1. Introduction

Excited-state absorption (ESA) is a process of excitation of a single atom, ion or molecule from a lower-lying excited-state |1 > to a higher-lying one |2 > with the absorption of a photon [1]. ESA can occur only if the system has been already excited to the |1 > state from the ground-state |0 > by ground-state absorption (GSA), Fig. 1. Therefore, the excitation to the |2 > state via ESA involves at least two pump photons. The energy differences between the (|0 > and |1>) and (|1 > and |2>) states can be either similar (resonant) or different. In the latter case, two pump beams with different wavelengths are required to observe ESA. ESA is commonly observed in molecules and materials doped with transition metal (TM) ions [2–4]. It is less common for rare-earth (RE³⁺) doped materials because of their narrow bandwidth transitions. Among RE³⁺ ions, a larger variety of ESA processes occurs for those exhibiting rich (ladder-like) energy-level schemes [5–7].

 

Fig. 1. Two-photon processes: (a) excited-state absorption, (b) energy-transfer upconversion and (c) two-photon absorption. |0>, ground-state, |1 > and |2>, excited-states, |1'>, virtual state.

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ESA should be distinguished from other two-ion or two-photon processes, such as energy-transfer upconversion (ETU) and two-photon absorption (TPA). For ETU, two neighboring centers are both excited to the |1 > state and transfer non-radiatively the excitation energy, so that one ion is further excited to the higher-lying |2 > state and another one returns to the ground-state |0> [8], Fig. 1. The probability of ETU depends on the concentration of centers and their spatial distribution [9,10]. TPA is a process of simultaneous absorption of two pump photons with the same frequency leading to the population of an excited state |1 > through a virtual state |1'>, Fig. 1. It is a third-order process, and its probability depends quadratically on the light intensity.

Frequently, ESA is used to obtain emission from higher-lying excited-states with a frequency ν_e being higher than the pump photon frequency ν_p (an anti-Stokes process). ESA can also assist in obtaining Stokes emissions from such higher lying states. In this way, ESA can be a key process for upconversion pumping of RE³⁺-doped laser materials resulting in lasing in the infrared [11,12] or visible [13,14] spectral ranges. Concerning the latter, upconversion pumping is known for several rare-earth ions, such as Er³⁺, Pr³⁺, Tm³⁺ [15].

If the transitions |0> → |1 > and |1> → |2 > are not resonant in energy, the higher-lying excited-state can be populated either by using two different pump beams with photon frequencies ν_p1 and ν_p2, or via the photon avalanche (PA) effect [16]. PA was first evidenced by Chivian et al. for Pr³⁺ ions [17]. Furthermore, it was used to demonstrate upconversion pumping of laser materials doped with Pr³⁺ [18], Nd³⁺ [19] and Tm³⁺ [20] ions. The typical scheme of PA is illustrated in Fig. 2. The first step is a non-resonant (phonon-assisted) GSA to the |1 > state followed by a resonant ESA, |1> → |2 > . The second step relies on the cross-relaxation (CR) process between two adjacent ions, one being in the higher-lying excited-state |2 > and one - in the ground-state |0 > . Because of the non-radiative energy transfer, the first one experiences a de-excitation to the intermediate state |1 > and the second one is promoted to the same state. As a consequence, the population of the |1 > state increases thus feeding the ESA process. In this way, the higher lying |2 > state is efficiently populated even if the first GSA transition is non-resonant.

 

Fig. 2. Photon avalanche effect scheme: GSA and ESA – ground- and excited-state absorption, respectively, CR – cross-relaxation, -hν_ph and + hν_ph – absorption / generation of phonons.

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Among the rare-earth ions, thulium (Tm³⁺) is attracting a lot of attention for coherent light generation in the short-wavelength-infrared (SWIR) spectral range. Tm³⁺ has an electronic configuration of [Xe]4f¹² and its simplified scheme of energy levels is shown in Fig. 3(a). The most widely exploited Tm³⁺ laser transition originates from the metastable ³F₄ state (³F₄ → ³H₆, at ∼2 µm) [21] and less common ones – from the higher-lying ³H₄ state (³H₄ → ³F₄, at ∼1.5 µm and ³H₄ → ³H₅, at ∼2.3 µm [22]).

