An Investigation of the Photonic Application of TeO₂-K₂TeO₃-Nb₂O₅-BaF₂ Glass Co-Doped with Er₂O₃/Ho₂O₃ and Er₂O₃/Yb₂O₃ at 1.54 μm Based on Its Thermal and Luminescence Properties

Abstract

A glass composition using TeO₂-K₂TeO₃-Nb₂O₅-BaF₂ co-doped with Er₂O₃/Ho₂O₃ and Er₂O₃/Yb₂O₃ was successfully fabricated. Its thermal stability and physical parameters were studied, and luminescence spectroscopy of the fabricated glasses was conducted. The optical band gap, E_opt, decreased from 2.689 to 2.663 eV following the substitution of Ho₂O₃ with Yb₂O₃. The values of the refractive index, third-order nonlinear optical susceptibility (χ⁽³⁾), and nonlinear refractive index (n₂) of the fabricated glasses were estimated. Furthermore, the Judd–Ofelt intensity parameters ${\Omega }_t (t=2,4,6)$, radiative properties such as transition probabilities (A_ed), magnetic dipole-type transition probabilities (A_md), branching ratios (β), and radiative lifetime (τ) of the fabricated glasses were evaluated. The emission cross-section and FWHM of the ⁴I_13/2→⁴I_15/2 transition around 1.54 μm of the glass were reported, and the emission intensity of the visible signal was studied under 980 nm laser excitation. The material might be a useful candidate for solid lasers and nonlinear amplifier devices, especially in the communications bands.

1. Introduction

Due to their unique optical characteristics, glasses are frequently utilized as optical materials. In recent years, there has been an obvious increase in interest in the possible applications of tellurite glasses due to their intriguing and significant optical features. Tellurite glasses have been used to fabricate a wide range of devices, such as planar waveguides, nanowires, and optic amplifiers [1,2,3,4]. Numerous studies have been conducted to examine the physical characteristics of several tellurite (TeO₂) glass combinations. TeO₂ glasses have attracted some technical and scientific interest because of their many applications [5]. Technological optical fiber devices, lasers, optical fibers, solar cells, sensors, memory-switching devices, gas sensors, optoelectronics, and optical waveguide applications have all demonstrated significant potential for using these glasses [5,6,7,8]. Furthermore, due to TeO_2’s excellent nonlinear optical characteristics, low melting point, superior chemical stability, and elevated index of refraction, it has drawn greater interest as a glass former than other glass formers such as silicates and phosphate [9,10,11]. Given their huge transparent window, great refractive index, and outstanding stability, TeO₂ glasses containing a heavy metal oxide—such as Nb₂O₅, which is a heavy metal tellurite glass—are appealing for the additional development of infrared lasers and amplifiers [12]. Furthermore, niobic TeO₂ glasses have an elevated third-order nonlinear optical susceptibility, which makes them a viable option for nonlinear fiber devices such as all-optical switches [12]. The contents of non-bridging oxygen (NBO) and TeO₃ units both rise when modifiers such as BaO₂ disrupt the random arrangement of glasses [13]. Moreover, the alkaline earth metal element barium (Ba) has a high basicity, a large ionic radius, and a higher atomic number. Consequently, the presence of Ba in the TeO₂ glass framework modifies the glass’s construction and enhances its chemical stability, density, gloss, and refractive index [14,15]. When fluorine ions are added to TeO₂ glass, the glass’s formation range is increased, its viscosity is decreased, its degree of transparency is enhanced, and its moisture resistance is increased [16]. Also, adding fluorine to TeO₂ glasses disintegrates the network of the glassy system due to fluorine’s electronegativity being greater than that of oxygen [16]. This implies that the network’s configuration of the glass in these kinds of glasses is impacted by the fluoride’s replacement of oxygen. When the framework is altered, a lot of fundamental characteristics also change. Unlike both oxygen and fluorine matrix structures, an oxide–fluoride glass matrix might offer rare earth ions a unique home. As a result, oxyfluoride glasses with a high proportion of rare earth elements are novel beneficial substances [17]. Fluoride glasses have a minimal cost limit and a wide spectrum range, which makes them ideal for optical fiber applications involving sensors. It is commonly known that increasing a glass matrix’s metal fluoride compounds improves its transparent nature and refractive index [9]. The prospective utilization of co-doping of glasses with rare earth ions in solid-state lasers, three-dimensional displays, and optical amplifiers has garnered significant interest in the last decade [18]. The rare earth ions erbium (Er³⁺), ytterbium (Yb³⁺), and holmium (Ho³⁺) have received the most attention. Er³⁺, Ho³⁺, and Yb³⁺ ion insertion into the matrix of glass allows for the production of up-conversion luminescence when it is exposed to mid-infrared (MIR) rays. A high-power laser diode energy of 980 nm might be used to actively excite Er³⁺ ions. According to the mutual concentration of these ions, a transfer of energy process among them might then alter the up-conversion emission intensity [19]. Moreover, erbium-doped glasses are extensively utilized in a diversity of optical implementations, mostly in the areas of eye-safe lasers and optical amplifiers for fiber networking [18,20,21]. Furthermore, the observable up-conversion emission can be amplified by co-doping the host substance with Ho₂O₃/Er₂O₃ or Yb₂O₃/Er₂O₃ couplings. This is because of a greater absorbing cross-section and greater energy transmission operations from Ho³⁺ to Er³⁺ ions or from Yb³⁺ to Er³⁺ ions, respectively [18,22,23]. The present work aimed to prepare TeO₂-based glass (70TeO₂-15K₂TeO₃-10Nb₂O₅-5BaF₂) that was co-doped with Er₂O₃/Ho₂O₃ and Er₂O₃/Yb₂O₃ and to investigate the impact of these rare earth oxides on the thermal and optical properties of the glasses. Differential scanning calorimetry and a double-beam spectrophotometer were utilized to study these characteristics. Furthermore, optical spectrum measurements were conducted to estimate the optical factors. The energy levels of the glass co-doped with Er₂O₃/Yb₂O₃ were determined, and the branching ratios (β), radiative lifetimes (τ_rad), electric dipole-type transition probabilities (A_ed), magnetic dipole-type transition probabilities (A_md), and Judd–Ofelt intensity factors Ω_t (t = 2, 4, 6) were calculated.

The primary goal of this work was to investigate how the amount of Yb³⁺ and Ho³⁺ affects the spectroscopic characteristics of Er³⁺-co-doped TeO₂ glasses. This will help maximize the transition’s gain and emission cross-section between ⁴I_13/2→⁴I_15/2 and will also determine whether or not these glasses are suitable as optical glasses for laser and fiber amplifiers. Utilizing measurements of the absorption spectra and McCumber theory, the absorbing, emitting, and gain cross-sections of the ⁴I_13/2→⁴I_15/2 transition were derived at approximately 1.54 µm. Finally, the emission intensity of the visible signal was studied under 980 nm laser excitation. The FWHM of the ⁴I_13/2→⁴I_15/2 transition of the glass was reported. Furthermore, we estimated the nonlinear refractive index and third-order susceptibility of fabricated glass which have good transmission with multiple absorption peaks in the near-infrared wavelength spectrum. Therefore, fabricated glass is a unique property that can used in nonlinear devices.

