Abstract

The influence of lead fluoride on the formation and luminescent properties of glass and glass-ceramics has been studied. Using the Judd-Ofelt theory, the intensity parameters Ωt have been obtained. The stimulated emission cross-section of the 4I13/24I15/2 transition has been determined. The energy transfer from ytterbium to erbium with 96% efficiency was found with an erbium laser level lifetime of ∼8 ms. The gain shape of the glass under study has been found to be smoother than for phosphate glasses, which makes the former very attractive gain material for ultra-short laser sources. For the first time, fluorophosphate glass-ceramics were obtained with nanocrystals doped with erbium and ytterbium ions.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Due to the strong emission, associated with the 4I13/24I15/2 transition, erbium-ytterbium co-doped glasses are widespread as a gain material for lasers and amplifiers working in the eye-safe region near 1.5 µm, which are used in telecommunication systems and medicine [13]. The development of wavelength demultiplexing (WDM) demands gain materials with a flat amplification profile over a wide frequency region for broadband erbium-doped fiber amplifiers and ultra-short pulse sources. Nevertheless, fused silica fibers with a narrow gain curve, which limits the transmission capacity of the WDM systems, still prevail in this field. In the last decade a great variety of active glasses, doped with Er3+ ions, such as phosphate [46], tellurite [7], germanate [8,9], fluoride [1014] or fluorophosphates [1517], have been studied to find more efficient optical amplifiers with broad and flat gain profiles.

Mixed-anion oxyfluoride glasses based on phosphates, silicates, or germanates allow to obtain materials with a higher degree of line broadening and smoother line shapes [15]. Thereby, oxyfluoride glasses [1518] and glass-ceramics [1931] represent a compromise between oxide and fluoride materials.

Transparent oxyfluoride glass-ceramics with the merits from both fluoride and oxide materials are considered to be a good choice for RE ions host, and thus have turned to be one of the most promising gain materials. The key factor for the efficient luminescence is high content of the optically active RE ions in the precipitated fluoride nanocrystals with low phonon energy. Still, there are few works on fluorophosphate glass-ceramics [2022,2431]. However, the glasses studied in most papers usually contain several crystalline phases simultaneously, including phases with no rare earth ions [21,22,30,31].

It should be noted that known fluorophosphate glasses commonly contain 10-20 mol. % of phosphates [15,16]. This leads to the increase of hydroxyl groups’ concentration in glass and to the up rise of the absorption intensity in the spectral region above ∼3.0 µm due to a phosphate component. When the phosphates content is lower than 5.0 mol. %, the concentration of hydroxyl groups and the average density of P–O vibrational states reduce dramatically, contributing to increased transmission in 2.0–5.0 µm spectral range.

The aim of the given research was to study the fluorophosphate glasses and glass-ceramics with low concentration of phosphates co-doped with erbium and ytterbium ions, their spectroscopic and lasing properties, such as absorption, luminescent cross-sections, Judd-Ofelt parameters, quantum efficiency, fluorescence lifetime, as well as chemical composition and structure. The effect of barium fluoride substitution for lead fluoride on the glass-ceramics formation and optical properties was additionally investigated.

2. Experimental section

The chemical composition of the glasses under study was 5 Ba(PO3)2– 84 (AlF3 – CaF2 –MgF2 – BaF2(– SrF2) – 10 YbF3 - x ErF3, where x = 0; 1.0 (glass B) and 5 Ba(PO3)2 - 84 (AlF3 –CaF2 –MgF2 – PbF2(– SrF2) – 10 YbF3 - x ErF3, where x = 0; 1.0 (glass P) (mol. %). The study of lead fluoride effect on the glass properties was motivated by the ability of lead to increase the refractive index, which was important to match the refractive indexes of the given glasses and widely used silica glass. Previously, neodymium-doped glasses of this system were studied in [32].

Batch materials from reagent-grade, ErF3, and YbF3 (Spectrum Materials) had purity above 99.99%. A series of glasses was weighed with 0.001% accuracy and mixed thoroughly. The raw materials were melted in a closed glassy carbon crucible in Ar atmosphere at temperatures 1000-1050°C. The Er3+/Yb3+ co-doped transparent glass-ceramics was prepared by bulk crystallization of the initial glasses during isothermal treatment. The glass transition temperature (Tg) was measured with STA 449F1 Jupiter Netzsch differential scanning calorimeter with 10°/min scanning rate and was found to be ∼436°C (B) and ∼420°C (P). The glass samples thermal treatment was carried out in Nabertherm muffle furnace with program control. Specimens were heat treated at 480°C within 60-240 minutes.

X-ray dispersive energy analysis (XDEA) with a scanning electron microscope (SEM) Vega 3SBH (Tescan) (EPMA) of energy dispersion type x-Act Energy (Oxford Instruments) was used for determination of the Yb3+ and Er3+ ions concentration and nanocrystals chemical composition in the glass-ceramics. The concentration estimation error was specific for each chemical element, but, in general, for Ba, Ca, Sr, Yb, Er, P, Mg, it was in the range of 0.1-0.2 wt.%. X-ray phase analysis was performed using a Rigaku X-ray diffractometer (XDR) with CuKα-radiation source and D/teX Ultra detector. The search of analogues was carried out within the ICDD PDF-2 database of powder X-ray diffraction patterns. The roentgenogram was taken in the Bragg-Brentano geometry. A 0.02-mm-thick Ni foil was applied to suppress CuKβ radiation. The measurement step was 0.05°. The mean size value estimation error was 0.5 nm.

The refractive index was measured by Abbe refractometer IRF −454 BМ working at D spectral line (λ=589.3 nm) and was calculated for the spectral region using equation from our previous work [32] with accuracy of ± 0.001.

The absorption spectra at room temperature were recorded in the range of 300-1700 nm by Perkin-Elmer Lambda 900 spectrophotometer. Emission spectra were obtained by excitation of the samples with 457 nm radiation from a 1 W cw solid-state laser. The luminescence was recorded in 500-1700 nm range using RED-Wave-NIRX-SR photodetector (Stellarnet Inc.). The lifetime of the excited state was determined using a second harmonic of Q-switched Nd3+:YAG laser LQ 529B (Solar Laser Systems) with emission wavelength 532.8 nm as an excitation source. The luminescence radiation was detected using a 0.300 Meter Triple Grating Monochromator (Spectra Pro 300i) (Princeton Instruments) with InGaAs- ID-441 photodetector. The duration of the excitation pulses was 10 ns, the pulse frequency was 10 Hz. The luminescence decay curves were recorded by an Infinium HP54830 digital storage oscilloscope (Agilent Technologies). The mean error in determining the lifetime was 10%. Judd–Ofelt (JO) analyses of Er3+ ions in glasses and glass-ceramics was performed to evaluate the intensity parameters with t = 2, 4, 6. Up-conversion luminescence of glass and glass-ceramics was studied under a 980-nm laser diode excitation.

3. Experimental results

3.1 Absorption cross-sections and Judd-Ofelt parameters

Composition of the fluorophosphate glasses doped with Yb3+ and Er3+ are presented in Table 1. Dopant ion concentration (i.e., number of ions per unit volume) was evaluated using the following relation:

$$N\left( {\frac{{ion}}{{c{m^3}}}} \right) = \frac{{\textrm{mole fraction of rare earth oxide } \times \textrm{Avogadro number}}}{{\textrm{average molecular weight}}}$$
The Yb3+ and Er3+ ions energy level diagram, the main transitions associated with the excitation of Yb3+ ions, direct and reverse energy transfer between Yb3+ and Er3+, and up-conversion transitions are presented in Fig. 1. Erbium sensitization by ytterbium ions is commonly used in bulk lasers to reduce significantly the pump threshold.

 figure: Fig. 1.

Fig. 1. (a)Yb3+ and Er3+ ions energy level diagram: the Er–Yb energy transfer process (dotted arrow), the nonradiative transitions (dashed arrow), and the radiative transitions (solid arrows); (b) the absorption cross-sections of the B and P glasses: absorption cross-sections of the 2B (dashed line) and 2P (solid line) glasses in a wide spectral range.

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Tables Icon

Table 1. Composition (mol. fraction) and physical properties of the fluorophosphate glasses doped with ErF3, YbF3, where NEr, NYb – dopant ions concentration

The nine major bands of absorption spectra were identified as transitions from the ground energy level 4I15/2 of Er3+ ions to upper ones (Fig. 2a). The JO intensity parameters were calculated using the JO theory from the obtained absorption cross-section spectra ${\sigma _{abs}}(\lambda )$ for 2B and 2P glasses (Fig. 2).

 figure: Fig. 2.

Fig. 2. The luminescence spectra of the 2B (dashed line) and 2P (solid line) glasses doped with Er3+ ions of (a) 4I13/24I15/2 transition and (b) in the visible spectral region (λexc = 980 nm).

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There is a contribution of a magnetic dipole transition for the transitions that comply with following selection rules: $\Delta S = \Delta L = 0,\Delta J = 0, \pm 1$ (Eq. 2) [33]. Wave functions in the intermediate-coupling approximation were used for the calculation of the oscillator strength associated with the magnetic dipole mechanism. The experimental oscillator strength considering electric dipole mechanism ($S_{ed}^{meas}$) was calculated using the integral values of the absorption cross-sections (σ(λ)), associated with transitions from the ground 4I15/2 state (J) to upper states (J’) (Eq. 2):

$$S_{ed}^{meas}(J \to J^{\prime}) = \frac{{27hc(2J + 1)n}}{{8{\pi ^3}\overline \lambda {e^2}{{({n^2} + 2)}^2}}}\int {\sigma (\lambda )d\lambda - \frac{{9{n^2}}}{{{{({n^2} + 2)}^2}}}{S_{md}}}$$
where h is the Planck constant, c is the light speed in vacuum, e is the electron charge, n is a refractive index, $\overline \lambda $ is an absorption band mean wavelength (nm), J and J’ are the quantum numbers of total angular momentum of ground and excited states respectively. Smd is the line strength of the magnetic component. According to the JO theory the theoretical line strengths of the electric $S_{ed}^{calc}$ and magnetic ${S_{md}}$ dipole components are given by (Eq. 3, 4):
$$S_{ed}^{calc}(J \to J^{\prime}) = \sum\limits_{t = 2,4,6} {{\Omega _t}{{|{\langle{{\Psi _J}||{{U^t}} ||\Psi {^{\prime}_{J^{\prime}}}} \rangle } |}^2}}$$
$${S_{md}}(J \to J^{\prime}) = {\left[ {\frac{h}{{4\pi mc}}} \right]^2}{|{\langle{{\Psi _J}||{L + 2S} ||\Psi {^{\prime}_{J^{\prime}}}} \rangle } |^2}$$
Experimental and calculated values of oscillator strengths for glasses under study are given in Table 2. For 2B (1) and 2P (2) glasses three parameters ${\Omega _t}$ (t = 2, 4, 6) were obtained from the absorption cross-sections using the least square method (Table 2). Calculations demonstrate that the ${\Omega _2}$ and ${\Omega _6}$ parameters of Er3+ ions in the 2P glass containing lead fluoride is higher than for the 2B glass.

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Table 2. The experimental Sexp (10−20 cm2), calculated Scal (10−20 cm2) line strengths, the root mean square deviation (Δrms) for transitions from 4I15/2 level to different excited levels, and the Judd-Ofelt parameters ${\Omega _2}$, ${\Omega _4}$, ${\Omega _6}$ and $\chi $ of the Er3+ ions in the 2B, 2P glasses

3.2 Emission spectra

Luminescence spectra of the 2B and 2P glasses in IR (Fig. 2a) and visible (Fig. 2b) range are very similar.

The main difference for the 4I13/24I15/2 transition is the total angular momentum (ΔJ = 1). Therefore, the magnetic dipole contribution should be taken into account for this transition [34]. The spontaneous emission probability of the 4I13/24I15/2 transition is given by:

$$A(J \to J^{\prime}) = \frac{{64{\pi ^4}{e^2}}}{{3h(2J + 1){{\overline \lambda }^3}}}\left[ {n{{\left( {\frac{{{n^2} + 2}}{3}} \right)}^2}S_{ed}^{calc} + {n^3}{S_{md}}} \right]$$
where h is the Planck constant, e is the electron charge, n is a refractive index at a mean wavelength $\bar{\lambda }$, $S_{ed}^{calc}$ and Smd are the line strengths of the electric dipole and magnetic dipole transitions, J and J’ are the quantum numbers of total angular momentum of ground and excited states respectively. The obtained emission parameters are presented in Table 3.

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Table 3. Emission parameters of the 2B and 2P glasses ($\overline \lambda $– the mean emission wavelength, Sed and Smd –the line strengths of the electric dipole and magnetic dipole transitions, $A(J \to J^{\prime})$ - spontaneous emission probability, QE - quantum efficiency, τrad and τexp - radiative and experimental lifetime by 532.8 nm excitation)

According to the calculations Smd is 0.7 for the glasses under study. The second term in parentheses (Eq. 5) is regardless of the ligand field, it depends only on the refractive index, and its contribution to spontaneous emission probability is about 25%. According to the JO theory, the line strength of the electric dipole components of the 1.55-µm transition is determined as follows [33].

$${S_{ed}}[{{}^4{I_{13/2}};{}^4{I_{15/2}}} ]= \sum\limits_{t = 2,4,6} {{\Omega _t}{{\langle{{}^4{I_{13/2}}||{{U^t}} ||{}^4{I_{15/2}}} \rangle }^2} = 0.19 \cdot {\Omega _2} + 0.118 \cdot {\Omega _4} + 1.462 \cdot {\Omega _6}}$$
where three coefficients of U(t) are the reduced matrix elements of the unit tensor operator U(t) [34], calculated in the intermediate-coupling approximation, and Ωt (t = 2, 4, 6) are the intensity JO parameters.

The radiative lifetime (τrad) of the excited 4I13/2 state was calculated using Eq. 7:

$${\tau _{rad}} = \frac{1}{{\sum {A(J \to J^{\prime})} }}$$
According to the JO calculations, the radiative lifetime of Er3+ 4I13/2 state was 8.0 ms for the 2B glass and 7.35 ms for the 2P glass (Table 3). The obtained radiative lifetime made it possible to calculate the stimulated emission spectra ${\sigma _{em}}(\lambda )$ for erbium ions at the laser transition 4I13/24I15/2. The potential performance of the erbium-glass laser is highly dependent on the radiation and absorption properties as well as their spectral forms since erbium lasers operates using the three-level scheme. There are two methods to calculate the stimulated emission cross-section spectrum: in the framework of the modified reciprocity method (MR) and by Fuchtbauer-Landenburg equation (FL). These calculations offer a possible way to validate the reciprocity results. The calculation of the stimulated emission cross-section spectrum according to the MR method for isotropic media [35] was made by Eq. 8.
$${\sigma _{em}} = \frac{{A(J \to J^{\prime})}}{{8\pi c{n^2}}} \cdot \frac{{\exp ({ - hc/({kT\lambda } )} )}}{{\int {{\lambda ^{ - 4}}{\sigma _{abs}}(\lambda )\exp ({ - hc/({kT\lambda } )} )d\lambda } }}{\sigma _{abs}}(\lambda )$$
The stimulated emission cross-section spectrum was calculated by the FL method for isotropic media [36] via following expression:
$${\sigma _{em}}(\lambda ) = \frac{{A(J \to J^{\prime}){\lambda ^5}}}{{8\pi c{n^2}}} \cdot \frac{{{I_{lum}}(\lambda )}}{{\int {\lambda {I_{lum}}(\lambda )d\lambda } }}$$
The stimulated emission cross-section spectra calculated by different methods are shown in Fig. 3. As it can be seen that the results of the calculations by two methods are close to each other.

 figure: Fig. 3.