 

Fig. 3. Thulium ions: (a) simplified energy-level scheme showing laser transitions in the near- and mid-infrared; (b-c) pumping schemes of Tm³⁺ ions: (b) conventional pumping; (c) upconversion pumping, GSA and ESA – ground- and excited-state absorption, CR - cross-relaxation, NR – multiphonon non-radiative relaxation.

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To achieve laser emission in all these channels, Tm³⁺ ions are typically excited to the ³H₄ state owing to the intense ³H₆ → ³H₄ GSA transition spectrally overlapping with the emission of commercial AlGaAs laser diodes and Ti:Sapphire lasers. The CR process, ³H₄ + ³H₆ → ³F₄ + ³F₄, Fig. 3(b), is very efficient in many Tm³⁺-doped materials even at moderate doping levels, resulting in efficient population of the ³F₄ metastable state with a pump quantum efficiency approaching 2 (a two-for-one pump process) [23] and giving rise to high laser efficiencies particularly at ∼2 µm well exceeding the Stokes limit [24].

Owing to the complex ladder-like structure of Tm³⁺ energy levels, the presence of efficient CR process and a long-living metastable level, the upconversion pumping of Tm³⁺-doped materials relying on the PA effect for lasing at ∼2.3 µm was also realized recently [20,25], Fig. 3(c). It is based on the non-resonant GSA ³H₆ → ³H₅ followed by multiphonon NR relaxation to the metastable level ³F₄, followed by a resonant ESA ³F₄ → ³F_2,3 and efficient CR. Altogether, this leads to efficient population of both the ³F₄ and ³H₄ states.

As explained above, excited-state absorption is a key effect for realization of upconversion pumping of Tm lasers emitting in the mid-infrared, at ∼2.3 µm. Such lasers find applications in sensing of atmospheric species such as HF, CO, CH₄ or H₂CO, non-invasive glucose (C₆H₁₂O₆) blood measurements, or frequency conversion further in the mid-infrared. Unfortunately, to date, the information about the near-infrared ESA properties of commonly used Tm³⁺-doped laser crystals is relatively scarce.

In the present work, we aimed to perform a systematic study of excited-state absorption of Tm³⁺ ions in the near-infrared for a series of laser host materials (fluorides and oxides).

2. Methodology

2.1 Excited-state absorption by thulium ions

Figure 4 summarized the excited-state absorption transitions of Tm³⁺ ions exploited so far. The corresponding ground-state absorption transitions are shown as well. Note that we consider the case of resonant energy differences between the (|0 > and |1>) and (|1 > and |2>) states, Fig. 1(a). The transition ³F₄ → ¹D₂ falls in the blue spectral range (∼0.45 µm) and is resonant with the ³H₆ → ¹G₄ GSA channel. The transitions ³H₄ → ¹D₂ and ³F₄ → ¹G₄ both in the red are nearly resonant with the ³H₆ → ³F_2,3 GSA channel; they are crucial for demonstrating upconversion pumping of blue thulium lasers [26–28]. The ESA properties of various Tm³⁺ doped materials in the red spectral range have been extensively studied in the past [5,29].

 

Fig. 4. Summary of excited-state absorption transitions of thulium ions exploited so far (bold arrows – this work), the corresponding ground-state absorption is also shown.

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In the near-IR, the ESA transitions ³F₄ → ³F_2,3 and ³H₄ → ¹G₄ at ∼1 µm are close in energy with the ³H₆ → ³H₅ GSA channel. The ³F₄ → ³H₄ ESA transition at ∼1.5 µm is close to the ³H₆ → ³H₅ GSA channel. In the present work, we focus on two near-infrared ESA transitions, ³F₄ → ³F_2,3 and ³F₄ → ³H₄, which are relevant for UC pumping of mid-infrared Tm lasers. So far, these ESA channels have been studied only for Tm:Y₃Al₅O₁₂ [30]. Some incomplete ESA data were also reported for Tm:KY₃F₁₀ and Tm:LiYF₄ [31].

2.2 Experimental setup

The measurements of the ESA spectra of Tm³⁺-doped materials corresponding to the ³F₄ → ³F_2,3 and ³F₄ → ³H₄ transitions were carried out using a pump-probe method [1,32]. Its principle is the following: the first intense laser beam (the pump) excites Tm³⁺ ions into the metastable ³F₄ state. The second weak beam (the probe) propagates through the pumped area of the sample revealing the additional absorption caused by ESA.

