1. Introduction

Tm³⁺-activated materials are extensively investigated because of their luminescent properties, which are important in two aspects: due to a broad infrared luminescence emission around 1.8 µm (³F₄ → ³H₆ transition), a relevant wavelength in fiber-based telecommunications [1–5], and due to their capabilities as ultraviolet and blue emitter (at 371 nm, 476 nm) following infrared (IR) excitation, i.e. through an up-conversion process, that makes these materials particularly interesting in nanomedicine and bioimaging [6, 7]. Besides, LiNbO₃:Tm³⁺combines the optical capabilities of Tm³⁺ with the electro-optic, acousto-optic and non-linear properties of LiNbO₃ making it an attractive material to develop optoelectronic devices. As a consequence of the outstanding properties of LiNbO₃:Tm³⁺, it has been recently used as active material for photon-echo quantum memories [8, 9] and laser action has been investigated and demonstrated in several configurations [1, 2], including π-polarized cw laser oscillation of Tm³⁺ ions in Zn-diffused channel waveguides operating at 1.76 µm under 795 nm pumping (to ³H₄ state of thulium) [10]. Nevertheless, the efficiency slope reported for these waveguide lasers is lower than that reported for other Tm³⁺ lasers [11, 12], a result that has been attributed to either a non-optimized cavity design or losses due energy transfer processes that affect Tm³⁺ luminescence.

This commonly used excitation wavelength is linked to one spectroscopic characteristic that makes Tm³⁺-doped materials outstanding for optical applications: the occurrence of an infrared cross-relaxation (CR) mechanism (³H₄ → ³F₄:³H₆ → ³F₄) that yields photoluminescence quenching of the near-IR emission at 795 nm, while doubly populates the 1.8 µm emitting level (³F₄ → ³H₆ transition). The knowledge of the depopulation mechanisms from ³H₄ state and of how structural changes in the host affect it, result crucial to unveil the efficiency of Tm³⁺ lasers and other characteristic optical properties of Tm³⁺-doped materials.

It is obviously possible to alter the probability of energy-transfer by doping with different concentrations of Tm³⁺ ions, since the distance between donor and acceptor ions in the host is different for different concentrations [13]. However, an increase in doping concentration introduces as well charge compensation defects in the lattice, and thus modifies the local environment of the ions. An alternative approach is the application of external hydrostatic pressure. Although applied pressure only produces a small variation in the distance between donor and acceptor for any practical range of pressures, it offers the advantage of being a highly controlled perturbation that affects the whole range of parameters influencing ion-ion interaction. Consequently, it offers a very rigorous test of energy transfer theoretical models.

Particularly interesting in this case, is the effect of pressure on the transition frequencies of Tm³⁺. It must be pointed out that the above mentioned CR scheme is a non-resonant process [13, 14], and therefore, the energy mismatch between the emission of the sensitizer and the absorption of the acceptor has to be energetically bridged by involving host phonons. Generally, the larger the mismatch is, the lower the efficiency of the CR process. In this way, it is expected that the application of high pressure increases the crystal-field strength on Tm³⁺ dopants producing a shift of the energy states, thus modifying the energy mismatch and CR probability, while keeping unaltered the chemical composition of the material.

The aim of this work is to study the energy shift of Tm³⁺ states and their dynamics under high-pressure conditions, in order to understand its effect over the infrared CR and also to establish the most advantageous conditions to enhance CR efficiency.

2. Materials and methods

The LiNbO₃:Tm³⁺ sample studied in this work has been grown by the Czochralski technique in air atmosphere using a platinum crucible and automatic diameter control by crucible weighting. The starting raw materials were congruent LiNbO₃ ([Li]/[Nb] = 0.945) and thulium oxide (purity 99.99%), both of them are commercially available and were used without further purification. An optically selected sample from the as-grown boule was analyzed through x-ray diffraction to check the crystal structure and to verify the [Li]/[Nb] ratio. The [Tm³⁺] concentration was determined both by x-ray fluorescence and optical absorption to conclude a value [Tm³⁺] = 2.5 mol% [15].

