Investigation of ion-ion interaction effects on Yb 3 + -doped fiber amplifiers

: Ytterbium (Yb 3 + )-doped materials have been widely used for high eﬃciency high energy laser sources at the 1 µm wavelength region because of their very low quantum defect and the unique simple energy level structure of Yb 3 + , resulting in no excited-state absorption and low occurrence probability of deleterious ion-ion interaction processes. It has been generally recognized that these ion-ion interaction processes have very little inﬂuence on the operation of Yb 3 + -doped ﬁber lasers at low and moderate power levels. However, our recent study shows that the performance of Yb 3 + -doped ﬁber ampliﬁers operating at low power levels is still inﬂuenced by the ion-ion interaction processes due to the large amount of population at the upper laser level 2 F 5 / 2 . In this paper, experimental evidences of the ion-ion interaction eﬀects in Yb 3 + -doped ﬁber ampliﬁers are presented and a new model including these eﬀects is developed for the numerical simulation. Our experimental and numerical investigations on the 976 nm and 1030 nm Yb 3 + -doped silica and phosphate ﬁber ampliﬁers show that ion-ion interaction has non-negligible impact on the performance of Yb 3 + -doped ﬁber ampliﬁers indeed, and compared to Yb 3 + -doped silica ﬁbers, Yb 3 + -doped phosphate ﬁbers


Introduction
Ytterbium (Yb 3+ )-doped crystals, ceramics, glasses, and fibers have become the most preferable gain media for high energy laser systems because of the unique properties of Yb 3+ , including simple energy level structure, broad absorption band, and high emission efficiency.Yb 3+ has the ground-state manifold ( 2 F 7/2 ) and only one excited state manifold ( 2 F 5/2 ) with an energy gap of ∼ 10000 cm −1 at near infrared.Its other upper energy levels are located in the deep-ultraviolet (UV) range.Therefore, an Yb 3+ laser system doesn't have excited-state absorption, cross-relaxation, energy upconversion transfer and other deleterious energy transfer processes that have shown constraints on most rare-earth-based laser systems, such as neodymium (Nd 3+ ), erbium (Er 3+ ) and thulium (Tm 3+ ).Owing to the outstanding heat dissipation capability of optical fibers and advances in fiber laser technology, Yb 3+ -doped fiber lasers are able to produce 10s-kW single-mode and 100s-kW multimode continuous-wave output powers [1,2].However, it has been found that the long-term operation of a high power Yb 3+ -doped fiber laser system, especially a pulsed laser system, is still influenced by photodarkening [3][4][5] when the doping level of the Yb 3+ -doped fiber is high.Photodarkening in the Yb 3+ -doped fiber lasers has been studied extensively and can be effectively mitigated by several techniques [6][7][8][9][10][11][12][13][14].However, the other ion-ion interaction effects, such as concentration quenching and cooperative luminescence, have not been thoroughly investigated due to their negligible impact on the Yb 3+ -doped fiber lasers operating at long wavelengths (> 1 µm).
In recent years, high power Yb 3+ -doped fiber laser sources operating below 1 µm, especially at 976 nm, have attracted great interest due to the great demand for high beam-quality, high brightness and high stability pump lasers for nonlinear wavelength convertors at the blue wavelength region and further at the deep-UV.However, for Yb 3+ -doped fiber lasers and amplifiers operating at 976 nm, where the emission and absorption cross-sections of Yb 3+ are close, intense pump is required to excite more than 50% population to the upper laser level to achieve a gain.Consequently, the ion-ion interaction effects in the 976 nm Yb 3+ -doped fiber lasers and amplifiers become significant and severely influence their performances because these effects are highly dependent on the excited populations in the upper laser level.Therefore, it is very necessary and valuable to investigate these effects in the 976 nm Yb 3+ -doped fiber lasers and amplifiers and obtain instructions for the design and development of high power 976 nm fiber laser sources.
It is well known that the ion-ion interaction becomes stronger with the increased doping level of rare-earth ions due to the ion clustering and reduced distance between single ions.Compared to silica glass, phosphate glass has much higher solubility and rare-earth doped phosphate fibers have exhibited much less quenching effect [14] and photodarkening [15].Therefore Yb 3+ -doped phosphate fibers are superior to counterpart silica fibers for high power 976 nm fiber lasers.Most recently, we have developed a 10-watt-level 976 nm single-frequency fiber amplifier with 45 cm 1.5 wt% Yb 3+ -doped phosphate fiber [16].During the experiments, we noticed that the performance of the 976 nm Yb 3+ -doped phosphate fiber amplifier was still influenced by the ion-ion interaction effects.Therefore, experimental and numerical investigations on the ion-ion interaction effects in Yb 3+ -doped phosphate fibers are very beneficial for further power scaling of the 976 nm fiber lasers.In this paper, ion-ion interaction effects in Yb 3+ -doped fiber amplifiers are confirmed with solid experiment evidences and a new model including these ion-ion interaction effects is developed for the simulation of Yb 3+ -doped fiber amplifiers.The experimental and simulation results of Yb 3+ -doped silica and phosphate fiber amplifiers are presented to verify the new model.