The scheme of the pump-probe set-up is shown in Fig. 5. The sample was prepared as a relatively thin (a few mm thick) plane-parallel plate with both faces polished to laser quality to avoid scattering of the pump. It was placed between two pinholes (diameter: 500 µm) placed close to the sample faces. This provided a good spatial overlap between the pump and the probe beams, as well as helped to limit the spot size of the probe in the sample.

 

Fig. 5. (a) Scheme of the pump-probe setup used for ESA measurements: AOM, acousto-optic modulator, M1 and M2, folding mirrors, L1, focusing lens, L2 – L4, wide-aperture lenses, ν_pump and ν_probe, pump and probe modulation frequencies, respectively, P, polarizer, F, filter; (b) Energy-level scheme of Tm³⁺ ions illustrating the principle of the pump-probe method.

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As a source of pump radiation, we used a continuous-wave Ti:Sapphire laser (Coherent 890) delivering up to 1.5 W at ∼0.8 µm in the fundamental transverse mode (M² ≈ 1). The pump wavelength was tuned to match precisely the maximum of the ³H₆ → ³H₄ Tm³⁺ absorption band of the studied material. The pump beam was modulated at low frequency (ν_pump ≈ 10 Hz) by an acousto-optic modulator (Isomet, model 1205C-2) and it was focused into the sample using a lens L1 (f = 150 mm) resulting in a spot size (diameter) of ∼300 µm. The single-pass pump absorption was >50%. This resulted in a relatively uniform distribution of inversion through the sample.

A 100 W halogen-tungsten white-light lamp (Oriel, model 66184) with a broad emission spectrum close to that of a black body was used as a probe source. Its output was modulated at a higher frequency (ν_probe ≈ 1 kHz) by a mechanical chopper. The probe beam was also focused into the sample using a pair of wide-aperture lenses L2 and L3 (f = 50 mm and 100 mm, respectively) and after passing the sample, it was reimaged on the input slit of the monochromator using the lens L4 (f = 100 mm).

For measuring the spectra, a 0.64 m monochromator (Jobin-Yvon, HRS2) and an InGaAs detector with two lock-in amplifiers (SR810 DSP, Stanford Research Systems) were used. The first one was locked with the pump modulation frequency ν_pump and measured the average intensity transmitted through the sample I and the second one was locked with the probe frequency ν_probe to measure the pump-induced variation ΔI of the abovementioned transmission. For the developed setup, the transmission variation ΔI/I was about ∼10⁻⁴…10⁻³ at the wavelengths of the ESA peaks.

A Glan-Taylor polarizer was placed before the input slit of the monochromator for polarization-resolved measurements. A long-pass filter (LP950, Spectrogon) was also used to filter out the residual pump radiation.

The wavelength calibration was performed with a mercury vapor lamp (Schwabe, Sp 60/U-V). The measurements were performed in the spectral range of 950-1575 nm with a resolution (spectral bandwidth, SBW) of 1-2 nm, depending on the sample. All the studies were performed at room temperature (RT, 20 °C).

2.3 Materials

Table 1 presents the list of the Tm³⁺-doped fluoride (ZBLAN glass, KY₃F₁₀, CaF₂, LiYF₄, LiLuF₄ and BaY₂F₈ crystals) and oxide (Y₃Al₅O₁₂ and YAlO₃ crystals) laser materials that have been studied.

Table 1. Summary of the Studied Thulium-Doped Materials

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To ensure that no ESA from the ³H₄ pump level (e.g., the ³H₄ → ¹G₄ transition at ∼1 µm, cf. Figure 4) can contribute to the measured spectra, we used highly Tm³⁺-doped materials (several at.% Tm, Table 1) providing efficient self-quenching of the ³H₄ lifetime by cross-relaxation. Under these conditions, a favorable ratio of the ³F₄ to ³H₄ lifetimes (τ_lum(³F₄) of about few ms and τ_lum(³F₄) of about few tens of µs) was observed.

For the optically isotropic glass and cubic crystals (KY₃F₁₀, CaF₂ and Y₃Al₅O₁₂), the studies were performed using unpolarized light. For the optically uniaxial crystals (LiYF₄ and LiLuF₄, the optical axis is parallel to the c-axis), the measurements were performed using a-cut samples for the two principal light polarizations, π (E || c) and σ (E ⊥ c). For the optically biaxial crystals (BaY₂F₈ and YAlO₃), several samples with different cuts were used giving access to three principal light polarizations. For monoclinic BaY₂F₈, they were denoted as E || X, Y and Z (X, Y and Z are the optical indicatrix axes; their assignment is given in [33]). For orthorhombic YAlO₃, the optical indicatrix frame coincides with the crystallographic one, so that the polarizations were denoted simply as E || a, b and c. Here, the crystallographic axes are selected according to the standard crystallographic setting Pnma [34].