2. Experimental Section

The TeO₂ glasses with the composition 70TeO₂-15K₂TeO₃-10Nb₂O₅-5BaF₂ (in mol%) co-doped with Er₂O₃/Ho₂O₃ and Er₂O₃/Yb₂O₃ were produced using a melt-quenching process. All the starting chemicals were procured from Aldrich and were 99.99% pure. The particulars of their compositions are displayed in

Table 1. The composition, density ρ, and molar volume V_m of the TKNB glassy system co-doped with Er₂O₃/Ho₂O₃ and Er₂O₃/Yb₂O₃.

Sample Code	Composition (mol%)	Ρ (g/cm3) ± 0.001	Vm (cm3/mol) ± 0.0056
TKNB1	70TeO2-15K2TeO3-10Nb2O5-5BaF2-30,000 ppm Ho2O3-30,000 ppm Er2O3	4.9585	35.373
TKNB2	70TeO2-15K2TeO3-10Nb2O5-5BaF2-30,000 ppm Yb2O3-30,000 ppm Er2O3	4.9622	35.353

.

After mixing, the mixture was heated for 25 min at 950 °C in a platinum crucible within a furnace. After that, a graphite mold was filled with the extremely viscous melt. Following two hours of annealing at 270 °C, the quenched glass was gradually cooled to room temperature (RT). Figure 1 shows pictures of the sample glasses as they were produced. The samples were cut and polished to 2.1 mm thick. To investigate the thermal properties of these glasses, a DSC Shimadzu 50 with a resolution ± 1.0 °C at a heating rate of 10 °C/min over a temperature range of 550 °C was employed. Toluene, a known-density immersing solution (0.8669 g/cm³), was employed using the Archimedes method with a resolution ± 0.001 g/cm³ to determine the glass sample’s density at RT. In the 190–2500 nm wavelength range with a resolution of 1nm, the optical absorbing and transmission spectra were attained utilizing a JASCO V-570 spectrophotometer.

3. Results and Discussion

3.1. Thermal Characteristics

The DSC thermograms for the 70TeO₂-15K₂TeO₃-10Nb₂O₅-5BaF₂ glass co-doped with Er₂O₃/Ho₂O₃ and Er₂O₃/Yb₂O₃ at a rate of 10 °C/min are shown in Figure 2. These graphs demonstrate that at the glass transition, distinct endothermic peaks are seen, followed by an exothermic crystallization peak.

Table 2. Thermal parameters of the TKNB glass system co-doped with Er₂O₃/Ho₂O₃ and Er₂O₃/Yb₂O₃.

Sample Code	Tg (°C) ± 1.0	Tc (°C) ± 0.1	Tp (°C) ± 1.0	∆T (°C) ± 1.0	H ± 1.0	KSP ± 1.0
TKNB1	333	422	447	89	0.267	8.56
TKNB2	332	419	442	87	0.262	7.62

lists the glass transition temperature (T_g), crystallization start temperature (T_c), and peak crystallization temperature (T_p) for the investigated glasses.

The glassy nature of the current samples was confirmed based on the curve forms. Additionally, T_g offers details regarding the glass network’s interconnectivity and binding strength. It is understood that T_g increases as the glass’s interconnectivity and binding strength grow [24]. These values of (T_g) are close to the tellurite-based glasses [25]. The distinction between T_c and T_g is employed to determine the thermal stability (ΔT) [26,27] of the produced glasses (ΔT = T_c − T_g). The glass’s excellent thermal stability is indicated by the significant difference between T_c and T_g. Another method for calculating the glass’s stability would be to use Sestak’s estimation of the Hruby index, H = ΔT/T_g [28,29]. The ∆T and H are shown in Table 2, and these are important for determining the glass devitrification process [27]. The next relation could be employed to evaluate the value of the factor K_SP, which is associated with glass’s stability versus crystallization [28,29,30,31]:

$$K_{\mathrm{S}\mathrm{P}}={\displaystyle \frac{\left(T_{\mathrm{p}}-T_{\mathrm{c}}\right)\left(T_{\mathrm{p}}-T_{\mathrm{g}}\right)}{T_{\mathrm{g}}}}$$

(1)

The K_SP magnitudes of the produced glasses are documented in Table 2. It can be seen that all these values of the thermal stability parameters (∆T, H, K_SP) of the TKNB glass co-doped with Er₂O₃/Yb₂O₃ (TKNB2) are slightly lower than those of the TKNB glass co-doped with Er₂O₃/Ho₂O₃ (TKNB1). This may suggest a decrease in rigidity in the glassy matrix due to the replacement of Ho₂O₃ with Yb₂O₃. On the other hand, this decrease in thermal stability due to the substitution of Ho₂O₃ with Yb₂O₃ might be credited to the following causes: (i) the low bond strengthening Yb-O (387.7 kJ mol⁻¹) in contrast to Ho-O (606 kJ mol⁻¹) [32]; or (ii) the cation radius of Yb³⁺ (2.28 Å) being slightly lower than the cation radius of Ho³⁺ (2.33 Å). In addition to reducing the length of bonds and producing a high coulombic force of attraction between opposing ions, cation’s radius is directly proportional to polarizability [33].

3.2. Density and Molar Volume

The densities of the examined glasses were determined by utilizing Archimedes’ principle. We used toluene (ρ₀ = 0.8669 g/cm³) as the immersing fluid. The density (ρ) was computed employing the equation presented below after the weights of the glasses were first measured separately in the air (W_a) and then in the previously mentioned liquid (W_l).

$$\rho ={\displaystyle \frac{W_a}{W_a-W_l}}{\rho }_0$$

(2)

Using the formula V_m = M_w/ρ, where M_w is the glass sample’s molecular weight, the molar volume (V_m) was computed. The values of ρ and V_m are listed in Table 1. It is worth mentioning that the substitution of Ho₂O₃ with Yb₂O₃ leads to a slight increase in glass density, while the V_m slightly decreases. This phenomenon could be ascribed to both the higher ρ and the higher M_w of Yb₂O₃ (9.17 g/cm³, 394.08 g/mol) compared with Ho₂O₃ (8.41 g/cm³, 377.86 g/mol). Given that the ρ is inversely correlated to the V_m and is proportionate to the average M_w, it is generally predicted that the two quantities will behave in opposition to one another.

