Optical Transitions and Excited State Absorption Cross Sections of SrLaGaO₄ Doped with Ho³⁺ Ions

Abstract

The spectroscopic properties of SrLaGaO₄ (SLO) crystal doped with Ho³⁺ ions were studied in this work. Absorption, emission spectra and decay dynamics of excited states have been measured and discussed using the Judd–Ofelt model. Photoluminescence emissions were attributed to transitions from the excited ³D₃, ⁵S₂, ⁵F₅, ⁵I₆ and ⁵I₇ multiplet manifolds. The experimental lifetimes for five excited states have been compared to the theoretical values, calculated using Judd–Ofelt theory, allowing for the determination of the multiphonon relaxation rates (W_nR) of the respective states. The experimental data were approximately on a line expressed by W_nR = W₀ exp(−αΔE) with W₀ = 0.5 × 10⁷ s⁻¹ and α = 2.6 × 10⁻³ cm. To discuss the excited state absorption (ESA) pathways, that originated from several excited levels, we used the Judd–Ofelt formalism allowing determination of the integrated cross section for ESA transitions.

1. Introduction

The strontium lanthanum gallate SrLaGaO₄ (SLO) and its solid solutions have been extensively investigated since they were considered as a promising substrate for high temperature superconducting (HTSc) films [1]. SLO doped with transition metal (TM) or rare earth (RE) ions was also considered as laser active media [2]. Recently, SLO has been investigated as a host lattice for phosphors, which exhibit unusual and interesting properties when activated by RE or TM ions [3,4]. In particular, when Tb³⁺ is introduced into SrLaGaO₄, long persistent green emission has been reported [3]. The duration of green afterglow was observed even after more than 3.5 h. A novel, far-red emitting SrLaGaO₄:Mn⁴⁺ phosphor with high brightness and excellent luminescent properties has been synthesized by the simple solid-state reaction and investigated in [5]. In addition, the luminescence spectra of SLO activated by Cr³⁺ were reported in [4]. It was shown that Cr³⁺ ions occupy more than four different sites in the SLO lattice, which is related to the random distribution of Sr²⁺ and La³⁺ ions.

The interest in studying RE³⁺ doped SLO strongly results both from its structural disorder, and the resulting inhomogeneous broadening of the optical transitions [6], as well as from the ability of this matrix to accept high concentrations of activator [1,2]. Such a broadening helps the generation of ultrashort pulses by mode locking. Thus, due to its good thermal properties and broad gain spectrum, Yb³⁺-doped strontium lanthanum aluminate crystal was reported as a promising material for power-scalable, broadly tunable CW and sub-100 fs mode-locked lasers at ~1 μm [7].

Holmium ion activated solids have been the subject of extensive investigations for the development of laser materials, phosphors and color displays [8,9,10]. The use of Ho³⁺ as a potential candidate for temperature sensing has been confirmed, and thermal-coupled levels of Ho³⁺ ions, ⁵F₄ and ⁵S₂, ⁵S₂ and ⁵F₅, have been investigated in temperature measurement applications [11].

In particular, much attention has been dedicated to the studies of up-conversion processes in Ho³⁺ doped low-energy phonon crystals, optical glasses [12] and glass ceramics [13]. Up-conversion materials have wide applications, such as for bio-imaging, photo-therapies, photo-catalysis, solar conversion, temperature sensing, etc. Generally, energy transfer up-conversion (ETU), excited state absorption (ESA), and photon avalanche (PA) are the three main channels for up-conversion fluorescence. The Ho³⁺ ion is very suitable for use in up-conversion processes because it has many long-lived intermediate metastable levels, from which ESA can take place resulting in a strong, green anti-Stokes luminescence from ⁵S₂ level which has been demonstrated in a number of Ho³⁺ doped materials after red and infrared excitation [14]. It was also demonstrated that ESA has an important influence on the performance of Ho-doped fiber lasers [15]. Thus, for the modeling of these up-conversion excitation processes the ESA cross sections have to be known.

