Abstract

We report on an extensive investigation of photodarkening in Yb-doped silica fibers. A set of similar fibers, covering a large Yb concentration range, was made so as to compare the photodarkening induced losses. Careful measurements were made to ensure equal and uniform inversion for all the tested fibers. The results show that, with the specific set-up, the stretching parameter obtained through fitting has a very limited variation. This gives more meaning to the fitting parameters. Results tend to indicate a square law dependence of the concentration of excited ions on the final saturated loss. We also demonstrate self-similarity of loss evolution when experimental curves are simply normalized to fitting parameters. This evidence of self-similarity also supports the possibility of introducing a preliminary figure of merit for Yb-doped fiber. This will allow the impact of photodarkening on laser/amplifier devices to be evaluated.

©2011 Optical Society of America

1. Introduction

Power scaling of 1-micron fiber lasers and laser amplifier stages is one of the most important issues in order to provide flexible new tools for manufacturing processes [1]. This often implies high concentration of Ytterbium ions that may induce extra losses due to the photodarkening (PD) process [2]. This process generates propagation losses when the active fiber is pumped and has been reported as well for fibers doped with other rare-earth such as Thulium for example [3]. To make the issue more complex, the preferred core material host for high-power applications, silica, exhibits far higher PD than other type of glass or more complex silica based multi-component glass [46]. Advantages of silica glass, such as compatibility with other optical components and excellent thermal stability and thermal conductivity, justify a huge amount of effort in order to understand, model and solve PD in Yb-doped silica fibers [712]. To our knowledge the mechanism is presently not fully understood and several theories have been proposed. So far there is a common agreement that PD depends on the Yb3+ ion concentration, particularly to the concentration of ions in the excited state, and glass composition. In the last few years, different mechanisms have been proposed to explain the formation of color centers among them charge transfer bands [5] and oxygen deficient centers [10]. However a deep understanding of the structural changes due to pump power irradiation is still far from being reached and is not the purpose of this paper. In this paper we report on an extensive investigation of PD in Yb-doped silica fibers in order to improve the understanding of this process. A large set of fibers was made with similar geometrical parameters so as to compare the PD induced losses over a large concentration range. The Yb3+ concentration range varies from 0.5 to 1.8 wt% and all fibers had similar geometry and aluminum based core composition. The aluminum concentration was measured from the refraction index profile and was about 3%. The lowest and highest doping level fibers had a very slightly lower Al content. We observed that having on hand a set of homogenous samples led to minimized dispersion in the results. In particular we show a strong self-similarity of all curves when we use the fitting parameters to scale the results. We therefore suggest the possibility of introducing a fitting law of induced PD losses and a preliminary figure of merit (FOM) to evaluate fiber quality. The results seem to indicate that there is a square law dependence of the saturated induced losses versus the concentration of excited ions. .

2. Set-up and numerical fitting

To test our fibers we used a standard configuration set-up [7]. A 980-nm pump diode was coupled through a wavelength-division-multiplexer to a 633-nm probe beam. The two combined beams were launched into the optical fiber and the output was analyzed by an optical-spectrum analyzer to monitor the transmitted probe power at 633-nm as a function of time. All our test fibers were spliced at both ends. The formula used to fit our data is the standard stretched exponential function [8]:

α(t)=αeq[1exp{(t/τ)β}],
where α(t) represents the loss induced at a time t after pump is switched on, αeq the loss at the final equilibrium state, τ a time scale and β the stretching parameter, respectively. The limitation of this formula lies in the fact that the stretching parameter combines itself with the time scale parameter and β values were found in the past ranging from 0.3 to 0.7 [9]. We will discuss this point later. One main concern regarding the set-up and its intrinsic capability of measuring a relevant PD level is the fact that the level of inversion of Yb ions strongly impacts the final value of PD induced losses [11]. This means that if the inversion is not uniform along the fiber, the results will not be easily comparable. To this aim we used a pump diode centered on the Yb absorption peak at 976 nm and a pumping power level of about 260 mW suitable to obtain saturation of the inversion level all along the fiber sample and throughout the test duration.

