Inverse Judd–Ofelt Formalism Based on Radiative Lifetime for Comparative Spectroscopy of RE³⁺ Ions in Glass

Abstract

This work shows that inverse Judd–Ofelt (JO) analysis of relative absorption spectra, anchored by a single lifetime, provides JO parameters and radiative rates without absolute calibration. The method is applied to Er³⁺, Dy³⁺, and Sm³⁺ in a compositionally identical oxyfluoride glass. Three well-resolved ground-state 4f–4f absorption bands were selected. After baseline removal and wavenumber-domain integration, their normalized strengths S_rel,k (k = 1, 2, 3; k∈S) define a 3 × 3 system solved by non-negative least squares to obtain the anchor-independent ordering (Ω₂:Ω₄:Ω₆). Absolute scaling uses a single lifetime anchor. We report lifetime-scaled Ω_t and A_rad, and the normalized fractions p_k within the selected triplets; as imposed by the method, the anchor-independent ordering (Ω₂:Ω₄:Ω₆) is analyzed, while absolute A_rad and Ω_t scale with τ_ref. The extracted parameters fall within the expected ranges for oxyfluoride hosts and reveal clear ion-specific trends: Ω₂ follows Dy³⁺ > Er³⁺ > Sm³⁺ (site asymmetry/hypersensitive response), while the ordering Ω₄ > Ω₆ holds across all ions (oxide-rich networks). Er³⁺ exhibits the largest Ω₄ and the smallest Ω₆, indicative of pronounced medium-range “rigidity” with suppressed long-range polarizability; Sm³⁺ shows the lowest Ω₂ (more symmetric/less covalent coordination); and Dy³⁺ the highest Ω₂ (strong hypersensitive behavior). Uncertainty was quantified by Monte Carlo resampling of the preprocessing steps, yielding compact 95% confidence intervals; the resulting JO-parameter trends (Ω₂, Ω₄, Ω₆) and normalized f_k fractions reproduce the characteristic spectroscopic behavior known for each ion. This method enables quantitative JO outputs from uncalibrated spectra, allowing direct spectroscopic comparisons and quick screening when only relative absorption data are available.

1. Introduction

Rare-earth (RE³⁺)-doped materials occupy a central position in photonics owing to their sharp intra-4f electronic transitions, partially shielded from the host environment by outer 5s and 5p orbitals. This shielding produces spectrally narrow, chemically stable absorption and emission lines—a combination difficult to match with other dopants—making RE³⁺ ions indispensable in lasers, optical amplifiers, phosphors, displays, and biomedical imaging systems [1,2]. Specifically relevant is that the 4f multiplet energies are nearly host-independent, while transition intensities remain strongly host-sensitive.

Technologically significant cases include erbium (Er³⁺), whose ⁴I_13/2 → ⁴I_15/2 transition near 1500 nm supports telecom-band amplification; dysprosium (Dy³⁺), which offers complementary blue–yellow emission channels for white-light generation; and samarium (Sm³⁺), valued for its intense orange–red emission in display and sensing technologies [3,4].

The optical performance of RE³⁺-doped systems is intimately linked to the local coordination environment of the dopant ions within the host lattice. In the 1960s, Judd and Ofelt [5,6] resolved the long-standing problem of intensity transfer by showing that odd-parity crystal-field components mix 4f states of opposite parity, thereby relaxing the Laporte rule and allowing electric–dipole transitions to acquire measurable intensity. The Judd–Ofelt (JO) theory [5,6] provides a unified framework for describing such behavior, expressing electric–dipole (ED) line strengths through three phenomenological parameters, Ω₂, Ω₄, and Ω₆. Beyond their role in calculating Einstein radiative coefficients A_rad (spontaneous emission probabilities), branching ratios (β), and radiative lifetimes, these parameters also serve as structural probes: Ω₂ reflects local asymmetry and 4f–ligand covalency; Ω₄ is linked to intermediate-range network rigidity; and Ω₆ captures long-range polarizability [7]. In this study, attention is restricted to electric–dipole (ED)-dominated transitions, since the selected bands in these glasses have negligible magnetic–dipole (MD) contributions. In most RE³⁺-doped oxide and oxyfluoride glasses, the 4f–4f transitions employed in JO analysis are predominantly ED in character, with magnetic–dipole (MD) contributions comparatively small in the relevant spectral regions [8,9,10].

In the selected windows, MD contributions are small; at such levels, they primarily affect the absolute scale of radiative probabilities and do not alter the Ω-ratios or the normalized fractions pk within our reported uncertainty.

Under these conditions, the experimental line strengths can be directly related to the squared reduced matrix elements $⟨J_g‖U^{\left(t\right)}‖{\mathrm{J}}_u⟩$ (t = 2, 4, 6), which quantify the coupling between the ground (g) and upper (u) 4f multiplets for each tensor rank (multipole order). Extensive tabulations of these quantities for RE³⁺ ions in the free-ion/LaF₃ benchmark [10] provide a practical baseline for Judd–Ofelt analysis. However, where reliable host-specific reduced matrix elements ⟨U^(t)⟩² are available, they should be preferred [11]. In this comparative study, we adopt a single reference set (Carnall’s LaF₃ tables) for Er³⁺, Dy³⁺, and Sm³⁺ solely to maintain cross-ion consistency within the same glass; the host dependence is then captured by the Ωₜ parameters rather than by mixing heterogeneous U-tables across ions.

The classical JO formalism requires absolute oscillator strengths from calibrated absorption spectra, together with the sample thickness (d), dopant number density (N), and refractive index (n). While this is feasible in benchmark systems [8,9], exploratory glass compositions and small-batch samples often lack reliable thickness and concentration measurements.

Several methodological adaptations have been developed to circumvent these limitations. Diffuse-reflectance spectroscopy, combined with fluorescence lifetime measurements, enables recovery of Ωₜ values without transmission data, even in powdered samples [12]. Emission-based strategies provide another route: the JOES (Judd–Ofelt Emission Spectroscopy) was developed to extract parameters directly from Eu³⁺ emission spectra [13], while a self-referenced excitation method has been proposed to internally normalize excitation intensities and thus avoid absolute calibration [14]. Modified formalisms have also been applied to Pr³⁺-doped fluoride glasses, where normalization against lifetime data improved consistency between calculated and experimental results [15]. Goldner and Auzel [16] carried out a systematic comparison between the standard JO framework and a modified version incorporating normalized oscillator strengths in Pr³⁺-doped fluorozirconate glasses. Their study showed that such normalization not only improved agreement between calculated and experimental lifetimes, but also mitigated the strong dependence of JO parameters on the set of transitions chosen for fitting—a recurrent difficulty in Pr³⁺ systems. This work remains an important reference, as it demonstrated that careful adaptation of this formalism can substantially enhance the reliability of JO analyses in hosts where absolute calibration of absorption spectra is challenging. Extensions of the methodology to nanocrystalline hosts further highlight its flexibility, as shown by Hien et al. [17] in their analysis of Dy³⁺-doped CoAl₂O₄, where spectroscopic parameters were explicitly linked to local structural features.

More recently, “inverse” JO methodologies have been developed, in which relative absorption data are first used to determine ${\Omega }_2:{\Omega }_4:{\Omega }_6$ ratios, and a single lifetime measurement is subsequently applied to fix the absolute scale. Notable examples include Xue et al. [18], who applied a diffuse-reflectance/Kubelka–Munk approach, extracting JO parameters from uncalibrated powder spectra by combining them with the radiative lifetime of a well-defined transition, thereby anchoring the absolute scale. Likewise, Vega et al. [19] applied a transmission-based strategy in Er³⁺/Yb³⁺-doped BaTiO₃, obtaining relative parameters from absorption spectra and enforcing absolute scaling through a predominantly radiative emission channel. Despite differing in experimental modality, both studies converge on the same underlying principle: the decoupling of relative parameter ratios from absolute scaling, a concept that provides the basis for the systematic multi-ion implementation presented here.

In this study, we implement a three-band, absorptance-only inverse Judd–Ofelt (JO) protocol and apply it to Er³⁺, Dy³⁺, and Sm³⁺ ions embedded in compositionally identical oxyfluoride glasses. For each ion, baseline-corrected integrated absorption band areas yield relative line strengths. These, combined with tabulated reduced matrix elements, provide (Ω₂:Ω₄:Ω₆) ratios via a linear system. A single radiative-lifetime constraint is then used to fix the absolute scale, enabling complete JO parameter sets and derived radiative descriptors to be obtained from relative spectra alone. This framework permits direct, anchor-independent comparison of JO properties across multiple RE³⁺ ions in a shared host environment. The goals are two-fold: (i) to assess the stability of the inverse JO method under fixed-lifetime scaling, and (ii) to enable consistent cross-ion spectroscopic comparisons under identical compositional and measurement conditions. In this way, the methodology supports rapid screening of new RE³⁺-doped materials when only relative absorption data are available.