2.4 Methodology

The relative variation of the transmitted probe signal ΔI/I is given by [32]:

There are two Tm³⁺ emissions falling in the studied spectral range, namely ³H₄ → ³F₄ (at ∼1.5 µm) and ³F₄ → ³H₆ (at ∼2 µm). The former emission originates from the ³H₄ state which is different from the probed one (³F₄). The spectral overlap of the ESA spectra extending from ∼1 to 1.6 µm with the second emission is very weak. Thus, for simplicity, we assume σ_SE ≈ 0 in Eq. (1).

For the ³F₄ → ³H₄ transition (at ∼1.5 µm), the ESA cross-sections can be independently calculated using the reciprocity method [35] from the σ_SE spectra of the ³H₄ → ³F₄ transition:

The ESA spectra for the ³F₄ → ³F_2,3 transition cannot be calculated by the reciprocity method because the ³F_2,3 states are depopulated by efficient multiphonon NR relaxation (the energy-gap to the lower-lying ³H₄ state is ∼1800cm^-1) and does not exhibit detectable luminescence.

2.5 Methodology: the case study of Tm:LiYF₄

To illustrate the analysis procedure of measuring and interpreting the Tm³⁺ ESA spectra, we have selected the Tm:LiYF₄ crystal. So far, it is the most widely used material for laser operation of Tm³⁺ ions in the mid-infrared. Figure 6 shows the raw spectrum, σ_GSA – σ_ESA, proportional to the ΔI/I ratio, the independently determined σ_GSA spectra for the ³H₆ → ³H₅ and ³H₆ → ³F₄ transitions (obtained from classical absorption studies) and the σ_SE spectra for the ³H₄ → ³H₅ and ³F₄ → ³H₆ transitions (obtained from emission spectra). The same figure shows the ESA spectra determined using Eq. (1). Following state-of-the-art studies [1], they are plotted in the range of negative cross-sections, i.e., as (–σ_ESA).

 

Fig. 6. Evaluation of the ESA cross-section spectra, σ_ESA, for the Tm:LiYF₄ crystal in the near-infrared: violet – calibrated raw spectrum ∼ΔI/I, black – σ_GSA spectra, green – σ_SE spectra, red - σ_ESA spectra. The light polarization is π.

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To confirm our assignment of the measured ESA spectral bands, we have calculated the full set of wavelengths corresponding to electronic transitions between the Stark sub-levels of the involved Tm³⁺ multiplets (³F₄ and ³H₄ + ³F_2,3). For this, the data on the crystal-field splitting for Tm³⁺ ions in LiYF₄ were used [36], Table 2. Here, we use empirical notations for Stark sub-levels (³F₄ = Y_i, ³H₄ = W_j, ³F₃ = V_k, ³F₂ = U_m, where the subscripts i, j, etc., number the sub-levels starting from 1) proposed by Lupei et al. [37], as well as indicate the irreducible representations (Г₁, Г₂ or Г_3,4) for each sub-level.

Table 2. Crystal-Field Splitting of Selected Tm³⁺ Multiplets in LiYF₄

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The polarization selection rules for electric-dipole (ED) and magnetic-dipole (MD) transitions for ions in S₄ sites are listed in Table 3. The MD transitions are those following the selection rules ΔJ = |J – J'| = 0, ±1, except of 0 ↔ 0’. In our case, the ESA transitions ³F₄ → ³F₃ and ³F₄ → ³H₄ can have a MD contribution. Indeed, the Judd-Ofelt calculations for Tm³⁺ ions in LiYF₄ predict the following ED and MD probabilities of spontaneous radiative transitions: A_ED = 51.35 s^-1 and A_MD = 18.95 s^-1 (for ³H₄ → ³F₄) and A_ED = 44.09 s^-1 and A_MD = 35.49 s^-1 (for ³F₃ → ³F₄) [38]. Thus, the MD contribution to the ESA probabilities is expected to be non-negligible, especially for the ³F₄ → ³F₃ transition. The extra MD lines should correspond to transitions between the Stark sub-levels having the same irreducible representations, i.e., Г₁ → Г₁ and Г₂ → Г₂, only for π-polarization, cf. Table 3.