3.3. Optical Properties

The spectra of optical transmission of the TKNB glasses co-doped with Er₂O₃/Ho₂O₃ and Er₂O₃/Yb₂O₃ are provided in Figure 3, which indicates that the fabricated glasses are highly transmissible in the range of UV–VIS–NIR wavelengths. Figure 4 demonstrates several peaks in the spectra, which are due to the presence of Er³⁺, Ho³⁺, and Yb³⁺ ions in the TKNB glass system co-doped with Er₂O₃/Ho₂O₃ and Er₂O₃/Yb₂O₃. The absorption bands corresponding to the transition from ⁴I_15/2 to the ground states ⁵I₇(Ho³⁺), ⁴I_13/2, ⁵I₆(Ho³⁺), ⁴I_11/2(Er³⁺) + ⁴F_5/2(Yb³⁺), ⁴I_9/2, ⁴F_3/2, ⁴S_3/2, ²H_11/2, ⁴F_7/2, ⁴F_3/2, and ²G_9/2 can be attributed to the following wavelengths: 1950, 1534, 1155, 970, 803, 655, 540, 520, 490, 450, and 420 nm. These correspond with the reported wavelengths in similar glasses [21,22,23].

The absorption coefficient (α) of the examined glasses is computed using the following formula [32,33,34]:

$$\alpha ={\displaystyle \frac{1}{d}}\mathit{ln}\left({\displaystyle \frac{I_0}{I_t}}\right)=2.303{\displaystyle \frac{OD}{L}}$$

(3)

where L is the thickness of the sample before and after the light traverses the sample, its intensities are I₀ and I_t, and OD is the optical density.

The spectrum of absorption of glasses with varying compositions is used to study changes in the optical band gap (E_opt) and refractive index (n). The E_opt, which is the difference between the highest energy in the valence band and the lowest energy in the conduction band, and temperature both affect how many electrons are excited toward the conduction band. When a material’s E_opt falls within the ranges of 0 to 4 eV, it is classified as a semiconductor. A substance is classified as an insulator if its E_opt value is substantial, falling between 4 and 12 eV. These statistics are crucial for designing semiconductor devices, since the energy gap determines the electrical and optical characteristics of the device. Clarifying the transformation and electronic band construction of substances has been shown to have been an extremely beneficial effect in research on the optical absorbing edge in the UV area. Using this connection, the optical band gap E_opt of the specimens is computed from their absorption patterns [35].

$$\alpha h\nu =b{\left(h\nu -E_{\mathrm{o}\mathrm{p}\mathrm{t}}\right)}^s$$

(4)

where b is a constant, hv represents the energy of the incoming photon, and α is the absorbing coefficient. An index called s, which is equivalent to ½, 2, 3/2, or 3 for directly allowed, indirectly allowed, directly forbidden, and indirectly forbidden transitions, respectively, is used to describe the optical absorption processes. In this instance, indirect transmissions in substances are covered by this formula with s = 2, which is employed to characterize experimental findings for amorphous substances [33,34,35]. The change in (αhν)^1/2 vs. hν for indirect transitions is shown in Figure 5. By means of extrapolation of the linear fitting of the curve in the UV–VIS nm range across the hν axis at (αhν)^1/2 = 0, the E_opt of the fabricated glasses was determined.

Table 3. The Sellmeier coefficients, molar refractivity R_m, molar polarizability α_m, metallization criterion M, and optical band gap E_opt of the TKNB glass system co-doped with Er₂O₃/Ho₂O₃ and Er₂O₃/Yb₂O₃.

Sellmeier Coefficients					Rm (Mol−1) ± 0.0465	αm (Å−3) ± 0.0064	M ± 0.0008	Eopt (eV) ± 0.001
Sample Code	A	B	C	D
TKNB1	3.195	0.7393	0.2552	0.01397	17.449	6.921	0.507	2.689
TKNB2	3.205	0.7512	0.2593	0.01695	17.558	6.964	0.503	2.663

lists each value of E_opt.

It is discovered that the E_opt values of the two glass samples are lower than the pristine TeO₂ glass’s 3.79 eV value, determined in a previous study [36]. The values of E_opt for the TKNB glass co-doped with Er₂O₃/Yb₂O₃ (TKNB2) are slightly lower than the TKNB glass doped with Er₂O₃/Ho₂O₃ (TKNB1). The BO in the glass-forming system is altered by the addition of rare earth ions, and any alteration in BO, including the creation of NBO, alters the absorption properties, which in turn reduces the optical band gap. The TeO₂ matrix’s coordination number varies when rare earth is added. The sample matrix’s structural organization and chemical makeup both have an impact on the optical band gap [37]. When Ho₂O₃ is replaced with Yb₂O₃, the slight decrease in E_opt is because of the substitution of the Ho-O bond with the Yb-O bond. The decrease in E_opt with the addition of Yb₂O₃ can be elucidated by the change in electron density [38]. When Yb₂O₃ is added to a glass sample instead of Ho₂O₃, the amount of NBO may increase, which consequently decreases the E_opt [37]. The value of the materials’ molar mass is the cause of the increase in NBO [39]. In comparison to the molar mass of Ho₂O₃, the molar mass of Yb₂O₃ is higher. Thus, there is an increase in NBO atoms, which lowers the value of E_opt for the Yb₂O₃-containing glass sample. Additionally, the E_opt falls under the category of semiconductor substances, ranging from 2.689 eV to 2.663 eV [40].

The following equations characterize the extinction coefficient (k) and n of the examined glasses [41]:

$$k={\displaystyle \frac{\alpha \lambda }{4\pi }}$$

(5)

$$R={\displaystyle \frac{{\left(n-1\right)}^2+k^2}{{\left(n+1\right)}^2+k^2}}; n=\left({\displaystyle \frac{1+R}{1-R}}\right)+{\left({\displaystyle \frac{4R}{{\left(1-R\right)}^2}}-k^2\right)}^{{\displaystyle \frac{1}{2}}}$$

(6)

where R is the reflectance. Figure 6 and Figure 7 show the computed values of the k and n of the examined glasses, respectively. Both the variations in structure and the fluctuation in the incident wavelength (λ) can affect the n and k. According to Figure 7, the n drops as the λ increases. The following figure shows the various optical factors that are determined using the n and k spectrum distributions vs. λ for the tested glasses.

The data for the refractive indexes in Figure 7 are fitted to a three-term Sellmeier equation:

$$n^2\left(\lambda \right)=A+B/\left(1-{\displaystyle \frac{C}{{\lambda }^2}}\right)+D/\left(1-{\displaystyle \frac{E}{{\lambda }^2}}\right)$$

(7)

where A, B, C, D, and E are the glass substance dispersion factors (Sellmeier coefficients), and λ is the wavelength in μm. The n is influenced by both smaller and greater energy gaps from electronic adsorption, as indicated by the first and second terms. The final term indicates the refractive index-lowering effect of network absorbing [42]. The square of twice the IR transmitting edge could be utilized for calculating the E [42,43]. Table 3 displays Sellmeier’s factors, which were obtained by fitting the experimental data [44] with the use of Equation (7). The Wemple and DiDomenico (WDD) single-oscillator model was utilized to study the n dispersion [45]. The relation between n and hν can be represented as follows utilizing this model:

$${\left(n_0^2-1\right)}^{-1}={\displaystyle \frac{E_o}{E_d}}-{\displaystyle \frac{1}{E_d{E}_o}}{\left(h\nu \right)}^2$$

(8)

where E_d is the dispersion energy, which is an estimate of the oscillator strength or average strength of the interband optical transition, and E_o is the oscillator energy. Equation (8) may be used to derive the E_o and E_d through graphing (n²−1)⁻¹ versus (hv)², as Figure 8 illustrates. Applying the fit of the linear parameters, E_o and E_d may be calculated based on the graph. The intercepts and slopes of the arcs yield the Ed and Eo values.

