Abstract

In this work, we studied the interlock requirements in a seed failure scenario for Er3+:Yb3+ doped fiber amplifiers (EYDFAs) pumped with high intensities in the MWcm−2 range at 9XX nm. We fed a time-dependent FEM-tool with the data from backwards directed amplified spontaneous emission (ASE) transients of different commercially available core-pumped single-mode fibers. In the FEM-tool, the Er3+:Yb3+ system is defined as a bi-directional energy transfer process and described by the corresponding rate equations. The power evolution of the pump, seed, and ASE signal is computed by differential equations taking into account the transient population densities of the relevant energy levels. With the model, we computed the temporal evolution of the corresponding energy levels after a seeder failure to take place within tens to hundreds of µs and calculated the associated gain. The fibers under test provide a critical total gain of 30 dB after ∼ 80 µs within the Yb3+ band and after ∼300 µs within the Er3+ band. This time decreases with increasing pump power and doping concentration. The results can be extrapolated to high-power cladding-pumped EYDFAs to meet the challenging requirements of engineering-level systems.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Er$^{3+}$ based fiber amplifiers at 1.5 µm are of significant importance in applications of telecommunication [1,2], light detection and ranging [3] or in the next generation of interferometric gravitational wave detectors [4]. In comparison to free-space solid-state laser systems, an all-fiber design can provide outstanding thermal management with robust beam quality, good compactness and the utilization of standard telecommunication equipment [2]. The detrimental tendency of Er$^{3+}$ to form clusters [5] and the low absorption of commercially available pump diodes can be compensated by sensitizing the corresponding fibers with Yb$^{3+}$ ions. The Yb$^{3+}$ ions absorb the pump light and transfer the energy to Er$^{3+}$ ions in close vicinity by a short-range Förster resonance energy transfer as illustrated in Fig. 1. But the co-doping with Yb$^{3+}$ can introduce detrimental aspects, e.g. the presence of the 1 µm amplified spontaneous emission (ASE), which can induce stability problems and degrade the energy transfer efficiency. However, Er$^{3+}$:Yb$^{3+}$ co-doped fiber amplifiers (EYDFAs) in master-oscillator and power-amplifier (MOPA) configuration have proven their capability to generate diffraction-limited output power levels of 100 W and kHz-linewidth [6]. The high pump power levels associated with this output power require an appropriate seeder input to control the ASE. A seeder loss at that power level introduces the risk of catastrophic failure due to parasitic lasing or Q-switching. Comprehensive knowledge of the time scales of the processes leading to such failures are critical in order to design proper interlock systems. Though, to the best of our knowledge, there is no literature covering the interlock requirements of high-power EYDFAs, i.e. required response time in the case of a seeder loss. In this work, we study commercially available single-mode fibers pumped in the core with low power but similar intensity to a typical high-power cladding-pumped EYDFA. We simplified the situation by leaving the fibers unseeded to obtain the total ASE rise times. We use the experimental data of the backwards directed ASE and a time-dependent model and derive the minimum requirements for a sufficient interlock system.

 figure: Fig. 1.

Fig. 1. Bottom left: Experimental setup. WDM: wavelength division multiplexer, FUT: fiber under test, PD: photo detector. Top left: Typical Er$^{3+}$ and Yb$^{3+}$ band ASE power traces of FUT1 at different pump intensities (see Fig. 3). Right: Er$^{3+}$:Yb$^{3+}$ energy levels linked by a dipole-dipole ($R_{ij}$) interaction with lifetimes $\tau _{ij}$ and stimulated transfer rates $W_{ij}$ between the $i$-th and $j$-th level.

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2. Numerical model

We used a self-written solver based on a finite-element-method (FEM) to simulate the coupled Yb$^{3+}$ and Er$^{3+}$ energy level scheme, where the ions are assumed to be linked by a dipole-dipole interaction that allows bi-directional energy transfer. We refer to the $^4$I$_{{15}/{2}}$, $^4$I$_{{13}/{2}}$, $^4$I$_{{11}/{2}}$ and the $^4$I$_{{9}/{2}}$ levels of Er$^{3+}$ as levels 1 to 4. The $^2$F$_{{7}/{2}}$ level of Yb$^{3+}$ is denoted as level $5$ and the $^2$F$_{{5}/{2}}$ level as level 6. The corresponding population densities are designated as $n_1$ to $n_6$ respectively and depicted in Fig. 1 (right).

