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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 12849–12864
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The three-electron bond =Si<O2:∙Yb absorption center of pre-darkened ytterbium-doped silica

Kent E. Mattsson  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 12849-12864 (2013)
http://dx.doi.org/10.1364/OE.21.012849


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Abstract

The formation and bleaching of color centers during annealing of pre-darkened ytterbium-doped silica fibers is modeled by three-electron bond (TEB) = Si<O2:∙Yb absorption centers. The nature of a center and underlying mechanism for its annealing in formation, shift and dissociation of chemical bonds is described in terms of a Markov statistical model with state change set by Bose-Einstein phonon statistics. The center hold one terminal and four active states with activation energies for transitions among these found to match bond energies of molecular oxygen in ionic character bonds of 1 and 1½ bond order. Experimentally observed in- and decrease in absorption during ramp and isothermal annealing of pre-darkened ytterbium co-doped silica fibers are hereby matched by a set of = Si<O2:∙Yb centers.

© 2013 OSA

1. Introduction

The long term stability of amplifiers and lasers is critically influenced by photodarkening (PD) with time dependent broadband loss centered at visible wavelengths and a long tail extending into the near infrared. Several mechanisms and species potentially responsible for PD and the herewith associated PD color center (CC) have been proposed as discussed in [1

1. Y.-S. Liu, T. C. Galvin, T. Hawkins, J. Ballato, L. Dong, P. R. Foy, P. D. Dragic, and J. G. Eden, “Linkage of oxygen deficiency defects and rare earth concentrations in silica glass optical fiber probed by ultraviolet absorption and laser excitation spectroscopy,” Opt. Express 20(13), 14494–14507 (2012). [CrossRef] [PubMed]

,2

2. K. E. Mattsson, “Photo darkening of rare earth doped silica,” Opt. Express 19(21), 19797–19812 (2011). [CrossRef] [PubMed]

] but a definitive explanation for the process remains a subject of debate.

In [1

1. Y.-S. Liu, T. C. Galvin, T. Hawkins, J. Ballato, L. Dong, P. R. Foy, P. D. Dragic, and J. G. Eden, “Linkage of oxygen deficiency defects and rare earth concentrations in silica glass optical fiber probed by ultraviolet absorption and laser excitation spectroscopy,” Opt. Express 20(13), 14494–14507 (2012). [CrossRef] [PubMed]

] a linear dependency for pristine Yb and Er-doped fibers between type II oxygen deficiency center (ODC(II)) absorption at 245 nm (5.06 eV) and rare earth number density is found. The direct ODC(II) to Yb emission excitation in these fibers is in [1

1. Y.-S. Liu, T. C. Galvin, T. Hawkins, J. Ballato, L. Dong, P. R. Foy, P. D. Dragic, and J. G. Eden, “Linkage of oxygen deficiency defects and rare earth concentrations in silica glass optical fiber probed by ultraviolet absorption and laser excitation spectroscopy,” Opt. Express 20(13), 14494–14507 (2012). [CrossRef] [PubMed]

] seen as either direct ODC(II) to Yb excitation transfer or that ODC(II) and Yb emission hold a common origin for UV excitation. The growing Tm3+/Er3+ trace impurity emission found to follow increase of PD in [2

2. K. E. Mattsson, “Photo darkening of rare earth doped silica,” Opt. Express 19(21), 19797–19812 (2011). [CrossRef] [PubMed]

] collaborate this.

In [3

3. 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]

] Yb/Al co-doped preform material show 220 nm (5.63 eV) absorption and excitation by 488 nm light which leads to broadband PD loss as well as increased 220 nm absorption. This result is in [3

3. 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]

] explained by Yb ions preferably occupying oxygen-surrounding voids created by Al with tetrahedral or octahedral coordination. As the number of Yb ions by co-doping goes beyond the available voids in the silica matrix the possibility of ill-valence bonds between cations (Yb-Al or Yb-Yb) increases. These ill-valence bonds resemble Ge-Ge or Ge-Si bonds in Ge-silicate glasses in that they originate from a deficiency of oxygen [3

3. 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]

]. In [4

4. 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]

] preform Yb/Al material is found to show ≈230 nm (≈5.4 eV) centered absorption proposed to be due to charge transfer bands of Yb.

The difference between observed centers of [1

1. Y.-S. Liu, T. C. Galvin, T. Hawkins, J. Ballato, L. Dong, P. R. Foy, P. D. Dragic, and J. G. Eden, “Linkage of oxygen deficiency defects and rare earth concentrations in silica glass optical fiber probed by ultraviolet absorption and laser excitation spectroscopy,” Opt. Express 20(13), 14494–14507 (2012). [CrossRef] [PubMed]

,3

3. 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]

] and [4

4. 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]

] is in [1

1. Y.-S. Liu, T. C. Galvin, T. Hawkins, J. Ballato, L. Dong, P. R. Foy, P. D. Dragic, and J. G. Eden, “Linkage of oxygen deficiency defects and rare earth concentrations in silica glass optical fiber probed by ultraviolet absorption and laser excitation spectroscopy,” Opt. Express 20(13), 14494–14507 (2012). [CrossRef] [PubMed]

] proposed to be due to drawing induced ODC(II) measured on drawn fibers whereas [3

3. 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]

,4

4. 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]

] are for preform material. It is further in [1

1. Y.-S. Liu, T. C. Galvin, T. Hawkins, J. Ballato, L. Dong, P. R. Foy, P. D. Dragic, and J. G. Eden, “Linkage of oxygen deficiency defects and rare earth concentrations in silica glass optical fiber probed by ultraviolet absorption and laser excitation spectroscopy,” Opt. Express 20(13), 14494–14507 (2012). [CrossRef] [PubMed]

] hinted at that ODC(II), the E’ center and non-bridging oxygen hole center (NBOHC) all are participants in UV-induced defect inter conversion processes [5

5. L. Skuja, “Optically active oxygen-deficiency-related centers in amorphous silicon dioxide,” J. Non-Cryst. Solids 239(1–3), 16–48 (1998). [CrossRef]

] that provides one potential avenue for PD in Yb/Al fibers. The E’ center and NBOHC hold, respectively, 5.6 – 5.7 eV and ≈4.8 eV in center absorption [5

5. L. Skuja, “Optically active oxygen-deficiency-related centers in amorphous silicon dioxide,” J. Non-Cryst. Solids 239(1–3), 16–48 (1998). [CrossRef]

].

In [6

6. S. Girard, Y. Ouerdane, B. Tortech, C. Marcandella, T. Robin, B. Dadier, J. Baggio, P. Paillet, V. Ferlet-Cavrois, A. Boukenter, J.-P. Meunier, J. R. Schwank, M. R. Shaneyfelt, P. E. Dodd, and E. W. Blackmore, “Radiation Effects on ytterbium- and Ytterbium/erbium-Doped Double-Clad Optical Fibers,” IEEE Trans. Nucl. Sci. 56(6), 3293–3299 (2009). [CrossRef]

] X-ray exposed Yb/Al, Yb/P and P co-doped silica fibers are characterized. Here it is found that the Yb/P fiber with 4 times higher Yb concentration and identical broadband 870 – 1100 nm absorption shows half the ≈5.1 eV absorption than the Yb/Al fiber. The Yb/P fiber show increased absorption at ≈1600 nm which is identified as the P1 phosphorous defect [6

6. S. Girard, Y. Ouerdane, B. Tortech, C. Marcandella, T. Robin, B. Dadier, J. Baggio, P. Paillet, V. Ferlet-Cavrois, A. Boukenter, J.-P. Meunier, J. R. Schwank, M. R. Shaneyfelt, P. E. Dodd, and E. W. Blackmore, “Radiation Effects on ytterbium- and Ytterbium/erbium-Doped Double-Clad Optical Fibers,” IEEE Trans. Nucl. Sci. 56(6), 3293–3299 (2009). [CrossRef]

]. In [4

4. 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]

] an Yb/P fiber with 4.5 times higher Yb concentration show identical slope and half the 700 – 1000 nm absorption than an Yb/Al fiber. The ODC(II) paved PD avenue may therefore also apply for Yb/P fibers – but with a changed ODC(II) to Yb number density relation.