Ion-ion interaction effects in Yb 3+ -doped fiber amplifiers
In this section, ion-ion interaction effects in Yb 3+ -doped glasses and fibers are confirmed by the experimental investigations on the lifetime quenching, photodarkening, and cooperative luminescence.

Lifetime quenching
It has been found that, in the rare-earth doped glasses and fibers, the fluorescence lifetimes of the excited ions always decrease with the increased concentration.The reduction of the decay time, namely lifetime quenching, results from several different mechanisms such as multi-phonon transitions, energy transfer between the doped rare earth ions, and energy transfer to the impurities or color centers.Due to the simple energy level structure of Yb 3+ , lifetime quenching usually occurs in an Yb 3+ -doped material at a doping level much higher than in other rare-earth-doped materials.Concentration quenching of Yb 3+ ions in silica fiber was first reported by R. Paschotta et al. in 1997 [3].Jetschke et al. found that the near infrared fluorescence lifetime of Yb 3+ in silica fiber decreases with the increased doping level as the concentration of Yb 3+ exceeds 2×10 26 m −3 [17].Due to the high solubility of phosphate glass, the lifetime quenching generally happens at much higher concentration of Yb 3+ compared to silica glass [14].We have made phosphate glasses doped with different Yb 3+ concentrations and measured their florescence lifetimes of the upper laser level ( 2 F 5/2 ) from the fluorescence decay by pumping 500-µm-thick Yb 3+ -doped phosphate slabs with a 10 ns pulsed laser source at 915 nm (SureLite TM OPO Plus).It should be noted that the effect of re-absorption or radiation trapping on the lifetime measurement was negligible due to the very small thickness of the glass slab and very short duration of the pump pulse.The measured lifetimes of Yb 3+ -doped phosphate glasses with concentrations of 2 wt%, 4 wt%, 6 wt% and 13.55 wt% are shown in Fig. 1.Lifetime quenching was not measured even when the concentration is as high as 6 wt% (6.69×10 26 m −3 ) and the fluorescence lifetime decreases to 0.986 ms when the concentration is 13.55 wt% (1.51×10 27 m −3 ).The resistance of concentration quenching for phosphate fiber is 7.5 times higher than that of silica fiber.Therefore, the effect of lifetime quenching on Yb 3+ -doped phosphate fiber laser is negligible when the concentration is 6 wt% or less.