This last point is particularly important because the losses induced by PD at the pumping side of the fiber under test can reduce the inversion level at the opposite end and therefore minimizing the measured PD. To evaluate our set-up, we performed a set of measurements of the fiber Y180 with the highest expected PD induced loss. We tested the same fiber with different sample lengths and Fig. 1 shows the equilibrium losses, the time scale and the stretching parameter normalized to the results obtained with the shortest samples of 1.8 cm.

 

Fig. 1 All three fitting parameters versus fiber length. All values are normalized to the measured values using the shortest samples (1.8 cm).

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Figure 1 shows that for samples longer than 6 cm the measured PD losses is decreasing; the 9.6 cm long samples exhibited a PD loss value that is only 70% of the one measured for sample with length less or equal to 6 cm. Similarly the stretching parameter increases and the time constant decreases for test fibers longer than 6 cm. We also repeated our measurements for the short (1.8 cm) and long (5.6 cm) samples and obtained repeatable results confirming this trend. We then state that when inversion saturation is reached, the pump absorption is proportional to fiber length times the Yb3+ ions concentration and we therefore scaled the length of all our test samples accordingly. The sample lengths were calculated in cm as L = 10/Nyb, where Nyb is the fiber doping level in wt%. Since we pumped at the Yb peak wavelength the above rule will guarantee a percentage of inverted ions of 0.46. The main limitation on the set-up resolution arises from the probe power variation over time (roughly 1dB over 24h). Since the length of the samples depends on Yb3+ concentration, as from the above discussed formula, we estimated our setup resolution to be 10 dB/m wt% since, e.g., a 1 wt% doped sample is 10 cm long and we may see a spurious 1 dB probe power variation corresponding to 10 dB when scaled to a length of 1 meter. On the contrary a 2 wt% doped sample would be only 5 cm long and the overall error could lead to 20 dB/m sensitivity limit. Measurement of low PD fiber, to be significant, will therefore need an improved probe power stability and higher pump power in order to reach uniform inversion over longer samples. In the future we plan a new round of measurements on fiber samples exhibiting low PD levels.

3. Experiments

Table 1 shows the fiber we tested. All fibers are single mode (SM) and all were core pumped. Figure 2 shows the temporal evolution of the induced losses.

Tables Icon

Table 1. List of tested fibers. All fiber are silica glass.

 

Fig. 2 Temporal evolution of the induced losses for fiber with same geometry. Symbols are experimental data. Line shows the fitting using Eq. (1).

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Triangles in Fig. 3 and Fig. 4 summarize the results for the three fitting parameters. Note that the horizontal is in wt% of the total Yb concentration but can be scaled by the inversion factor 0.46 to be read as concentration of excited Yb ions. Figure 3 shows that quadratic fitting gives a reasonable result.

 

Fig. 3 Equilibrium loss versus Yb-doping level. Solid black line is a square fitting with exponent 2.08. The horizontal scale is also proportional to the concentration of excited Yb ions (scale factor is 0.46). The initial point refers to zero induced loss in an undoped fiber.

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Fig. 4 Stretching parameter (circles) and time scale parameter (triangles) versus Yb-doping level. The right picture shows the inverse of time scale parameter in a double log scale.

