1. Introduction

The Erbium ion (Er³⁺) is well known for its emission in the eye-safe spectral range of ∼1.5 - 1.6 µm originating from the ⁴I_13/2 → ⁴I_15/2 electronic transition. The applications of eye-safe erbium lasers are in the fields of range-finding, environmental sensing, aerial navigation and telecom. So far, Er³⁺,Yb³⁺-codoped phosphate glasses represent the state-of-the-art erbium laser media [1]. Here, the Yb³⁺ ion acts as a sensitizer to enhance the pump absorption at ∼0.96 - 0.98 µm, the spectral range addressed by commercial InGaAs laser diodes [2]. Such glasses feature attractive spectroscopic behavior, i.e., broadband Yb³⁺ absorption, efficient Yb³⁺ → Er³⁺ energy transfer and long Er³⁺ upper laser lifetime. Er,Yb:phosphate glass lasers can generate mJ-level nanosecond pulses making them ideal for portable range-finders [3]. At the same time, glasses suffer from poor thermo-mechanical properties limiting the power scaling capabilities.

As an alternative to glassy gain media, single crystals can be used. However, so far, only a few materials were found to be suitable for efficient lasers based on the Yb³⁺,Er³⁺ codoping scheme. These include vanadates (YVO₄ [4]), borates (REAl₃(BO₃)₄ [5,6], RECa₄O(BO₃)₃ [7]), and silicates (Y₂SiO₅ [8]). Tolstik et al. demonstrated an Er,Yb:YAl₃(BO₃)₄ (Er,Yb:YAB) laser delivering an output power of 1 W at 1555 nm with a slope efficiency of 35% [5]. Still, further power scaling of such lasers is limited by severe thermo-optic effects originating from upconversion losses. Moreover, the growth technology of REAl₃(BO₃)₄ borate crystals is far from being mature.

An alternative approach is the use of singly Er³⁺-doped crystals which can be pumped at ∼1.54 µm. This corresponds to excitation directly to the upper laser level (in-band or resonant pumping). Owing to the relatively high absorption cross-sections for the ⁴I_15/2 → ⁴I_13/2 Er³⁺ transition, high pump efficiencies can be achieved even for low Er³⁺ doping levels (∼1 at.%). This leads to weak energy-transfer upconversion, which, together with the low quantum effect inherent to the resonant pumping scheme, determines weak heat loading and high laser slope efficiencies. Contrary to the Er³⁺, Yb³⁺ codoping scheme for which only a few particular host matrices are suitable, single Er³⁺ doping is more advantageous from the material point of view: efficient laser operation can be achieved in many singly Er³⁺-doped laser materials (crystals) under in-band pumping. In-band pumped Er:Y₃Al₅O₁₂ (Er:YAG) lasers have delivered multi-watt output at 1617 nm and 1645 nm [9,10]. However, one disadvantage of cubic Er:YAG crystals are the depolarization losses appearing at high pump powers in lasers containing polarization-selective elements. Thus, it is still important to search for novel singly Er³⁺-doped materials offering intrinsic optical anisotropy, broad and intense spectral bands for polarized light, and good thermo-mechanical properties.

Magnesium monotungstate (MgWO₄) has recently emerged as a promising host crystal for doping with laser-active rare-earth ions [11–13]. This crystal is monoclinic and optically biaxial offering strong optical anisotropy. It was found to exhibit good thermo-mechanical properties, i.e., high thermal conductivity of ∼8.7 Wm^-1K^-1 [14] and low anisotropy of thermal expansion [13]. High-power laser operation was demonstrated using such crystals: Loiko et al. developed a diode-pumped Yb³⁺:MgWO₄ laser delivering 18.2 W at ∼1056 nm with a high slope efficiency of ∼89% and a linearly polarized output [12]. The low-symmetry distorted coordination of rare-earth ions in MgWO₄ leads to broad emission bands [15]. Because of the broadband emission properties, such crystals have been implemented in femtosecond mode-locked lasers [16,17]. Although Er³⁺:MgWO₄ has never been studied so far, other monoclinic crystals doped with Er³⁺ ions appear promising for the development of eye-safe lasers: Serres et al. reported on a diode-pumped 1 at.% Er:KLu(WO₄)₂ laser generating 268 mW at 1610 nm with a slope efficiency of 30% [18].