Fig. 3. The absorption cross-section spectra (1), stimulated emission obtained by MR method (2) and FL method (3) for the4I13/24I15/2 transition of the 2P glass.

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3.3 Glass-ceramics

It is known, that the luminescent properties of the RE-ions-doped glass-ceramic materials are better than of the amorphous materials due to RE-ions entering into crystalline environment. Such regularities were observed previously for oxyfluoride silicate and fluorophosphate glasses [27,37]. Heat treatment of fluorophosphate glasses can lead to the formation of the fluoride nanocrystals in the glass host. The heat treatment temperatures for 2B and 2P glass-ceramics formation were derived from DSC curves (Fig. 4).

 figure: Fig. 4.

Fig. 4. The DSC curves of the 2B and 2P glasses

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Analysis of the DSC curves shows that the substitution of barium fluoride for lead fluoride causes radical changes in glass crystallization process. The DSC curve of the 2B glass shows a single narrow intense exothermal peak (1 mV/mg). The DSC curve of the 2P glass has two crystallization peaks (0.2 mV/mg). It is clearly that crystallization ability of the 2P glass is lower comparing with the 2B glass. This results are in good agreement with the glass-forming ability of lead fluoride [9,37]. Onset crystallization temperature Tx is 520°C for both the 2B and 2P glasses. For glass-ceramics synthesis the 2B and 2P glasses were heat treated at 480°C (within 120 and 180 min), which is 40 °C below the onset crystallization temperature. The choice of temperature is due to the need to obtain crystals smaller than 50 nm.

X-ray diffraction was obtained for the 2B and 2P glass-ceramics (Fig. 5). Figure 5a shows X-ray diffraction results on the 2B glass after the heat treatment at 480°C during 180 min, where two crystalline phases were found. One of them corresponds to crystal phase Usovite (JCPDS card № 010722129). In addition to Usovite peaks, a set of different intense peaks is present in the diffraction pattern. The location and ratio of these peaks indicate the existence of cubic crystal phase in the glass-ceramics; however, there is no card with corresponding reflexes in the database. X-ray diffraction pattern of the 2P glass after 180-min heat treatment at 480°C shows single crystalline phase (Fig. 5b), which is identical to the unknown cubic phase as in the 2B glass-ceramics. The nanocrystals’ mean sizes were estimated as ∼40 nm based on Debye-Scherer formula.

 figure: Fig. 5.

Fig. 5. X-ray diffraction pattern of the 2B (a) and 2P (b) glasses after the heat treatment for 180 min at T = 480°C: Usovit crystal phase (JCPDS card, No. 010722129) denoted by x and cubic unknown phase (denoted by o)

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The crystal phase distribution in the glass-ceramics was studied by scanning electron microscopy (SEM). Electron microscope images of the 2P glass after 180-min heat treatment are shown in Fig. 6. The micrograph demonstrates that nanocrystals locate close to each other and form dendritic particles. According to the XRD data, the mean size of nanocrystals is about 40 nm. The chemical composition of the glass-ceramics was determined in two points: near the nanocrystals, surrounded by glass host (1), and in pure glass region with no nanocrystals (2) (Fig. 6). The obtained compositions (numbered as in Fig. 6) are presented in Table 4. Analysis of the compositions 1 and 2 shows that erbium ions were mainly included in the crystalline phase and were absent in the residual glass.

 figure: Fig. 6.

Fig. 6. (a, b). SEM micrographs of the 2P glass (after the heat treatment) at different scale. The chemical composition of the glass was determined in two points: near the nanocrystals, surrounded by glass host (1), and in pure glass region without the nanocrystals (2)

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Table 4. The chemical composition of the 2P glass-ceramics obtained by XDEA: composition 1 – in the region of nanocrystals surrounded by glass; composition 2 - at pure glass region (at.%) (according to Fig. 8)

The absorption and luminescence spectra of the initial glass and glass-ceramics are compared in Fig. 7. Figure 7a shows that the absorption spectra of erbium ions in the region of 1500-1550 nm of the initial and heat-treated (at 480 °C for 2 and 3 hours) 2P glasses are the same (similar situation occurred in case of 2B glass). The luminescence spectra in the NIR region were also not affected by the heat treatment within two and three hours. With an increase in the heat treatment duration up to four hours the luminescence spectra of erbium ions in the NIR region change, however, the glass becomes almost opaque. In contrast, the luminescence bands of the erbium ions in the visible region are sensitive to the heat treatment (Fig. 7b). In Fig. 7b the luminescence spectra of transitions 2H11/24I15/2 (∼525 nm) and 4S3/24I15/2 (∼545 nm) of the 2P glass are presented. As the heat treatment duration increases, the spectra become more structured, which may be due to the ordering of the nearest environment of the erbium ions, as they are introduced in the crystal phase in the glass host. A similar pattern is observed for the samples of the B series.

 figure: Fig. 7.

Fig. 7. The absorption (a) and luminescence (b) spectra of the 2P glass before and after the heat treatment with different duration

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4. Discussion

4.1 Glasses

The JO parameters are important for the study of local structure and bonding of RE ions. Among the three JO Ωt parameters (t = 2, 4, 6), which determine the f–f transition probabilities of Er3+ ions, parameter Ω2 is the most sensitive to the asymmetry degree of the crystal field, in which the RE ion locates. Ω6 shows a correlation with a covalence ratio of the activator–ligand bond [38].

Considering these regularities, we can draw some conclusions on the glasses under study. The comparison of Ωt parameters of the studied 2P and 2B glasses and fluoroaluminate glass from [38] reveals the coincidence of the Ω2 and Ω4 parameters in these glasses and difference in Ω6. The values of Ω6 are 0.98, 1.34, and 1.63 for fluoride glass, the 2B glass, and the 2P glass respectively (Table 2). It should be noted that among the f–f transitions of Er3+, the radiative transition probabilities of 4S3/24I15/2 (0.55 μm) and 4I13/24I15/2 (1.5 μm) depend strongly on Ω6, since the reduced matrix elements 〈||U(2)||〉 and 〈||U(4)||〉 of these transitions are equal to zero or negligibly small. The Ω6 parameter is the most sensitive to electron density changes.

The Ω2 parameter of the 2B glass coincides with the one of the fluoride glass (Table 2). This coincidence indicates that the introduction of phosphates does not affect the nearest environment of the erbium ions in the glass host. The Ω2 parameter is higher in the 2P glass than in the 2B glass, which means that the Er3+ ions occupy less symmetric sites in the 2P glasses. Usually the change of Ω2 with introduction of phosphates or increase in their content is attributed to the change in the content and polarizability of the oxygen ions, which leads to a change in electron donation properties of ligands at the RE sites. The introduction of lead fluoride into the glass composition causes the increase of the asymmetry degree at the RE sites, which is evidenced by the increase of the Ω2 parameter.

It was shown in [38] that in oxide glasses the Ω6 parameter was the most sensitive to the overlap integral of the 4f and 5d orbitals. The Ω6 parameter has a larger value owing to a larger ionicity of the Er–O bond in glasses. However, it was shown [38] that in the fluoride glasses the Ω6 parameter increased with a decrease in the Coulomb interaction, which could be a measure of the crystal field strength. Additionally, Ω6 increased with increase in a distance between erbium and fluorine ions. It was shown in [38] that the Ω6 changes with phosphate content were caused by changes in the p-electron donation in case of fluorophosphate glasses with various modifier ions. The Ω6 parameter was also affected by the rigidity and long-range effects of the glass host [3944]. The increase of this parameter with the introduction of lead evidences the significant effect of this component on the formation of the local erbium environment. The change in the glass structure is also evidenced by the DSC curves change.

It is important that the obtained effect of lead introduction in glasses under study differs from the fluorophosphates glasses with high phosphate concentration. The significant decrease in all Ωt parameters was observed for glasses with 20 mol.% of phosphates with the introduction of lead ions [9]. On the contrary, our calculations have shown that in the 2B and 2P glasses the increase in all JO parameters is observed. This result can be explained by different role of lead fluoride in the glass formation. In glass with high concentration of phosphates, lead fluoride acts as a modifier, while in the 2P glass lead fluoride acts as a glass-forming element, which can be seen from the DSC curves (Fig. 4).

The JO parameters are useful for calculation of emission properties such as radiative lifetimes of transitions. The comparison of calculated and measured lifetimes gives information about non-radiative decay rates. From the comparison of the 2P and 2B glasses it could be seen, that the introduction of lead fluoride into the glass composition significantly affects the luminescent properties of glasses with low phosphate concentration (Table 3).

An efficient energy transfer process from Yb3+ ions to Er3+ ions, make co-doped fluorophosphate glass an excellent active material for improving the gain of the Er3+ 4I13/24I15/2 transition near the 1536 nm. The paper assessed the energy transfer efficiency based on the Burshtein theory [45]. The Burstein model takes into account the effect of energy migration between sensitizing ions. It is assumed that energy migrates between sensitizers (Yb in our case) until it finds an activator ion (Er).

The choice of the theoretical approach was due to the high concentration of ytterbium being a sensitizer (s) and small concentration of Er3+ being an activator ion (a). Taking into account the absorption σabsx and emission σems cross-sections (where x = a, s), and the fluorescence decay time τs = 1.3 ms (measured for the 1B and 1P glasses doped only with ytterbium), we calculated that Rss = 1.55 nm and Rsa = 1.21 nm for the 2B and 2P glasses. The probability of the energy transfer process was determined by the following equation:

$${W_{DA}} = \frac{{\pi {{(2\pi /3)}^{5/2}}R_{sa}^3R_{ss}^3}}{{{\tau _s}}}{N_s}{N_a}$$
where Nx is the number of ions per unit volume (cm−3).

The transfer efficiency ηt was defined by Eq. 11:

$${\eta _t} = \frac{{{W_{DA}}{\tau _s}}}{{1 + {W_{DA}}{\tau _s}}}$$
Using the measured value for single ytterbium-doped 1B (1P) glass, τs, and fluorescence decay values for ytterbium-erbium co-doped 2B (2P) glass, τsa, one can estimate the transfer efficiency ηt = 96%.

One of the important characteristics of the laser material is the gain band $g(\lambda )$. A broad emission spectrum of Er3+ 1.55 µm transition can be obtained in the glass compositions under study.

For a quasi-three-level active medium:

$$g(\lambda )= N{\sigma _l}(\lambda )({{\beta_2} - {\beta_l}(\lambda )} )$$
where N is the activator concentration, ${\sigma _l}(\lambda )= {\sigma _{abs}}(\lambda )+ {\sigma _{em}}(\lambda )$, ${\beta _2} = {{{N_2}} \mathord{\left/ {\vphantom {{{N_2}} N}} \right.} N}$ is the relative population of the upper multiplet states, $N_2^{}$ is the population of the upper multiplet states, ${\beta _l}(\lambda )= {{{\sigma _{abs}}(\lambda )} \mathord{\left/ {\vphantom {{{\sigma_{abs}}(\lambda )} {{\sigma_l}(\lambda )}}} \right.} {{\sigma _l}(\lambda )}}$. For characterization of the gain band shape, the gain section is often used, and defined as:
$${\Sigma _{gain}} = {{g(\lambda )} \mathord{\left/ {\vphantom {{g(\lambda )} N}} \right.} N} = {\sigma _l}(\lambda )({{\beta_2} - {\beta_l}(\lambda )} )$$
The gain bandwidth of the laser material $\Delta \lambda$ characterizes the spectral range of the generated radiation and the duration of the pulses ($\Delta \tau$) that can be generated or amplified by this laser material:
$$\Delta \tau = \frac{{\lambda _{\max }^2}}{{c\Delta \lambda }}$$
Here c is the light velocity in vacuum, $\lambda _{\max }^{}$ is the wavelength corresponding to the gain band maximum.

Fig. 8 demonstrates the spectra of the gain cross-section ${\Sigma _{gain}}$ of the 2B and 2P glasses. Comparison of the gain sections shows that lead introduction causes an increase in the gain cross-section by 15%. The fluoride phosphate is expected to have a flat gain shape between 1525–1565 nm for a normal population inversion of 50–60%. The Fig. 8 shows that at ${\beta _2}$ = 0.35-0.70 the full width at half maximum of the gain spectra varies from ∼50 to 80 nm, which corresponds to the duration of the generated or amplified spectrally limited light pulses of 140-290 fs.

 figure: Fig. 8.

Fig. 8. Spectra of amplification cross-sections of the 2B (a) and 2P (b) glasses at different values of ${\beta _2}$

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4.2 Glass-ceramics

The main points in the formation of glass-ceramics are:

  • • What crystal phase forms during the heat treatment?
  • • How to control the mean size of nanocrystals and the content of RE ions in crystal phase?
  • • What is the impact of glass-ceramics formation on the luminescent properties of the material?
The first point could be addressed by the combined results on the electron microscopy, X-ray diffraction, and luminescence before and after the heat treatment. The XRD data suggested that the nanocrystals formation began at the lower temperature than the maximum crystallization temperature (Fig. 4, 5). With no lead fluoride in the glass composition, we have observed the formation of two different crystal phases. One of them was Usovite and the other one was an unknown cubic crystal phase. The nanocrystal sizes were determined from the Debye–Scherrer formula and were equal to ∼40 nm. Unfortunately, no data has been found on such compounds, which prevented the precise determination of the crystal phase composition. The introduction of lead fluoride causes formation of glass-ceramics with no Usovite, but with a single cubic crystal phase doped with erbium and ytterbium. The XRD data clearly showed that the cubic crystal phase was the same in the glass-ceramics obtained from glasses with addition of lead fluoride and without it.

SEM analysis has been performed to study the morphology and chemical composition of the crystal phase. It was shown that small nanocrystals formed dendritic structure in the glass-ceramics based on the 2P glass. The comparison of chemical composition in the region of the nanocrystal formation and the pure glass host allowed determination of the components, which formed the crystal phase: strontium, calcium, ytterbium, and erbium. Analysis of the compositions 1 and 2 (Table 4) showed that the erbium ions were mainly incorporated in the crystalline phase and were absent in the residual glass. Spontaneous crystallization lead to the formation of the dendrites in the glass host (Fig. 4), which were composed from smaller nanocrystals. It has been suggested that the cubic fluoride phase composition could be CaxSr1-xYb1-yEryF5 (d1 = 3.24, d2 = 2.815, d3 = 1.95), which is structurally isomorphous to the known compound BaGdF5 (d1 = 3.47, d2 = 3.015, d3 = 2,135). If we assume that the peak d1 = 3.24 corresponds to the (111) plane, then the lattice constant of the nucleated phase will be a = 5.61 Å. Unfortunately, there is no data on such compounds that prevents making an unambiguous conclusion on the nature of the cubic crystalline phase, which determines the properties of the glass-ceramics under study.

The assumptions about introduction of the active ion in the nanocrystals are supported by the data on up-conversional luminescence. The up-conversion luminescence spectra obtained with 0.975–µm pump are shown in Fig. 7. The intense peaks originates from the 4S3/24I15/2 and 4F9/24I15/2 transitions, which are observed near 550 and 660 nm respectively. The up-conversion spectrum of the glass-ceramics (glass after the heat treatment) is more structured; a larger number of Stark sublevels are observed (Fig. 7b). This also confirms the incorporation of the erbium ions into the crystalline phase during the heat treatment of the initial glass.

However, the close match of the erbium absorption cross-section and the JO parameters in the glasses and glass-ceramics indicates small changes of the local erbium ions environment. Unlike the earlier studied oxyfluoride glass-ceramics on the base of silicate and phosphate glasses [9,27,32,46], in the fluorophosphates glass-ceramics the energy of phonons is very close both in the glass host and in the crystal phase. Therefore, the luminescent characteristics of the main laser transition remain the same in the glass and glass-ceramics. Noticeable changes are observed only in the up-conversional luminescence.