The absorption and emission/excitation spectra at ambient conditions were obtained with a spectrophotometer (Cary 6000i, Varian) and a fluorometer (Jobin-Yvon FluoroMax-2), respectively. For blue up-conversion luminescence experiments, the sample was excited with a Ti-Sapphire laser operating at 795 nm (100 mW).

Hydrostatic pressure experiments in the 0-12 GPa range were carried out on a membrane-type diamond anvil cell (DAC) and a Meryl-Basset-type MALTA DAC. Inconel gaskets of 200 μm thickness were pre-indented down to 40 μm thickness. Suitable 150 μm-diameter holes were drilled with a Betsa motorized electrical discharge machine. The DAC was loaded with suitable LiNbO₃:Tm³⁺ single crystal and ruby microspheres (<10 µm diameter) using either methanol-ethanol-water mixture (16:3:1) or silicon oil as pressure transmitters. In all cases, experiments were performed in nearly-hydrostatic regime and the pressure was calibrated from the ruby photoluminescence [16, 17]. For cw excitation inside the DAC, the 476.5 nm line of a Krypton laser (Coherent CR-500K) has been used, while for pulsed excitation, the tunable beam of an OPO (Vibrant B 355 II) has been focalized backward on the sample with a 20 × microscope objective. The emission was collected with a microscope objective that allows the selection of the measured signal (sample or ruby probes). For lifetimes and time-resolved emission and excitation spectra with pulsed excitation, the emission light was dispersed by a 0.32 m-focal monochromator (Horiba-Jobin-Yvon Triax HR 320) and detected with a fast intensified charge-coupled device (iCCD) with 5 ns time resolution. The excitation spectra were corrected from the pulsed energy using a beam-splitter together with a power-meter. For lifetime measurements, the photoluminescence decay signal was recorded with the iCCD. In continuous mode excitation, the fluorescence signal was carried out by optical fiber for spectral analysis in the 600 – 1700 nm range using Ocean spectrometers (HR2000 + and NIRQuest512) equipped with Si and InGaAs array detectors, respectively, using a prototype microscope [18].

3. Infrared cross-relaxation mechanism

Tm³⁺ ions have a rich energy level structure arising from their 4f¹² electronic configuration. Some of these levels are equally spaced, allowing energy transfer between Tm³⁺ ions. Particularly, a near-infrared CR process, that can be found in a wide variety of hosts [14, 19–21], takes place between the ³H₄ excited state and the ground state (³H₆) and populates the intermediate state ³F₄ (³H₄ → ³F₄:³H₆ → ³F₄) following the scheme of Fig. 1 (right). The activation of this mechanism yields a reduction of the ³H₄→³H₆ infrared emission (795 nm) and an increase of the population of ³F₄ state, and thus it eventually produces an intensity enhancement around 1700 nm (³F₄ → ³H₆). This CR process has important consequences also for the upper excited states of Tm³⁺_, directly responsible for ultraviolet (¹D₂ → ³H₆ at 371 nm) and blue (¹G₄ → ³H₆ at 476 nm) light emissions, since they can be populated through upconversion mechanisms. These mechanisms start on ³H₄ and ³F₄ infrared excited states, and thus the CR process that links these two states and modify their populations affects the intensity and dynamics of the ultraviolet and blue emissions [22–25]. A complete characterization of these processes implies thus to study all involved emissions that can be relevant or affect CR.

 

Fig. 1 (A) Optical absorption and (B) up-conversion photoluminescence via excitation at 795 nm of LiNbO₃:Tm³⁺ at ambient conditions. On the right a partial scheme of Tm³⁺ energy states and the selected excitation scheme is added.