In the present work, we report on the growth, structure refinement and a polarization-resolved spectroscopic study of an Er³⁺-doped MgWO₄ crystal, for the first time, to the best of our knowledge.

2. Crystal growth and structure

2.1 Crystal growth

Er³⁺:MgWO₄ was grown by the top-seeded solution growth (TSSG) method [11,19] from the flux with a composition of MgWO₄: Na₂WO₄ = 5:7 mol (sodium tungstate, Na₂WO₄, was used as a solvent) in a vertical tubular furnace. The starting materials were Na₂CO₃, MgO, WO₃ (purity: analytical grade) and Er₂O₃ (dopant, purity: 99.99%). They were weighed according to the above-mentioned composition with the initial Er³⁺ concentration of 10 at.% (with respect to Ca²⁺). The weighed materials were mixed, ground and then put into a Pt crucible with dimensions of Ø55 × 60 mm³. The crucible was placed into a resistive furnace equipped with a programmable temperature controller, nickel-chrome heating wires and a Pt–Rh/Pt thermocouple. A [010]-oriented seed from an undoped MgWO₄ crystal was used. The solution was kept at 980°C for 2 days to ensure that the starting materials melted completely and homogeneously. The saturation temperature was determined to be 953°C by repeated seeding trials. The crystal was grown at a cooling rate of 0.6 - 1°C/day and a rotation speed of 10 rpm in the temperature range of 953 - 930°C. Once the growth was completed, the crystal was slowly pulled out of the solution and cooled down to room temperature (RT, 293 K) at a rate of 10 K/h. Figure 1 shows an as-grown Er³⁺:MgWO₄ crystal with dimensions of 20 × 8 × 6 mm³. It has a rose coloration due to erbium doping.

 

Fig. 1. A photograph of the as-grown 1.01 at.% Er³⁺:MgWO₄ crystal, the growth direction is along the [010] axis.

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The actual Er³⁺ doping level was determined by inductively coupled plasma atomic emission spectrometry (ICP-AES, Ultima2, Jobin-Yvon) to be 1.01 at.% (ion density: N_Er = 1.419 × 10²⁰ at/cm³). No significant variation of the Er³⁺ doping level across the crystal boule was observed. The segregation coefficient of Er³⁺ K_Er was then 0.1. Such a low value is due to a significant difference of ionic radii of Mg²⁺ and Er³⁺ (see below). Despite the low K_Er value for MgWO₄, it is relatively easy to access the desired doping level typical for singly Er³⁺-doped crystals (∼1 at.%). Higher doping levels can be accessed by increasing the initial content of Er³⁺ ions in the growth charge or probably by using other charge compensators (e.g., Li⁺ cations). However, in in-band-pumped Er lasers, the doping levels above 1 at.% are usually not used due to the enhanced energy-transfer upconversion.

2.2 Structure refinement and electronic structure

The X-ray powder diffraction (XRD) data were collected using a Rigaku MiniFlex 600 X-ray diffractometer with CuKα radiation (λ = 1.5418 Å) in an angular range of 2θ = 10-80° with a scan step of 0.02° and a scan speed of 5°/min, as shown in Fig. 2(a). The measured XRD pattern was well assigned using a standard diffraction pattern of undoped MgWO₄ (JCPDS card #96-101-0643) and no other phases were found.

 

Fig. 2. Structural study of Er³⁺:MgWO₄: (a) X-ray powder diffraction (XRD) analysis: observed (black), calculated (red) and residual (blue) patterns, dashes – Bragg positions, (hkl) – Miller’s indices; (b,c) projection of the crystal structure on (b) the a-c plane and (c) the b-c plane, black lines – unit-cell.

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The crystal structure was refined by the Rietveld method (Match3 software was used). The crystallographic data of undoped MgWO₄ [20] were used as a starting model. Er³⁺:MgWO₄ crystallizes in the monoclinic system with space group C⁴_2h - P2/c, No. 13, and centrosymmetric point group 2/m. The lattice parameters are a = 4.6939(6) Å, b = 5.6747(4) Å, c = 4.9316(6) Å, monoclinic angle of β = 90.7858(4)° (number of formula units per unit-cell Z = 2), unit-cell volume of V = 131.351 ų and calculated density ρ_calc = 6.078 g/cm³. The reliability factors are R_p = 10.5%, R_wp = 14.5%, R_exp = 10.29% and χ² = (R_wp/R_exp)² = 1.98 indicating good convergence of the fit.

