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Tm3+ doped lead silicate glass single mode fibers for 2.0 μm laser applications

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Abstract

Tm3+ doped lead silicate glasses with good thermal stability were prepared by the melt-quenching method. Based on the absorption and emission spectra, Judd-Ofelt intensity parameters, absorption and emission cross sections, gain spectra, and σe × FWHM were calculated and analyzed. These results suggest that Tm3+ doped lead silicate glasses are promising as mid-infrared laser materials. Tm3+ doped lead silicate glass single mode (SM) fibers with cladding diameter of 125 μm and core diameter of 8.5 μm were then fabricated by the rod-in-tube technique. The Tm3+ doping concentration reached as high as 4.545 × 1020 ions/cm3. ~2.0 μm amplified spontaneous emission (ASE) was realized in a 3.5-cm-long as-drawn SM fiber when pumped by a homemade single mode 1560 nm fiber laser. The results indicate that these Tm3+ doped lead silicate glass single mode fibers are promising fiber material for 2.0 μm fiber laser applications.

© 2016 Optical Society of America

1. Introduction

Rear-earth-doped glasses and fibers have attracted much attention to develop mid-infrared lasers operating in the eye-safe 2.0 μm wavelength region due to their wide applications such as laser surgery, remote sensing, laser imaging, environmental monitoring, coherent laser radar systems and pump sources for mid-infrared lasers and optical parametric oscillators (OPOs) [1–6]. Among the rare-earth ions, Tm3+ and Ho3+ are well-known for the generation of 2.0 μm lasers. However, Ho3+ cannot be pumped directly by readily available commercial 808 or 980 nm laser diodes (LDs) because of the lack of an appropriate ground absorption band. Tm3+ is considered particularly appropriated to generate 2.0 μm lasers, because it can be efficiently pumped by 808 nm LD. Moreover, it is an ideal mid-infrared luminescent center which has broadest wavelength tuning range from 1800 to 2100 nm among all the rare earth ions [7].

To date, 2.0 μm lasers have been realized in Tm3+ doped various glass fibers such as fluoride glasses [8], silica glasses [9], tellurite glasses [10], and germanate glasses [11]. The obvious defects of fluoride glasses are inferior mechanical and chemical properties. And tellurite glasses are limited in power values due to the low laser damage threshold. As for germanate glasses, the rare-earth oxides tend to rapidly cluster in host and therefore promote microscopic segregation and crystallization, which will limit the doping concentration of rare earth [12]. In addition, the raw materials of germanate glasses are comparatively high-cost. Silicate glasses have excellent thermal and mechanical characteristics, which are promising material in realizing 2.0 μm laser. Compared to silica glasses, they have lower phonon energy, higher solubility of rare earth ions due to their less defined structure [13]. What is more, the slope efficiency in the silicate fiber lasers can be much higher than that in other glass fibers [14,15].

In 2009, J. Geng et al. have reported single-frequency laser operation near 2 μm region by using the Tm3+ doped multicomponent silicate double cladding fiber [14]. Compared with the traditional silicate glasses, which compose of silica, alumina, alkalis or alkaline earth ions, the heavy metal silicate glasses have attracted much attention. The heavy metal ions such as Bi2O3, PbO are added to improve the luminescent properties [16]. More recently, X. Liu et al. reported 2 μm laser output in Tm3+ doped lead silicate double cladding fiber [17]. The highly Tm3+ doped single mode fibers are promising in applications that require high gain and high power from a short piece of active optical fiber. However, there is no literature report the Tm3+ doped lead silicate single mode optical fibers. In this work, Tm3+ doped lead silicate glasses were fabricated by the melt-quenching method. The thermal and Raman properties, fluorescence properties, optical parameters, and gain properties were systematically investigated. Then Tm3+ doped lead silicate glass single mode fibers were fabricated by the rod-in-tube technique. The Tm3+ doping concentration reaches as high as 4.545 × 1020 ions/cm3. ~2.0 μm amplified spontaneous emission was realized in a 3.5-cm-long as-drawn SM fiber when pumped by a home-made 1560 nm fiber laser.

