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Superbroadband near-infrared emission in Tm-Bi codoped sodium-germanium-gallate glasses

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Abstract

Superbroadband emission from 1.0 to 1.7 μm wavelength was observed in thulium-bismuth (Tm-Bi) codoped sodium-germanium-gallate (NGG) glasses under 793 nm excitation. Efficient energy transfer process from Bi to Tm ions, with value as high as 67.7%, was achieved which is beneficial in achieving flat broadband lineshape. The large stimulated emission cross-section and measured lifetime confirm the potentials of Tm-Bi codopants as luminescence sources for superbroadband near-infrared (NIR) optical amplifiers and tunable lasers. Planar optical waveguides were fabricated successfully in the codoped NGG glasses using K+-Na+ ion-exchange process.

©2011 Optical Society of America

1. Introduction

Substantial progress in the production of hydroxyl-free silica fibers (dry optical fibers) has led to an expansion of the transmission wavelength region from 1.2 to 1.7 μm wavelenth, thus it is attractive to explore luminescence sources for superbroadband near-infrared (NIR) optical amplifiers and tunable lasers operating in this entire low loss wavelength region [1,2]. Rare earth ions (e.g., erbium, thulium, holmium, and praseodymium) doped material systems play a significant role as optical amplifiers and laser sources at separate C-, L-, S-, E-, and O-band wavelength regions [3]. Novel oxide glass systems, including gallate, tellurite, and germanate materials, have been investigated to improve further the bandwidth and the quantum efficiency of specific rare earth luminescence [48]. Also, the wavelength/frequency source located in the third window (1.45-1.65 μm) has been explored and obtained from Tm-Er codoped configuration due to their compensatory NIR emission characteristic in this region [913]. Up to now, however, it remains a challenge to obtain superbroadband luminescence/amplification covering the entire expanded transmission window from rare earth ions doped materials due to their restricted gain bandwidth characteristics.

Recently, we have observed broadband NIR luminescence at around 1.20 μm based on the radiative Tm3+ 1G43H4 transition from tellurite glasses [14]. This wavelength region is near the zero-dispersion (~1.3 μm) of silica fibers, and optical amplification at this wavelength region can increase considerably the information capacity by utilizing dense wavelength division multiplexer (DWDM) [15]. Unfortunately a wavelength gap at around 1.3 μm remains for the Tm doped and Tm-Er codoped systems. Luminescence at 1.3 μm wavelength region can be obtained from Pr3+: 1G43H5, Nd3+: 4F3/24I13/2, or Dy3+: (6H9/2,6F11/2)→3H5 transition, but these transitions exhibit very low efficiency in oxide glasses resulting from the strong non-radiative decay, thus restricting their applications in oxide-based materials and devices [16]. Active bismuth ions show potentials in overcoming this shortcoming due to the broadband NIR emissions [17], and 1.2~1.3 μm wavelength operations have been demonstrated in glasses, crystals, and fiber/waveguide devices [1822]. However, it is shown that the emission line-shape and bandwidth are sensitive to the host matrix and the excitation wavelength, and further understanding of the luminescence origins is desirable [23].

In this work, we report the broadband NIR luminescence from Tm, Bi singly doped and Tm-Bi codoped sodium-germanium-gallate (NGG) glasses under argon laser and laser diode excitations. NGG glass is selected as the host primarily due to its relative lower phonon energy, broad transmission window, good mechanical properties and chemical durability. Low phonon energy is critical in obtaining efficient NIR emissions from Tm ions. Higher transparency compared to tellurite based glasses in the visible and the NIR regions mean better pump efficiency. The energy transfer processes involved are also proposed and discussed, and the optical amplification is evaluated. Planar waveguides were also fabricated using K+-Na+ ion-exchange process to demonstrate the practicality of the glass host.

