406/473 nm Pump-Band Absorption Cross Sections and Derivative-Based Line-Shape Descriptors in Er³⁺/Ho³⁺:Y₃Ga₅O₁₂

Abstract

We establish a general, device-oriented procedure to extract absolute pump-band metrics from room-temperature UV–Vis (ultraviolet–visible) absorbance—including the absorption coefficient α(λ), per-active-ion cross-section ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$, the effective per-active-ion absorption cross section ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ and derivative-based line-shape descriptors. As a representative case study, the procedure is applied to nanocrystalline Er³⁺/Ho³⁺:Y₃Ga₅O₁₂ over the 350–700 nm spectral range. After baseline correction and line-shape inspection assisted by the numerical second derivative of the absorbance, we extract conservative peak positions and the full width at half maximum across the visible 4f–4f manifolds. At the technologically relevant pump wavelengths near 406 nm (Er-addressing) and 473 nm (Ho-addressing) bands, resulting absorption coefficients are α = 0.313 ± 0.047 cm⁻¹ and α = 0.472 ± 0.071 cm⁻¹, respectively. The corresponding per-active-ion ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ of (3.62 ± 0.54) × 10⁻²² cm² and (5.46 ± 0.82) × 10⁻²² cm², referenced to the measured optical path length L = 0.22 ± 0.03 mm (approximately 15% propagated relative uncertainty; explicit 1/L rescaling). Cross sections are reported per total active-ion density (Er³⁺ + Ho³⁺). The spectra exhibit Stark-type substructure only partially resolved at room temperature; the second derivative highlights hidden components, and we report quantitative descriptors (component count, mean spacing, curvature-weighted prominence, and pump detuning) that link line-shape structure to absolute pump response. These device-grade metrics enable rate-equation modelling (pump thresholds, detuning tolerance), optical design choices (path length, single/multi-pass or cavity coupling), and host-to-host benchmarking at 295 K. The procedure is general and applies to any rare-earth-doped material given an absorbance spectrum and path length.

1. Introduction

Rare-earth-doped garnets are model solid-state systems in which localized 4f electrons are strongly shielded by filled 5s/5p orbitals and therefore couple only weakly to the lattice [1]. The resulting intra-configurational 4f–4f transitions are comparatively narrow and chemically robust; their energies and widths encode crystal-field symmetry, the magnitude of Stark splitting, and electron–phonon coupling. In nanocrystalline Er³⁺/Ho³⁺:Y₃Ga₅O₁₂ (yttrium gallium garnet, YGG), the trivalent rare earths substitute on the low-symmetry dodecahedral site of the garnet host, producing manifold structures where Kramers ions (Er³⁺) display Stark doublets and non-Kramers ions (Ho³⁺) split into singlets [2,3]. These Stark-split manifolds are only partially resolved at room temperature (RT, 295 K), due to inhomogeneous broadening and phonon coupling, but nonetheless play a crucial role in understanding optical transitions and energy transfer processes in devices [4].

In garnets like Y₃Ga₅O₁₂, the low-symmetry dodecahedral site and relatively rigid lattice set the scale of Stark splitting and electron–phonon coupling, shaping both the visibility and width of 4f–4f bands at 295 K [5,6]. This host-field environment underpins the pump-band contrast we quantify at 406/473 nm and explains why band crowding differs across visible manifolds [5]. In Er³⁺/Ho³⁺:Y₃Ga₅O₁₂, trivalent rare earths occupy the dodecahedral (Y) site of the garnet host [7,8]. Derivative-aided inspection using the numerical second derivative of the ultraviolet–visible (UV–Vis) absorbance spectrum is a suitable approach to reveal substructure without over-decomposition at 295 K [9,10,11,12].

Y₃Ga₅O₁₂ is a cubic garnet where trivalent rare earths substitute at a low-symmetry dodecahedral site [13,14]. In the blue/blue-violet range the host is essentially transparent [15], enabling efficient 4f–4f pumping with compact diode/DPSS (diode pumped solid state) sources. This combination of site symmetry and low background provides a favourable stage for channel-specific pumping at 406 nm (Er-addressing) and 473 nm (Ho-addressing).

From an applications standpoint, the blue–violet (about 406 nm) and blue (about 473 nm) pump bands are particularly valuable because these spectral bands coincide with compact diode-laser wavelengths [16] and address distinct manifolds of Er³⁺ and Ho³⁺, enabling selective pumping, energy-transfer pathways, and up-conversion schemes [17]. Visible-band excitation of Ho³⁺ is technologically relevant, with blue/green/red stimulated emission demonstrated in oxide and oxyfluoride hosts [18]. Because the garnet host is essentially transparent in this spectral range, these pump bands allow efficient excitation with minimal background absorption. In Er³⁺/Ho³⁺:Y₃Ga₅O₁₂, the approximately 406 and 473 nm pump bands address distinct manifolds, enabling channel-specific pumping as shown by prior excitation and photoluminescence studies [19], with analogous wavelength-choice trade-offs documented in other mixed-ion systems (for example, Er/Yb at 940 versus 976 nm and Tm/Ho at 790–805 nm) [20,21,22].

For materials selection and device design, qualitative band assignments are not sufficient: one requires absolute absorption metrics—the absorption coefficient α(λ) and the effective per-active-ion cross section ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$, with λ the wavelength—along with conservative line-shape descriptors (peak energies and full width at half maximum (FWHM)). These quantities feed directly into rate-equation models and pump-threshold estimates, enable host-to-host benchmarking at fixed dopant density, and constrain energy-transfer efficiencies when combined with emission spectra. Reporting α(λ) and ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ with propagated uncertainties and explicit 1/L rescaling ensures transferability across geometries and datasets. Accordingly, this study emphasizes absolute scaling of the absorption spectrum, a measured optical path, and derivative-aided inspection to inform line-shape analysis, including structure-aware descriptors—defined here as metrics responsive to component spacing and local curvature in the pump band—derived from numerical derivatives (component count N_c, mean spacing $\overline{\Delta \lambda }$, curvature-weighted prominence K, and detuning measure Δ). To the best of our knowledge, this is the first time the 406/473 nm bands of Er³⁺/Ho³⁺:Y₃Ga₅O₁₂ are placed on such an absolute, uncertainty-propagated footing at 295 K, bridging laboratory-grade metrology and device-level design inputs.