2. Experimental

Lead silicate glasses with the molar compositions of (58-x)SiO2-32PbO-5K2O-5BaO-xTm2O3 (x = 0.5, 1.0, 1.5), which are hereafter demoted as SPKB-x (x = 0.5, 1.0, 1.5), were prepared by the conventional melting-quenching technique. The extra pure anhydrous reagents (99.99% minimum) of SiO2, Pb3O4, K2CO3, BaCO3, Tm2O3 were used. For each glass, well-mixed raw materials (30 g) were melted at 1240 °C for 1 h in a covered alumina crucible and bubbled with high purity oxygen, maintained at 0.1 L/h for 20 min. Subsequently, the melt was cast into a preheated steel mold and annealed at 390 °C before they were cooled to room temperature. The well annealed samples were cut and optically polished to the size of 20 mm × 10 mm × 1.5 mm for optical property measurements.

Bulk core glass (SPKB-1.0) was melted with well-mixed raw materials (600 g) at 1240 °C for 5 h in a covered alumina crucible and bubbled with high purity oxygen, maintained at 0.5 L/h for 1.5 h. After which the melt was stirred and clarified to remove bubbles and stripes. Bulk cladding glass was also fabricated by using the same method with composition of 63SiO2-27PbO-5K2O-5BaO (SPKB). The rod-in-tube fiber drawing technique was used for fiber fabrication with a preform designed for the requirement of the SM optical fiber. A diameter about 2 mm of core glass rod was first drawn from the polished cylindrical core glass in a drawing tower. The designed cylindrical cladding glass with a hole in axial direction was mechanically polished, followed by a chemical etching process to obtain a high surface quality. Thereafter, the core glass rod was also etched and inserted into the cladding glass hole. The assembled preform was handed in the furnace of the fiber drawing tower. Then continuously lead silicate glass SM fibers were drawn around 660 °C inside the drawing tower under N2 controlled atmosphere.

The characteristic temperatures of glass transition (Tg), onset crystallization peak (Tx), and top crystallization (Tp) were investigated by using a Netzsch STA 449C Jupiter differential scanning calorimeter (DSC) under Ar atmosphere at a heating rate of 5 °C/min. The Raman spectrometer (Bruker, Switzerland) was used with a 532 nm laser as the excitation sourece. The refractive indexes of samples were recorded on a prism coupling apparatus (Metricon Model 2010). The density was tested by Archimedes’ liquid-immersion method in distilled water. Absorption spectra of the glass samples were performed on a Perkin-Elmer Lambda 900 UV-Vis-NIR spectrophotometer. IR transmittance was measured using a Vector-33 FTIR spectrometer (Bruker, Switzerland). Fluorescence spectra of the glass samples were measured by a computer controlled Triax 320 type spectro-fluorimeter (Jobin-YvonCorp) with a lock-in amplifier upon excitation of an 808 nm LD. The lifetime of Tm3+: 3F4 level was obtained from the first e-folding time of emission intensity in the decay curve recorded with a digitizing oscilloscope. The amplified spontaneous emission spectrum was measured under excitation of a home-made SM 1560 nm fiber laser. All measurements were carried out at room temperature.

3. Results and discussions

3.1 Thermal stability, structure, optical absorption, and Judd-Ofelt analysis

Figure 1 shows the DSC curves of SPKB-1.0 and SPKB glasses. It can be noted that no crystallization peak appears in cladding glass (SPKB) as the temperature increases to 800 °C, indicting the high stability of the host glass against crystallization. Thermal stability is one of the most important properties for glasses and fiber drawing. Based on the DSC curves, ΔT = Tx-Tg is often used as an important parameter to evaluate glass thermal stability. Generally, a larger ΔT above 100 °C will facilitate the fiber drawing [18]. It is found that the ΔT is 210 °C in core glass (SPKB-1.0), which is much larger than 100 °C, indicating a wide operating temperature range and excellent glass stability against crystal nucleation and growth during the fiber drawing. Moreover, the Saad-Poulain criterion referred as S = (Tp-Tx)(Tx-Tg)/Tg was applied to further evaluate the thermal glass stability [19]. The S value of core glass is up to 30.73, which is larger than that of barium gallo-germanate glass (8.41) [5], fluogermanate glass (9.45) [19], tellurite glass (4.40) [6], and fluorophosphates glass (1.89) [20]. The larger ΔT and S values of the core glass make it show a lower crystal nucleation and growth rate during a reheating process, which is significantly essential to draw crystal-free glass fibers.

 figure: Fig. 1

Fig. 1 DSC curves of the SPKB and SPKB-1.0 glasses.