2. Experimental

NGG glass samples were prepared by melting well-mixed batches of high-purity Na2CO3, GeO2, Ga2O3, BaCO3, and La2O3 powders following the procedures described in [7]. BaCO3 and La2O3 were added to improve the glass formation and the thermal properties. Dopants used were xTm2O3-yBi2O3 where x=0, y=0.1, 0.5, 1.0, 2.0; x=0.1, 0.5, 1.0, 2.0, y=0; and x=0.3, 0.5, 1.0, 2.0, y=1.0 in wt%. The as-prepared glasses were polished optically for optical measurements. Ion-exchange process was carried out in pure KNO3 molten bath at 380 °C for 4 h. The refractive indices of the glass samples and the refractive index change in the surface layer after ion-exchange were measured using a Metricon 2010 prism coupler at three different wavelengths (473, 632.8, and 1536 nm). The dependence of refractive index on wavelength is determined using Cauchy formula [14]. The absorption spectra were recorded using a Perkin Elmer UV-VIS-NIR Lambda 19 double beam spectrophotometer. The infrared photoluminescence (PL) spectra were recorded using a HORIBA Jobin Yvon Fluorolog-3 Spectrophotometer equipped with a near infrared photomultiplier tube (NIRPMT) and a PbS detector, under argon laser excitation wavelength of 476.5 nm, and laser diode wavelengths of 793 and 980 nm. The photoluminescence excitation (PLE) spectra were recorded using the same setup with a continuous wavelength xenon lamp as the excitation source. The emission decay curves were recorded using the same setup with a flash xenon lamp (3 μs) as the excitation source. All the measurements were carried out at room temperature.

3. Results and discussion

Figure 1 shows the optical absorption spectra of the Tm, Bi singly doped and Tm-Bi codoped NGG glass samples. Absorption bands located at 355, 470, 684, 791, 1211, and 1641 nm from Tm doped sample correspond to the transitions from ground state to specific excited states as indicated in Fig. 1. There is a broad absorption band at around 0.8 μm from the Bi doped sample which means that 793 nm laser diode is a suitable excitation source and, in particular, efficient Tm emission/amplification has been reported [5,6,24]. The result is in agreement with the PLE characteristic as shown in Fig. 1 inset, and an intense PLE band is indeed observed in this wavelength region. The strong absorption bands at short-wavelength side can be ascribed to the intrinsic absorption of the NGG matrix, and a red-shift as well as an enhanced absorption coefficient at around 793 nm wavelength is observed by adding Bi.

 figure: Fig. 1

Fig. 1 Optical absorption spectra of (A) Bi (0.5 wt%), (B) Tm (0.5 wt%) singly doped and (C) Tm(0.5 wt%)-Bi(1.0 wt%) codoped NGG glasses. Inset shows the normalized PLE spectra of (a) Bi doped NGG monitored at 1326 nm, (b) Tm doped NGG at 1460 nm, and (c) Tm-Bi codoped NGG at 1460 nm.

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Based upon the absorption spectra, the spontaneous transition properties of Tm in NGG samples were investigated using the Judd-Ofelt calculations [25,26]. The intensity parameters Ωt (t=2,4,6) was determined using a least-square-fitting of the theoretical and experimental oscillator strength values [27]. In this process, five (1G4, 3F2,3, 3H4, 3H5, and 3F4) and four (3F2,3, 3H4, 3H5, and 3F4) absorption bands were used for the calculations of Tm doped and Tm-Bi codoped NGG samples, respectively. It should be noted that the magnetic-dipole contribution to the absorption 3H53H6 was removed. The Ωt (t=2,4,6) obtained are 5.67×10−20, 0.19×10−20, 1.02×10−20 cm2 for Tm doped sample and 5.50×10−20, 0.61×10−20, 0.76×10−20 cm2 for Tm-Bi codoped sample, respectively. The larger Ω2 values in both samples compared to the values in other Tm doped host materials indicate a stronger asymmetry and covalent environment between RE ions and ligand in NGG glass matrix [14]. A minimal decrease of Ω2 with the addition of Bi was observed in our samples. This may be due to the presence of Bi with strong covalent Bi-O bond characteristic resulting in a reduced covalency of Tm ions and hence the Ω2 value [28]. The important parameters including spontaneous transition probabilities (Aed and Amd), branch ratio (β), radiative lifetime (τrad) were calculated and shown in Table 1 . The total transition probabilities for both 1.20 and 1.45 μm emissions are smaller than those in tellurite glasses are mainly due to the relative lower glass refractive index [14]. Both Aed and Amd show a slight decrease after the incorporation of Bi, and this is caused by the energy transfer from Tm to nearby Bi ions, since Bi ions exhibit intense absorptions at wavelength regions corresponding to the Tm 1G4 and 3H4 states.

Tables Icon

Table 1. Spectroscopic Parameters of Tm Doped and Tm-Bi Codoped NGG Glasses.