Beyond classical calibrated-absorption JO fits, calibration-light approaches include emission-based extraction (e.g., JOES) [13], self-referenced excitation normalization [14], and diffuse-reflectance/Kubelka–Munk combined with lifetimes [12,18]. Recent overviews—e.g., Hrabovský et al. [20]—also recommend using the mean emission wavelength (band barycenter) in the λ⁻³ factor and summarize practical sources of bias. Against that background, our contribution is an absorbance-only, three-band inverse protocol that: (i) determines Ω₂:Ω₄:Ω₆ from uncalibrated transmission data without d, N, or absolute intensity calibration; (ii) applies a single lifetime anchor only after inversion, making Ω-ratios and within-set fractions p_k anchor-independent; (iii) uses three well-resolved, non-overlapping bands to avoid de-overlap and arbitrary weighting inherent to multi-band fits; and (iv) reports uncertainties via resampling. This yields like-for-like cross-ion comparisons from minimal inputs in a shared host—capabilities not delivered together by existing calibration-light methods.

Regarding the scope of the inverse Judd–Ofelt framework, when absolute calibration is available (reliable d, N, n), classical JO is the preferred route; in this calibrated limit, the inverse framework recovers the same Ω-ratios and normalized descriptors. When calibration is incomplete or uncertain (archival/relative spectra, exploratory compositions, cross-lab heterogeneity), classical JO becomes under-determined or assumption-laden, whereas the inverse method provides reproducible, JO-consistent Ω-ratios and pk from relative spectra under explicit assumptions and quantified uncertainties.

2. Theory and Notation

The Judd–Ofelt (JO) theory provides a rigorous quantum mechanical description of ED transitions between 4f levels of RE³⁺ ions in glassy hosts. In practice, absorption spectra display many overlapping 4f–4f bands of varying intensity. For the inverse JO procedure adopted here, we restrict the analysis to three well-resolved ground-state absorption bands per ion, chosen for their spectral prominence, minimal overlap, and well-established assignments. To avoid ambiguity, we denote these bands by the index k and define the restricted set S = {1, 2, 3}. All sums ∑_k∈S are confined to this subset. Within this framework, the normalized ED fractions pₖ are defined only across S and must not be confused with branching ratios β, which apply when all possible radiative channels are considered.

2.1. JO Relations (ED Line Strength, Radiative Rate, Branching)

For an electric–dipole (ED) 4f–4f transition from the upper (emitting) manifold $\left|J_u⟩\right.$ to a lower manifold $\left|J_{\mathcal{l}}⟩\right.$, the line strength is:

$$S_{\mathrm{E}\mathrm{D}}\left(J_u\to {\mathrm{J}}_{\mathcal{l}}\right)={\sum }_{t=\mathrm{2,4},6}{\Omega }_t.{\left|⟨J_u‖U^{\left(t\right)}‖{\mathrm{J}}_{\mathcal{l}}⟩\right|}^2,$$

(1)

where ${\Omega }_t$(t = 2,4,6) are the JO intensity parameters and $⟨J_u‖U^{\left(t\right)}‖{\mathrm{J}}_{\mathcal{l}}⟩$ are reduced matrix elements tabulated for RE³⁺ ions [10]. Equivalently, for absorption $\left(J_g\to {\mathrm{J}}_u\right)$, the same S_ED enters with ${\left|J_g‖U^{\left(t\right)}‖{\mathrm{J}}_u⟩\right|}^2$; the two forms are identical in modulus-squared.

The corresponding Einstein radiative rate (spontaneous emission probability) follows as:

$$A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_u\to J_{\mathcal{l}}\right)={\displaystyle \frac{64{\mathrm{e}}^2{\pi }^4}{3h(2J_u+1)}}L\left(n\right) . {\lambda }^{-3} . S_{\mathrm{E}\mathrm{D}}\left(J_u\to J_{\mathcal{l}}\right), L\left(n\right)={\displaystyle \frac{n{(n^2+2)}^2}{9}}$$

(2)

which links the ED line strength to the spontaneous-emission probability: given $S_{\mathrm{E}\mathrm{D}}$ and $\lambda$, it returns the physical $A_{\mathrm{r}\mathrm{a}\mathrm{d}}$. Here, e is the elementary charge, h is Planck’s constant, n is the host refractive index, $\lambda$ is the emission wavelength of the transition (approximated here by the absorption maximum, λ_em ≈ λ_abs), the factor $1/(2J_u+1)$ and the Lorentz (virtual-cavity) local-field factor $L\left(n\right)=n{\left({\displaystyle \frac{n^2+2}{3}}\right)}^2$, appropriate for dipoles embedded in a homogeneous dielectric [19,21]. Magnetic–dipole contributions are neglected for the selected bands, since their estimated fraction in these oxyfluoride hosts is <10% in the relevant windows [8,9,10].

The total radiative lifetime ${\tau }_{\mathrm{r}\mathrm{a}\mathrm{d}}$ and branching ratio $\beta$ (for a single emitting manifold $J_u$) are, respectively:

$${\tau }_{\mathrm{r}\mathrm{a}\mathrm{d}}={\left({\sum }_{\mathcal{l}}{A}_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_u\to J_{\mathcal{l}}\right)\right)}^{-1}$$

(3)

$$\beta (J_u\to J_{\mathcal{l}})={\displaystyle \frac{A_{\mathrm{r}\mathrm{a}\mathrm{d}}(J_u\to J_{\mathcal{l}})}{\sum _{\mathcal{l}}{A}_{\mathrm{r}\mathrm{a}\mathrm{d}}(J_u\to J_{\mathcal{l}})}}$$

(4)

We use $\beta$ (branching ratio) only for decays from a single emitting manifold. For the three ground-state channels per ion (set S), we report normalized ED shares p_k

$$p_k=A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,k}\to J_g\right)/{\sum }_{m\in S}{A}_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,m}\to J_g\right)$$

(5)

which are dimensionless, scale-invariant, and defined only within S (not β).

For clarity, for S = {1, 2, 3}:

$$p_1={\displaystyle \frac{A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,1}\to J_g\right)}{A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,1}\to J_g\right)+A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,2}\to J_g\right)+A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,3}\to J_g\right)}}$$

(5a)

$$p_2={\displaystyle \frac{A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,2}\to J_g\right)}{A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,1}\to J_g\right)+A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,2}\to J_g\right)+A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,3}\to J_g\right)}}$$

(5b)

$$p_3={\displaystyle \frac{A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,3}\to J_g\right)}{A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,1}\to J_g\right)+A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,2}\to J_g\right)+A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,3}\to J_g\right)}}$$

(5c)

Importantly, $p_k$ is not the oscillator strength of atomic spectroscopy.

2.2. Inverse JO from Relative Absorption: Ratios and Absolute Scaling

Since only relative absorbance spectra are typically available (arbitrary units), the corresponding band strengths cannot yield absolute oscillator strengths directly. We therefore define S_rel,k (k = 1, 2, 3) as the baseline-corrected, wavenumber-integrated band area normalized within the ion (so any common Nd factor cancels). For each absorption channel $J_g\to {\mathrm{J}}_{u,k}$, ${U_{k,t}=\left|⟨J_g‖U^{\left(t\right)}‖{\mathrm{J}}_{u,k}⟩\right|}^2$(t = 2, 4, 6) [10].