5. Summary

The Er3+ fluorescence and laser properties at 1.5 µm were studied in the fluorophosphate glass. The energy transfer from ytterbium to erbium with 96% efficiency was found in the 2P and 2B glasses, with lifetimes for the erbium laser level of ∼8 ms. The smooth gain shape of the fluorophosphate glasses resulted in a considerably larger laser tuning range of 1510–1600 nm with a maximum near 1560 nm. The glasses with addition of lead fluoride had the higher spontaneous emission probability of the 4I13/24I15/2 transition and gain cross-section and more resistant to crystallization. The introduction of lead fluoride caused formation of glass-ceramics without Usovite crystal phase, with a single cubic crystal phase doped with erbium and ytterbium. The XRD data clearly showed that the cubic crystal phase was the same in the glass-ceramics obtained from glasses with and with no addition of lead fluoride. The broad and flattened laser tuning spectra makes the 2P and 2B glasses a very attractive candidate for the broadband amplifiers in wavelength-division multiplexing systems or as gain material for ultra-short laser sources.

Funding

Russian Science Foundation (19-13-00343).

References

1. E. Tanguy, C. Larat, and J. P. Pocholle, “Modelling of the erbium - ytterbium laser,” Opt. Commun. 153(1-3), 172–183 (1998). [CrossRef]  

2. S. Taccheo, P. Laporta, S. Longhi, O. Svelto, and C. Svelto, “Diode-pumped bulk erbium-ytterbium lasers,” Appl. Phys. B: Lasers Opt. 63(5), 425–436 (1996). [CrossRef]  

3. W. C. Wang, B. Zhou, S. H. Xu, Z. M. Yang, and Q. Y. Zhang, “Recent advances in soft optical glass fiber and fiber lasers,” Prog. Mater. Sci. 101, 90–171 (2019). [CrossRef]  

4. S. Taccheo, P. Laporta, and C. Svelto, “Widely tunable single-frequency erbium-ytterbium phosphate glass laser,” Appl. Phys. Lett. 68(19), 2621–2623 (1996). [CrossRef]  

5. S. Jiang, T. Luo, B.-C. Hwang, G. Nunzi-Conti, M. Myers, D. Rhonehouse, S. Honkanen, and N. Peyghambarian, “New Er3+-doped phosphate glass for ion-exchanged waveguide amplifiers,” Opt. Eng. 37(12), 3282–3286 (1998). [CrossRef]  

6. S. Jiang, T. Luo, B. Hwang, F. Smekatala, K. Seneschal, J. Lucas, and N. Peyghambarian, “Er3+-doped phosphate glasses for fiber amplifiers with high gain per unit length,” J. Non-Cryst. Solids 263-264, 364–368 (2000). [CrossRef]  

7. Y. Ohishi, A. Mori, M. Yamada, H. Ono, Y. Nishida, and K. Oikawa, “Gain characteristics of tellurite-based erbium-doped fiber amplifiers for 1.5-µm broadband amplification,” Opt. Lett. 23(4), 274–276 (1998). [CrossRef]  

8. B. M. Walsh, N. P. Barnes, D. J. Reichle, and S. Jiang, “Optical properties of Tm3+ ions in alkali germanate glass,” J. Non-Cryst. Solids 352(50-51), 5344–5352 (2006). [CrossRef]  

9. A. J. Stevenson, H. Serier-Brault, P. Gredin, and M. Mortier, “Fluoride materials for optical applications: Single crystals, ceramics, glasses, and glass-ceramics,” J. Fluorine Chem. 132(12), 1165–1173 (2011). [CrossRef]  

10. G. Dantelle, M. Mortier, G. Patriarche, and D. Vivien, “Er3+-doped PbF2: Comparison between nanocrystals in glass-ceramics and bulk single crystals,” J. Solid State Chem. 179(7), 1995–2003 (2006). [CrossRef]  

11. M. Mortier, P. Goldner, P. Féron, G. M. Stephan, H. Xu, and Z. Cai, “New fluoride glasses for laser applications,” J. Non-Cryst. Solids 326-327, 505–509 (2003). [CrossRef]  

12. K. Soga, H. Inoue, and A. Makishima, “Calculation and simulation of spectroscopic properties for rare earth ions in chloro-fluorozirconate glasses,” J. Non-Cryst. Solids 274(1-3), 69–74 (2000). [CrossRef]  

13. G. Krieke and A. Sarakovskis, “Crystallization and upconversion luminescence of distorted fluorite nanocrystals in Ba2+ containing oxyfluoride glass ceramics,” J. Eur. Ceram. Soc. 36(7), 1715–1722 (2016). [CrossRef]  

14. M. Yamada, H. Ono, T. Kanamori, S. Sudo, and Y. Ohishi, “Broadband and gain-flattened amplifier composed of a 1.55 µm-band and a 1.58 µm-band Er3+-doped fibre amplifier in a parallel configuration,” Electron. Lett. 33(8), 710–711 (1997). [CrossRef]  

15. H. Ebendorff-Heidepriem, D. Ehrt, M. Bettinelli, and A. Speghini, “Effect of glass composition on Judd-Ofelt parameters and radiative decay rates of Er3+ in fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 240(1-3), 66–78 (1998). [CrossRef]  

16. J. F. Philipps, T. Töpfer, H. Ebendorff-Heidepriem, D. Ehrt, and R. Sauerbrey, “Spectroscopic and lasing properties of Er3+:Yb3+-doped fluoride phosphate glasses,” Appl. Phys. B: Lasers Opt. 72(4), 399–405 (2001). [CrossRef]  

17. J. H. Choi, A. Margaryan, A. Margaryan, and F. G. Shi, “Judd-Ofelt analysis of spectroscopic properties of Nd3+ -doped novel fluorophosphate glass,” J. Lumin. 114(3-4), 167–177 (2005). [CrossRef]  

18. Z. H. Jiang and Q. Y. Zhang, “The structure of glass: A phase equilibrium diagram approach,” Prog. Mater. Sci. 61, 144–215 (2014). [CrossRef]  

19. V. Nazabal, M. Poulain, M. Olivier, P. Pirasteh, P. Camy, J. L. Doualan, S. Guy, T. Djouama, A. Boutarfaia, and J. L. Adam, “Fluoride and oxyfluoride glasses for optical applications,” J. Fluorine Chem. 134, 18–23 (2012). [CrossRef]  

20. V. A. Aseev, V. V. Golubkov, A. V. Klementeva, E. V. Kolobkova, and N. V. Nikonorov, “Spectral luminescence properties of transparent lead fluoride nanoglassceramics doped with erbium ions,” Opt. Spectrosc. 106(5), 691–696 (2009). [CrossRef]  

21. H. Rahimian, Y. Hatefi, A. Dehghan Hamedan, S. P. Shirmardi, and H. Mokhtari, “Structural and optical investigations on Eu3+ doped fluorophosphate glass and nano glass-ceramics,” J. Non-Cryst. Solids 487, 46–52 (2018). [CrossRef]  

22. R. Zheng, J. Wang, L. Zhang, C. Liu, and W. Wei, “Preparation and properties of Nd3+:SrAlF5 nanocrystals embedded fluorophosphate transparent glass–ceramic with long fluorescence lifetime,” Appl. Phys. A: Mater. Sci. Process. 122(1), 1–6 (2016). [CrossRef]  

23. J. Wang, X. Qiao, X. Fan, and M. Wang, “Judd-Ofelt analysis and upconversion emission of Er3+-Yb3+ co-doped oxyfluoride glass ceramics containing LaF3 nanocrystals,” Philos. Mag. 85(32), 3755–3766 (2005). [CrossRef]  

24. E. V. Kolobkova, V. G. Melekhin, and A. N. Penigin, “Optical glass-ceramics based on fluorine-containing silicate glasses doped with rare-earth ions,” Glass Phys. Chem. 33(1), 8–13 (2007). [CrossRef]  

25. V. A. Aseev, E. V. Kolobkova, K. S. Moskaleva, Y. A. Nekrasova, N. V. Nikonorov, and R. K. Nuryev, “Luminescent properties of ytterbium-erbium nanostructured lead fluoride silicate glass-ceramics at low temperatures,” Opt. Spectrosc. 114(5), 751–755 (2013). [CrossRef]  

26. I. Gugov, M. Müller, and C. Rüssel, “Transparent oxyfluoride glass ceramics co-doped with Er3+ and Yb3+ Crystallization and upconversion spectroscopy,” J. Solid State Chem. 184(5), 1001–1007 (2011). [CrossRef]  

27. P. A. Burdaev, V. A. Aseev, E. V. Kolobkova, N. V. Nikonorov, and A. O. Trofimov, “Nanostructured glass ceramics based on fluorophosphate glass with a high content of rare-earth ions,” Glass Phys. Chem. 41(1), 132–136 (2015). [CrossRef]  

28. V. A. Aseev, P. A. Burdaev, E. V. Kolobkova, and N. V. Nikonorov, “Fluorophosphate nanostructured glass ceramics activated by erbium ions,” Glass Phys. Chem. 39(2), 174–181 (2013). [CrossRef]  

29. Y. Xu, Y. Tian, Q. Liu, X. Sheng, W. Tang, J. Zhang, and S. Xu, “Effect of the heat treatment conditions on the structure and 2 micron luminescence of thulium-doped oxyfluoride silicate glass-ceramics,” J. Lumin. 211, 418–425 (2019). [CrossRef]  

30. S. Cui, J. Massera, M. Lastusaari, L. Hupa, and L. Petit, “Novel oxyfluorophosphate glasses and glass-ceramics,” J. Non-Cryst. Solids 445-446, 40–44 (2016). [CrossRef]  

31. J. Fan, S. Chen, X. Yuan, Y. Jiang, L. Pan, B. Jiang, X. Mao, R. Li, X. Jiang, L. Zhang, and J. Ballato, “Recrystallization of Er3+:CaF2 in Transparent Fluorophosphate Glass-Ceramics with the Co-Firing Method,” J. Am. Ceram. Soc. 99(9), 2971–2976 (2016). [CrossRef]  

32. E. Kolobkova, A. Alkhlef, B. M. Dinh, A. S. Yasukevich, O. P. Dernovich, N. V. Kuleshov, and N. Nikonorov, “Spectral properties of Nd 3+ ions in the new fluoride glasses with small additives of the phosphates,” J. Lumin. 206, 523–529 (2019). [CrossRef]  

33. S. Tanabe, “Rare-earth-doped glasses for fiber amplifiers in broadband telecommunication,” C. R. Chim. 5(12), 815–824 (2002). [CrossRef]  

34. W. T. Carnall, P. R. Fields, and K. Rajnak, “Spectral Intensities of the Trivalent Lanthanides and Actinides in Solution. II. Pm 3+, Sm 3+, Eu 3+, Gd 3+, Tb 3+, Dy 3+, and Ho 3+,” J. Chem. Phys. 49(10), 4412–4423 (1968). [CrossRef]  

35. A. S. Yasyukevich, V. G. Shcherbitskii, VÉ Kisel’, A. V. Mandrik, and N. V. Kuleshov, “Integral Method of Reciprocity in the Spectroscopy of laser crystals with impurity,” J. Appl. Spectrosc. 71(2), 202–208 (2004). [CrossRef]  

36. S. A. Payne, L. K. Smith, W. L. Kway, J. B. Tassano, and W. F. Krupke, “The mechanism of Tm-Ho energy transfer in LiYF4,” J. Phys.: Condens. Matter 4(44), 8525–8542 (1992). [CrossRef]  

37. V. K. Tikhomirov, V. D. Rodriguez, J. Méndez-Ramos, P. Núñez, and A. B. Seddon, “Comparative spectroscopy of (ErF3)(PbF2) alloys and Er3+-doped oxyfluoride glass-ceramics,” Opt. Mater. (Amsterdam, Neth.) 27(3), 543–547 (2004). [CrossRef]  

38. S. Tanabe, K. Takahara, M. Takahashi, and Y. Kawamoto, “Spectroscopic studies of radiative transitions and upconversion characteristics of Er3+ ions in simple pseudoternary fluoride glasses MFn–BaF2–YF3 (M : Zr, Hf, Al, Sc, Ga, In, or Zn),” J. Opt. Soc. Am. B 12(5), 786–793 (1995). [CrossRef]  

39. E. W. J. L. Oomen and A. M. A. van Dongen, “Europium (III) in Oxide Glasses: Dependence of the emission spectrum upon glass composition,” J. Non-Cryst. Solids 111(2-3), 205–213 (1989). [CrossRef]  

40. Y. Nageno, H. Takebe, and K. Morinaga, “Correlation between radiative transition probabilities of Nd3+ and composition in silicate, borate, and phosphate glasses,” J. Am. Ceram. Soc. 76(12), 3081–3086 (1993). [CrossRef]  

41. H. Takebe, K. Morinaga, and T. Izumitani, “Correlation between radiative transition probabilities of rare-earth ions and composition in oxide glasses,” J. Non-Cryst. Solids 178, 58–63 (1994). [CrossRef]  

42. H. Takebe, Y. Nageno, and K. Morinaga, “Effect of Network Modifier on Spontaneous Emission Probabilities of Er3+ in Oxide Glasses,” J. Am. Ceram. Soc. 77(8), 2132–2136 (1994). [CrossRef]  

43. H. Takebe, Y. Nageno, and K. Morinaga, “Compositional Dependence of Judd-Ofelt Parameters in Silicate, Borate, and Phosphate Glasses,” J. Am. Ceram. Soc. 78(5), 1161–1168 (1995). [CrossRef]  

44. H. Ebendorff-Heidepriem and D. Ehrt, “Spectroscopic properties of Eu3+ and Tb3+ ions for local structure investigations of fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 208(3), 205–216 (1996). [CrossRef]  

45. A. I. Burshtein, “Hopping Mechanism of Energy Transfer,” Sov. J. Exp. Theor. Phys. 35(5), 882–885 (1972).

46. S. F. Li, Q. Y. Zhang, and Y. P. Lee, “Absorption and photoluminescence properties of Er-doped and Er/Yb codoped soda-silicate laser glasses,” J. Appl. Phys. 96(9), 4746–4750 (2004). [CrossRef]  

References

  • View by:

  1. E. Tanguy, C. Larat, and J. P. Pocholle, “Modelling of the erbium - ytterbium laser,” Opt. Commun. 153(1-3), 172–183 (1998).
    [Crossref]
  2. S. Taccheo, P. Laporta, S. Longhi, O. Svelto, and C. Svelto, “Diode-pumped bulk erbium-ytterbium lasers,” Appl. Phys. B: Lasers Opt. 63(5), 425–436 (1996).
    [Crossref]
  3. W. C. Wang, B. Zhou, S. H. Xu, Z. M. Yang, and Q. Y. Zhang, “Recent advances in soft optical glass fiber and fiber lasers,” Prog. Mater. Sci. 101, 90–171 (2019).
    [Crossref]
  4. S. Taccheo, P. Laporta, and C. Svelto, “Widely tunable single-frequency erbium-ytterbium phosphate glass laser,” Appl. Phys. Lett. 68(19), 2621–2623 (1996).
    [Crossref]
  5. S. Jiang, T. Luo, B.-C. Hwang, G. Nunzi-Conti, M. Myers, D. Rhonehouse, S. Honkanen, and N. Peyghambarian, “New Er3+-doped phosphate glass for ion-exchanged waveguide amplifiers,” Opt. Eng. 37(12), 3282–3286 (1998).
    [Crossref]
  6. S. Jiang, T. Luo, B. Hwang, F. Smekatala, K. Seneschal, J. Lucas, and N. Peyghambarian, “Er3+-doped phosphate glasses for fiber amplifiers with high gain per unit length,” J. Non-Cryst. Solids 263-264, 364–368 (2000).
    [Crossref]
  7. Y. Ohishi, A. Mori, M. Yamada, H. Ono, Y. Nishida, and K. Oikawa, “Gain characteristics of tellurite-based erbium-doped fiber amplifiers for 1.5-µm broadband amplification,” Opt. Lett. 23(4), 274–276 (1998).
    [Crossref]
  8. B. M. Walsh, N. P. Barnes, D. J. Reichle, and S. Jiang, “Optical properties of Tm3+ ions in alkali germanate glass,” J. Non-Cryst. Solids 352(50-51), 5344–5352 (2006).
    [Crossref]
  9. A. J. Stevenson, H. Serier-Brault, P. Gredin, and M. Mortier, “Fluoride materials for optical applications: Single crystals, ceramics, glasses, and glass-ceramics,” J. Fluorine Chem. 132(12), 1165–1173 (2011).
    [Crossref]
  10. G. Dantelle, M. Mortier, G. Patriarche, and D. Vivien, “Er3+-doped PbF2: Comparison between nanocrystals in glass-ceramics and bulk single crystals,” J. Solid State Chem. 179(7), 1995–2003 (2006).
    [Crossref]
  11. M. Mortier, P. Goldner, P. Féron, G. M. Stephan, H. Xu, and Z. Cai, “New fluoride glasses for laser applications,” J. Non-Cryst. Solids 326-327, 505–509 (2003).
    [Crossref]
  12. K. Soga, H. Inoue, and A. Makishima, “Calculation and simulation of spectroscopic properties for rare earth ions in chloro-fluorozirconate glasses,” J. Non-Cryst. Solids 274(1-3), 69–74 (2000).
    [Crossref]
  13. G. Krieke and A. Sarakovskis, “Crystallization and upconversion luminescence of distorted fluorite nanocrystals in Ba2+ containing oxyfluoride glass ceramics,” J. Eur. Ceram. Soc. 36(7), 1715–1722 (2016).
    [Crossref]
  14. M. Yamada, H. Ono, T. Kanamori, S. Sudo, and Y. Ohishi, “Broadband and gain-flattened amplifier composed of a 1.55 µm-band and a 1.58 µm-band Er3+-doped fibre amplifier in a parallel configuration,” Electron. Lett. 33(8), 710–711 (1997).
    [Crossref]
  15. H. Ebendorff-Heidepriem, D. Ehrt, M. Bettinelli, and A. Speghini, “Effect of glass composition on Judd-Ofelt parameters and radiative decay rates of Er3+ in fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 240(1-3), 66–78 (1998).
    [Crossref]
  16. J. F. Philipps, T. Töpfer, H. Ebendorff-Heidepriem, D. Ehrt, and R. Sauerbrey, “Spectroscopic and lasing properties of Er3+:Yb3+-doped fluoride phosphate glasses,” Appl. Phys. B: Lasers Opt. 72(4), 399–405 (2001).
    [Crossref]
  17. J. H. Choi, A. Margaryan, A. Margaryan, and F. G. Shi, “Judd-Ofelt analysis of spectroscopic properties of Nd3+ -doped novel fluorophosphate glass,” J. Lumin. 114(3-4), 167–177 (2005).
    [Crossref]
  18. Z. H. Jiang and Q. Y. Zhang, “The structure of glass: A phase equilibrium diagram approach,” Prog. Mater. Sci. 61, 144–215 (2014).
    [Crossref]
  19. V. Nazabal, M. Poulain, M. Olivier, P. Pirasteh, P. Camy, J. L. Doualan, S. Guy, T. Djouama, A. Boutarfaia, and J. L. Adam, “Fluoride and oxyfluoride glasses for optical applications,” J. Fluorine Chem. 134, 18–23 (2012).
    [Crossref]
  20. V. A. Aseev, V. V. Golubkov, A. V. Klementeva, E. V. Kolobkova, and N. V. Nikonorov, “Spectral luminescence properties of transparent lead fluoride nanoglassceramics doped with erbium ions,” Opt. Spectrosc. 106(5), 691–696 (2009).
    [Crossref]
  21. H. Rahimian, Y. Hatefi, A. Dehghan Hamedan, S. P. Shirmardi, and H. Mokhtari, “Structural and optical investigations on Eu3+ doped fluorophosphate glass and nano glass-ceramics,” J. Non-Cryst. Solids 487, 46–52 (2018).
    [Crossref]
  22. R. Zheng, J. Wang, L. Zhang, C. Liu, and W. Wei, “Preparation and properties of Nd3+:SrAlF5 nanocrystals embedded fluorophosphate transparent glass–ceramic with long fluorescence lifetime,” Appl. Phys. A: Mater. Sci. Process. 122(1), 1–6 (2016).
    [Crossref]
  23. J. Wang, X. Qiao, X. Fan, and M. Wang, “Judd-Ofelt analysis and upconversion emission of Er3+-Yb3+ co-doped oxyfluoride glass ceramics containing LaF3 nanocrystals,” Philos. Mag. 85(32), 3755–3766 (2005).
    [Crossref]
  24. E. V. Kolobkova, V. G. Melekhin, and A. N. Penigin, “Optical glass-ceramics based on fluorine-containing silicate glasses doped with rare-earth ions,” Glass Phys. Chem. 33(1), 8–13 (2007).
    [Crossref]
  25. V. A. Aseev, E. V. Kolobkova, K. S. Moskaleva, Y. A. Nekrasova, N. V. Nikonorov, and R. K. Nuryev, “Luminescent properties of ytterbium-erbium nanostructured lead fluoride silicate glass-ceramics at low temperatures,” Opt. Spectrosc. 114(5), 751–755 (2013).
    [Crossref]
  26. I. Gugov, M. Müller, and C. Rüssel, “Transparent oxyfluoride glass ceramics co-doped with Er3+ and Yb3+ Crystallization and upconversion spectroscopy,” J. Solid State Chem. 184(5), 1001–1007 (2011).
    [Crossref]
  27. P. A. Burdaev, V. A. Aseev, E. V. Kolobkova, N. V. Nikonorov, and A. O. Trofimov, “Nanostructured glass ceramics based on fluorophosphate glass with a high content of rare-earth ions,” Glass Phys. Chem. 41(1), 132–136 (2015).
    [Crossref]
  28. V. A. Aseev, P. A. Burdaev, E. V. Kolobkova, and N. V. Nikonorov, “Fluorophosphate nanostructured glass ceramics activated by erbium ions,” Glass Phys. Chem. 39(2), 174–181 (2013).
    [Crossref]
  29. Y. Xu, Y. Tian, Q. Liu, X. Sheng, W. Tang, J. Zhang, and S. Xu, “Effect of the heat treatment conditions on the structure and 2 micron luminescence of thulium-doped oxyfluoride silicate glass-ceramics,” J. Lumin. 211, 418–425 (2019).
    [Crossref]
  30. S. Cui, J. Massera, M. Lastusaari, L. Hupa, and L. Petit, “Novel oxyfluorophosphate glasses and glass-ceramics,” J. Non-Cryst. Solids 445-446, 40–44 (2016).
    [Crossref]
  31. J. Fan, S. Chen, X. Yuan, Y. Jiang, L. Pan, B. Jiang, X. Mao, R. Li, X. Jiang, L. Zhang, and J. Ballato, “Recrystallization of Er3+:CaF2 in Transparent Fluorophosphate Glass-Ceramics with the Co-Firing Method,” J. Am. Ceram. Soc. 99(9), 2971–2976 (2016).
    [Crossref]
  32. E. Kolobkova, A. Alkhlef, B. M. Dinh, A. S. Yasukevich, O. P. Dernovich, N. V. Kuleshov, and N. Nikonorov, “Spectral properties of Nd 3+ ions in the new fluoride glasses with small additives of the phosphates,” J. Lumin. 206, 523–529 (2019).
    [Crossref]
  33. S. Tanabe, “Rare-earth-doped glasses for fiber amplifiers in broadband telecommunication,” C. R. Chim. 5(12), 815–824 (2002).
    [Crossref]
  34. W. T. Carnall, P. R. Fields, and K. Rajnak, “Spectral Intensities of the Trivalent Lanthanides and Actinides in Solution. II. Pm 3+, Sm 3+, Eu 3+, Gd 3+, Tb 3+, Dy 3+, and Ho 3+,” J. Chem. Phys. 49(10), 4412–4423 (1968).
    [Crossref]
  35. A. S. Yasyukevich, V. G. Shcherbitskii, VÉ Kisel’, A. V. Mandrik, and N. V. Kuleshov, “Integral Method of Reciprocity in the Spectroscopy of laser crystals with impurity,” J. Appl. Spectrosc. 71(2), 202–208 (2004).
    [Crossref]
  36. S. A. Payne, L. K. Smith, W. L. Kway, J. B. Tassano, and W. F. Krupke, “The mechanism of Tm-Ho energy transfer in LiYF4,” J. Phys.: Condens. Matter 4(44), 8525–8542 (1992).
    [Crossref]
  37. V. K. Tikhomirov, V. D. Rodriguez, J. Méndez-Ramos, P. Núñez, and A. B. Seddon, “Comparative spectroscopy of (ErF3)(PbF2) alloys and Er3+-doped oxyfluoride glass-ceramics,” Opt. Mater. (Amsterdam, Neth.) 27(3), 543–547 (2004).
    [Crossref]
  38. S. Tanabe, K. Takahara, M. Takahashi, and Y. Kawamoto, “Spectroscopic studies of radiative transitions and upconversion characteristics of Er3+ ions in simple pseudoternary fluoride glasses MFn–BaF2–YF3 (M : Zr, Hf, Al, Sc, Ga, In, or Zn),” J. Opt. Soc. Am. B 12(5), 786–793 (1995).
    [Crossref]
  39. E. W. J. L. Oomen and A. M. A. van Dongen, “Europium (III) in Oxide Glasses: Dependence of the emission spectrum upon glass composition,” J. Non-Cryst. Solids 111(2-3), 205–213 (1989).
    [Crossref]
  40. Y. Nageno, H. Takebe, and K. Morinaga, “Correlation between radiative transition probabilities of Nd3+ and composition in silicate, borate, and phosphate glasses,” J. Am. Ceram. Soc. 76(12), 3081–3086 (1993).
    [Crossref]
  41. H. Takebe, K. Morinaga, and T. Izumitani, “Correlation between radiative transition probabilities of rare-earth ions and composition in oxide glasses,” J. Non-Cryst. Solids 178, 58–63 (1994).
    [Crossref]
  42. H. Takebe, Y. Nageno, and K. Morinaga, “Effect of Network Modifier on Spontaneous Emission Probabilities of Er3+ in Oxide Glasses,” J. Am. Ceram. Soc. 77(8), 2132–2136 (1994).
    [Crossref]
  43. H. Takebe, Y. Nageno, and K. Morinaga, “Compositional Dependence of Judd-Ofelt Parameters in Silicate, Borate, and Phosphate Glasses,” J. Am. Ceram. Soc. 78(5), 1161–1168 (1995).
    [Crossref]
  44. H. Ebendorff-Heidepriem and D. Ehrt, “Spectroscopic properties of Eu3+ and Tb3+ ions for local structure investigations of fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 208(3), 205–216 (1996).
    [Crossref]
  45. A. I. Burshtein, “Hopping Mechanism of Energy Transfer,” Sov. J. Exp. Theor. Phys. 35(5), 882–885 (1972).
  46. S. F. Li, Q. Y. Zhang, and Y. P. Lee, “Absorption and photoluminescence properties of Er-doped and Er/Yb codoped soda-silicate laser glasses,” J. Appl. Phys. 96(9), 4746–4750 (2004).
    [Crossref]

2019 (3)

W. C. Wang, B. Zhou, S. H. Xu, Z. M. Yang, and Q. Y. Zhang, “Recent advances in soft optical glass fiber and fiber lasers,” Prog. Mater. Sci. 101, 90–171 (2019).
[Crossref]

Y. Xu, Y. Tian, Q. Liu, X. Sheng, W. Tang, J. Zhang, and S. Xu, “Effect of the heat treatment conditions on the structure and 2 micron luminescence of thulium-doped oxyfluoride silicate glass-ceramics,” J. Lumin. 211, 418–425 (2019).
[Crossref]

E. Kolobkova, A. Alkhlef, B. M. Dinh, A. S. Yasukevich, O. P. Dernovich, N. V. Kuleshov, and N. Nikonorov, “Spectral properties of Nd 3+ ions in the new fluoride glasses with small additives of the phosphates,” J. Lumin. 206, 523–529 (2019).
[Crossref]

2018 (1)

H. Rahimian, Y. Hatefi, A. Dehghan Hamedan, S. P. Shirmardi, and H. Mokhtari, “Structural and optical investigations on Eu3+ doped fluorophosphate glass and nano glass-ceramics,” J. Non-Cryst. Solids 487, 46–52 (2018).
[Crossref]

2016 (4)

R. Zheng, J. Wang, L. Zhang, C. Liu, and W. Wei, “Preparation and properties of Nd3+:SrAlF5 nanocrystals embedded fluorophosphate transparent glass–ceramic with long fluorescence lifetime,” Appl. Phys. A: Mater. Sci. Process. 122(1), 1–6 (2016).
[Crossref]

S. Cui, J. Massera, M. Lastusaari, L. Hupa, and L. Petit, “Novel oxyfluorophosphate glasses and glass-ceramics,” J. Non-Cryst. Solids 445-446, 40–44 (2016).
[Crossref]

J. Fan, S. Chen, X. Yuan, Y. Jiang, L. Pan, B. Jiang, X. Mao, R. Li, X. Jiang, L. Zhang, and J. Ballato, “Recrystallization of Er3+:CaF2 in Transparent Fluorophosphate Glass-Ceramics with the Co-Firing Method,” J. Am. Ceram. Soc. 99(9), 2971–2976 (2016).
[Crossref]

G. Krieke and A. Sarakovskis, “Crystallization and upconversion luminescence of distorted fluorite nanocrystals in Ba2+ containing oxyfluoride glass ceramics,” J. Eur. Ceram. Soc. 36(7), 1715–1722 (2016).
[Crossref]

2015 (1)

P. A. Burdaev, V. A. Aseev, E. V. Kolobkova, N. V. Nikonorov, and A. O. Trofimov, “Nanostructured glass ceramics based on fluorophosphate glass with a high content of rare-earth ions,” Glass Phys. Chem. 41(1), 132–136 (2015).
[Crossref]

2014 (1)

Z. H. Jiang and Q. Y. Zhang, “The structure of glass: A phase equilibrium diagram approach,” Prog. Mater. Sci. 61, 144–215 (2014).
[Crossref]

2013 (2)

V. A. Aseev, E. V. Kolobkova, K. S. Moskaleva, Y. A. Nekrasova, N. V. Nikonorov, and R. K. Nuryev, “Luminescent properties of ytterbium-erbium nanostructured lead fluoride silicate glass-ceramics at low temperatures,” Opt. Spectrosc. 114(5), 751–755 (2013).
[Crossref]

V. A. Aseev, P. A. Burdaev, E. V. Kolobkova, and N. V. Nikonorov, “Fluorophosphate nanostructured glass ceramics activated by erbium ions,” Glass Phys. Chem. 39(2), 174–181 (2013).
[Crossref]

2012 (1)