Figure 2(a) shows the emission spectra of Tm singly doped NGG glass samples under 476.5 nm wavelength excitation. Broadband NIR emissions located at 1.20 and 1.45 μm originated from the Tm3+: 1G43H4 and 3H43F4 transitions, respectively, were recorded. The emission located at around 1.64 μm can be attributed to the high energy side of Tm3+: 3F43H6 transition [23]. As the Tm concentration increases, the 1.20 μm emission intensity increases initially before decreasing, and the 1.45 μm emission also shows a similar tendency. The intensity enhancement for both emissions is primarily due to the increasing Tm concentration. At high dopant concentration, the intensity decrease for the 1.20 μm emission can be ascribed to the two cross relaxation processes [1G4-3F2]:[3H6-3F4] and [1G4-3H4]:[3H6-3H5]. The closer Tm-Tm separation at high doping level leads to easy occurrence of both processes and a depopulation of the Tm3+ 1G4 level [14] due to the matched energy difference. For the 1.45 μm emission, it can be ascribed to the cross relaxation process [3H4-3F4]:[3H6-3F4], Tm (3H4) is depleted at the expense of a pair of Tm populated at the 3F4 level. This process has been investigated in many host materials [6,2931], and the given explanations are confirmed by the lifetime results for both emissions; as shown in Fig. 2(a) inset the lifetime decrease with increasing Tm concentration. A normalized emission spectra of Tm doped NGG glasses with respect to the 1.20 μm emission is shown in the inset in Fig. 2(b), and it can be seen that higher Tm dopant concentration leads to flatter broadband luminescence. This means that in order to obtain a flat spectrum based on both emissions it is necessary to optimize the Tm dopant concentration. Figure 2(b) shows the emission spectra measured under 793 nm wavelength excitation and the results have similar characteristics to those in Fig. 2(a). This can be ascribed to the forward discussed cross relaxation process [3H4-3F4]:[3H6-3F4]. No emission at around 1.20 μm wavelength was observed, meaning that it is impossible to realize 1.20 μm emission by upconversion excitation. This is probably due to more ions residing at lower states and resulting in difficulty in achieving population inversion.

 figure: Fig. 2

Fig. 2 Near-infrared emission spectra of Tm doped NGG glasses under (a) 476.5 nm and (b) 793 nm wavelength excitation. Inset of Fig. 2 (a) shows the lifetimes of Tm 1.20 and 1.45 μm emissions as a function of Tm concentration. Inset of Fig. 2(b) shows the normalized (with respect to the 1.20 μm emission) near-infrared emission spectra of Tm singly doped NGG glasses under 476.5 nm wavelength excitation.

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Figure 3(a) shows the emission spectra from Bi doped samples under 793 nm wavelength excitation. An intense emission peak at 1326 nm with a broad 300 nm full-width at half-maximum (FWHM) bandwidth was observed. As Bi dopant concentration is increased the intensity increases rapidly initially before decreasing, and the lifetime decreases from 351 to 273, 226, and 206 μs, as shown in Fig. 3(a) inset. This can be ascribed to the cooperative upconversion process; one active Bi in the emission level decays non-radiatively to the ground state with another one from the emission level to the upper excited state, and the closer Bi-Bi distance at high concentration leads to easy occurrence of this process [32,33]. In addition, the diffusion towards the abnormal-valence ion impurities and/or defects in the host caused by high Bi doping concentration would raise the non-radiative rate of the active Bi emission state, suppressing the Bi 1.3 μm emission and reducing the lifetime of the emission level [3436]. Further studies on the possible origins are underway. The emission spectrum lineshape is similar to those under 476.5 nm wavelength argon laser and 980 nm laser diode excitations, as shown in Fig. 3(b). This indicates that the origin for the broadband luminescence is unchanged, and it is possible to develop compact superbroadband source from Bi-Tm codopant under 793 nm wavelength laser diode pumping.

 figure: Fig. 3

Fig. 3 (a) Near-infrared emission from Bi doped NGG glasses under 793 nm wavelength excitation. Inset shows the lifetime as a function of Bi concentration. (b) Normalized NIR emission with respect to the emission peak intensity under different excitation wavelengths.