The determined fractional atomic coordinates, site occupancy factors (O.F.) and isotropic displacement parameters B_iso are listed in Table 1. Figures 2(b) and (c) present a fragment of the crystal structure calculated according to the determined atomic positions. Er³⁺:MgWO₄ exhibits a wolframite [(Fe,Mn)WO₄] type structure (MgWO₄ is called huanzalaite in mineral form). The Mg²⁺|Er³⁺ cations occupy the 2f Wyckoff positions with VI-fold O^2- coordination; in the distorted [Mg|ErO₆] octahedra, there are two shorter [1.9392(0) Å], two intermediate [2.1732(0) Å] and two longer [2.2199(4) Å] Mg-O distances. The W⁶⁺ cations (2e Wyckoff positions) are also located in distorted octahedra with W-O distances in the range 1.7747(7) - 2.1673(4) Å. The network of Er³⁺:MgWO₄ is made up of alternating zig-zag chains of edge-sharing [Mg|ErO₆] and [WO₆] polyhedra along the c-axis. The shortest distance Mg|Er - Mg|Er is 3.3323(4) Å is observed along the vector [u v w]:[0 - 0.3950 0.5].

Table 1. Fractional Atomic Coordinates (x, y, z), Site Occupancy Factors (O.F.) and Isotropic Displacement Parameters B_iso (Ų) for Er³⁺:MgWO₄

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In MgWO₄, erbium ions are expected to replace for the host-forming cations Mg²⁺ in a single type of sites (2f, site symmetry: C₂). The local charge compensation is most probably ensured by univalent Na⁺ cations entering from the Na₂WO₄ solvent (Er³⁺ + Na⁺ ↔ 2Mg²⁺). The corresponding ionic radii for VI-fold oxygen coordination are: R_Mg = 0.72 Å, R_Er = 0.89 Å and R_Na = 1.02 Å [21]. Large ionic radii of the dopant and charge compensation cations determine the expansion of the unit-cell [for MgWO₄, a = 4.68892(2) Å, b = 5.67529(3) Å, c = 4.92891(2) Å and β = 90.726(1) Å] [20]. The difference in the charge and ionic radii of Mg²⁺, Er³⁺ and Na⁺ is expected to distort the crystal field around the Er³⁺ ions leading to additional spectral broadening.

The electronic structure of the host matrix, MgWO₄, was analyzed by the density functional theory in which the generalized gradient approximation with the Perdew–Burke–Ernzerhof functional was used to study the exchange–correlation effects. The calculations were performed using the CASTEP code. The energy cutoff was set to 517 eV. The criterion for the self-consistent field was eigenenergy convergence within 2.0×10⁻⁶ eV per atom. K-space sampling was performed using a Monkhorst–Pack grid of 4 × 4 × 4 atoms with respect to k-points in the irreducible Brillouin zone.

The results are shown in Fig. 3(a). The top of the valence band (VB) and the bottom of the conduction band (CB) are located at different points in the Brillouin zone (B and Y, respectively), indicating an indirect bandgap of E_g,calc = 3.42 eV. For Er³⁺-doped MgWO₄, E_g,calc is higher, 3.47 eV, in agreement with the optical bandgap derived from the absorption spectrum using the Tauc plot [22], E_g = 3.53 eV. The assignment of electronic bands was performed with the help of calculated total and partial densities of states, Fig. 3(b). The VB band in the range from the Fermi energy level around 0 eV to -6.26 eV is mainly derived from the p-states of O. The CB extends from 3.42 to 22 eV and it is mainly due to the p- and s-states of Mg.

 

Fig. 3. Electronic structure of MgWO₄: (a) calculated band structure, E_g – bandgap energy; (b) calculated total and partial densities of states (TDOS and PDOS).

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3. Polarized optical spectroscopy

3.1 Experimental

MgWO₄ is an optically biaxial crystal. For the polarization-resolved study, we have prepared a six-side polished rectangular sample oriented in the crystallographic frame by means of single-crystal XRD having the dimensions 4.39(a) × 4.15(b) × 4.35(c*) mm³. Here, c* is a direction being orthogonal to the a-axis and lying in the a-c plane (as the monoclinic angle of Er³⁺:MgWO₄ is close to 90°, c* ≈ c below in the text).