Although the optical properties of Nd³⁺, Pr³⁺, Eu³⁺, Yb³⁺ and Tm³⁺ doped SLO crystals have been reported [3,6,7,16], the spectroscopic data on Ho³⁺ doped SLO have not, to our knowledge, been presented in detail. In [17] we reported on the nature of strong spectral lines broadening in Ho³⁺ doped SrLaGa₃O₇ and SrLaGaO₄ crystals which have been investigated by using the non-resonant fluorescence line narrowing (FLN) technique. Also, up-conversion ultraviolet emission from ³D₃ level in these compounds was observed in [18]. The partial energy diagram of Ho³⁺ ions in the SLO host, together with the strongest observed emission transitions, is illustrated in Figure 1.

Figure 1.

The partial energy diagram of Ho³⁺ ions in SrLaGO₄ crystal. Arrows indicate the emission transitions.

Recently, we studied ESA processes that involve the ⁵I₆ and ⁵I₇ energy levels of Ho³⁺ in singly Ho³⁺-doped ZrF₄–BaF₂–LaF₃–AlF₃–NaF (ZBLAN) glass [12] and also YAlO₃ (YAP), Y₃Al₅O₁₂ (YAG), LiYF₄ (YLF) and SrLaGa₃O₇ (SLG) crystals [14]. In the present investigation, we extend this work by identifying and quantifying a range of pump excited state absorption ESA processes in SLO that may affect the functioning of Ho³⁺-doped SLO green phosphor or laser.

2. Materials and Experiment

A SrLaGaO₄:Ho³⁺ crystal with activator concentrations of 0.3 at % was grown using the Czochralski technique at ITME Laboratory in Warsaw. The samples used were monocrystalline of good optical quality which, after orientation, were cut into 10 × 5 × 2 mm³ plates and polished carefully in order to meet the requirements for spectroscopic measurements. Polarized absorption spectra were measured in the 200–2900 nm range of wavelength at room temperature on a Cary-50 UV–Vis-NIR spectrophotometer (Varian, Inc., Palo Alto, CA, USA) equipped with Glan–Taylor polarizers.

Luminescence from the sample was excited using both pulsed and CW laser sources. For the pulsed measurements an optical parametric oscillator Surelite OPO (Continuum, Santa Clara, CA, USA) with YAG:Nd laser (Surelite II, Continuum Santa Clara, CA, USA) as a pumping source was used. CW excitation was performed with tunable Ti-sapphire laser (Coherent 899 Ring Laser, (Santa Clara, CA, USA)) pumped by argon laser (Coherent Innova 300, (Santa Clara, CA, USA). Optical signal from the sample was analyzed using DK480 monochromator (CVI Laser Corporation, Albuquerque, NM, USA) and detected by cooled photomultiplier tubes (EMI C1034-02 GaAs (EMI Electronics Ltd., London, England) or Hamamatsu 7102 (Hamamatsu Photonics K.K., Hamamatsu, Japan)) and PbS detector for visible and infrared range, respectively. Stanford SR 400 photon-counting system (Stanford Research Systems Sunnyvale, CA, USA) controlled by PC was used to perform data acquisition. Measurements of luminescence decays were recorded using Stanford SR 430 multichannel analyzer (Stanford Research Systems Sunnyvale, California, USA). Low temperatures of the samples were obtained with a Displex Model CSW-202 cryostat (Apd Cryogenics Inc., Allentown, PA, USA).

SrLaGaO₄ belongs to a large family of compounds of general chemical formula ABCO₄ where A = Ca, Sr, Ba; B = Y, La-Gd; C = Al, Ga. SLO crystallizes in a perovskite-like, tetragonal structure, space group I4/mmm with unit cell parameters of a = b = 0.38437(3) nm and c = 1.26880(15) nm, the volume of the unit-cell V = 178.27(1) ų and the number of structural units Z = 2 [19,20]. The crystal structure is built from CO₆ (GaO₆) layers formed in the ab plane. The Sr²⁺ and La³⁺ cations are distributed randomly between the layers, taking positions in nine coordinated sites with distorted C4v symmetry. The C cations have a coordination of six and are in a slightly distorted octahedral environment, while Sr and La ions are surrounded by nine oxygens. The elementary cell of SrLaGaO₄ is shown in Figure 2 with the crystal built along the c-axis.