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In this work we choose to directly populate the excited state ¹G₄ with 476 nm laser excitation, a strategy that allows the study of all the emissions and associated dynamics arising from this and lower excited states. This excitation provides the most efficient pumping for ¹G₄ multiplet (Fig. 1) and, subsequently, after relaxation via radiative and/or non-radiative paths, also populates the ³H₄ level, from which the cooperative energy transfer mechanism under study starts. In addition, this 476 nm excitation scheme is advantageous to characterize radiative emissions associated with ³H₄ state, since it is far from all the emission bands of interest and thus avoids unwanted overlap with the excitation beam.

With the aim of analysing the effect of external pressure in energy transfer efficiency, a single crystal of LiNbO₃:Tm³⁺ (90 × 80 × 20 μm³) was loaded in a membrane DAC for time-resolved spectroscopy. The selected pulsed-excitation wavelength (476 nm at ambient pressure) was finely re-tuned at every pressure to ensure the maximum emission intensity.

Incidentally, under the selected excitation wavelength, other Tm³⁺ luminescence emissions arising from some intermediate levels that also become sufficiently populated can also be monitored. All these emissions can affect not only CR processes in LiNbO₃:Tm³⁺ (altering other excited states populations of levels participating in the CR path, such as ³F₄, for instance) but also other important optical processes observed in the material. In fact, the excitation scheme adopted here is somehow the reversal of multiphoton up-conversion processes that lead to blue luminescence via infrared excitation at 795 nm (Fig. 1(b)), and are observed in LiNbO₃ and other Tm³⁺-doped materials of interest. Therefore we shall present a complete characterization of pressure dependence of all the luminescence emissions observed under 476 nm pumping.

3.1. Pressure dependence of the peak position and intensity

The pressure dependence of the emission spectrum associated to all the observable transitions from 600 nm up to 1700 nm has been investigated under both continuous and pulsed excitation, and from ambient pressure (P₀) up to 12 GPa. To ensure suitable noise-to-signal ratios, different integration times have been considered. Figure 2(a) shows the pressure dependence of the time-resolved emission spectra associated with the main emissions from ³H₄ state (³H₄ → ³H₆) as well as ¹G₄ → ³H₅. In the figure, vertical dashed lines have been added to indicate the position of the main peak of each transition at the starting pressure. The observed variations indicate that high pressure triggers red shift and broadening of the different peaks. Furthermore, the overall peak intensity strongly decreases with pressure, which could be connected with the activation of non-radiative mechanisms and CR processes. In particular, for ³H₄ → ³H₆ this modification would be related to a decrease of the photoluminescence lifetime, as will be discussed later on.

 

Fig. 2 (A) Pressure dependence of the emission spectrum corresponding to ³H₄→³H₆ and ¹G₄→³H₅ transitions after 476 nm excitation. (B) Decomposition of the emission spectrum at 2.2 GPa into four Lorentzian peaks. The inset shows the variation of the emission peaks with pressure.

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Each peak in the emission band can be associated to transitions from different Stark levels of LiNbO₃:Tm³⁺ as assigned elsewhere [22]. Following this assignment, to analyze the intensity, position and widths of the different peaks, each spectrum has been fitted to four Lorentzian profiles. Figure 2(b) shows an example of the resulting fit and the variation of the position of the emission peaks with pressure is summarized in the figure inset.