The dynamic behavior of the PD loss makes its characterization challenging. In [9

9. 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]

] the PD process is found to demand 5 – 6 eV in activation energy (EA) or what correspond to 4 - 5 excited Yb ions. In [10

10. S. Yoo, A. J. Boyland, R. J. Standish, and J. K. Sahu, “Measurement of photodarkening in Yb-doped aluminosilicate fibres at elevated temperature,” Electron. Lett. 46(3), 233–244 (2010). [CrossRef]

] dynamic color center bleaching is found to show 0.07 eV in EA with full recovery of the induced PD for fibers operating at 573 K. The photochemical reaction mechanisms for the active pumped fibers relies on the actual level of pump absorption, temperature, fiber geometry and Yb ion concentration. To distinguish the different mechanisms involved in the PD process it appears therefore advantageous to analyze the bleaching of pre-darkened fibers by annealing as the relaxation reaction hereby only relies on fiber temperature. This opens for the possibility to directly relate observed change in absorption with activation energy. For pre-darkened fibers an unexpected increase in PD absorption is hereby observed for annealing ramps from room temperature to about 350-450 K followed by the expected bleaching that initiate at 470-600 K [11

11. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Heat-induced darkening and spectral broadening in photodarkened ytterbium-doped fiber under thermal cycling,” Opt. Express 17(12), 9940–9946 (2009). [CrossRef] [PubMed]

,12

12. M. Leich, U. Röpke, S. Jetschke, S. Unger, V. Reichel, and J. Kirchhof, “Non-isothermal bleaching of photodarkened Yb-doped fibers,” Opt. Express 17(15), 12588–12593 (2009). [CrossRef] [PubMed]

]. Thermally induced PD absorption is not observed for pristine fibers [12

12. M. Leich, U. Röpke, S. Jetschke, S. Unger, V. Reichel, and J. Kirchhof, “Non-isothermal bleaching of photodarkened Yb-doped fibers,” Opt. Express 17(15), 12588–12593 (2009). [CrossRef] [PubMed]

] while PD bleaching shows EA of ≈1.3 eV [12

12. M. Leich, U. Röpke, S. Jetschke, S. Unger, V. Reichel, and J. Kirchhof, “Non-isothermal bleaching of photodarkened Yb-doped fibers,” Opt. Express 17(15), 12588–12593 (2009). [CrossRef] [PubMed]

,13

13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

] to 1.5 eV [13

13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

] dependent on fiber composition and manufacturing condition.

In this paper the CC in Yb/Al and Yb/Al/P material is proposed to be due to a TEB formed between dioxasilirane and Yb ( = Si<O2:∙Yb). The CC is described through an analogy between polarized TEBs of = Si<O2:∙Yb and polarized TEBs found as an intermediate bond between ½ (1) bond order molecular oxygen and 1 (1½) bond order super-oxide. The interpolated bond energies found hereby are along with Bose-Einstein phonon statistics applied to develop a Markov chain statistical model for annealing of pre-darkened fibers. This model gives good match to the annealing data from [11

11. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Heat-induced darkening and spectral broadening in photodarkened ytterbium-doped fiber under thermal cycling,” Opt. Express 17(12), 9940–9946 (2009). [CrossRef] [PubMed]

13

13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

] for a distributed set of CC sites of individual TEB polarization.

2. The ionic character of PD Color centers

The precursor state ii↑ with one polarized TEB↑ and no bond directly between the oxygen is shown in Fig. 1(c). A further precursor state 0 is in Fig. 1(d) shown for the part where two non-bridging oxygen (NBO) present anti-bond electrons to the valence electron of Yb in a polarized TEB↑. With respect to annealing both state ii and 0 is terminal as transitions among these hold no influence on absorption in the near infrared. As such they can be regarded as one state. The difference between state ii and 0 is of importance to which type of centers that may be regarded as precursors for PD – a discussion which is beyond the scope of this paper.

The introduction of Yb3+ to corner shared SiO4 tetrahedral networks is according to [16

16. C. C. Robinson and J. T. Fournier, “Co-ordination of Yb3+ in phosphate, silicate, and germinate glasses,” J. Phys. Chem. Solids 31(5), 895–904 (1970). [CrossRef]

] as six-fold coordinated YbO6. In Fig. 2(a)
Fig. 2 (a) Six-fold coordinated Yb3+ with three (AlO4, PO6 and = Si<O2 center) edge matches, (b) four-fold coordinated Yb3+ with one edge match = Si<O2 center, (c) AlO4 tetrahedron with four shared corners internally linked by two σ- and one TEB bond, (d) PO4 tetrahedron with three shared corners and one double bond oxygen internally linked by four σ- and one π-bond, (e) PO4 tetrahedron with two shared and two non-shared corners internally linked by four σ-bonds and the 5th electron in TEB (as in POHC of [17]).
are the three valence electrons of Yb3+ represented by three TEBs (grey shaded areas) where each TEB is anti-bond against a pair of oxygen hereby leading to six-fold coordination in an octahedron. The D3 symmetry with trigonal fields of YbO6 leads to the F7/2 and F5/2 manifold splitting [16

16. C. C. Robinson and J. T. Fournier, “Co-ordination of Yb3+ in phosphate, silicate, and germinate glasses,” J. Phys. Chem. Solids 31(5), 895–904 (1970). [CrossRef]

]. A trigonal field symmetry may, however, also be found in an YbO4 tetrahedron comprising two σ - bond oxygen corners and one TEB linked edge as shown in Fig. 2(b). This as the field follows the bond axes of σ-bonds and is for the TEB given by the direction from the electron donor to the midpoint between the oxygen. This is taken to show that YbO6 and YbO4 are optically active in silica.

The number of shared TEB edges that may be matched by = Si<O2 for each Yb3+ is 1 for YbO4 (denoted by YO2) and 3 for YbO6 (denoted by |Y|). Single Yb ions are incorporated into silica as YbO4 and clusters of Yb and Al and/or P convert to YbO6/AlO6 for the central part with interface layer by either octahedral or tetrahedral (YbO4/AlO4/PO4) coordination to the silica network. Here AlO6 (|A|) is the result of two Al-O to TEB conversions in going from Fig. 2(b) by adding one oxygen for each interaction and hereby reach Fig. 2(a). The PO4 edge match P| is represented by the room temperature stable POHC structure of [17

17. D. L. Griscom, E. J. Friebele, K. J. Long, and J. W. Fleming, “Fundamental defect centers in glass: Electron spin resonance and optical absorption studies of irradiated phosphorous-doped silica glass and optical fibers,” J. Appl. Phys. 54(7), 3743–3762 (1983). [CrossRef]

] in Fig. 2(e). Here the P = O double bond is converted to a TEB between the oxygen leaving the overall unit in neutral charge. The Y or |Y| share preferentially edge(s) (cluster) with other Y, |Y|, A, |A|, or P| in lowering the number of edges in need of coordination to the silica network.

In Table 1

Table 1. The Pauling ionic character of single bond interactions [19] and corresponding resonant TEB ionic character κ found as the geometric average of bonds where YO2 (AO2) are edge connections between Yb (Al) and = Si<O2 in tetrahedral units (Al is to share one corner with Yb). The |Y|/|A| in octahedral units hold one edge connection to = Si<O2 and two TEB linked edges to Y, A, |Y|, |A| or P|.