Stacking the three normalized band strengths $S_{\mathrm{r}\mathrm{e}\mathrm{l},\mathrm{k}}$ and the three Ω_t into vectors yields the linear JO system:

$$\underset{S_{\mathrm{r}\mathrm{e}\mathrm{l}}}{\underbrace{\left[\begin{array}{c}{S}_{\mathrm{r}\mathrm{e}\mathrm{l},1}\\ S_{\mathrm{r}\mathrm{e}\mathrm{l},2}\\ S_{\mathrm{r}\mathrm{e}\mathrm{l},3}\end{array}\right]}}=\underset{\mathrm{U}}{\underbrace{\left[\begin{array}{ccc}{\left|⟨J_g‖U^{\left(2\right)}‖J_{u,1}⟩\right|}^2& {\left|⟨J_g‖U^{\left(4\right)}‖J_{u,1}⟩\right|}^2& {\left|⟨J_g‖U^{\left(6\right)}‖J_{u,1}⟩\right|}^2\\ {\left|⟨J_g‖U^{\left(2\right)}‖J_{u,2}⟩\right|}^2& {\left|⟨J_g‖U^{\left(4\right)}‖J_{u,2}⟩\right|}^2& {\left|⟨J_g‖U^{\left(6\right)}‖J_{u,2}⟩\right|}^2\\ {\left|⟨J_g‖U^{\left(2\right)}‖J_{u,3}⟩\right|}^2& {\left|⟨J_g‖U^{\left(4\right)}‖J_{u,3}⟩\right|}^2& {\left|⟨J_g‖U^{\left(6\right)}‖J_{u,3}⟩\right|}^2\end{array}\right]}}·\underset{{\Omega }_{\mathrm{r}\mathrm{e}\mathrm{l}}}{\underbrace{\left[\begin{array}{c}{\Omega }_2\\ {\Omega }_4\\ {\Omega }_6\end{array}\right]}},$$

(6)

solved by non-negative least squares (NNLS), $\hat{\Omega }={\arg \mathrm{m}\mathrm{i}\mathrm{n}}_{\Omega \ge 0}{‖S_{\mathrm{r}\mathrm{e}\mathrm{l}}-U\Omega ‖}_2$, which returns scale-free JO ratios Ω₂:Ω₄:Ω₆. If a selected feature contains two partially resolved channels (J_u,a, J_u,b), linearity is preserved by summing areas and rows: ${\mathrm{S}}_{\mathrm{r}\mathrm{e}\mathrm{l},\mathrm{k}}\leftarrow {\mathrm{S}}_{\mathrm{r}\mathrm{e}\mathrm{l},\mathrm{a}}+{\mathrm{S}}_{\mathrm{r}\mathrm{e}\mathrm{l},\mathrm{b}}$, and the corresponding row of U is $U_{\mathrm{k},\mathrm{t}}\leftarrow U_{\mathrm{a},\mathrm{t}}+U_{\mathrm{b},\mathrm{t}}$.

2.3. Fixing the Absolute Scale by a Lifetime Constraint

Let ${\Omega }_{\mathrm{r}\mathrm{e}\mathrm{l}}=\left({\Omega }_2^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)},{\Omega }_4^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)},{\Omega }_6^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)}\right)$ be the (column) vector returned by Equation (6). Using Equation (2), we first form relative radiative rates.

$$A_{\mathrm{r}\mathrm{a}\mathrm{d}}^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)}\left(J_{u,k}\to J_g\right)=\kappa \left(n, { J}_{u,k}\right){\lambda }_k^{-3} .{ S}_{\mathrm{E}\mathrm{D}}^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)}\left(J_{u,k}\to J_g\right), { S}_{\mathrm{E}\mathrm{D}}^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)}={\sum }_t{\Omega }_t^{\left(rel\right)}{U}_{k,t}$$

(7)

i.e., the same functional form as Equation (2), but built from the relative line strengths ${ S}_{\mathrm{E}\mathrm{D}}^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)}$ implied by Equation (6). Here, the wavelength factor in Equation (2) is evaluated, preferentially, at the band’s mean emission wavelength ⟨λ_em⟩ (barycenter), which is the recommended choice for the λ⁻³ dependence [11]. In our absorption-only workflow, emission barycenters are not available for all samples, so we use λ_max as a practical proxy; for intra-4f transitions in oxide/oxyfluoride glasses the Stokes shift is small, so the impact in A_rad is at the few-percent level (ΔA/A ≈ −3 Δλ/λ), well within our reported uncertainties. The local-field factor is included uniformly within $\kappa \left(n, { J}_{u,k}\right)={\displaystyle \frac{64{\mathrm{e}}^2{\pi }^4}{3h(2J_u+1)}}L\left(n\right)$. Thus, $A_{\mathrm{r}\mathrm{a}\mathrm{d}}^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)}$ has the correct ratios across the three bands, but remains in arbitrary units.

To set the absolute scale, we impose a single lifetime anchor across the triplet,

$$A_{\mathrm{r}\mathrm{a}\mathrm{d},\mathrm{t}\mathrm{o}\mathrm{t}}={\sum }_{k=1}^3{A}_{\mathrm{r}\mathrm{a}\mathrm{d},\mathrm{k}}={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\tau }_{\mathrm{r}\mathrm{e}\mathrm{f}}$}\right.} , {\tau }_{\mathrm{r}\mathrm{e}\mathrm{f}}=2 \mathrm{m}\mathrm{s}$$

(8)

which defines a global factor α, such that:

$$\alpha ={\displaystyle \frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\tau }_{\mathrm{r}\mathrm{e}\mathrm{f}}$}\right.}{\sum _{k\in S}{A}_{\mathrm{r}\mathrm{a}\mathrm{d},\mathrm{k}}^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)}}}, {\Omega }_t=\mathsf{\alpha }{\Omega }_t^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)}, A_{\mathrm{r}\mathrm{a}\mathrm{d}}\left(J_{u,k}\to J_g\right)=\mathsf{\alpha }{A}_{\mathrm{r}\mathrm{a}\mathrm{d}}^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)}\left(J_{u,k}\to J_g\right)$$

(9)

(note that $p_k$ is invariant to ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{f}}$).

The chosen anchor τ_ref = 2 ms lies within the ms-scale lifetimes reported for ⁴I_13/2 (Er³⁺), ⁴F_9/2 (Dy³⁺), and ⁴G_5/2 (Sm³⁺) in related glasses [22,23,24,25,26] and is used solely to set the absolute scale.

This constraint is purely operational: it does not represent the true τ_rad of a single emitting manifold but enforces a uniform absolute scale. As a direct consequence of the scaling procedure, the calculated radiative lifetime τ_rad from the Einstein coefficients equals the imposed anchor τ_ref. Thus, $A_{\mathrm{r}\mathrm{a}\mathrm{d},\mathrm{k}}^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)}$ denote scale-free relative coefficients, while $A_{\mathrm{r}\mathrm{a}\mathrm{d},\mathrm{k}}$ are their absolute counterparts after applying α.

Summary of Scaling Logic

Starting from relative absorption areas, Equation (6) is solved by NNLS to obtain the scale-free JO parameters ${\Omega }_{\mathrm{r}\mathrm{e}\mathrm{l}}$. Using Equation (2) with the relative line strengths implied by Equation (6), we form the relative rates $A_{\mathrm{r}\mathrm{a}\mathrm{d}}^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)}\left(J_{u,k}\to J_g\right)$ (Equation (7)), which carry the correct inter-band ratios but remain in arbitrary units. A single lifetime anchor is then imposed across the triplet, ${\sum }_{k=1}^3{A}_{\mathrm{r}\mathrm{a}\mathrm{d},\mathrm{k}}={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\tau }_{\mathrm{r}\mathrm{e}\mathrm{f}}$}\right.}$ (Equation (8)), which defines the global scale α converting ${\Omega }_{\mathrm{r}\mathrm{e}\mathrm{l}}$ and $A_{\mathrm{r}\mathrm{a}\mathrm{d}}^{\left(\mathrm{r}\mathrm{e}\mathrm{l}\right)}$ into their absolute counterparts (Equation (9)). As imposed by the method, the τ_rad computed from the scaled A_rad equals τ_ref. The JO ratios (Ω₂:Ω₄:Ω₆) and the normalized fractions p_k are anchor-independent; only absolute magnitudes rescale.