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The lower value for the highest concentrated sample is under investigation, nevertheless, a quadratic fitting implies that, although small, PD process should occur even at very low doping values were clusters should likely not occur. Since Fig. 3 can be read as PD losses versus number of excited ions it may also suggest a contribution of the ion-ion distance. In fact it was previously reported a linear dependence of PD losses versus number of Yb excited ions when a single fiber is tested (same ion-ion distance) [12]. We may therefore assume the difference with our results is due to the impact of ion-ion distance. This has a strong implication on modeling PD. We do not enter into this question in this paper but we believe that this issue should be further investigated. Figure 4 shows that the stretching parameter is close to a constant and that this gives a first order weight to the time evolution parameter reported in the same figure. To compare with previous results we transform the time constant values as decay rate (1/τ) values and we obtain a third power growth versus Yb doping level as shown in Fig. 4 right. This is a bit lower that recently reported testing a single fiber with variable inversion level [8,9]. The discrepancy will be further investigated since the implication on the physical process of the power law fitting is not clear and considering also that PD time evolution is also governed by the stretched exponential parameter and type of pumping [12].

4. Self-similarity and figure of merit

Figure 5 shows the loss and the time scale of curves of Fig. 2 normalized using the fitting parameter of Fig. 3 and Fig. 4, i.e. we scale all curves to their own equilibrium loss, αeq, and time parameters, τ, and we plot α(t)/αeq, versus time divided by τ.Figure 5 shows a reasonable self-similarity in the evolution of PD losses when we compare the five fibers under test. Previously self-similarity was reported comparing two inversion levels in the same fiber but questioning the possibility of using the fitting parameter to scale the curves [8]. We believe that since in our case the variance of the stretching parameter is limited the equilibrium loss and time scale parameters have indeed a physical meaning and they can be used to describe the PD loss evolution. The above finding also suggests that we can use the fitting parameter to compare the fiber behavior. This is very important since the industry is in great need of a tool to assess the different fibers available on the market. This evidence of self-similarity allows us to introduce a FOM for Yb-doped fiber, a very useful tool to evaluate the impact of PD on laser/amplifier devices. Since the PD generate a given amount of loss in a given time PD effect can be characterized by two parameters, one giving the impact of extra losses and the other related to the time needed to approach the final equilibrium state. First we can define a figure of merit (PDFOM) to assess the loss. A complete analytical approach is out of the aim of this paper but we would like to make a few practical considerations. The impact of PD loss is twofold: first it reduces the pump power due to extra absorption and second it increases the loss at laser wavelength. Both effects depend on the extra loss αeq, depending on Nyb2, times the fiber length L. However, an increase of the Yb concentration reduces linearly the length of fiber needed, proportionally to Nyb −1 which benefits total PD losses and, in some cases, minimizes some of the undesirable nonlinear effects. An overall PDFOM could be

PDFOM=(αeqL)2(αeqNYb)2,
where we assumed that required length would be proportional to the inverse of Yb3+ concentration. Since equilibrium loss grows as the square of Yb3+ concentration the PDFOM is again a quadratic function with respect Yb3+ concentration. We note that FOM could be modified to take into account particular laser design issues of which we can state the two main families: pulsed and CW laser. Fiber laser designers do take into account the length of the active fiber to create compact sources or to minimize the detrimental nonlinear effects. In such a case the FOM could also be normalized with respect to length and therefore decrease linearly with the concentration. The time scale of the PD process is the second issue. To describe the time evolution we could take a time-related parameter. For example 5 times the time scale parameter of Fig. 4 would guarantee that 90% of the final loss value is reached, as can be seen in Fig. 5. Further experiments to correlate PD parameters for different inversion level in the same fiber and using different pump levels will provide proper FOM scaling laws and will be the aim of future works.

 

Fig. 5 Data of Fig. 2 normalized using the fitting parameters reported in Fig. 4. Losses are normalized to the equilibrium value and time to the time constant.

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5. Conclusion

In conclusions we reported on an extensive investigation of photodarkening in Yb-doped silica fibers. A set of fibers was made in order to be able to compare the photodarkening induced losses over a large concentration range. Careful measurements were made to ensure equal and uniform inversion in all tests and we propose a rule of thumbs to calculate the length of fiber to be tested. Results seem to indicate a quadratic law dependence of final losses on the concentration of excited ions. The decay rate, inverse of fitting time constant parameter, grows as the third power of the concentration. Results also show that the fitting stretching parameter has a very limited variation. This gives more meaning to the fitting decay time parameter. Self-similarity among the evolution of photodarkening induced losses is also demonstrated and we propose a preliminary figure of merit for Yb-doped fibers in order to evaluate the impact of photodarkening on laser/amplifier devices.