Table 3. Polarization Selection Rules for Electric- and Magnetic-Dipole Transitions in S₄ Sites

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The results on the assignment of the measured ESA spectra of Tm³⁺ ions to ED and MD allowed electronic transitions are shown in Fig. 7. Several conclusions can be drawn from this figure. First, the observed ESA lines are well assigned only to transitions originating from the ³F₄ state and terminating at the ³H₄ and ³F_2,3 ones (i.e., extra lines due to possible ESA from the ³H₄ state are not observed). Second, the polarization selection rules well explain the polarization anisotropy of ESA spectra. Third, the prominent ESA peaks can be in most cases assigned to single electronic transitions which determine their narrow-linewidth nature. Moreover, the ESA transitions terminating at the closely located ³F₃ and ³F₂ states are well resolved.

 

Fig. 7. Interpretation of the polarized ESA cross-section, σ_ESA, spectra for Tm³⁺ ions in the LiYF₄ crystal in the near-infrared: (a,c) the ³F₄ → ³F_2,3 transitions, (b,d) the ³F₄ → ³H₄ transition, light polarizations are (a,b) π and (c,d) σ. Curves – measured spectra, in (c,d), spectra calculated using the reciprocity method (RM) are shown for comparison, vertical dashes – ED electronic transitions of Tm³⁺ ions according to the polarization selection rules.

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Figure 7(b,d) also contains the ESA cross-section spectra for the ³F₄ → ³H₄ transition calculated by means of the reciprocity method. They are in good agreement with the measured ones confirming the correctness of the pump-probe-based approach.

3. Excited-state absorption spectra

3.1 Fluoride materials

The ESA cross-section spectra corresponding to the ³F₄ → ³F_2,3 and ³F₄ → ³H₄ transitions for Tm³⁺ ions in fluoride materials (ZBLAN glass, cubic KY₃F₁₀ and CaF₂, tetragonal LiYF₄ and LiLuF₄ and monoclinic BaY₂F₈ crystals) are shown in Fig. 8. Let us briefly discuss them.

 

Fig. 8. Polarized (where applicable) ESA cross-section, σ_ESA, spectra for Tm³⁺ ions in fluoride materials: (a, b) isotropic hosts: ZBLAN glass, cubic KY₃F₁₀ and CaF₂ crystals; (c-f) uniaxial crystals: (c,d) tetragonal LiYF₄ and (e,f) LiLuF₄ and (g,h) biaxial crystal: monoclinic BaY₂F₈. Transitions: (a,c,e,g) ³F₄ → ³F_2,3 and (b,d,f,h) ³F₄ → ³H₄. The spectral resolution is indicated on the graphs (it is the same in left and right panels).

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Due to its amorphous nature, the Tm:ZBLAN glass shows smooth and broad bands at the expense of low peak ESA cross-sections. For the ³F₄ → ³F_2,3 ESA transition which is of interest for UC pumping at ∼1 µm, σ_ESA is 0.23×10⁻²⁰ cm² at 1056 nm and the corresponding absorption bandwidth (full width at half maximum, FWHM) Δλ_ESA is 39 nm. As Tm:ZBLAN glasses are used in the form of fibers, relatively high absorption via UC pumping is easily achievable. A ∼2.3 µm Tm:ZBLAN fiber lasers with UC pumping at ∼1 µm was recently reported [25].

Among the studied Tm³⁺-doped fluoride crystals, Tm:CaF₂ is standing apart: its spectra are similar to those of the glass. Such a “glassy-like” behavior is due to the strong ion clustering even at low doping levels leading to significant inhomogeneous spectral broadening [39]. For the ³F₄ → ³F_2,3 ESA transition, σ_ESA is low, only 0.07×10⁻²⁰ cm² at 1064 nm with broad Δλ_ESA of 38 nm. The origin of the ESA peak at 1089.7 nm is not clear; absorption of residual isolated Tm³⁺ ions in cubic symmetry sites (O_h) can explain this sharp line.

