Polarizabilities and emission cross-sections of lanthanide laser crystals

1 Introduction

The trivalent lanthanide ions (Ln³⁺) are unavoidable in research and application of solid-state lasers, either doped in insulating single crystals, in fibers, in waveguide/clad structures or as crystals of stoichiometric compounds. Their ground-state electronic configuration is _Z Ln³⁺ = [₅₄Xe] 4f ^N , where Z varies from 58 (Ce) to 71 (Lu), with N changing from 1 to 14, respectively. The atomic number Z and the number N of 4f electrons predetermine the complexity of the energy-level schemes including all types of interactions. ¹ The most studied in the mentioned scope among Ln³⁺ ions are ₆₀Nd³⁺ = [Xe] 4f³, ₆₇Ho³⁺ = [Xe] 4f¹⁰, ₆₈Er³⁺ = [Xe] 4f¹¹, and ₆₉Tm³⁺ = [Xe] 4f¹². Stimulated emission transitions in these ions occur via mechanisms comprising three or four energy levels of 4f electrons, including appropriate absorption of light from optical source: LED, diode-, dye- or ion laser. The emission 4f–4f transitions often initiate after and terminate with non-radiative transitions. The basic characteristics include generation wavelength λ _e, absorption σ _a and emission σ _e cross-sections, lifetime τ of the stimulated-emission transition, optical-pump power threshold, and output/input efficiency. Most of these quantities have been interrelated for Nd³⁺ miniature lasers. ² It is noteworthy that laser transitions with Nd³⁺ and Er³⁺ ions can be obtained at room temperature, mostly quoted T = 300 K, while the active media with Ho³⁺ and Tm³⁺ ions often require cryogenic temperatures, e.g. T = 77–100 K, in order to achieve inverse population of the excited level relative to the level at which the stimulated emission terminates. The concentration of Ln³⁺ ions in doped crystals is low, usually 0.5–5 mol.%, which is alternatively expressed as a number of ions per unit of volume, 10¹⁹–10²⁰ cm⁻³. The generation wavelengths for the mentioned Ln³⁺ ions are located within 1–3 μm for most of the transitions.

The band widths of Russell–Saunders terms in Nd³⁺: Y₃Al₅O₁₂ have been experimentally verified to depend on the doping concentration and growth process. ³ It was pointed out in the same study that the increase of concentration reduces the stimulated-emission cross-section of the activated crystal more than twice per 1 mol.% Nd³⁺ proceeding from emission at 1.06 μm to that at 1.32 μm. Er³⁺ cross-sections for the transitions ⁴I_13/2 → ⁴I_15/2 have been analyzed with the aid of McCumber’s theory. ^{4 , 5}

Another issue of primary importance for the operation of a solid-state laser resonator is the change of polarizability of the ions in the active media causing optical inhomogeneity. For this purpose, the indices of refraction n for various solid-state laser materials have been measured during pumping. ⁶

A good linear dependence between polarizabilities and molecular volumes of 25 molecules of various size, shapes and degrees of polarity has been found. ⁷ In the same article, the authors obtained a better correlation between the same quantities by including the tightness of binding of electrons on the molecular surface. The proportionality of the atomic polarizability to the volumes of real atoms has also been confirmed. ⁸ It has been suggested that this correlation could be enhanced by the introduction of a slowly varying periodic function of the nuclear charge.

Since the active volume of a laser crystal is significant not only for the stoichiometric compounds containing Ln³⁺ ions as constituents of the crystal lattice, ⁹ it is important to study the dependence between emission cross-sections and molar volumes of lanthanide laser crystals. This is the purpose of the present work. The polarizability volumes are associated with: (i) a specific feature directly linked with the volume of the laser resonator since it may contain crystal lattice with unit cell of different symmetry and number of molecules, (ii) a more systematic dependence than separately exerted by the relative dielectric permittivity or by the index of refraction. The difference in both cases is accumulated in the ratio M/ρ that reflects the chemical composition via molar mass and a basic physical quantity, respectively.

2 Methods

At frequencies lower than those for the VIS range of the electromagnetic spectrum, i.e. at ν < 372 THz, the molar polarizability α may be revealed in the right-hand side of Clausius – Mossotti equation for molar polarization P _m: ¹⁰

where the symbols designate: M/10⁻³ kg mol⁻¹ – molar mass, ρ/10³ kg m⁻³ – mass density, N _A = 6.02214 × 10²³ mol⁻¹ – Avogadro’s number, ε _r – relative dielectric permittivity, ε ₀ = 8.85419 × 10⁻¹² J⁻¹ C² m⁻¹ – electric permittivity of vacuum. It is noteworthy that Equation (1) does not contain thermally averaged electric dipole moment μ = (μ ² E/3kT) in applied electric field E.