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the average ionic character of the TEBs within a tetrahedral or octahedral unit is given by the geometric average of the individual links. This is a consequence of the two states for each TEB which leads to formation of resonant spin paired TEB↓ to TEB↑ bonds for Yb|P, Yb|Al or Yb|Yb edge connections while the TEB to Si<O2 (or its precursor) is “non-bond”. The valence electrons of (Yb3+, Al3+ or P5+) is in Table 1 taken to form resonant bonds to their surroundings such that the valence electron density is influenced by the individual ionic character links. Here TEB↓ and TEB↑ are assumed to show identical ionic character.

3. The phonon and photon absorptions in = Si<O2:∙Yb

In the following the dissociation of bonds formed between oxygen of = Si<O2:∙Yb (state iii and II) by phonons and photons is described. An estimate for the bond energy is given in Eq. (1):
EB=Vc(r,κ)+κEA(O)EIP(κ),
(1)
where Vc(r, κ) is the covalent bond energy (between oxygen in bond length r) added ionic character κ for the TEB in electron affinity to oxygen EA(O) = 1.479 eV [18

18. P. H. Krupenie, “The spectrum of molecular oxygen,” J. Phys. Chem. Ref. Data 1(2), 423–534 (1972). [CrossRef]

] balanced by ionization potential EIP(κ) ≈κ ∙ EX ∙ exp(aX∙(κ-1)) of X = Yb. Here aYb = 0.157 is found for ionization potential EYb = 6.254 eV [19

19. Table of periodic properties of the Elements, Sargent-Welch, Skokie, IL, (1980).

] and κYb-O = 0.746 [19

19. Table of periodic properties of the Elements, Sargent-Welch, Skokie, IL, (1980).

]. This yields a TEB ionic character energy (κYb-O∙EA(O) - EIPYb-O)) in balance for Eionic = 1.02 eV. The balanced ionic character energy for a TEB to Al is likewise determined to be 0.83 eV.

The covalent bond energy Vc(r, κ) is a function of oxygen to oxygen bond length and the degree of polarization (ionic character κ) of the TEB link. This is a result of the “extra” electron charge partially shielding the oxygen to oxygen nuclei repulsion. Taken that covalent bond energy is additive and that a σ-bond on = Si<O2 show a similar bond potential as single σ-bond molecular O2 (c1Σu) and a 100% polarized bond of = Si<O2 show a similar bond potential as “1½” bond order superoxide O2 (4Σu and 2Πu) the covalent bond energy Vc(r, κ) of a σ-bond in parallel to a partially polarized TEB of ionic character κ is given by their interpolation. Here the covalent bond energy Vb(r) from spectroscopic data of molecular oxygen [18

18. P. H. Krupenie, “The spectrum of molecular oxygen,” J. Phys. Chem. Ref. Data 1(2), 423–534 (1972). [CrossRef]

] is fitted by the Morse potential function:
Vb(r)=DE[exp(2a±(rr0)2exp(a±(rr0)],
(2)
where DE is dissociation energy, r0 the relaxed bond length and a ± the potential well shape in a+, r > r0 and a, r ≤ r0. The fitted parameters to [18

18. P. H. Krupenie, “The spectrum of molecular oxygen,” J. Phys. Chem. Ref. Data 1(2), 423–534 (1972). [CrossRef]

] are given in Table 2

Table 2. The potential energy for O-O bond, with spectroscopic term for molecular oxygen of [18] by: DE dissociation energy, r0 relaxed bond length and potential well shape parameters (a and a+).

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.

The interpolated potential for κ degree of polarization is hereby given as:
Vc(r,κ)=Vσ(r)+κESZ(r),ESZ(r)=Vσ+SZ(r)Vσ(r),
(3)
where Vσ(r) is the σ-bond (c1Σu) potential by Eq. (2) and ESZ(r) is the shield set by potential difference Vσ + TEB↑ – Vσ(r) and Vσ + TEB↓ – Vσ(r) for a parallel TEB↑ and TEB↓, respectively.

Excitation of = Si<O2:∙Yb (state II or iii) by a phonon will dissociate the bond between oxygen for phonon energy in excess of the covalent bond energy of Eq. (1). This excitation is without change to the resonant TEB ionic character.

For an incoming photon electric field vector E of angular frequency ω will the dipole moment µ of the = Si<O2:∙Yb interaction initiates a torque on TEB↑ (force in opposite directions for Yb and the TEB↑ electron). For TEB↓ either of the anti-bond electrons ↑ (of O) initiates an elastic ↑↓ torque exchange with the bond ↓ electron (of Yb) which is without absorption of energy. The torque on TEB↑ does temporary increase the degree of ionization of Yb with perturbation given by the energy V(t) as function of time t in Eq. (4):
V(t)=μEexp(iωt),
(4)
where the dipole strength |µ|2 = |∫ψTEB(r) µ ψTEB*(r)dr|2 with ψTEB(r) and ψTEB*(r) wave functions for the relaxed and distorted TEB electron field, respectively. The torque leads to absorption of the electric field for invariant O to O bond length in σ→ σ* (bond to anti-bond) excitation for photon energy in excess of:
EII=Vb*(rII)Vb(rII)κESZ(rII),
(5)
where Vb*(rII↑) is the σ*-bond potential and Vb(rII↑) the σ-bond potential for CC bond length rII with ESZ(rII↑) shields as given by Eq. (3). The anti-bond Vb*(rII↑) is R*(κ,T) more anti-bonding than the σ-bond as result of the low electron density between the nuclei:
Vb*(rII)=(1+R*(κ,T))DE[exp(2a±(rIIr0)],
(6)
where DE is dissociation energy, r0 the relaxed bond length and a ± the potential well shape in a+, r > r0 and a, r ≤ r0 as specified for the σ-bond in Table 2. The R*(κ,T) is given by the “missing” electron density (relative to a free atom) between the nuclei R* that partly is interfered by the κ ionic character TEB electron (1-κ∙R*) and by phonon interference on the charge distribution (δR* ∙T) – both adding electron density between the nuclei. The relative (decrease) to the anti-bond increase as function of κ and T is hereby:
R*(κ,T)=[1κR*](R*δR*T),
(7)
where the experimental data of section 4 are fitted by R* = 0.58 and δR* = 10−3 K–1.

The CC absorption EII↑ at room temperature is derived by insertion of Eq. (6) and (7) into Eq. (5) as given in Table 3 for the TEB sites of Table 1. Increase (decrease) in temperature will linearly shift the absorption center to lower (higher) center energy. The spread to the σ→ σ* excitation absorption σII↑ is set by the TEB ionic character energy of the Yb-O and Al-O interactions. Here the ionic character perturbation during photon excitation opens for ionic character energy to add or subtract the excitation as kinetic ( + ) or potential (–) energy released by the Yb-O or Al-O interactions. Taken the spread in ionic character energy to represent 3.5 standard deviations (>99.95%) of a Gaussian distribution σII = Eionic/3.5 the spread for TEBs to Yb is 1.02 eV/3.5 = 0.29 eV and to Al is 0.83 eV/3.5 = 0.237 eV as given in Table 3.

For state iii↑ the “½” bond (TEB↓) excitation to anti-bond (TEB↑) require excitation of a 1.55 to ≈0.5 bond order Si-O interaction along with a spin flip. This TEB↓ + TEB↑→ TEB↑ + TEB↑ transition demands collaboration of a phonon why it is taken to show insignificant optical absorption for the Eiii↑ transitions given in Table 3 at room temperature. Strain to the material or increase in temperature may, however, lead to that the photon absorption become observable but for the present model this is not included.