Other Tm³⁺-doped fluoride crystals exhibit much narrower ESA peaks owing to electronic transitions of ions located in a single type of sites with reduced inhomogeneous broadening (cf. Table 1). For cubic Tm:KY₃F₁₀, σ_ESA reaches 0.91×10⁻²⁰ cm² at 1067.7 nm with Δλ_ESA = 3.4 nm. Another intense ESA line is observed at 1048.7 nm corresponding to a σ_ESA value of 0.73×10⁻²⁰ cm² and a Δλ_ESA of 3.6 nm. This line was recently used for demonstrating UC pumping of a ∼2.3 µm Tm:KY₃F₁₀ laser [40].

For optically uniaxial and biaxial fluoride crystals, a noticeable polarization-anisotropy of ESA spectra is observed. For the ³F₄ → ³F_2,3 ESA transition of Tm³⁺ ions in tetragonal LiYF₄ and LiLuF₄ crystals, higher ESA cross-sections are measured for σ-polarized light: σ_ESA reaches 0.62×10⁻²⁰ cm² at 1042.0 nm and 0.36×10⁻²⁰ cm² at 1055.9 nm with Δλ_ESA = 2.5 nm and 3.2 nm, respectively (Tm:LiYF₄, σ-polarization). Both these ESA lines were used for UC pumping of a ∼2.3 µm Tm:LiYF₄ laser [20]. The ESA properties of the isostructural LiY/LuF₄ crystals are similar, while the ESA cross-sections seem to be slightly higher for the Lu-compound. Finally, for monoclinic Tm:BaY₂F₈, the preferable light polarization for UC pumping at ∼1 µm is E || Z, as the corresponding σ_ESA reaches 0.27×10⁻²⁰ cm² at 1047.2 nm with Δλ_ESA = 5.5 nm.

Among the studied fluoride materials, Tm:KY₃F₁₀ and Tm:LiY/LuF₄ crystals offer the most intense ESA lines around ∼1 µm making them attractive for UC pumping.

3.2 Oxide materials

The ESA cross-section spectra corresponding to the ³F₄ → ³F_2,3 and ³F₄ → ³H₄ transitions for Tm³⁺ ions in oxide crystals (cubic Y₃Al₅O₁₂ and orthorhombic YAlO₃) are shown in Fig. 9.

 

Fig. 9. Polarized (where applicable) ESA cross-section, σ_ESA, spectra for Tm³⁺ ions in oxide crystals: (a,b) cubic Y₃Al₅O₁₂ and (c,d) orthorhombic YAlO₃ crystals. Transitions: (a,c) ³F₄ → ³F_2,3 and (b,d) ³F₄ → ³H₄. The spectral resolution is indicated on the graphs.

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For cubic Tm:Y₃Al₅O₁₂, the ³F₄ → ³F_2,3 ESA transition, σ_ESA reaches 0.48×10⁻²⁰ cm² at 1031.0 nm with a relatively narrow Δλ_ESA of 1.5 nm.

For orthorhombic Tm:YAlO₃, the ESA spectra exhibit a significant polarization anisotropy. The most attractive polarization is E || a for which σ_ESA is 1.55×10⁻²⁰ cm² (the highest value among the studied materials) at 1044.2 nm corresponding to Δλ_ESA of 2.7 nm.

3.3 Towards upconversion pumping at ∼1 µm

Table 4 lists the ESA peaks slightly above 1 µm (the ³F₄ → ³F₂ transition) which are of practical interest for realization of upconversion pumping of ∼2.3 µm Tm lasers. The corresponding pump sources can be commercially available Yb bulk or fiber lasers offering power-scalable output as well as wavelength tunability in this spectral range.

Table 4. Excited-State Absorption Peaks* of Tm³⁺ Ions in Fluoride and Oxide Materials Suitable for Upconversion Pumping at ∼1 µm

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4. Energy-transfer upconversion

The ESA spectra in the IR spectral range can be also used for evaluating the concentration-independent parameter of one of the energy-transfer upconversion (ETU) processes for Tm³⁺ ions, i.e., ³F₄ + ³F₄ → ³H₄ + ³H₆. This process acts against the cross-relaxation, ³H₄ + ³H₆ → ³F₄ + ³F₄, refilling the higher-lying ³H₄ excited-state and depopulating the metastable ³F₄ state. It can significantly affect the performance of Tm lasers. In particular, it is detrimental for Tm lasers operating on the ³F₄ → ³H₆ transition (at ∼1.9 µm) [41]: strong ETU in highly Tm³⁺-doped crystals is responsible for the increased laser threshold and additional heat generation possibly leading to higher risk of thermal fracture. In opposite, this ETU process could be useful for Tm lasers operating on the ³H₄ → ³H₅ transition (at ∼2.3 µm) as it refills the upper laser level at the expense of the metastable intermediate ³F₄ state [42]. In this way, pump quantum efficiencies up to 2 could be reached leading to laser slope efficiencies exceeding the Stokes limit. Thus, knowledge of the ETU parameter is of practical importance. Below, we describe the general calculation procedure and apply it to the particular case of Tm:LiYF₄.