When hv→0, the Wemple–DiDomenico dispersion relationship, Equation (8), is extrapolated to obtain the static refractive index (n₀) of the as-prepared glasses, which yields the following formula:

$$n_0=\sqrt{1+E_d/E_o}$$

(9)

The determined values of E_o, E_d, and n_o are recorded in

Table 4. Dispersion and nonlinear optical parameters of the TKNB glass system co-doped with Ho₂O₃/Er₂O₃ and Yb₂O₃/Er₂O₃.

Sample Code	Dispersion Parameters			Nonlinearity Parameters
	Eo (eV)	Ed (eV)	no ± 0.001	χ(1)	χ(3) × 10−13 (esu)	n2 × 10−12 (esu)
TKNB1	8.59	25.28	1.98	0.23407	5.1	9.68
TKNB2	8.75	25.95	1.99	0.23596	5.26	9.97

. It can be seen that the value of n_o for TKNB glass doped with Er₂O₃/Yb₂O₃ (TKNB2) is higher than that of TKNB glass doped with Er₂O₃/Ho₂O₃ (TKNB1) due to the higher density of the TKNB2 sample.

Important factors for the production of optical equipment, such as fiber optic and laser substances, are the n, molar polarizability (α_m), and molar refraction (R_m). Consequently, the following formulas [34] are applied to calculate these properties of the examined glasses:

$$R_m=\left({\displaystyle \frac{n_o^2-1}{n_o^2+2}}\right)V_m$$

(10)

$${\alpha }_m=\left({\displaystyle \frac{3}{4\pi N_A}}\right)R_m$$

(11)

where N_A is Avogadro’s number. The R_m and α_m values are recorded in Table 3. These values for TKNB glass doped with Er₂O₃/Yb₂O₃ (TKNB2) are higher than those of glass doped with Er₂O₃/Ho₂O₃ (TKNB1). This provides a qualitative explanation for the rise in refractive indices observed when Yb₂O₃ replaces Ho₂O₃. We conclude that a rise in molar refraction may be the cause of the observed pattern, which is a rise in the oxide ion polarizability, accompanied by an elevation in n. This relationship [31] was utilized to estimate the metallization criterion (M) for the as-prepared glasses:

$$M=1-{\displaystyle \frac{R_m}{V_m}}$$

(12)

The metallic or non-metallic character of the substance is indicated by the M. M < 0 suggests a metallic nature of the substances, whilst a value of M > 0 indicates an insulating nature. The M values are listed in Table 3 and were between 0.503 and 0.507. Therefore, the produced glasses demonstrated an insulating nature [30,31]. On the other hand, the exchange of Ho₂O₃ with Yb₂O₃ causes an upsurge in the valance band’s width and a decrease in M and, thus, a decrease in E_opt. As shown in Table 3, the glass doped with Er₂O₃/Ho₂O₃ had higher values of M and E_opt, while the glass co-doped with Er₂O₃/Ho₂O had smaller values of M and E_opt.

The nonlinear optical parameters can be determined based on Miller’s rule. The dispersion of the optical linear susceptibility χ⁽¹⁾ and the third-order nonlinear optical susceptibility χ₍₃₎ are, respectively, deduced based on their empirical relations [46].

$${\chi }^{\left(1\right)}={\displaystyle \frac{n_o^2-1}{4\pi }}$$

(13)

$${\chi }^{\left(3\right)}={\left[{\chi }^{\left(1\right)}\right]}^4\times 1.7\times {10}^{-10}$$

(14)

The nonlinear refractive index n₂ of the examined glasses relates to the third-order nonlinear optics χ⁽³⁾ and static refractive index n_o, which can be determined using the following equation [47,48]:

$$n_2={\displaystyle \frac{12\pi }{n_o}}{\chi }^{\left(3\right)}$$

(15)

The calculated values of χ⁽¹⁾, χ⁽³⁾, and n₂ are listed in Table 4. It can be observed that these values are higher for the TKNB glass doped with Yb₂O₃/Er₂O₃ (TKNB2) than for the TKNB glass doped with Ho₂O₃/Er₂O₃ (TKNB1). The reported results demonstrate an increase in the nonlinear susceptibility, and the examined glasses show high values of optical linear susceptibility, χ⁽¹⁾, replacing an atom with an atomic radius that is smaller than the removed one. This also applies when replacing Ho³⁺ (atomic radii = 2.33 Å) with Yb³⁺ (atomic radius = 2.28 Å) in the TKNB glass doped with Yb₂O₃/Er₂O₃ (TKNB2), which causes an increase in the χ⁽³⁾ value. This is dependent on the value of the third-order nonlinear optical susceptibility, χ⁽³⁾, and nonlinear refractive index, n₂, which suggest that these glasses can be used in nonlinear optical devices, especially in the communication bands.

3.4. Absorption Spectra and Judd and Ofelt Analysis

Figure 9 displays the Er³⁺/Yb³⁺-co-doped glasses and their absorption spectra in the UV–VIS–NIR wavelength between 400 and 1800 nm. The 4f¹¹-4f¹¹ transitions of Er³⁺ and Yb³⁺ are responsible for all of the bands. Three bands at 800, 975, and 1530 nm are detectable in the infrared section of the spectrum (Figure 9a). The first and third bands result from Er³⁺ transitions between ⁴I_9/2 and ⁴I_13/2, which are ground states, and ⁴I_15/2. With an increased optical density, the second band represents an overlapping absorption among the two transmissions, ²F_7/2→²F_5/2 (originating from Yb³⁺ ion) and ⁴I_15/2→⁴I_11/2 (originating from Er³⁺ ion). As seen in Figure 9b, the UV–visible region of the absorbed spectrum demonstrates well-resolved lines, which are ascribed to the transition of Er³⁺ ions in the ground state, ⁴I_15/2 to the ⁴F_3/2, ⁴F_3/2, ⁴F_5/2, ⁴F_7/2, ⁴H_11/2, ⁴S_3/2, and ⁴F_9/2. These absorption bands were assigned and located in accordance with Carnall et al. [49] and all relevant research.

