Mathematical Model of Photodarkening in Rare-Earth-Doped Fiber

Round 1

Reviewer 1 Report

The manuscript by Sun et al. proposed an improved mathematical model to explain and analyze the PD phenomenon by taking the factors of high-energy photons and temperature into consideration. The mathematical derivation of the model is rigorous and the experimental data are solid. The reliability of the model is also carefully verified by fitting some experimental results. In summary, this work is valuable for understanding and suppress the PD phenomenon. It can be considered for publication in Photonics.

Some minor comments should be considered:

1. Some format errors in the texts and graphics should be carefully checked, i.e., inconsistent uppercase of “oxygen-deficient Center (ODC)” in line 72, superscript error of “4.953×10-3

” in line 324.

2. It’s suggested to carefully adjust the position of the texts in some figures to avoid the overlapping, i.e., Fig.10, 14.
3. I think the graphics in Fig. 6 have different sizes, i.e., Fig.6a is obviously larger than Fig.6b. Besides, the character “(b)” is bold, while the character “(a)” is not. Same questions for Fig. 3.

Author Response

We thank the reviewer for their very thorough review of the paper. We are very sorry for making such a low-level mistake, and we have corrected the writing mistake you pointed out. 

Reviewer 2 Report

  In this paper, the authors proposed an improved mathematical model to explain the mechanism of the photodarkening phenomenon. The authors thought that the excitation of color centers by high-energy photons is the main reason for photodarkening. The fitting analysis showed that the photodarkening rate was related to the high-energy photons generated by pump power upconversion. Also, the maximum additional loss was obviously affected by temperature. The fitting experimental data could be well consistent to the mathematical model.

  Recently, photodarkening phenomenon was also observed in rare-earth ions doped transparent ceramics (Pump laser induced photodarkening in ZrO2-doped Yb:Y2O3 laser ceramics, 2019, 39(2-3), 635-640). Pumped by the 940 nm LD, a broad band absorption (color center) was seen in the Zr4+-doped Yb:Y2O3 transparent ceramics. However, in the Yb:Y2O3 ceramics doped without Zr4+, photodarkening was not detected. In this respect, the high energy photon was not likely the cause for the photodarkening of Yb:Y2O3 transparent ceramics. The Zr4+ induced point defects (cation vacancies) were the main cause for the photodarkening of Yb:Y2O3 ceramics.

  In this paper, the authors found that the photodarkening rate is directly proportional to the high energy photon power. So what is the origin of the photodarkening in the fiber lasers? Defects, Yb2+, impurities or other factors?

Author Response

We thank the reviewer for their very thorough review of the paper. We feel that in responding to the reviewer’s comments, we have made the paper much stronger. The following is our response to your concerns.

Point 1: Recently, the photodarkening phenomenon was also observed in rare-earth ions doped transparent ceramics (Pump laser-induced photodarkening in ZrO₂-doped Yb: Y₂O₃ laser ceramics, 2019, 39(2-3), 635-640). Pumped by the 940 nm LD, a broadband absorption (color center) was seen in the Zr⁴⁺-doped Yb: Y₂O₃ transparent ceramics. However, in the Yb:Y₂O₃ ceramics doped without Zr⁴⁺, photodarkening was not detected. In this respect, the high-energy photon was not likely the cause for the photodarkening of Yb: Y₂O₃ transparent ceramics. The Zr⁴⁺ induced point defects (cation vacancies) were the main reason for the photodarkening of Yb: Y₂O₃ ceramics.

Response 1: 

We are very sorry for missing such important literature, and thank you for pointing it out. Because there are many kinds of color centers that lead to photodarkening in fiber, and the ways of different color centers generation are also different. Zr³⁺ color center is only produced in zirconia fiber, and our research mostly focuses on silica fiber and ZBLAN fiber. In order not to complicate the model, we did not take the color center of Zr³⁺ into consideration. In the paper you mentioned, the Zr⁴⁺ crystal sites capture the excited electrons to form the Zr³⁺ color center, and the electrons are excited by a 940nm laser, which is not inconsistent with the theme of our paper. In the process of PD, photons are the excitation source. The higher the photon energy, the more prone for PD process to happen.

In addition, there are other explanations for the formation process of Zr³⁺ color center, as shown in Fig. 1 [1].The upconversion of Tm trace impurities in Yb-doped pumped by near-infrared laser is an important reason for the generation of short-wavelength light [2,3]. 

Figure 1.  Schematic of conversion processes in Yb / Tm doped fiber [1].

Point 2: In this paper, the authors found that the photodarkening rate is directly proportional to the high energy photon power. So what is the origin of the photodarkening in the fiber lasers? Defects, Yb²⁺, impurities or other factors?

Response 2: The essential cause of photodarkening is the formation of color centers. Some of the factors you mentioned above such as defects, Yb²⁺, impurities and so on are color centers themselves, and some of them play the role of precursors that produce color centers, and some others are the incentives for producing color centers. In the process of color centers generation, the identities of these factors are not fixed and they will cooperate with each other. Please refer to my other paper (T. R. Sun, X. Y. Su, Y. H. Zhang, H. W. Zhang, and Y. Zheng, "Progress and Summary of Photodarkening in Rare Earth Doped Fiber," Applied Sciences-Basel 11 (2021).) for details.

Reference

[1] R. Peretti, C. Gonnet, A.M. Jurdyc, Revisiting literature observations on photodarkening in Yb3+doped fiber considering the possible presence of Tm impurities, Journal of Applied Physics 112 (2012) 093511. 10.1063/1.4761973.

