{array}{l} 0 = p_{E r} - n_{1} - n_{2} \\ \frac{\partial n_{2}}{\partial t} = W_{12} n_{1} - W_{21} n_{2} - \frac{n_{2}}{τ_{21}} + R n_{6} n_{1} \\ 0 = p_{Y b} - n_{5} - n_{6} \\ \frac{\partial n_{6}}{\partial t} = W_{56} n_{5} - W_{65} n_{6} - \frac{n_{6}}{τ_{65}} - R n_{6} n_{1} \end{array}

and the evolution of the pump light P_p(z,t) and seed light P_s(z,t) along the fiber for any time t is given by

\frac{d P_{p}}{d z} = Γ_{p} (n_{6} σ_{65} - n_{5} σ_{56}) P_{p} = A_{c} (W_{65} n_{6} - W_{56} n_{5})

\frac{d P_{s}}{d z} = Γ_{s} (n_{2} σ_{21} - n_{1} σ_{12}) P_{s} = A_{c} (W_{21} n_{2} - W_{12} n_{1}) .

In case of a forward pumped amplifier, Eqs. (7), (8) and (9) constitute to a system of partial differential equations (PDE) with boundary conditions

\begin{array}{l} P_{p} (z = 0, t) = P_{p, 0} (t) \\ P_{s} (z = 0, t) = P_{s, 0} (t) . \end{array}

In order to simplify this PDE system further, we follow an approach which was already used to analyze the gain dynamics of purely Er³⁺ or Yb³⁺ doped fiber amplifiers [7]. We are interested here in the response of the amplified seed signal to pump or seed power modulations, thus we focus on the corresponding equations for the upper Er³⁺ state only. The response of the residual pump power to a modulation of the seed or pump power can be treated similarly. First, substitution of Eq. (9) into the corresponding rate equation for the upper Er³⁺ state and integration along the fiber leads to

\frac{\partial N_{2}}{\partial t} = P_{s} (z = 0) - P_{s} (z = L) - \frac{N_{2}}{τ_{21}} + X

where N₂ corresponds to the total length-integrated ion population of the upper Er³⁺ state and

X = \frac{R}{A_{c}} \int_{z = 0}^{z = L} A_{c} n_{1} A_{c} n_{6}

corresponds to the total energy transfer along the fiber. In addition, by partial integration of Eq. (9) one can relate the seed power at the end of the fiber (z = L) with the seed power at z = 0

P_{s} (z = L, t) = P_{s} (z = 0, t) e^{B_{s} N_{2} - C_{s}}

with

B_{s} = \frac{Γ_{s}}{A_{c}} = (σ_{12} + σ_{21})

C_{s} = Γ_{s} σ_{12} L p_{Er} .

Substitution of Eq. (13) into Eq. (11) leads to

\frac{\partial N_{2}}{\partial t} = P_{s} (z = 0) (1 - e^{B_{s} N_{2} - C_{s}}) - \frac{N_{2}}{τ_{21}} + X

which is one of the key equations for describing dynamic effects in Er³⁺:Yb³⁺ co-doped fiber amplifiers and for deriving the corresponding transfer functions.

Any sinusoidal modulation of the input pump or seed power

P_{p, s} (z = 0, t) = P_{p, s, 0} (1 + m_{p, s} e^{i ω t})

with a small modulation depth m_p,s will lead to a sinusoidal modulation of the total population of the upper Er³⁺ state

N_{2} (t) = N_{2, 0} (1 + m_{2}^{(p, s)} e^{i (ω t + ϕ_{2}^{(p, s)})})

and of the total energy transfer

X (t) = X_{0} (1 + m_{X}^{(p, s)} e^{i (ω t + ϕ_{X}^{(p, s)})})

where the superscripted (p) or (s) indicates either pump or seed power modulation. By substituting Eq. (18), Eq. (19), and in case of a seed power modulation also Eq. (17) into Eq. (16) and by neglecting higher order terms with respect to the modulation depths one obtains in case of a pump power modulation