2.4. Assumptions and Physical Validity

• Physical grounding of the inverse JO approach: The inverse route does not modify JO physics; it only changes the data normalization. The integrated absorbance areas S_k of ground-state bands are proportional to the same electric–dipole line strengths that underlie classical JO, i.e., S_k ∝ ${\sum }_{t\in \{\mathrm{2,4},6\}}{\Omega }_t{⟨U^{\left(t\right)}⟩}_k^2$, up to a single global prefactor (thickness, dopant density, detector gain, local-field). That prefactor multiplies all bands equally and therefore sets only the absolute scale of (Ω₂, Ω₄, Ω₆). Ratios such as h = Ω₂/(Ω₄ + Ω₆) and χ = Ω₄/Ω₆, and within-set normalized fractions p_k, remain invariant and retain their physical meaning (e.g., hypersensitivity, medium-range rigidity, branching trends). Selection rules, local-field modeling, and Einstein relations are exactly those of classical JO; we simply exploit the scale-independence of the ratios to work with relative spectra.
• Electric–dipole (ED) dominance: The three bands per ion are ED-allowed in these hosts; magnetic–dipole (MD) terms are neglected within the chosen windows. This is standard for the selected manifolds in oxide/oxyfluoride glasses and within our uncertainty budget [8,9,10,25]. For the selected bands in these oxyfluoride glasses, magnetic–dipole (MD) contributions are estimated to remain below ~10% [8,9,10,25]. Within this uncertainty budget they are neglected here, although the framework allows their inclusion whenever significant.
• Small Stokes shift: For intra-4f transitions we take λ_em ≈ λ_abs in the λ⁻³ factor of Equation (2); the resulting bias is sub-dominant relative to baseline/window choices used to form S_rel,k (Section 3.1.1). The Einstein-rate scaling with λ⁻³ follows standard radiative-rate formulas [27].
• Local-field model: We adopt the virtual-cavity factor $L\left(n\right)=n{\left({\displaystyle \frac{n^2+2}{3}}\right)}^2$, with a representative visible-wavelength refractive index n = 1.52 for these oxyfluoride glasses [28], consistent with typical values in related hosts and borosilicate-crown standards [29,30]. Varying n within 1.50–1.55 changes L (n) by ≈ 11% (about +4% from 1.50→1.52), which only rescales absolute A_rad; the normalized fractions $p_k$ and the JO ratios are unaffected.
• Minimal determinacy and stability: With three bands and three Ω_t, Equation (6) is minimally determined. Stability relies on the linear independence of the rows of U (each defined by $U_{k,t}$); conditioning is verified before scaling. Solver choices and diagnostics (non-negative least squares, residual norms, condition numbers) are detailed in Section 3.1.

The workflow, summarized in Figure 1, converts relative absorption areas into Ω-ratios via tabulated reduced matrix elements, and scales them through a single lifetime anchor. This yields physically meaningful JO parameters and radiative descriptors directly from uncalibrated spectra.

3. Experimental

3.1. Computational Methodology

3.1.1. Spectral Preprocessing and Band Integration

The workflow in Figure 1 was applied to the Er³⁺, Sm³⁺, and Dy³⁺ absorption spectra acquired as (λ, Abs(λ)) in relative absorbance. Each spectrum was used on its native sampling; only when spacing irregularities compromised numerical stability did we apply a light linear interpolation to a 1–2 nm grid.

For every selected band, two short side windows bracketing the maximum defined a local linear baseline $\hat{b}(\lambda )$ (robust fit); this baseline was subtracted so that the subsequent integration reflects the physical line shape rather than slow background variations.

Because oscillator strengths are proportional to the wavenumber integral, band areas were computed in $\tilde{\nu }=1/\lambda$ rather than in wavelength. Practically, we evaluate the wavenumber integral (implemented on the wavelength grid) using ${\mathcal{A}}_k={\int }_{{\tilde{\nu }}_1}^{{\tilde{\nu }}_2}\left[Abs\left(\tilde{\nu }\right)-\hat{b}\left(\tilde{\nu }\right)\right]d\tilde{\nu }={\int }_{{\lambda }_2}^{{\lambda }_1}\left[Abs\left(\lambda \right)-\hat{b}\left(\lambda \right)\right]{\lambda }^{-2}d\lambda$, where $d\tilde{\nu }=-{\lambda }^{-2}d\lambda$, hence the reversed wavelength limits (λ₂ < λ₁). The integral was evaluated over a symmetric window centered at the maximum, using the composite trapezoidal rule, yielding a baseline-corrected band area ${\mathcal{A}}_k$ (a.u. × nm⁻¹).

To remove the arbitrary vertical scale of each spectrum, the three areas for a given ion were normalized by the largest:

$$S_{\mathrm{r}\mathrm{e}\mathrm{l},k}={\displaystyle \frac{{\mathcal{A}}_k}{{max}_m{\mathcal{A}}_m}},k=\mathrm{1,2},3.$$

(10)

Thus, S_rel,k is normalized integrated absorption areas per ion. Although derived from relative areas, they play the role of normalized strengths in the JO inversion and preserve the ${\Omega }_2:{\Omega }_4:{\Omega }_6$ ratios.

3.1.2. Inverse JO Analysis from Normalized Areas

The squared reduced matrix elements ${\left|⟨J_g‖U^{\left(t\right)}‖J_u⟩\right|}^2$(t = 2, 4, 6) [10] populate the 3 × 3 design matrix U in Equation (6). The spectral weights are the three S_rel,k. If a feature comprises two partially resolved channels, both the baseline-corrected sub-areas are summed, and the corresponding JO rows are added (Section 2.2).

Robustness was assessed via Monte Carlo on S_rel with ±7% multiplicative jitter and total-area renormalization; [2.5%, 97.5%] intervals for h and χ were computed from n = 1000 resamples.

In all cases, we verify identifiability via full rank and report the 3 × 3 condition number of U. Robustness is quantified for the anchor-independent descriptors h = Ω₂/(Ω₄ + Ω₆) and χ = Ω₄/Ω₆ under light multiplicative jitter of the band areas S_rel (Gaussian, σ ≈ 7%) and small baseline/window perturbations. The resulting ranges [min, max] are reported in Supplementary Table S1, with graphical summaries in Supplementary Figures S1 and S2.

The 3 × 3 condition number is computed from the tabulated ⟨U^(t)⟩² of the selected ground-state transitions only; it does not depend on absolute scaling or on the lifetime anchor. It therefore quantifies the physics-imposed information content of the band set (selection rules and intensity pattern) rather than introducing an adjustable mathematical parameter.

The system $S_{rel}=\mathrm{U}{\Omega }_{rel}$ was solved by non-negative least squares (NNLS), enforcing Ω_t ≥ 0. Conditioning of U was monitored; if the system was ill-conditioned or the residual was dominated by a single equation, integration limits were slightly adjusted—remaining within the same physical band and without violating the selection criteria—or, where necessary, a different well-resolved band was chosen to improve linear independence. This step yields Ω_rel, which encodes the ratios ${\Omega }_2:{\Omega }_4:{\Omega }_6$, but remains undetermined by a common scale. As in classical and combinatorial JO analysis, band choice influences the extracted Ω_t. To mitigate this, we restrict the inversion to three well-resolved, non-overlapping ground-state transitions, and quantify uncertainty by Monte Carlo resampling.

In terms of band selection and conditioning, we restrict the inversion to three well-resolved ground-state bands per ion to ensure a full-rank system and a numerically well-conditioned 3 × 3 inversion. In relative absorbance, adding partially overlapping bands requires explicit de-overlap and per-band weighting; in the absence of a calibrated noise model, this can bias Ω_t and degrade conditioning. Our three-band design is therefore a robustness choice for identifiability with relative spectra. Where absolutely calibrated spectra and defensible weights are available, the framework can be extended to a weighted multi-band fit.

For definiteness, the ground-state bands entering Equation (6) were: for Er³⁺ (⁴I_15/2 → ⁴F_7/2, ²H_11/2, ⁴I_11/2); for Sm³⁺ (⁶H_5/2 → ⁶F_7/2, ⁶F_5/2, ⁶F_3/2); and for Dy³⁺ (⁶H_15/2 → ⁶F_5/2, ⁶F_7/2 + ⁶H_5/2, ⁶F_11/2+⁶H_9/2). For Dy³⁺, two of the selected bands involve close pairs of upper levels; each feature was integrated as a single line, and the corresponding JO row in U was taken as the sum of the two-level rows (⁶F_7/2 and ⁶H_5/2; ⁶F_11/2 and ⁶H_9/2), which preserves Equation (6) exactly.

3.1.3. Scaling and Uncertainty Analysis

Band sufficiency and identifiability: Three well-resolved ground-state bands per ion suffice because the 3 × 3 JO system is identifiable when the rows of U are linearly independent and the 3 × 3 condition number is moderate.

Zero/near-zero entries in ⟨U^(t)⟩²: Zero or near-zero reduced-matrix elements in one or two bands do not preclude inversion. Identifiability is governed by full rank and conditioning of U; if U is full rank (e.g., a non-zero 2 × 2 minor among the U₄ and U₆ columns together with at least one non-zero U₂ entry), the solution remains well posed. This is reflected in the reported condition numbers and the jitter-based robustness ranges (Supplementary Table S1; Figures S1 and S2).

Diagnostics and robustness: Complete numerical diagnostics are provided in Supplementary Table S1 and summarized in Supplementary Figures S1 and S2: 3 × 3 condition number, h, χ and [2.5%, 97.5%] robustness intervals obtained under ±7% multiplicative jitter of S_rel, with renormalization of the total band area and small baseline/window perturbations modeled by the jitter. All robustness results were computed with n = 1000 Monte Carlo resamples. An analytical proof that the ratios and p_k are invariant to global intensity rescaling is given in Appendix A.