[2] S. Jetschke, S. Unger, A. Schwuchow, M. Leich, J. Fiebrandt, M. J?Ger, J. Kirchhof, Evidence of Tm impact in low-photodarkening Yb-doped fibers, Optics Express 21 (2013) 7590-7598.

[3] D. Simpson, K. Gibbs, S. Collins, W. Blanc, B. Dussardier, G. Monnom, P. Peterka, G. Baxter, Visible and near infra-red up-conversion in Tm3+/Yb3+ co-doped silica fibres under 980 nm excitation, Optics Express 16 (2008) 13781-13799.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments for author File: Comments.pdf

Author Response

We thank the reviewer for their very thorough review of the paper. We feel that in responding to the reviewer’s comments, we have made the paper much stronger. The following is our response to your concerns.

Point 1: Page 8, Row 252: Why the authors think in fiber A-C, the WPD is independent of the pump power?

Response 1: According to our fitting results, the PD rate  of A-C fiber does not change with the change of power.

Point 2:Page 9, Row 273: ‘Furthermore, the probe power is very low (2 μW), which just explainsof Fiber D is much higher than that of Fiber A-C.’ How much probe power did fiber A-C use? Why the authors got this conclusion?

Response 2: We are very sorry for explaining unclearly and inaccurately here, this time we rewrite this section as follows,

Because the fitting results of A-C fiber show that no matter how the pump power varies, and  is almost unchanged. Therefore, we speculate that the low doping concentration of thulium in A-C fiber makes the upconversion of the 1.12 μm pump laser impossible to produce high-energy photons leading to photodarkening. However, according to [4,47], a 488 nm laser can lead to photodarkening. Therefore, we speculate that the photodarkening is caused by the 488 nm probe laser, and the very low power (2 μW) of the 488 nm probe laser just explains the very small and constant photodarkening rateof A-C fiber. 

Point 3:Page 9, figure 7: There are 4 power levels used in the experiment shown in Figure 6(d). Why only 3 power levers were used here?

Response 3: We are very sorry for explaining unclearly and inaccurately here. In the experiment in Fig. 6, the data measured at the minimum pump power of all four groups of experiments are measured without photodarkening. They are almost flat, so we do not take them into account.

Point 4:Page 11, Row 328: ‘It is thought that because the ����� is too high, the �� obtained by upconversion has reached saturation, so the ��� is almost unchanged.’ I do not think this presume is very convincing. Why the authors did not try low power to prove it which should be easy to implement?

Response 4: Due to the limitation of our experimental conditions, we can not find the power range where photodarkening can occur without upconversion saturation. We speculate that it may be because the particularity of our sample fiber makes the power range satisfying the conditions so small that it can not be found.

Point 5:There are a lot of typo errors in this manuscript. For example, in Table 5 there are 2 errors without proper use of superscript and subscript. In Page 4, Equation 3, is N=N0(0) right?

Response 5: Thank you for pointing out the mistake. We have corrected it. For equation 3, N = N0(0) is redundant, so we modify this equation as follows:

Point 6:The style of figures is not consistent and some legends are difficult to read.

Response 6: We have optimized the image

Author Response File: Author Response.docx

Reviewer 4 Report

In this paper, the authors proposed a mathematical model about photodarkening (PD) with taking the factors of high-energy photons and temperature into account. They also validated the model throughout the work by providing quantitative results from numerical analysis. Correlations between the power of high-energy photons and PD rate and between temperature and saturated PD loss are presented. Fitting analysis shows temperature can significantly affect maximum additional loss. These are valuable findings in the photodarkening research. Therefore, I think this manuscript could be acceptable for publication in Photonics with a minor revision if the following concerns could be solved.

1. How to define high-energy photons? Is there a critical wavelength of the photon for someone to judge?
2. In line 59, the character “s” is different from the surrounding words, please check.
3. In line 148, please check usage of the word “respectively”.
4. In line 171, line 317, line 319, line 330, and ect… the word should be “powers” rather than “power”. Please carefully check them and some other words in the manuscript.
5. In Figure 8, it would be better to enlarge the figure since some of the words in the pictures are overlapped with the lines.
6. In line 302, the word “The” in the bracket should be written as “the”
7. In line 177 and 243, all the word “line” should be rewritten as plural form.

Author Response

We thank the reviewer for their very thorough review of the paper. We feel that in responding to the reviewer’s comments, we have made the paper much stronger. The following is our response to your concerns.

Point 1: How to define high-energy photons? Is there a critical wavelength of the photon for someone to judge?

Response 1: We cannot clearly define how short the wavelength is as high-energy photons. According to the existing theory, the shorter the wavelength, the easier it is to produce photodarkening.

Point 2:

In line 59, the character “s” is different from the surrounding words, please check.

In line 148, please check usage of the word “respectively”.

In line 171, line 317, line 319, line 330, and ect… the word should be “powers” rather than “power”. Please carefully check them and some other words in the manuscript.

In Figure 8, it would be better to enlarge the figure since some of the words in the pictures are overlapped with the lines.

In line 302, the word “The” in the bracket should be written as “the”

In line 177 and 243, all the word “line” should be rewritten as plural form.

Response 2: We are very sorry for making such a low-level mistake, and we have corrected the writing mistake you pointed out.

Author Response File: Author Response.docx