Figure 2.

Schematic crystal structure of SrLaGaO4.

Since the radius of the Ho³⁺ ion (0.172 Å) is comparable to those of the La³⁺ (0.126 Å) and Sr²⁺ (0.131 Å), it can be assumed that both La³⁺ and Sr²⁺ sites can be occupied by Ho³⁺ in SrLaGaO₄. It has been shown that the random distribution of the A and B cations over different lattice positions leads to a disordered structure of these crystals resulting in the strong inhomogeneous broadening of the spectral lines.

3. Results and Discussion

3.1. Absorption Spectra and Optical Transition Intensity Analysis

The structure makes SrLaGaO₄ uniaxially anisotropic where the strength of the absorption lines is dependent on the direction of the incident light. Bearing in mind that the optical axis is parallel to the crystallographic c-axis, two orthogonal light polarizations, π (E || c) and σ (E ⊥ c), should be taken into account. Thus, in our spectroscopic study, we have measured the absorption characteristics for two principal light polarizations for a-cut 0.3 at% Ho³⁺ doped SrLaGaO₄ crystal. The measured spectra in the 300–2200 nm range are presented in Figure 3.

Figure 3.

(a) Visible and (b) Infrared part of the polarized absorption spectrum of SrLaGaO₄:Ho³⁺ crystal recorded at room temperature.

Once the polarized absorption spectra had been measured, integrated absorbances Γ were calculated according to Equation (1) [21]:

$$\overline{\Gamma }={\displaystyle {\int }_{band}\overline{\alpha }\left(\lambda \right)d\left(\lambda \right)}=-\frac{1}{L}{\displaystyle {\int }_{band}\ln [\frac{1}{3}\exp (-{\alpha }_{\pi }(\lambda )L)+\frac{2}{3}\exp (-}{\alpha }_{\sigma }(\lambda )L)]d(\lambda )$$ (1)

where L is the distance the light has traveled inside the sample, α_π(λ) and α_σ(λ) are the absorption coefficients for π and σ polarizations and λ is wavelength. Integrated absorbances determined in this way were used to obtain the experimental line strengths needed to perform the Judd–Ofelt [22,23] intensity analysis. Because the drawbacks and precision of the Judd–Ofelt theory have been analyzed in detail elsewhere [24], only the essential results have been included in this article. The main point of this approach is that the oscillator strength f_calc of the electric dipole transition between multiplets of lanthanide ion (J → J’) is described as:

$$f_{calc}=\frac{8{\pi }^{2}mc}{3h\left(2J+1\right)}\frac{1}{\lambda }\overline{\chi }{\displaystyle \sum }_{t=2,4,6}{\left|U_{J{J}^{\copyright }}^{\left(t\right)}\right|}^2{\mathsf{\Omega }}_t$$ (2)

where h is Planck’s constant, J is the angular momentum of the initial level, $\overline{\chi }=\frac{1}{3}({\chi }_{\pi }+2{\chi }_{\sigma })$ is an average local field correction factor, where for electric dipole transition ${\chi }_{\pi ,\sigma }=\frac{{(n_{\pi ,\sigma }^2+2)}^2}{9n_{\pi ,\sigma }}$, U^(t) are the doubly reduced matrix elements and Ω_t are empirically determined parameters. The value of oscillator strength can be also obtained from experimentally measured absorption or emission spectra. For the absorption line, the experimental oscillator strength f_exp, can be defined as

$$f_{exp}=\frac{m{c}^2{\epsilon }_0}{\pi e^{2}N}\overline{\Gamma }$$ (3)

where e and m are the electron charge and mass respectively, $\overline{\Gamma }$ is the integrated absorbance obtained from Equation (1) and c represents the light velocity.

From the least-square fit of calculated (f_calc) and measured (f_exp) oscillator strengths the three Ω_t intensity parameters were obtained. The reduced matrix elements needed for this calculation were taken from [25]. The values of calculated and measured oscillator strengths together with the average wavelengths for all measured absorption lines are presented in Table 1. The Judd–Ofelt intensity parameters were evaluated to be Ω₂ = 1.25 × 10⁻²⁰ cm⁻¹, Ω₄ = 0.42 × 10⁻²⁰ cm⁻¹, Ω₆ = 1.80 × 10⁻²⁰ cm⁻¹ with RMS deviation between the calculated and measured oscillator strength values of 5.3 × 10⁻⁶. The obtained measure of the quality of the fit is closed to values found by applying the Judd–Ofeld approach to other materials doped with holmium ions [26].