Co-doping Er³⁺/Ho³⁺ is also pursued for wide-range optical thermometry, where channel-specific pumping helps decouple pathways. While thermometry is not the focus of this study, we make explicit what is added relative to prior practice: a traceable, device-oriented basis at 295 K for the 406/473 nm bands, with absolute α(λ) and ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ and the above noted structure-aware descriptors, so that pump choice and detuning tolerance are specified rather than assumed.

1.1. Physical and Mathematical Framework

In transmission geometry, the decadic absorbance and transmittance are related by

$$A\left(\lambda \right)=-{\log }_{10}\left(T\left(\lambda \right)\right)\mathrm{with} T\left(\lambda \right)=I\left(\lambda \right)/I_0\left(\lambda \right),$$

(1)

where $I$ and $I_0$ denote the transmitted and incident intensities, respectively. Using a measured optical path L, the absorption coefficient is

$$\alpha \left(\lambda \right)=\ln \left(10\right)A\left(\lambda \right)/L=2.303A\left(\lambda \right)/L.$$

(2)

Dividing by the active-ion number density N yields the species-specific per-ion cross section $\sigma \left(\lambda \right)$:

$$\sigma \left(\lambda \right)=\alpha \left(\lambda \right)/N.$$

(3)

When multiple species contribute within an unresolved band, we report the per-active-ion cross section referenced to the total active-ion density ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$:

$${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)=\alpha \left(\lambda \right)/N_{\mathrm{t}\mathrm{o}\mathrm{t}},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} N_{\mathrm{t}\mathrm{o}\mathrm{t}}=N_{\mathrm{E}\mathrm{r}}+N_{\mathrm{H}\mathrm{o}}.$$

(4)

As ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ distributes α(λ) over all active ions, ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ provides a lower bound to the species-specific cross section when one ion dominates; if the dominant species has fraction x_s = N_s/N_tot, then σ_s ≈ ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$/x_s.

For isolated lines in frequency ν, the integrated cross section relates to the oscillator strength f via

$$\int \sigma \left(\nu \right)d\nu ={\displaystyle \frac{\pi e^2}{m_{e}c{n}_{\mathrm{r}}}}f{L}_{\mathrm{L}\mathrm{F}}\left(n_r\right) ,$$

(5)

where $n_{\mathrm{r}}$ is the refractive index, $e$ the electron charge, $m_e$ the electron mass, $c$ the speed of light, and $L_{\mathrm{L}\mathrm{F}}\left(n_r\right)$ an appropriate local-field factor. In this study, we use ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)=\alpha \left(\lambda \right)/N_{\mathrm{t}\mathrm{o}\mathrm{t}}$ directly (independent of $n_r\left(\lambda \right)$); only conversions to local-field-corrected quantities (for example, absolute oscillator strengths) require a dispersion model and a specific local-field choice.

The same absorption data enter the dipole–dipole (Förster) spectral-overlap [23,24],

$$J=\int F_D\left(\lambda \right){\epsilon }_A\left(\lambda \right){\lambda }^{4}d\lambda , R_0^6\propto {\kappa }^2{\mathsf{\Phi }}_D{n}_r^{-4}J,$$

(6)

where $J$ is the spectral-overlap integral quantifying the wavelength-dependent overlap between the normalized donor emission $F_D\left(\lambda \right)$ and the acceptor molar absorptivity ${\epsilon }_A\left(\lambda \right)$; ${\kappa }^2$ is the dipole–dipole orientation factor and ${\mathsf{\Phi }}_D$ is the donor quantum yield and R₀ is the Förster radius (the donor–acceptor separation at which the energy-transfer efficiency is 50%).

For completeness, when emission spectra and lifetimes are available at the same temperature, the McCumber relation links emission and absorption cross sections under quasi-thermal equilibrium. We do not apply it here (absorption-only, co-doped sample at 295 K), but we report absolute $\alpha \left(\lambda \right)$ and per-ion ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ in a form directly compatible with subsequent McCumber-based consistency checks [25].

1.2. Kramers Versus Non-Kramers Ions and Stark Multiplicities

Time-reversal symmetry enforces Kramers degeneracy in systems with an odd number of electrons: each crystal-field level is at least doubly degenerate (a Kramers doublet) at zero magnetic field. In trivalent lanthanides, this maps onto the 4f electron count: ions with odd 4fⁿ (half-integer J) are Kramers ions and each level remains a doublet; ions with even 4fⁿ (integer J) are non-Kramers and their levels can split fully into singlets [26]. In practice, the extent of this splitting depends not only on the ion type but also on the symmetry of the crystal field and the degree of inhomogeneous broadening.

A multiplet of total angular momentum J splits under the garnet’s low-symmetry dodecahedral field into (2J + 1)/2 doublets for Kramers ions and up to (2J + 1) singlets for non-Kramers ions. In the co-doped YGG considered here, Er³⁺ (4f¹¹; half-integer J) is Kramers, whereas Ho³⁺ (4f¹⁰; integer J) is non-Kramers. At room temperature, however, the expected number of components is often only partially observed, because inhomogeneous broadening (nano-scale disorder) and homogeneous phonon coupling merge weaker features. Cooling typically sharpens the lines and increases the number of resolvable Stark components [27]. The derivative-aided view used below is intended to highlight hidden components without over-decomposition, providing guidance for line-shape interpretation at 295 K [11,12].

1.3. Selective Pumping in Mixed-Ion Rare-Earth Systems: Why Wavelength Choice Matters

In co-doped hosts, selective pumping is often required to partition excitation pathways and avoid cross-excitation: one chooses spectral ranges where the target ion’s absorption dominates while the partner’s is weak. This is possible because 4f–4f manifold envelopes differ in energy and strength for different ions and hosts; picking the right pump minimizes spectral overlap and stabilizes device behaviour (gain competition, thermal load, parasitic transfer).

