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Study on optical properties and upconversion luminescence of Er3+/Yb3+ co-doped tellurite glass for highly sensitive temperature measuring

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Abstract

Er3+/Yb3+ co-doped tellurite glass is prepared by conventional melt-quenching method. The thermal and optical properties of the glass are investigated for temperature sensing. The Judd-Ofelt theory is applied for spectral analysis and the obtained intensity parameters (Ω2 = 6.48 × 10−20 cm2, Ω4 = 1.82 × 10−20 cm2, Ω6 = 1.27 × 10−20 cm2) are used to estimate the spectroscopic parameters of Er3+ ions in the glass. The upconversion (UC) luminescence of the glass is investigated under the laser diode (LD) excitation at 976 nm. The absolute quantum yield for UC luminescence is determined to be 0.0049% when the pump power density is 10 W/cm2. The dependences of green and red UC emission intensities on pump power indicate that both of the red and green UC emissions of Er3+ ions are mostly contributed by the co-existing two and three-photon involved energy transfer processes between the Yb3+ and Er3+ ions. Furthermore, the temperature-dependent green UC emissions of the glass are studied and the results show that the glass is an excellent candidate for construction of temperature sensors based on the self-referenced fluorescence intensity ratio (FIR) technique.

© 2017 Optical Society of America

1. Introduction

Rare earth doped glasses are promising in applications in numerous fields such as fiber amplifiers and lasers, white-LED, and displays [1–5]. Taking advantage of the abundant energy level structures of rare earth ions and the high transparency of glass hosts to develop the novel luminescent glasses, especially the high performance upconversion (UC) glasses have attracted considerable attention in recent years. These glasses can generate high-energy photons upon low-energy excitation, which have some special potential applications in the solar cells for raising light harvest and optical thermometry for self-referenced temperature sensing [6, 7].

The tellurite based glasses are well known as excellent hosts for the rare earth luminescence. They possess very low cut-off phonon energy (~750 cm−1) among oxide materials and exhibit extraordinary optical properties, low melting temperature, decent thermal stability, chemical durability, as well as broad homogeneity range [8–10]. Many tellurite based glasses with different major components have been used for the UC hosts such as the TeO2-WO3, TeO2-ZnO, TeO2-PbO, and so on [11–13]. Among all the rare earth ions, the Er3+/Yb3+ combined systems are mostly attractive for the green and red emissions. The Yb3+ ions act as sensitizers that absorb strongly at around 980 nm, a wavelength that matches the emission wavelength of currently available economic commercial LD. The enhanced UC luminescence of Er3+ can be produced with the help of highly efficient resonant energy transfer from Yb3+ to Er3+ [14]. This is very essential for many applications such as the temperature sensing based on the fluorescence intensity ratio (FIR) of the thermally coupled energy levels of Er3+, i.e., 2H11/2 and 4S3/2 [15–17]. Although the temperature sensing behaviors based on the FIR technique have been demonstrated in many Er3+/Yb3+ co-doped tellurite glasses [11, 18–20], the UC glasses for temperature sensing with good thermal stability and high sensor sensitivity are still need to be carefully investigated.

In this work, the TeO2-LaF3-NaF-TiO2 glass co-doped with Er3+ and Yb3+ ions was prepared and its thermal and optical properties were studied in detail. When excited at around 980 nm, the intense UC luminescence located in the green and red regions was observed and the possible energy transfer mechanisms were analyzed. The UC green emissions at varying temperatures were measured. Based on the FIR of the two thermally coupled levels of Er3+ ions, this glass has potential application in the construction of optical temperature sensors.

2. Experimental

The tellurite glass with nominal composition of (in molar ratio) 79TeO2-10LaF3-10NaF-1TiO2 and extra additions of 10YbF3-1ErF3 (named TLNT glass) was prepared by using the conventional melt-quenching method. All the raw materials of TeO2, TiO2, LaF3, NaF, YbF3, and ErF3, with high chemical purity (>99%), were well weighted and mixed altogether, and then were heated at 900 °C for 30 minutes in an alumina crucible in an electric furnace. The homogeneous melt was obtained and quickly cast into a brass mould to form a glass bulk. In order to remove the thermal stress, the glass bulk was annealed at 350 °C for 5h. Finally, the transparent glass was obtained and the samples were cut and polished for measurements.