Table 1.

Oscillator strengths (f_exp—measured, f_theor—calculated) of absorption transitions for Ho³⁺ ion in SrLaGaO₄ crystal.

Transition 5I8 →	Wavelength λ [nm]	Oscillator Strengths
		${\overline{\mathit{f}}}_{\mathit{e}\mathit{x}\mathit{p}}$ [10−6]	${\overline{\mathit{f}}}_{\mathit{t}\mathit{h}\mathit{e}\mathit{o}\mathit{r}}$ [10−6]
5I7	2011	2.19	1.73
5I6	1169	1.13	1.34
5I5	902	0.33	0.24
5I4	748	0.03	0.02
5F5	647	3.37	2.29
5S2 + 5F4	541	3.86	4.04
5F3	488	0.92	1.58
5F2	476	0.10	0.92
3K8	470	0.29	0.84
5G6 + 5F1	454	20.7	6.68
RMS = 5.3 × 10−6

Comparison of Judd–Ofelt parameters between Ho³⁺:SLO and other Ho³⁺—doped hosts is shown in Table 2.

Table 2.

Calculated Judd–Ofelt intensity parameters for SrLaGaO₄:Ho³⁺ compared with several Ho³⁺ ion activated solids.

Material	Ω2 [10−20 cm2]	Ω4 [10−20 cm2]	Ω6 [10−20 cm2]	Ref.
SrLaGaO4 (SLO)	1.25	0.42	1.80	This work
SrLaGa3O7 (SLG)	2.23	0.85	1.81	[27]
ZBLAN glass	2.46	2.02	1.71	[12]
Y3Al5O12 (YAG)	0.10	2.09	1.72	[28]
LiYF4 (YLF)	1.16	2.24	2.09	[29]

From the calculated set of Ω_t intensity parameters the electric dipole transition probabilities A_JJ’ for emissions between J and J’ manifolds of Ho³⁺ were calculated using the equation:

$$A_{J{J}^{’}}=\frac{64{\pi }^4{e}^2}{3h(2J+1)}\frac{1}{{\lambda }^3}\overline{\chi }{{\displaystyle \sum _{t=2,4,6}\left|U_{J{J}^n}^{(t)}\right|}}^2{\Omega }_t$$ (4)

Evaluated probabilities A_JJ’ of the radiative transitions from the excited states, the branching ratios β_calc and resulting radiative lifetimes are presented in Table 3. The values of radiative lifetimes of the ³D₃, ⁵F₃, ⁵S₂, ⁵F₅, ⁵I₄, ⁵I₅, ⁵I₆ and ⁵I₇ were calculated to be: 477 μs, 217 μs, 211 μs, 372 μs, 5.34 ms, 3.37 ms, 2.83 ms and 8.02 ms, respectively.

Table 3.

Probabilities of radiative transitions, branching ratios and radiative lifetimes of excited states of Ho³⁺ in SrLaGaO₄ crystal.

Transition		SrLaGaO4:Ho3+
		A [1/s]	β	τR
3D3 → 3G5		22	0.011
	5G5	70	0.034
	3K7	32	0.015
	5G6	20	0.009
	3K8	821	0.374
	5F3	25	0.011
	5F4	35	0.016
	5F5	11	0.005
	5I4	50	0.023
	5I6	106	0.141
	5I7	794	0.361
	5I8	87	0.039
∑A = 2097			1	477 μs
5F3 → 5F4		0	0.000
	5S2	0	0.000
	5F5	1	0.000
	5I4	112	0.024
	5I5	50	0.011
	5I6	265	0.057
	5I7	761	0.165
	5I8	3427	0.742
∑A = 4615			1	217 μs
5S2 → 5F5		0	0.000
	5I4	64	0.013
	5I5	64	0.014
	5I6	256	0.054
	5I7	1835	0.388
	5I8	2511	0.531
∑A = 4731			1	211 μs
5F5 → 5I4		0	0.000
	5I5	11	0.004
	5I6	140	0.052
	5I7	471	0.175
	5I8	2068	0.769
∑A = 2690			1	372 μs
5I4 → 5I5		9	0.050
	5I6	70	0.371
	5I7	90	0.478
	5I8	19	0.100
∑A = 187			1	5.34 ms
5I5 → 5I6		9	0.031
	5I7	173	0.583
	5I8	115	0.386
∑A = 297			1	3.37 ms
5I6 → 5I7		30	0.084
	5I8	324	0.916
∑A = 353			1	2.83 ms
5I7 → 5I8		125	1	8.02 ms