The pressure dependence of the overall visible-infrared emission spectrum has been studied under cw excitation at 476.5 nm, with the aim of exploring changes induced in the emission spectra under the same excitation/pressure conditions (Fig. 3). The main emission peaks shown in the figure have been analyzed following the same procedure used in Fig. 2(b). The results obtained for the shift of the different peaks are given in Table 1.It must be noted that the intensity of ³H₄ → ³H₆ peak (not shown in Fig. 3 since it is in the detection limit of the spectrometer and has been analyzed separately in Fig. 2) decreases with pressure, but its variation is slower than that observed in the time-resolved spectra of Fig. 2(a). This difference in the intensity pressure dependence is entirely due to pressure-induced excitation detuning when the excitation is achieved at the fixed wavelength of 476.5 nm, and not retuned at every pressure as in Fig. 2. In fact, taking into account that the red-shift of the ³H₆ → ¹G₄ absorption peak at 476 nm (Fig. 2(a)), derived from the ³H₄ → ³H₆ and ¹G₄→³H₄ pressure shifts, is about −2 cm⁻¹/GPa and the FWHM is 100 cm⁻¹, the pumping capability at 476.5 nm increases by about 70% from 0 to 12 GPa due to pressure tuning enhancement of the absorption band. Therefore, the associated increase in ¹G₄ population, and correspondingly of ³H₄ state with increasing pressure, palliates the loss of emission intensity due to non-radiative de-excitation. Consequently, as mentioned before, the pressure-induced intensity decrease is less rapid here than in Fig. 2(a).

 

Fig. 3 (A) Variation of the emission spectra of LiNbO₃:Tm³⁺ (2.5 mol%) in the visible and infrared ranges upon cw excitation at 476.5 nm. The infrared ³H₄→ ³H₆ emission peak at 12579 cm⁻¹ (795.0 nm) shown in Fig. 2(a) is missed here because it appears at the spectral limit of the visible spectrometer. (B) Magnification of the variation with pressure of the ³F₄→³H₆ and ¹G₄→³F₃ emission bands, and (C) pressure shifts of levels involved in the CR (³H₆→³F₄: ³H₄→³F₄).

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Table 1. Peak position at ambient pressure, ω_i(P₀), and linear pressure shifts, (∂ω_i/∂P), of the emission peaks associated to all the measured transitions. The most intense ones are highlighted in bold type.

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Different Stark sublevels experience different shifts under pressure, as shown in Table 1. Particularly interesting, is the case of ³H₄ and ³F₄ states, since they are involved in the CR process we are aiming to analyze. There is a noteworthy pressure-induced blue shift undergone by ³F₄ → ³H₆ peak ( +1.6 cm⁻¹/GPa) and a red shift of the most intense energy peak linked to ³H₄ state (³H₄ → ³H₆, −2.1 cm⁻¹/GPa). As a consequence, the corresponding ³H₄ and ³F₄ sublevels reduce significantly their energy distance (see Fig. 3(c)) at higher pressures, which would imply a strong red shift for the ³H₄ → ³F₄ transition (−3.7 cm⁻¹/GPa) related to the same Stark sublevels that gave rise to the same exact ³H₄ and ³F₄ transitions commented before (It must be noted that this ³H₄ transition is overlapped with ¹G₄ → ³F₃ and cannot be accurately measured). The opposite shifts of ³F₄ → ³H₆ and ³H₄ → ³F₄ transitions, directly involved in the CR process (see Fig. 1), reduce substantially the energy mismatch between both transitions, thus enhancing CR probability as pressure increases. In this way the CR energy-mismatch is 509 cm⁻¹ at P₀ and reduces to 445 cm⁻¹ at 12 GPa; i.e. there is a reduction of ca. 12%. This result is crucial to explain the CR enhancement under high-pressure conditions yielding reduction of blue and infrared emission intensities.

Finally, the relative intensity variations of the main emission peaks with pressure are presented in Fig. 4. For that purpose, the intensities have been normalized taking the ³H₄ → ³H₆ emission band at 795 nm as reference. Comparing the variation of the relative intensities of Fig. 4 with the pressure dependence of the ³H₄ → ³H₆ absolute intensity shown in Fig. 2(a), it can be concluded that all the emissions reduce their intensity with pressure, although each transition shows a different decrease rate. It must be noted that under cw excitation, the ³H₄ → ³H₆ peak experiences the strongest intensity reduction with respect to ¹G₄ → ³H₄, ³H₄ → ³F₄, and ³F₄ → ³H₆ peaks, whose relative intensities are reduced less than about 50% of ³H₄ → ³H₆ peak intensity. This result is important to interpret and justify the CR processes as the main de-excitation path responsible for the loss of intensity with pressure.