Interpreting the robustness intervals: The reported [2.5%, 97.5%] bounds are percentiles of the Monte Carlo distribution (central 95% interval) rather than percentages of the nominal value. Narrow intervals indicate good numerical conditioning and low sensitivity to spectral perturbations; wider intervals reflect a more sensitive yet still identified 3 × 3 system under ±7% multiplicative jitter of S_rel with total-area renormalization.

The absolute scale follows Section 2.3, enforcing, for the three selected ground-state absorption transitions per ion (set S), the constraint A_rad,tot = 500 s⁻¹ (τ_ref = 2 ms), which defines the single factor α given by Equation (8). With the JO parameters Ω_t (t = 2, 4, 6) scaled, the absolute S_ED, A_rad and τ_rad follow from Equations (1)–(3). As terminology, we report $f_k$ within S (β reserved for single-manifold decays; see Section 2.1).

Uncertainties were estimated by perturbing only the steps that affect relative areas:

(i)

baseline fits were jittered within the local noise;

(ii)

integration half-widths were randomly varied by ±(5–10) nm;

(iii)

peak centers were dithered by ±2 nm to emulate discrete sampling.

In addition, we performed Monte Carlo resampling of S_rel by applying multiplicative Gaussian noise (σ ≈ 7%, capturing baseline/window/peak variability). For each ion, 1000 realizations were propagated through Equations (2) and (5)–(8), and the spreads of Ω_t and p_k were summarized as percentile-based 95% confidence intervals.

Under these controlled steps, the inverse JO workflow converts relative absorption data into scaled Ω_t, Einstein coefficients A_rad, and normalized ED fractions $f_k$ suitable for cross-ion comparison in this specific oxyfluoride glass host.

All numerical processing was performed in Python 3.11 using NumPy (array handling and trapezoidal integration), SciPy (linear interpolation, NNLS solver, and condition-number analysis), and pandas (data organization and export). Random perturbations for uncertainty analysis were generated with NumPy.random, and the full pipeline (preprocessing → S_rel → NNLS → lifetime scaling) was executed reproducibly. Diagnostic plots and spectral checks were produced with Matplotlib.

3.2. Glass Preparation and Absorption Measurements

Oxyfluoride glasses (mol%: 60 SiO₂, 20 Al₂O₃, 10 Na₂O, 10 BaF₂) were prepared by melt–quenching. Three independent batches were produced, each doped with a single rare-earth oxide (Er₂O₃, Dy₂O₃, or Sm₂O₃) at a nominal number density N ≈ 5 × 10²⁰ cm⁻³. Raw materials were of high purity: SiO₂ (≥99.5%, Sigma-Aldrich, St. Louis, MO, USA), Al₂O₃ (≥99.9%, Alfa Aesar, Ward Hill, MA, USA), Na₂CO₃ (≥99.5%, Merck, Darmstadt, Germany), BaF₂ (≥99.9%, Sigma-Aldrich), and rare-earth oxides (99.9%, Alfa Aesar). Batches of ~30–50 g were melted in a platinum crucible at 1450 °C for 1 h, cast onto a stainless-steel plate, and annealed at 500 °C for 2 h to relieve internal stress.

Specimens had thickness d = 2.0 ± 0.1 mm. Room-temperature absorption spectra were recorded with a PerkinElmer Lambda 9 UV–Vis–NIR spectrophotometer (PerkinElmer, Waltham, MA, USA) over 390–1700 nm, with 2 nm spectral resolution. Data were acquired as relative absorbance (a.u.) vs. wavelength (no absolute calibration).

Radiative-rate calculations use the optical model described in Section 2.4. For each ion, three ground-state 4f→4f bands were selected based on prominence and minimal overlap (band-selection details and treatment of partially mixed features are in Section 2.2 and Section 3.1).

4. Results and Discussion

The inverse Judd–Ofelt (JO) analysis yields a self-consistent set of JO (intensity) parameters (Ω₂, Ω₄, Ω₆) and derived radiative quantities for Er³⁺, Sm³⁺, and Dy³⁺ embedded in the same oxyfluoride glasses under uniform optical assumptions (see Section 2.4). Only the normalized strengths S_rel,k enter the inversion (NNLS); the absolute scale is fixed as described in Section 2.3. Uncertainties are reported as 95% confidence intervals from resampling of S_rel (σ ≈ 7%, Section 3.1.3). Normalized quantities—the JO ordering (Ω₂:Ω₄:Ω₆) and p_k—are anchor-independent (see Section Summary of Scaling Logic). We also use anchor-independent JO metrics (normalized Ω vectors and the indices χ = Ω₄/Ω₆, h = Ω₂/(Ω₄ + Ω₆)) to aid cross-ion interpretation.

In what follows, we first comment on the absorption features (Figure 2, Figure 3 and Figure 4,

Table 1. Er³⁺: integrated areas ${\mathcal{A}}_k$, normalized strengths S_rel,k, and squared reduced matrix elements ∣⟨U^(t)⟩∣² of the three selected peaks [10].

Peak	l (nm)	Electronic Transition	$\mathbf{Integrated} \mathbf{Area} ({\mathcal{A}}_{\mathit{k}}$) (×10−6 a.u./nm)	Srel,k	⟨U(2)⟩2	⟨U(4)⟩2	⟨U(6)⟩2
1	487	4I15/2→4F7/2	3.224	0.467	0.0	0.1465	0.6272
2	521	4I15/2→2H11/2	6.908	1.00	0.7158	0.4138	0.0927
3	974	4I15/2→4I11/2	1.169	0.169	0.0276	0.0002	0.3942

,

Table 2. Sm³⁺: integrated areas ${\mathcal{A}}_k$, normalized strengths S_rel,k, and squared reduced matrix elements ∣⟨U^(t)⟩∣² of the three selected peaks [10].

Peak	l (nm)	Electronic Transition	$\mathbf{Integrated} \mathbf{Area} ({\mathcal{A}}_{\mathit{k}}$) (×10−6 a.u./nm)	Srel,k	⟨U(2)⟩2	⟨U(4)⟩2	⟨U(6)⟩2
1	1232	6H5/2→6F7/2	1.843	1.0	0.0020	0.1429	0.4301
2	1380	6H5/2→6F5/2	0.822	0.446	0.0331	0.2844	0.0
3	1487	6H5/2→6F3/2	0.706	0.383	0.1444	0.1365	0.0

and

Table 3. Dy³⁺: integrated areas ${\mathcal{A}}_k$, normalized strengths S_rel,k, and squared reduced matrix elements ∣⟨U^(t)⟩∣² of the three selected peaks [10].

Peak	l (nm)	Electronic Transition	$\mathbf{Integrated} \mathbf{Area} ({\mathcal{A}}_{\mathit{k}}$) (×10−6 a.u./nm)	Srel,k	⟨U(2)⟩2	⟨U(4)⟩2	⟨U(6)⟩2
1	794	6H15/2→6F5/2	0.154	0.092	0.0	0.0	0.3446
2	881	6H15/2→6F7/2+6H5/2	0.412	0.246	0.0	0.1352	0.7164
3	1257	6H15/2→6F11/2+6H9/2	1.673	1.0	0.9394	0.8465	0.4078

), then discuss the extracted JO parameters (

Table 4. Inverse JO intensity parameters (Ω₂, Ω₄, Ω₆).

Ion	Ω2 (×10−20 cm2)	Ω4 (×10−20 cm2)	Ω6 (×10−20 cm2)
Dy3+	4.910	3.368	2.246
Sm3+	0.193	2.040	1.388
Er3+	2.257	3.380	0.700

) and radiative outputs (

Table 5. JO-derived electric–dipole (ED) radiative rates and normalized ED fractions for the three selected ground-state transitions per ion. Calculations use optical model as in Section 2.4; absolute scaling as in Section 2.3. p_k are normalized within set S (not single-manifold branching ratios; see Section 2.1), and 95% CIs via Monte Carlo resampling of S_rel (see Section 3.1.3). As imposed by the method, f₁ + f₂ + f₃ = 1.000 (differences only from rounding).

Ion	Electronic Transition	λ (nm)	SED (×10−20 cm2)	Arad (s−1)	95% CI on Arad	pk	95% CI on pk	Arad,tot (s−1)
Er3+	4I15/2→4F7/2	487	0.934	133.9	116.3–153.2	0.268	0.233–0.306	500
Er3+	4I15/2→2H11/2	521	3.079	360.1	340.2–378.0	0.720	0.680–0.756
Er3+	4I15/2→4I11/2	974	0.339	6.0	4.5–7.6	0.012	0.009–0.015
Sm3+	6H5/2→6F7/2	1232	0.889	102.3	91.1–113.6	0.205	0.182–0.227	500
Sm3+	6H5/2→6F5/2	1380	0.587	258.7	246.0–270.9	0.517	0.492–0.542
Sm3+	6H5/2→6F3/2	1487	0.306	139.0	124.0–154.0	0.278	0.248–0.308
Dy3+	6H15/2→6F5/2	794	0.774	88.3	75.0–102.3	0.177	0.150–0.205	500
Dy3+	6H15/2→6F7/2+6H5/2	881	2.064	171.7	152.6–190.8	0.343	0.305–0.382
Dy3+	6H15/2→6F11/2+6H9/2	1257	8.379	240.0	218.7–260.9	0.480	0.437–0.522

), and finally assess robustness, limitations, and implications.