For both state iii and II the TEB dipole initiate an excitation transfer to the ≈0.5 bond order Si to ≡O interaction dipole. According to [15

15. V. O. Sokolov and V. B. Sulimov, “Threefold coordinated oxygen atom in silica glass,” J. Non-Cryst. Solids 217(2–3), 167–172 (1997). [CrossRef]

] neutral ≡O in Si to O interactions hold doublet-to-doublet transitions at 2.7 eV and 2.9 eV with oscillator strengths of ≈10−1 and another at 4.4 eV with oscillator strength of ≈10−3. This neutral ≡O (of three Si-O interactions) hold a single electron in (doublet) ground state which may be excited to either of the doublet states. In the present context the ≡O center is surrounded by a Si-O (∙∙∙), an O-O (either σ- or ½-bond) and a TEB to Yb interaction which reduces the electron density of the TEB to Yb interaction relative to the Si-O (∙∙∙) interactions in [15

15. V. O. Sokolov and V. B. Sulimov, “Threefold coordinated oxygen atom in silica glass,” J. Non-Cryst. Solids 217(2–3), 167–172 (1997). [CrossRef]

]. The reduced electron density (increase in localization of a hole) is taken to down-shift the optical absorption energy in proportion to the ionic character of the TEB in analogy to what is found for self-trapped holes in [20

20. A. V. Kimmel, P. V. Sushko, and A. L. Shluger, “Structure and spectroscopic properties of trapped holes in silica,” J. Non-Cryst. Solids 353(5-7), 599–604 (2007). [CrossRef]

]. The experimental absorption spectra of section 5 are matched by applying:

E(κ)=2.980.6κ
(8)

The parameter choice in Eq. (8) yields 2.7 eV in peak absorption for a Si-O interaction ionic character [19

19. Table of periodic properties of the Elements, Sargent-Welch, Skokie, IL, (1980).

]. It is further taken that all ≡O centers show spread identical to state II: σ≡O = σII↑ as both excitations are by dipole-to-dipole excitation transfer from the TEB dipole.

For state 0 the precursor is not expected to result in absorption in the visible or near infrared while state ii, iii and II may show additional centers created in parallel to = Si<O2 during the PD process. The ODC(II) hold ≈5 eV and 3.15 eV absorption where the latter show 1.6∙10−7 in absorption strength [5

5. L. Skuja, “Optically active oxygen-deficiency-related centers in amorphous silicon dioxide,” J. Non-Cryst. Solids 239(1–3), 16–48 (1998). [CrossRef]

] which is several orders of magnitude below the in [15

15. V. O. Sokolov and V. B. Sulimov, “Threefold coordinated oxygen atom in silica glass,” J. Non-Cryst. Solids 217(2–3), 167–172 (1997). [CrossRef]

] estimated absorption strength of neutral ≡O. Further may Al (≈2.9 eV [20

20. A. V. Kimmel, P. V. Sushko, and A. L. Shluger, “Structure and spectroscopic properties of trapped holes in silica,” J. Non-Cryst. Solids 353(5-7), 599–604 (2007). [CrossRef]

]) and P (≈3.1 eV, ≈2.5 eV and ≈2.2 eV [17

17. D. L. Griscom, E. J. Friebele, K. J. Long, and J. W. Fleming, “Fundamental defect centers in glass: Electron spin resonance and optical absorption studies of irradiated phosphorous-doped silica glass and optical fibers,” J. Appl. Phys. 54(7), 3743–3762 (1983). [CrossRef]

]) contribute with oxygen hole centers. However, taken the dispute on the actual nature of the PD process such additional centers are not included.

The absorption is hereby due to PD generated CCs (state II↑) and ≡O (state II↑ with two ≡O and state iii↑ with one ≡O) for photon frequency ν taken to be proportional to the dipole strength (TEB ionic character κz) in the sum over all CC sites given by Eq. (9):
α(ν)=zκzρ(z)(σa,g(ν,κz)(Niii+2NII)+σa,gII(ν,κz)NII),
(9)
where κz is the TEB average ionic character of CC site type z (as given in Table 3) with ρ(z) distribution of CC site type z (where Σ ρ(z) = 1), σa,≡ and σa,↑ are the absorption cross section of the TEB↑ to σ→ σ* and ≡O electronic excitations, Niii↑ and NII↑ the population number in state iii↑ and II↑ while g(ν, κz) and gII↑(ν, κz) are given by Eq. (10):
gX(ν,κz)=exp(½[hνEX(κz)σX(κz)]2),
(10)
where Ez) is given by Eq. (8) and σz) = σII↑z), while σII↑z) and EII↑z) are given in Table 3 for CC type z and h is the Planck’s constant.

4. Markov chain description of PD color center anneal

The annealing of pre-darkened material is in this section described by a Markov chain for the different bonds of section 2 and 3 with phonon activated state transitions by displacive transitions of 4-member SiO4 rings carrying the CC precursor as proposed in [2

2. K. E. Mattsson, “Photo darkening of rare earth doped silica,” Opt. Express 19(21), 19797–19812 (2011). [CrossRef] [PubMed]

].

The CCs (state II) hold two three-fold coordinated oxygen (≡O) in = Si<O2 as shown in Fig. 1(a). The ≡O is in [15

15. V. O. Sokolov and V. B. Sulimov, “Threefold coordinated oxygen atom in silica glass,” J. Non-Cryst. Solids 217(2–3), 167–172 (1997). [CrossRef]

] found to hold ≤20% weaker Si-O bonds than regular Si-O bonds (444 kJ/mol or 4.6 eV/bond [21

21. A. F. Holleman, E. Wiberg, and N. Wiberg, Inorganic Chemistry (Academic Press, 2001) p.131.

]) of pure silicon dioxide giving E ≈0.9 eV for its formation. The transition from state ii (with two ordinary Si-O bonds) to state II (with two ≡O) is through state iii (with one ≡O) as shown in Fig. 4
Fig. 4 State diagram for annealing of PD material with activation energies for transitions, note EB1 is II↑ and EB2 is II↓ covalent bond energy of Table 3 for the respective CC site ionic character.
. State iii↓ hold a “½” bond (TEB↓) in parallel to TEB↓ while state iii↑ is a “½” bond in parallel to TEB↑. Here ii→ iii transitions is by ≡O generation in overcoming O to O repulsion (≈0.78 eV and ≈0.86 eV as found in Table 3) for activation energy E≡1 and E≡3, respectively, as given in Table 4

Table 4. Transition activation energies

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. The ii→ 0 is by EDT = 1.53 eV [2

2. K. E. Mattsson, “Photo darkening of rare earth doped silica,” Opt. Express 19(21), 19797–19812 (2011). [CrossRef] [PubMed]

] in a displacive transition that facilitate the oxygen transfer. As E≡3 > E≡1 > EDT both state ii and 0 are terminal with respect to annealing (when monitored by ≈600 nm absorption).

The phonon activation of CC sites is given by the heat dissipated in a unit volume of fiber core. The phonons are carried by SiO4 rings (“fixated” gas molecules) with expected number density of phonons of energy E given by the Bose-Einstein distribution [exp(E/(kB T)) –1]−1. Here kB is Boltzmann constant and T fiber core absolute temperature. From Avogadros number, atomic masses [16

16. C. C. Robinson and J. T. Fournier, “Co-ordination of Yb3+ in phosphate, silicate, and germinate glasses,” J. Phys. Chem. Solids 31(5), 895–904 (1970). [CrossRef]

] and ≈2.2 g/cm3 mass density the silica hold 6.61∙1022 cm−3 atoms. With each SiO2 ring element to bring 2 atoms to the ring and to hold 2 degree of freedom for average 6-member rings the total number of rings is NT ≈1.1∙1022 cm−3. The number of CC sites NS is given by the method of manufacture and it is assumed that all sites are on 4-member planar rings in a relative small number compared to the total ring population NT. The 4-member planar rings hold an inscribed diameter Λ of ≈3 Å in SiO4 units of 2.65 Å between edge oxygen. This leads for speed of sound in fused silica vs ≈5980 m/s to a phonon “collision” frequency for the 4-member rings of ν0 = vs/Λ ≈20 THz.