The ETU rate was evaluated using the hopping (Burshtein) model describing a migration-assisted energy transfer [9]. This model treats the transfer of energy between donors as a random process until the excitation is transferred to an acceptor ion. First, two microparameters C_DD and C_DA were calculated from the overlap integrals between the SE and absorption (GSA or ESA) cross-section spectra [43]:

 

Fig. 10. Energy-transfer upconversion (ETU), ³F₄ + ³F₄ → ³H₄ + ³H₆, for Tm³⁺ ions. Two elementary processes are shown: (a) energy-migration and (b) phonon-assisted “direct” ETU.

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The macroscopic ETU rate in (s^-1) was then determined as [44]:

The considered ETU process is non-resonant and is phonon-assisted. The calculation of the overlap integrals between the SE and absorption (GSA and ESA) spectra should account for the multiphonon sidebands. The shape of the latter can be expressed as [45,46]:

Using Eqs. (4)–(6) and the spectroscopic data for Tm:LiYF₄, as shown in Fig. 11(a), we calculated the concentration-independent ETU parameter, C_ETU = 2.56×10⁻⁴⁰ cm⁶s^-1. In the previous studies, the ETU parameter (K_ETU) was typically reported for given Tm³⁺ doping concentrations, Fig. 11(b). In this work, the result of our calculation can be expressed as a linear dependence of K_ETU on N_Tm with a slope equal to C_ETU. Let us briefly review the methods used in previous studies to obtain the K_ETU values. So far, they were obtained from (i) luminescence-decay measurements [47,48], (ii) stationary luminescence-intensity measurements [42], (iii) theoretical calculations [49], and (iv) modeling of Tm laser performance [50,51]. Our results are in good agreement with the previously reported data, especially with the experimental ones from [42].

 

Fig. 11. Evaluation of the ETU parameter for Tm:LiYF₄: (a) polarization-averaged SE (³F₄ → ³H₆), GSA (³H₆ → ³F₄) and ESA (³F₄ → ³H₄) spectra plotted in semi-log scale, solid curves – measured spectra with phonon sidebands calculated using Eq. (7); (b) summary of the ETU parameters K_ETU reported so far: symbols – literature data, line – this work, calculation using Eqs. (4)–(6).

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The determined C_ETU value should be compared with that for the cross-relaxation process, i.e., C_CR = 0.25 ± 0.03×10⁻³⁷ s⁻¹cm⁶ for Tm:LiYF₄ [42]. One can see that the considered ETU process is by two orders of magnitude weaker that the CR one (for each particular Tm³⁺ doping level).

5. Conclusion

To conclude, we have studied excited-state absorption of Tm³⁺ ions in the near-infrared spectral range for a series of fluoride and oxide laser crystals using a dedicated pump-probe method. The considered ESA transitions originate from the metastable ³F₄ Tm³⁺ state and terminate at the higher lying ³F_2,3 and ³H₄ levels. The ³F₄ → ³F_2,3 ESA channel is particularly attractive for upconversion pumping of Tm lasers via a photon avalanche mechanism. This pumping scheme can be used for obtaining laser emission at ∼2.3 µm corresponding to the ³H₄ → ³H₅ transition with commercial, power scalable and wavelength tunable Yb fiber lasers emitting slightly above 1 µm. The ESA line positions, cross-sections and spectral linewidths which are required for the design of upconversion pumped Tm lasers were measured in detail. Among the studied crystals, Tm:KY₃F₁₀, Tm:LiY/LuF₄ and Tm:YAlO₃ are identified as promising candidates for upconversion pumping. Regarding the second studied ESA channel, ³F₄ → ³H₄, we show a route for calculating another key spectroscopic parameter for the development of Tm lasers, namely, the energy-transfer upconversion parameter. The ETU parameters are calculated from the absorption and emission overlap integrals using the hopping model.