The stimulated transfer rates $W_{ij}(z,t)$ between the $i$-th and $j$-th energy level are expressed as

$$W_{ij}(z,t) = \frac{\Gamma\lambda\sigma_{ij}}{A_{\mathrm{c}}hc}P(z,t)$$
with $A_c$ the area of the fiber core, $P(z,t)$ the power at a given time $t$ and position $z$ of the fiber, $\lambda$ the wavelength, $h$ the Planck constant, and $c$ the speed of light [7]. The parameters $\sigma _{ij}$ represent the associated absorption and emission cross-sections and have been compiled from [8,9] (see Fig. 3). The factor $\Gamma$ accounts for the overlap of the LP$_{01}$ mode with the doped core area [7] and can be defined as
$$\Gamma = 1 - \exp\left(-2\frac{r_{\mathrm{c}}^2}{r_{\mathrm{m}}^2}\right).$$
Here, $r_c$ is the core radius and $r_m$ the LP$_{01}$ mode radius as explained in [10]. In the case of cladding pumping, the pump overlap is rescaled by the core-to-cladding area ratio. We calculated the forwards directed energy transfer coefficient $R_{63}$ from the Yb$^{3+}$ doping concentration as shown in [11] and chose $R_{36}$ to be the same in accordance to McCumber theory [12]. The lifetimes are approximated to $\tau _{21}=8\,$ms [9,13], $\tau _{32}=10\,$µs [13], $\tau _{65}=1\,$ms [6,9,14], $\tau _{43}$ is assumed to be $0.2$ µs [15] and the up-conversion coefficient $C_{\mathrm {up}}$ is computed from the total Er$^{3+}$ doping concentration as in [16]. The time derivatives of the population densities $n_i$ are determined by the transition rates along the fiber and are calculated at a given time by:
$$\frac{\partial n_1}{\partial t} = W_{21}n_{2}+W_{31}n_{3} - W_{12}n_{1} - W_{13}n_{1}+R_{36}N_\mathrm{Yb}n_3n_5 - R_{63}N_\mathrm{Yb}n_6n_1 + C_\mathrm{up}N_\mathrm{Er}n_2^2+\frac{n_2}{\tau_\mathrm{21}}$$
$$\frac{\partial n_2}{\partial t} = W_{12}n_{1} - W_{21}n_{2} - 2C_\mathrm{up}N_\mathrm{Er}n_2^2 + \frac{n_3}{\tau_\mathrm{32}} - \frac{n_2}{\tau_\mathrm{21}}$$
$$\frac{\partial n_3}{\partial t} = W_{13}n_{1} - W_{31}n_{3} + R_{63}N_\mathrm{Yb}n_6n_1 - R_{36}N_\mathrm{Yb}n_5n_3+ \frac{n_4}{\tau_\mathrm{43}} - \frac{n_3}{\tau_\mathrm{32}}$$
$$\frac{\partial n_4}{\partial t} = C_\mathrm{up}N_\mathrm{Er}n_2^2 - \frac{n_4}{\tau_\mathrm{43}}$$
$$\frac{\partial n_5}{\partial t} = W_{65}n_{6} - W_{56}n_{5} + R_{63}N_\mathrm{Er}n_6n_1 - R_{36}N_\mathrm{Er}n_5n_3 + \frac{n_6}{\tau_\mathrm{65}}$$
$$\frac{\partial n_6}{\partial t} = W_{56}n_{5}-W_{65}n_{6} - R_{63}N_\mathrm{Er}n_6n_1 + R_{36}N_\mathrm{Er}n_5n_3 - \frac{n_6}{\tau_\mathrm{65}}.$$
Here, $N_{\textrm {Er}}$ and $N_{\textrm {Yb}}$ are the total Er$^{3+}$ and Yb$^{3+}$ doping concentrations and are calculated from the small signal absorption of the test fibers. The model includes the forwards and backwards directed propagation of pump, seed and ASE signal, whereas the differential power evolution along the fiber is given by the equation:
$$\Delta P(z,t) =~ W_{\mathrm{net}}(z,t) \cdot \frac{hc}{\lambda} + \beta W_{\mathrm{SE}}(z,t) \cdot \frac{hc}{\lambda}$$
with $W_{\mathrm {net}}$ the net stimulated transition rate between the relevant energy levels including emission and absorption and $hc/\lambda$ the corresponding photon energy. The last summand is only applied for ASE computation and represents photons spontaneously emitted at a rate $W_{\mathrm {SE}}$ and guided in the core of the fiber. The pre-factor $\beta$ is the ratio of the partial sphere area given by the NA incident angle and accounts for the probability of having such ASE photons guided in the core. From the model, we numerically extract a function $f$, which maps the backwards directed ASE power $P_{\mathrm {ASE}}$ to the spatially integrated population $N_i$ of the corresponding laser-active or ground level:
$$N_i(t) = \int_0^L\partial z~n_i(t, z) = f_i(P_{\mathrm{ASE}}),$$
e.g. the function $f$ maps the Er$^{3+}$-ASE power to $N_1(t)$ and $N_2(t)$ or the Yb$^{3+}$-ASE power to $N_5(t)$ and $N_6(t)$. The mapping function was tested to be reasonably robust against deviation of the experimental parameters by systematically varying these within the respective error intervals, e.g. $\Delta L=\pm 2\,\mathrm {mm}$ and fiber parameters within the manufacturer’s specifications. However, only relative deviations of $f_i$ less than $\sim 0.5\,\%$ were found. We also ensured by parameter sweeping that the function is bijective, so that it gives an appropriate measure of $N_i(t)$ when $P_{\mathrm {ASE}}$ is given.