The absorption spectra of this glass medium (TKNB2) were subjected to a Judd and Ofelt (JO) investigation in order to ascertain their spectroscopic characteristics. Below is a quick synopsis of the JO investigation.

The well-known theory developed by JO in 1962 makes it possible to compute the probability of an electric dipole transition between rare earth ion energy levels in diverse contexts [50]. Theoretically, the electric $S_{ed}^{cal}$ and magnetic ($S_{md}$) dipole line strengths can be utilized for describing the radiative transmissions of the first J level to the succeeding J′ level in the 4fn conformation of rare earth ions [51,52].

The calculated line strength $S_{ed}^{calc}\left(SLJ,S^{\prime }{L}^{\prime }{J}^{\prime }\right)$ between the initial state $J,$ described by $\left(S,L,J\right),$ and the final state $J^{\prime },$ given by ($S^{\prime },L^{\prime },J^{\prime }),$ can be expressed using the following relation [53,54]:

$$S_{ed}^{calc}\left(SLJ,S^{\prime }{L}^{\prime }{J}^{\prime }\right)=\sum _{t=\mathrm{2,4},6}{\mathsf{\Omega }}_t{\left|〈SLJ‖U^{\left(t\right)}‖S^{\prime }{L}^{\prime }{J}^{\prime }〉\right|}^2$$

(16)

The influence of the host glasses on the luminescence intensity is represented by the JO intensity factors, ${\mathsf{\Omega }}_{\mathrm{t}} (t=2, 4, 6)$, and the doubly reduced matrix elements of grade t among the statuses with the quantum numbers of $(S,L,J)$ and $\left(S^{\prime },L^{\prime },J^{\prime }\right)$ are ${\left|〈SLJ‖U^{\left(t\right)}‖S^{\prime }{L}^{\prime }{J}^{\prime }〉\right|}^2$ [51]. The reduced matrix elements, which may be obtained in the literature [52,53,54,55,56,57,58], are only dependent on the angular momentum of the Er³⁺ states. The reduced matrices are defined as the total of the relevant matrix elements for two or more manifolds.

Table 5. Reduced matrix element values for Er³⁺ absorption transitions.

Transition from 4I15/2 to	${‖{\mathit{U}}^{\left(2\right)}‖}^2$	${‖{\mathit{U}}^{\left(4\right)}‖}^2$	${‖{\mathit{U}}^{\left(6\right)}‖}^2$
4I13/2	0.0194	0.1173	1.4332
4I9/2	0.0	0.1694	0.0101
4F9/2	0.0	0.548	0.4769
4S3/2	0.0	0.0	0.2369
2H11/2	0.8120	0.4669	0.0951
4F7/2	0.0	0.1447	0.6348

provides the matrix element values for each Er³⁺ absorption band.

Meanwhile, the measured electric dipole line strength $S_{ed}^{meas}\left(SLJ,S^{\prime }{L}^{\prime }{J}^{\prime }\right)$ can also be calculated from the absorption spectra (Figure 1) [56,57,58] by using the following equation:

$$S_{ed}^{meas}=\left[{\displaystyle \frac{9n}{{\left(n^2+2\right)}^2}}4\pi {ϵ}_0{\displaystyle \frac{3ch\left(2J+1\right)}{8{\pi }^3{e}^2}}\times {\displaystyle \frac{2.303}{LN\overline{\lambda }}}{\int }_{J\to J^{\prime }}^{j}OD\left(\lambda \right)d\lambda \right]-n·S_{md}\left(SLJ,SL{J}^{\prime }\right)$$

(17)

where J is the total angular momentum quantum number of the ground state $J={\displaystyle \raisebox{1ex}{$15$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$, e is the charge of the electron, and N is the ion content (ions/cm³). The average wavenumber of the absorbing band is denoted by $\overline{\lambda } \left(\mathrm{nm}\right)$, the refractive index of the host with respect to $\overline{\lambda }$ is represented by n, L is the thickness of the sample under study (L = 1 mm), and ${\int }_{J\to J^{\prime }}^{j}OD\left(\lambda \right)d\lambda$ symbolizes the experimentally determined integrated optical density in the respective wavelength ranges.

The magnetic dipole transitions $S_{md}\left(SLJ,SL{J}^{\prime }\right)$ contribute as follows [59]:

$$S_{md}\left(SLJ,SL{J}^{\prime }\right)={\left({\displaystyle \frac{h}{4\pi mc}}\right)}^2{\left|〈SLJ‖L+2S‖SL{J}^{\prime }〉\right|}^2$$

(18)

The symbols m and c stand for the electron mass and light velocity, respectively. The magnetic dipole matrix elements between the LS-coupled states are represented by $〈S‖L+2S‖SL{J}^{\prime }〉$ [60].

Table 6. Values of average wavelengths, refractive index, and integrated absorption coefficients’ electric and magnetic dipole line strengths for Er³⁺/Yb³⁺-co-doped tellurite glass (TNBK2).

Transition from 4I15/2 to	$\overline{\mathit{\lambda }}$ $\left(\mathbf{n}\mathbf{m}\right)$	$\mathit{n}$	$\int \mathit{O}\mathit{D}\left(\mathit{\lambda }\right)\mathit{d}\mathit{\lambda }$ $\left(\mathbf{n}\mathbf{m}\right)$	${\mathit{S}}_{\mathit{e}\mathit{d}}^{\mathit{c}\mathit{a}\mathit{l}}$ $\left(\mathbf{p}{\mathbf{m}}^2\right)$	${\mathit{S}}_{\mathit{e}\mathit{d}}^{\mathit{m}\mathit{e}\mathit{s}}$ $\left(\mathbf{p}{\mathbf{m}}^2\right)$	${\mathit{S}}_{\mathit{m}\mathit{d}}$ $\left(\mathbf{p}{\mathbf{m}}^2\right)$
4I13/2	1530	1.9975	17.88414	1.2087	1.1855	0.7148
4I9/2	800	2.0155	1.06715	0.3254	0.2938	0
4F9/2	650	2.0286	4.02307	1.3446	1.3483	0
4S3/2	545	2.0480	0.46898	0.1558	0.1845	0
2H11/2	520	2.0554	7.03041	2.8796	2.8801	0
4F7/2	490	2.0669	1.70483	0.6949	0.7341	0
				${\delta }_{rms}=0.0361\times {10}^{-20} {\mathrm{c}\mathrm{m}}^2$

demonstrates that the only transition with a contribution of the magnetic dipole, $S_{md}=0.7148\times {10}^{-20} {\mathrm{c}\mathrm{m}}^2$, in the case of Er³⁺ ions is the ⁴I_15/2→⁴I_13/2 ($selection rules: ∆J=1)$.