1.4. Implications for This Study

This is the first report to provide absolute, uncertainty-propagated ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ for Er³⁺/Ho³⁺:Y₃Ga₅O₁₂ at 295 K under diode-accessible blue/blue-violet pumping, bridging laboratory-grade metrology and device-level design inputs.

The methodology below turns such qualitative “pick the right range” guidance into device-grade, absolute numbers at 295 K: we place the 406/473 nm bands on an absolute footing (α, per-active-ion ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$, with propagated uncertainty and explicit 1/L rescaling) and quantify structure-aware usability via second-derivative descriptors: the component count N_c, the mean component spacing $\overline{\Delta \lambda }$, the curvature-weighted prominence K, and the pump-band detuning Δ. We also introduce two aggregate indices: a dimensionless selectivity S, and a curvature-overlap figure of merit (FOM). For context, IR (infrared) Ho³⁺ bands can be an order of magnitude stronger than visible 4f–4f lines (for example, Ho:YGG σ_abs ≈ (2.6–3.7) × 10⁻²¹ cm² near 1.9 µm), underscoring why absolute visible ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ at 406/473 nm are needed for quantitative pump design rather than heuristic extrapolation from IR data [28].

1.5. Scope and Contributions

In this paper, we present a general methodology to obtain absolute pump-band metrics and structure-aware descriptors from room-temperature absorbance and reanalyze a published, room-temperature UV–Vis absorbance spectrum of nanocrystalline Er³⁺/Ho³⁺:Y₃Ga₅O₁₂ as a case study (the raw absorbance for this material was reported previously in Ref. [19]), applying a physically transparent pipeline—light median plus mean smoothing, rolling fifth-percentile baseline correction, derivative-aided feature detection, and bounded Gaussian fitting—to extract (i) absolute absorption metrics α(λ) and per-active-ion ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ referenced to a measured pellet thickness of L = 0.22 ± 0.03 mm, with propagated uncertainty (about 15% dominated by δL/L) and explicit 1/L rescaling; (ii) a conservative catalogue of peak centres and FWHM across the visible 4f–4f manifolds; and (iii) quantitative derivative-based line-shape descriptors in the pump bands—N_c, $\overline{∆\lambda }$, K, and Δ—along with two aggregate metrics: S and FOM (see Section 2.2 below for defining fprmulas). All α and ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ values are reported per total active-ion density N_tot. Operationally, within the analysis settings in Section 2.2 and at 295 K, a higher FOM in the data used here is consistent with greater tolerance to small diode detuning at a fixed optical path and is intended only as a practical indicator for pump stability and path-length sizing.

We treat the technologically relevant pump channels at 406 nm (Er-selective) and 473 nm (Ho-selective), which coincide with widely available diode and DPSS laser lines, while the Y₃Ga₅O₁₂ host remains essentially transparent in this spectral range. Because the spectrum is co-doped (composite) and the lines are broadened at 295 K, the second derivative is used to reveal subcomponents without over-decomposition rather than to assign ions unambiguously. Ion-selective excitation at 406/473 nm has been demonstrated independently [19]. The study is quantitative and transferable in scope: we report uncertainty-annotated pump-band cross sections and derivative-based descriptors at 295 K, with explicit 1/L rescaling and a reproducible processing workflow, aimed at informing pump-scheme design, detuning-tolerance appraisal, and host-to-host benchmarking. Low-temperature or singly doped follow-ups can refine Stark assignments without altering the pump-relevant descriptors established here. The methodology is general; Er/Ho:YGG is used as a case study. Unlike the FIR (fluorescence intensity ratio) thermometry focus in Ref. [19], the current study provides absolute pump-band metrics and derivative-based descriptors to support device-level modelling at 295 K.

2. Data, Method and Definitions

2.1. Samples and Prior Characterization

Structural and microstructural verification follow Ref. [19] (Soler et al., 2020) for the same nanocrystalline Y₃Ga₅O₁₂ specimen co-doped with Er³⁺/Ho³⁺; no additional structural work was repeated here because the present contribution is an absolute scaling and line-shape analysis of the visible-band absorption recorded on that specimen. Er³⁺/Ho³⁺ co-doped Y₃Ga₅O₁₂ (nominal Y_2.8Er₀.₁Ho_0.1Ga₅O₁₂) was prepared by a citrate sol–gel route and heat-treated as in Ref. [19]. We analyzed the room-temperature UV–Vis absorbance dataset acquired in that earlier study with the same specimen in straight-through transmission; its optical path length was $L=0.22\pm 0.03$ mm and is adopted here for absolute scaling (consistent with the thickness range typically used for transmission measurements on garnet ceramics [29]; see Section 2.4). Details of reagents, thermal schedules, and phase/microstructure verification follow that previous work. No additional structural characterization was performed in the present study, as our analysis reprocesses the published transmission spectrum from the same specimen. Beyond absolute absorption (α, σ), we also quantified line-shape structure near the 406 and 473 nm pump bands using the second derivative of the absorbance. We reported component count, mean spacing, curvature-weighted prominence, and laser detuning, and we summarized structure versus absolute response through a dimensionless selectivity index and a curvature-overlap figure of merit. Formal definitions are given in Section 2.2 and used in Section 3 and Section 4.

The UV–Vis absorbance dataset analyzed here is the same as the one reported in Ref. [19] (see Section 3.2, Figure 2 in Ref. [19]). No new absorption measurements were performed. We re-processed that published dataset to obtain absolute α(λ) and per-active-ion ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$, adopting the same specimen and optical path length L as in Ref. [19]. Full experimental details (instrument model, double-beam configuration and spectral-response correction) can be found [19] (Section 3.2).