Photodarkening
Photodarkening, the transmission of an optical material in the visible and near-infrared decreases as it is irradiated by high power lasers, especially at UV and visible, has been widely observed in optical fibers and laser crystals.Due to the small diameters of optical fibers, the light intensity in the fiber core is always very high and consequently photodarkening can occur even in Yb 3+ -doped fiber lasers at 1 µm and influence their efficiency and long-time operation stability.Since Koponen et al. reported the photodarkening in an Yb 3+ -doped fiber laser in 2005 [4], photodarkening has been extensively investigated and different mechanisms have been proposed to explain this deleterious effect [18][19][20].It is widely agreed that photodarkening in Yb 3+ -doped fiber lasers is related to ion clustering and started with the energy transfer between the Yb 3+ ions [21,22].Therefore, photodarkening can be used to characterize the ion clustering in an optical fiber.Since the effect of photodarkening depends on the irradiation laser wavelength and it has been mostly investigated in Yb 3+ -doped fiber lasers operating at > 1 µm, it should be significant for us to study the photodarkening in Yb 3+ -doped fiber amplifiers and investigate its effect on fiber lasers at 976 nm.
An experimental setup shown in Fig. 2 was used to measure the photodarkening effect in Yb 3+ -doped fiber amplifiers.A 976 nm distributed feedback (DFB) laser diode (LD) and a 1030 nm single-frequency fiber laser (Rock, NP Photonics) were used as the signal laser source alternatively.A 915 nm LD was used as the pump laser.The signal laser and the pump laser were coupled together and launched into the gain fiber via a 915/976 nm wavelength division multiplexer (WDM).An Yb 3+ -doped silica fiber (Nufern PM-YDF-5/130-VIII) with a cladding absorption of about 0.6 dB at 915 nm and an Yb 3+ -doped phosphate fiber (6/125 µm) with a concentration of 6.69×10 26 m −3 were tested.Both fibers have a length of 1 cm.The transmission spectrum of the fresh 1-cm Yb 3+ -doped active fiber was measured by using a white light source (Yokogawa AQ4305) and an optical spectrum analyzer (Yokogawa AQ6315A) before launching any laser into the active fiber.Then a pump laser of 200 mW at 915 nm, or the pump laser and a signal laser of 1 mW together were launched into the active fiber for one hour.After that, the transmission spectrum of the irradiated Yb 3+ -doped fiber was measured.The photodarkening effect can be characterized by the difference between the transmission spectra of the fresh fiber and the irradiated fiber, i.e. the change of the transmission before and after the laser illumination.The photodarkening effects on the transmission spectra of the 1-cm Yb 3+ -doped silica fiber amplifiers and phosphate fiber amplifiers are shown in Figs.3(a) and 3(b), respectively.Clearly, the transmission change of the Yb 3+ -doped silica fibers [Fig.3(a)] at the visible is significant while that of Yb 3+ -doped phosphate fibers [Fig.3(b)] is negligible, indicating that phosphate fiber has much higher resistant to photodarkening due to the much less ion clustering.The ion clustering in the Yb 3+ -doped phosphate fiber with a concentration of 6.69×10 26 m −3 is even much less than that in the Yb 3+ -doped silica fiber with a concentration of 9.7×10 25 m −3 .As shown in Fig. 3(a), the magnitude of the photodarkening effect in the Yb 3+ -doped silica fiber illuminated only by the 915 nm pump laser is larger than that in the two Yb 3+ -doped silica fiber amplifiers and that in the 1030 nm fiber amplifier is larger than that in the 976 nm fiber amplifier.It is worthwhile to note that the output signal powers of the 976 nm and 1030 nm 1-cm silica fiber amplifiers were measured to be 1.6 mW and 1.1 mW, respectively, indicating that more excited Yb 3+ ions decay to the ground state in 976 nm fiber amplifier than in the 1030 nm fiber amplifier.Thus, these experimental results of Yb 3+ -doped silica fibers tell us that the photodarkening effect is highly related to the population in the upper laser level ( 2 F 5/2 ) in addition to the concentration of the clustered ions.When the Yb 3+ -doped silica fiber was illuminated only by the pump laser at 915 nm, the excited ions decay spontaneously to the ground state and thus the population in the 2 F 5/2 level is much more than that of the fiber amplifiers, in which the excited ions decay to the ground stage more quickly due to the stimulated emission.As a result, the photodarkening-induced loss is the largest in this case.For the 976 nm and 1030 nm fiber amplifiers, since the gain fiber length was only 1 cm, the whole gain fiber was pumped sufficiently.Because the emission cross-section of Yb 3+ at 976 nm is larger than that at 1030 nm, the population in the 2 F 5/2 level of the 1030 nm fiber amplifier is more than that of the 976 nm fiber amplifier and consequently the photodarkening-induced loss is larger as validated by experimental results shown in Fig. 3(a).
Photodarkening is one of the major constraints for power scaling of Yb 3+ -doped fiber lasers.Various techniques have been used to mitigate photodarkening in Yb 3+ -doped silica fibers by optimizing the core glass composition.Co-doping with aluminum (Al) [9], phosphorous (P) [10], cerium (Cr) [11], sodium (Na) [12] and alkaline earth metals [13] has been proven effective to improve the photodarkening resistant.It is worth to note that the commercial Yb 3+ -doped silica fiber used in our experiment was co-doped with Al [23], which can improve the solubility of the SiO 2 glass and reduce the formation of Yb 3+ ion clustering.Therefore, compared to Yb 3+ -doped pure silica fiber, remarkable reduction of photodarkening was achieved in Yb 3+ -Al co-doped silica fiber.Nevertheless, our experiment shows that the photodarkening in Yb 3+ -doped phosphate fiber is much less than in Yb 3+ -doped silica fiber co-doped with Al.Therefore Yb 3+ -doped phosphate fibers are more suitable for high power fiber lasers and fiber amplifiers operating below 1 µm.