The density of the TLNT glass was measured to be 5.09 g/cm3 by using the buoyancy method based on the Archimedes principle with distilled water as immersion liquid. The differential scanning calorimetry (DSC) curve of the glass was measured in a thermal analysis apparatus (STA449C, Netzsch) and the linear thermal expansion behavior was recorded by a dilatometer (DIL 402CL, Netzsch). Both of the thermal measurements were carried out at the same heating rate of 5 K/min. The refractive index was measured by using a prism coupler (Model 2010, Metricon) at varying wavelengths. The absorption spectrum was recorded by a UV/VIR/NIR spectrophotometer (Cary5000, Agilent) over the wavelength range of 400−1800 nm. The UC spectra were recorded by a spectrometer (LS55, PerkinElmer) with a continuous wave (CW) fiber coupled LD excitation at 976 nm. The absolute UC quantum yield was measured at room temperature by employing a barium sulfate-coated integrating sphere as the sample chamber that was mounted on a spectrometer (FLS920, Edinburgh), with the entry and output port of the sphere located in 90° geometry from each other in the plane of the spectrometer. The measurement was conducted deliberately according to the protocols reported by van Veggel and U. Resch-Genger et al. [21, 22]. All the spectral data collected were corrected for the spectral response of both the spectrometer and the integrating sphere. For the temperature-dependent UC measurements, the sample was heated in a homemade small furnace in the temperature range from 323 K to 573 K, and measured with the help of a thermocouple located close to it.

3. Results and discussion

3.1 Thermal properties

Figure 1 shows the DSC and linear thermal expansion curves of the TLNT glass. Both of the curves demonstrate that the temperature of glass transition Tg is about 395 °C. After this temperature the atoms in the glass start to move and the fluidity of the glass begin to increase, which can be verified by the considerably increasing of the thermal expansion. The temperature Ts corresponding to the maximum of the thermal expansion suggests that the glass softening and is about 413 °C. The coefficient of thermal expansion (CTE) is also determined and displayed in the inset of Fig. 1. The CTE at temperatures below the Tg remains approximately constant and the value is about 2.04 × 10−5 K−1. The thermal expansion coefficient of TLNT glass is very close to that of some other tellurite glasses such as the TeO2-PbO (2.10 × 10−5 K−1) and TeO2-ZnF2 (2.07 × 10−5 K−1) [23, 24].

 figure: Fig. 1

Fig. 1 DSC and linear thermal expansion curves of the TLNT glass recorded at a heating rate of 5 K/min. The inset displays the curve of CTE versus temperature.

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3.2 Evaluation of refractive index

The refractive index is measured at wavelengths of 516, 638, and 1540 nm and the dispersion curve of the glass can be obtained by a nonlinear fit using the empirical Cauchy dispersion equation [25],

n=A+Bλ2+Cλ4
where A, B, and C are known as Cauchy coefficients which characterize the material. Figure 2 demonstrates the variation of refractive index versus wavelength, and the fitting to the measure data yields the Cauchy coefficients of A = 1.853, B = 0.018 μm2, and C = 9.839 × 10−5 μm4, respectively.

 figure: Fig. 2

Fig. 2 Refractive indices of the TLNT glass and its dispersion curve fitted by the empirical Cauchy dispersion equation.

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3.3 Absorption spectrum and J-O analysis

Figure 3 shows the absorption spectrum of TLNT glass in the wavelength range from 400 nm to 1800 nm. The strongest absorption band peaked at 976 nm is mainly assigned to the 2F7/22F5/2 transition of Yb3+ ions and the peak absorption cross section σabs can be determined to be 2.08 × 10−20 cm2, according to σAbs=α/NYbwhere the α is absorption coefficient and the NYb is concentration of Yb3+ ions (9.38 × 1020 cm−3), which can be calculated from the density and batch composition of the glass. The peak absorption cross section of Yb3+ in the TLNT glass is larger than those of some other tellurite glasses such as the TeO2-ZnO-Nb2O5 glass [26]. In addition, the full width at half maximum (FWHM) of the absorption band of Yb3+ ions is about 8 nm, which can ensure the TLNT glass absorb the NIR excitation light very effectively. The other well-resolved absorption bands at 451 nm, 489 nm, 520 nm, 544 nm, 652 nm, 922 nm, and 1525 nm can be assigned to the transitions of Er3+ ions from the ground energy level of 4I15/2 to the different excited energy levels of 4F5/2, 4F7/2, 2H11/2, 4S3/2, 4F9/2, 4I11/2 and 4I13/2, respectively, according to the energy level distribution of Er3+ ion [27]. In addition, the absorption band of 4I15/24I11/2 of Er3+ was partially overlapped with the absorption band of 2F7/22F5/2 of Yb3+ [28].

 figure: Fig. 3

Fig. 3 Absorption spectrum of the TLNT glass.