2.2. Derivative-Based Spectral Descriptors

• Band choice and robustness

The 15 nm spread was chosen to encompass room-temperature Stark substructure and typical diode linewidth plus drift. Sensitivity tests with the spread of 10/20 nm change S by less than 0.03 and FOM by less than 5%; a parallel 25 nm spread test yields ΔS < 0.05 and ΔFOM < 8%, consistent with the stated robustness (Figure S1 in Supplementary Materials), which also includes the 25 nm spread case). Here, “structure-aware” denotes descriptors that respond to local curvature (K) and intra-band component spacing $\left(\overline{∆\lambda }\right)$ around the pump, quantifying subcomponent contrast and crowding. FOM combines this structure term with the absolute ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left({\lambda }_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}\right)$, where ${\lambda }_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}$ denotes the pump wavelength at the window center, to reflect both absorption magnitude and tolerance to detuning or drift. A single global prominence threshold—set from the spectrum-wide noise floor—was applied to both bands; 20% spread adjustments did not alter the qualitative rankings of N_c, K, or $\overline{∆\lambda }.$Figure S1 demonstrates that the derivative-based descriptors remain stable across 10–25 nm spread analysis (ΔS < 0.05, ΔFOM < 8%), confirming that the chosen 15 nm spread provides a robust, parameter-insensitive basis for the structure-aware metrics used throughout.

• Interpretation

FOM (${\lambda }_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}$) combines absolute response and local structure density; larger ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ increases absorption per pass, while larger K and wider $\overline{∆\lambda }$ indicate stronger curvature-weighted contrast and lower congestion near ${\lambda }_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}$. Comparable FOM values across the bands reflect a trade-off between magnitude and structure-based robustness.

• Operational definition

Let A(λ) be the baseline-corrected absorbance and D₂(λ) = d²A/dλ² its numerical second derivative (central differences after light median plus mean smoothing; identical parameters for all analyses) [30]. For each pump band W_λ (15 nm spread around 406 or 473 nm bands), local maxima of −D₂ above a fixed prominence threshold are taken as candidate subcomponent centres {λ_i} (sorted ascending). Zero crossings of −D₂ delimit widths.

Component set and spacing are defined as

$$N_{\mathrm{c}}=\left\{{\lambda }_i\in W_{\lambda }\right\} \mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{s}, \overline{\Delta \lambda }={\displaystyle \frac{1}{N_{\mathrm{c}}-1}}\sum _{i=1}^{N_c-1}\left({\lambda }_{i+1}-{\lambda }_i\right) ,$$

(7)

respectively.

Curvature-weighted prominence is defined as

$$K_W={\sum }_{{\lambda }_i\in W_{\lambda }}\left[-D_2\left({\lambda }_i\right)\right]w_i,$$

(8)

with $w_{i }$ the local width estimated from the nearest zero crossings of −D₂, increases with the number and sharpness of resolvable subcomponents (arbitrary units; same thresholding for both bands).

Pump detuning is defined as

$$\Delta =\underset{i}{\min }\left|{\lambda }_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}-{\lambda }_i\right|.$$

(9)

Selectivity index (dimensionless) is defined as

$$\mathrm{S}={\displaystyle \frac{K_{406}}{K_{406}+K_{473}}}\in \left[\mathrm{0,1}\right],$$

(10)

so that larger S favours the 406 nm band; (1 − S) favours 473 nm band.

Curvature-overlap FOM which links structure to absolute response is defined as

$$\mathrm{F}\mathrm{O}\mathrm{M}\left({\lambda }_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}\right)={\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left({\lambda }_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}\right){\displaystyle \frac{K_W}{{\overline{\Delta \lambda }}_w}}.$$

(11)

Here ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is reported per total active-ion density, and $W_{\lambda }$ is the corresponding 15 nm spread. FOM scales as 1/L via σ_eff, whereas S is L-invariant.

For reproducibility, the same smoothing, prominence threshold and 15 nm half-width spread is applied to both bands; derivatives use the baseline-corrected A(λ). We report on N_c, $\overline{\Delta \lambda }$, Δ, K, S, and FOM. Uncertainties in ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ propagate into FOM. S is dominated by relative uncertainty in K₄₀₆ and K₄₇₃.

2.3. UV–Vis Spectrum and Pre-Processing

We reprocessed the UV–Vis absorbance dataset of Ref. [19] to obtain absolute α(λ) and per-active-ion ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ and to extract derivative-based line-shape descriptors, adopting the same specimen and optical path length L = 0.22 ± 0.03 mm as in Ref. [19]. Full instrumental and acquisition details are given in Ref. [19]. We analyze that published room-temperature absorbance spectrum (about 350–700 nm) as the input to the processing pipeline used here. Figure 1 shows the raw absorbance used as the starting dataset; rescaling details are provided in Section 2.4.

Figure 1. Raw room-temperature absorbance A(λ) of nanocrystalline Er³⁺/Ho³⁺:Y₃Ga₅O₁₂. No baseline correction or smoothing has been applied.

The slowly varying diffuse-scattering background was removed by subtracting a rolling fifth-percentile baseline. To stabilize numerical derivatives while avoiding artificial line narrowing, we applied two light smoothing steps in sequence: a short-band median (to suppress impulsive spikes) followed by a short moving average (to reduce residual high-frequency noise). Band lengths were intentionally kept minimal. The resulting working spectrum is the baseline-corrected, somewhat smoothed $A\left(\lambda \right)$; Figure 2 shows the smoothed trace together with the rolling baseline, whose difference defines the working spectrum. A raw-versus-working overlay (Figure S2) shows no resolvable shifts in peak centres and no measurable change in FWHM at the plotting resolution here.

Figure 2. Somewhat smoothed absorbance $A\left(\lambda \right)$ (solid yellow curve) along with the rolling 5th-percentile baseline (dashed blue line). Subtracting this baseline defines the working (baseline-corrected) spectrum, which underpins the absolute $\alpha \left(\lambda \right)$ and ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ reported using $L=0.22\pm 0.03$ mm from Ref. [19].