The emission characteristic of the crystal was measured at room temperature in the 250 nm–2.5 μm spectral range after CW excitation. All experimentally observed emission lines corresponding to transitions from the ³D₃, ⁵F₃, ⁵S₂, ⁵F₅, ⁵I₆ and ⁵I₇ excited states of SrLaGaO₄:Ho³⁺ are presented in Figure 4. The emission spectrum is dominated by ⁵S₂ → ⁵I₈ transition, resulting in the strongest luminescence line at about 550 nm.

Figure 4.

(a) UV-Vis and (b) IR part of polarized emission spectrum of SrLaGaO₄: Ho³⁺ recorded at room temperature.

3.2. Luminescence Dynamics

Fluorescence decay curves were registered after wavelength selective pulsed excitation. Figure 5 shows decay profiles of the ³D₃, ⁵S₂, ⁵F₅, ⁵I₆ and ⁵I₇ emissions in 0.3% Ho³⁺ doped SrLaGaO₄ measured at 300 K. The fluorescence lifetimes of ³D₃, ⁵S₂, ⁵F₅, ⁵I₆ and ⁵I₇ manifolds were determined with estimated relative errors of 1% to be of 35 μs, 98 μs, 6 μs, 1 ms and 4 ms, respectively. Due to the large energy gap to the next lying lower level of about 4000 cm⁻¹ the ⁵I₆ and ⁵I₇ decays could be considered as being predominantly radiative which was confirmed by its temperature independence.

Figure 5.

Decay profiles of the ³D₃, ⁵S₂, ⁵F₅, ⁵I₆ and ⁵I₇ emission in Ho³⁺ doped SrLaGaO₄ crystal measured at 300 K.

Knowing values of radiative and measured lifetimes, the rates of multiphonon relaxations can be simply calculated using the equation:

$$W_{nr}=\frac{1}{{\tau }_{obs}}-\frac{1}{{\tau }_R}$$ (5)

Calculated radiative lifetimes, measured at low temperature lifetimes, the energy gaps to the next lower level and the multiphonon transition rate for excited states of Ho³⁺ in SrLaGaO₄ are shown in Table 4.

Table 4.

Calculated radiative lifetimes τ_R, lifetimes observed at T = 10 K τ_obs, energy gaps ΔE, and obtained multiphonon transition rates W_nr for excited states of Ho³⁺ in SrLaGaO₄ crystal.

Manifold	SrLaGaO4:Ho3+
	τR [µs]	τobs [µs]	ΔEi(i−1) [cm−1]	Wnr [s−1]
3D3	477	35	2100	26,500
5S2	211	151	2794	1883
5F5	372	7	~1950	140,169
5I6	2833	1200	3426	480
5I7	8015	4000	4225	125

Dependence of the multiphonon transition probability W_nr on the energy gap to the next lower level ΔE can be described as an exponential function [30]:

W_nr = W₀ × exp(−αΔE) (6)

The W_nr(ΔE) dependence, together with the data for low phonon ZBLAN glass [31], is shown in Figure 6. The values of W₀ and α in Equation (6) were calculated as 0.5 × 10⁷ s⁻¹ and 2.6 × 10⁻³ cm, respectively.

Figure 6.

The multiphonon relaxation rate W_nr as a function of energy gap ΔE for the excited states of Ho³⁺ ion in SrLaGaO₄—red line. Open circles denote experimental points. For comparison the data for ZBLAN glass are also presented—blue solid line.