4.1. Absorption Features

The room-temperature absorption spectra across the UV–NIR window of the three samples, each doped with 5 × 10²⁰ cm⁻³ of the respective RE³⁺, are shown in Figure 2 (Er³⁺), Figure 3 (Sm³⁺), and Figure 4 (Dy³⁺). Spectra are presented in relative absorbance. Transition assignments follow the standard free-ion multiplet scheme (Dieke’s diagram [31]) and the reduced-matrix-element compilation of Carnall et al. [10], and are consistent with prior reports for related glass hosts [32,33,34,35,36]. Comparing the three spectra, a clear pattern emerges: Er³⁺ concentrates oscillator strength in the UV/visible, whereas Sm³⁺ and Dy³⁺ display comparatively stronger near-IR manifolds—again in line with the literature [32,33,34,35,36]. Band integration followed the methodology in Section 3.1.1.

Guided by the criteria in Section 3, we retained three ground-state bands per ion (see Section 2 for the definition of S). The integrated areas ${\mathcal{A}}_k$, S_rel,k, and matrix elements ∣⟨U^(t)⟩∣² (t = 2, 4, 6) appear in Table 1, Table 2 and Table 3 as input to Equation (6). The selected peaks are explicitly highlighted in Figure 2, Figure 3 and Figure 4.

For Er³⁺ (Figure 2), the visible features near ∼487, ∼521, and ∼974 nm are assigned to ⁴I_15/2→⁴F_7/2, ²H_11/2, and ⁴I_11/2, respectively, consistent with oxide/fluoride-containing glass spectra [33,37,38,39]. These transitions are well separated, enabling reliable baseline subtraction and integration.

Table 1 lists the integrated areas, normalized strengths, and corresponding matrix elements.

For Sm³⁺ (Figure 3), the spectrum from the ⁶H_5/2 ground state shows the expected ladder of near-IR ⁶F_J transitions. The three retained bands, centered at ∼1232, 1380, and 1487 nm, align with previous glass studies [34,35]. Table 2 summarizes the integrated areas, normalized strengths (${\mathcal{A}}_k$, S_rel,k), and ∣⟨U^(t)⟩∣².

For Dy³⁺ (Figure 4), from ⁶H_15/2 the three selected bands lie at ∼794 nm, 881 nm, and 1257 nm. The ∼881 nm feature is commonly assigned to a close pair ⁶F_7/2 + ⁶H_5/2, and the ∼1257 nm feature to ⁶F_11/2 + ⁶H_9/2 in glass hosts [40,41,42,43]; here, it is integrated as a single band (Section 3.2), with linearity preserved by summing sub-areas and the corresponding rows of U. The numerical inputs (${\mathcal{A}}_k$, S_rel,k, and ∣⟨U^(t)⟩∣²) are given in Table 3.

For each set of squared reduced matrix elements ⟨U^(t)⟩² used in the JO analysis, the numerical values were taken directly from Carnall’s tabulations for LaF₃ [10]. These tabulated values provide the U_k,t coefficients that populate the design matrix U in Equation (6).

Taken together, the bands highlighted in Figure 2, Figure 3 and Figure 4 and quantified in Table 1, Table 2 and Table 3 define, for each ion, the set S used in the inversion. The qualitative contrast observed here—visible-dominated Er³⁺ versus near-IR-dominated Sm³⁺—propagates quantitatively to the JO parameters and, ultimately, to the normalized radiative fractions p_k.

For completeness, we note that the selected sets include ΔJ = 2 channels (e.g., Dy³⁺: ⁶F_11/2; Er³⁺: ²H_11/2, ⁴I_11/2), which are classically regarded as hypersensitive [44,45]; their relative weights are discussed in Section 4.2.

4.2. JO Intensity Parameters: Trends and Host Interpretation

The lifetime-scaled JO parameters obtained by the inverse procedure (Table 4) lie within customary ranges for RE-doped oxide/oxyfluoride hosts and show the expected sensitivity to local structure [9,35,36,42,44]. The obtained Ω_t values fall within ranges reported for oxyfluoride hosts and reproduce the known ordering trends across ions [37,38,39,42,45]. While combinatorial JO analysis provides an alternative route [13,14], our protocol is complementary, tailored for cases where only relative spectra are available.

Two salient features are observed (structural trends). First, Ω₂—widely used as an indicator of site asymmetry and 4f–ligand covalency—follows Dy³⁺ > Er³⁺ > Sm³⁺, with values Ω₂ = 4.910, 2.257, and 0.193 (×10⁻²⁰ cm²), respectively. This ranking is consistent with stronger symmetry breaking and/or greater RE–O character around Dy³⁺, and a comparatively more centrosymmetric, more ionic (RE–F-weighted) environment around Sm³⁺ in these glasses [9,36,42,43,44,45]. Second, the empirical rule Ω₄ > Ω₆ holds for all three ions (Er³⁺: 3.380 > 0.700; Sm³⁺: 2.040 > 1.388; Dy³⁺: 3.368 > 2.246, in ×10⁻²⁰ cm²), in line with reports for oxide-leaning networks where the medium-range rigidity sensed by Ω₄ exceeds the longer-range polarizability captured by Ω₆ [35,36,42,44]

For anchor-independent comparison, we also consider anchor-independent metrics. With sum-normalization (Ω₂, Ω₄, Ω₆)/∑Ω_t, the resulting normalized vectors are Er³⁺ (0.356, 0.533, 0.110), Sm³⁺ (0.053, 0.563, 0.383), and Dy³⁺ (0.467, 0.320, 0.213). The quality factor χ = Ω₄/Ω₆ is 4.83 (Er³⁺), 1.50 (Dy³⁺), 1.47 (Sm³⁺), while the hypersensitivity index h = Ω₂/(Ω₄ + Ω₆) is 0.875 (Dy³⁺), 0.553 (Er³⁺), 0.056 (Sm³⁺). Together, these descriptors substantiate that Dy³⁺ exhibits the strongest site-asymmetry response (Ω₂ and h largest), Er³⁺ displays a comparatively rigid intermediate-range response (χ largest, Ω₆ smallest), and Sm³⁺ retains the most symmetric/least covalent local environment (Ω₂ smallest) in this oxyfluoride host [42].

The present composition (mol%: 60 SiO₂, 20 Al₂O₃, 10 Na₂O, 10 BaF₂) is oxide-leaning but fluorine-bearing. In such networks, (i) F⁻ lowers covalency and often reduces Ω₂ relative to purely oxide hosts, especially when RE–F contacts prevail [36,44,45]; (ii) Na₂O introduces non-bridging oxygens (NBOs) that can increase site asymmetry and promote RE–O coordination when RE³⁺ resides near modifier-rich regions [4,35]; (iii) Al₂O₃ increases cross-linking/field strength and contributes to intermediate-range rigidity, which tends to elevate Ω₄ over Ω₆ [4,35]; and (iv) BaF₂ supplies F while its heavy cation enhances local field, but only modestly affects polarizability compared with high-index heavy-metal oxides. Within this framework, the very small Ω₂ of Sm³⁺ suggests that Sm³⁺ sites in this glass are on average more symmetric and more ionic, plausibly with a higher F share in the first coordination shell and/or fewer nearby NBOs. By contrast, the large Ω₂ of Dy³⁺ points to stronger symmetry breaking and greater RE–O mixing, consistent with Dy³⁺ sampling more modifier-rich, O-bearing motifs. Er³⁺ lies in between for Ω₂ but shows the largest χ, indicating that the network’s intermediate-range stiffness (to which Ω₄ is sensitive) is particularly effective around Er³⁺—a behavior often reported for aluminosilicate-rich matrices [4,35,36,37], where Al–O cross-linking is significant. Therefore, the pattern Ω₄ > Ω₆ across all ions is fully consistent with an oxide-dominated backbone modulated by fluorine, i.e., an oxyfluoride with lower covalency than pure oxides, yet greater rigidity than fluoride-rich glasses [35,42,44].