The equation of Lorenz–Lorentz, an analogue of the Clausius–Mossotti equation for the optical region of the electromagnetic spectrum, has been used here for determination of the molar polarizability α and the polarizability volume α _v:

where P _m/10⁻⁶ m³ mol⁻¹, with dimension of molar volume, and α _v/10⁻³⁰ m³ pertain to the transparent host crystal. Both Equations (1) and (2) relate macroscopic quantities ε _r and n _λ respectively, to the microscopic, molar quantity α and this link has been explicitly included in this study.

The stimulated-emission transitions are in fact internal 4f–4f transitions in the lanthanide (3+) ions. For this reason, the specific Ln³⁺ ion polarizability, α (Ln³⁺), need to be included to the overall polarizability of the doped crystal, α v + c α L n 3 + . The latter provides an opportunity to assess the polarizability volume of a crystal matrix in respect to that of the doped crystal. Thus, the relative contribution due to Ln³⁺ ion polarizability has been accounted:

The values of α (Ln³⁺) are known from electric polarizability computations for the same ions in free state; ¹¹ those obtained in the approximation (LC + CPP), i.e. of large-core and contracted pseudopotentials, have been used here. The quantities α (Ln³⁺) have the same order of magnitude and dimension as α _v, namely 10⁻³⁰ m³ or ų. The contributions from Ln³⁺ ions to β are proportional to the molar concentration c of the Ln³⁺ ions, as noted in Equation (4).

The index of refraction n _λ in Equation (2) should be considered dependent on the wavelength λ _e to which it is referred, in this case – those of laser transitions. The latter for each of the studied Ln³⁺ ions are shown in Table 1.

Molar masses M and mass densities ρ of the crystals comprised in this work are listed in Table 2, first fluorides (simple and complex) followed by the complex oxides. Molar masses of the host crystals are according to the stoichiometries with molar masses of the elements, ¹² the latter have been compiled in a source on optical materials, along with indices of refraction for the same substances, including isotropic, uniaxial and biaxial crystals. ¹³ The refraction indices have been evaluated in the present work for the wavelengths of stimulated (or rarely, spontaneous) emissions through the Sellmeier equations n ² = f(λ ²), available for most of the listed laser crystals except for certain stoichiometric neodymium compounds. ¹³

3 Results and discussion

Tables 3–6 contain values of relevant quantities included in Equation (2) for Nd³⁺, Ho³⁺, Er³⁺, and Tm³⁺ ions doped in crystals, respectively, as well as for three stoichiometric neodymium compounds. The crystals are listed in all tables in order of increasing polarizability volumes. The quantities comprise experimentally determined emission cross-sections σ _e, peak wavelengths λ _e of emission transitions, interpolated indices of refraction n _λ at λ _е, molar concentrations c of Ln³⁺ ions, as well as calculated here polarizability volumes α _v and intrinsic indices of refraction for the wavelength of generation. References in all tables pertain to σ _e, λ _e, and c (Ln³⁺) for corresponding crystal. The input data are contained in the three columns after that with the empirical formula of each crystal following the column with the respective reference. Our results are in the last two columns in Tables 3–6. The influence of σ-, π- and other types (x, y, z) of polarization for anisotropic crystals on the polarizability volumes α _v in the observed laser transitions with Ln³⁺ -doped single crystals has been reflected in this work by means of the Sellmeier equations. ¹³ For example, these are of the type n _o ² = f(λ _e ²), n _e ² = f(λ _e ²) – for CaWO₄, LaF₃, LiNbO₃, LiYF₄, YVO₄, etc. or n _a ² = f(λ _e ²), n _b ² = f(λ _e ²), n _c ² = f(λ _e ²) – for YAlO₃, where n _o and n _e are ordinary and extraordinary indices of refraction, respectively, a, b, and c are axes of the unit cell, λ _e is in μm. The isotropic crystals such as garnets and M^IIF₂ exhibit only one Sellmeier equation for n _λ ².

Simple and complex fluorides and complex oxides in crystalline state have attracted considerable research interest for the following reasons. The fluorides are transparent in a wider wavelength range, have lower refraction indices that reduce non-linear optic effects during pumping and exhibit lower phonon frequencies limiting the losses due to non-radiative transitions in the active media. ¹⁴ On the other side, the complex oxides provide higher power output remaining thermo-mechanically stable. ¹⁵ These important properties for stimulated emission are very sensitive to the energy-level scheme of the particular Ln³⁺ ion. The results for Nd³⁺ ions are presented in Table 1. It should be noted that the n _λ ² values for YVO₄ have been determined after Sellmeier equations for each polarization at the indicated wavelengths.