The spectroscopic studies were performed at RT and low temperature (LT, 10 K). The absorption spectra were measured using a spectrophotometer (Lambda 1050, Perkin Elmer) and the luminescence spectra - by an optical spectrum analyzer (AQ6375B, Yokogawa). For polarization-resolved studies, a Glan-Taylor polarizer was implemented. The luminescence decay curves were measured using a nanosecond optical parametric oscillator (Horizon, Continuum), a 1/4 m monochomator (Oriel 77200), a PMT tube or an InGaAs detector and an 8 GHz digital oscilloscope (DSA70804B, Tektronix). For LT studies, the crystal was mounted on an APD DE-202 closed-cycle cryo-cooler equipped with an APD HC 2 Helium vacuum cryo-compressor and a Laceshore 330 temperature controller.

3.1 Absorption spectra and Judd-Ofelt analysis

The RT polarized absorption spectra of Er³⁺ in MgWO₄ are shown in Fig. 4. The crystal exhibits a significant anisotropy of the absorption properties. For the ⁴I_15/2 → ⁴I_11/2 transition which can be addressed by commercial InGaAs diode lasers, the maximum absorption cross-section σ_abs is 0.77 × 10⁻²⁰ cm² at 983.7 nm and the corresponding absorption bandwidth (full width at half maximum, FWHM) is 4.1 nm (for E || b). Slightly lower σ_abs = 0.74 × 10⁻²⁰ cm² at 980.8 nm is observed for E || c at a much narrower bandwidth, 1.4 nm.

 

Fig. 4. (a-f) RT (293 K) absorption cross-section, σ_abs, spectra of Er³⁺:MgWO₄ crystal, the light polarizations are E || a, b, c.

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Er³⁺:MgWO₄ is also attractive for in-band pumping (directly into the ⁴I_13/2 manifold): the maximum σ_abs is 1.70×10⁻²⁰ cm² at 1482.9 nm and the absorption bandwidth is as broad as ∼10 nm (for E || b) and at longer wavelengths addressed by Er fiber lasers, σ_abs = 1.51×10⁻²⁰ cm² at 1528.0 nm with still broad absorption bandwidth of ∼11 nm (for E || c).

The 4fⁿ transition intensities of Er³⁺ were analyzed using the Judd-Ofelt (J-O) formalism based on the measured absorption spectra. Both the standard J-O theory [23,24] and its modification accounting for an intermediate configuration interaction (ICI) with an excited configuration of the opposite parity 4f^n-15d¹ [25,26] were implemented to determine the electric dipole (ED) contributions. The set of squared reduced matrix elements U^(k) (k = 2, 4, 6) for Er³⁺ was calculated in the present work based on the free-ion parameters reported in [27]. The magnetic dipole (MD) contributions to transition intensities (for transitions following the selection rule ΔJ = 0, ±1, except of 0 ↔ 0’) were calculated here within the Russell–Saunders approximation on wave functions of Er³⁺ under an assumption of a free-ion. The refractive index data from [13] were used. More details can be found elsewhere [28].

Table 2 presents the experimental <f_exp> and calculated f_calc absorption oscillator strengths of Er³⁺ in MgWO₄. The obtained intensity parameters are Ω₂ = 11.111, Ω₄ = 3.394, Ω₆ = 0.598 [10⁻²⁰ cm²] for the J-O theory and Ω₂ = 11.480, Ω₄ = 3.782, Ω₆ = 0.703 [10⁻²⁰ cm²], R₂ = -0.154, R₄ = 0.518, R₆ = 0.064 [10⁻⁴ cm] for the ICI approximation. Note that these intensity parameters are derived from polarization-averaged absorption oscillator strengths. The ICI model provided lower root mean square (r.m.s.) deviation between <f_exp> and f_calc values, as well as better agreement between the radiative and measured lifetimes of the lowest-lying excited-state, ⁴I_13/2. Thus, it was selected for further calculations.

Table 2. Judd-Ofelt Analysis^a of Transitions in Absorption for Er3+ in MgWO₄

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The probabilities of spontaneous radiative transitions (A^Σ_JJ’, where ^Σ indicates both ED + MD contributions), the mean luminescence wavelength <λ> calculated from the barycenter energies of the corresponding multiplets (cf. Table 2), the luminescence branching ratios B_JJ’ and the radiative lifetimes τ_rad are listed in Table 3 (all the values were calculated within the ICI approximation). For the upper laser level ⁴I_13/2, τ_rad = 4.78 ms.