\frac{N_{2, 0} m_{2}^{(p)}}{m_{p}} e^{i ϕ_{2}^{(p)}} = \frac{X_{0} m_{X}^{(p)}}{m_{p}} e^{i ϕ_{X}^{(p)}} \frac{1}{ω_{eff} + i ω}

and in case of a seed power modulation

\frac{N_{2, 0} m_{2}^{(s)}}{m_{s}} e^{i ϕ_{2}^{(s)}} = \frac{P_{s, 0} (z = 0) - P_{s, 0} (z = L)}{ω_{eff} + i ω} + \frac{\frac{X_{0} m_{x}^{(s)}}{m_{s}} e^{i ϕ_{x}^{(s)}}}{ω_{eff} + i ω}

with

ω_{eff} = B_{s} P_{s, 0} (z = L) + \frac{1}{τ_{21}} .

Equations (20) and (21) can be used to calculate the direct response of the amplified seed signal to a pump power modulation

\frac{m_{s}^{(p)}}{m_{p}} e^{i ϕ_{s}^{(p)}} = \frac{X_{0} m_{X}^{(p)}}{m_{p}} e^{i ϕ_{X}^{(p)}} \frac{B_{s}}{ω_{eff} + i ω}

or to a seed power modulation

\frac{m_{s}^{(s)}}{m_{s}} e^{i ϕ_{s}^{(s)}} = \frac{ω_{0} + i ω}{ω_{eff} + i ω} + \frac{\frac{X_{0} m_{X}^{(s)}}{m_{s}} e^{i ϕ_{X}^{(s)}}}{ω_{eff} + i ω}

with

ω_{0} = B_{s} P_{s, 0} (z = 0) + \frac{1}{τ_{21}} .

Thus, in case of a pump power modulation (see Eq. (23)) the transfer function consists of a low pass $\frac{B_{s}}{ω_{eff} + i ω}$ with a corresponding corner frequency ω_eff multiplied with an unknown transfer function of the total energy transfer $\frac{X_{0} m_{X}^{(p)}}{m_{p}} e^{i ϕ_{X}^{(p)}}$ for which no analytical solution exists. In case of a seed power modulation (see Eq. (24)) the transfer functions involves two terms, one is a damped high pass $\frac{ω_{0} + i ω}{ω_{eff} + i ω}$ with two corner frequencies ω₀ < ω_eff and the other one is again a product of a low pass with another unknown transfer function of the total energy transfer $\frac{X_{0} m_{X}^{(s)}}{m_{s}} e^{i ϕ_{X}^{(s)}}$ .

Based on these transfer functions several conclusions can be made and some open questions remain. First, Eq. (23) predicts that any modulation of the pump power gets filtered by the unknown transfer function of the energy transfer before it has any influence on the Er³⁺ populations. This will have impact on the development of low-noise amplifiers. In addition, if the corner frequency ω_eff can be determined experimentally, the corresponding low pass in Eq. (23) can be canceled out in order to obtain the transfer function of the total energy transfer. Here, it is interesting to analyze the behavior of the corresponding transfer function and if it is somehow universal (for example a low pass) or differs for different amplifiers setups. For the case of a seed power modulation it seems reasonable to question if the second term in Eq. (24) can be neglected in comparison with the first one, especially at high modulation frequencies for which it is likely that the energy transfer will strongly suppress any transfer of modulation from the Er³⁺ to the Yb³⁺ ions due to its own finite transfer rate and corresponding timescale. Thus, if the energy transfer would have no influence on both corner frequencies or at least on the upper corner frequency, they would be given by analytical expressions (see Eqs. (25) and (22)) and would be determined only by fiber parameters like in the case of a purely Er³⁺ or Yb³⁺ doped fiber amplifier [7]. In addition, the upper corner frequency could then be measured by modulating the seed power and Eq. (23) could be used to determine directly the transfer function of the total energy transfer, as explained above.