Cooperative luminescence
Cooperative luminescence is an upconversion radiative process that two neighboring excited ions decay simultaneously to the ground state and emit a photon with their combined energy.Cooperative luminescence was first observed in YbPO 4 by Nakazawa and Shionoya, in 1970 [24] and thereafter has been investigated in various Yb 3+ -doped glasses and crystals [25][26][27][28][29].The cooperative luminescence in Yb 3+ -doped fibers has also been studied to characterize the level of ion clustering [30].Cooperative luminescence in Yb 3+ -doped phosphate glass was investigated by Bell et al in 2003 [31].However, there is no report on the cooperative luminescence in Yb 3+ -doped phosphate fibers.On the other hand, most of the previous research works focused on underlying mechanism of cooperative luminescence [31,32].Most recently, Vallés et al, reported their investigation of cooperative luminescence on the performance of Yb 3+ -doped fiber lasers operating at 1090-1100 nm and found that the impact is negligible [33].However, the cooperative luminescence in Yb 3+ -doped fiber amplifiers should be very different from that in Yb 3+ -doped fiber lasers operating at 1090-1100 nm, in which a very small fraction of ions (∼ 5%) are excited.In a 976 nm Yb 3+ -doped fiber amplifier, more than 50% of the population needs to be excited to provide gain throughout the gain fiber.The large number of population in the 2 F 5/2 level makes it prone to the cooperative luminescence.Therefore, it is very valuable to investigate the cooperative luminescence in the 976 nm Yb 3+ -doped fiber amplifiers.
The cooperative luminescence in an Yb 3+ -doped silica fiber (Nufern PM-YDF-5/130-VIII) and a 6/125 µm 6 wt% Yb 3+ -doped phosphate fiber were investigated with the experimental setup shown in Fig. 4. The 915 nm pump laser and the signal laser at 976 nm or 1030 nm were coupled together by a WDM and launched into 1-cm active fiber.A spectrometer (Ocean 2000+) was used to measure the cooperative luminescence side emitting from the active fibers at the same position.The output gain fiber output end was angle-cleaved to suppress possible parasitic lasing.
The spectra of the cooperative luminescence at the range of 480-550 nm from the Yb 3+ -doped silica fiber amplifiers at 976 nm and 1030 nm were measured by the spectrometer with an integration time of 100 ms and are shown in Figs.5(a) and 5(b), respectively.Clearly, the intensity of the cooperative luminescence increases with the increased pump power for both fiber amplifiers.When there was no pump laser, cooperative luminescence was still measured from the 976 nm fiber amplifier while no cooperative luminescence was measured from the 1030 nm fiber amplifier.This is because Yb 3+ ions have strong absorption at 976 nm and a large number of Yb 3+ ions are excited by the 976 nm signal laser.When both fiber amplifiers were pumped by the 915 nm pump laser at the same power, the intensity of the cooperative luminescence of the 1030 nm Yb 3+ -doped fiber amplifier was larger than that of the 976 nm Yb 3+ -doped fiber amplifier.This is because the population in the 2 F 5/2 level of the 1030 nm fiber laser is more than that of the 976 nm fiber amplifier as analyzed above.All these experimental results tell us that the intensity of cooperative luminescence depends on the population density of the excited ions.The spectra of the cooperative luminescence from the Yb 3+ -doped phosphate fiber amplifiers at 976 nm and 1030 nm were measured by the spectrometer with an integration time of 1 s and are shown in Fig. 5(c) and 5(d), respectively.Similar to the observations of the Yb 3+ -doped silica fiber amplifiers, the cooperative luminescence of the Yb 3+ -doped phosphate fiber amplifiers increases with the increased pump power and that of the 1030 nm fiber amplifier is stronger than that of the 976 nm fiber amplifier.It is worth to note that the cooperative luminescence from the Yb 3+ -doped phosphate fiber amplifiers is much weaker than that from the Yb 3+ -doped silica fiber amplifiers because the integration time for Figs.5(c) and 5(d) is only one tenth of that for Figs.5(a) and 5(b).Because cooperative luminescence is attributed to the energy migration among the clustered Yb 3+ ions, the weak cooperative luminescence indicates that the energy migration is very low in Yb 3+ -doped phosphate glass due to the well-dispersed Yb 3+ ions.It is interesting that the cooperative luminescence of Yb 3+ -doped phosphate fiber amplifiers exhibits different features from that of Yb 3+ -doped silica fiber amplifiers.The cooperative luminescence of the silica fiber amplifier has a major peak at 502 nm and a secondary peak at 491 nm while that of the phosphate fiber amplifier has only one peak at 500 nm, which is attributed to the different features of the emission cross-sections of Yb 3+ doped in silica and phosphate glasses.
The spectra of the cooperative luminescence of the 1-cm Yb 3+ -doped silica and phosphate fiber amplifiers were measured with different signal powers at 976 nm and the same pump power of 200 mW at 915 nm and are shown in Figs.6(a) and 6(b), respectively.Both of them show that the intensity of the cooperative luminescence decreases with the increased signal laser power.This is because the population in the 2 F 5/2 level becomes smaller as the input signal power becomes higher and consequently the stimulated emission becomes stronger.The experimental results tell us that the energy migration among the excited ions plays an important role in cooperative luminescence and the intensity of the cooperative luminescence is related to the population density in the 2 F 5/2 level.Because cooperative luminescence depletes two excited Yb 3+ ions via radiation at visible not at the signal wavelength, it indeed influences the performance of Yb 3+ -doped fiber amplifiers.