 

Fig. 4 Variation of the relative intensity of the main emission peaks of LiNbO₃:Tm³⁺ with pressure. The relative variation for each peak is compared with the variation of the ³H₄ → ³H₆ peak taken as reference.

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3.2. Pressure dependence of the emission lifetime

Generally, energy transfer processes shorten the lifetime of the donor state, in our case ³H₄. Eventually, it is possible that the acceptor state (here ³F₄) accounts as well for some modification, since the energy transfer population path can be active at the same time the acceptor state is being depopulated, producing some lengthening of the decay. However, in our case the acceptor state ³F₄ accounts for a lifetime more than one order of magnitude longer than the donor state, and thus no effect is expected on its decay time [14]. Consequently, the following study will be focused only on ³H₄ lifetimes.

The temporal dependence I(t) of the emission intensity of the 795 nm emission associated to ³H₄ → ³H₆ transition has been studied as a function of pressure in the range 0 - 10 GPa. Figure 5(a) depicts the temporal evolution, in a semilogarithmic scale, of the intensity at three particular pressure values. As it can be observed I(t) behaves as a single exponential in the explored pressure range, and its lifetime decreases from 50 μs at ambient pressure to 34 μs at 10 GPa, following an approximate linear dependence with pressure (Fig. 5(b)).

 

Fig. 5 (A) Intensity decay curves corresponding to the ³H₄ → ³H₆ emission (795 nm) after pulsed excitation into the¹G₄ state (475 nm) in LiNbO₃:Tm³⁺ (2.5 mol%) at different pressures. The decays can be fitted to single exponential functions (blue solid lines). (B) Pressure dependence of the associated lifetime calculated by fitting the experimental data I(t) to a single-exponential function. The grey dotted line is a linear fit to the data, added to guide the eye.

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All the experimental results presented above show the strong influence of high pressure on the spectroscopic properties of Tm³⁺ ions in LiNbO₃. Particularly, the intensity of ³H₄ → ³H₆ emission at 795 nm, especially in comparison to ³F₄ → ³H₆, and its lifetime behaviors indicate a strong dependence of the emission quantum efficiency. Consequently, the characteristic infrared CR process of Tm³⁺is behind the pressure induced changes, becoming, possibly, one of the main de-excitation paths of ³H₄ state.

In general, the pressure-induced intensity reduction, as well as the lifetime shortening, can be related to different pressure dependent effects, such as variations in the phonon energy and/or density of states, energy gaps, oscillator strengths or energy transfer probabilities. Bearing in mind all these effects, we have analyzed the different contributions to the relaxation of ³H₄ state. Therefore, the main objective of the present section is to quantify the possible incidence of these different variables, how they depend on pressure and to unveil which one is the most relevant.

There are two major ways in which pressure can influence the de-excitation transition probability, i.e. the reciprocal lifetime. On one hand by modifying the single-ion experimental lifetime (limit of low doping concentrations), either through radiative probability, 1/τ_rad, or through non-radiative multiphonon probability, 1/τ_NR, on the other hand by modifying the energy-transfer CR probability, W_ET. In terms of transition probability, it can be written as:

The possibility of an increase of 1/τ_rad as main effect for the experimental lifetime decrease is ruled out, since this would represent an increase in transition probability and therefore an increase of the emission intensity, contrary to observations (Fig. 2(a)). Also the effect of pressure over τ_NR, which is analyzed in the next section, is negligible. Finally, we must consider that if the effect of high pressure on W_ET makes it larger, it would also make lifetime shorter, in agreement with experimental observations.