To answer these questions we developed a numerical simulation, which is using a ready-made finite volume solver [17] for the PDE system defined by Eqs. (7), (8) and (9). While the input pump or seed power is modulated, the solver starts at steady state conditions and steps forward in time until a stable solution emerges. This solution is then used to calculate the magnitude and phase of the relevant modulation transfer functions. Depending on the amplifier setup and due to the stiffness of the PDE system a maximum allowed time step exists even for small modulation frequencies. Thus, depending on the actual number of modulation frequencies in the frequency range of interest and depending on the number of steps along the fiber, several hours were needed on a standard PC for the calculation of the transfer functions for a given amplifier setup.

3. Experiment and simulation: Setup and results

The setup we used in our experiment and which also defined the parameters for our numerical simulations is depicted in Fig. 2. A single mode laser diode emitting at 976 nm was used to pump the amplifier. Due to the high doping concentrations of Er³⁺:Yb³⁺ co-doped fibers core-pumping schemes typically lead to extreme short and bulky fiber lengths. Thus, in order to set up a cladding pumped amplifier a piece a of multimode fiber was used to convert the single mode radiation of the pump diode into multimode radiation. Modulation of the pump power was realized by a modulation of the corresponding diode driving current. A DFB laser diode emitting at 1555 nm was used to seed the amplifier and an additional fiber-coupled AOM was integrated to modulate the seed power, as a modulation of the driving current led to unwanted wavelength shifts. The pump and seed radiation were combined in an in-house made pump combiner. As a reference port for the pump power modulation a low-ratio tap coupler behind the pump diode was used. To set up a reference port for the seed power modulation a 1064/1550 nm WDM was spliced behind the seed diode, which was additionally used to shield the seed diode from any potentially backward propagating Yb-ASE. Right behind the pump combiner a maximum pump power of 300 mW and a seed power of around 5 mW could be used for the amplifier. Around 1.8 m of a 6/130 double-clad Er³⁺:Yb³⁺ fiber from Nufern was used as active fiber. High index paste and a short piece of a matching single mode fiber was used to remove the cladding light at the end of the amplifier, because we discovered that the presence of cladding light distorted the measured transfer functions. After filtering out any residual pump light with a dichroic mirror, the amplified seed power was measured and a small fraction was used for the modulation measurements. A maximum output power of 30 mW could be achieved at a maximum pump power of 300 mW. All reference ports within the setup were connected to photo diodes, each with a bandwidth of 150 MHz. Sweeping of the pump or seed power modulation frequency and recording of the corresponding transfer functions was performed with a commercial dynamic signal analyzer.

Fig. 2 Schematic overview of the used experimental setup to measure the transfer functions of the Er³⁺:Yb³⁺ co-doped fiber amplifier.

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In what follows we will first introduce our numerical results in order to answer some of the questions formulated at the end of the last section. Subsequently we will present our experimental results and make a qualitative comparison with the numerical results.

3.1. Numerical results

The parameters for our numerical simulations are listed in Table 1. As far as possible, parameters similar to those defined by the experimental setup were used. However, in contrast to the doping concentrations the cross sections were not known and, thus, cross sections from a commercially available software (Liekki Application Designer) were used. These cross-sections have been determined by Liekki (now nLight) from purely Yb³⁺ and Er³⁺ doped aluminosilicate fibers with a methodology similar to the one presented in [18]. Indeed, the cross sections of Er³⁺:Yb³⁺ co-doped fibers might certainly be different compared to purely Yb³⁺ and Er³⁺ doped fibers. However, in the following we will not perform a quantitatively but a qualitatively comparison between experiment and numerical simulations. Thus, with respect to the gain dynamics, it is sufficient that the cross-sections are of the correct magnitude. At steady state conditions the computed ideal efficiency of the amplifier was two times higher then the measured one, which seems acceptable as our model is rather simple and does not include any upconversion, energy back-transfer or similar loss mechanisms. Transfer functions were computed for six different pump power levels comparable to those used in the experiment. A maximum time step of 1 × 10⁻⁴ s was necessary to prevent any overshooting during the computation of the steady state solution. For any sinusoidally time-varying boundary conditions, i.e. for the computation of the transfer functions, either the mentioned time step of 1 × 10⁻⁴ s or smaller time steps were used, in order to have a sufficiently large number of points per oscillation for the later evaluation. Especially at high modulation frequencies it was necessary to compute several (> 10) oscillations in order to allow the system to reach a stable solution, whereas at low modulation frequencies a stable solution emerged already after a few (< 5) oscillations.