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The Raman spectrum was utilized to analyze the structure of core glass, and the vibrational spectrum of core glass was shown in Fig. 2. It can be found that the largest phonon energy only extends to 955 cm−1, much lower than that of silicate glass (1100 cm−1) [13], and lead silicate glass (1020 cm−1) [21]. The lower phonon energy could reduce the probability of non-radiative relaxation and thus be helpful to Tm3+ 1.8 μm luminescence. The inset of Fig. 2 shows the Raman bands in the frequency region from 800 to 1200 cm−1. Five peaks centered at 840, 882, 949, 1010, and 1048 cm−1 are observed by Gaussian fitting, which are attributed to the SiO4 tetrahedral with four, three, two, one non-bridging oxygen ion and with four bridging oxygen, respectively [22]. The centre of the strong Raman band of the lead silicate glass at 955 cm−1 seems to be too high for isolated SiO4 tetrahedra and some degree of polymerization of the tetrahedral is required [23]. Therefore, the silicate network was depolymerized with the PbO doped, which is favor to improve the solubility of rare-earth ions.

 figure: Fig. 2

Fig. 2 Raman spectrum of SPKB-1.0 glass. The inset shows the Raman bands in the frequency region from 800 to 1200 cm−1 with five peaks are observed by Gaussian fitting.

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Figure 3 presents the absorption spectra of SPKB-x glasses in the wavelength range from 400 to 2200 nm. The characteristic absorption bands corresponding to the transitions from the ground state to excited states of Tm3+ ions are labeled in the spectra. The absorption transitions centered at 1650, 1209, 795, 682, and 472 nm are attributed to the transition from the ground state 3H6 to the exited states of 3F4, 3H5, 3H4, 3F2,3, and 1G4, respectively. The broad absorption bands around 795 and 1650 nm indicate that commercially available 808 LD or a ~1.5 μm fiber laser can be used as the pump sources. Meanwhile, there is no obvious change in the position of the Tm3+ absorption peaks, but the intensity enhances with the increment of Tm2O3. The inset of Fig. 3 shows the transmittance spectrum of core glass (SPKB-1.0) from 2.5 to 5.5 μm. It can be seen, the maximum transmittance reaches as high as 84% and the infrared transparency extends up to 5.0 μm. The broad absorption near 3 μm is corresponding to the stretching vibration of free OH- groups, and the glass absorption coefficient, αOH (cm−1) can be obtained by using the Eq. (1) [24]:

αOH=1llnT0T
where l is the thickness of the sample, T0 and T are the transmission at 2600 and 3000 nm, respectively. The OH- absorption coefficient of core glass is 1.04 cm−1, lower than that of Tm3+ doped lead silicate glass fiber (1.70 cm−1) [17].

 figure: Fig. 3

Fig. 3 Absorption spectra of Tm3+ doped lead silicate glasses. The inset shows the transmittance spectrum of the SPKB-1.0 glass.

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Judd-Ofelt (J-O) theory is used to calculate spectroscopic parameters of Tm3+ doped lead silicate glass based on absorption spectra [25,26]. The experimental oscillator strength (fexp) can be determined by the following expression (2):

fexp=2.303mc2πe2Nlλ¯2OD(λ)dλ
where m and e are the mass and charge of the electron, respectively, c is the velocity of light in vacuum, N is the number density of rare-earth ions, l is the sample thickness,λ¯is the central wavelength, and OD(λ) is the optical density. The theoretical oscillator strength (fexp) of an electron-dipole transition from the initial state(S,L)J to the final state (S,L)Jcan be calculated by the following Eq. (3):
fcal=8π2mc3hλ¯(2J+1)(n2+2)29nΩt|(S,L)JU(t)(S,L)J)|2
where h is the Planck constant, n is the refractive index of the host glass, and J is the total angular momentum quantum number. The term (S,L)JU(t)(S,L)J)stands for the reduced matrix element of the tensor operators, which is generally insensitive to the host materials and these values of Tm3+ used in this work was quoted from Ref [27]. Then the J-O intensity parameters Ωt (t = 2, 4, 6) of Tm3+ can be calculated by a least squares fit to the values of fexp using Eq. (3). The Ωt (t = 2, 4, 6) of Tm3+ in various lead silicate glass hosts doped with different alkali and alkaline earth ions are listed in Table 1. The root-mean-square derivation (δrms) between fexp and fcal of Tm3+ is 8 × 10−8, indicating the results are reliable. Generally, Ω2 strongly depends on the ligand symmetry of the local environment at the Tm3+ ion sites and it will increase with the decrement of the symmetry between Tm3+ ions and the ligand fields [28]. As the cation field strength of alkali and alkaline earth ions rises, the distortion around Tm3+ ion increases and induces a larger Ω2 [29]. The Ω2 in the present glass is larger than that in the other silicate glasses, indicating lower ligand symmetry due to the high polarizability of Pb2+ and higher covalency of Tm-O bond in the present glass, resulted from the co-existence of various polyhedrons such as [SiO4], [PbO3], and [PbO4] [30]. Ω4 and Ω6 decrease but Ω2 increases with increased covalency between the rare-earth and oxygen ions in silicate glasses.