3. Experimental results

The experimental setup is shown in Fig. 1 (bottom left): a single-mode diode at $0.98\,$µm is used to pump the fiber under test (FUT). Three different commercially available fibers were tested: TC1500Y(6/125)HD (Fibercore Ltd.), SM-EYDF-6/125-HE (Nufern Inc.) doped with Er$^{3+}$:Yb$^{3+}$ and a purely Er$^{3+}$ doped fiber Er30-4/125 (nLight Inc.) to remove the influence of Yb$^{3+}$. The fibers are designated as FUT1, FUT2 and FUT3 respectively and their respective doping concentrations are summarized in Fig. 3. In every case, the FUT is 4 cm-long and the endface is cleaved with 8$^\circ$-angle to prevent Fresnel reflections. The FUTs are pumped with pulses at 976 nm with a maximum peak power of 350 mW, a fixed frequency of 1 Hz and 50 % duty cycle to ensure full excited state depletion between pump pulses. A set of two wavelength division multiplexers (WDMs) retrieves the back-propagating light and separates the components corresponding to the Er$^{3+}$ and Yb$^{3+}$ ASE emission bands. All signals are measured using InGaAs photodiodes with 140 MHz bandwith and an oscilloscope with 400 MHz bandwidth at a sampling rate of 100 kHz. Additionally a sample of the pump pulse is detected as a rise time reference. $N_1(t)$, $N_2(t)$ and $N_5(t)$, $N_6(t)$ have been calculated from the ASE transients mapped with $f$ and are depicted in Fig. 2 color-coded for different pump intensities and solid or dashed to distinguish between the energy levels. When the pump pulse arrives, the ions are excited from their ground state into their respective higher energy levels, so $N_1(t=0\,\mathrm {s})=N_5(t=0\,\mathrm {s})=1$ applies. In the process, the Yb$^{3+}$ ions are primarily excited via the direct channel ($W_{56}$) instead of the backwards directed energy transfer ($R_{63}$). The reason for that is the high probability of Yb$^{3+}$ to absorb a pump photon due to the doping concentration ratio $N_{\mathrm {Yb}}/N_{\mathrm {Er}} \approx 10$ and also cross-sections being larger than for Er$^{3+}$. As a consequence, an Yb$^{3+}$ ion can be described as a 2-level-system that does not allow more depletion of the $^2$F$_{{7}/{2}}$ level than 50 %. The energy transitions of the Yb$^{3+}$ ions take place in the first few $100\,$µs to $500\,$µs, whereby the span decreases with increasing pump intensity. The Er$^{3+}$ ions are excited to the short-lived energy level $^4$I$_{{11}/{2}}$ via direct pump absorption ($W_{13}$) or forwards energy transfer ($R_{63}$), from where they pass to the laser-active $^4$I$_{{13}/{2}}$ state through multi-phonon relaxation. However, the population of the $^4$I$_{{11}/{2}}$ and $^4$I$_{{9}/{2}}$ levels are negligibly small due to the short lifetimes. Accordingly, the curves of the $^4$I$_{{15}/{2}}$ and $^4$I$_{{13}/{2}}$ levels show a mirror-symmetrical course. As a consequence of the quasi-3-level-structure with the forwards directed energy transfer also providing additional inversion, the ground state can be depleted entirely. Here, the dynamics take place on a time scale from $500\,$µs to several ms. The transient populations allow conclusions to be drawn about the transient gain coefficients $g_{\mathrm {Yb}}(\lambda ,t)$ and $g_{\mathrm {Er}}(\lambda ,t)$ within the Yb$^{3+}$ and Er$^{3+}$ ASE bands respectively by using the following equations:

$$g_{\mathrm{Yb}}(\lambda, t) = ~\sigma_{65}(\lambda)\cdot N_6(t) - \sigma_{56}(\lambda)\cdot N_5(t)$$
and
$$g_{\mathrm{Er}}(\lambda, t) = ~\sigma_{21}(\lambda)\cdot N_2(t) - \sigma_{12}(\lambda)\cdot N_1(t).$$
In the case of a high-power cladding-pumped EYDFA, an interlock system must be able to turn off the pumping process before the total gain exceeds the internal losses and uncontrolled ASE or parasitic lasing occurs. For our calculations, we assumed that this requires the fiber to provide a gain of at least 30 dB. However, the actual critical gain value is difficult to estimate and can vary for different configurations. Thus, we verified that the following conclusions and in particular the time scales of our results are unaffected by that choice, so that the core-pump experiments accurately mimic the high-power scenario. By setting Eq. (11) and Eq. (12) to 30 dB gain, we obtain the equation
$$g_{\mathrm{Yb/Er}}(\lambda, t)= 3\ln{10}.$$
Equation (13) contains the implicit relationship $t_{\mathrm {Yb/Er}}(\lambda )$ that can be interpreted as time needed for a gain build-up of 30 dB at a specific wavelength $\lambda$ and that is plotted in Fig. 3 for FUT1.

 figure: Fig. 2.

Fig. 2. Time traces of $N_5(t)$, $N_6(t)$ (left) and $N_1(t)$, $N_2(t)$ (right) for representative pump intensities. The levels $N_5(t)$ and $N_1(t)$ are depicted with a dashed line.

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 figure: Fig. 3.

Fig. 3. Bottom left: Visualization of $t_{\mathrm {Yb/Er}}(\lambda )$ for FUT1 and different pump intensities. Inset: Evolution of backwards directed Yb$^{3+}$ and Er$^{3+}$ ASE for a) pumping without seed and b) pumping with seed but switching off seed at $t=0\,$s. Top left: Corresponding Yb$^{3+}$ and Er$^{3+}$ emission and absorption cross-sections. Right: Minimum of $t_{\mathrm {Yb/Er}}(\lambda )$ versus doping concentration of corresponding dopant (Yb$^{3+}$: red, Er$^{3+}$: blue). The dashed lines are a guide to the eye.

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As a striking result, Fig. 3 (left) shows that FUT1 already provides a gain of $30\,$dB in the Yb$^{3+}$ band after $\sim 80$ µs. This time span is an order of magnitude smaller than for wavelengths in the Er$^{3+}$ band, here $\sim 300\,$µs. The reason for that is the larger population rate of $^2$F$_{{5}/{2}}$ due to the higher Yb$^{3+}$ doping concentration and the cross-sections being larger than for Er$^{3+}$. However, $t_{\mathrm {Yb}}(\lambda )$ and $t_{\mathrm {Er}}(\lambda )$ are not fixed but scale with the pump intensity, which can enhance the population rate of the the laser-active Yb$^{3+}$ and Er$^{3+}$ levels and therefore decreases the period until the fiber shows a positive net gain. Above $\sim 1\,\mathrm {MWcm}^{-2}$ pump intensity, $t_{\mathrm {Yb}}(\lambda )$ and $t_{\mathrm {Er}}(\lambda )$ quench significantly given by the inverse relationship between the excitation rate and the time to saturate the fiber. The FUTs show mutually similar results with the time being minimal at 1030 nm with $81 \pm 5$ µs (FUT1) and $77 \pm 5$ µs (FUT2) and being minimal at 1530 nm with $237 \pm 30$ µs (FUT1), $386 \pm 50$ µs (FUT2) and $278 \pm 50$ µs (FUT3), here at $\sim 1\,\mathrm {MWcm}^{-2}$. The pronounced shoulder around $990\,$nm is given by the minimum of the Yb$^{3+}$ emission cross-section. The results are summarized in Fig. 3 (right) and illustrate the dependence on the doping concentration of the corresponding dopant (see dashed line). The reason lies in the link between increased pump light absorption and doping concentration as in heavily doped fibers more ions will be excited within a given period of time. In turn, more gain is available on a shorter time scale and moreover higher maximum gain levels are enabled. We also note that codoping with Yb$^{3+}$ shows no significant influence on the Er$^{3+}$ ASE dynamics.