The CC sites of ionic character z given in Table 3 are distributed by ρ(z) with ρ(z)∙NS rings to carry CC type z. Each of the CC sites is in either of the states U = (0, ii, iii↓, iii↑, II↓, II↑) to time t where ρ(z)∙NS = ΣNU,z(t) when summed over all states. The probability for a randomly chosen ring to hold CC site z in state U is hereby NU,z(t)/NT. This leads to the rate of phonons with activation energy EU,Z in rings carrying a CC site type z in state U by Eq. (11):

kU,z(T,t)=NU,z(t)NTν0[exp(EU,zkBT)1]1
(11)

The process of state transition by phonon activation energy EU,z for a CC site type z in state U is given by the single element reaction:
dNU,z(t)dt=kU,z(T,t)NU,z(t)ΔNU,z(t)=(1exp(kU,z(T,t)Δt))NU,z(t),
(12)
where dN/dt ≈ΔN/Δt and Taylor expansion (ε = 1-(1-ε) ≈1 – exp(-ε) for ε << 1) is applied in Eq. (12) for division of time into Δt transition steps.

For CC site type z the transitions of the Markov chain hold several transition routes as shown in Fig. 4 why change rate for state U is a sum of rates kU(T,t) = Σkj(T,t). This results in exp[-Σkj(T,t)∙Δt] decrease in state U population and kj(T,t)/Σkj(T,t) ∙ (1- exp[-Σkj(T,t)∙Δt]) increase in state j population with activation energy Ej in the U→ j transition. As there is no return path from terminal state 0 the transitions are handled in the 6x5 Markov matrix MZ:
M¯¯Z=(A1kDT(T,t)/kΣ100001A10A3kR(T,t)/kΣ300A1k3(T,t)/kΣ11A2A3kS1(T,t)/kΣ3A40A1k1(T,t)/kΣ1A2kS2(T,t)/kΣ21A30A50A2k4(T,t)/kΣ201A4000A3k2(T,t)/kΣ301A5),
(13)
where Aj = [1 – exp(-kΣj∙Δt)] for j = [1

1. Y.-S. Liu, T. C. Galvin, T. Hawkins, J. Ballato, L. Dong, P. R. Foy, P. D. Dragic, and J. G. Eden, “Linkage of oxygen deficiency defects and rare earth concentrations in silica glass optical fiber probed by ultraviolet absorption and laser excitation spectroscopy,” Opt. Express 20(13), 14494–14507 (2012). [CrossRef] [PubMed]

,5

5. L. Skuja, “Optically active oxygen-deficiency-related centers in amorphous silicon dioxide,” J. Non-Cryst. Solids 239(1–3), 16–48 (1998). [CrossRef]

] and kΣ1 = kDT(T,t) + k≡3(T,t) + k≡1(T,t), kΣ2 = kS2(T,t) + k≡4(T,t), kΣ3 = kR(T,t) + kS1(T,t) + k≡2(T,t), kΣ4 = kB2(T,t) and kΣ5 = kB1(T,t) in Eq. (13) are for the PD process of CCs type z in TEB ionic character.

For each CC site population z vectors pz,1 = (N0, Nii, Niii↓, Niii↑, NII↓, NII↑) and pz,2 contains the populations before and after the transition in time step Δt for the matrix multiplication:

pz,2=M¯¯Zpz,1
(14)

The rate coefficients for the matrix multiplication of Eq. (14) are recalculated after each time step as the concentration changes for the various states. The absorption as function of time is determined by the Niii↑,z and NII↑,z populations by insertion into Eq. (9).

5. Annealing of pre-photo darkened material

In this section the kinetic model for annealing of PD material is applied on experimental data from thermal cycles [11

11. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Heat-induced darkening and spectral broadening in photodarkened ytterbium-doped fiber under thermal cycling,” Opt. Express 17(12), 9940–9946 (2009). [CrossRef] [PubMed]

], thermal ramps [12

12. M. Leich, U. Röpke, S. Jetschke, S. Unger, V. Reichel, and J. Kirchhof, “Non-isothermal bleaching of photodarkened Yb-doped fibers,” Opt. Express 17(15), 12588–12593 (2009). [CrossRef] [PubMed]

] and isothermal anneals [13

13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

].

In [11

11. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Heat-induced darkening and spectral broadening in photodarkened ytterbium-doped fiber under thermal cycling,” Opt. Express 17(12), 9940–9946 (2009). [CrossRef] [PubMed]

] a short section of pre-darkened commercial 20 µm core diameter large mode area (LMA) fiber is subjected to temperature cycles with 60–180 K/min heating/cooling rates. Absorption data measured at 600 nm [from Fig. 2 and Fig. 1(b) in [11

11. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Heat-induced darkening and spectral broadening in photodarkened ytterbium-doped fiber under thermal cycling,” Opt. Express 17(12), 9940–9946 (2009). [CrossRef] [PubMed]

]] are given by solid and open dots in Fig. 5
Fig. 5 Experimental temperature cycles (dots) from [11] with model (solid lines) anneal of pre-darkened fiber at 60-180 K/min rates for CC type/ state distribution as given in Table 5.
. These are from two experiments on the same fiber (F1) with PD at identical conditions. The solid dots are for a 296 K→ 598 K→ 296 K→ 700 K (→ 296 K) cycle and the open dots are a 296 K→ 905 K → 296 K cycle. The core material is not specified except by ≈0.5 dB/m pump absorption at 915 nm [11

11. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Heat-induced darkening and spectral broadening in photodarkened ytterbium-doped fiber under thermal cycling,” Opt. Express 17(12), 9940–9946 (2009). [CrossRef] [PubMed]

] which suggest 0.15 mol % Yb2O3 by Al2O3 co-doping and yield NYb = 6.5∙1019 cm−3 for evenly distributed Yb ions.

This leads for fiber F1 to the absorption cross sections σa,≡ = σa,↑ = 303 pm2. Here it is noted that a uniform reduction in either cross section or NAS followed by a corresponding increase in the other will yield equal results. As there in addition is no model available for the distribution of Yb and Al atoms on tetrahedral and octahedral sites this is done by trial and error as given in Table 5

Table 5. The fiber F1 distribution on XO2 and |X| CCs of ionic character κ, number of active sites relative to number of Yb (NAS/NYb) and initial distribution of tetrahedral and octahedral CCs on states (II and iii).

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. Here shift among nearby ionic character CCs lead only to small changes in the overall form of the anneal ramp curve why the distribution in Table 5 present only a rough estimate. However, for this choice of distribution and absorption cross sections the solid lines in Fig. 5 are found.

The thermally induced increase in absorption of Fig. 5 is met by ≈70% of the tetrahedral CCs in state II↑ and ≈30% distributed to II↓, iii↓ and iii↑ while nearly all octahedral CCs are in II↑. The annealing cycles of Fig. 5 are calculated for 60, 120 and 180 K/min rates with the best overall fit for 120 K/min.

In Fig. 6
Fig. 6 Experimental absorption spectra (dots) from [11] with (solid curves) model anneal of pre-darkened fiber F1. The measures are after PD at 296 K and after a 296K→ 598K→ 296K thermal cycle and further →419 K and →577 K.
the spectral distribution in the 550 nm to 900 nm range is shown for the pre-darkened fiber F1 at 296 K and after thermal cycle 296 K→ 598 K→ 296 K and further to 419 K and 577 K as indicated by the dots [from Fig. 3 in [11

11. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Heat-induced darkening and spectral broadening in photodarkened ytterbium-doped fiber under thermal cycling,” Opt. Express 17(12), 9940–9946 (2009). [CrossRef] [PubMed]

]]. The spectra are in [11

11. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Heat-induced darkening and spectral broadening in photodarkened ytterbium-doped fiber under thermal cycling,” Opt. Express 17(12), 9940–9946 (2009). [CrossRef] [PubMed]

] found to show no change in absorption at 612 nm upon the annealing cycle in the further increase in temperature from 296 K to 577 K. The state model finds no such spectral pivot point but the spectral shifts meet a region of low variation in absorption around 600 nm identified by the dotted line in Fig. 6. The spectral shape for temperatures <450 K is independent on ramp rate while for >450 K it shows dependence on ramp rate. I.e. if it takes more than ≈1 min to obtain the spectrum at 577 K the recorded form is skewed in relation to its “true” spectral form.