In Table 1, the concentrations of Nd³⁺ ions doped in YAlO₃, as well as in Y₃Al₅O₁₂ and Y₂O₃, have not been explicitly given in Ref. ²⁰ . It is seen in the same table that while fluorides exhibit the lowest values of polarizability volumes α _v, the highest values are those of garnets, especially gadolinium gallium garnet. This may be attributed mainly to the ratio M/ρ and, to a lesser extent, to the indices of refraction. The polarizability of the active ions α (Nd³⁺) = 0.955 × 10⁻³⁰ m³. ¹¹ LiNbO₃ and LiTaO₃ single crystals doped with Nd³⁺ ions present special interest due to high electrooptical and linear coefficients of the crystal hosts providing frequency doubling in the gain material. ²¹

Further, it is useful to normalize the polarizability volumes by means of Equation (4). Such a procedure provides an estimate of the deviations for each crystal host while the dopant ion is kept constant. The dashed lines in Figures 1–4 represent the upper limit at β = 1 with polarizability of the crystal matrices only, i.e. excluding contributions from polarizabilities of the dopant, cα (Ln³⁺). Except crystals of stoichiometric Nd compounds in Figure 1, the deviations are below β = 1 and small for all doped crystal lattices and for all of the studied ions. These results are mainly due to the low concentration of the doped Ln³⁺ ions which in turn is a proportionality coefficient to the electric polarizability of the corresponding lanthanide ion, α (Ln³⁺). The stoichiometric Nd crystals exhibit emission cross-sections nearly in the same range of values but at lower relative polarizability volumes. This fact may be added to their advantages compared to the other transparent crystals doped with Nd³⁺ ions.

Figure 1:

Correspondence between relative polarizability volumes β and emission cross-sections σ _e in crystals containing Nd³⁺ ions (the lowest three points, from left to right, pertain to NdP₅O₁₄, LiNdP₄O₁₂, and NdAl₃(BO₃)₄, respectively).

Figure 2:

Relative polarizability volumes β versus emission cross-sections σ _e in Ho³⁺ activated crystals (the lowest point is of LaF₃ crystal).

Figure 3:

Dependence of relative polarizability volumes β on emission cross-sections σ _e in Er³⁺ doped crystals (the lowest two points pertain to CaF₂, that above – to LiYF₄).

Figure 4:

Relative polarizability volumes β as a function of the emission cross-sections σ _e in Tm³⁺ doped crystals (the lowest 4 points present LiLuF₄ and YAlO₃ (3) crystals).

For Ho³⁺ -doped crystals in Table 4, YVO₄ takes an intermediate position in relative polarizability volumes β between fluorides and garnets due to high M/ρ ratio and high indices of refraction. The variation of the normalized polarizability volumes versus emission cross-sections for the same ion is shown in Figure 2. It is seen that the reductions from β = 1 are small mainly because of the low concentrations c(Ho³⁺) with α(Ho³⁺) = 0.702 × 10⁻³⁰ m³. ¹¹

It is known that the emission cross-section of Ho³⁺: YVO₄ is higher than that for Ho³⁺: Y₃Al₅O₁₂ ³⁶ (in Table 4), whereas the comparison between the polarizability volumes α _v for the same crystals is vice versa, with emission transitions of the same type (⁵I₇ → ⁵I₈) and at close wavelengths, ca. 2.0 μm. As the same situation occurs with Nd³⁺ ions in the same matrices, it may be concluded that relatively soft crystals, in terms of lower polarizability volumes, produce higher emission cross-sections.

The crystals doped with Er³⁺ ions (Table 5), lasing at 1.55 μm and near 2.9 μm reveal the same small deviations from β = 1; here, only Er³⁺: BaF₂ crystal may be exemplified with a stimulated emission at 0.55 μm. With α (Er³⁺) = 0.678 × 10⁻³⁰ m³, ¹¹ the most distinct reductions in Figure 3 pertain to CaF₂ and LiYF₄ crystals doped with Er³⁺ ions. In this regard, it should be noted the deviation seen in Figure 3 for the heavily doped single crystal of Er³: CaF₂ ³⁸ and Er³⁺: LiYF₄. ³¹ It is noteworthy the discrepancy in the emission cross-sections data of Er³⁺: Y₃Al₅O₁₂ from two sources ^{15 , 43} in Table 5. Those with almost identical wavelengths (2.9 μm) of the transition ⁴I_11/2 → ⁴I_13/2 differ six times in magnitude, being reported in one and the same year. The origin of this discrepancy may be assigned either to the method and techniques for measurements in the mid-IR of the electromagnetic spectrum, to certain optical inhomogeneity or to traces of other Ln³⁺ ions in the crystal samples.