For line-shape analysis, derivative-aided peak finding and bounded Gaussian fits were used (see Section 2.2), yielding conservative peak centres and FWHM across the visible 4f–4f manifolds. Although $\alpha \left(\lambda \right)$ and ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ at the pump wavelengths can be obtained directly from the raw absorbance, smoothing and baseline correction were applied to (i) suppress high-frequency noise for reliable derivative analysis, (ii) enforce a physically meaningful zero-absorption reference, and (iii) ensure reproducible peak positions and widths.

• Robustness and quantitative choices

Replacing the rolling baseline with a spline baseline yields $\alpha$ at 406/473 nm within the stated ~15% uncertainty (dominated by the thickness term $\delta L/L$), indicating that conclusions are insensitive to the baseline model. For quantitative extraction at 406/473 nm, we used a 15 nm analysis spread, which balances local signal-to-noise and line-shape fidelity while avoiding neighbouring manifolds; using 10–20 nm spread gives indistinguishable peak centres within plotting resolution (Figure S2 (Supplementary Material)). At normal incidence no Fabry–Pérot fringes are resolved; any residual slowly varying contributions are captured by the rolling baseline, and Fresnel-related corrections are expected at the few-percent level, i.e., within the stated uncertainty. Index-dispersion effects may somewhat perturb absolute ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$, but remain small compared with the stated uncertainty and do not affect relative peak positions or widths.

2.4. Absolute Quantities, Measured Path, and Rescaling

• Thickness metrology

The optical path length used here is that of the same specimen whose UV–Vis spectrum was reported in Ref. [19]: $L=0.22\pm 0.03$ mm. We adopt the same value and uncertainty for the present reprocessing; the fractional thickness term $\mathsf{\delta }L/L\approx 0.14$ dominates the overall error budget.

• Absolute scaling

Absolute coefficients follow from Equation (2) and the effective per-active-ion cross section from Equation (4), with N_tot the total (Er³⁺ + Ho³⁺) number density from stoichiometry (Table 1). All reported absolute values refer to the considered path length; for other geometries, quantities rescale linearly with $1/L$:

$${\alpha }_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}}\left(\lambda \right)={\alpha }_{\mathrm{a}\mathrm{b}\mathrm{s}}\left(\lambda \right){\displaystyle \frac{L_{\mathrm{a}\mathrm{b}\mathrm{s}}}{L_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}}}}, {\mathsf{\sigma }}_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}}\left(\lambda \right)={\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{a}\mathrm{b}\mathrm{s}}\left(\lambda \right){\displaystyle \frac{L_{\mathrm{a}\mathrm{b}\mathrm{s}}}{L_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}}}}$$

(12)

Table 1. Constants used for absolute scaling (stoichiometry, density, measured L). See text more for details.

Quantity	Value	Comment
Molar mass, M (g·mol−1)	822.76	for Y2.8Er0.1Ho0.1Ga5O12
Bulk density, ρ (g·cm−3)	5.90	assumed for YGG
Unit density, nfu (cm−3)	4.32 × 1021	ρ × NA/M *
Er number density, NEr (cm−3)	4.32 × 1020	0.1 × nfu
Ho number density, NHo (cm−3)	4.32 × 1020	0.1 × nfu
Total active-ion density, Ntot (cm−3)	8.64 × 1020	NEr + NHo
Measured optical path, L (mm)	0.22 ± 0.03	used in all absolute scaling **

* N_A = 6.02214076 × 10²³ mol⁻¹ is the Avogadro’s number. ** Unless stated otherwise, α and ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ values include an overall 15% relative uncertainty dominated by δL/L ≈ 0.14. Values for other paths rescale with 1/L as above. The shallow V-shaped dependence of S on the band width (Figure S1) reflects the balance between truncation and spectral spill-over, confirming 15 nm as the optimum spread for stable structure-aware descriptors.

• Uncertainty propagation

Relative uncertainties were propagated in quadrature:

$${\displaystyle \frac{\delta \alpha }{\alpha }}\cong \sqrt{{\left({\displaystyle \frac{\delta A}{A}}\right)}^2+{\left({\displaystyle \frac{\delta L}{L}}\right)}^2,} {\displaystyle \frac{\delta \left(\sigma \right)}{\sigma }}\cong \sqrt{{\left({\displaystyle \frac{\delta A}{A}}\right)}^2+{\left({\displaystyle \frac{\delta L}{L}}\right)}^2+{\left({\displaystyle \frac{\delta N}{N}}\right)}^2 .}$$

(13)

In the case under study, δL/L ≈ 0.14 dominates, leading to about 15% overall relative uncertainty in α and σ; this uncertainty is reported throughout (see Tables 1–4).

3. Results

3.1. Spectrum Overview at 295 K

The room-temperature absorbance A(λ) of the co-doped Er³⁺/Ho³⁺:Y₃Ga₅O₁₂ sample spans about 350–700 nm (see Figure 1 for the raw spectrum and Figure 2 for the smoothed spectrum t along with the rolling baseline used for subtraction). The working dataset for all quantitative analysis is the baseline-corrected, somewhat smoothed absorbance spectrum (Section 2.3), which preserves weak 4f–4f features while suppressing high-frequency noise and low-frequency background. An overlay (Figure S2) shows no resolvable peak shift or measurable narrowing at the plotting resolution.

3.2. Pump-Band Absorption Metrics (Absolute)

The 15 nm spread choice is robust: varying within 10/20 nm spread keeps $S$ within 0.03 and FOM within 5%; an additional 25 nm spread test gives $\Delta S<0.05$ and $\Delta$FOM < 8% (Figure S1).

Absolute coefficients at the technologically relevant pumps follow from Equations (2)–(4) using the measured optical path L = 0.22 ± 0.03 mm. At 406 nm (approximately 3.054 eV) the smoothed absorbance is A = 0.00299, yielding α = 0.313 ± 0.047 cm⁻¹ and ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ = (3.62 ± 0.54) × 10⁻²² cm²; at 473 nm (approximately 2.622 eV) A = 0.00451, giving α = 0.472 ± 0.071 cm⁻¹ and σ_eff = (5.46 ± 0.82) × 10⁻²² cm². Uncertainties (about 15%) are dominated by δL/L ≈ 0.14 and propagate according to Section 2.4. Values for other geometries rescale with 1/L (see Equation (12)). Figure 3 shows absorbance zooms (within 15 nm spread) around each pump. The raw view resolves only the broad envelopes; the unresolved substructure motivates the derivative analysis below. Table 2 summarizes the metrics concisely. Sensitivity tests with 10, 20, and 25 nm spread leave the qualitative rankings of $S$ and FOM unchanged (Figure S1). The retrieved visible-band ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ values (a few of 10⁻²² cm²) are consistent with 4f–4f transitions in garnet hosts at 295 K, i.e., in the expected weak-line regime.