V. Nazabal, M. Poulain, M. Olivier, P. Pirasteh, P. Camy, J. L. Doualan, S. Guy, T. Djouama, A. Boutarfaia, and J. L. Adam, “Fluoride and oxyfluoride glasses for optical applications,” J. Fluorine Chem. 134, 18–23 (2012).
[Crossref]

2011 (2)

I. Gugov, M. Müller, and C. Rüssel, “Transparent oxyfluoride glass ceramics co-doped with Er3+ and Yb3+ Crystallization and upconversion spectroscopy,” J. Solid State Chem. 184(5), 1001–1007 (2011).
[Crossref]

A. J. Stevenson, H. Serier-Brault, P. Gredin, and M. Mortier, “Fluoride materials for optical applications: Single crystals, ceramics, glasses, and glass-ceramics,” J. Fluorine Chem. 132(12), 1165–1173 (2011).
[Crossref]

2009 (1)

V. A. Aseev, V. V. Golubkov, A. V. Klementeva, E. V. Kolobkova, and N. V. Nikonorov, “Spectral luminescence properties of transparent lead fluoride nanoglassceramics doped with erbium ions,” Opt. Spectrosc. 106(5), 691–696 (2009).
[Crossref]

2007 (1)

E. V. Kolobkova, V. G. Melekhin, and A. N. Penigin, “Optical glass-ceramics based on fluorine-containing silicate glasses doped with rare-earth ions,” Glass Phys. Chem. 33(1), 8–13 (2007).
[Crossref]

2006 (2)

G. Dantelle, M. Mortier, G. Patriarche, and D. Vivien, “Er3+-doped PbF2: Comparison between nanocrystals in glass-ceramics and bulk single crystals,” J. Solid State Chem. 179(7), 1995–2003 (2006).
[Crossref]

B. M. Walsh, N. P. Barnes, D. J. Reichle, and S. Jiang, “Optical properties of Tm3+ ions in alkali germanate glass,” J. Non-Cryst. Solids 352(50-51), 5344–5352 (2006).
[Crossref]

2005 (2)

J. H. Choi, A. Margaryan, A. Margaryan, and F. G. Shi, “Judd-Ofelt analysis of spectroscopic properties of Nd3+ -doped novel fluorophosphate glass,” J. Lumin. 114(3-4), 167–177 (2005).
[Crossref]

J. Wang, X. Qiao, X. Fan, and M. Wang, “Judd-Ofelt analysis and upconversion emission of Er3+-Yb3+ co-doped oxyfluoride glass ceramics containing LaF3 nanocrystals,” Philos. Mag. 85(32), 3755–3766 (2005).
[Crossref]

2004 (3)

A. S. Yasyukevich, V. G. Shcherbitskii, VÉ Kisel’, A. V. Mandrik, and N. V. Kuleshov, “Integral Method of Reciprocity in the Spectroscopy of laser crystals with impurity,” J. Appl. Spectrosc. 71(2), 202–208 (2004).
[Crossref]

V. K. Tikhomirov, V. D. Rodriguez, J. Méndez-Ramos, P. Núñez, and A. B. Seddon, “Comparative spectroscopy of (ErF3)(PbF2) alloys and Er3+-doped oxyfluoride glass-ceramics,” Opt. Mater. (Amsterdam, Neth.) 27(3), 543–547 (2004).
[Crossref]

S. F. Li, Q. Y. Zhang, and Y. P. Lee, “Absorption and photoluminescence properties of Er-doped and Er/Yb codoped soda-silicate laser glasses,” J. Appl. Phys. 96(9), 4746–4750 (2004).
[Crossref]

2003 (1)

M. Mortier, P. Goldner, P. Féron, G. M. Stephan, H. Xu, and Z. Cai, “New fluoride glasses for laser applications,” J. Non-Cryst. Solids 326-327, 505–509 (2003).
[Crossref]

2002 (1)

S. Tanabe, “Rare-earth-doped glasses for fiber amplifiers in broadband telecommunication,” C. R. Chim. 5(12), 815–824 (2002).
[Crossref]

2001 (1)

J. F. Philipps, T. Töpfer, H. Ebendorff-Heidepriem, D. Ehrt, and R. Sauerbrey, “Spectroscopic and lasing properties of Er3+:Yb3+-doped fluoride phosphate glasses,” Appl. Phys. B: Lasers Opt. 72(4), 399–405 (2001).
[Crossref]

2000 (2)

K. Soga, H. Inoue, and A. Makishima, “Calculation and simulation of spectroscopic properties for rare earth ions in chloro-fluorozirconate glasses,” J. Non-Cryst. Solids 274(1-3), 69–74 (2000).
[Crossref]

S. Jiang, T. Luo, B. Hwang, F. Smekatala, K. Seneschal, J. Lucas, and N. Peyghambarian, “Er3+-doped phosphate glasses for fiber amplifiers with high gain per unit length,” J. Non-Cryst. Solids 263-264, 364–368 (2000).
[Crossref]

1998 (4)

Y. Ohishi, A. Mori, M. Yamada, H. Ono, Y. Nishida, and K. Oikawa, “Gain characteristics of tellurite-based erbium-doped fiber amplifiers for 1.5-µm broadband amplification,” Opt. Lett. 23(4), 274–276 (1998).
[Crossref]

S. Jiang, T. Luo, B.-C. Hwang, G. Nunzi-Conti, M. Myers, D. Rhonehouse, S. Honkanen, and N. Peyghambarian, “New Er3+-doped phosphate glass for ion-exchanged waveguide amplifiers,” Opt. Eng. 37(12), 3282–3286 (1998).
[Crossref]

E. Tanguy, C. Larat, and J. P. Pocholle, “Modelling of the erbium - ytterbium laser,” Opt. Commun. 153(1-3), 172–183 (1998).
[Crossref]

H. Ebendorff-Heidepriem, D. Ehrt, M. Bettinelli, and A. Speghini, “Effect of glass composition on Judd-Ofelt parameters and radiative decay rates of Er3+ in fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 240(1-3), 66–78 (1998).
[Crossref]

1997 (1)

M. Yamada, H. Ono, T. Kanamori, S. Sudo, and Y. Ohishi, “Broadband and gain-flattened amplifier composed of a 1.55 µm-band and a 1.58 µm-band Er3+-doped fibre amplifier in a parallel configuration,” Electron. Lett. 33(8), 710–711 (1997).
[Crossref]

1996 (3)

S. Taccheo, P. Laporta, S. Longhi, O. Svelto, and C. Svelto, “Diode-pumped bulk erbium-ytterbium lasers,” Appl. Phys. B: Lasers Opt. 63(5), 425–436 (1996).
[Crossref]

S. Taccheo, P. Laporta, and C. Svelto, “Widely tunable single-frequency erbium-ytterbium phosphate glass laser,” Appl. Phys. Lett. 68(19), 2621–2623 (1996).
[Crossref]

H. Ebendorff-Heidepriem and D. Ehrt, “Spectroscopic properties of Eu3+ and Tb3+ ions for local structure investigations of fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 208(3), 205–216 (1996).
[Crossref]

1995 (2)

1994 (2)

H. Takebe, K. Morinaga, and T. Izumitani, “Correlation between radiative transition probabilities of rare-earth ions and composition in oxide glasses,” J. Non-Cryst. Solids 178, 58–63 (1994).
[Crossref]

H. Takebe, Y. Nageno, and K. Morinaga, “Effect of Network Modifier on Spontaneous Emission Probabilities of Er3+ in Oxide Glasses,” J. Am. Ceram. Soc. 77(8), 2132–2136 (1994).
[Crossref]

1993 (1)

Y. Nageno, H. Takebe, and K. Morinaga, “Correlation between radiative transition probabilities of Nd3+ and composition in silicate, borate, and phosphate glasses,” J. Am. Ceram. Soc. 76(12), 3081–3086 (1993).
[Crossref]

1992 (1)

S. A. Payne, L. K. Smith, W. L. Kway, J. B. Tassano, and W. F. Krupke, “The mechanism of Tm-Ho energy transfer in LiYF4,” J. Phys.: Condens. Matter 4(44), 8525–8542 (1992).
[Crossref]

1989 (1)

E. W. J. L. Oomen and A. M. A. van Dongen, “Europium (III) in Oxide Glasses: Dependence of the emission spectrum upon glass composition,” J. Non-Cryst. Solids 111(2-3), 205–213 (1989).
[Crossref]

1972 (1)

A. I. Burshtein, “Hopping Mechanism of Energy Transfer,” Sov. J. Exp. Theor. Phys. 35(5), 882–885 (1972).

1968 (1)

W. T. Carnall, P. R. Fields, and K. Rajnak, “Spectral Intensities of the Trivalent Lanthanides and Actinides in Solution. II. Pm 3+, Sm 3+, Eu 3+, Gd 3+, Tb 3+, Dy 3+, and Ho 3+,” J. Chem. Phys. 49(10), 4412–4423 (1968).
[Crossref]

Adam, J. L.

V. Nazabal, M. Poulain, M. Olivier, P. Pirasteh, P. Camy, J. L. Doualan, S. Guy, T. Djouama, A. Boutarfaia, and J. L. Adam, “Fluoride and oxyfluoride glasses for optical applications,” J. Fluorine Chem. 134, 18–23 (2012).
[Crossref]

Alkhlef, A.

E. Kolobkova, A. Alkhlef, B. M. Dinh, A. S. Yasukevich, O. P. Dernovich, N. V. Kuleshov, and N. Nikonorov, “Spectral properties of Nd 3+ ions in the new fluoride glasses with small additives of the phosphates,” J. Lumin. 206, 523–529 (2019).
[Crossref]

Aseev, V. A.

P. A. Burdaev, V. A. Aseev, E. V. Kolobkova, N. V. Nikonorov, and A. O. Trofimov, “Nanostructured glass ceramics based on fluorophosphate glass with a high content of rare-earth ions,” Glass Phys. Chem. 41(1), 132–136 (2015).
[Crossref]

V. A. Aseev, P. A. Burdaev, E. V. Kolobkova, and N. V. Nikonorov, “Fluorophosphate nanostructured glass ceramics activated by erbium ions,” Glass Phys. Chem. 39(2), 174–181 (2013).
[Crossref]

V. A. Aseev, E. V. Kolobkova, K. S. Moskaleva, Y. A. Nekrasova, N. V. Nikonorov, and R. K. Nuryev, “Luminescent properties of ytterbium-erbium nanostructured lead fluoride silicate glass-ceramics at low temperatures,” Opt. Spectrosc. 114(5), 751–755 (2013).
[Crossref]

V. A. Aseev, V. V. Golubkov, A. V. Klementeva, E. V. Kolobkova, and N. V. Nikonorov, “Spectral luminescence properties of transparent lead fluoride nanoglassceramics doped with erbium ions,” Opt. Spectrosc. 106(5), 691–696 (2009).
[Crossref]

Ballato, J.

J. Fan, S. Chen, X. Yuan, Y. Jiang, L. Pan, B. Jiang, X. Mao, R. Li, X. Jiang, L. Zhang, and J. Ballato, “Recrystallization of Er3+:CaF2 in Transparent Fluorophosphate Glass-Ceramics with the Co-Firing Method,” J. Am. Ceram. Soc. 99(9), 2971–2976 (2016).
[Crossref]

Barnes, N. P.

B. M. Walsh, N. P. Barnes, D. J. Reichle, and S. Jiang, “Optical properties of Tm3+ ions in alkali germanate glass,” J. Non-Cryst. Solids 352(50-51), 5344–5352 (2006).
[Crossref]

Bettinelli, M.

H. Ebendorff-Heidepriem, D. Ehrt, M. Bettinelli, and A. Speghini, “Effect of glass composition on Judd-Ofelt parameters and radiative decay rates of Er3+ in fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 240(1-3), 66–78 (1998).
[Crossref]

Boutarfaia, A.

V. Nazabal, M. Poulain, M. Olivier, P. Pirasteh, P. Camy, J. L. Doualan, S. Guy, T. Djouama, A. Boutarfaia, and J. L. Adam, “Fluoride and oxyfluoride glasses for optical applications,” J. Fluorine Chem. 134, 18–23 (2012).
[Crossref]

Burdaev, P. A.

P. A. Burdaev, V. A. Aseev, E. V. Kolobkova, N. V. Nikonorov, and A. O. Trofimov, “Nanostructured glass ceramics based on fluorophosphate glass with a high content of rare-earth ions,” Glass Phys. Chem. 41(1), 132–136 (2015).
[Crossref]

V. A. Aseev, P. A. Burdaev, E. V. Kolobkova, and N. V. Nikonorov, “Fluorophosphate nanostructured glass ceramics activated by erbium ions,” Glass Phys. Chem. 39(2), 174–181 (2013).
[Crossref]

Burshtein, A. I.

A. I. Burshtein, “Hopping Mechanism of Energy Transfer,” Sov. J. Exp. Theor. Phys. 35(5), 882–885 (1972).

Cai, Z.

M. Mortier, P. Goldner, P. Féron, G. M. Stephan, H. Xu, and Z. Cai, “New fluoride glasses for laser applications,” J. Non-Cryst. Solids 326-327, 505–509 (2003).
[Crossref]

Camy, P.

V. Nazabal, M. Poulain, M. Olivier, P. Pirasteh, P. Camy, J. L. Doualan, S. Guy, T. Djouama, A. Boutarfaia, and J. L. Adam, “Fluoride and oxyfluoride glasses for optical applications,” J. Fluorine Chem. 134, 18–23 (2012).
[Crossref]

Carnall, W. T.

W. T. Carnall, P. R. Fields, and K. Rajnak, “Spectral Intensities of the Trivalent Lanthanides and Actinides in Solution. II. Pm 3+, Sm 3+, Eu 3+, Gd 3+, Tb 3+, Dy 3+, and Ho 3+,” J. Chem. Phys. 49(10), 4412–4423 (1968).
[Crossref]

Chen, S.

J. Fan, S. Chen, X. Yuan, Y. Jiang, L. Pan, B. Jiang, X. Mao, R. Li, X. Jiang, L. Zhang, and J. Ballato, “Recrystallization of Er3+:CaF2 in Transparent Fluorophosphate Glass-Ceramics with the Co-Firing Method,” J. Am. Ceram. Soc. 99(9), 2971–2976 (2016).
[Crossref]

Choi, J. H.

J. H. Choi, A. Margaryan, A. Margaryan, and F. G. Shi, “Judd-Ofelt analysis of spectroscopic properties of Nd3+ -doped novel fluorophosphate glass,” J. Lumin. 114(3-4), 167–177 (2005).
[Crossref]

Cui, S.

S. Cui, J. Massera, M. Lastusaari, L. Hupa, and L. Petit, “Novel oxyfluorophosphate glasses and glass-ceramics,” J. Non-Cryst. Solids 445-446, 40–44 (2016).
[Crossref]

Dantelle, G.

G. Dantelle, M. Mortier, G. Patriarche, and D. Vivien, “Er3+-doped PbF2: Comparison between nanocrystals in glass-ceramics and bulk single crystals,” J. Solid State Chem. 179(7), 1995–2003 (2006).
[Crossref]

Dehghan Hamedan, A.

H. Rahimian, Y. Hatefi, A. Dehghan Hamedan, S. P. Shirmardi, and H. Mokhtari, “Structural and optical investigations on Eu3+ doped fluorophosphate glass and nano glass-ceramics,” J. Non-Cryst. Solids 487, 46–52 (2018).
[Crossref]

Dernovich, O. P.

E. Kolobkova, A. Alkhlef, B. M. Dinh, A. S. Yasukevich, O. P. Dernovich, N. V. Kuleshov, and N. Nikonorov, “Spectral properties of Nd 3+ ions in the new fluoride glasses with small additives of the phosphates,” J. Lumin. 206, 523–529 (2019).
[Crossref]

Dinh, B. M.