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The standard Judd-Ofelt (J-O) theory has been applied to analyze the absorption spectrum so as to predict the spectroscopic parameters including the spontaneous emission rates, radiative lifetimes, and fluorescence branching ratios of Er3+ in the TLNT glass [29, 30]. Five intense absorption bands corresponding to the transitions of 4I15/24I13/2, 4F9/2, 4S3/2, 2H11/2, and 4F7/2 were chosen to calculate the J-O intensity parameters (Ωt, t = 2, 4, 6).

The experimental oscillator strength fexp of absorption transition is determined by the equation,

fexp(JJ)=mc2πN0λ¯2e2Г
where J and are the total angular momentum quantum number of the initial and final states, respectively, m is the mass of electron, c is the velocity of light in vacuum, N0 is the Er3+ ion concentration, λ¯ is the mean wavelength of each chosen absorption band, e is the charge of electron, and Γ is the integrated absorption coefficient as a function of wavelength λ, which is given by,

Г=α(λ)dλ

According to the J-O theory, the calculated oscillator strength can be given as,

fcal(JJ)=8π2mc3h(2J+1)λ¯[(n2+2)29nSed+nSmd]
where h is Planck's constant, n is the refractive index, Sed and Smd are the electric dipole (ED) line strength and magnetic dipole (MD) line strength, respectively, which take the forms,
Sed(JJ)=t=2,4,6Ωt|<(S,L)JU(t)(S,L)J>|2
Smd(JJ)=h216π2m2c2|<(S,L)JL+2S(S,L)J>|2
where the reduced matrix elements of |<(S,L)JU(t)(S,L)J>|2and |<(S,L)JL+2S(S,L)J>|2are calculated following Carnall et al. [31]. It is worth mentioning that only the 4I15/24I13/2 transition contained both the contributions of ED and MD interactions among the absorption bands of Er3+. The MD line strength is calculated to be 0.71 × 10−20 cm2, which should be excluded before the J-O analysis.

The J-O intensity parameters were calculated using the least-squares fitting of fexp and fcal [32]. Table 1 summarizes the values of some parameters used in calculation, and the experimental and calculated absorption oscillator strengths. The root-mean-square deviation δrms for the best-fitted oscillator strengths,

δrms=(fexpfcal)2qp
where q is the number of analyzed spectral bands and p = 3 is the number of the parameters determined, is1.48 × 10−7. The error is relatively small in comparison with those obtained in some other glasses [20, 33–37], which indicates that the deduced J-O intensity parameters are reliable.

Tables Icon

Table 1. Values of the mean wavelengths, refractive indices, integrated absorption coefficients, experimental and calculated oscillator strengths of Er3+ in TLNT glass.

The three J-O intensity parameters of Er3+ for the TLNT glass were obtained to be Ω2 = 6.48 × 10−20 cm2, Ω4 = 1.82 × 10−20 cm2, and Ω6 = 1.27 × 10−20 cm2. These values are compared to those obtained for other glass hosts. Table 2 lists the J-O intensity parameters of Er3+ ions in some other typical glasses. In general, the intensity parameter Ω2 is sensitive to the asymmetry at the rare earth sites and it will be strongly enhanced by the covalent bonding between the rare earth ions and the ligand ions, and in contrast, the Ω6 parameter is related to the rigidity of the medium in which the ions are situated [38]. It can be seen that, compared with other glasses, the J-O intensity parameters Ω2,6 of Er3+ ions in the TLNT glass exhibit relatively high magnitude, which suggests that the chemical bonds between Er3+ and oxygen have a very high degree of covalence characteristics and the environment around the sites of Er3+ is of low symmetry and high rigidity.

Tables Icon

Table 2. J-O intensity parameters Ωt= 2, 4, 6 (10−20 cm2) and RMS deviations for the best-fitted oscillator strengths of Er3+ in different glass hosts, and their maximum sensor sensitivities for thermometry.