Tables Icon

Table 1. J-O intensity parameters of Tm3+ in various lead silicate glass systems

3.2 Fluorescence spectra, cross sections, and gain properties

Figure 4 presents near-infrared fluorescence spectra in SPKB-x glasses pumped by 808 nm LD. The broadband 1.8 μm emission from 3F43H6 transition can be obtained. Moreover, with the increment of Tm2O3 concentration, 1.8 μm emission intensity increases gradually and reaches the maximum value, when 1.0 mol% Tm2O3 is doped. This can be accounted for shorting the distance between Tm3+ ions, which increases the cross relaxation energy transfer (3H4 + 3H63F4 + 3F4) probability. And then the 1.8 μm emission intensity reduces sharply with further increasing Tm2O3 concentration because of the enhanced energy transfer rate from 3F4 to OH- or impurities [31]. The inset of Fig. 4 shows the lifetime of Tm3+: 3F4 level in the SPKB-1.0 glass. It was measured to be 0.36 ms, which is larger than that in Tm3+ doped silicate glass (0.25 ms) [32]. The higher lifetime is favorable to achieve laser action.

 figure: Fig. 4

Fig. 4 Emission spectra of Tm3+ doped lead silicate glasses pumped by an 808 nm LD. The inset shows fluorescence decay curve of 1850 nm emission in the SPKB-1.0 glass.

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In order to further estimate the 1.8 μm emission properties of Tm3+ doped lead silicate glass (SPKB-1.0), the absorption and emission cross sections are calculated. Based on measured absorption spectra, the absorption cross section (σa) can be determined by the Beer-Lambert law [28]:

σa(λ)=2.303Nllog(I0I)
where log(I0/I) is absorptivity from absorption spectra, N is the rare-earth doping concentration, and l is the glass sample thickness. The emission cross section (σeFL) can be calculated from the fluorescence spectra by using the Fuchtbauer-Ladenburg equation [33]:
σeFL=Ar8πn2cλ5I(λ)λI(λ)d(λ)
where Ar is the spontaneous emission probability of the transition, c is the speed of light, n is the refractive index, I(λ) is the emission intensity, and λ is the wavelength.

Figure 5 shows the calculated absorption and emission cross sections of the core glass (SPKB-1.0). It is found that the maximum values of σa andσeFLare 2.8 × 10−21 cm2 at 1650 nm and 5.67 × 10−21 cm2 at 1865 nm, respectively. The core glass (SPKB-1.0) has higher emission cross section than that of Tm3+ doped ZBLAN glass (2.4 × 10−21 cm2) [34], other lead silicate glass (5.35 × 10−21 cm2) [16], silicate glass (3.89 × 10−21 cm2) [35], and tellurite glass (5.40 × 10−21 cm2) [36]. In addition, the full-width at half-maximum (FWHM) of 1.8 μm emission is approximately 215 nm. FWHM × σepeakis an important parameter for estimating gain properties, its larger value generally representing a wider bandwidth and a higher gain character [19]. The FWHM × σepeakof Tm3+:3F43H6 in core glass is 12.19 × 10−26 cm3, which is larger than that of fluoride glass (7.68 × 10−26 cm3), fluorophosphate glass (10.53 × 10−26 cm3), germanate glass (11.08 × 10−26 cm3), silicate glass (9.69 × 10−26 cm3), and tellurite glass (11.34 × 10−26 cm3) [32,36].

 figure: Fig. 5

Fig. 5 (a) Absorption and emission cross sections and (b) the calculated gain coefficient of the SPKB-1.0 glass.