Acknowledgments

This project was funded by the FP7 LIFT (Leadership in Fiber Technology) Project (Grant #228587). Authors are indebted with Chris Pannell for valuable support. Stefano Taccheo is indebted with Dr. Bolla for inspiring discussions.

References and links

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

2. 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]  

3. M. M. Broer, D. M. Krol, and D. J. Digiovanni, “Highly nonlinear near-resonant photodarkening in a thulium-doped aluminosilicate glass fiber,” Opt. Lett. 18(10), 799–801 (1993). [CrossRef]   [PubMed]  

4. Y. W. Lee, S. Sinha, M. J. F. Digonnet, R. L. Byer, and S. Jiang, “Measurement of high photodarkening resistance in heavily Yb3+-doped phosphate fibres,” Electron. Lett. 44(1), 14–15 (2008). [CrossRef]  

5. M. Engholm, L. Norin, and D. Aberg, “Strong UV absorption and visible luminescence in ytterbium-doped aluminosilicate glass under UV excitation,” Opt. Lett. 32(22), 3352–3354 (2007). [CrossRef]   [PubMed]  

6. M. Engholm and L. Norin, “Preventing photodarkening in ytterbium-doped high power fiber lasers; correlation to the UV-transparency of the core glass,” Opt. Express 16(2), 1260–1268 (2008). [CrossRef]   [PubMed]  

7. J. J. Koponen, M. J. Söderlund, H. J. Hoffman, and S. K. T. Tammela, “Measuring photodarkening from single-mode ytterbium doped silica fibers,” Opt. Express 14(24), 11539–11544 (2006). [CrossRef]   [PubMed]  

8. S. Jetschke and U. Röpke, “Power-law dependence of the photodarkening rate constant on the inversion in Yb doped fibers,” Opt. Lett. 34(1), 109–111 (2009). [CrossRef]   [PubMed]  

9. S. Jetschke, S. Unger, U. Röpke, and J. Kirchhof, “Photodarkening in Yb doped fibers: experimental evidence of equilibrium states depending on the pump power,” Opt. Express 15(22), 14838–14843 (2007). [CrossRef]   [PubMed]  

10. S. Yoo, C. Basu, A. J. Boyland, C. Sones, J. Nilsson, J. K. Sahu, and D. Payne, “Photodarkening in Yb-doped aluminosilicate fibers induced by 488 nm irradiation,” Opt. Lett. 32(12), 1626–1628 (2007). [CrossRef]   [PubMed]  

11. J. Koponen, M. Söderlund, H. J. Hoffman, D. A. V. Kliner, J. P. Koplow, and M. Hotoleanu, “Photodarkening rate in Yb-doped silica fibers,” Appl. Opt. 47(9), 1247–1256 (2008). [CrossRef]   [PubMed]  

12. S. Jetschke, U. Röpke, S. Unger, and J. Kirchhof, “Characterization of photodarkening processes in Yb doped fibers,” Proc. SPIE 7195, 71952B, 71952B-12 (2009). [CrossRef]  