Although allowing higher pump power levels due to its pump architecture, classical cladding-pumped EYDFAs show pump intensities in the core that are similar to the experiment presented in this work. In the case of [6], pump intensity and doping concentrations are consistent with the fibers used in this work being 0.34 MWcm$^{-2}$, $N_{\mathrm {Er}}=3.7\times 10^{25}\,\mathrm {m}^{-3}$ and $N_{\mathrm {Yb}}=1.3\times 10^{26}\,\mathrm {m}^{-3}$.

In real amplifier operation, the excited levels $N_6$ and $N_2$ are already populated at the time of the seeder failure. However, this does not significantly affect the respective rise times of the ASE levels. We verified this fact by means of simulations. As example, the evolution of the backwards directed Yb$^{3+}$ and Er$^{3+}$ ASE power is illustrated in the inset in Fig. 3 (left) for the case scenarios a) pumping without seed and b) pumping with seed but switching off seed at $t=0\,$s. Therefore, our results indicate that a suitable interlock system must also be able to react to a seeder shutdown within the same time range as indicated by our core-pump experiments, i.e. in a few tens of µs to avoid the generation of excessive Yb$^{3+}$ ASE whilst the Er$^{3+}$ ASE only occurs after several hundreds of µs.

4. Summary

We separated the backwards directed ASE components of different commercially available Er$^{3+}$ and Yb$^{3+}$ doped fibers core-pumped at $976\,$nm. We applied the data to a time-dependent numerical model and computed the transient integrated populations of the relevant energy levels. We then calculated the period of time during which the FUT show less than 30 dB gain, here $\sim 80\,$µs within the Yb$^{3+}$-band and $\sim 300$ µs within the Er$^{3+}$-band. We noticed that this time scales with the pump intensity and doping concentration. An interlock system must be able to stop the pumping process within these time windows, especially within the Yb$^{3+}$ time window, thus preventing the generation of excessive ASE to meet the challenging requirements of an engineering-level, reliable EYDFA.

Funding

Max-Planck-Institute for Gravitational Physics; Deutsche Forschungsgemeinschaft (EXC 2123 QuantumFrontiers 390837967).

Acknowledgement

This research was funded by the Max-Planck-Institute for Gravitational Physics (Hanover, Germany). This work was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2123 QuantumFrontiers 390837967.

Disclosures

The authors declare no conflicts of interest.

References

1. J. M. Senior, Optical Fiber Communications (Pearson Education Ltd., 2009).

2. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27(11), B63–B92 (2010). [CrossRef]  

3. W. Lee, J. Geng, S. Jiang, and A. W. Yu, “1.8mJ, 3.5kW single-frequency optical pulses at 1572nm generated from an all-fiber MOPA system,” Opt. Lett. 43(10), 2264–2267 (2018). [CrossRef]  

4. Science Team, “Einstein gravitational wave Telescope conceptual design study,” Tor Vergata University of Rome (2011).

5. J. L. Wagener, P. F. Wysocki, M. J. F. Digonnet, H. J. Shaw, and D. J. DiGiovanni, “Effects of concentration and clusters in erbium-doped fiber lasers,” Opt. Lett. 18(23), 2014–2016 (1993). [CrossRef]  

6. O. de Varona, W. Fittkau, P. Booker, T. Theeg, M. Steinke, D. Kracht, J. Neumann, and P. Wessels, “Single-frequency fiber amplifier at 1.5 µm with 100 W in the linearly-polarized TEM00 mode for next-generation gravitational wave detectors,” Opt. Express 25(21), 24880–24892 (2017). [CrossRef]  

7. Q. Han, J. Ning, and Z. Sheng, “Numerical Investigation of the ASE and Power Scaling of Cladding-Pumped Er-Yb Codoped Fiber Amplifiers,” IEEE J. Quantum Electron. 46(11), 1535–1541 (2010). [CrossRef]  

8. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997). [CrossRef]  

9. M. J. F. Digonnet, Rare-Earth-Doped Fiber Lasers and Amplifiers (Marcel Dekker Inc., 2001).

10. D. Marcuse, “Gaussian approximation of the fundamental modes of graded-index fibers,” J. Opt. Soc. Am. 68(1), 103–109 (1978). [CrossRef]  