Theoretical model including ion-ion interaction effects
Our experimental results have shown that the ion-ion interaction effects are highly related to the population density in the 2 F 5/2 level and influence the performance of Yb 3+ -doped fiber amplifiers, especially the 976 nm fiber amplifiers in which more than 50% Yb 3+ ions need to be excited along the whole gain fiber.In order to understand and predict the performance of the 976 nm Yb 3+ -doped fiber amplifiers, a new theoretical model including the ion-ion interaction effects is needed.The schematic of energy migrations among the Yb 3+ ions and acceptors such as color centers and impurities and the transitions of radiative decays are shown in Fig. 7.The near-infrared (NIR) light is generated by the singly Yb 3+ ions through the radiative decay from the excited state 2 F 5/2 to the ground state 2 F 7/2 (orange arrow).The visible light is generated by the interaction of Yb 3+ -dimers (ion-pair) through a radiative decay from an upper virtual level to the ground state (green arrow).Although cooperative luminescence at UV was not observed in our experiment due to the strong absorption of UV light of the optical glass fiber and the induced color centers in the Yb 3+ -doped fiber core, UV radiations could be produced by the interaction of the clusters of more than two Yb 3+ ions, i.e.Yb 3+ -trimers and Yb 3+ -tetramers, through the virtual transitions and absorbed by the acceptors represented by the light and deep purple arrows [34].In addition to the radiative decays, there exists nonradiative processes (black arrows) in which the energies of the excited Yb 3+ ions are transferred to the acceptors including color centers, defects, impurities, etc, and they decay to the ground state without emitting any photons.
We have developed a model including all of these ion-ion interactions.The rate equations are written as follows: (1) where R 12 = σ a,λP P P /(hν P A eff ,λP ) and R 21 = σ e,λP P P /(hν P A eff ,λP ) are the absorption and emission rates of the pump laser.σ a,λP and σ e,λP are the absorption and emission cross-sections at the pump wavelength.P P is the pump power.h is the Plank constant.ν P is the frequency of the pump.A eff ,λP is the effective mode area of the pump light.W 12 = σ a,λi P S,λi /(hν λi A eff ,λi ) is the total absorption rate of the signal light and amplified spontaneous emission (ASE) at other wavelengths while W 21 = σ e,λi P S,λi /(hν λi A eff ,λi ) is the total emission rate of all the emission light.σ a,λi and σ e,λi are the absorption and emission cross sections at the wavelength λ i , respectively.P S,λi is the signal or ASE power at the wavelength λ i .ν λi are the frequency of the emission light at the wavelength λ i .A eff ,λi is the effective mode area of the emission light at the wavelength λ i .A 21 = 1/τ 0 is the spontaneous emission rate where τ 0 is the radiative lifetime of the excited Yb 3+ ions.W nr = 1/τ -1/τ 0 [31,[35][36] is the nonradiative rate, where τ is the measured lifetime of the excited Yb 3+ ions.Here, it includes all the non-radiative processes including the multi-phonon relaxation, the energy exchange between Yb 3+ ions due to the clustering effect, energy transfer to impurities.D (m 3 /s) is the cooperative decay rate of the Yb 3+ -pairs, T (m 6 /s) is that of the Yb 3+ -trimers, and Q (m 9 /s) is that of the Yb 3+ -tetramers.As shown in our simulation results below, the influence of the cooperative decay of Yb 3+ -trimers is very small and that of Yb 3+ -tetramers is negligible.Therefore, the cooperative decays of Yb 3+ -tetramers and larger Yb 3+ -clusters can be neglected in the simulation.N 1 and N 2 are the total population densities in the states 2 F 7/2 and 2 F 5/2 .N total is the total concentration of Yb 3+ ions.The evolution equations of the pump, signal and ASE powers along the gain fiber are expressed as: where P P is the pump power and P S,λi is the signal or ASE power at the wavelength λ i ."+" and "-" represent the forward and backward directions, respectively.Γ P is the spatial overlap of the pump with the gain fiber core, which is 1-exp(−2×A core /A eff ,λP ) for single-mode core pumping and is A core /A clad for multimode cladding-pumping, respectively.A core is the area of the gain fiber core.Γ λi = 1-exp(-2×A core /A eff ,λi ) is the spatial overlap of the signal or ASE with the gain fiber core.α P and α S,λi are the propagation losses of the light at the pump wavelength and at the wavelength λ i , respectively.P 0,λi = 2hc 2 /λ i 3 is the power of the spontaneous emission at the wavelength λ i .c is the light velocity in vacuum.
The boundary conditions for the powers of the signal and the ASE can be given by where P input , λi is the input power at the wavelength λ i .L is the length of the gain fiber.The rate equations for steady states were solved to obtain population inversion along the gain fiber and the power propagation equations were solved with the 4th-order Runge-Kutta method.The output power of the fiber amplifier at the wavelength λ i can be obtained by