As can be observed from Figure 6 the nonradiative decay rates in SLO are about one order of magnitude higher than in ZBLAN. The difference in results from higher in the case of SLO of the the phonon frequency distribution function, which is approximately hω_max = 707 cm⁻¹ [32], than in ZBLAN matrix, which is 580 cm⁻¹ [3].

3.3. Excited State Absorption Transitions

The high number of long-lived multiplets present in Ho³⁺ acting as good population reservoirs leads to the possibility of ESA transitions into higher lying energy levels. The intermediate levels for ESA are therefore ⁵I_J states (J = 5, 6, 7). Using the Judd–Ofelt formalism it is possible to estimate the ESA integrated cross sections by means of the following expression:

$${\sigma }_{int}=\underset{\lambda 1}{\overset{\lambda 2}{{\displaystyle \int }}}\sigma (\lambda )d\lambda =\frac{8{\pi }^3{e}^2}{3hc \left(2J+1\right)}\chi {\displaystyle \sum }_{t=2,4,6}{\left|U_{JJ’}^{\left(t\right)}\right|}^2{\Omega }_t$$ (7)

where λ is the mean wavelength corresponding to the ESA transition.

With this method, the ESA transition cross sections in the spectral range between 400 and 5000 nm in Ho³⁺-doped SLO were calculated and are presented in Figure 7, together with the GSA cross sections.

Figure 7.

Spectrum of calculated ground state absorption (GSA) and excited state absorption (ESA) integrated cross sections σ in SrLaGaO₄:Ho³⁺ crystal.

Figure 7 also illustrates that several infrared excitation wavelengths could lead to population of the ⁵S₂ state from which green anti-Stokes emission results. These processes for excitation in the 720–750, 870–920 and 960–1000 nm spectral ranges are interpreted in Figure 8 and are confirmed by the excitation spectrum of up-converted ⁵S₂ green emission presented in Figure 9. Similar spectra were also reported for other Ho³⁺ activated systems [14].

Figure 8.

Energy level diagram of Ho³⁺ presenting different paths of populating the ⁵S₂ state of Ho³⁺ involving GSA and ESA processes.

Figure 9.

Excitation spectra of SrLaGaO₄:Ho³⁺ crystal when monitoring the strongest ⁵S₂ → ⁵I₈ emission transition at 550 nm, T = 300 K.

On the other hand, the energy level diagram in Figure 8 illustrates that in the up-conversion excitation of ⁵S₂ emission several near-resonant transitions, with participation of low energy phonons, are involved. It can also be observed that energies of GSA and ESA transitions are in close proximity, so it is difficult to indicate one responsible process. We have focused our attention on the 750 nm excitation band, as it was reported to be responsible for the efficient photon avalanche process that occurs in Ho³⁺ activated solids [33].

As results from Figure 8, after 750 nm excitation, the first step is the weak quasi-resonant GSA ⁵I₈ → ⁵I₄ absorption, followed by non-radiative de-excitation to the ⁵I₆ and ⁵I₇ metastable levels which act as intermediate levels. The next steps are: a strong resonant ESA ⁵I₇ → ⁵S₂, ⁵F₄ and non-resonant ESA ⁵I₆ → ⁵G₆, ⁵F₁ transitions followed by non-radiative de-excitation to the ⁵S₂, ⁵F₄ levels from which an intense green emission is obtained.

Calculated GSA and ESA absorption cross section values for transitions involved in the 750 nm excitation process for SrLaGaO₄ and several other Ho³⁺ activated materials are summarized in Table 5. The cross-sections of the non-resonant transitions from ⁵I₇ and ⁵I₆ to the multi-phonon side bands were calculated in the framework of formalism developed by Auzel [34] who showed that the probability for Stokes excitation with respect to an energy gap ΔE to the electronic level is given by:

W_S = W_S(0) exp(−α_S ΔE) (8)

where α_S is related to the multi-phonon nonradiative decay parameter α_NR by

α_NR = α_S − (hω) − 1/Nln(N/S₀) (9)

where N = E/hω is the average order of the multi-phonon process, S₀ is the Huang–Rhys electron–phonon coupling parameter at 0 K and hω is the highest phonon frequency of the host.