11. M. Karasek, “Optimum Design of Er3+-Yb3+ Codoped Fibers for Large-Signal High-Pump-Power Applications,” IEEE J. Quantum Electron. 33(10), 1699–1705 (1997). [CrossRef]  

12. D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev. 136(4A), A954–A957 (1964). [CrossRef]  

13. O. de Varona, M. Steinke, D. Kracht, J. Neumann, and P. Wessels, “Influence of the third energy level on the gain dynamics of EDFAs: analytical model and experimental validation,” Opt. Express 24(22), 24883–24895 (2016). [CrossRef]  

14. R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5-6), 375–378 (1997). [CrossRef]  

15. J. Thøgersen, N. Bjerre, and J. Mark, “Multiphoton absorption and cooperative upconversion excitation in Er3+-doped fibers,” Opt. Lett. 18(3), 197–199 (1993). [CrossRef]  

16. M. Federighi and F. D. Pasquale, “The effect of pair-induced energy transfer on the performance of silica waveguide amplifiers with high Er3+/Yb3+ concentrations,” IEEE Photonics Technol. Lett. 7(3), 303–305 (1995). [CrossRef]  

References

  • View by:

  1. J. M. Senior, Optical Fiber Communications (Pearson Education Ltd., 2009).
  2. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27(11), B63–B92 (2010).
    [Crossref]
  3. W. Lee, J. Geng, S. Jiang, and A. W. Yu, “1.8mJ, 3.5kW single-frequency optical pulses at 1572nm generated from an all-fiber MOPA system,” Opt. Lett. 43(10), 2264–2267 (2018).
    [Crossref]
  4. Science Team, “Einstein gravitational wave Telescope conceptual design study,” Tor Vergata University of Rome (2011).
  5. J. L. Wagener, P. F. Wysocki, M. J. F. Digonnet, H. J. Shaw, and D. J. DiGiovanni, “Effects of concentration and clusters in erbium-doped fiber lasers,” Opt. Lett. 18(23), 2014–2016 (1993).
    [Crossref]
  6. O. de Varona, W. Fittkau, P. Booker, T. Theeg, M. Steinke, D. Kracht, J. Neumann, and P. Wessels, “Single-frequency fiber amplifier at 1.5 µm with 100 W in the linearly-polarized TEM00 mode for next-generation gravitational wave detectors,” Opt. Express 25(21), 24880–24892 (2017).
    [Crossref]
  7. Q. Han, J. Ning, and Z. Sheng, “Numerical Investigation of the ASE and Power Scaling of Cladding-Pumped Er-Yb Codoped Fiber Amplifiers,” IEEE J. Quantum Electron. 46(11), 1535–1541 (2010).
    [Crossref]
  8. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997).
    [Crossref]
  9. M. J. F. Digonnet, Rare-Earth-Doped Fiber Lasers and Amplifiers (Marcel Dekker Inc., 2001).
  10. D. Marcuse, “Gaussian approximation of the fundamental modes of graded-index fibers,” J. Opt. Soc. Am. 68(1), 103–109 (1978).
    [Crossref]
  11. M. Karasek, “Optimum Design of Er3+-Yb3+ Codoped Fibers for Large-Signal High-Pump-Power Applications,” IEEE J. Quantum Electron. 33(10), 1699–1705 (1997).
    [Crossref]
  12. D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev. 136(4A), A954–A957 (1964).
    [Crossref]
  13. O. de Varona, M. Steinke, D. Kracht, J. Neumann, and P. Wessels, “Influence of the third energy level on the gain dynamics of EDFAs: analytical model and experimental validation,” Opt. Express 24(22), 24883–24895 (2016).
    [Crossref]
  14. R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5-6), 375–378 (1997).
    [Crossref]
  15. J. Thøgersen, N. Bjerre, and J. Mark, “Multiphoton absorption and cooperative upconversion excitation in Er3+-doped fibers,” Opt. Lett. 18(3), 197–199 (1993).
    [Crossref]
  16. M. Federighi and F. D. Pasquale, “The effect of pair-induced energy transfer on the performance of silica waveguide amplifiers with high Er3+/Yb3+ concentrations,” IEEE Photonics Technol. Lett. 7(3), 303–305 (1995).
    [Crossref]

2018 (1)

2017 (1)

2016 (1)

2010 (2)

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27(11), B63–B92 (2010).
[Crossref]