Figure 3. Smoothed absorbance $A\left(\lambda \right)$ in 15 nm spread around the pump wavelengths for (left) 406 nm and (right) 473 nm. The raw spectra resolve only the broad envelopes, whereas the somewhat smoothed working spectrum (solid gold curve) makes the local substructure more evident. Vertical dashed red line marks the nominal pump wavelengths. For comparison between raw and working spectra, see Figure S2 (Supplementary Materials).

Table 2. Pump-band absorption metrics at 406 and 473 nm bands. See text for details.

Wavelength, λ (nm)	Energy (eV)	Absorbance, A (Smoothed)	α (cm−1) *	${\mathit{\sigma }}_{\mathbf{e}\mathbf{f}\mathbf{f}}\left(\mathit{\lambda }\right)$ (cm2) *
406	3.054	0.00299	0.313 ± 0.047	(3.62 ± 0.54) × 10−22
473	2.622	0.00451	0.472 ± 0.071	(5.46 ± 0.82) × 10−22

* Optical path L = 0.22 ± 0.03 mm is used; uncertainties (about 15%) dominate by δL/L. For other thicknesses, rescale via Equation (12) is applied. Values are given per total active-ion density N_tot.

3.3. Peak Catalogue Across 350–700 nm

Local maxima on the baseline-corrected spectrum were identified using the detection procedure described in Section 2.2, and those maxima were used as robust seeds for the bounded multi-Gaussian fits. We adopt a fixed prominence threshold to avoid over-decomposition at 295 K. Figure 4 overlays the detected maxima on the working spectrum; the top twelve peaks (centers and estimated FWHM) are listed in Table 3. Because the spectrum originates from a co-doped sample and the lines are broadened at room temperature, assignments remain conservative; the catalogue is intended as a reproducible baseline for later refinement (for example, at low temperature).

Figure 4. Working absorbance spectrum, A(λ) (baseline-corrected, somewhat smoothed) at 295 K with detected local maxima. The twelve most prominent peaks within 350–700 nm (marked 1 to 12) are indicated and listed in Table 3 along with their centres and full widths at half maximum.

Table 3. Peak catalogue (top twelve maxima between 350 and 700 nm) extracted from the working spectrum (baseline-corrected, somewhat smoothed) from Figure 4. The full widths at half maximum (FWHMs) estimated from bounded Gaussian fits. A single, conservative global prominence threshold—common to both pump bands—was used; varying the threshold by 20% did not change the qualitative rankings (see Section 2.2).

Peak *	Peak Centre, λ0 (nm) **	Amplitude, A ***	FWHM (nm, Estimated) **
1	381.00	0.01993	7.47
2	418.50	0.01697	5.93
3	449.50	0.02628	12.41
4	454.50	0.03038	11.58
5	487.50	0.02308	6.01
6	518.50	0.02736	11.41
7	524.00	0.03840	9.94
8	538.00	0.04540	8.71
9	638.50	0.04102	21.93
10	647.50	0.03135	23.23
11	654.50	0.03060	23.34
12	664.50	0.01074	37.67

* Peaks were detected on the working spectrum using derivative-aided finding (Figure 4). ** Bounded multi-Gaussian fits were seeded by second-derivative maxima. FWHM is estimated from the fitted component. *** Amplitude is the baseline-corrected absorbance $A\left(\lambda \right)$ at ${\lambda }_0$ (dimensionless). Assignments remain conservative for a co-doped, room-temperature spectrum. The findings are as robust as of 20% change in the global prominence threshold keeping the qualitative ordering of $N_{\mathrm{c}}$, $\overline{∆\lambda }$ and $K$ unchanged (Section 2.2).

3.4. Second-Derivative Analysis and Quantitative Descriptors

The global second derivative D₂(λ) = d²A/dλ² (Figure 5) enhances substructure that is otherwise unresolved at 295 K due to inhomogeneous broadening, consistent with Stark splitting. To connect line shape to pump performance, we use the derivative-based descriptors defined in Section 2.2: $N_{\mathrm{c} }$ (see Equation (7)), $\overline{∆\lambda }$ (7), $K$ (8), $\Delta$ (9), $S=K_{406}/(K_{406}+K_{473})$ (10) and FOM (11). All derivatives and peak finding operate on the working spectrum (baseline-corrected, somewhat smoothed $A\left(\lambda \right)$; Section 2.3).

Figure 5. Second-derivative spectrum D₂ $=d^{2}A/d{\lambda }^2$ over about 350–700 nm at 295 K. The crosses indicate detected subcomponent centres (local maxima of −D_2.); vertical red dashed lines denote the pump wavelengths (406 and 473 nm). The derivative emphasizes substructure consistent with Stark splitting while the raw absorbance remains envelope-dominated at room temperature. Computed on the baseline-corrected, somewhat smoothed working spectrum (Section 2.3); smoothing stabilizes ${-D}_2$ without shifting peak centres or changing FWHM (Figure S2).