The J-O intensity parameters can be applied to calculate the emission line strengths corresponding to the transitions from the upper energy levels, 4I13/2, 4I11/2, 4I9/2, 4F9/2, 4S3/2, 2H11/2, and 4F7/2, to their corresponding lower-lying energy levels. Using these line strengths, the spontaneous emission rate A, can be calculated by,

A(JJ)=Aed+Amd=64π4e23h(2J+1)λ¯3[n(n2+2)2Sed+n3Smd]
where Aed and Amd are the electric-dipole and magnetic-dipole spontaneous emission rates, respectively. The radiative lifetime τrad of an energy level is related to the spontaneous emission rates for all transitions from this level and can be determined by,
τrad=1A(JJ)
where the sum runs over all terminal state . The fluorescence branching ratio β can be determined by,

β(JJ)=A(JJ)A(JJ)

Table 3 summarizes the values of calculated spontaneous emission rates, fluorescence branching ratios, and radiative lifetimes of the Er3+ ions in TLNT glass, which corresponding to the transitions from the upper energy levels of 4I13/2, 4I11/2, 4F9/2, 4S3/2, 2H11/2, and 4F7/2 to their inferior lying energy levels, respectively. Usually, the Yb3+ and Er3+ co-doped system is promising for green and red UC emissions under NIR excitation. The branching ratios of 2H11/24I15/2 (green), 4S3/24I15/2 (green), and 4F9/24I15/2 (red) transitions are estimated to be 95%, 66%, and 90%, respectively. These results show that, in theory, the transition channels for the green and red emissions are dominant among all their corresponding upper level radiative transitions.

Tables Icon

Table 3. Calculated spontaneous emission rate A, fluorescence branching ratio β, and radiative lifetime τrad of Er3+ ions in TLNT glass.

3.4 UC emission spectra and energy transfer mechanisms

Figure 4 displays the UC emission spectra of the TLNT glass when excited at 976 nm with different pump powers. The spot diameter of pump light is maintained at about 5 mm and the power density estimated varies from 2 to 14 W/cm2. Three emission bands corresponding to the 2H11/24I15/2 (green), 4S3/24I15/2 (green), and 4F9/24I15/2 (red) transitions of Er3+ ions, respectively, were measured in the visible region. The UC luminescence is very bright to the naked eyes and the digital photograph is taken and shown in the inset of Fig. 4. For evaluating the UC efficiency, the absolute quantum yield,defined as the ratio of the number of emitted photons to the number of absorbed photons, was measured to be 0.0030% for the VIS UC emission upon 976 nm NIR laser excitation at a power density of 5 W/cm2, and this value increases to 0.0049% when the pump power is raised to 10 W/cm2, which are higher than that of germanate glass (0.0027%@539W/cm2, excitation at 975 nm) [39]. Generally, the UC emission intensity I and the pump power P follow the relation of IPn [40], where n represents the number of photons involved in the process to populate the UC upper level. Figure 5 shows the double logarithmic plot for the green and red UC emissions at varying pump powers. The slopes of the green and red emissions are fitted to be 2.45 and 2.34, respectively, which indicates some two and three-photon processes are co-responsible for both the red and green emissions.

 figure: Fig. 4

Fig. 4 UC spectra of the TLNT glass excited by LD at around 976 nm under different pump powers.

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 figure: Fig. 5

Fig. 5 Double logarithmic plot of pump power versus UC emission intensity under 976 nm LD excitation.

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Figure 6 depicts the energy transfer (ET) mechanisms responsible for the UC emissions based on the energy level diagrams of Er3+ and Yb3+ ions. When excited at around 976 nm, the Yb3+ ions are primarily excited to 2F5/2 from the ground energy level of 2F7/2 by absorbing the pump photon, and then transfer the energy to the nearby Er3+ ions via ET1: 4I15/2(Er3+) + 2F5/2(Yb3+)→4I11/2(Er3+) + 2F7/2 (Yb3+). After that the Er3+ ions are populated onto the energy level of 4I11/2.Thereafter, for the two-photon UC emissions, the Er3+ ions on the energy level of 4I11/2 will continue to acquire the energy transferred from the excited Yb3+ ions to arrive onto the energy level of 4F7/2 via ET2: 4I11/2(Er3+) + 2F5/2(Yb3+) →4F7/2(Er3+) + 2F7/2(Yb3+). Finally the green emitting energy levels of 2H11/2 and 4S3/2 can be populated by multi-photon relaxation (MPR) from 4F7/2. Similarly, the red emitting energy level of 4F9/2 can be populated by the energy transfer of ET2*: 4I13/2(Er3+) + 2F5/2(Yb3+) →4F9/2(Er3+) + 2F7/2(Yb3+), with the help of MPR from 4I11/2 to 4I13/2.

 figure: Fig. 6

Fig. 6 Energy level diagrams of Er3+ and Yb3+ ions and UC luminescence mechanism of the TLNT glass.