Three internal checks reinforce this interpretation: (i) all Ω_t are positive and fall within customary ranges for oxide/oxyfluoride glasses; (ii) the ordering Ω₂ (Dy³⁺) > Ω₂ (Er³⁺) > Ω₂ (Sm³⁺) mirrors the prominence of hypersensitive (DJ = 2) channels noted in Section 4.1 [42,43]; and (iii) the normalized JO vectors and the indices χ = Ω₄/Ω₆, h = Ω₂/(Ω₄ + Ω₆) are invariant to the lifetime anchor and to the local-field factor (Section 2.4), providing an anchor-independent basis for cross-ion comparison under uniform optical assumptions.

4.3. Radiative Probabilities and Normalized ED Fractions P_k

Table 5 compiles the electric–dipole (ED) radiative rates and the corresponding normalized fractions for the three ground-state absorption transitions selected per ion. The ED line strengths S_ED were computed from the JO parameters (Table 4) and the squared reduced matrix elements (Table 1, Table 2 and Table 3), and the Einstein coefficients A_rad follow from Equation (2) under the optical model of Section 2.4; the absolute scale is set as in Section 2.3. In the present oxyfluoride host, these patterns should be interpreted against an oxide-leaning backbone modulated by F⁻ coordination and modifier-induced non-bridging oxygens (NBOs): in aluminosilicate/oxyfluoride systems, Ω₂ generally decreases with increasing fluoride character, whereas Ω₄/Ω₆ tracks intermediate-range rigidity [45]. We reiterate that the lifetime anchor τref sets only the global scale; Ω-ratios and normalized fractions pk remain invariant. Where measured lifetimes exist, they can directly replace τref, yielding ion-specific absolute values without altering cross-ion comparisons.

For Er³⁺, the partition is dominated by the blue–green ⁴I_15/2 →²H_11/2 line at 521 nm (f ≈ 0.72), with a secondary ⁴I_15/2 → ⁴F_7/2 channel at 487 nm (f ≈ 0.27); the NIR ⁴I_11/2 branch at 974 nm is minor (f ≈ 0.01). This reflects both the larger S_ED of the 521 nm band and the λ⁻³ dependence in Equation (2) [25,45], consistent with the comparatively large Ω₄/Ω₆ ratio for Er³⁺. For Sm³⁺, the near-IR set favors ⁶F_5/2 at 1380 nm (f ≈ 0.52), followed by ⁶F_3/2 at 1487 nm (f ≈ 0.28) and ⁶F_7/2 at 1232 nm (f ≈ 0.21). For the very small Ω₂ together with moderate Ω₄, Ω₆ (Table 4) is compatible with more ionic, more symmetric local fields—plausibly a higher RE–F share and/or fewer nearby NBOs typical of oxyfluoride environments [45]. Dy³⁺ exhibits the most uneven split: the mixed ~1257 nm feature (⁶F_11/2 + ⁶H_9/2) carries ∼48% of the ED return, the 881 nm mixed line (⁶F_7/2 + ⁶H_5/2) contributes ∼34%, and the 794 nm ⁶F_5/2 line accounts for ∼18%. The coexistence/mixing of the ⁶F_11/2 and ⁶H_9/2 upper terms around 1.3 µm in glasses is well documented [38,39,40,41] (integration of mixed features follows Section 3.1.2.). The prominence of Dy³⁺ at shorter wavelengths aligns with its larger Ω₂, consistent with Dy³⁺ sampling more asymmetric, O-richer modifier environments (NBO-bearing) despite the λ⁻³ penalty at longer wavelengths [38,39,45].

Uncertainties are reported in Table 5 as 95% confidence intervals (see Section 3.1.3). Typical half-widths are ~5–12% for A_rad and ~0.02–0.04 for p_k. Because p_k depend only on relative weights, both the p_k patterns and the JO-ratio interpretations of Section 4.2 are insensitive to the lifetime anchor and to the local-field model (Section 2.3).

4.4. Cross-Ion Comparison Under Uniform Optical Assumptions

Our extracted Ωt values and relative fractions are consistent with systematic studies across different RE³⁺ ions, including oxide and fluoride hosts [37], and more recent works on fluorotellurite glasses [46] and RE-doped ceramics [47], as well as the compositional surveys of silicate, borate, and phosphate glasses by Takebe et al. [48]. These comparisons reinforce that the observed Ω₂ suppression and Ω₄/Ω₆ trends in oxyfluoride hosts are fully aligned with the broader literature.

Because all three ions were analyzed with the same optical model (see Section 2.4), the same lifetime anchor (see Section 2.3), and the same preprocessing/inversion pipeline (Section 3.1 and Section 3.2), the cross-ion trends are directly comparable and, for normalized quantities, anchor-independent [19,26,27,36,39].

(i)

Site asymmetry/covalency: Ω₂ follows Dy³⁺ > Er³⁺ > Sm³⁺ (see Section 4.2; Table 4);

(ii)

Intermediate- vs. long-range response: The empirical rule Ω₄ > Ω₆ holds for all ions (see Section 4.2; Table 4);

(iii)

Radiative partitioning within the selected triplets: The normalized ED fractions p_k (Table 5) summarize how each ion distributes ED strength under identical assumptions. In brief, Er³⁺ is dominated by the blue–green ⁴I_15/2 →²H_11/2 branch with ⁴I_15/2  → ⁴F_7/2 secondary and ⁴I_11/2 minor; Sm³⁺ partitions more evenly across the near-IR set, led by ⁶F_5/2 with contributions from ⁶F_3/2 and ⁶F_7/2; Dy³⁺ is most asymmetric, with the longest-wavelength mixed (⁶F_11/2 + ⁶H_9/2) feature leading, followed by the mixed (⁶F_7/2 + ⁶H_5/2) line and ⁶F_5/2. For numerical values and uncertainties, see Section 4.3 and Table 5.

(iv)

Link to glass composition: In the 60SiO₂–20Al₂O₃–10Na₂O–10BaF₂ matrix, an oxide-leaning backbone (Si–O–Al cross-linking) promotes Ω₄ > Ω₆, while F⁻ coordination tends to lower covalency and suppress Ω₂. The very small Ω₂ of Sm³⁺ is therefore consistent with more ionic, more symmetric local fields (greater RE–F share/fewer nearby NBOs), whereas the large Ω₂ of Dy³⁺ suggests O-richer, asymmetry-enhancing neighborhoods (modifier/NBO influence). Er³⁺ combines moderate Ω₂ with the largest χ, consistent with stiffer intermediate-range environments in this aluminosilicate-rich glass [30,35,36,44,45].

(v)

Robustness: These qualitative rankings persist within the 95% confidence intervals reported in Table 5. Treating Dy³⁺ mixed features as single bands (with summed ⟨U^(t)⟩²) preserves the linear model and does not change the ordering of Ω_t or the p_k patterns; changes to n or τ_ref rescale A_rad, but leave Ω-ratios and p_k unchanged [19,38,39,40,41,42].

4.5. Practical Implications

The inverse JO route gives two things straight from uncalibrated absorbance, the intensity set Ω_t and the within-set ED fractions p_k—precisely the quantities that are often available in early glass screening or when mining legacy data [12,13,17,36,42]. Because all ions were treated using the same optical model (local-field model and lifetime anchor), the JO ordering (Ω₂:Ω₄:Ω₆) and p_k support like-for-like comparisons before absolute calibration. In practice, p_k provides a compact preview of the emission partition within the chosen triplets—blue/green share for Er³⁺, orange–red for Sm³⁺, and blue–yellow balance for Dy³⁺—without requiring absolute emission measurements [3,18,30,40,49].

From a host-tuning perspective, the present oxyfluoride composition (60SiO₂–20Al₂O₃–10Na₂O–10BaF₂) offers clear levers that map onto Ω_t and hence onto p_k. In oxide-leaning networks, Ω₄ > Ω₆ is expected and reflects intermediate-range rigidity [35,42,44]; increasing Al₂O₃ (cross-linking) typically raises Ω₄ relative to Ω₆ [35,36]. Adding F⁻ (via BaF₂ or other fluorides) tends to lower Ω₂ by reducing 4f–ligand covalency and site asymmetry [36,37,44,45,50], whereas Na₂O creates non-bridging oxygens (NBOs) that can increase Ω₂ by enhancing asymmetry and RE–O mixing [4]. Within this composition domain, our results suggest that Dy³⁺, with the largest Ω₂, benefits from O-richer, more asymmetric neighborhoods—useful for pronounced blue/yellow pairs or high-contrast sensing [30,40,43]; Sm³⁺, with very small Ω₂ and moderate Ω₄ and Ω6, suits smoother, more evenly shared orange–NIR partitions for broadband or multi-line emission [24,46,51]; Er³⁺ combines moderate Ω₂ with the largest χ = Ω₄/Ω₆, indicating stiffer intermediate-range environments and stable blue/green partitions—attributes aligned with narrowband applications and, from other manifolds, communications-band applications [18,35,36,52].