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Based on the calculated σa andσeFL, it is valuable to calculate the wavelength dependence of net gain as a function of population inversion for the upper laser level so as to determine the gain property qualitatively [33]:

G(λ)=N[pσe(λ)(1p)σa(λ)]
where population inversion p is assigned to the concentration ration of Tm3+ in the 3F4 level and N stands for the total concentration of Tm3+ ions. The gain coefficients with various p values ranging from 0 to 1 are calculated for 3F43H6 transition of the core glass, which is shown in Fig. 5(b). Obviously, the gain becomes positive when p is more than 0.2, indicating that a low pumping threshold will be required for ~2.0 μm laser operation in these Tm3+ doped lead silicate SM fibers. In addition, the maximum gain coefficient reaches 2.57 cm−1 at 1866 nm, which is larger than that of Tm3+ doped silicate glass (1.5 cm−1) [13], showing promising applications for efficient 2.0 μm fiber lasers.

3.3 Single-mode fiber parameters and ASE spectrum

The measured and fitted refractive indices of the core and cladding glasses were shown in Fig. 6(a). The experimental data are used to calculate the coefficients in the Sellmeier dispersion equation by using a least-squares fitting code to establish S and λ0 in the following expression [37]:

 figure: Fig. 6

Fig. 6 (a) The refractive indices of the core and cladding glasses and numerical aperture of the fiber as a function of wavelength; (b) Cutback measurement of Tm3+ doped lead silicate glass fiber.

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n2(λ)=1+Sλ2λ2λ02

The Sellmeier coefficients S and λ0 are calculated to be 1.958 and 159.8 nm for the cladding glass, and 1.968 and 158.7 nm for the core glass. Then a continuous refractive indices data as a function of wavelength was calculated based on the fitted Sellmeier parameters, as shown in Fig. 6(a).

The fiber preform was designed for the requirement of SM optical fibers by the typical rod-in-tube technique based on the bulk core glass (SPKB-1.0) and well-matched bulk cladding glass (SPKB). Continuous Tm3+ doped lead silicate glass fibers were successfully drawn in-house with cladding diameter of 125 μm and core diameter of 8.5 μm by the rod-in-tube technique. The numerical aperture (NA) of the fiber can be calculated by the Eq. (8) [18]:

NA=ncore2ncladding2
where ncore and ncladding are the refractive indices of the core and cladding glasses, respectively. The NA of this fiber is 0.100 at 1958 nm. When the normalized frequency V of a fiber is less than or equal to 2.405, only the fundamental mode (LP01) can be propagated in the active fiber and the single-mode operation can occur. The V of a fiber is given as [31]:
V=2πaλNA
where a is the radius of the fiber, λ is the wavelength, NA is the numerical aperture. Assuming that V is equal to 2.405, the cut-off wavelength for the as-drawn fiber is calculated to be 1.59 μm, indicating that once the operation wavelength is larger than 1.59 μm, the condition for single-mode operation is satisfied [31]. The wavelength of the Tm3+ fiber laser is usually around 1.8 μm, giving a result of V < 2.405 which indicates that the as-drawn Tm3+ doped lead silicate optical fibers can be single mode operated. Figure 6(b) shows the propagation loss measurement of the as-drawn fibers. The propagation loss value at 1310 nm is 4.17 dB/m by the cutback method, which is lower than that of Tm3+ doped lead silicate double cladding fiber (7 dB/m) [17]. Considering that the fibers were fabricated by the rod-in-tube technique which might create inherent problems such as core-clad interface irregularities, the propagation loss of these fibers is relatively low.

Figure 7(a) presents the schematic diagram of the Tm3+ doped lead silicate SM fiber ASE source using the pump excitation at 1560 nm. The inset of Fig. 7(a) shows the photomicrograph of Tm3+ doped lead silicate SM fiber. The core of the fiber was uniformly doped with 1.0 mol % Tm2O3 (4.545 × 1020 ions/cm3). The high doping concentration of Tm3+ ions provides sufficient gain in a short active fiber length, which can also be favorable for avoiding optical nonlinearity. A 3.5-cm-long active fiber was pumped by a homemade watt-class 1560 nm single mode fiber laser. An optical spectrum analyzer was employed to monitor the output characteristic. Figure 7(b) shows the ASE spectrum of Tm3+ doped lead silicate SM fiber upon excitation of 1560 nm fiber laser. An intense emission peak at 1958 nm with a FWHM of 23 nm originated from Tm3+: 3F43H6 transition is realized in the as-drawn fiber. With the pump power of 1.0 W, 0.05 μW power was obtained inside the ASE. And the residual pump power is 500 mW. It can be clearly observed that the ASE spectrum is much smaller than that of fluorescence (seen in Fig. 4). The spectral narrowing effect is very obvious especially in this high gain fiber [38]. The results suggest that the as-drawn highly Tm3+ doped lead silicate SM fibers are promising active medium for 2.0 μm fiber laser.