Table 3. Probabilities^a of Spontaneous Radiative Transitions of Er3+ in MgWO₄

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3.2 Emission (spectra and lifetimes)

The stimulated-emission (SE) cross-sections for the ⁴I_13/2 → ⁴I_15/2 transition of Er³⁺ were calculated using two complementary methods: the Füchtbauer–Ladenburg (F-L) equation [29] and the reciprocity method (RM) [30]. The combined SE cross-section spectra are shown in Fig. 5(a). The maximum σ_SE is 1.42×10⁻²⁰ cm² at 1533.9 nm (for light polarization E || c) corresponding to the zero-phonon line (ZPL) transition at RT (see below). At the wavelengths exceeding ZPL corresponding to the expected Er³⁺ laser emission (see the gain spectra), σ_SE is lower, namely 0.26×10⁻²⁰ cm² at 1640 nm (E || a), 0.31×10⁻²⁰ cm² at 1637 nm (E || b) and 0.26×10⁻²⁰ cm² at 1633 nm (E || c).

 

Fig. 5. Stimulated-emission (SE) cross-sections, σ_SE, for the ⁴I_13/2 → ⁴I_15/2 Er³⁺ transition in MgWO₄: (a) combined spectra for E || a, b, c; (b) a comparison of spectra calculated by the Füchtbauer – Ladenburg (F-L) equation and the reciprocity method (RM), for E || c.

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A relatively good agreement between the σ_SE values calculated by both methods is achieved for a radiative lifetime of the ⁴I_13/2 state τ_rad = 4.85 ± 0.05 ms, Fig. 5(b). This value is in line with the J-O analysis. Lower SE cross-sections achieved by the F-L method at shorter wavelengths are due to the reabsorption affecting the measured luminescence spectra.

Erbium ions represent a quasi-three-level laser scheme with intrinsic reabsorption (the ⁴I_13/2 ↔ ⁴I_15/2 transition). Gain cross-sections, σ_gain = βσ_SE – (1 – β)σ_abs, where β = N₂(⁴I_13/2)/N_Er is the inversion ratio, are thus calculated to conclude about the expected laser wavelength and polarization. The calculated gain profiles for light polarizations E || a and E || c are shown in Fig. 6. For the high-gain polarization E || a, a local peak at 1640 nm dominates in the spectra. The gain bandwidth (FWHM) is ∼17 nm.

 

Fig. 6. RT gain cross-section, σ_gain, profiles for the ⁴I_13/2 ↔ ⁴I_15/2 transition of Er³⁺ in MgWO₄: the light polarization is (a) E || a and (b) E || c, β – inversion ratio.

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RT luminescence decay curves from the ⁴I_13/2 and ⁴I_11/2 Er^3 +multiplets are shown in Fig. 7. They were measured using a powdered crystal sample to reduce the effect of radiation trapping (reabsorption). The luminescence decay is single exponential in agreement with a single type of sites for Er³⁺ ions in MgWO₄ (C₂ symmetry). The luminescence lifetimes τ_lum are 4.93 ms and 50.4 µs, respectively. The luminescence lifetime of the ⁴I_13/2 metastable level is close to the radiative one obtained from the J-O calculations (4.78 ms) and from the evaluation of SE cross-sections (4.85 ms), indicating a luminescence quantum efficiency close to unity. The slightly longer value of τ_lum is probably due to residual reabsorption effect.

 

Fig. 7. RT luminescence decay curves from the ⁴I_13/2 (a) and ⁴I_11/2 (b) multiplets of Er³⁺ in MgWO₄: circles – experimental data, lines – single-exponential fits.

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The relatively long lifetime of the upper laser level (for oxide crystals) makes Er³⁺:MgWO₄ attractive for passively Q-switched lasers.