Table 1. Parameters used in the numerical simulations

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Computed transfer functions of the amplified seed for the case of a seed power modulation and for different pump power levels are presented in Figs. 3(a) and 3(b). Shown are each time the magnitude and the corresponding phase. At first glance the transfer functions appear to be damped high passes with a flat magnitude at low and high frequencies and two distinctive corner frequencies in between. As we are interested in the influence of both terms in Eq. (24) on the actual transfer functions, we performed some further numerical investigations. Fig. 3(c) shows the numerical results for the highest and lowest pump power in comparison with the magnitude of the first term of Eq. (24), which is actually a damped high pass and can be calculated directly from the steady state solution. As already presumed at the end of the last section, the second term of Eq. (24) gets small for high modulation frequencies, so that the numerical solution is mainly determined by the first term. In order to investigate how the corner frequencies are influenced by the second term in Eq. (24) we performed fits of a damped high pass to the numerical solutions. The corresponding corner frequencies are compared in Fig. 3(d) with the corner frequencies $f_{0} = \frac{ω_{0}}{2 π}$ and $f_{eff} = \frac{ω_{eff}}{2 π}$ of the first term in Eq. (24). Both corner frequencies are influenced by the second term, depending on the actual pump power. In addition, we also verified numerically that the influence of the second term also depends on the launched seed power. Thus, in a measurement the transfer functions for a seed power modulation will show a damped high pass behavior, but setting the corner frequencies equal to ω_eff and ω₀ should only be used as an approximation and its suitability depends on the actual amplifier configuration. Note that the fact that the phase shifts in Fig. 3(b) do not converge to zero at high frequencies, as one would expect for a pure damped high pass, is another indication for the residual influence of the energy transfer on the total transfer function even at frequencies above the second corner frequency ω_eff.

Fig. 3 Computed transfer functions for the case of a seed power modulation: Magnitude (a) and phase (b). (c): Comparison of the numerically obtained magnitude and the magnitude of the first term in Eq. (24) for the lowest and highest pump power. (d): Comparison of the corner frequencies ω₀ and ω_eff of the first term in Eq. (24) and the corner frequencies of a damped high pass fit to the numerically obtained transfer functions.

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The relevant results for the case of a pump power modulation are presented in Figs. 4(a) and 4(b). Shown are the transfer functions of the energy transfer (dashed lines) and the transfer functions of the amplified seed (solid lines) for different pump power levels. Each transfer function of the energy transfer appears to be a single low pass, as indicated by the behavior of the magnitude (maximum inclination of −10 dB per decade) and by the maximum phase shift of −90 degree. In addition, fits of a low pass to the data revealed that the corresponding corner frequency does not shift with increasing pump power. Note, that the magnitude at low modulation frequencies actually depends weakly on the pump power, i.e. the low-frequency magnitude is not the same for all pump power levels, as this may appear to be not the case in Fig. 4(a).

Fig. 4 Computed transfer functions of the energy transfer (dashed) and amplified seed (solid) for the case of a pump power modulation: Magnitude (a) and phase (b). Computed transfer function of the energy transfer and amplified seed for a 915 nm pumped amplifier (see text for further explanation): Magnitude (c) and phase (d).