Q. Han, J. Ning, and Z. Sheng, “Numerical Investigation of the ASE and Power Scaling of Cladding-Pumped Er-Yb Codoped Fiber Amplifiers,” IEEE J. Quantum Electron. 46(11), 1535–1541 (2010).
[Crossref]

1997 (3)

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997).
[Crossref]

M. Karasek, “Optimum Design of Er3+-Yb3+ Codoped Fibers for Large-Signal High-Pump-Power Applications,” IEEE J. Quantum Electron. 33(10), 1699–1705 (1997).
[Crossref]

R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5-6), 375–378 (1997).
[Crossref]

1995 (1)

M. Federighi and F. D. Pasquale, “The effect of pair-induced energy transfer on the performance of silica waveguide amplifiers with high Er3+/Yb3+ concentrations,” IEEE Photonics Technol. Lett. 7(3), 303–305 (1995).
[Crossref]

1993 (2)

1978 (1)

1964 (1)

D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev. 136(4A), A954–A957 (1964).
[Crossref]

Barber, P. R.

R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5-6), 375–378 (1997).
[Crossref]

Bjerre, N.

Booker, P.

Caplen, J. E.

R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5-6), 375–378 (1997).
[Crossref]

Clarkson, W. A.

de Varona, O.

DiGiovanni, D. J.

Digonnet, M. J. F.

Federighi, M.

M. Federighi and F. D. Pasquale, “The effect of pair-induced energy transfer on the performance of silica waveguide amplifiers with high Er3+/Yb3+ concentrations,” IEEE Photonics Technol. Lett. 7(3), 303–305 (1995).
[Crossref]

Fittkau, W.

Geng, J.

Han, Q.

Q. Han, J. Ning, and Z. Sheng, “Numerical Investigation of the ASE and Power Scaling of Cladding-Pumped Er-Yb Codoped Fiber Amplifiers,” IEEE J. Quantum Electron. 46(11), 1535–1541 (2010).
[Crossref]

Hanna, D. C.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997).
[Crossref]

R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5-6), 375–378 (1997).
[Crossref]

Jiang, S.

Karasek, M.

M. Karasek, “Optimum Design of Er3+-Yb3+ Codoped Fibers for Large-Signal High-Pump-Power Applications,” IEEE J. Quantum Electron. 33(10), 1699–1705 (1997).
[Crossref]

Kracht, D.

Lee, W.

Marcuse, D.

Mark, J.

McCumber, D. E.

D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev. 136(4A), A954–A957 (1964).
[Crossref]

Neumann, J.

Nilsson, J.

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27(11), B63–B92 (2010).
[Crossref]

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997).
[Crossref]

R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5-6), 375–378 (1997).
[Crossref]

Ning, J.

Q. Han, J. Ning, and Z. Sheng, “Numerical Investigation of the ASE and Power Scaling of Cladding-Pumped Er-Yb Codoped Fiber Amplifiers,” IEEE J. Quantum Electron. 46(11), 1535–1541 (2010).
[Crossref]

Paschotta, R.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997).
[Crossref]

R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5-6), 375–378 (1997).
[Crossref]

Pasquale, F. D.

M. Federighi and F. D. Pasquale, “The effect of pair-induced energy transfer on the performance of silica waveguide amplifiers with high Er3+/Yb3+ concentrations,” IEEE Photonics Technol. Lett. 7(3), 303–305 (1995).
[Crossref]

Richardson, D. J.

Senior, J. M.

J. M. Senior, Optical Fiber Communications (Pearson Education Ltd., 2009).

Shaw, H. J.

Sheng, Z.

Q. Han, J. Ning, and Z. Sheng, “Numerical Investigation of the ASE and Power Scaling of Cladding-Pumped Er-Yb Codoped Fiber Amplifiers,” IEEE J. Quantum Electron. 46(11), 1535–1541 (2010).
[Crossref]

Steinke, M.

Theeg, T.

Thøgersen, J.

Tropper, A. C.

R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5-6), 375–378 (1997).
[Crossref]

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997).
[Crossref]

Wagener, J. L.

Wessels, P.

Wysocki, P. F.

Yu, A. W.