E. Kolobkova, A. Alkhlef, B. M. Dinh, A. S. Yasukevich, O. P. Dernovich, N. V. Kuleshov, and N. Nikonorov, “Spectral properties of Nd 3+ ions in the new fluoride glasses with small additives of the phosphates,” J. Lumin. 206, 523–529 (2019).
[Crossref]

Djouama, T.

V. Nazabal, M. Poulain, M. Olivier, P. Pirasteh, P. Camy, J. L. Doualan, S. Guy, T. Djouama, A. Boutarfaia, and J. L. Adam, “Fluoride and oxyfluoride glasses for optical applications,” J. Fluorine Chem. 134, 18–23 (2012).
[Crossref]

Doualan, J. L.

V. Nazabal, M. Poulain, M. Olivier, P. Pirasteh, P. Camy, J. L. Doualan, S. Guy, T. Djouama, A. Boutarfaia, and J. L. Adam, “Fluoride and oxyfluoride glasses for optical applications,” J. Fluorine Chem. 134, 18–23 (2012).
[Crossref]

Ebendorff-Heidepriem, H.

J. F. Philipps, T. Töpfer, H. Ebendorff-Heidepriem, D. Ehrt, and R. Sauerbrey, “Spectroscopic and lasing properties of Er3+:Yb3+-doped fluoride phosphate glasses,” Appl. Phys. B: Lasers Opt. 72(4), 399–405 (2001).
[Crossref]

H. Ebendorff-Heidepriem, D. Ehrt, M. Bettinelli, and A. Speghini, “Effect of glass composition on Judd-Ofelt parameters and radiative decay rates of Er3+ in fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 240(1-3), 66–78 (1998).
[Crossref]

H. Ebendorff-Heidepriem and D. Ehrt, “Spectroscopic properties of Eu3+ and Tb3+ ions for local structure investigations of fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 208(3), 205–216 (1996).
[Crossref]

Ehrt, D.

J. F. Philipps, T. Töpfer, H. Ebendorff-Heidepriem, D. Ehrt, and R. Sauerbrey, “Spectroscopic and lasing properties of Er3+:Yb3+-doped fluoride phosphate glasses,” Appl. Phys. B: Lasers Opt. 72(4), 399–405 (2001).
[Crossref]

H. Ebendorff-Heidepriem, D. Ehrt, M. Bettinelli, and A. Speghini, “Effect of glass composition on Judd-Ofelt parameters and radiative decay rates of Er3+ in fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 240(1-3), 66–78 (1998).
[Crossref]

H. Ebendorff-Heidepriem and D. Ehrt, “Spectroscopic properties of Eu3+ and Tb3+ ions for local structure investigations of fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 208(3), 205–216 (1996).
[Crossref]

Fan, J.

J. Fan, S. Chen, X. Yuan, Y. Jiang, L. Pan, B. Jiang, X. Mao, R. Li, X. Jiang, L. Zhang, and J. Ballato, “Recrystallization of Er3+:CaF2 in Transparent Fluorophosphate Glass-Ceramics with the Co-Firing Method,” J. Am. Ceram. Soc. 99(9), 2971–2976 (2016).
[Crossref]

Fan, X.

J. Wang, X. Qiao, X. Fan, and M. Wang, “Judd-Ofelt analysis and upconversion emission of Er3+-Yb3+ co-doped oxyfluoride glass ceramics containing LaF3 nanocrystals,” Philos. Mag. 85(32), 3755–3766 (2005).
[Crossref]

Féron, P.

M. Mortier, P. Goldner, P. Féron, G. M. Stephan, H. Xu, and Z. Cai, “New fluoride glasses for laser applications,” J. Non-Cryst. Solids 326-327, 505–509 (2003).
[Crossref]

Fields, P. R.

W. T. Carnall, P. R. Fields, and K. Rajnak, “Spectral Intensities of the Trivalent Lanthanides and Actinides in Solution. II. Pm 3+, Sm 3+, Eu 3+, Gd 3+, Tb 3+, Dy 3+, and Ho 3+,” J. Chem. Phys. 49(10), 4412–4423 (1968).
[Crossref]

Goldner, P.

M. Mortier, P. Goldner, P. Féron, G. M. Stephan, H. Xu, and Z. Cai, “New fluoride glasses for laser applications,” J. Non-Cryst. Solids 326-327, 505–509 (2003).
[Crossref]

Golubkov, V. V.

V. A. Aseev, V. V. Golubkov, A. V. Klementeva, E. V. Kolobkova, and N. V. Nikonorov, “Spectral luminescence properties of transparent lead fluoride nanoglassceramics doped with erbium ions,” Opt. Spectrosc. 106(5), 691–696 (2009).
[Crossref]

Gredin, P.

A. J. Stevenson, H. Serier-Brault, P. Gredin, and M. Mortier, “Fluoride materials for optical applications: Single crystals, ceramics, glasses, and glass-ceramics,” J. Fluorine Chem. 132(12), 1165–1173 (2011).
[Crossref]

Gugov, I.

I. Gugov, M. Müller, and C. Rüssel, “Transparent oxyfluoride glass ceramics co-doped with Er3+ and Yb3+ Crystallization and upconversion spectroscopy,” J. Solid State Chem. 184(5), 1001–1007 (2011).
[Crossref]

Guy, S.

V. Nazabal, M. Poulain, M. Olivier, P. Pirasteh, P. Camy, J. L. Doualan, S. Guy, T. Djouama, A. Boutarfaia, and J. L. Adam, “Fluoride and oxyfluoride glasses for optical applications,” J. Fluorine Chem. 134, 18–23 (2012).
[Crossref]

Hatefi, Y.

H. Rahimian, Y. Hatefi, A. Dehghan Hamedan, S. P. Shirmardi, and H. Mokhtari, “Structural and optical investigations on Eu3+ doped fluorophosphate glass and nano glass-ceramics,” J. Non-Cryst. Solids 487, 46–52 (2018).
[Crossref]

Honkanen, S.

S. Jiang, T. Luo, B.-C. Hwang, G. Nunzi-Conti, M. Myers, D. Rhonehouse, S. Honkanen, and N. Peyghambarian, “New Er3+-doped phosphate glass for ion-exchanged waveguide amplifiers,” Opt. Eng. 37(12), 3282–3286 (1998).
[Crossref]

Hupa, L.

S. Cui, J. Massera, M. Lastusaari, L. Hupa, and L. Petit, “Novel oxyfluorophosphate glasses and glass-ceramics,” J. Non-Cryst. Solids 445-446, 40–44 (2016).
[Crossref]

Hwang, B.

S. Jiang, T. Luo, B. Hwang, F. Smekatala, K. Seneschal, J. Lucas, and N. Peyghambarian, “Er3+-doped phosphate glasses for fiber amplifiers with high gain per unit length,” J. Non-Cryst. Solids 263-264, 364–368 (2000).
[Crossref]

Hwang, B.-C.

S. Jiang, T. Luo, B.-C. Hwang, G. Nunzi-Conti, M. Myers, D. Rhonehouse, S. Honkanen, and N. Peyghambarian, “New Er3+-doped phosphate glass for ion-exchanged waveguide amplifiers,” Opt. Eng. 37(12), 3282–3286 (1998).
[Crossref]

Inoue, H.

K. Soga, H. Inoue, and A. Makishima, “Calculation and simulation of spectroscopic properties for rare earth ions in chloro-fluorozirconate glasses,” J. Non-Cryst. Solids 274(1-3), 69–74 (2000).
[Crossref]

Izumitani, T.

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V. A. Aseev, V. V. Golubkov, A. V. Klementeva, E. V. Kolobkova, and N. V. Nikonorov, “Spectral luminescence properties of transparent lead fluoride nanoglassceramics doped with erbium ions,” Opt. Spectrosc. 106(5), 691–696 (2009).
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S. Jiang, T. Luo, B. Hwang, F. Smekatala, K. Seneschal, J. Lucas, and N. Peyghambarian, “Er3+-doped phosphate glasses for fiber amplifiers with high gain per unit length,” J. Non-Cryst. Solids 263-264, 364–368 (2000).
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S. Jiang, T. Luo, B. Hwang, F. Smekatala, K. Seneschal, J. Lucas, and N. Peyghambarian, “Er3+-doped phosphate glasses for fiber amplifiers with high gain per unit length,” J. Non-Cryst. Solids 263-264, 364–368 (2000).
[Crossref]

S. Jiang, T. Luo, B.-C. Hwang, G. Nunzi-Conti, M. Myers, D. Rhonehouse, S. Honkanen, and N. Peyghambarian, “New Er3+-doped phosphate glass for ion-exchanged waveguide amplifiers,” Opt. Eng. 37(12), 3282–3286 (1998).
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J. Fan, S. Chen, X. Yuan, Y. Jiang, L. Pan, B. Jiang, X. Mao, R. Li, X. Jiang, L. Zhang, and J. Ballato, “Recrystallization of Er3+:CaF2 in Transparent Fluorophosphate Glass-Ceramics with the Co-Firing Method,” J. Am. Ceram. Soc. 99(9), 2971–2976 (2016).
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V. K. Tikhomirov, V. D. Rodriguez, J. Méndez-Ramos, P. Núñez, and A. B. Seddon, “Comparative spectroscopy of (ErF3)(PbF2) alloys and Er3+-doped oxyfluoride glass-ceramics,” Opt. Mater. (Amsterdam, Neth.) 27(3), 543–547 (2004).
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V. A. Aseev, P. A. Burdaev, E. V. Kolobkova, and N. V. Nikonorov, “Fluorophosphate nanostructured glass ceramics activated by erbium ions,” Glass Phys. Chem. 39(2), 174–181 (2013).
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V. A. Aseev, E. V. Kolobkova, K. S. Moskaleva, Y. A. Nekrasova, N. V. Nikonorov, and R. K. Nuryev, “Luminescent properties of ytterbium-erbium nanostructured lead fluoride silicate glass-ceramics at low temperatures,” Opt. Spectrosc. 114(5), 751–755 (2013).
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V. Nazabal, M. Poulain, M. Olivier, P. Pirasteh, P. Camy, J. L. Doualan, S. Guy, T. Djouama, A. Boutarfaia, and J. L. Adam, “Fluoride and oxyfluoride glasses for optical applications,” J. Fluorine Chem. 134, 18–23 (2012).
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G. Dantelle, M. Mortier, G. Patriarche, and D. Vivien, “Er3+-doped PbF2: Comparison between nanocrystals in glass-ceramics and bulk single crystals,” J. Solid State Chem. 179(7), 1995–2003 (2006).
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S. A. Payne, L. K. Smith, W. L. Kway, J. B. Tassano, and W. F. Krupke, “The mechanism of Tm-Ho energy transfer in LiYF4,” J. Phys.: Condens. Matter 4(44), 8525–8542 (1992).
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E. V. Kolobkova, V. G. Melekhin, and A. N. Penigin, “Optical glass-ceramics based on fluorine-containing silicate glasses doped with rare-earth ions,” Glass Phys. Chem. 33(1), 8–13 (2007).
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S. Cui, J. Massera, M. Lastusaari, L. Hupa, and L. Petit, “Novel oxyfluorophosphate glasses and glass-ceramics,” J. Non-Cryst. Solids 445-446, 40–44 (2016).
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S. Jiang, T. Luo, B. Hwang, F. Smekatala, K. Seneschal, J. Lucas, and N. Peyghambarian, “Er3+-doped phosphate glasses for fiber amplifiers with high gain per unit length,” J. Non-Cryst. Solids 263-264, 364–368 (2000).
[Crossref]

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E. Tanguy, C. Larat, and J. P. Pocholle, “Modelling of the erbium - ytterbium laser,” Opt. Commun. 153(1-3), 172–183 (1998).
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H. Rahimian, Y. Hatefi, A. Dehghan Hamedan, S. P. Shirmardi, and H. Mokhtari, “Structural and optical investigations on Eu3+ doped fluorophosphate glass and nano glass-ceramics,” J. Non-Cryst. Solids 487, 46–52 (2018).
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[Crossref]

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S. Jiang, T. Luo, B.-C. Hwang, G. Nunzi-Conti, M. Myers, D. Rhonehouse, S. Honkanen, and N. Peyghambarian, “New Er3+-doped phosphate glass for ion-exchanged waveguide amplifiers,” Opt. Eng. 37(12), 3282–3286 (1998).
[Crossref]

Rodriguez, V. D.

V. K. Tikhomirov, V. D. Rodriguez, J. Méndez-Ramos, P. Núñez, and A. B. Seddon, “Comparative spectroscopy of (ErF3)(PbF2) alloys and Er3+-doped oxyfluoride glass-ceramics,” Opt. Mater. (Amsterdam, Neth.) 27(3), 543–547 (2004).
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Rüssel, C.

I. Gugov, M. Müller, and C. Rüssel, “Transparent oxyfluoride glass ceramics co-doped with Er3+ and Yb3+ Crystallization and upconversion spectroscopy,” J. Solid State Chem. 184(5), 1001–1007 (2011).
[Crossref]

Sarakovskis, A.

G. Krieke and A. Sarakovskis, “Crystallization and upconversion luminescence of distorted fluorite nanocrystals in Ba2+ containing oxyfluoride glass ceramics,” J. Eur. Ceram. Soc. 36(7), 1715–1722 (2016).
[Crossref]

Sauerbrey, R.

J. F. Philipps, T. Töpfer, H. Ebendorff-Heidepriem, D. Ehrt, and R. Sauerbrey, “Spectroscopic and lasing properties of Er3+:Yb3+-doped fluoride phosphate glasses,” Appl. Phys. B: Lasers Opt. 72(4), 399–405 (2001).
[Crossref]

Seddon, A. B.

V. K. Tikhomirov, V. D. Rodriguez, J. Méndez-Ramos, P. Núñez, and A. B. Seddon, “Comparative spectroscopy of (ErF3)(PbF2) alloys and Er3+-doped oxyfluoride glass-ceramics,” Opt. Mater. (Amsterdam, Neth.) 27(3), 543–547 (2004).
[Crossref]

Seneschal, K.

S. Jiang, T. Luo, B. Hwang, F. Smekatala, K. Seneschal, J. Lucas, and N. Peyghambarian, “Er3+-doped phosphate glasses for fiber amplifiers with high gain per unit length,” J. Non-Cryst. Solids 263-264, 364–368 (2000).
[Crossref]

Serier-Brault, H.

A. J. Stevenson, H. Serier-Brault, P. Gredin, and M. Mortier, “Fluoride materials for optical applications: Single crystals, ceramics, glasses, and glass-ceramics,” J. Fluorine Chem. 132(12), 1165–1173 (2011).
[Crossref]

Shcherbitskii, V. G.

A. S. Yasyukevich, V. G. Shcherbitskii, VÉ Kisel’, A. V. Mandrik, and N. V. Kuleshov, “Integral Method of Reciprocity in the Spectroscopy of laser crystals with impurity,” J. Appl. Spectrosc. 71(2), 202–208 (2004).
[Crossref]

Sheng, X.

Y. Xu, Y. Tian, Q. Liu, X. Sheng, W. Tang, J. Zhang, and S. Xu, “Effect of the heat treatment conditions on the structure and 2 micron luminescence of thulium-doped oxyfluoride silicate glass-ceramics,” J. Lumin. 211, 418–425 (2019).
[Crossref]

Shi, F. G.

J. H. Choi, A. Margaryan, A. Margaryan, and F. G. Shi, “Judd-Ofelt analysis of spectroscopic properties of Nd3+ -doped novel fluorophosphate glass,” J. Lumin. 114(3-4), 167–177 (2005).
[Crossref]

Shirmardi, S. P.