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For the three-photon green emissions, one possibility is that after excited onto the energy level of 4F9/2 via ET1 and ET2*, part of the Er3+(4F9/2) ions may continue to jump to the upper-lying energy level of 2H9/2 via ET3: 4F9/2(Er3+) + 2F5/2(Yb3+) →2H9/2(Er3+) + F7/2(Yb3+), and then relaxed to the green emitting energy levels of 2H11/2 and 4S3/2. As was seen in Fig. 4, the TLNT glass also exhibited strong red UC emission. This means the energy level of 4F9/2 can be very efficiently populated under the 976 nm excitation. A large amount of population on the 4F9/2 should be an important contributor to the three-photon green UC emission. Since the red emitting energy level of 4F9/2 could also be populated by the MPR from the nearest upper-lying energy level of 4S3/2, if the green UC emissions are realized through a three-photon involved energy transfer process, certainly the red UC emission should also be contributed by such a process. Similar UC luminescence containing both contributions of the two and three-phonon involved energy transfer processes have also been discussed in some other studies including glass and nanocrystals [41–43].

3.5 Temperature-dependence of UC emissions

It is well known that the green emissions of Er3+ are produced by the thermally coupled energy levels of 2H11/2 and 4S3/2 and the fluorescence intensity ratio (FIR) of the two green UC emissions is closely related to the temperature [20, 44]. The relation can be described as,

FIR=I523I545=Cexp(-ΔEkT)
where I523 and I545 are the peak intensity at the wavelengths of 523 and 545 nm, C is a constant, ΔE is the energy gap separating the two energy levels of 2H11/2 and 4S3/2, k is the Boltzmann constant, and T is temperature. On the other hand, in the framework of the J-O theory, the spontaneous emission intensity can be expressed as a function of the three intensity parameters without considering the nonradiative de-excitation. The FIR involving the 2H11/2 and 4S3/2 to the 4I15/2 ground energy level obeys [45],
FIR=I523I5450.7158Ω2+0.4138Ω4+0.0927Ω60.2225Ω6
In this way, it is necessary to selected host matrix to obtain large values of Ω2 and Ω4 and a small value of Ω6. In this sense, as compared in Table 2, the TLNT glass possesses a good parametric distribution with Ω2>Ω4>Ω6 and it will be one of the best temperature sensitive materials satisfying the above requirement.

Figure 7 shows the experimental dependence of logarithm of FIR (I523/I545) on the reciprocal temperature for the TLNT glass in the temperature range from 323 K to 573 K. The UC emission spectra are displayed in the inset. In the temperature-dependent UC measurement, the pump power density is estimated to be 10 W/cm2 and the duration time is about 1 minute at fixed measuring temperature. Dry air is used to purge the sample chamber to avoid the thermal effect from the excitation source and thus stabilize the temperature of sample. It can be seen that the FIR of Er3+ ions increases significantly with increasing temperature and the experimental data is well fitted to a straight line. According to Eq. (11), the fitting of experimental data gives the slope about −1117.14 ± 7.78 and the intercept about 2.89 ± 0.02, resulting the energy gap ΔE is about 776.43 cm−1 and the coefficient C is about 17.99, respectively. The energy gap is very close to the value reported in the fluoro-tellurite glass (766 cm−1) [46] and tellurite-germanate glass (781 cm−1) [47].

 figure: Fig. 7

Fig. 7 The monolog plot of the FIR of green UC emissions as a function of inverse absolute temperature for the TLNT glass. The inset displays the UC green emissions normalized at 545 nm for the varying temperatures.

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In addition, the sensor sensitivity S which defined as the rate at which FIR changes with a change in temperature can be given by [48],

S=dFIRdT=FIRΔEkT2
Figure 8 shows the sensitivity of the TLNT glass at varying temperatures. Apparently, the value of sensitivity keeps above 6.0 × 10−3 K−1 over the experimental temperature region and the maximum theoretical sensitivity is up to 8.72 × 10−3 K−1 at 558 K. As compared in Table 2, the maximum sensitivity is much higher than the typical maximum values (3~8 × 10−3) reported in some other oxide glass systems for optical thermometry based on the FIR technique of Er3+. In theory, the TLNT glass can be used for self-referenced temperature sensing as long as the temperature to be measured does not exceed the glass transition. That means the TLNT glass could be in favor of application in measuring temperature as high as about 590 K with very high sensitivity.

 figure: Fig. 8

Fig. 8 The sensitivity as a function of the temperature for the TLNT glass.