Figure 6 shows the 15 nm spread around each pump with the detected subcomponent centres (local maxima of −D_2. The corresponding descriptors are summarized in Table 4. For the dataset considered, $N_{\mathrm{c}}=4$ near 406 nm and $N_{\mathrm{c}}=3$ near 473 nm; $\overline{∆\lambda } \approx 8.5$ nm (for 406 nm) and about 5.8 nm (for 473 nm); and $\Delta \approx 1.0$ nm for both bands. Curvature weights give $K_{406}=0.0179$ and $K_{473}=0.00705$ (arbitr.), hence $S\approx 0.72$, indicating denser/sharper structure near 406 nm. Combining with the absolute cross sections yields ${\mathrm{F}\mathrm{O}\mathrm{M}}_{406}\approx 7.6\times {10}^{-25}$ and ${\mathrm{F}\mathrm{O}\mathrm{M}}_{473}\approx 6.7\times {10}^{-25}$, i.e., comparable “structure times absorption” merits at 295 K despite ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}(473 \mathrm{n}\mathrm{m})>{\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}(406 \mathrm{n}\mathrm{m})$. Absolute magnitudes rescale with $1/L$ via ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ (Section 2.4); $S$ is $L$-invariant. Robustness of the 15 nm spread (10/20/25 nm spread tests) is documented in Figure S1.

Figure 6. Second-derivative spectrum D₂ $=d^{2}A/d{\lambda }^2$ in 15 nm spread around the pump wavelengths (406 nm and 473 nm). The crosses indicate the maxima of −D₂ (detected subcomponent centres); vertical red dashed lines indicate the nominal pump wavelengths. The spread underpins the quantitative descriptors listed in Table 4.

Table 4. Derivative-based descriptors in the pump band spread of 15 nm (see Figure 6).

Spread (15 nm)	${\mathit{N}}_{\mathbf{c}}$ (Components)	Mean Spacing $\overline{\mathbf{\Delta }\mathit{\lambda }}$ (nm)	Curvature-Weighted Prominence, K (Arbitrary Units)	Detuning Δ to Pump (nm)	Selectivity S	${\mathit{\sigma }}_{\mathbf{e}\mathbf{f}\mathbf{f}}\left(\mathit{\lambda }\right)$ (cm2)	Figure of Merit (FOM)
406 nm	4	8.50	0.017884	1.00	0.717	3.62 × 10−22	7.62 × 10−25
473 nm	3	5.75	0.007047	1.00	0.283	5.46 × 10−22	6.69 × 10−25

Ion-specific selectivity (about 406 nm predominantly addressing Er³⁺ manifolds and about 473 nm predominantly addressing Ho³⁺) follows from independent excitation studies reported earlier [19]; the present absorption analysis constrains the spectral structure that underpins that behaviour.

3.5. Uncertainty and Sensitivity

All $\alpha$ and ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ values include the propagated effects of instrumental absorbance noise (below 3%) and path-length uncertainty ($\delta L/L\approx 0.14$), giving an overall about 15% relative uncertainty dominated by the thickness term inherited from Ref. [19]. Derivative-based descriptors were computed using identical smoothing parameters, prominence thresholds, and 15 nm spread for both pumps. The selectivity index $S$ is $L$-invariant, whereas the figure of merit FOM scales linearly with $1/L$ through its dependence on ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$.

4. Discussion

Since the present analysis uses linear ground-state absorption, inter-ionic energy transfer (for example., Er → Ho) does not alter the extracted $\alpha \left(\lambda \right)$ or ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$. Processes such as excited-state absorption or cross-relaxation require excitation/emission data and are outside the scope of the present study.

The per-active-ion cross sections retrieved at 406 and 473 nm bands fall in the 10⁻²² cm² range, as expected for visible 4f–4f transitions (parity-forbidden, crystal-field-allowed), and are one–two orders below the strongest Er³⁺ near-infrared bands (about 980/1530 nm) [31,32]. Single-ion Ho:YGG values near 1.9 μm (approximately 10⁻²¹ cm²) likewise illustrate that IR 4f–4f lines can exceed their visible counterparts [22]. What was missing for this co-doped host is supplied here: absolute, uncertainty-annotated α and ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ at 406/473 nm with explicit 1/L rescaling (Table 2), referenced to a measured path length [19]. These quantities provide the ground-state inputs for rate-equation modelling and enable quantitative comparison across geometries.

4.1. Single-Pass Implications

A first practical implication follows from the single-pass absorption at the measured thickness. With α = 0.313 cm⁻¹ (406 nm) and 0.472 cm⁻¹ (473 nm), a specimen of $L=0.22 \mathrm{mm}$ absorbs only $1-e^{-\alpha L}\approx 0.69\%$ and about $1.0\%$ per pass, respectively; the corresponding $1/\alpha$ lengths are approximately 3.2 cm (406 nm) and 2.1 cm (473 nm). Thus, short-path devices must rely on multi-pass or cavity coupling to achieve meaningful pump absorption at 295 K. Conversely, to reach about 10% single-pass absorption $L\approx 3.4 \mathrm{mm}$ at 406 nm and $L\approx 2.2 \mathrm{mm}$ at 473 nm are needed.

Due to $\alpha \propto N_{\mathrm{tot}}\hspace{0.17em}{\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$, increasing active-ion density raises absorption linearly until concentration-dependent broadening or energy transfer offsets the gain, a design trade-off that remains transparent when reporting per-ion quantities. For visible-band absorption in related garnets, reported per-ion cross sections are typically 10⁻²²–10⁻²¹ cm² at room temperature, consistent with parity-forbidden, crystal-field-allowed 4f–4f transitions; the obtained ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ values at 406/473 nm lie within that range.

4.2. Channel-Specific Implications at 406 nm (Er) and 473 nm (Ho)

In practice, pump deployment is ion-directed: one about 406 nm addresses Er³⁺ manifolds, whereas that about 473 nm addresses Ho³⁺ [19]. The ranged second-derivative analysis therefore provides channel-specific specifications rather than a binary choice. In the data used, the Ho-selective 473 nm channel provides with the larger ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ (Table 2), favouring absorption per pass, but its mean spacing is smaller ($\overline{\Delta \lambda }$ ≈ 5.8 nm) and the curvature-weighted prominence is lower (K₄₇₃ = 0.00705 arbitrary units). In contrast, Er-selective 406 nm channel shows larger K (0.0179 arb.) and a larger $\overline{\Delta \lambda }$ ≈ 8.5 nm, i.e., fewer but more contrasted subcomponents. Detuning to the nearest component is comparable in both bands (Δ ≈ 1.0 nm). The selectivity index S ≈ 0.72 quantifies this structural bias towards 406 nm, while the combined FOM values are comparable (Table 4).