H. Rahimian, Y. Hatefi, A. Dehghan Hamedan, S. P. Shirmardi, and H. Mokhtari, “Structural and optical investigations on Eu3+ doped fluorophosphate glass and nano glass-ceramics,” J. Non-Cryst. Solids 487, 46–52 (2018).
[Crossref]

Smekatala, F.

S. Jiang, T. Luo, B. Hwang, F. Smekatala, K. Seneschal, J. Lucas, and N. Peyghambarian, “Er3+-doped phosphate glasses for fiber amplifiers with high gain per unit length,” J. Non-Cryst. Solids 263-264, 364–368 (2000).
[Crossref]

Smith, L. K.

S. A. Payne, L. K. Smith, W. L. Kway, J. B. Tassano, and W. F. Krupke, “The mechanism of Tm-Ho energy transfer in LiYF4,” J. Phys.: Condens. Matter 4(44), 8525–8542 (1992).
[Crossref]

Soga, K.

K. Soga, H. Inoue, and A. Makishima, “Calculation and simulation of spectroscopic properties for rare earth ions in chloro-fluorozirconate glasses,” J. Non-Cryst. Solids 274(1-3), 69–74 (2000).
[Crossref]

Speghini, A.

H. Ebendorff-Heidepriem, D. Ehrt, M. Bettinelli, and A. Speghini, “Effect of glass composition on Judd-Ofelt parameters and radiative decay rates of Er3+ in fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 240(1-3), 66–78 (1998).
[Crossref]

Stephan, G. M.

M. Mortier, P. Goldner, P. Féron, G. M. Stephan, H. Xu, and Z. Cai, “New fluoride glasses for laser applications,” J. Non-Cryst. Solids 326-327, 505–509 (2003).
[Crossref]

Stevenson, A. J.

A. J. Stevenson, H. Serier-Brault, P. Gredin, and M. Mortier, “Fluoride materials for optical applications: Single crystals, ceramics, glasses, and glass-ceramics,” J. Fluorine Chem. 132(12), 1165–1173 (2011).
[Crossref]

Sudo, S.

M. Yamada, H. Ono, T. Kanamori, S. Sudo, and Y. Ohishi, “Broadband and gain-flattened amplifier composed of a 1.55 µm-band and a 1.58 µm-band Er3+-doped fibre amplifier in a parallel configuration,” Electron. Lett. 33(8), 710–711 (1997).
[Crossref]

Svelto, C.

S. Taccheo, P. Laporta, S. Longhi, O. Svelto, and C. Svelto, “Diode-pumped bulk erbium-ytterbium lasers,” Appl. Phys. B: Lasers Opt. 63(5), 425–436 (1996).
[Crossref]

S. Taccheo, P. Laporta, and C. Svelto, “Widely tunable single-frequency erbium-ytterbium phosphate glass laser,” Appl. Phys. Lett. 68(19), 2621–2623 (1996).
[Crossref]

Svelto, O.

S. Taccheo, P. Laporta, S. Longhi, O. Svelto, and C. Svelto, “Diode-pumped bulk erbium-ytterbium lasers,” Appl. Phys. B: Lasers Opt. 63(5), 425–436 (1996).
[Crossref]

Taccheo, S.

S. Taccheo, P. Laporta, S. Longhi, O. Svelto, and C. Svelto, “Diode-pumped bulk erbium-ytterbium lasers,” Appl. Phys. B: Lasers Opt. 63(5), 425–436 (1996).
[Crossref]

S. Taccheo, P. Laporta, and C. Svelto, “Widely tunable single-frequency erbium-ytterbium phosphate glass laser,” Appl. Phys. Lett. 68(19), 2621–2623 (1996).
[Crossref]

Takahara, K.

Takahashi, M.

Takebe, H.

H. Takebe, Y. Nageno, and K. Morinaga, “Compositional Dependence of Judd-Ofelt Parameters in Silicate, Borate, and Phosphate Glasses,” J. Am. Ceram. Soc. 78(5), 1161–1168 (1995).
[Crossref]

H. Takebe, Y. Nageno, and K. Morinaga, “Effect of Network Modifier on Spontaneous Emission Probabilities of Er3+ in Oxide Glasses,” J. Am. Ceram. Soc. 77(8), 2132–2136 (1994).
[Crossref]

H. Takebe, K. Morinaga, and T. Izumitani, “Correlation between radiative transition probabilities of rare-earth ions and composition in oxide glasses,” J. Non-Cryst. Solids 178, 58–63 (1994).
[Crossref]

Y. Nageno, H. Takebe, and K. Morinaga, “Correlation between radiative transition probabilities of Nd3+ and composition in silicate, borate, and phosphate glasses,” J. Am. Ceram. Soc. 76(12), 3081–3086 (1993).
[Crossref]

Tanabe, S.

Tang, W.

Y. Xu, Y. Tian, Q. Liu, X. Sheng, W. Tang, J. Zhang, and S. Xu, “Effect of the heat treatment conditions on the structure and 2 micron luminescence of thulium-doped oxyfluoride silicate glass-ceramics,” J. Lumin. 211, 418–425 (2019).
[Crossref]

Tanguy, E.

E. Tanguy, C. Larat, and J. P. Pocholle, “Modelling of the erbium - ytterbium laser,” Opt. Commun. 153(1-3), 172–183 (1998).
[Crossref]

Tassano, J. B.

S. A. Payne, L. K. Smith, W. L. Kway, J. B. Tassano, and W. F. Krupke, “The mechanism of Tm-Ho energy transfer in LiYF4,” J. Phys.: Condens. Matter 4(44), 8525–8542 (1992).
[Crossref]

Tian, Y.

Y. Xu, Y. Tian, Q. Liu, X. Sheng, W. Tang, J. Zhang, and S. Xu, “Effect of the heat treatment conditions on the structure and 2 micron luminescence of thulium-doped oxyfluoride silicate glass-ceramics,” J. Lumin. 211, 418–425 (2019).
[Crossref]

Tikhomirov, V. K.

V. K. Tikhomirov, V. D. Rodriguez, J. Méndez-Ramos, P. Núñez, and A. B. Seddon, “Comparative spectroscopy of (ErF3)(PbF2) alloys and Er3+-doped oxyfluoride glass-ceramics,” Opt. Mater. (Amsterdam, Neth.) 27(3), 543–547 (2004).
[Crossref]

Töpfer, T.

J. F. Philipps, T. Töpfer, H. Ebendorff-Heidepriem, D. Ehrt, and R. Sauerbrey, “Spectroscopic and lasing properties of Er3+:Yb3+-doped fluoride phosphate glasses,” Appl. Phys. B: Lasers Opt. 72(4), 399–405 (2001).
[Crossref]

Trofimov, A. O.

P. A. Burdaev, V. A. Aseev, E. V. Kolobkova, N. V. Nikonorov, and A. O. Trofimov, “Nanostructured glass ceramics based on fluorophosphate glass with a high content of rare-earth ions,” Glass Phys. Chem. 41(1), 132–136 (2015).
[Crossref]

van Dongen, A. M. A.

E. W. J. L. Oomen and A. M. A. van Dongen, “Europium (III) in Oxide Glasses: Dependence of the emission spectrum upon glass composition,” J. Non-Cryst. Solids 111(2-3), 205–213 (1989).
[Crossref]

Vivien, D.

G. Dantelle, M. Mortier, G. Patriarche, and D. Vivien, “Er3+-doped PbF2: Comparison between nanocrystals in glass-ceramics and bulk single crystals,” J. Solid State Chem. 179(7), 1995–2003 (2006).
[Crossref]

Walsh, B. M.

B. M. Walsh, N. P. Barnes, D. J. Reichle, and S. Jiang, “Optical properties of Tm3+ ions in alkali germanate glass,” J. Non-Cryst. Solids 352(50-51), 5344–5352 (2006).
[Crossref]

Wang, J.

R. Zheng, J. Wang, L. Zhang, C. Liu, and W. Wei, “Preparation and properties of Nd3+:SrAlF5 nanocrystals embedded fluorophosphate transparent glass–ceramic with long fluorescence lifetime,” Appl. Phys. A: Mater. Sci. Process. 122(1), 1–6 (2016).
[Crossref]

J. Wang, X. Qiao, X. Fan, and M. Wang, “Judd-Ofelt analysis and upconversion emission of Er3+-Yb3+ co-doped oxyfluoride glass ceramics containing LaF3 nanocrystals,” Philos. Mag. 85(32), 3755–3766 (2005).
[Crossref]

Wang, M.

J. Wang, X. Qiao, X. Fan, and M. Wang, “Judd-Ofelt analysis and upconversion emission of Er3+-Yb3+ co-doped oxyfluoride glass ceramics containing LaF3 nanocrystals,” Philos. Mag. 85(32), 3755–3766 (2005).
[Crossref]

Wang, W. C.

W. C. Wang, B. Zhou, S. H. Xu, Z. M. Yang, and Q. Y. Zhang, “Recent advances in soft optical glass fiber and fiber lasers,” Prog. Mater. Sci. 101, 90–171 (2019).
[Crossref]

Wei, W.

R. Zheng, J. Wang, L. Zhang, C. Liu, and W. Wei, “Preparation and properties of Nd3+:SrAlF5 nanocrystals embedded fluorophosphate transparent glass–ceramic with long fluorescence lifetime,” Appl. Phys. A: Mater. Sci. Process. 122(1), 1–6 (2016).
[Crossref]

Xu, H.

M. Mortier, P. Goldner, P. Féron, G. M. Stephan, H. Xu, and Z. Cai, “New fluoride glasses for laser applications,” J. Non-Cryst. Solids 326-327, 505–509 (2003).
[Crossref]

Xu, S.

Y. Xu, Y. Tian, Q. Liu, X. Sheng, W. Tang, J. Zhang, and S. Xu, “Effect of the heat treatment conditions on the structure and 2 micron luminescence of thulium-doped oxyfluoride silicate glass-ceramics,” J. Lumin. 211, 418–425 (2019).
[Crossref]

Xu, S. H.

W. C. Wang, B. Zhou, S. H. Xu, Z. M. Yang, and Q. Y. Zhang, “Recent advances in soft optical glass fiber and fiber lasers,” Prog. Mater. Sci. 101, 90–171 (2019).
[Crossref]

Xu, Y.

Y. Xu, Y. Tian, Q. Liu, X. Sheng, W. Tang, J. Zhang, and S. Xu, “Effect of the heat treatment conditions on the structure and 2 micron luminescence of thulium-doped oxyfluoride silicate glass-ceramics,” J. Lumin. 211, 418–425 (2019).
[Crossref]

Yamada, M.

Y. Ohishi, A. Mori, M. Yamada, H. Ono, Y. Nishida, and K. Oikawa, “Gain characteristics of tellurite-based erbium-doped fiber amplifiers for 1.5-µm broadband amplification,” Opt. Lett. 23(4), 274–276 (1998).
[Crossref]

M. Yamada, H. Ono, T. Kanamori, S. Sudo, and Y. Ohishi, “Broadband and gain-flattened amplifier composed of a 1.55 µm-band and a 1.58 µm-band Er3+-doped fibre amplifier in a parallel configuration,” Electron. Lett. 33(8), 710–711 (1997).
[Crossref]

Yang, Z. M.

W. C. Wang, B. Zhou, S. H. Xu, Z. M. Yang, and Q. Y. Zhang, “Recent advances in soft optical glass fiber and fiber lasers,” Prog. Mater. Sci. 101, 90–171 (2019).
[Crossref]

Yasukevich, A. S.

E. Kolobkova, A. Alkhlef, B. M. Dinh, A. S. Yasukevich, O. P. Dernovich, N. V. Kuleshov, and N. Nikonorov, “Spectral properties of Nd 3+ ions in the new fluoride glasses with small additives of the phosphates,” J. Lumin. 206, 523–529 (2019).
[Crossref]

Yasyukevich, A. S.

A. S. Yasyukevich, V. G. Shcherbitskii, VÉ Kisel’, A. V. Mandrik, and N. V. Kuleshov, “Integral Method of Reciprocity in the Spectroscopy of laser crystals with impurity,” J. Appl. Spectrosc. 71(2), 202–208 (2004).
[Crossref]

Yuan, X.

J. Fan, S. Chen, X. Yuan, Y. Jiang, L. Pan, B. Jiang, X. Mao, R. Li, X. Jiang, L. Zhang, and J. Ballato, “Recrystallization of Er3+:CaF2 in Transparent Fluorophosphate Glass-Ceramics with the Co-Firing Method,” J. Am. Ceram. Soc. 99(9), 2971–2976 (2016).
[Crossref]

Zhang, J.

Y. Xu, Y. Tian, Q. Liu, X. Sheng, W. Tang, J. Zhang, and S. Xu, “Effect of the heat treatment conditions on the structure and 2 micron luminescence of thulium-doped oxyfluoride silicate glass-ceramics,” J. Lumin. 211, 418–425 (2019).
[Crossref]

Zhang, L.

R. Zheng, J. Wang, L. Zhang, C. Liu, and W. Wei, “Preparation and properties of Nd3+:SrAlF5 nanocrystals embedded fluorophosphate transparent glass–ceramic with long fluorescence lifetime,” Appl. Phys. A: Mater. Sci. Process. 122(1), 1–6 (2016).
[Crossref]

J. Fan, S. Chen, X. Yuan, Y. Jiang, L. Pan, B. Jiang, X. Mao, R. Li, X. Jiang, L. Zhang, and J. Ballato, “Recrystallization of Er3+:CaF2 in Transparent Fluorophosphate Glass-Ceramics with the Co-Firing Method,” J. Am. Ceram. Soc. 99(9), 2971–2976 (2016).
[Crossref]

Zhang, Q. Y.

W. C. Wang, B. Zhou, S. H. Xu, Z. M. Yang, and Q. Y. Zhang, “Recent advances in soft optical glass fiber and fiber lasers,” Prog. Mater. Sci. 101, 90–171 (2019).
[Crossref]

Z. H. Jiang and Q. Y. Zhang, “The structure of glass: A phase equilibrium diagram approach,” Prog. Mater. Sci. 61, 144–215 (2014).
[Crossref]

S. F. Li, Q. Y. Zhang, and Y. P. Lee, “Absorption and photoluminescence properties of Er-doped and Er/Yb codoped soda-silicate laser glasses,” J. Appl. Phys. 96(9), 4746–4750 (2004).
[Crossref]

Zheng, R.

R. Zheng, J. Wang, L. Zhang, C. Liu, and W. Wei, “Preparation and properties of Nd3+:SrAlF5 nanocrystals embedded fluorophosphate transparent glass–ceramic with long fluorescence lifetime,” Appl. Phys. A: Mater. Sci. Process. 122(1), 1–6 (2016).
[Crossref]

Zhou, B.