(a) Multiphonon relaxation process: W_NR

One of the simplest options to account for ³H₄ lifetime reduction is an increase in the multiphonon relaxation probability (Eq. (1)). It is known that pressure modifies the phonon frequency and density of states, an effect that has been extensively studied in LiNbO₃ [26, 27]. For lanthanide ions, it is well established that the probability of multiphonon relaxation (W_NR) between two states depends exponentially on the energy gap between both levels (ΔE), following the well-known “energy gap law” [28–30]:

In Eq. (2) the more sensitive parameters to pressure are ħω and ΔE. For LiNbO₃, ħω shifts with pressure at a rate of 3.3 cm⁻¹/GPa [27], while the energy gap (³H₄ - ³H₅) increases with pressure at a rate of 2.2 cm⁻¹/GPa (Table 1). Bearing these values in mind, the increase in the multiphonon probability with pressure can be easily calculated through Eq. (2). We obtain a value of W_NR = 8.2 × 10⁻⁴ µs⁻¹ at ambient pressure and 9.7 × 10⁻⁴ µs⁻¹ at 11 GPa. This implies a total variation, ∆W_NR = 1.5 × 10⁻⁴ µs⁻¹, which is an order of magnitude smaller than the total CR transfer probability experimentally observed, ∆W_ET = 10.5 × 10⁻³ µs⁻¹ (Fig. 6(a)). In fact this non-radiative probability contribution (1% of the total variation) is within the experimental accuracy, and thus can be considered as a negligible contribution to data of Fig. 5.

 

Fig. 6 Dependence of the measured transfer probability on the hydrostatic pressure. The experimental W_ET data (blue points), have been obtained from the measured lifetime using Eq. (1), taking τ_W0 = 230 μs. The blue dashed-dotted line has been drawn to guide the eye. The figure includes the probabilities, calculated from Eqs. (3) and (5), for the three main orders of the interaction (d-d, d-q and q-q). (A) The calculated probabilities consider only variations in R_DA due to pressure-induced volume compression by means of the LiNbO₃ equation of state. (B) Calculated energy-transfer probabilities due to both variations in R_DA and refractive index of LiNbO₃.

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(b) Energy transfer mechanism: W_ET

Energy transfer processes, apart from triggering intensity changes on the emissions of the levels involved in the mechanism, also influence the lifetime of the donor level (³H₄ in the present case) since it is an additional relaxation path for that excited state (Eq. (1)). Considering the mentioned changes in the intensity of the emission bands (Fig. 3), the results obtained from the lifetime evolution of ³H₄ state (Fig. 5) together with the irrelevance of non-radiative multiphonon relaxation processes analyzed previously, it turns out that energy transfer should eventually increase the transition probability.

In order to make quantitative estimations of W_ET(P), it is useful to look at the microscopic (ion to ion) definition of W_ET. Within a multipolar expansion of the ion-ion interaction, W_ET is a sum of different contributions that can be written as [34, 35]:

The strong dependence of W_ET on the donor-acceptor distance makes it very sensitive to variations in doping concentration or external pressure. This behavior is consistent with the observed lifetime variation as a function of Tm³⁺ concentration, reported at ambient pressure elsewhere [14]. The lifetime reduction from τ_exp = 230 μs for [Tm³⁺] = 0.06 mol% to τ_exp = 50 μs for [Tm³⁺] = 2.5 mol% observed experimentally in LiNbO₃:Tm³⁺ is ascribed to de-excitation via energy-transfer CR, thus showing the relevance of this mechanism.

It has been already demonstrated in several hosts that in an accurate description of the CR process between Tm³⁺ ions ³H₄ → ³F₄:³H₆ → ³F₄, the dipole-dipole order is unable to describe it, and thus dipole-quadrupole and quadrupole-quadrupole orders must be taken into consideration. Particularly, the quadrupole-quadrupole order interaction is very relevant since it usually provides the best CR description as a function of the [Tm³⁺] concentration (i.e. R_DA) [31, 37–40]. The microparameters corresponding to different order parameters $C_{DA}^{(S)}$ are defined as [35, 36]:

Looking at Eqs. (3) and (4), it turns out that W_ET will be sensitive to pressure through different magnitudes. For instance, pressure compresses the material leading to a reduction in the donor-acceptor distance (R_DA). Consequently, the energy transfer probability would be larger at higher pressures as far as the R_DA dependence is concerned. We can quantify this effect using the Murnaghan’s equation of state for LiNbO₃:

If we now rewrite Eq. (3) as W_ET = W₀(R₀/R)^S, with W₀ = C_DA/(R₀)^S, and replace the distance dependence by the Murnaghan equation of state, it is easy to calculate how the pressure will modify the energy transfer probability due to R_DA shortening (which is about −20% change for a pressure of 10 GPa). At this point W₀ is considered as a constant, and is selected to allow every curve to have the W_ET value found experimentally at ambient pressure. For comparison purposes, the experimental values, expressed as an energy transfer probability, have been derived from τ_exp through Eq. (1) using τ_W0 = 230 μs [14]. The results for the three main orders of the interaction (S = 6, 8 and 10) are shown in Fig. 6(a).

It can be deduced from Fig. 6(a) that the shortening of R_DA has a relevant effect in the pressure dependence of energy transfer probability, and as expected from Eq. (3) this effect is stronger for higher orders of the interaction. As previously mentioned, the quadrupole-quadrupole interaction provides the best description of this specific CR process between Tm³⁺ ions. However, even though its contribution is larger than the dipole-dipole and dipole-quadrupole ones, it is not enough to explain the experimental energy-transfer probability shown in Fig. 6(a). Although higher orders of the interaction could be added to explain the observed W_ET(P) variation, this option is not consistent given that the probability of such orders is negligible [31, 37, 38, 40].

Next we focus on the energy-transfer microparameter defined in Eq. (4), the pressure dependence of which was assumed to be constant in our previous analysis. It can be seen that in every case, its value is dependent to the fourth power of the refractive index. Due to the elasto-optic effect in LiNbO₃, the refractive index increases with pressure, and its variation can be estimated through the differential Clausius-Mossoti equation by:

The last pressure-sensitive magnitude affecting microparameters in Eq. (4) is the overlap integral, since the pressure-induced shifts of the donor-emission and acceptor-absorption bands can substantially modify the energy transfer probability. According to the results shown in Table 1, donor emission and acceptor absorption approach each other at higher pressures (Fig. 3(c)), yielding increase of the overlap integral, and thus of the energy-transfer probability. It must be observed (Fig. 7(a)) that because the CR process is non-resonant, the overlap mainly takes place in a small energy range around the tails of the involved bands. By considering their respective peak shifts of −3.7 cm⁻¹ (6544 cm⁻¹) and +1.6 cm⁻¹ (6035 cm⁻¹) (see Fig. 2(c) and Table 1), the variation of the overlap integral with pressure can be calculated on the basis of fitted Lorentzian line-shapes that reproduce the spectra at atmospheric pressure in the overlap range. Then, the obtained line shapes of the donor and acceptor are normalized following the standard procedure to calculate the spectral overlap [35]. It has to be mentioned that the shift of the phonon energy can be considered as well in the overlap, since as it has been mentioned, the process is non-resonant and it could slightly modify the cross-relaxation probability. Nevertheless, as previously stated, the effect of pressure on the energy of the phonons is mostly negligible compared to other contributions, and no differences have been observed introducing this correction to the calculations.

 

Fig. 7 (A) Spectral overlap of the donor emission and acceptor absorption in the considered cross-relaxation process. (B) Experimental W_ET (blue points) and calculated (orange dotted line) predictions for quadrupole–quadrupole interaction considering all possible contributions: R_DA variations, refractive index changes and spectral overlap modification. The grey dashed lines have been calculated considering a ±0.1 cm⁻¹/GPa error estimation in the shift coefficients of the peaks in order to illustrate the uncertainty of the calculation.