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Figure 4(a) compares the emission spectra of Tm-Bi codoped NGG samples under 793 nm wavelength excitation. When Tm increases the active Bi relevant emission intensity decreases although the Bi concentration is constant, whilst the emission intensity of Tm 1.45 μm emission increases initially and then decreases as Tm concentration further increases. Superbroadband emission with a maximum FWHM of ~450 nm is achieved when Bi-Tm concentration ratio equal to 1:0.3. This result suggests that the presence of Tm quench not only the active Bi relevant luminescence but also the 1.45 μm emission at high Tm dopant concentration in Tm-Bi codoped NGG glasses. In order to further understand this phenomenon, the lifetime was measured that shows a decrease from 226 to 205, 194, 141, and 73 μs with the addition/increase of Tm, as shown in Fig. 4(a) inset.

 figure: Fig. 4

Fig. 4 (a) Near-infrared emission from Tm-Bi codoped NGG glasses with different concentration of Tm with Bi constant at 1.0 wt%. Inset shows the lifetime of Bi 1.3 μm emission as a function of Tm codopant concentration. (b) Normalized emission spectra of Tm (2.0 wt%), Bi (1.0 wt%) doped and Tm (2.0 wt%)-Bi(1.0 wt%) codoped NGG glasses with respect to the emission peak intensity under 980 nm wavelength excitation. Inset compares the emission spectra in wavelength region from 1.6 to 2.2 μm as recorded using a PbS detector.

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The results indicate that energy transfer takes place among active Bi and Tm ions. One leading energy transfer channel responsible for the decrease of the Bi emission intensity is the energy from Bi 1.30 μm emission level (the first excited state, ES1) which is transferred resonantly to nearby Tm in 3H5 state, and lead to a population of Tm (3F4) by rapid non-radiative decay. Another possible energy transfer channel is that the energy can be transferred directly from Bi (ES1) to Tm (3F4) with access energy dispersed in the NGG matrix in the form of phonons, since there remains a small overlap between the absorption band of Tm 3F43H6 transition and the active Bi 1.30 μm emission. To demonstrate and confirm the above energy transfer processes, the emission spectra of both Tm and Bi singly doped, and Tm-Bi codoped NGG samples under 980 nm wavelength excitation were measured and shown in Fig. 4(b) and inset. The observation of Tm 1.86 μm emission (3F43H6 transition) from Tm-Bi codoped NGG confirms the occurrence of both energy transfer processes, because under 980 nm wavelength excitation no Tm relevant emissions are available for the Tm singly doped sample. In addition, energy transfer Bi [ES1-GS]:Tm [3F4-3H4] may still contribute to the Tm relevant emissions; the energy from the Bi 1.30 μm emission level may be further absorbed by a nearby Tm3+ in the 3F4 state due to the long lifetime, and then populate the Tm3+(3H4) through the 3F43H4 transition. This energy transfer is supported by the observation of a weak enhancement in the emssion around 1.45 μm when Bi is codoped. The Tm 1.45 μm emission intensity decreases at high Tm dopant concentration in Tm-Bi codoped NGG samples [Fig. 4(a)]. One possible mechanism is the forward cross relaxation process [3H4-F4]:[3H6-3F4]. This self-quenching effect was also observed in Tm-Er codoped scheme [37,38]. Another possible mechanism is the resonant energy transfer from Tm (3H4) to Bi (the third excited state, ES3), since active Bi has matched energy level ES3 similar to that of Tm (3H4), as shown in Fig. 1. Also, in the back energy transfer process of Bi [ES1-GS]:Tm [3F4-3H4] one Tm 1.45 μm wavelength photon can be reabsorbed by a nearby Bi from the ground state to the first excited state ES1, because more Tm ions surround one Bi ion at high Tm-Bi concentration ratio [38]. Using the lifetime data measured in the above processes, the total energy transfer rate (WET) from active Bi to Tm can be calculated by WET=τxy1τx1, where τxy and τx are the lifetime of Bi 1.30 μm emission with and without Tm codopant [27]. WET increases from 453, 730, 2667, to 9274 s–1 as the Tm concentration increases from 0.3, 0.5, 1.0, to 2.0 wt%, respectively. The energy transfer efficiency (ηET) can be obtained using ηET=1(τxy/τx) [39], and the value increases from 9.3%, to 14.2%, 37.6%, and 67.7%, indicating that the energy transfer from active Bi to Tm is efficient. In addition, the increased Tm absorption coefficient of Tm-Bi codoped sample at 793 nm wavelength, as shown in Fig. 1, means improved pump efficiency. A simplified energy-level diagram of the active Bi and Tm is shown in Fig. 5 , where possible energy transfer processes are schematically illustrated.