W. C. Wang, B. Zhou, S. H. Xu, Z. M. Yang, and Q. Y. Zhang, “Recent advances in soft optical glass fiber and fiber lasers,” Prog. Mater. Sci. 101, 90–171 (2019).
[Crossref]

Appl. Phys. A: Mater. Sci. Process. (1)

R. Zheng, J. Wang, L. Zhang, C. Liu, and W. Wei, “Preparation and properties of Nd3+:SrAlF5 nanocrystals embedded fluorophosphate transparent glass–ceramic with long fluorescence lifetime,” Appl. Phys. A: Mater. Sci. Process. 122(1), 1–6 (2016).
[Crossref]

Appl. Phys. B: Lasers Opt. (2)

S. Taccheo, P. Laporta, S. Longhi, O. Svelto, and C. Svelto, “Diode-pumped bulk erbium-ytterbium lasers,” Appl. Phys. B: Lasers Opt. 63(5), 425–436 (1996).
[Crossref]

J. F. Philipps, T. Töpfer, H. Ebendorff-Heidepriem, D. Ehrt, and R. Sauerbrey, “Spectroscopic and lasing properties of Er3+:Yb3+-doped fluoride phosphate glasses,” Appl. Phys. B: Lasers Opt. 72(4), 399–405 (2001).
[Crossref]

Appl. Phys. Lett. (1)

S. Taccheo, P. Laporta, and C. Svelto, “Widely tunable single-frequency erbium-ytterbium phosphate glass laser,” Appl. Phys. Lett. 68(19), 2621–2623 (1996).
[Crossref]

C. R. Chim. (1)

S. Tanabe, “Rare-earth-doped glasses for fiber amplifiers in broadband telecommunication,” C. R. Chim. 5(12), 815–824 (2002).
[Crossref]

Electron. Lett. (1)

M. Yamada, H. Ono, T. Kanamori, S. Sudo, and Y. Ohishi, “Broadband and gain-flattened amplifier composed of a 1.55 µm-band and a 1.58 µm-band Er3+-doped fibre amplifier in a parallel configuration,” Electron. Lett. 33(8), 710–711 (1997).
[Crossref]

Glass Phys. Chem. (3)

P. A. Burdaev, V. A. Aseev, E. V. Kolobkova, N. V. Nikonorov, and A. O. Trofimov, “Nanostructured glass ceramics based on fluorophosphate glass with a high content of rare-earth ions,” Glass Phys. Chem. 41(1), 132–136 (2015).
[Crossref]

V. A. Aseev, P. A. Burdaev, E. V. Kolobkova, and N. V. Nikonorov, “Fluorophosphate nanostructured glass ceramics activated by erbium ions,” Glass Phys. Chem. 39(2), 174–181 (2013).
[Crossref]

E. V. Kolobkova, V. G. Melekhin, and A. N. Penigin, “Optical glass-ceramics based on fluorine-containing silicate glasses doped with rare-earth ions,” Glass Phys. Chem. 33(1), 8–13 (2007).
[Crossref]

J. Am. Ceram. Soc. (4)

J. Fan, S. Chen, X. Yuan, Y. Jiang, L. Pan, B. Jiang, X. Mao, R. Li, X. Jiang, L. Zhang, and J. Ballato, “Recrystallization of Er3+:CaF2 in Transparent Fluorophosphate Glass-Ceramics with the Co-Firing Method,” J. Am. Ceram. Soc. 99(9), 2971–2976 (2016).
[Crossref]

H. Takebe, Y. Nageno, and K. Morinaga, “Effect of Network Modifier on Spontaneous Emission Probabilities of Er3+ in Oxide Glasses,” J. Am. Ceram. Soc. 77(8), 2132–2136 (1994).
[Crossref]

H. Takebe, Y. Nageno, and K. Morinaga, “Compositional Dependence of Judd-Ofelt Parameters in Silicate, Borate, and Phosphate Glasses,” J. Am. Ceram. Soc. 78(5), 1161–1168 (1995).
[Crossref]

Y. Nageno, H. Takebe, and K. Morinaga, “Correlation between radiative transition probabilities of Nd3+ and composition in silicate, borate, and phosphate glasses,” J. Am. Ceram. Soc. 76(12), 3081–3086 (1993).
[Crossref]

J. Appl. Phys. (1)

S. F. Li, Q. Y. Zhang, and Y. P. Lee, “Absorption and photoluminescence properties of Er-doped and Er/Yb codoped soda-silicate laser glasses,” J. Appl. Phys. 96(9), 4746–4750 (2004).
[Crossref]

J. Appl. Spectrosc. (1)

A. S. Yasyukevich, V. G. Shcherbitskii, VÉ Kisel’, A. V. Mandrik, and N. V. Kuleshov, “Integral Method of Reciprocity in the Spectroscopy of laser crystals with impurity,” J. Appl. Spectrosc. 71(2), 202–208 (2004).
[Crossref]

J. Chem. Phys. (1)

W. T. Carnall, P. R. Fields, and K. Rajnak, “Spectral Intensities of the Trivalent Lanthanides and Actinides in Solution. II. Pm 3+, Sm 3+, Eu 3+, Gd 3+, Tb 3+, Dy 3+, and Ho 3+,” J. Chem. Phys. 49(10), 4412–4423 (1968).
[Crossref]

J. Eur. Ceram. Soc. (1)

G. Krieke and A. Sarakovskis, “Crystallization and upconversion luminescence of distorted fluorite nanocrystals in Ba2+ containing oxyfluoride glass ceramics,” J. Eur. Ceram. Soc. 36(7), 1715–1722 (2016).
[Crossref]

J. Fluorine Chem. (2)

V. Nazabal, M. Poulain, M. Olivier, P. Pirasteh, P. Camy, J. L. Doualan, S. Guy, T. Djouama, A. Boutarfaia, and J. L. Adam, “Fluoride and oxyfluoride glasses for optical applications,” J. Fluorine Chem. 134, 18–23 (2012).
[Crossref]

A. J. Stevenson, H. Serier-Brault, P. Gredin, and M. Mortier, “Fluoride materials for optical applications: Single crystals, ceramics, glasses, and glass-ceramics,” J. Fluorine Chem. 132(12), 1165–1173 (2011).
[Crossref]

J. Lumin. (3)

J. H. Choi, A. Margaryan, A. Margaryan, and F. G. Shi, “Judd-Ofelt analysis of spectroscopic properties of Nd3+ -doped novel fluorophosphate glass,” J. Lumin. 114(3-4), 167–177 (2005).
[Crossref]

E. Kolobkova, A. Alkhlef, B. M. Dinh, A. S. Yasukevich, O. P. Dernovich, N. V. Kuleshov, and N. Nikonorov, “Spectral properties of Nd 3+ ions in the new fluoride glasses with small additives of the phosphates,” J. Lumin. 206, 523–529 (2019).
[Crossref]

Y. Xu, Y. Tian, Q. Liu, X. Sheng, W. Tang, J. Zhang, and S. Xu, “Effect of the heat treatment conditions on the structure and 2 micron luminescence of thulium-doped oxyfluoride silicate glass-ceramics,” J. Lumin. 211, 418–425 (2019).
[Crossref]

J. Non-Cryst. Solids (10)

S. Cui, J. Massera, M. Lastusaari, L. Hupa, and L. Petit, “Novel oxyfluorophosphate glasses and glass-ceramics,” J. Non-Cryst. Solids 445-446, 40–44 (2016).
[Crossref]

H. Rahimian, Y. Hatefi, A. Dehghan Hamedan, S. P. Shirmardi, and H. Mokhtari, “Structural and optical investigations on Eu3+ doped fluorophosphate glass and nano glass-ceramics,” J. Non-Cryst. Solids 487, 46–52 (2018).
[Crossref]

H. Ebendorff-Heidepriem, D. Ehrt, M. Bettinelli, and A. Speghini, “Effect of glass composition on Judd-Ofelt parameters and radiative decay rates of Er3+ in fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 240(1-3), 66–78 (1998).
[Crossref]

M. Mortier, P. Goldner, P. Féron, G. M. Stephan, H. Xu, and Z. Cai, “New fluoride glasses for laser applications,” J. Non-Cryst. Solids 326-327, 505–509 (2003).
[Crossref]

K. Soga, H. Inoue, and A. Makishima, “Calculation and simulation of spectroscopic properties for rare earth ions in chloro-fluorozirconate glasses,” J. Non-Cryst. Solids 274(1-3), 69–74 (2000).
[Crossref]

S. Jiang, T. Luo, B. Hwang, F. Smekatala, K. Seneschal, J. Lucas, and N. Peyghambarian, “Er3+-doped phosphate glasses for fiber amplifiers with high gain per unit length,” J. Non-Cryst. Solids 263-264, 364–368 (2000).
[Crossref]

B. M. Walsh, N. P. Barnes, D. J. Reichle, and S. Jiang, “Optical properties of Tm3+ ions in alkali germanate glass,” J. Non-Cryst. Solids 352(50-51), 5344–5352 (2006).
[Crossref]

H. Ebendorff-Heidepriem and D. Ehrt, “Spectroscopic properties of Eu3+ and Tb3+ ions for local structure investigations of fluoride phosphate and phosphate glasses,” J. Non-Cryst. Solids 208(3), 205–216 (1996).
[Crossref]

E. W. J. L. Oomen and A. M. A. van Dongen, “Europium (III) in Oxide Glasses: Dependence of the emission spectrum upon glass composition,” J. Non-Cryst. Solids 111(2-3), 205–213 (1989).
[Crossref]

H. Takebe, K. Morinaga, and T. Izumitani, “Correlation between radiative transition probabilities of rare-earth ions and composition in oxide glasses,” J. Non-Cryst. Solids 178, 58–63 (1994).
[Crossref]

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

J. Phys.: Condens. Matter (1)

S. A. Payne, L. K. Smith, W. L. Kway, J. B. Tassano, and W. F. Krupke, “The mechanism of Tm-Ho energy transfer in LiYF4,” J. Phys.: Condens. Matter 4(44), 8525–8542 (1992).
[Crossref]

J. Solid State Chem. (2)

I. Gugov, M. Müller, and C. Rüssel, “Transparent oxyfluoride glass ceramics co-doped with Er3+ and Yb3+ Crystallization and upconversion spectroscopy,” J. Solid State Chem. 184(5), 1001–1007 (2011).
[Crossref]

G. Dantelle, M. Mortier, G. Patriarche, and D. Vivien, “Er3+-doped PbF2: Comparison between nanocrystals in glass-ceramics and bulk single crystals,” J. Solid State Chem. 179(7), 1995–2003 (2006).
[Crossref]

Opt. Commun. (1)

E. Tanguy, C. Larat, and J. P. Pocholle, “Modelling of the erbium - ytterbium laser,” Opt. Commun. 153(1-3), 172–183 (1998).
[Crossref]

Opt. Eng. (1)

S. Jiang, T. Luo, B.-C. Hwang, G. Nunzi-Conti, M. Myers, D. Rhonehouse, S. Honkanen, and N. Peyghambarian, “New Er3+-doped phosphate glass for ion-exchanged waveguide amplifiers,” Opt. Eng. 37(12), 3282–3286 (1998).
[Crossref]

Opt. Lett. (1)

Opt. Mater. (Amsterdam, Neth.) (1)

V. K. Tikhomirov, V. D. Rodriguez, J. Méndez-Ramos, P. Núñez, and A. B. Seddon, “Comparative spectroscopy of (ErF3)(PbF2) alloys and Er3+-doped oxyfluoride glass-ceramics,” Opt. Mater. (Amsterdam, Neth.) 27(3), 543–547 (2004).
[Crossref]

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[Crossref]

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[Crossref]

Philos. Mag. (1)

J. Wang, X. Qiao, X. Fan, and M. Wang, “Judd-Ofelt analysis and upconversion emission of Er3+-Yb3+ co-doped oxyfluoride glass ceramics containing LaF3 nanocrystals,” Philos. Mag. 85(32), 3755–3766 (2005).
[Crossref]

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[Crossref]

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[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. (a)Yb3+ and Er3+ ions energy level diagram: the Er–Yb energy transfer process (dotted arrow), the nonradiative transitions (dashed arrow), and the radiative transitions (solid arrows); (b) the absorption cross-sections of the B and P glasses: absorption cross-sections of the 2B (dashed line) and 2P (solid line) glasses in a wide spectral range.
Fig. 2.
Fig. 2. The luminescence spectra of the 2B (dashed line) and 2P (solid line) glasses doped with Er3+ ions of (a) 4I13/24I15/2 transition and (b) in the visible spectral region (λexc = 980 nm).
Fig. 3.
Fig. 3. The absorption cross-section spectra (1), stimulated emission obtained by MR method (2) and FL method (3) for the4I13/24I15/2 transition of the 2P glass.
Fig. 4.
Fig. 4. The DSC curves of the 2B and 2P glasses
Fig. 5.
Fig. 5. X-ray diffraction pattern of the 2B (a) and 2P (b) glasses after the heat treatment for 180 min at T = 480°C: Usovit crystal phase (JCPDS card, No. 010722129) denoted by x and cubic unknown phase (denoted by o)
Fig. 6.
Fig. 6. (a, b). SEM micrographs of the 2P glass (after the heat treatment) at different scale. The chemical composition of the glass was determined in two points: near the nanocrystals, surrounded by glass host (1), and in pure glass region without the nanocrystals (2)
Fig. 7.
Fig. 7. The absorption (a) and luminescence (b) spectra of the 2P glass before and after the heat treatment with different duration
Fig. 8.
Fig. 8. Spectra of amplification cross-sections of the 2B (a) and 2P (b) glasses at different values of ${\beta _2}$

Tables (4)

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Table 1. Composition (mol. fraction) and physical properties of the fluorophosphate glasses doped with ErF3, YbF3, where NEr, NYb – dopant ions concentration

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Table 2. The experimental Sexp (10−20 cm2), calculated Scal (10−20 cm2) line strengths, the root mean square deviation (Δrms) for transitions from 4I15/2 level to different excited levels, and the Judd-Ofelt parameters Ω 2 , Ω 4 , Ω 6 and χ of the Er3+ ions in the 2B, 2P glasses

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Table 3. Emission parameters of the 2B and 2P glasses ( λ ¯ – the mean emission wavelength, Sed and Smd –the line strengths of the electric dipole and magnetic dipole transitions, A ( J J ) - spontaneous emission probability, QE - quantum efficiency, τrad and τexp - radiative and experimental lifetime by 532.8 nm excitation)

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Table 4. The chemical composition of the 2P glass-ceramics obtained by XDEA: composition 1 – in the region of nanocrystals surrounded by glass; composition 2 - at pure glass region (at.%) (according to Fig. 8)

Equations (14)

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N ( i o n c m 3 ) = mole fraction of rare earth oxide  × Avogadro number average molecular weight
S e d m e a s ( J J ) = 27 h c ( 2 J + 1 ) n 8 π 3 λ ¯ e 2 ( n 2 + 2 ) 2 σ ( λ ) d λ 9 n 2 ( n 2 + 2 ) 2 S m d
S e d c a l c ( J J ) = t = 2 , 4 , 6 Ω t | Ψ J | | U t | | Ψ J | 2
S m d ( J J ) = [ h 4 π m c ] 2 | Ψ J | | L + 2 S | | Ψ J | 2
A ( J J ) = 64 π 4 e 2 3 h ( 2 J + 1 ) λ ¯ 3 [ n ( n 2 + 2 3 ) 2 S e d c a l c + n 3 S m d ]
S e d [ 4 I 13 / 2 ; 4 I 15 / 2 ] = t = 2 , 4 , 6 Ω t 4 I 13 / 2 | | U t | | 4 I 15 / 2 2 = 0.19 Ω 2 + 0.118 Ω 4 + 1.462 Ω 6
τ r a d = 1 A ( J J )
σ e m = A ( J J ) 8 π c n 2 exp ( h c / ( k T λ ) ) λ 4 σ a b s ( λ ) exp ( h c / ( k T λ ) ) d λ σ a b s ( λ )
σ e m ( λ ) = A ( J J ) λ 5 8 π c n 2 I l u m ( λ ) λ I l u m ( λ ) d λ
W D A = π ( 2 π / 3 ) 5 / 2 R s a 3 R s s 3 τ s N s N a
η t = W D A τ s 1 + W D A τ s
g ( λ ) = N σ l ( λ ) ( β 2 β l ( λ ) )
Σ g a i n = g ( λ ) / g ( λ ) N N = σ l ( λ ) ( β 2 β l ( λ ) )
Δ τ = λ max 2 c Δ λ

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