The model predicts that at 120 K/min rate a 598 K → 296 K→ 598 K temperature cycling of fiber F1 result in unchanged number of state II and iii population. This leads to that either is the observed spectral shift in Fig. 6 (< 450 K) a result of thermally induced change in full width half maximum (FWHM) of the absorption cross section [11

11. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Heat-induced darkening and spectral broadening in photodarkened ytterbium-doped fiber under thermal cycling,” Opt. Express 17(12), 9940–9946 (2009). [CrossRef] [PubMed]

], temperature dependent absorption cross sections [12

12. M. Leich, U. Röpke, S. Jetschke, S. Unger, V. Reichel, and J. Kirchhof, “Non-isothermal bleaching of photodarkened Yb-doped fibers,” Opt. Express 17(15), 12588–12593 (2009). [CrossRef] [PubMed]

] or as here applied a temperature dependent anti-bond level R*(κ,T) as given by Eq. (7). The latter R*(κ,T) results in linear shift to CC peak absorption for unchanged peak absorption cross section.

In [12

12. M. Leich, U. Röpke, S. Jetschke, S. Unger, V. Reichel, and J. Kirchhof, “Non-isothermal bleaching of photodarkened Yb-doped fibers,” Opt. Express 17(15), 12588–12593 (2009). [CrossRef] [PubMed]

] are short sections of pre-darkened fibers (F2) of 0.6 mol % Yb2O3, 3.9 mol % Al2O3 and 0.6 mol% P2O5 core composition subjected to 5 – 20 K/min ramp rates. The absorption data [Fig. 3(a) in [12

12. M. Leich, U. Röpke, S. Jetschke, S. Unger, V. Reichel, and J. Kirchhof, “Non-isothermal bleaching of photodarkened Yb-doped fibers,” Opt. Express 17(15), 12588–12593 (2009). [CrossRef] [PubMed]

]] measured at 633 nm is shown in Fig. 7
Fig. 7 Experimental temperature ramps (dots) from [12] with model anneal of pre-darkened fiber F2 at 5 – 20 K/min rates for CC type/ state distribution of Table 6.
as dots. The solid lines in Fig. 7 are for a CC type distribution on octahedral CCs where nearly all are in state II↑ for the pre-darkened fiber as given in Table 6

Table 6. The fiber F2 distribution on XO2 and |X| CCs of ionic character κ, number of active CC sites relative to number of Yb (NS/NYb) and initial distribution on states (II and iii) for octahedral CCs.

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. By applying identical absorption cross sections as for F1 the absorption level of Fig. 6 suggest that NAS = 0.0069∙NYb or that F2 performs ≈45% better than the Yb/Al material of F1 with respect to PD induced absorption per Yb ion.

In [22

22. K. Arai, H. Namikawa, K. Kumata, T. Honda, Y. Ishii, and T. Handa, “Aluminum or phosphorous co-doping effects on the fluorescence and structural properties of neodymium-doped silica glass,” J. Appl. Phys. 59(10), 3430 (1986). [CrossRef]

] it is proposed that Al and P form salvation shells to Yb and it is found that P in contrast to Al distribute evenly within the silica matrix. This fits the NAS applied for F1 and F2 if P offers nucleation sites for Yb that replaces tetrahedral and Al dominated octahedral CCs. The absorption between 400 nm and 750 nm [from Fig. 2 of [12

12. M. Leich, U. Röpke, S. Jetschke, S. Unger, V. Reichel, and J. Kirchhof, “Non-isothermal bleaching of photodarkened Yb-doped fibers,” Opt. Express 17(15), 12588–12593 (2009). [CrossRef] [PubMed]

]] is shown in Fig. 8
Fig. 8 Experimental absorption spectra (dots) from [12] with (solid curves) model anneal of pre-darkened fiber F2. The measures are for pre-darkened fiber at 296 K and after ramp to 453 K, 518 K, 547 K and 583 K for 5 K/min ramps.
by dots for the F2 fiber pre-darkened at 296 K and annealed at 453 K, 518 K, 547 K and 583 K. In [12

12. M. Leich, U. Röpke, S. Jetschke, S. Unger, V. Reichel, and J. Kirchhof, “Non-isothermal bleaching of photodarkened Yb-doped fibers,” Opt. Express 17(15), 12588–12593 (2009). [CrossRef] [PubMed]

] the thermal history of the spectra is not specified but the 633 nm absorption (identified by the dotted line) find a good fit to 5 K/min ramps as indicated in Fig. 7 by stars. The solid lines in Fig. 8 are for 5 K/min rates found to give good match to the spectra above ≈600 nm while “missing” parts can be observed at shorter wavelengths. These “missing” parts correspond for the presently applied parameters for ≡O absorption in Eq. (8) to a center with peak absorption near ≈400 nm (3.1 eV) and another near ≈540 nm (2.3 eV). Here the ≈2.3 eV peak absorption and strength is directly influenced by choice of ≡O parameters. However, the “missing” parts could be due to phosphorous oxygen hole-center absorption at ≈3.1 eV, 2.5 eV and 2.2 eV [17

17. D. L. Griscom, E. J. Friebele, K. J. Long, and J. W. Fleming, “Fundamental defect centers in glass: Electron spin resonance and optical absorption studies of irradiated phosphorous-doped silica glass and optical fibers,” J. Appl. Phys. 54(7), 3743–3762 (1983). [CrossRef]

] that follows the CC formation and bleaching. Here ≈3.1 eV absorption decreases in strength with annealing while ≈2.3 eV absorption first grows up to ≈518 K and then decreases for higher temperatures. This difference may be due to that part of the ≈3 eV absorption is from the second doublet-to-doublet transitions of ≡O which for a Si-O interaction would lead to ≈2.9 eV absorption. This second transition is, however, not included in the present model.

In [13

13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

] three isothermal anneals on the same pre-darkened 20 µm alumino-silicate ytterbium doped LMA fiber (F3) are fitted by Δαn = [1 + (t/τ)β]−1. The dots in Fig. 9
Fig. 9 Absorption as function of time for isothermal anneals at 510 K, 556 K and 676 K with dots from [13] with solid curve model calculations for F3 by CC type and state distribution as given in Table 7.
represent these fits scaled to 1750 dB/m (as specified in Table 1 of [13

13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

]). The core material composition is not presented in [13

13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

] but by applying identical CC absorption cross sections as for F1 (and F2) the number of CC sites is fitted by ≈7.3 times the number of sites in F1. The distribution on CC type and states in Table 7

Table 7. Fiber F3 distribution on XO2 and |X| CCs of ionic character κ, number of active CC sites relative to number of sites in F1 and initial distribution on states (II and iii) for tetrahedral and octahedral CCs.

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is found by match to the temporal data for the 556 K anneal [from Fig. 7 of [23

23. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening in ytterbium-doped fibers,” Proc. SPIE 7580, 75800B, 75800B-8 (2010). [CrossRef]

]] shown as open dots in Fig. 10
Fig. 10 Absorption as function of time for 296 K→ 556 K ramp, 556 K anneal and 556K→ 920 K ramp given by open dots from [23], solid dots from [13] and with model solid curve for F3 by CC type and state distribution as given in Table 7.
. The solid dots in Fig. 10 are from the [13

13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

] fits. The annealing in Fig. 10 is by the shown temperature profile where identical ramps are applied for the 510 K and 676 K anneals of Fig. 9.

The initial increase in absorption for 296 K→ 556 K in Fig. 10 is met by 58% tetrahedral CC population in state II↑ and 42% in state II↓, iii↑ and iii↓ for the pre-darkened fiber. This distribution is applied for all isothermal anneals of F3. The actual performance of the F3 fiber is due to missing data on the level of Yb ion doping in the material not possible to assess.