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The effect of variation of the overlap integral together with contributions from R_DA and refractive index to the energy-transfer probability is shown in Fig. 7(b). The results clearly indicate that the overlap integral contribution definitively accounts for the experimental energy-transfer probability W_ET(P). The calculated W_ET fits fairly well to the experimental data by considering mainly the quadrupole-quadrupole (S = 10) interaction (orange solid line in the figure), while other interaction orders (d-d or d-q, not shown in the figure for clarity) give results substantially lower than the experimental values. This conclusion is in agreement with previous studies in which CR is analyzed as a function of Tm³⁺ concentration [37, 38, 40].

As a further analysis to understand the accuracy of the calculations, we have estimated the outcome that a ± 0.1 cm⁻¹/GPa experimental error in each peak shift would have on the final result. It is clear that an error in the calculated shift would only affect to the contribution coming from the overlap integral, and that it would affect twice, since two different peaks are involved (Fig. 7(a)). The obtained result, shown between dashed lines for the quadrupole-quadrupole interaction in Fig. 7(b), accounts for the observed data within the experimental accuracy of the experiments.

Present results provide fully consistency on the microscopic mechanisms describing the CR process in LiNbO₃:Tm³⁺ by analyzing the effects of volume reduction but keeping constant the Tm³⁺ content. For this purpose the use of high-pressure spectroscopy is crucial to achieve this goal. This demonstrates that the main reason for the reduction of ³H₄ lifetime and the concomitant loss of emission intensity at 795 nm when pressure increases, is caused by an increase of the energy-transfer CR probability through three main combined mechanisms: reduction of donor-acceptor distances, increase of refractive index and increase of the overlap integral between donor emission and acceptor absorption bands.

Up to that point, W₀ has been used as a fitting parameter at zero-pressure to separately study the influence of each magnitude on the total W_ET. However, quantitative estimations on the basis of quadrupole-quadrupole interaction using W₀ = C_DA/(R₀)^S and Eqs. (3) and (4), enable us to ensure that the quadrupole-quadrupole interaction is mainly responsible for CR. In fact, the experimental value W_ET(0) = 1.5 × 10⁻² µs⁻¹ (Fig. 7) can be accounted for within this model using Kushida’s equations for energy transfer, S = 10 and R₀ = 8.4 Å [43]. So that we obtain W₀ = 1.5 × 10⁻² μs⁻¹. The employed R₀ value deviates only 5% of the actual value 8.0 Å derived on the assumption of a homogeneous distribution of Tm³⁺ (2.5 mol%) in LiNbO₃, and thus confirms the adequacy of quadrupole-quadrupole interaction as most likely to CR. Instead, the next term contributing to the probability, i.e. the dipole-quadrupole (S = 8) is, according to Eqs. (3) and (4), two orders of magnitude smaller than the quadrupole-quadrupole interaction (S = 10) using R₀ = 8.4 Å, thus ruling out this order interaction in CR.

4. Conclusions

We have demonstrated that the calculated energy-transfer probability fits fairly well to the experimental data by considering quadrupole-quadrupole interaction. This conclusion is in agreement with previous works in which CR is analyzed as a function of the Tm³⁺ concentration. The present work provides fully consistency on the microscopic mechanisms describing the CR in LiNbO₃:Tm³⁺ by analyzing the effects of volume reduction while keeping constant the Tm³⁺ content. We also show that the main reason for the strong reduction of the main 795nm (³H₄→³H₆) emission intensity of Tm³⁺ and the corresponding lifetime decrease with pressure is caused by an increase of the energy-transfer CR probability through three combined mechanisms: reduction of donor-acceptor distances, increase of overlap integral favored by enhancement of the resonant conditions induced by pressure, and increase of the refractive index. All these magnitudes can be modified at the same time by applying an external pressure at variance with variations of Tm³⁺ concentration, mainly providing variation of R_DA. The present analysis indicates that high-pressure spectroscopy is crucial to unveil the quadrupole-quadrupole interaction as responsible for CR in LiNbO₃:Tm³⁺.