 figure: Fig. 7

Fig. 7 (a) Schematic diagram of the Tm3+ doped lead silicate SM fiber ASE source using the pump excitation at 1560 nm; inset: the photomicrograph of Tm3+ doped lead silicate SM fiber cross section. (b) ASE spectra of as-drawn fiber in the wavelength range of 1900–2000 nm.

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4. Conclusion

In conclusion, we systematically investigated the thermal properties, structure, and optical parameters of Tm3+ doped lead silicate glasses. Then the bulk core glass and well-matched cladding glass were prepared. The continuous Tm3+ doped lead silicate single mode fibers were drawn by the rod-in-tube technique. The propagation loss of the as-drawn fibers at 1310 nm was measured to be 4.17 dB/m by the cutback method. The calculated maximum gain coefficient of the as-drawn fiber reaches 2.57 cm−1 at 1866 nm. The ASE at ~2.0 μm in a 4.545 × 1020 ions/cm3 Tm3+ doped lead silicate SM fiber was realized when pumped by a homemade single-mode 1560 nm fiber laser. The results indicate that the as-drawn Tm3+ doped lead silicate glass SM fibers are promising fiber materials for 2.0 μm fiber laser applications.

Acknowledgments

This research was supported by China National Funds for Distinguished Young Scientists (61325024), the High-level Personnel Special Support Program of Guangdong Province (2014TX01C087), Fundamental Research Funds for the Central Universities (2015ZP019), the Science and Technology Project of Guangdong (2015B090926010), China State 863 Hi-tech Program (2014AA041902) and NSFC (61535014 and 51302086).

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Figures (7)

Fig. 1
Fig. 1 DSC curves of the SPKB and SPKB-1.0 glasses.
Fig. 2
Fig. 2 Raman spectrum of SPKB-1.0 glass. The inset shows the Raman bands in the frequency region from 800 to 1200 cm−1 with five peaks are observed by Gaussian fitting.
Fig. 3
Fig. 3 Absorption spectra of Tm3+ doped lead silicate glasses. The inset shows the transmittance spectrum of the SPKB-1.0 glass.
Fig. 4
Fig. 4 Emission spectra of Tm3+ doped lead silicate glasses pumped by an 808 nm LD. The inset shows fluorescence decay curve of 1850 nm emission in the SPKB-1.0 glass.
Fig. 5
Fig. 5 (a) Absorption and emission cross sections and (b) the calculated gain coefficient of the SPKB-1.0 glass.
Fig. 6
Fig. 6 (a) The refractive indices of the core and cladding glasses and numerical aperture of the fiber as a function of wavelength; (b) Cutback measurement of Tm3+ doped lead silicate glass fiber.
Fig. 7
Fig. 7 (a) Schematic diagram of the Tm3+ doped lead silicate SM fiber ASE source using the pump excitation at 1560 nm; inset: the photomicrograph of Tm3+ doped lead silicate SM fiber cross section. (b) ASE spectra of as-drawn fiber in the wavelength range of 1900–2000 nm.

Tables (1)

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Table 1 J-O intensity parameters of Tm3+ in various lead silicate glass systems

Equations (9)

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α O H = 1 l ln T 0 T
f exp = 2.303 m c 2 π e 2 N l λ ¯ 2 O D ( λ ) d λ
f c a l = 8 π 2 m c 3 h λ ¯ ( 2 J + 1 ) ( n 2 + 2 ) 2 9 n Ω t | ( S , L ) J U ( t ) ( S , L ) J ) | 2
σ a ( λ ) = 2.303 N l log ( I 0 I )
σ e F L = A r 8 π n 2 c λ 5 I ( λ ) λ I ( λ ) d ( λ )
G ( λ ) = N [ p σ e ( λ ) ( 1 p ) σ a ( λ ) ]
n 2 ( λ ) = 1 + S λ 2 λ 2 λ 0 2
N A = n c o r e 2 n c l a d d i n g 2
V = 2 π a λ N A
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