References

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  1. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspective,” J. Opt. Soc. Am. B 27(11), B63–B92 (2010).
    [Crossref]
  2. 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]
  3. M. M. Broer, D. M. Krol, and D. J. Digiovanni, “Highly nonlinear near-resonant photodarkening in a thulium-doped aluminosilicate glass fiber,” Opt. Lett. 18(10), 799–801 (1993).
    [Crossref] [PubMed]
  4. Y. W. Lee, S. Sinha, M. J. F. Digonnet, R. L. Byer, and S. Jiang, “Measurement of high photodarkening resistance in heavily Yb3+-doped phosphate fibres,” Electron. Lett. 44(1), 14–15 (2008).
    [Crossref]
  5. M. Engholm, L. Norin, and D. Aberg, “Strong UV absorption and visible luminescence in ytterbium-doped aluminosilicate glass under UV excitation,” Opt. Lett. 32(22), 3352–3354 (2007).
    [Crossref] [PubMed]
  6. M. Engholm and L. Norin, “Preventing photodarkening in ytterbium-doped high power fiber lasers; correlation to the UV-transparency of the core glass,” Opt. Express 16(2), 1260–1268 (2008).
    [Crossref] [PubMed]
  7. J. J. Koponen, M. J. Söderlund, H. J. Hoffman, and S. K. T. Tammela, “Measuring photodarkening from single-mode ytterbium doped silica fibers,” Opt. Express 14(24), 11539–11544 (2006).
    [Crossref] [PubMed]
  8. S. Jetschke and U. Röpke, “Power-law dependence of the photodarkening rate constant on the inversion in Yb doped fibers,” Opt. Lett. 34(1), 109–111 (2009).
    [Crossref] [PubMed]
  9. S. Jetschke, S. Unger, U. Röpke, and J. Kirchhof, “Photodarkening in Yb doped fibers: experimental evidence of equilibrium states depending on the pump power,” Opt. Express 15(22), 14838–14843 (2007).
    [Crossref] [PubMed]
  10. S. Yoo, C. Basu, A. J. Boyland, C. Sones, J. Nilsson, J. K. Sahu, and D. Payne, “Photodarkening in Yb-doped aluminosilicate fibers induced by 488 nm irradiation,” Opt. Lett. 32(12), 1626–1628 (2007).
    [Crossref] [PubMed]
  11. J. Koponen, M. Söderlund, H. J. Hoffman, D. A. V. Kliner, J. P. Koplow, and M. Hotoleanu, “Photodarkening rate in Yb-doped silica fibers,” Appl. Opt. 47(9), 1247–1256 (2008).
    [Crossref] [PubMed]
  12. S. Jetschke, U. Röpke, S. Unger, and J. Kirchhof, “Characterization of photodarkening processes in Yb doped fibers,” Proc. SPIE 7195, 71952B, 71952B-12 (2009).
    [Crossref]

2010 (1)

2009 (2)

S. Jetschke and U. Röpke, “Power-law dependence of the photodarkening rate constant on the inversion in Yb doped fibers,” Opt. Lett. 34(1), 109–111 (2009).
[Crossref] [PubMed]

S. Jetschke, U. Röpke, S. Unger, and J. Kirchhof, “Characterization of photodarkening processes in Yb doped fibers,” Proc. SPIE 7195, 71952B, 71952B-12 (2009).
[Crossref]

2008 (3)

2007 (3)

2006 (1)

1997 (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]

1993 (1)

Aberg, D.

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]

Basu, C.

Boyland, A. J.

Broer, M. M.

Byer, R. L.

Y. W. Lee, S. Sinha, M. J. F. Digonnet, R. L. Byer, and S. Jiang, “Measurement of high photodarkening resistance in heavily Yb3+-doped phosphate fibres,” Electron. Lett. 44(1), 14–15 (2008).
[Crossref]

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.

Digiovanni, D. J.

Digonnet, M. J. F.

Y. W. Lee, S. Sinha, M. J. F. Digonnet, R. L. Byer, and S. Jiang, “Measurement of high photodarkening resistance in heavily Yb3+-doped phosphate fibres,” Electron. Lett. 44(1), 14–15 (2008).
[Crossref]

Engholm, M.

Hanna, D. 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]

Hoffman, H. J.

Hotoleanu, M.

Jetschke, S.

Jiang, S.

Y. W. Lee, S. Sinha, M. J. F. Digonnet, R. L. Byer, and S. Jiang, “Measurement of high photodarkening resistance in heavily Yb3+-doped phosphate fibres,” Electron. Lett. 44(1), 14–15 (2008).
[Crossref]

Kirchhof, J.