The measured line strengths $(S_{ed}^{meas}$) of the electric dipoles were utilized to calculate the values of the JO intensity factors ${\mathsf{\Omega }}_2,{\mathsf{\Omega }}_3, \mathrm{a}\mathrm{n}\mathrm{d} {\mathsf{\Omega }}_6$. If the JO factors produce a column vector $\mathsf{\Omega }$, and the double-decreased matrix elements produce an $n\times 3$ matrix $A$, where $n$ is the number of transmissions to fit and $3$ matches the three JO factors, then the measured electric dipole line strength can be transcribed as a $1\times n$ column vector. The equality among Equations (13) and (14) can be articulated as $S_{ed}^{meas}=A.\mathsf{\Omega }$. The group of JO factors was obtained from the matrix $\mathsf{\Omega }={(A^T.A)}^{-1}.A^T.S_{ed}^{meas}$, where $A^T$ is the transposition of matrix $A$. This matrix-based process is perfect for computations that are carried out on a computer.

The optimal modification was performed, accounting for the initial six transitions and yielding the following values: ${\mathsf{\Omega }}_2=2.387\times {10}^{-20} {\mathrm{c}\mathrm{m}}^2$, ${\mathsf{\Omega }}_4=1.881\times {10}^{-20} {\mathrm{c}\mathrm{m}}^2$, and ${\mathsf{\Omega }}_6=0.657\times {10}^{-20} {\mathrm{c}\mathrm{m}}^2$. These computed ${\mathsf{\Omega }}_t$ values were utilized in Equation (12) to obtain the values of $S_{ed}^{calc}$. Table 6 provides an overview of the outcomes of some of the calculated parameters, as well as the $S_{ed}^{calc}$ and $S_{ed}^{meas}$ absorption line strengths for the Er³⁺/Yb³⁺-co-doped TKNB2 sample.

Table 6 presents the values of some of the factors that were employed in computation, as well as the $S_{ed}^{meas}$ and $S_{ed}^{calc}$ absorbing line strengths for the Er³⁺/Yb³⁺-co-doped TKNB2 sample.

The formula below indicates the root mean square ${\delta }_{rms}$ difference between the predicted and observed line strengths of the transitions, which serves as a gauge for the fit’s accuracy:

$${\delta }_{rms}={\left(\sum _{i=1}^{N_{trans}}{\displaystyle \frac{\left(S_{ed}^{meas}\left(i\right)-S_{ed}^{cal}\left(i\right)\right)}{\left(N_{trans}-3\right)}}\right)}^{1/2}$$

(19)

where $N_{trans}$ is the number of strong absorption transitions.

The value found in this study, ${\delta }_{rms}=0.0361\times {10}^{-20} {\mathrm{c}\mathrm{m}}^2$, is tiny when compared with values found in other kinds of glasses [61,62,63,64,65,66].

The calculated ${\mathsf{\Omega }}_t$ parameters for different glasses accord well with those reported in existing works. The ${\mathsf{\Omega }}_{\mathrm{t}}$ measurements of Er³⁺ ions in several other common glasses [67,68,69,70,71,72,73] are shown in

Table 7. Judd–Ofelt parameters and spectroscopic quality factor ($Q={\mathsf{\Omega }}_4/{\mathsf{\Omega }}_6$) in different host materials, along with reported Er³⁺-doped systems.

Glass Host	${\mathsf{\Omega }}_2\times {10}^{-20} {\mathbf{c}\mathbf{m}}^2$	${\mathsf{\Omega }}_4\times {10}^{-20} {\mathbf{c}\mathbf{m}}^2$	${\mathsf{\Omega }}_6\times {10}^{-20} {\mathbf{c}\mathbf{m}}^2$	$\mathit{Q}={\mathsf{\Omega }}_4/{\mathsf{\Omega }}_6$	Reference
Er3+/Yb3+	2.387	1.881	0.657	2.8630	[Present work]
ZBLAN	2.73	1.40	1.10	1.2727	[77,80]
Boro-tellurite	4.232	0.779	0.612	1.2729	[78,81]
Tellurite	5.6	2.11	0.73	2.8904	[79,82]
TLNT	6.48	1.82	1.27	1.4331	[80,83]

. The intensity parameters comprise two terms, based on the JO theory [74]. Firstly, the symmetry and distortions associated with the constructional alteration in the presence of rare earth ions are described by the crystal field parameter. The other represents the covalency between the ligand anions and doped rare earth ions, which is connected to the excited opposite parity electronic states and 4f radial integral states of wave functions. Further, the covalent bonding of binder anions and rare earth ions (less ionic in nature) in the host and the symmetry of the immediate environment surrounding them are correlated with the intensity parameter ${\mathsf{\Omega }}_2$. The degree of asymmetry surrounding rare earth sites increases with an increasing value of ${\mathsf{\Omega }}_2$, indicating a greater covalency between the metal and ligand link. The bulk characteristics of the glass framework, such as stiffness, viscosity, and basicity, are primarily described by the intensity factors ${\mathsf{\Omega }}_4$ and ${\mathsf{\Omega }}_6$, which are also influenced by the acidity and alkalinity of the host substance [75]. The host material’s hardness and basicity increase and decrease, respectively, with higher values of ${\mathsf{\Omega }}_4$ and ${\mathsf{\Omega }}_6$. Table 7 demonstrates that all three intensity parameters exhibit an upward trend of ${\mathsf{\Omega }}_2$ > ${\mathsf{\Omega }}_4$ > ${\mathsf{\Omega }}_6$, which aligns with the majority of results from previous studies. A greater asymmetry and superior covalence are indicated by the higher ${\Omega }_2$ value compared with ${\Omega }_4$ and ${\Omega }_6$ [76]. In the meantime, all three Er³⁺ intensity characteristics that were measured in the current study were reduced to within the values of other glass hosts, suggesting a local asymmetry associated with the Er³⁺/Yb³⁺ ions and moderate covalency of the Er-O/Yb-O bonds in the current TeO₂ glass. The manufactured glass’s spectroscopic quality factor, $Q={\mathsf{\Omega }}_4/{\mathsf{\Omega }}_6$, has a value of 2.8630. When compared with the glass hosts ZBLAN [77], Boro-tellurite [78], and TLNT [79], the generated glass had a higher value of $Q={\mathsf{\Omega }}_4/{\mathsf{\Omega }}_6$. The glass that was developed had higher spectroscopic quality factor values, $Q={\mathsf{\Omega }}_4/{\mathsf{\Omega }}_6$, than those found for TeO₂ glasses in earlier investigations, indicating that it is a better fit for optical devices. For optical amplifier and laser usage, the glass used in the present research (TKNB2) is a great option [80].