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In agreement with Eq. (23), the transfer function of the amplified seed appears to be a double low pass, as indicated again by the shape of the magnitude (maximum inclination of −20 dB per decade) and a maximum phase shift of −180 degree. One corner frequency is equal to the corner frequency of the transfer function of the energy transfer and the second one is equal to ω_eff (see Eq. (22)). Figures 4(c) and 4(d) show the computed transfer functions of the energy transfer and amplified seed for an amplifier pumped at 915 nm, as this is the secondary absorption wavelength for Er³⁺:Yb³⁺ co-doped amplifiers. The fiber length was set to 0.5 m, the seed power was 0.1 W and the pump power was 5 W. In addition, the cross-sections have been changed to σ₅₆ = 6.7 × 10⁻²⁵ m² and σ₆₅ = 2.3 × 10⁻²⁶ m². In particular in the magnitude of the energy transfer, in contrast to a simple low pass, a non-uniform inclination at high frequencies can be observed. Furthermore, also the phase shows a behavior different to a single low pass (compare for example to Fig. 4(b)). Thus, although in the given case the deviation from a low pass appears to be small, no universal shape for the transfer functions of the energy transfer exists. This can be understood taking into account the nonlinear appearance of the energy transfer process in the rate equations.

3.2. Experimental results

Transfer functions were measured for pump power levels similar to those used in the numerical simulations. In Figs. 5(a) and 5(b) the measured transfer functions for the case of a seed power modulation are presented. Like in our numerical simulation the transfer functions show a behavior similar to a damped high pass, but, as stated earlier, the corresponding corner frequencies are not necessarily equal to ω_eff and ω₀. Nevertheless, in particular for a later comparison with transfer functions for the case of a pump power modulation, fits of a damped high pass to the measured data were performed in order to determine both corner frequencies.

Fig. 5 Measured transfer functions (not normalized) for the case of a seed power modulation: Magnitude (a) and phase (b). Measured transfer functions (not normalized) for the case of a pump power modulation: Magnitude (c) and phase (d).

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The corresponding transfer functions for pump power modulation are presented in Figs. 5(c) and 5(d). As predicted by our numerical simulation they show a double low pass behavior, as indicated by a maximum inclination of the magnitude of −20 dB per decade and by the maximum phase shift of almost −180 degree. In addition, in contrast to our numerical results the two kinks in the magnitude (corresponding to the two corner frequencies) at 0.1–1 kHz and around 8.5 kHz can be observed better. This reflects the fact that the two corner frequencies are more clearly separated as it was the case in our numerical simulations. Such behavior could be reproduced in our simulation by setting the energy transfer coefficient around 100 times higher than in Table 1. In particular compared to the experimentally obtained amplifier slopes, this led to significantly higher amplifier efficiencies. However, as our model does not include any loss mechanisms, this might be compensated by implementing upconversion, energy back-transfer or similar processes. However, as Eq. (15) in [14] is to the best of our knowledge the only result of an experimental investigation that links the energy transfer rate and the doping concentration, our results request further investigations on this subject in the future.

To obtain the two corner frequencies of the measured transfer functions we performed fits of a double low pass to the data and compared the corresponding corner frequencies with the corner frequencies that were obtained earlier for the case of a seed power modulation. The corresponding result is presented in Fig. 6. In case of a seed power modulation, both corner frequencies increase linearly with the amplifier output power like it was already the case in our numerical simulation. In case of a pump power modulation, the upper corner frequency is almost constant for different amplifier output power levels. Thus, based on our numerical results, it is determined by the energy transfer process. The lower corner frequency (which is then ω_eff, see Eq. (23)) increases linearly with amplifier output power and is, in contrast to our numerical results, equal to the upper corner frequency obtained for the case of a seed power modulation. Here, it seems that the additive term of the energy transfer (see Eq. (24)) has only a very weak influence on the transfer function at high frequencies. Thus, the corresponding upper corner frequency is equal to ω_eff. Furthermore, Eq. (22) states that for a vanishing amplifier output power, ω_eff is given by the inverse of the lifetime of the upper Er³⁺ state. By a linear fit to the measured data this lifetime was determined to be 3 ms, which is close to the value of 1–10 ms that is used throughout the literature.