Table 5.

Calculated GSA and ESA absorption cross section values for transitions involved in the 750 nm excitation process for SrLaGaO₄ and several other Ho³⁺ activated materials.

Material	5I8 → 5I4	5I7 → 5S2 + 5F4	5I6 → 5G6 + 5F1	Ref.
	σ (GSA) [10−20 cm2]	σ (ESA) [10−20 cm2]	σ (ESA) [10−20 cm2]
SLO	0.0053	1.242	0.903	This work
SLG	0.0037	1.318	0.116	[27]
ZBLAN	0.0025	1.220	0.942	[12]
YAG	0.0036	1.309	1.202	[28]
YLF	0.0037	1.405	1.080	[29]

The values of the integrated cross sections calculated according to Equation (7) are proportional to the line strength of the electric-dipole transition $Sed\left(J{J}^{\prime }\right)-{\displaystyle \sum }_{t=2,4,6}{\left|U_{JJ’}^{\left(t\right)}\right|}^2{\Omega }_t$. Considering the 750 nm excitation conversion to ⁵S₂ green emission following values of squared matrix elements, |U_λ|² should be considered. In Figure 10 the S_ed values’ dependence on Ω_t is presented.

Figure 10.

S_ed values’ dependence on Ω_t parameters for transitions involved in 750 nm up-conversion excitation.

It thus results that in the considered 750 nm excitation, the ESA (⁵I₇ → ⁵S₂ + ⁵F₄) is the most important process and that for its cross-section to be large, materials with a high value of the Ω₆ parameter should be chosen. Thus, according to the Judd–Ofelt parameters presented in Table 2, the highest (⁵I₇ → ⁵S₂ + ⁵F₄) ESA cross-sections could be expected for Ho³⁺:LiYF₄ crystal. This is confirmed by the data reported by Kuck et al. [35]. From Table 6 and Figure 10 it can also be seen that the ⁵I₈ → ⁵I₄ GSA is only Ω₆ dependent.

Table 6.

Doubly reduced matrix elements for selected transitions in Ho³⁺ ions.

| Ut | 2	5I8 → 5I4	5I7 → 5F4	5I7 → 5S2	5I7 → 5S2 + 5F4	5I6 → 5G6	5I6 → 5F1	5I6 → 5G6 + 5F1
|U2|2	0	0	0	0	0.0093	0	0.0093
|U4|2	0	0.1965	0	0.1965	0.0837	0	0.0837
|U6|2	0.0082	0.0349	0.4195	0.4544	0.1094	0.2383	0.3477

The ratio between the GSA and ESA cross sections (β = GSA/ESA) is an important parameter for establishing the photon avalanche process. It was proposed in [36] that a β value lower than 10⁻⁴ is necessary to expect avalanche up-conversion. Above this limit, the up-conversion emission does not exhibit any pump threshold. Evaluated here β = GSA/ESA is about 4 × 10⁻³ and is beyond the limit for photon avalanche up-conversion to occur.

4. Conclusions

The spectroscopy and the fluorescence dynamics of SrLaGaO₄ crystal doped with Ho³⁺ ions have been studied in detail and analyzed. Based on polarized absorption spectra, the Judd–Ofelt intensity parameters were determined and radiative transition probabilities for several Ho³⁺ excited states were calculated. This, together with the measured fluorescence lifetimes, allowed evaluation of the multiphonon relaxation parameters for SLO matrix.

Several ESA mechanisms which generate the green emission from the ⁵S₂ level in SLO under 750, 900 and 980 nm band excitations have been proposed. The Judd–Ofelt analysis, which is usually performed for transitions from the ground-state manifold, was used to determine the integrated cross-sections of the ESA transitions and to predict the possibility of photon avalanche up-conversion occuring under 750 nm pumping.

Author Contributions

Conceptualization, M.M. and M.K.; methodology M.M. and M.K.; formal analysis, M.M. and M.K.; investigation, M.M. and M.K.; writing—original draft preparation, M.M.; writing—review and editing, M.K. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on reasonable request from the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.