 figure: Fig. 5

Fig. 5 Energy-level diagram of Tm-Bi in NGG glasses showing possible energy transfer processes. GS and ES represent ground state and each excited state of active Bi, respectively; ET1, ET2, ET3, and RET denote energy transfer processes Bi [ES1-GS]:Tm [3H6-3H5], Bi [ES1-GS]:Tm [3H6-3F4], Bi [ES1-GS]:Tm [3F4-3H4], Tm (3H4)→Bi (ES3), respectively; (i), (ii), (iii), (iv), and (v) stand for cross relaxation processes [1G4-3H4]:[3H6-3H5], [1G4-3F2]:[3H6-3F4], [3H4-3F4]:[3H6-3F4] that occur among Tm ions, [ES1-GS]:[ES1- ES3] among Bi ions, and Tm [3F4-3H4]:Bi [ES1-GS] between Tm and Bi ions, respectively.

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To further evaluate the gain performance of the NGG glass system it is important to determine the stimulated emission cross-section σem. By considering the Tm emissions that occur between two excited states, σem can be determined by scaling the emission spectrum from the following equation [14]:

σem(λ)=Aλ5I(λ)8πcn2λI(λ)dλ,
where I(λ), A, n, and c are the measured spectrum intensity, the radiative transition probability, the refractive index, and the light speed in vacuum, respectively. The peak value of σem equals 3.37×10−21 and 2.45×10−21 cm2 for Tm 1.20 and 1.45 μm emissions, respectively. σem dependence on active Bi emission can be described by Eq. (1), and the value at the band center can be estimated by Eq. (2) by assuming a Gaussian shaped emission band:
σem(λ0)=Aλ02η4πn2(ln2π)1/21Δν,
where λ0, η, and Δν represent the band center wavelength, the quantum efficiency, and the FWHM of the emission, respectively. The radiative transition probability A=τr1τmlow1 (τmlow is the measured lifetime at low dopant concentration), and the quantum efficiency can be estimated to be ~1 under very low doping concentration (such as 0.1 wt%) assuming that the non-radiative decay channel is sufficiently low due to the very large Bi-Bi distance. σem is obtained to be 4.59×10−20 cm2 at 1321 nm. This value is larger than the reported values in bismuth doped silicate glasses (1.52×10−20 cm2) [39], bismuth doped aluminum germanium oxide glasses (4.435×10−20 cm2) [40], and bismuth- tantalum codoped germanium oxide glasses (1.59×10−20 cm2) [32], and the large product σemτm (1.61×10−23 cm2 s), which is proportional to the amplification gain and inversed laser oscillation threshold, suggests that the NGG host matrix is an excellent gain medium for the realization of broadband amplifiers and tunable lasers.

Planar optical waveguides in Tm-Bi codoped NGG glasses were fabricated using thermal K+-Na+ ion-exchange process. Figure 6 shows the measurement results of the reflected light intensity dependence on refractive index, and the observed peaks represent the detected modes of the planar waveguides. Here, three, double, and single modes were demonstrated at 473, 632.8, and 1536 nm wavelengths, respectively. Figure 6 inset shows the calculated refractive index profile as a function of the diffusion depth at 473 nm wavelength using the inverse Wentzel-Kramer-Brillouin method [7]. The surface refractive index of the waveguide n 0 is calculated to be 1.746, and the indeed profile can be represented by a complementary error function n(x)=nsub+Δnerfc(x/d), where nsub is the substrate refractive index, x is the depth, ∆n (=n 0n sub) is the maximum surface index change, and d is the effective diffusion depth [41]. The index change ∆n can be controlled by either the ion-exchange temperature or the time. The results confirm that the codoped glass material is promising for the fabrication of broadband waveguide amplifiers and lasers in the 1.0-1.7 μm wavelength region.

 figure: Fig. 6

Fig. 6 Intensity of reflected light versus refractive index at 1536, 632.8, and 473 nm laser wavelengths. Inset shows the index profile at 473 nm of the K+-Na+ ion-exchanged glass waveguide carried out at 380 °C for 4 h.