Fig. 6 Comparison of the different corner frequencies obtained from the transfer functions for the case of a pump or seed power modulation.

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Knowing both corner frequencies for the case of a pump power modulation, one can multiply the measured transfer functions with the inverse of a low pass with ω_eff as corner frequency to determine the transfer functions of the energy transfer (see Eq. (23)). The corresponding results are presented in Fig. 7. The transfer functions behave, as already suggested by our numerical simulation, like low passes (inclination of −10 dB per decade) with corner frequencies at around 8.5 kHz that are independent of the amplifier output power. In addition, like in our numerical simulation, the magnitude at low modulation frequencies depends on the actual pump power. Indeed, the differences are more pronounced, but this can be explained by a missing normalization for the measured data and the fact that not all experimental parameters like the energy transfer rate were known exactly.

Fig. 7 Magnitude of the transfer function of the energy transfer for a pump power modulation.

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4. Conclusion

We report on theoretical and experimental investigations of gain dynamics in Er³⁺:Yb³⁺ co-doped fiber amplifiers. It is shown that due to the energy transfer between the Er³⁺ and Yb³⁺ ions a full analytical solution of the problem is, in contrast to purely Er³⁺ or Yb³⁺ doped fiber amplifiers, not possible. Thus, the underlying PDE system was solved numerically with a finite volume solver to gain further insights. Numerical results for a 976 nm cladding pumped Er³⁺:Yb³⁺ co-doped fiber amplifier well below the threshold of parasitic processes at 1.0 μm suggest that, in case of a seed power modulation, the transfer function of the amplified signal is a damped high pass. However, in the worst case both corner frequencies are influenced by the unknown transfer function of the energy transfer. In case of a pump power modulation, the energy transfer behaves like a low pass and, in accordance with the analytical treatment, the transfer function of the amplified signal shows a double low pass behavior. However, it is also shown that the transfer function of the energy transfer of a 915 nm pumped Er³⁺:Yb³⁺ co-doped fiber amplifier can not be described by a low pass. Thus, no universal shape for the transfer function of the energy transfer exists.

Experimental transfer functions were recorded with an amplifier with similar parameters as used in the numerical simulations. The results show good qualitative agreement with the numerical results. Indeed, the transfer function for a pump power modulation is a double low pass and the transfer function for a seed power modulation is a damped high pass. However, in contrast to our numerical results, only the upper corner frequency of the damped high pass was influenced by the transfer function of the energy transfer. Experimental results suggest a lifetime of the upper Er³⁺ state in the range of 3 ms, which is close to the value of 1–10 ms used in the literature. Although the simple low pass behavior of the energy transfer was also observed in the experiment, the experimental results suggest that the energy transfer coefficient is higher than expected. Here, further investigations on the actual rate of the energy transfer have to be carried out in the future.

Acknowledgments

This work was supported by the German Research Foundation (DFG) through funding the Cluster of Excellence ”Centre for Quantum Engineering and Space-Time Research” (QUEST).

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Parameter	Value	Note
p_Er	1.11 × 10²⁵ m⁻³
p_Yb	2.42 × 10²⁶ m⁻³
r_c	6.0 × 10⁻⁶ m
NA_c	0.18
R	1.93 × 10⁻²² m³ s⁻¹	Calculated from Eq. (15) in [14]
τ₂₁	1.0 × 10⁻³ s
τ₆₅	0.7 × 10⁻³ s
σ₁₂	2.78 × 10⁻²⁵ m²
σ₂₁	4.36 × 10⁻²⁵ m²
σ₅₆	2.5 × 10⁻²⁴ m²
σ₆₅	2.44 × 10⁻²⁴ m²

Gain dynamics in Er³⁺:Yb³⁺ co-doped fiber amplifiers

Abstract

1. Introduction

2. Analytical model

3. Experiment and simulation: Setup and results

3.1. Numerical results

3.2. Experimental results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (25)

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