S. Jetschke, U. Röpke, S. Unger, and J. Kirchhof, “Characterization of photodarkening processes in Yb doped fibers,” Proc. SPIE 7195, 71952B, 71952B-12 (2009).
[Crossref]

S. Jetschke, S. Unger, U. Röpke, and J. Kirchhof, “Photodarkening in Yb doped fibers: experimental evidence of equilibrium states depending on the pump power,” Opt. Express 15(22), 14838–14843 (2007).
[Crossref] [PubMed]

Kliner, D. A. V.

Koplow, J. P.

Koponen, J.

Koponen, J. J.

Krol, D. M.

Lee, Y. W.

Y. W. Lee, S. Sinha, M. J. F. Digonnet, R. L. Byer, and S. Jiang, “Measurement of high photodarkening resistance in heavily Yb3+-doped phosphate fibres,” Electron. Lett. 44(1), 14–15 (2008).
[Crossref]

Nilsson, J.

Norin, L.

Paschotta, 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]

Payne, D.

Richardson, D. J.

Röpke, U.

Sahu, J. K.

Sinha, S.

Y. W. Lee, S. Sinha, M. J. F. Digonnet, R. L. Byer, and S. Jiang, “Measurement of high photodarkening resistance in heavily Yb3+-doped phosphate fibres,” Electron. Lett. 44(1), 14–15 (2008).
[Crossref]

Söderlund, M.

Söderlund, M. J.

Sones, C.

Tammela, S. K. T.

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]

Unger, S.

S. Jetschke, U. Röpke, S. Unger, and J. Kirchhof, “Characterization of photodarkening processes in Yb doped fibers,” Proc. SPIE 7195, 71952B, 71952B-12 (2009).
[Crossref]

S. Jetschke, S. Unger, U. Röpke, and J. Kirchhof, “Photodarkening in Yb doped fibers: experimental evidence of equilibrium states depending on the pump power,” Opt. Express 15(22), 14838–14843 (2007).
[Crossref] [PubMed]

Yoo, S.

Appl. Opt. (1)

Electron. Lett. (1)

Y. W. Lee, S. Sinha, M. J. F. Digonnet, R. L. Byer, and S. Jiang, “Measurement of high photodarkening resistance in heavily Yb3+-doped phosphate fibres,” Electron. Lett. 44(1), 14–15 (2008).
[Crossref]

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 (3)

Opt. Lett. (4)

Proc. SPIE (1)

S. Jetschke, U. Röpke, S. Unger, and J. Kirchhof, “Characterization of photodarkening processes in Yb doped fibers,” Proc. SPIE 7195, 71952B, 71952B-12 (2009).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 All three fitting parameters versus fiber length. All values are normalized to the measured values using the shortest samples (1.8 cm).
Fig. 2
Fig. 2 Temporal evolution of the induced losses for fiber with same geometry. Symbols are experimental data. Line shows the fitting using Eq. (1).
Fig. 3
Fig. 3 Equilibrium loss versus Yb-doping level. Solid black line is a square fitting with exponent 2.08. The horizontal scale is also proportional to the concentration of excited Yb ions (scale factor is 0.46). The initial point refers to zero induced loss in an undoped fiber.
Fig. 4
Fig. 4 Stretching parameter (circles) and time scale parameter (triangles) versus Yb-doping level. The right picture shows the inverse of time scale parameter in a double log scale.
Fig. 5
Fig. 5 Data of Fig. 2 normalized using the fitting parameters reported in Fig. 4. Losses are normalized to the equilibrium value and time to the time constant.

Tables (1)

Tables Icon

Table 1 List of tested fibers. All fiber are silica glass.

Equations (2)

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α ( t ) = α e q [ 1 exp { ( t / τ ) β } ]
P D F O M = ( α e q L ) 2 ( α e q N Y b ) 2

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