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4. Conclusion

In summary, the Er3+/Yb3+ co-doped tellurite glass of TLNT was prepared by the melt-quenching method. Thermal properties of the glass have been investigated using DSC and DIL methods. The results indicate that the TLNT glass is of good thermal quality. Based on the standard J-O theory, the J-O intensity parameters were fitted from the absorption spectrum and applied to predict the spontaneous emission rates, branching ratios, and radiative lifetimes. Furthermore, the dependences of UC emissions on the pump power and temperature for the TLNT glass were investigated in detail. Both of the red and green UC emissions of the glass can be explained by the co-contribution of the two and three-photon involved energy transfer processes. The absolute quantum yield for UC emissions at room temperature is determined to be 0.0049% when the pump power density is 10 W/cm2. Moreover, the FIR (I523/I545) of green UC emissions increases significantly with increasing temperature in the range from 323 K to 573 K, and the sensor sensitivity was determined to be as high as 8.72 × 10−3 K−1 at 558 K. These results indicate that the TLNT glass is of good performance in UC luminescence and may be used for temperature measuring with high sensitivity.

Funding

National Natural Science Foundation of China (51302228); the Chongqing Research Program of Basic Research and Frontier Technology (cstc2016jcyjA0306); and the Fundamental Research Funds for the Central Universities of Southwest University (XDJK2015B019 and XDJK2017D014).

Acknowledgments

The authors gratefully thank Dr. En Ma (Fujian Institute of Research on the Structure of Matter, Chinese Academy of Sciences, Fuzhou, China), for the measurement of the absolute quantum yields.

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Figures (8)

Fig. 1
Fig. 1 DSC and linear thermal expansion curves of the TLNT glass recorded at a heating rate of 5 K/min. The inset displays the curve of CTE versus temperature.
Fig. 2
Fig. 2 Refractive indices of the TLNT glass and its dispersion curve fitted by the empirical Cauchy dispersion equation.
Fig. 3
Fig. 3 Absorption spectrum of the TLNT glass.
Fig. 4
Fig. 4 UC spectra of the TLNT glass excited by LD at around 976 nm under different pump powers.
Fig. 5
Fig. 5 Double logarithmic plot of pump power versus UC emission intensity under 976 nm LD excitation.
Fig. 6
Fig. 6 Energy level diagrams of Er3+ and Yb3+ ions and UC luminescence mechanism of the TLNT glass.
Fig. 7
Fig. 7 The monolog plot of the FIR of green UC emissions as a function of inverse absolute temperature for the TLNT glass. The inset displays the UC green emissions normalized at 545 nm for the varying temperatures.
Fig. 8
Fig. 8 The sensitivity as a function of the temperature for the TLNT glass.

Tables (3)

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Table 1 Values of the mean wavelengths, refractive indices, integrated absorption coefficients, experimental and calculated oscillator strengths of Er3+ in TLNT glass.

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Table 2 J-O intensity parameters Ωt = 2, 4, 6 (10−20 cm2) and RMS deviations for the best-fitted oscillator strengths of Er3+ in different glass hosts, and their maximum sensor sensitivities for thermometry.

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Table 3 Calculated spontaneous emission rate A, fluorescence branching ratio β, and radiative lifetime τrad of Er3+ ions in TLNT glass.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

n = A + B λ 2 + C λ 4
f exp ( J J ) = m c 2 π N 0 λ ¯ 2 e 2 Г
Г = α ( λ ) d λ
f c a l ( J J ) = 8 π 2 m c 3 h ( 2 J + 1 ) λ ¯ [ ( n 2 + 2 ) 2 9 n S e d + n S m d ]
S e d ( J J ) = t =2,4,6 Ω t | < ( S , L ) J U ( t ) ( S , L ) J > | 2
S m d ( J J ) = h 2 16π 2 m 2 c 2 | < ( S , L ) J L + 2 S ( S , L ) J > | 2
δ r m s = ( f exp f c a l ) 2 q p
A ( J J ) = A e d + A m d = 64 π 4 e 2 3 h ( 2 J + 1 ) λ ¯ 3 [ n ( n 2 + 2 ) 2 S e d + n 3 S m d ]
τ r a d = 1 A ( J J )
β ( J J ) = A ( J J ) A ( J J )
F I R = I 523 I 545 = C exp ( - Δ E k T )
F I R = I 523 I 545 0.7158 Ω 2 + 0.4138 Ω 4 + 0.0927 Ω 6 0.2225 Ω 6
S = d F I R d T = F I R Δ E k T 2
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