Experimental and simulation results of Yb 3+ -doped fiber amplifiers
In section 2, our experiments have shown solid evidences of ion-ion interaction effects in Yb 3+doped fiber amplifiers.In this section, the experimental and simulation results of Yb 3+ -doped fiber amplifiers are presented.A good agreement between the experimental results and the simulation results validates of the new modeling including the cooperative decays and confirm the non-negligible influence of ion-ion interaction effects on the performance of Yb 3+ -doped fiber amplifiers.
4.1.Core-pumped Yb 3+ -doped silica fiber amplifiers The experimental setup of core-pumped Yb 3+ -doped fiber amplifiers is depicted in Fig. 8.A 915 nm single-mode laser diode was used as the pump source.A 976 nm DFB laser diode and a 1030 nm single-frequency fiber laser (NP Photonics Rock) were used as the signal laser source alternatively.The signal laser and pump laser are coupled together by a filter-type WDM and launched into the Yb 3+ -doped fiber.The splicing loss between the WDM and the silica gain fiber was measured to be 0.1 dB.The signal laser power launched into the gain fiber was 10 mW.The output end of the gain fiber was angle cleaved.A 976 nm bandpass filter with a transmission bandwidth of 10 nm and a 1000 nm longpass filter were used after the gain fiber to remove the residual pump power and ASE power for the 976 nm and 1030 nm signal laser power measurement, respectively.The output signal laser power was measured by a thermal power sensor (Thorlabs S401C).The measured and simulated output signal laser powers as a function of the 915 nm pump power for the silica fiber (Nufern PM-YDF-5/130-VIII) amplifiers at 976 nm and 1030 nm are shown in Fig. 9(a) and 9(b), respectively.The experimental results are plotted with the solid black dots and the simulation results are plotted with solid lines.In the simulation, the propagation losses were set to be 0.015 dB/m.The radiative lifetime of Yb 3+ in the silica was measured to 0.945 ms [3] and that the Yb 3+ -doped silica fiber was measured to be 0.826 ms.Therefore, the nonradiative rate W nr was set to be 152 s −1 .It is clear that there is large discrepancy between the simulation results and the experimental results (red lines) when the nonradiative decays and cooperative decays were not included in the modeling (W nr = 0, D = 0, T = 0).When W nr is included in the simulation, the calculated threshold of the 976 nm fiber amplifier increases and approaches to the experimental threshold as shown by the blue line in Fig. 9(a).However, the calculated output powers are still larger than the experimental results.Therefore, the model only including nonradiative decay of excited singly Yb 3+ ions cannot simulate the performance of the fiber amplifier accurately.When both W nr and cooperative decay of excited Yb 3+ -dimers D are included in the modeling, the simulation results shown by the green line are in a good agreement with the experimental results and a cooperative decay rate of 1×10 −23 m 3 /s is obtained.When the cooperative decay of excited Yb 3+ -trimers T is also included in the modeling, the simulation results plotted by the orange line have little difference from that of the green line, indicating that the cooperative decays of excited Yb 3+ -trimers or excited larger Yb 3+ -ion-clusters have negligible influence on the performance of the 976 nm Yb 3+ -doped silica fiber amplifiers.
The influences of nonradiative and cooperative decays on the 1030 nm Yb 3+ -doped silica fiber amplifier are shown in Fig. 9(b).When W nr is included, the output powers of the 1030 nm fiber amplifier decrease but they are still larger than the experimental results.A good agreement between the simulation results and the experimental results can be achieved when the cooperative decay of excited Yb 3+ -dimers D is also included.When the cooperative decay of excited Yb 3+ -trimers T is also included in the modeling, the simulation results plotted by the orange line are in a good agreement with the experimental results at low pump powers (< 50 mW) but deviate at high pump power, indicating the cooperative decay rate of excited Yb 3+ -trimers of 2×10 −50 m 6 /s is overestimated.Therefore, the model including W nr and D is accurate enough for the simulation.
To further verify the model, the output powers of the 976 nm and 1030 nm Yb 3+ -doped silica fiber amplifiers with different gain fiber lengths were measured and simulated and are shown in Fig. 10(a) and 10(b), respectively.Clearly, the simulation results are larger than the experimental Fig. 9. Measured and calculated output power as a function of the 915 nm pump power for (a) a core-pumped 12-cm Yb 3+ -doped silica fiber amplifier at 976 nm and (b) a core-pumped 22-cm Yb 3+ -doped silica fiber amplifier.results when D is not included in the modeling, while the simulation results are smaller than the experimental results when T of 2×10 −50 m 6 /s is included the modeling.Therefore, the term T cannot be neglected in the modeling and the term D influences the performance of the Yb 3+ -doped fiber amplifiers indeed.