At the device level, Equation (2) gives A_rad ∝ λ⁻³ S_ED, with S_ED = ∑Ω_t⟨U^(t)⟩² [5,6,25,42]. For fixed matrix elements, the combination “larger Ω₂” + “shorter λ” tends to yield larger A_rad, providing a simple screening rule for faster radiative channels; conversely, smaller Ω₂ (as for Sm³⁺) typically flattens the ED partition within the set when a smoother spectrum is desired. Two practical considerations are worth noting: (i) the lifetime anchor rescales all A_rad (but not p_k or Ω-ratios), so absolute brightness still depends on the true (radiative + non-radiative) lifetime and pump overlap [42]; (ii) host phonon energy and concentration effects (multiphonon relaxation, cross-relaxation) ultimately limit usable quantum efficiency—especially in oxide-leaning glasses and partially mitigated in oxyfluorides [4,22,37,50].

In practice, this method supports rapid screening: three well-chosen ground-state bands per ion suffice to recover Ω-ratios, S_ED, and p_k under a common optical model [12,13,36]. If the goal is to boost Ω₂ (e.g., stronger hypersensitive response for Dy³⁺), compositions with more NBOs (higher Na₂O) and less F⁻ are indicated; if a rigid network is desired (enhanced Ω₄, suppressed Ω₆) for stable partitions as in Er³⁺, raising Al₂O₃ at fixed SiO₂ and moderating the fluoride content is a rational path [35,36,37,50]. These composition→ Ω_t → p_k links support decision-making for composition selection while keeping measurements minimal and comparable across iterations.

4.6. Limitations and Sensitivity

The absolute Einstein coefficients scale linearly with the imposed lifetime anchor (A_rad ∝ τ_ref⁻¹); as noted in Section 2.3, JO ratios and p_k are anchor-independent. Accordingly, our discussion emphasizes ratio-based quantities and cross-ion comparisons under uniform optical assumptions rather than absolute A_rad values [42].

Replacing λ_max by a barycentric wavelength in the λ⁻³ factor changes individual A_rad by at most a few percent (ΔA/A ≈ −3 Δλ/λ) and leaves the JO ratios and the within-set fractions p_k within the reported 95% confidence intervals (Section 3.1.3).

Potential biases can arise from several sources. (i) Band windowing and baseline: finite windows and local backgrounds control integrated areas; we used robust baselines and Monte Carlo resampling (see Section 3.1.3) in line with best practice for JO data treatment [17,42]. (ii) Magnetic–dipole admixtures: MD terms were neglected within the selected windows; small leakage would mainly perturb A_rad, but has limited impact on p_k because the latter renormalize within the triplet [9,42]. (iii) Stokes shift: using λ_em ≈ λ_abs in the λ⁻³ factor introduces a small systematic; for intra-4f transitions in glasses this is typically subdominant to baseline/window uncertainty and is encompassed by our resampling intervals (Section 3.1.3) [36]. (iv) Local-field model and refractive index: adopting the virtual-cavity factor at a representative n rescales A_rad and Ω_t together but leaves Ω-ratios and p_k unchanged; moderate dispersion or n-uncertainty therefore does not alter qualitative trends [19,26,27,36]. (v) Matrix-element provenance: the squared reduced elements ⟨U^(t)⟩² come from standard tabulations [10]; modest host-dependent J-mixing cannot be excluded [10,28,36]. (vi) Linear determinacy and conditioning: with three bands and three Ω_t, Equation (6) is minimally determined; accuracy relies on the linear independence of the three U-rows, motivating our choice of well-separated bands [42].

Our Section 3.1.3 resampling analysis indicates that these effects are captured by the 95% confidence intervals in Table 4 and Table 5 and do not overturn the qualitative trends (ordering of Ω_t; p_k patterns) [17,42].

4.7. Where This Approach Helps—And Where It Does Not

The inverse JO route is most useful when (a) spectra are available only in relative units, (b) three ground-state bands per ion are well resolved with stable baselines, and (c) a reasonable lifetime anchor can be adopted for scaling. It is not a replacement for a full absolute JO analysis when calibrated absorbance, thickness, number density, and n (λ) are known; in that regime, the classical fit remains the reference method [5,6,36]. In practice, the present workflow complements modified-JO toolkits by providing a minimal-input, physically anchored way to compare ions in a common host and to triage compositions efficiently, with the understanding that absolute brightness still depends on the true (radiative + non-radiative) lifetime, pump overlap, and concentration-dependent quenching [14,15,17,22,42].

Equivalence in the calibrated limit and complementarity: When absolute calibration is available, both classical and inverse JO solve the same linear system AΩ = S; any absolute rescaling multiplies S by a global factor and leaves Ω-ratios and p_k unchanged. Thus, the inverse framework coincides with classical JO for ratios in the calibrated limit, while extending JO to the widely encountered case of relative, uncalibrated spectra. All cross-ion conclusions emphasized here are based on these anchor-independent descriptors, with conditioning and robustness explicitly reported (Supplementary Table S1; Figures S1 and S2). The formal ratio equivalence in the calibrated limit is given in Appendix A.

In our data, Sm³⁺ exhibits a low 3 × 3 condition number (cond ≈ 4), and correspondingly tight robustness intervals for h and χ (Supplementary Table S1; Figures S1 and S2). In contrast, Dy³⁺ shows a higher 3 × 3 conditioning of U (cond ≈ 31), and the resulting intervals are wider—as expected from the lower independent information across the selected transitions (set by the ⟨U^(t)⟩² pattern in this host, not by the inversion scheme). Interval width therefore tracks the information content of the JO matrix rather than any heuristic band-counting rule.

5. Conclusions

We present an inverse application of the Judd–Ofelt (JO) formalism that operates on relative absorption spectra and uses a single lifetime anchor to set the absolute scale. Applied to Er³⁺, Dy³⁺, and Sm³⁺ embedded in the same oxyfluoride glass, the workflow returns the JO parameters; the anchor-independent ordering (Ω₂:Ω₄:Ω₆) and—after scaling—absolute Ω_t magnitudes and Einstein coefficients A_rad. For radiative partitioning within the three selected ground-state lines per ion, we report normalized electric–dipole fractions p_k (not single-manifold branching ratios). As imposed by the method, p_k and the relative ordering of Ω₂, Ω₄, Ω₆ are anchor-independent, whereas absolute A_rad and Ω_t scale linearly with the chosen τ_ref.

Despite using uncalibrated spectra, the extracted parameters fall within established ranges for oxide-leaning/oxyfluoride hosts and reproduce ion-specific trends: Ω₂ follows Dy³⁺ > Er³⁺ > Sm³⁺, and Ω₄ > Ω₆ holds for all three ions. The within-triplet p_k patterns (Er³⁺ dominated by 521 nm; Sm³⁺ more evenly split across 1232/1380/1487 nm; Dy³⁺ most asymmetric with 1257 nm > 881 nm > 794 nm) are consistent with the measured absorption and with literature expectations under similar hosts. Interpreted against the glass composition (SiO₂–Al₂O₃–Na₂O–BaF₂), these trends align with an oxide-leaning backbone (favoring Ω₄ > Ω₆), F⁻ coordination that suppresses Ω₂, and modifier-induced NBOs that enhance site asymmetry—yielding very small Ω₂ for Sm³⁺, large Ω₂ for Dy³⁺, and an intermediate, rigidity-biased response for Er³⁺.

Uncertainty was quantified by Monte Carlo resampling on the relative band strengths (baseline, window, and peak perturbations). The resulting 95% confidence intervals for A_rad (typically ~5–12%) and for p_k (~0.02–0.04) indicate that our qualitative conclusions are robust to the dominant processing choices. Two caveats apply: (i) absolute metrics inherit the external lifetime and local-field model; (ii) the λ⁻³ factor uses λ_em ≈ λ_abs, appropriate for intra-4f bands in glasses and sub-dominant to baseline/window uncertainties.

Practically, the method offers a minimal-input, uniform route to obtain JO descriptors and p_k maps from uncalibrated data, enabling like-for-like materials screening across ions and compositions before full calibration is available. Future work will extend the approach to additional ions and hosts, incorporate independent emission lifetimes and absolute absorbance to tighten the scale, and combine with structural/vibrational probes to deepen the link between local environment and optical response—thereby supporting rational host design for targeted photonic functions.