IEEE J. Quantum Electron. (3)

Q. Han, J. Ning, and Z. Sheng, “Numerical Investigation of the ASE and Power Scaling of Cladding-Pumped Er-Yb Codoped Fiber Amplifiers,” IEEE J. Quantum Electron. 46(11), 1535–1541 (2010).
[Crossref]

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997).
[Crossref]

M. Karasek, “Optimum Design of Er3+-Yb3+ Codoped Fibers for Large-Signal High-Pump-Power Applications,” IEEE J. Quantum Electron. 33(10), 1699–1705 (1997).
[Crossref]

IEEE Photonics Technol. Lett. (1)

M. Federighi and F. D. Pasquale, “The effect of pair-induced energy transfer on the performance of silica waveguide amplifiers with high Er3+/Yb3+ concentrations,” IEEE Photonics Technol. Lett. 7(3), 303–305 (1995).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

R. Paschotta, J. Nilsson, P. R. Barber, J. E. Caplen, A. C. Tropper, and D. C. Hanna, “Lifetime quenching in Yb-doped fibres,” Opt. Commun. 136(5-6), 375–378 (1997).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. (1)

D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev. 136(4A), A954–A957 (1964).
[Crossref]

Other (3)

J. M. Senior, Optical Fiber Communications (Pearson Education Ltd., 2009).

Science Team, “Einstein gravitational wave Telescope conceptual design study,” Tor Vergata University of Rome (2011).

M. J. F. Digonnet, Rare-Earth-Doped Fiber Lasers and Amplifiers (Marcel Dekker Inc., 2001).

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Figures (3)

Fig. 1.
Fig. 1. Bottom left: Experimental setup. WDM: wavelength division multiplexer, FUT: fiber under test, PD: photo detector. Top left: Typical Er$^{3+}$ and Yb$^{3+}$ band ASE power traces of FUT1 at different pump intensities (see Fig. 3). Right: Er$^{3+}$:Yb$^{3+}$ energy levels linked by a dipole-dipole ($R_{ij}$) interaction with lifetimes $\tau _{ij}$ and stimulated transfer rates $W_{ij}$ between the $i$-th and $j$-th level.
Fig. 2.
Fig. 2. Time traces of $N_5(t)$, $N_6(t)$ (left) and $N_1(t)$, $N_2(t)$ (right) for representative pump intensities. The levels $N_5(t)$ and $N_1(t)$ are depicted with a dashed line.
Fig. 3.
Fig. 3. Bottom left: Visualization of $t_{\mathrm {Yb/Er}}(\lambda )$ for FUT1 and different pump intensities. Inset: Evolution of backwards directed Yb$^{3+}$ and Er$^{3+}$ ASE for a) pumping without seed and b) pumping with seed but switching off seed at $t=0\,$s. Top left: Corresponding Yb$^{3+}$ and Er$^{3+}$ emission and absorption cross-sections. Right: Minimum of $t_{\mathrm {Yb/Er}}(\lambda )$ versus doping concentration of corresponding dopant (Yb$^{3+}$: red, Er$^{3+}$: blue). The dashed lines are a guide to the eye.

Equations (13)

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W i j ( z , t ) = Γ λ σ i j A c h c P ( z , t )
Γ = 1 exp ( 2 r c 2 r m 2 ) .
n 1 t = W 21 n 2 + W 31 n 3 W 12 n 1 W 13 n 1 + R 36 N Y b n 3 n 5 R 63 N Y b n 6 n 1 + C u p N E r n 2 2 + n 2 τ 21
n 2 t = W 12 n 1 W 21 n 2 2 C u p N E r n 2 2 + n 3 τ 32 n 2 τ 21
n 3 t = W 13 n 1 W 31 n 3 + R 63 N Y b n 6 n 1 R 36 N Y b n 5 n 3 + n 4 τ 43 n 3 τ 32
n 4 t = C u p N E r n 2 2 n 4 τ 43
n 5 t = W 65 n 6 W 56 n 5 + R 63 N E r n 6 n 1 R 36 N E r n 5 n 3 + n 6 τ 65
n 6 t = W 56 n 5 W 65 n 6 R 63 N E r n 6 n 1 + R 36 N E r n 5 n 3 n 6 τ 65 .
Δ P ( z , t ) =   W n e t ( z , t ) h c λ + β W S E ( z , t ) h c λ
N i ( t ) = 0 L z   n i ( t , z ) = f i ( P A S E ) ,
g Y b ( λ , t ) =   σ 65 ( λ ) N 6 ( t ) σ 56 ( λ ) N 5 ( t )
g E r ( λ , t ) =   σ 21 ( λ ) N 2 ( t ) σ 12 ( λ ) N 1 ( t ) .
g Y b / E r ( λ , t ) = 3 ln 10 .

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