The luminescence lifetimes of five Er³⁺ excited states from ⁴I_13/2 to ⁴S_3/2 were measured to determine the rates of multiphonon non-radiative relaxation W_NR = (1/τ_lum) – (1/τ_rad), cf. Table 4. The values of W_NR were plotted vs. the energy gap between the emitting multiplet and the lower-lying manifold, ΔE, as shown in Fig. 8. The experimental points were fitted using the equation W_NR = Ce^-αΔE, where C and α are constants characteristic of the material [31,32]. C has the meaning of a rate constant at the limit of zero energy gap (ΔE → 0), and α = –ln(ε)/(hν_ph), where ε is the ratio between the probabilities of m-phonon and m – 1-phonon relaxation and hν_ph is the dominant (maximum) phonon energy of the host matrix. The experimental points in Fig. 8 are well fitted with the above-mentioned equation yielding the values of C= 6.5 ± 0.3×10⁹ s^-1 and α = 3.44 ± 0.1×10⁻³ cm. The multiphonon relaxation in MgWO₄ is slightly stronger than that in the monoclinic KLu(WO₄)₂ crystal [31] owing to the high maximum phonon energy of the former material (hν_ph = 916 cm^-1 [11]).

 

Fig. 8. The rate of multiphonon non-radiative relaxation W_NR vs. the energy gap to the lower-lying manifold ΔE for Er³⁺ in MgWO₄: circles – data obtained from luminescence lifetime measurements, red solid line – their fit, blue dashed line – fit for KLu(WO₄)₂ [31].

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Table 4. Evaluation^a of Non-Radiative Relaxation Rates for Er3+ in MgWO₄

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3.3 Low-temperature spectroscopy

LT absorption and luminescence spectra were measured to determine the Stark splitting of the upper (⁴I_13/2) and lower (⁴I_15/2) laser multiplets of Er³⁺, cf. Fig. 9(a,b). The assignment of the electronic transitions followed previous work on Er³⁺-doped KLu(WO₄)₂ crystal [33]. For C₂ symmetry sites, each ^2S+1L_J multiplet with non-integer J is split into J + ½ Stark sub-levels. All the sub-levels are identified from the measured spectra leading to the energy-level scheme shown in Fig. 9(c). The partition functions for the lower and upper laser manifolds are Z_l = 4.534 and Z_u = 4.343, respectively, and their ratio Z_l/Z_u = 1.044 (these data were used for the calculation of SE cross-sections via RM).

 

Fig. 9. (a,b) LT (10 K) (a) absorption and (b) luminescence spectra of Er³⁺ in the MgWO₄ crystal corresponding to the ⁴I_15/2 ↔ ⁴I_13/2 transition, “+” - assigned electronic transitions, dashes – electronic transitions for Er³⁺ in KLu(WO₄)₂ crystal [33]; (c) experimental Stark splitting of the ⁴I_15/2 and ⁴I_13/2 Er³⁺ multiplets, numbers indicate the sub-level energies in cm^-1.

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The energy of the ZPL transition occurring between the lowest Stark sub-levels of both multiplets E_ZPL is 6520 cm^-1 (1533.7 nm). The total Stark splitting of the lower laser level (⁴I_15/2) is 435 cm^-1 which is larger than that for Er³⁺ in KLu(WO₄)₂, 361 cm^-1 [33]. This is attributed to the stronger crystal field for distorted C₂ sites in MgWO₄. Large total Stark splitting of the ground-state explains the relatively long emission wavelengths (cf. Fig. 5) and is beneficial for low threshold laser operation.

4. Conclusion

To conclude, Er³⁺-doped MgWO₄ is a promising gain material for eye-safe lasers emitting at ∼1.6 µm. It exhibits (i) relatively intense and broad absorption bands around ∼1.54 µm which is attractive for in-band pumping, (ii) high stimulated-emission cross-sections for polarized light potentially leading to naturally polarized laser emission, (iii) broad emission bands making it feasible to access laser wavelengths at ∼1.62 - 1.64 µm similarly to Er:YAG and (iv) relatively long upper laser level lifetime and high luminescence quantum efficiency which is attractive for low-threshold operation, as well as generation of giant pulses in Q-switched lasers. In addition, as shown in previous work, MgWO₄ features good thermal properties. The observed broad, intense and polarized spectral bands for Er³⁺:MgWO₄ are assigned to low-symmetry (C₂) coordination of Er³⁺ ions replacing for the host-forming Mg²⁺ cations which is distorted by the heteroallene doping mechanism involving univalent Na⁺ cations, i.e., the difference in the charge and the ionic radii of Mg²⁺, Er³⁺ and Na⁺.