Core-pumped Yb 3+ -doped phosphate fiber amplifiers
The experimental setup of core-pumped 6/125 µm 6 wt% Yb 3+ -doped phosphate fiber amplifiers is similar to that shown in Fig. 8.The splicing loss between the silica fiber of the WDM and the Yb 3+ -doped phosphate fiber is about 0.5 dB.The propagation loss of the Yb 3+ -doped phosphate fiber was measured to be 2 dB/m by using the 1310 nm DFB laser source (Thorlabs S3FC1310).The lifetime τ 0 of Yb 3+ -doped phosphate was set to be 1.45 ms [34] and the nonradiative rate of the exited singly Yb 3+ ions is 74 s −1 .
The experimental results and the simulation results of the core-pumped 976 nm and 1030 nm Yb 3+ -doped phosphate fiber amplifiers are shown in Fig. 11(a) and 11(b), respectively.Similar to the results of the core-pumped Yb 3+ -doped silica fiber amplifiers, a good agreement between the simulation results and the experimental results is achieved only when the terms W nr and D have been included in the modeling.A cooperative decay rate of 3×10 −25 m 3 /s was obtained.Clearly, the term D of excited Yb 3+ -dimers in phosphate is much smaller than that in silica even though the Yb 3+ doping level of the phosphate fiber is almost 7 times of that of the silica fiber.This attributes to the very high solubility of phosphate glass and thus clustered ions in the phosphate fiber is much lower than that in the silica fiber.The output powers of the 976 nm and 1030 nm Yb 3+ -doped phosphate fiber amplifiers with different gain fiber lengths were also measured and simulated and are shown in Fig. 12(a) and 12(b), respectively.These results are consistent with the results and analyzes above.