Relatively short optical paths favour the Ho channel (473 nm), whereas multi-pass/cavity geometries or broader/less-stable sources can make the Er channel (406 nm) competitive despite its smaller ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$, exactly because its larger K and wider spacing reduce sensitivity to fine jitter and spectral congestion within the 15 nm spread.

4.3. What the Peak Catalogue Adds

The derivative-aided peak list (Table 3) provides reproducible seeds for bounded Gaussian fits and a conservative count of resolvable components at 295 K, extracted from the working spectrum (baseline-corrected, somewhat smoothed). This quantifies spectral congestion in the pump bands used for compact diode pumping, and links naturally to tolerance analysis (via $\Delta$ and $\overline{\Delta \lambda }$) when specifying pump linewidth/stability.

4.4. Relation to Prior Practice

Blue/blue-violet pumping near about 406/473 nm is commonly used to steer excitation pathways in Er/Ho platforms. In this study, we placed those bands on an absolute footing at 295 K and add structure-aware usability metrics in the visible 4f–4f manifolds, where robust experimental numbers have been scarce, complementing IR-centred datasets (for example, McCumber analyses [25] in related gallate garnets [33,34,35]). These ‘structure-aware’ descriptors (Section 2.2) depend on local curvature and component spacing in the pump band, not only on ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$.

4.5. Conversion Notes for Mixed-Ion Hosts

Since the spectrum is co-doped, the derivative-aided analysis is used here to measure structure, not to apportion species. The peak catalogue and second-derivative views quantify spectral congestion and detuning tolerance at 295 K without over-decomposition (Figure 4, Figure 5 and Figure 6; Table 3). Ion-selective excitation at 406/473 nm, i.e., Er- versus Ho-dominated channels, has been demonstrated by independent excitation/emission studies [19]; the present results complement that picture by placing the visible pump band on an absolute basis and by supplying structure-aware metrics where robust experimental data have been scarce [33].

Two conversion notes aid interpretation in mixed-ion hosts.

(i)

Reported ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ values are per total active-ion density N_tot = N_Er + N_Ho; if a given range is dominated by one species with fraction x_s = N_s/N_tot, a species-specific estimate follows as σ_s ≈ ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$/x_s (for example, for equal Er/Ho contents x_s = 0.5, σ_s ≈ 2 ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$).

(ii)

For device layouts it is convenient to work with decadic absorbance $A$ or with the product $\alpha L$. Using Equation (2), $A=\alpha L/2.303$ and $T={10}^{-A}$, so the single-pass absorbed fraction is ${\eta }_{\mathrm{abs}}=1-T=1-{10}^{-A}=1-e^{-\alpha L}$. Values reported here map directly to $A$, $\alpha L$, and ${\eta }_{\mathrm{abs}}$ for any path length $L$ via the stated $1/L$ rescaling (Section 2.4).

4.6. Limitations and Scope

Diffuse-scattering corrections (for example, integrating-sphere calibration) were not applied. Baseline alternatives (for example, spline) do not shift $\alpha$ or ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ at 406/473 nm beyond the ~15% uncertainty, which is dominated by the thickness term $\delta L/L\approx 0.14$ (Section 2.4). Refractive-index dispersion in the blue/blue-violet spectral range modifies local-field factors at the few-percent level and is sub-dominant to the stated uncertainty. Multiple internal reflections/etalon fringes were not modelled explicitly; no resolvable fringes were observed, and quite slowly varying components are mitigated by the rolling-quantile baseline. Thin-slab Fresnel corrections at normal incidence are expected to be at the few-percent level, again sub-dominant to the thickness uncertainty.

The uncertainty budget is dominated by $\delta L/L\approx 0.14$; tighter path-length metrology directly reduces the uncertainties in $\alpha$ and ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$. Descriptor values depend on fixed analysis parameters (light smoothing, rolling fifth-percentile baseline, prominence threshold); we held them constant across bands for fair comparison. Finally, this is an absorption-only analysis; integrating emission and lifetimes (for example, McCumber-based consistency checks or pump–probe gain) enables end-to-end predictions and deeper comparison with IR-centred datasets in gallate garnets.

5. Conclusions

In this paper, we reported room-temperature, absolute UV–Vis absorption data for nanocrystalline Er³⁺/Ho³⁺:Y₃Ga₅O₁₂: $\alpha \left(\lambda \right)$ and per-active-ion ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\lambda \right)$ (referenced to $N_{\mathrm{tot}}=N_{\mathrm{Er}}+N_{\mathrm{Ho}}$), a conservative peak catalogue, and structure-aware descriptors (for definitions, see Section 2.2) that link line shape to pump performance. The aggregate metrics—the selectivity index $S$ and the curvature–overlap FOM—indicate comparable FOMs at 406 and 473 nm: the Ho-selective 473 nm channel provides higher single-pass absorption, whereas the Er-selective 406 nm channel shows greater robustness to detuning and spectral congestion. The preferred option therefore depends on optical path/geometry and source stability at 295 K.

Absolute magnitudes are reported with propagated uncertainty (about 15%, dominated by $\delta L/L\approx 0.14$) and with explicit $1/L$ rescaling (Section 2.4), making the results directly usable for rate-equation modelling and for optical-geometry choices (single/multi-pass or cavity coupling).

Under reatively-small-signal conditions, the single-pass absorbed fractions at $L=0.22$ mm are found to be of about 0.69% (406 nm) and about 1.0% (473 nm); reaching approximately 10% requires paths of about 3.4 mm and about 2.2 mm, respectively.

This study quantifies not only where to pump (≈406 nm for Er, ≈473 nm for Ho), but also how strongly (absolute $\alpha$, ${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$), with what certainty (propagated errors and $1/L$ rescaling), and how robustly (structure-aware descriptors and FOM) at 295 K, turning qualitative guidance into channel-specific, quantitative design criteria.