The relationship between the emission line strengths related to the transmissions from the highest energy levels, ⁴I_15/2, ⁴I_13/2, ⁴I_11/2, ⁴I_9/2, ⁴F_9/2, ⁴S_3/2, ²H_11/2, and ⁴F_7/2, and their related lower-lying energy levels can be computed utilizing the JO intensity parameters. The spontaneous emission probabilities $A_{rad}(J\to J^{\prime })$ can be calculated utilizing these line strengths as follows:

$$A_{rad}\left(J\to J^{\prime }\right)=A_{ed}+A_{md}={\displaystyle \frac{e^2}{4\pi {ϵ}_0}}{\displaystyle \frac{64{\pi }^4}{3h\left(2J+1\right){\overline{\lambda }}^4}}\left({\displaystyle \frac{{n\left(n^2+2\right)}^2}{9}}{S}_{ed}+n^3{S}_{md}\right)$$

(20)

where $A_{ed}$ and $A_{md}$ denote the respective radiative transmission probabilities for electric and magnetic dipoles. Equations (13) and (15) are employed to calculate $S_{ed}$ and $S_{md}$, respectively.

The relative transmission probability, or branching ratio ${\beta }_{rad}\left(J\to J^{\prime }\right)$, can be obtained using the following equation if there are many transitions from the level:

$${\beta }_{rad}\left(J\to J^{\prime }\right)={\displaystyle \frac{A_{rad}\left(J\to J^{\prime }\right)}{\sum A_{rad}\left(J\to J^{\prime }\right)}}$$

(21)

where all terminal states J′ are covered by the total.

An energy level’s radiative lifetime ${\tau }_{rad}$, which is based on the rates of spontaneous emission across all transitions from this level, can be calculated utilizing the following equation:

$${\tau }_{rad}={\displaystyle \frac{1}{\sum A_{rad}\left(J\to J^{\prime }\right)}}$$

(22)

Table 8. Calculated radiative parameters of the different states of Er³⁺ ion-doped tellurite glass (TKNB2).

Transitions	$\overline{\mathit{\nu }}\left({\mathbf{c}\mathbf{m}}^{-1}\right)$	$\mathit{n}$	${\mathit{S}}_{\mathit{e}\mathit{d}} \left(\mathbf{p}{\mathbf{m}}^2\right)$	${\mathit{S}}_{\mathit{m}\mathit{d}} \left(\mathbf{p}{\mathbf{m}}^2\right)$	${\mathit{A}}_{\mathit{e}\mathit{d}} \left({\mathbf{s}}^{-1}\right)$	${\mathit{A}}_{\mathit{m}\mathit{d}} \left({\mathbf{s}}^{-1}\right)$	$\mathit{\beta }(\%)$=	${\mathit{\tau }}_{\mathit{r}}\left(\mathbf{m}\mathbf{s}\right)$
4I13/2→4I15/2	6536	1.998	1.2079	0.7000	277	80.4906	100.00	2.7956
4I11/2→4I15/2	10,256	2.007	0.3205	0	339	0	89.617	2.6416
4I11/2→4I13/2	3720	1.995	0.4040	0.7910	20	19.4855	10.383
4I9/2→4I15/2	12,500	2.016	0.3253	0	762	0	85.548	1.1221
4I9/2→4I13/2	5964	1.997	0.5063	0	123	0	13.838
4I9/2→4I11/2	2244	1.995	0.2490	0.3480	3	2.2576	0.6143
4F9/2→4I15/2	15,385	2.029	1.3441	0	605.5	0	92.589	0.1529
4F9/2→4I13/2	8849	2.003	0.3787	0	306	0	4.6771
4F9/2→4I11/2	5129	1.996	1.0018	0.1720	155	13.3420	2.5720
4F9/2→4I9/2	2885	1.995	0.3303	0.1120	9	1.5441	0.1623
4S3/2→4I15/2	18,349	2.048	0.1556	0	310.9	0	69.047	0.2221
4S3/2→4I13/2	11,813	2.013	0.2298	0	113.0	0	25.086
4S3/2→4I11/2	8093	2.001	0.0646	0	99	0	2.2064
4S3/2→4I9/2	5849	1.996	0.2848	0	164	0	3.6314
4S3/2→4F9/2	2964	1.995	0.0181	0	1	0	0.0299
2H3/2→4I15/2	19,231	2.055	2.8790	0	2245.2	0	94.859	0.0423
2H3/2→4I13/2	12,695	2.016	0.2473	0.1160	507	117.2763	2.6367
2H3/2→4I11/2	8975	2.003	0.3695	0.0860	260	30.1353	1.2244
2H3/2→4I9/2	6731	1.998	0.8182	0.0300	239	4.3980	1.0303
2H11/2→4F9/2	3846	1.995	1.0819	0.0090	59	0.2450	0.2489
2H11/2→4S3/2	882	1.995	0.4219	0	0	0	0.0012
4F7/2→4I15/2	20,408	2.067	0.6945	0	996.8	0	71.627	0.0719
4F7/2→4I13/2	13,872	2.021	0.6294	0	255.4	0	18.355
4F7/2→4I11/2	10,152	2.0070	0.6296	0	969	0	6.9622
4F7/2→4I9/2	7908	2.0005	0.5105	0.051	366	18.2576	2.7589
4F7/2→4F9/2	5023	1.9955	0.1131	0.212	21	19.3050	0.2862
4F7/2→4S3/2	2059	1.9947	0.0087	0	0	0	0.0008
4F7/2→4H11/2	1177	1.9947	0.6207	0	1	0	0.0104

presents the electric dipole transmission probability $A_{ed}$, magnetic dipole transmission probability $A_{md}$, spontaneous emission transmission probability $A_{rad}$, fluorescence branch ratio ${\beta }_{rad}$, and radiation lifetime ${\tau }_{rad}$ of the Er³⁺ ions in TKNB2 glass, which relate to the transmissions that occur from the higher energy levels of ⁴I_13/2, ⁴I_11/2, ⁴I_9/2, ⁴F_9/2, ⁴S_3/2, ⁴H_11/2, and $F_{7/2}$ to their lower-lying energy levels, respectively.

In general, the green and red UV emissions were below NIR excitement for the Er³⁺ and Yb³⁺-co-doped glass (see Figure 10). According to estimates, the branched ratios of the transitions ²H_11/2→⁴I_15/2 (green), ⁴S_3/2→⁴I_15/2 (green), and ⁴F_9/2→⁴I_15/2 (red) were 94.8585%, 69.0467%, and 92.5887%, respectively. These findings indicate that the green and red emission transition channels predominate over all related higher-level radiative transitions.