Cladding-pumped Yb 3+ -doped fiber amplifiers
To further verify the modeling, we did the simulations for two cladding-pumped Yb 3+ -doped phosphate fiber amplifiers [16].One used a 6 wt% Yb 3+ -doped fiber with the core/cladding diameters of 18/135 µm.The other used a 1.5 wt% Yb 3+ -doped fiber with the core/cladding diameters of 20/130 µm.The lengths of the 6 wt% and 1.5 wt% Yb 3+ -doped fibers are 10 cm and 45 cm, respectively.The output powers of two double-clad phosphate fiber amplifiers as a function of the launched 915 nm pump power were measured and the corresponding values of W nr and D of the two fibers are obtained by fitting with the experimental results.
The experimental results and simulations are shown in Fig. 13.It is clear that the nonradiative rate of single excited Yb 3+ W nr and cooperative decay parameter of excited Yb 3+ -dimer D increase with the increased doping level.The cooperative decay rate of the 6 wt% Yb 3+ -doped phosphate fiber is 3 times of that of the 1.5 wt% Yb 3+ -doped phosphate fiber, indicating that the number of Yb 3+ -dimers increases with the increased Yb 3+ concentration.The term W nr , however, doesn't increase much as the Yb 3+ concentration increases by 4 times.It should be noted that the two phosphate fiber cores' material compositions are almost same except the Yb 3+ concentration.Therefore, the concentrations of the glass defects (OH − , color center, etc.) in the two phosphate fibers should be close and the increase of W nr may only attribute to the increase of Yb 3+ -defect cluster as the Yb 3+ doping level increases.Our experimental and simulation results have confirmed the impact of ion-ion interaction effects on the performance of Yb 3+ -doped fiber amplifiers.Compared to Yb 3+ -doped silica fiber amplifiers, Yb 3+ -doped phosphate fiber amplifiers suffer much less ion-ion interaction effects and thus are preferable platforms for high power 976 nm fiber laser sources.

Conclusion
In conclusion, the ion-ion interaction effects in the Yb 3+ -doped fibers and their influences on the performance of Yb 3+ -doped fiber amplifiers are systematically investigated and analyzed.A new theoretical model including the ion-ion interaction effects has been developed and the simulation results have shown good agreement with the experimental results.Our experimental and simulation results confirm that the ion-ion interaction effects have non-negligible influence on the performance of Yb 3+ -doped fiber amplifiers.Compared to Yb 3+ -doped silica fibers, Yb 3+ -doped phosphate fibers have shown much less ion-ion interaction effects due to their much higher solubility and thus are more preferable for high power 976 nm fiber amplifiers.This study provides the new insight for the design and development of high power 976 nm Yb 3+ -doped fiber laser sources.

Funding
National Science Foundation Engineering Research Center for Integrated Access Networks (EEC-0812072); Technology Research Initiative Fund (TRIF) Photonics Initiative of the University of Arizona.

Fig. 5 .
Fig. 5. Measured spectra of the cooperative luminescence from an Yb 3+ -doped silica fiber when the signal laser is at (a) 976 nm and (b) 1030 nm and from an Yb 3+ -doped phosphate fiber when the signal laser is at (c) 976 nm and (d) 1030 nm for different 915 nm pump powers.The input 976 nm and 1030 nm signal powers were 1 mW.The integration time of the spectrometer for (a) and (b) is 100 ms while that for (c) and (d) is 1 s.

Fig. 6 .
Fig. 6.Measured spectra of the cooperative luminescence from (a) Yb 3+ -doped silica and (b) Yb 3+ -doped phosphate fiber amplifiers for different input 976 nm signal laser powers.The 915 nm pump powers were 200 mW.

Fig. 7 .
Fig. 7. Schematic of energy migration among Yb 3+ ions and acceptors and the transitions of radiative decays.

Fig. 10 .
Fig. 10.Measured and calculated signal laser power as a function of the gain fiber length for (a) 976 nm and (b) 1030 nm Yb 3+ -doped silica fiber amplifier.The input signal power is 10 mW and the launched pump power is 200 mW.

Fig. 11 .
Fig. 11.Output powers of as a function of the 915 nm pump power for (a) a core-pumped 2.8-cm Yb 3+ -doped phosphate fiber amplifier at 976 nm and (b) a core-pumped 4-cm Yb 3+ -doped phosphate fiber amplifier at 1030 nm.

Fig. 12 .
Fig. 12. Measured and calculated signal power as a function of the gain fiber length for (a) 976 nm and (b) 1030 nm Yb 3+ -doped phosphate fiber amplifiers.The input signal power is 10 mW and the launched pump power is 200 mW.

Fig. 13 .
Fig.13.Measured and calculated output power as a function of the 915 nm pump power for a 10-cm 6 wt% Yb 3+ -doped phosphate fiber amplifier and a 45-cm 1.5 wt% Yb 3+ -doped phosphate fiber amplifier, respectively.