In [13

13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

] demarcation energy ED = kBT ln(νDt) is applied to convert the time t of isothermal anneals into an energy scale. For constant characteristic frequency νD it is assumed that the trap release is by a 1st order reaction with rate k = νD ∙ exp(-EA/kBT) for activation energy EA. After time t at temperature T the fiber holds 1/e of the population trapped so k∙t = 1 divides the population by demarcation energy ED in that for EA ≤ ED the traps are open and for EA > ED the traps are closed. The absorption data as function of time is hereby transformed into a cumulative distribution of trap activation energy which brings the data of isothermal anneals onto a single curve. This may be observed in [13

13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

] by leaving out the initial values of the highest annealing temperatures. The first derivative of the curve d(Δα)/dED is then the probability frequency for the trap release.

For the present state model the ED technique is still applicable even though the reaction rate is concentration dependent and hereby of 2nd order. For the loss induced at 1000 nm only state II↑ is taken to induce absorption so the characteristic frequency depends here on time as: νD = ΣNII↑,z(t) ∙ ν0/NT, where ΣNII↑(t) is the population in state II↑ summed over all CC sites. In Fig. 11(a)
Fig. 11 (a) Probability frequency as function of demarcation energy ED for pre-darkened fibers F1, F2 and F3 by 2 K/min anneal ramps by induced loss at 1000 nm, and hereto corresponding (b) characteristic frequency νD as function of demarcation energy ED
is the probability frequency d[NII↑(t)]/dED shown against ED for F1, F2 and F3 fiber anneals by 2 K/min ramps based on the induced absorption at 1000 nm. The corresponding characteristic frequency νD as function of ED is shown in Fig. 11(b). This brings the activation energy for F1, F2 and F3 on a common scale.

It is for the ED technique to be observed that if absorption of the individual CC types scale with average ionic character the activation energy determined from number of sites will not in all cases agree with the value found directly from absorption. Here mixed tetrahedral and octahedral sites show larger difference between the estimates based on number of sites and absorption which makes it difficult to exactly relate a measured isothermal or ramp anneal profile to activation energy. It is further to be noted that the choice of absorption cross sections influences the values for activation energy in that a change in number of CC sites lead to change in the characteristic frequency.

The ED analysis in [13

13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

] for fiber F3 gives 1.32 eV in activation energy with 0.31 eV in FWHM by static νD = 2 GHz. For the present state model with a dynamic νD and from the number of sites the F3 fiber is found to exhibit peak activation energy at ≈1.37 eV with 0.2 eV FWHM and shoulders at 1.2 eV and 1.5 eV, respectively. The finding of activation energy in excess of O-O bond energy for the CC sites (in Table 3) is a result of the dynamic PD bleaching and creation process. This may also be observed through the negative probability frequency (thermally induced PD) peak at ≈1.1 eV for F1 and F3.

The F1 fiber shows peak activation energy at ≈1.6 eV by ≈0.1 eV FWHM which hold a similar trend relative to F2, F3 and the unspecified fiber in [13

13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

] of 1.5 eV in activation energy.

For F2 a peak at 1.36 eV and ≈0.2 eV FWHM is found which is to be compared to 1.3 eV and 0.5 eV FWHM in [12

12. M. Leich, U. Röpke, S. Jetschke, S. Unger, V. Reichel, and J. Kirchhof, “Non-isothermal bleaching of photodarkened Yb-doped fibers,” Opt. Express 17(15), 12588–12593 (2009). [CrossRef] [PubMed]

] where reaction rates for the various ramp anneals is applied to extract activation energy in relative relations among the various rates.

From Fig. 11(a) and Fig. 11(b) it appears that for Yb/Al material (F1, F3) the amount of thermally induced centers is larger than for Yb/Al/P material (F2). This is in the present implementation a result of different initial distributions on state II↑ and II↓, iii↑ or iii↓ for tetrahedral and octahedral CCs and that the P co-doped material not is expected to hold tetrahedral sites. The apparent increase in absorption for wavelengths in excess of 600 nm for Yb/Al/P in Fig. 8 is therefore mainly due to shift of the CC absorption center peaks with temperature as given by R*(κ,T) in Eq. (7).

The difference in pre-darkened distribution of states among the different fibers may be met by the model if it is assumed that there are internal temperature differences between tetrahedral and octahedral sites. Here a higher temperature is to be assumed for octahedral sites for a model with initially 50% population in state iii↓ and iii↑, respectively, such that after an isothermal anneal at temperatures from ≈350 K to ≈400 K the applied distributions may be found.

6. Conclusion

In this paper annealing of photo darkened rare earth co-doped silica material is described by means of a kinetic state model. It is proposed that the PD color centers are due to a σ-bond (“1”) in parallel to a three electron bond (TEB) to Yb as “1+TEB” formed on = Si<O2:∙Yb. The bleaching of pre-darkened material goes reversibly through a “½” bond in parallel to a TEB to Yb and irreversible to TEB to Yb terminal states. The dissociation energies for resonant TEBs shared between = Si<O2 and Yb3+ (or Al3+) are determined by applying an analogy to ionic character bonds of molecular oxygen and superoxide.

The bleaching of “1+TEB” bonds formed on = Si<O2 is modeled by a “fixated” phonon gas (SiO4 ring structure) with characteristic frequency given by ring size and relative number of CC sites for Bose-Einstein phonon statistics. This results for fixed number of sites to a single element second order chemical reaction for phonon activation. Based on the kinetic state model cycles, ramps and isothermal anneal data are analyzed and it is shown that a common set of ionic character bond energies for “1+TEB” and “½+TEB” bonds formed on = Si<O2 match the experimental data.

References and links

1.

Y.-S. Liu, T. C. Galvin, T. Hawkins, J. Ballato, L. Dong, P. R. Foy, P. D. Dragic, and J. G. Eden, “Linkage of oxygen deficiency defects and rare earth concentrations in silica glass optical fiber probed by ultraviolet absorption and laser excitation spectroscopy,” Opt. Express 20(13), 14494–14507 (2012). [CrossRef] [PubMed]

2.

K. E. Mattsson, “Photo darkening of rare earth doped silica,” Opt. Express 19(21), 19797–19812 (2011). [CrossRef] [PubMed]

3.

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]

4.

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]

5.

L. Skuja, “Optically active oxygen-deficiency-related centers in amorphous silicon dioxide,” J. Non-Cryst. Solids 239(1–3), 16–48 (1998). [CrossRef]

6.

S. Girard, Y. Ouerdane, B. Tortech, C. Marcandella, T. Robin, B. Dadier, J. Baggio, P. Paillet, V. Ferlet-Cavrois, A. Boukenter, J.-P. Meunier, J. R. Schwank, M. R. Shaneyfelt, P. E. Dodd, and E. W. Blackmore, “Radiation Effects on ytterbium- and Ytterbium/erbium-Doped Double-Clad Optical Fibers,” IEEE Trans. Nucl. Sci. 56(6), 3293–3299 (2009). [CrossRef]

7.

T. Deschamps, H. Vezin, C. Gonnet, and N. Ollier, “Evidence of AlOHC responsible for the radiation-induced darkening in Yb doped fiber,” Opt. Express 21(7), 8382–8392 (2013). [CrossRef] [PubMed]

8.

R. Schnadt and J. Schneider, “The Electronic Structure of the Trapped-Hole Center in Smoky Quartz,” Phys. Kondens. Materie 11, 19–42 (1970).

9.

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]

10.

S. Yoo, A. J. Boyland, R. J. Standish, and J. K. Sahu, “Measurement of photodarkening in Yb-doped aluminosilicate fibres at elevated temperature,” Electron. Lett. 46(3), 233–244 (2010). [CrossRef]

11.