The ions’ capacity for absorbing or emitting light is measured using the absorption and emission cross-sections. A significant emission cross-section indicates a large gain coefficient and lower pump laser threshold energy. Utilizing the Beer–Lambert equation, the absorbing cross-section $\left({\sigma }_a\right)$ of the ⁴I_13/2→⁴I_15/2 transmission (1540 nm) of Er³⁺ was found based on the absorbing patterns of Er³⁺/Yb³⁺-co-doped TeO₂ glass [41]:

$${\sigma }_a\left(\lambda \right)={\displaystyle \frac{2.303\times OD\left(\lambda \right)}{N\times L}}$$

(23)

where N is the content of Er³⁺ ions (ions/cm³), L is the thickness of the sample, and OD(λ) is the optical density of the realistic absorbing patterns of the manufactured glass. The McCumber theory allows for the extraction of the stimulated emission cross-section (${\sigma }_e)$ utilizing the following relationship [42,43,44]:

$${\sigma }_e\left(\lambda \right)={\sigma }_a\left(\lambda \right){\displaystyle \frac{Z_l}{Z_u}}exp\left[{\displaystyle \frac{hc}{k_{B}T}}\left({\displaystyle \frac{1}{{\lambda }_{ZL}}}-{\displaystyle \frac{1}{\lambda }}\right)\right]$$

(24)

The absorption cross-section and partition functions of the lesser and higher states engaged in the optical transmission under consideration are represented by ${\sigma }_a\left(\lambda \right)$, $Z_l$, and $Z_u$. The Planck constant $6.63\times {10}^{-34} J S$, Boltzmann constant $(1.38\times {10}^{-23}$, and temperature (RT in this case) are denoted by the parameters $h$, $k_B$, and $T$. Additionally, the wavelength at which the Stark sublevels of the emission are lower and multiple transmissions are achieved is denoted by ${\lambda }_{ZL}$. Figure 11 displays the computed values of the absorbing and emission cross-sections of Er³⁺/Yb³⁺-co-doped TeO₂ glass (between 1430 and 1630).

The stimulation-emitted cross-section peak, ${\sigma }_e^{peak}$, is located at $6.86\times {10}^{-21 }{\mathrm{c}\mathrm{m}}^2$. This value is in good agreement with those of other TeO₂ glasses reported in the literature [77,78,79,80]. Moreover, the emitted cross-section ${\sigma }_e^{peak}$ and the full width at half-maximum $\left(FWHM\right)$ are crucial factors in obtaining high gain and wideband amplification in optical amplifiers. The resulting $FWHM\times {\sigma }_e^{peak}$ could be employed to determine an optical amplifier’s bandwidth characteristics. A broader gain bandwidth and lower pump threshold power are implied by the higher values of the $FWHM\times {\sigma }_e^{peak}$ produced and the higher radiative lifetime ${\tau }_{rad}$.

Table 9. Obtained values of $FWHM \left(\mathrm{n}\mathrm{m}\right)$, ${\sigma }_e \left(\mathrm{c}{\mathrm{m}}^2\right)$ 1.54 μm, and ${\sigma }_e\times FWHM (\mathrm{n}\mathrm{m}·\mathrm{c}{\mathrm{m}}^2)$ at 1.53 μm.

	$\mathit{F}\mathit{W}\mathit{H}\mathit{M} \left(\mathbf{n}\mathbf{m}\right)$	${\mathit{\sigma }}_{\mathit{e}} \left(\mathbf{c}{\mathbf{m}}^2\right)$	${\mathit{\sigma }}_{\mathit{e}}\times \mathit{F}\mathit{W}\mathit{H}\mathit{M} (\mathbf{n}\mathbf{m}·\mathbf{c}{\mathbf{m}}^2)$
Present work Er3+/Yb3+	55	6.86 $\times {10}^{-21} \mathrm{c}{\mathrm{m}}^2$	$377.3\times {10}^{-21} \mathrm{n}\mathrm{m}·\mathrm{c}{\mathrm{m}}^2$
PBGG [35]	38	8.9	$338.5\times {10}^{-21} \mathrm{n}\mathrm{m}·\mathrm{c}{\mathrm{m}}^2$
Phosphate tellurite glass [36]	44.842	7.7	$337.5\times {10}^{-20} \mathrm{n}\mathrm{m}·\mathrm{c}{\mathrm{m}}^2$

lists the $FWHM\times {\sigma }_e^{peak}$ values of Er³⁺/Yb³⁺-co-doped glasses. It can be seen that the $FWHM\times {\sigma }_e^{peak}$ in TKNB2 has a comparable bandwidth characteristic to other glass hosts such as phosphate tellurite glass ($337.5\times {10}^{-21} \mathrm{n}\mathrm{m}·\mathrm{c}{\mathrm{m}}^2)$ and PBGG $(338.5\times {10}^{-21} \mathrm{n}\mathrm{m}·\mathrm{c}{\mathrm{m}}^2)$ [35]. Finally, we found that the effect of Er³⁺/Ho³⁺ was slightly changed compared with Er³⁺/Yb³⁺ in the same host glass; therefore, we only reported the emission cross-section and increased spectroscopic values for a transition energy level of ⁴I_3/2→⁴I_5/2 of TKNB2.

When assessing the effectiveness of laser mediums, the optical gain coefficient is an essential factor to consider. Once the absorbed and emitted cross-sections for the changes between the two working laser levels are known, the following equation can be applied to determine the optical gain coefficient $G\left(\lambda \right)$ [41]:

$$G\left(\lambda \right)=N\left[P{\sigma }_e\left(\lambda \right)-(1-P){\sigma }_a\left(\lambda \right)\right]$$

(25)

where P is the rate of population inversion.

The electron population ratio across the two energy levels is represented by the p value, which progressively increases from 0 to 1, as seen in Figure 12. This figure demonstrates that a positive gain can be achieved when p ≥ 0.6, suggesting that TKNB2 glass may find a use as a matrix substance for 1.54 μm fiber lasers.

4. Conclusions

The thermal stability parameters for the Er₂O₃/Yb₂O₃-co-doped glass were slightly lower than those for Er₂O₃/Ho₂O₃ due to the substitution of Ho₂O₃ with Yb₂O₃ in the glassy matrix. The value of E_opt reduced from 2.689 to 2.663 eV following the substitution of Ho₂O₃ with Yb₂O₃, because this substitution increased the amount of NBO in the glass network. The physical parameters (α_m, R_m, E_d, E_o, and n_o) were correlated to the existence of rare earth oxides (Er₂O₃/Ho₂O₃ and Er₂O₃/Yb₂O₃) in the glassy matrix. The spectroscopic characteristics of the Er³⁺/Yb³⁺-co-doped glasses were evaluated based on their intensity factors, radiative rates, branching ratios, and radiative lifetimes. The maximum emission cross-section reported was $6.8\times {10}^{-21} {\mathrm{c}\mathrm{m}}^2$, and the gain coefficient of Er³⁺/Yb³⁺ for the transition of ⁴I_13/2$\to$⁴I_15/2 was 6.0 cm⁻¹ with a high quality factor (${\sigma }_e\times FWHM=377.3\times {10}^{-21} \mathrm{n}\mathrm{m}·\mathrm{c}{\mathrm{m}}^2)$. When the Ho³⁺ ions are replaced with Yb³⁺ ions in the same host glass, it leads to a slight change in the spectroscopic properties, viz, the emission cross-section and gain and intensity parameters of the transition energy level, ⁴I_13/2$\to$⁴I_15/2. These findings suggest that the co-doping of tellurite glass is a viable choice for optical amplifier and laser manufacturing in the communication bands.