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4. Conclusions

In conclusion, superbroadband emission in the 1.0-1.7 μm wavelength region has been demonstrated in Tm-Bi codoped NGG glasses under 793 nm wavelength excitation. The addition of Bi causes a decrease in the spontaneous transition probability of the associated Tm emissions. However, efficient energy transfer from active Bi to Tm ions compensates not only the shortcomings but also results in an improvement in the emission in the entire 1.0-1.7 μm wavelength region. Planar optical waveguides were also fabricated using ion-exchange process. Our results indicate that this glass system is promising for superbroadband luminescence sources and optical waveguide amplifiers/lasers operating in the entire telecommunication transmission window. Further investigations and experiments are underway to realize its full potential.

Acknowledgements

The authors are grateful to Prof Jianrong Qiu and Prof Mingying Peng for useful discussions. This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, under project CityU 119708.

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Figures (6)

Fig. 1
Fig. 1 Optical absorption spectra of (A) Bi (0.5 wt%), (B) Tm (0.5 wt%) singly doped and (C) Tm(0.5 wt%)-Bi(1.0 wt%) codoped NGG glasses. Inset shows the normalized PLE spectra of (a) Bi doped NGG monitored at 1326 nm, (b) Tm doped NGG at 1460 nm, and (c) Tm-Bi codoped NGG at 1460 nm.
Fig. 2
Fig. 2 Near-infrared emission spectra of Tm doped NGG glasses under (a) 476.5 nm and (b) 793 nm wavelength excitation. Inset of Fig. 2 (a) shows the lifetimes of Tm 1.20 and 1.45 μm emissions as a function of Tm concentration. Inset of Fig. 2(b) shows the normalized (with respect to the 1.20 μm emission) near-infrared emission spectra of Tm singly doped NGG glasses under 476.5 nm wavelength excitation.
Fig. 3
Fig. 3 (a) Near-infrared emission from Bi doped NGG glasses under 793 nm wavelength excitation. Inset shows the lifetime as a function of Bi concentration. (b) Normalized NIR emission with respect to the emission peak intensity under different excitation wavelengths.
Fig. 4
Fig. 4 (a) Near-infrared emission from Tm-Bi codoped NGG glasses with different concentration of Tm with Bi constant at 1.0 wt%. Inset shows the lifetime of Bi 1.3 μm emission as a function of Tm codopant concentration. (b) Normalized emission spectra of Tm (2.0 wt%), Bi (1.0 wt%) doped and Tm (2.0 wt%)-Bi(1.0 wt%) codoped NGG glasses with respect to the emission peak intensity under 980 nm wavelength excitation. Inset compares the emission spectra in wavelength region from 1.6 to 2.2 μm as recorded using a PbS detector.
Fig. 5
Fig. 5 Energy-level diagram of Tm-Bi in NGG glasses showing possible energy transfer processes. GS and ES represent ground state and each excited state of active Bi, respectively; ET1, ET2, ET3, and RET denote energy transfer processes Bi [ES1-GS]:Tm [3H6-3H5], Bi [ES1-GS]:Tm [3H6-3F4], Bi [ES1-GS]:Tm [3F4-3H4], Tm (3H4)→Bi (ES3), respectively; (i), (ii), (iii), (iv), and (v) stand for cross relaxation processes [1G4-3H4]:[3H6-3H5], [1G4-3F2]:[3H6-3F4], [3H4-3F4]:[3H6-3F4] that occur among Tm ions, [ES1-GS]:[ES1- ES3] among Bi ions, and Tm [3F4-3H4]:Bi [ES1-GS] between Tm and Bi ions, respectively.
Fig. 6
Fig. 6 Intensity of reflected light versus refractive index at 1536, 632.8, and 473 nm laser wavelengths. Inset shows the index profile at 473 nm of the K+-Na+ ion-exchanged glass waveguide carried out at 380 °C for 4 h.

Tables (1)

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Table 1 Spectroscopic Parameters of Tm Doped and Tm-Bi Codoped NGG Glasses.

Equations (2)

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σ e m ( λ ) = A λ 5 I ( λ ) 8 π c n 2 λ I ( λ ) d λ ,
σ e m ( λ 0 ) = A λ 0 2 η 4 π n 2 ( ln 2 π ) 1 / 2 1 Δ ν ,
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