M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Heat-induced darkening and spectral broadening in photodarkened ytterbium-doped fiber under thermal cycling,” Opt. Express 17(12), 9940–9946 (2009). [CrossRef] [PubMed]

12.

M. Leich, U. Röpke, S. Jetschke, S. Unger, V. Reichel, and J. Kirchhof, “Non-isothermal bleaching of photodarkened Yb-doped fibers,” Opt. Express 17(15), 12588–12593 (2009). [CrossRef] [PubMed]

13.

M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett. 34(17), 2637–2639 (2009). [CrossRef] [PubMed]

14.

L. Pauling, “The nature of silicon-oxygen bonds,” Am. Mineral. 65, 321 (1980).

15.

V. O. Sokolov and V. B. Sulimov, “Threefold coordinated oxygen atom in silica glass,” J. Non-Cryst. Solids 217(2–3), 167–172 (1997). [CrossRef]

16.

C. C. Robinson and J. T. Fournier, “Co-ordination of Yb3+ in phosphate, silicate, and germinate glasses,” J. Phys. Chem. Solids 31(5), 895–904 (1970). [CrossRef]

17.

D. L. Griscom, E. J. Friebele, K. J. Long, and J. W. Fleming, “Fundamental defect centers in glass: Electron spin resonance and optical absorption studies of irradiated phosphorous-doped silica glass and optical fibers,” J. Appl. Phys. 54(7), 3743–3762 (1983). [CrossRef]

18.

P. H. Krupenie, “The spectrum of molecular oxygen,” J. Phys. Chem. Ref. Data 1(2), 423–534 (1972). [CrossRef]

19.

Table of periodic properties of the Elements, Sargent-Welch, Skokie, IL, (1980).

20.

A. V. Kimmel, P. V. Sushko, and A. L. Shluger, “Structure and spectroscopic properties of trapped holes in silica,” J. Non-Cryst. Solids 353(5-7), 599–604 (2007). [CrossRef]

21.

A. F. Holleman, E. Wiberg, and N. Wiberg, Inorganic Chemistry (Academic Press, 2001) p.131.

22.

K. Arai, H. Namikawa, K. Kumata, T. Honda, Y. Ishii, and T. Handa, “Aluminum or phosphorous co-doping effects on the fluorescence and structural properties of neodymium-doped silica glass,” J. Appl. Phys. 59(10), 3430 (1986). [CrossRef]

23.

M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening in ytterbium-doped fibers,” Proc. SPIE 7580, 75800B, 75800B-8 (2010). [CrossRef]

OCIS Codes
(060.2290) Fiber optics and optical communications : Fiber materials
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: March 25, 2013
Revised Manuscript: May 7, 2013
Manuscript Accepted: May 10, 2013
Published: May 17, 2013

Citation
Kent E. Mattsson, "The three-electron bond =Si<O2:∙Yb absorption center of pre-darkened ytterbium-doped silica," Opt. Express 21, 12849-12864 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12849


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References

  1. Y.-S. Liu, T. C. Galvin, T. Hawkins, J. Ballato, L. Dong, P. R. Foy, P. D. Dragic, and J. G. Eden, “Linkage of oxygen deficiency defects and rare earth concentrations in silica glass optical fiber probed by ultraviolet absorption and laser excitation spectroscopy,” Opt. Express20(13), 14494–14507 (2012). [CrossRef] [PubMed]
  2. K. E. Mattsson, “Photo darkening of rare earth doped silica,” Opt. Express19(21), 19797–19812 (2011). [CrossRef] [PubMed]
  3. 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]
  4. M. Engholm and L. Norin, “Preventing photodarkening in ytterbium-doped high power fiber lasers; correlation to the UV-transparency of the core glass,” Opt. Express16(2), 1260–1268 (2008). [CrossRef] [PubMed]
  5. L. Skuja, “Optically active oxygen-deficiency-related centers in amorphous silicon dioxide,” J. Non-Cryst. Solids239(1–3), 16–48 (1998). [CrossRef]
  6. S. Girard, Y. Ouerdane, B. Tortech, C. Marcandella, T. Robin, B. Dadier, J. Baggio, P. Paillet, V. Ferlet-Cavrois, A. Boukenter, J.-P. Meunier, J. R. Schwank, M. R. Shaneyfelt, P. E. Dodd, and E. W. Blackmore, “Radiation Effects on ytterbium- and Ytterbium/erbium-Doped Double-Clad Optical Fibers,” IEEE Trans. Nucl. Sci.56(6), 3293–3299 (2009). [CrossRef]
  7. T. Deschamps, H. Vezin, C. Gonnet, and N. Ollier, “Evidence of AlOHC responsible for the radiation-induced darkening in Yb doped fiber,” Opt. Express21(7), 8382–8392 (2013). [CrossRef] [PubMed]
  8. R. Schnadt and J. Schneider, “The Electronic Structure of the Trapped-Hole Center in Smoky Quartz,” Phys. Kondens. Materie11, 19–42 (1970).
  9. 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]
  10. S. Yoo, A. J. Boyland, R. J. Standish, and J. K. Sahu, “Measurement of photodarkening in Yb-doped aluminosilicate fibres at elevated temperature,” Electron. Lett.46(3), 233–244 (2010). [CrossRef]
  11. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Heat-induced darkening and spectral broadening in photodarkened ytterbium-doped fiber under thermal cycling,” Opt. Express17(12), 9940–9946 (2009). [CrossRef] [PubMed]
  12. M. Leich, U. Röpke, S. Jetschke, S. Unger, V. Reichel, and J. Kirchhof, “Non-isothermal bleaching of photodarkened Yb-doped fibers,” Opt. Express17(15), 12588–12593 (2009). [CrossRef] [PubMed]
  13. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening-induced loss in ytterbium-doped fibers,” Opt. Lett.34(17), 2637–2639 (2009). [CrossRef] [PubMed]
  14. L. Pauling, “The nature of silicon-oxygen bonds,” Am. Mineral.65, 321 (1980).
  15. V. O. Sokolov and V. B. Sulimov, “Threefold coordinated oxygen atom in silica glass,” J. Non-Cryst. Solids217(2–3), 167–172 (1997). [CrossRef]
  16. C. C. Robinson and J. T. Fournier, “Co-ordination of Yb3+ in phosphate, silicate, and germinate glasses,” J. Phys. Chem. Solids31(5), 895–904 (1970). [CrossRef]
  17. D. L. Griscom, E. J. Friebele, K. J. Long, and J. W. Fleming, “Fundamental defect centers in glass: Electron spin resonance and optical absorption studies of irradiated phosphorous-doped silica glass and optical fibers,” J. Appl. Phys.54(7), 3743–3762 (1983). [CrossRef]
  18. P. H. Krupenie, “The spectrum of molecular oxygen,” J. Phys. Chem. Ref. Data1(2), 423–534 (1972). [CrossRef]
  19. Table of periodic properties of the Elements, Sargent-Welch, Skokie, IL, (1980).
  20. A. V. Kimmel, P. V. Sushko, and A. L. Shluger, “Structure and spectroscopic properties of trapped holes in silica,” J. Non-Cryst. Solids353(5-7), 599–604 (2007). [CrossRef]
  21. A. F. Holleman, E. Wiberg, and N. Wiberg, Inorganic Chemistry (Academic Press, 2001) p.131.
  22. K. Arai, H. Namikawa, K. Kumata, T. Honda, Y. Ishii, and T. Handa, “Aluminum or phosphorous co-doping effects on the fluorescence and structural properties of neodymium-doped silica glass,” J. Appl. Phys.59(10), 3430 (1986). [CrossRef]
  23. M. J. Söderlund, J. J. Montiel i Ponsoda, J. P. Koplow, and S. Honkanen, “Thermal bleaching of photodarkening in ytterbium-doped fibers,” Proc. SPIE7580, 75800B, 75800B-8 (2010). [CrossRef]

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