The polarizability volume of YVO₄ (Table 6) is intermediate between those of fluorides, certain complex oxides and garnets whereas the lowest values of relative polarizability volumes β correspond to LiLuF₄ and YAlO₃ crystals doped with Tm³⁺ ions (Figure 4); the polarizability α (Tm³⁺) = 0.658 × 10⁻³⁰ m³. ¹¹ Tunable stimulated emissions have been observed at room temperature in Tm³⁺: CaF₂ crystal. ⁴⁴ It should be noted again the inverse dependence between emission cross-sections and polarizability volumes of simple fluorides (CaF₂, LaF₂) and garnets (Y₃Al₅O₁₂, Gd₃Ga₅O₁₂) at close wavelengths, around 1.9 μm of the transition ³F₄ → ³H₆. As far as the emission cross section σ _e is inversely proportional to the lifetime τ of the stimulated-emission transition. ² It may be concluded that increased polarizability volumes may serve as an additional operational characteristic in this respect, indicating higher lifetime of laser generation.

In order to examine the dependences between emission cross-sections σ _e and polarizability volumes α _v for given crystal lattice, we have verified the corresponding variations in Ln³⁺: Y₃ Al₅O₁₂ (Figure 5), Ln³⁺: YAlO₃ (Figure 6), and Ln³⁺: LiYF₄ (Figure 7). It is noteworthy that the mentioned crystal hosts differ in basic properties: structural, optical, and thermo-mechanical. YAG is cubic and isotropic while YAlO₃ is biaxial and anisotropic, ^{13 , 35 , 54} LiYF₄ is also anisotropic but exhibiting negative temperature dependence of the refraction index. ¹⁹ It has been found in the present work that despite these differences, doping with Nd³⁺ ions in each of the crystal matrices results in the largest values of stimulated emission cross-sections σ _e or polarizability volumes α _v or both, compared with Ho³⁺, Er³⁺, and Tm³⁺ ions. This fact may be explained with the energy-level scheme in Nd³⁺ -doped crystals: stimulated-emission transitions in the near-infrared region, with the largest energy-level differences at low terminal levels compared to the other mentioned Ln³⁺ ions.

Figure 5:

Correspondence between emission cross-sections σ _e and polarizability volumes α _v for Ln³⁺ ions (Ln = Nd, Ho, Er, Tm) doped in Y₃Al₅O₁₂ (both points on the top pertain to Nd³⁺).

Figure 6:

Emission cross-sections σ _e dependence on polarizability volumes α _v for Ln³⁺ ions (Ln = Nd, Ho, Er, Tm) doped in YAlO₃ (the two highest points are those of Nd³⁺).

Figure 7:

Dependence of emission cross-sections σ _e on polarizability volumes α _v for Ln³⁺ ions (Ln = Nd, Ho, Er, Tm) doped in LiYF₄ (the two points on the right pertain to Nd³⁺).

Whereas the absorption cross-sections are related to channels for optical pumping and to those for re-absorption, ³¹ the cross-sections σ _e of stimulated-emission transitions present key spectroscopic quantities directly employed in characterization of laser transitions. ³⁰ The latter are linked with the line-shape or spectral density-of-states function, F(ν), as evident in numerous studies applying either method for determination of σ _e: the reciprocity method or the Fuchtbauer–Ladenburg method. ^{15 , 22 , 30 , 31 , 32 , 34 , 39 , 41 , 43} On the other side, the bands of the same 4f–4f transitions are observed in insulating transparent crystals. Hence, both phenomena, emission transitions and polarizability, are complementary. Molecular polarizability has been included in unified treatment of nephelauxetic parameters within the framework of the dielectric screening model in single crystals containing Pr³⁺, ⁵⁵ Nd³⁺, ⁵⁶ and Tm^{3+ 57} ions.

4 Conclusions

Following the above considerations, the polarizability volumes have been determined for host dielectric crystals of simple and complex fluorides and complex oxides, as well as of certain stoichiometric laser crystals containing Nd³⁺ as constituents of the respective lattices. The introduced relative polarizability volumes have revealed that the dopant Nd³⁺, Ho³⁺, Er³⁺, or Tm³⁺ ions at low molar concentrations give small contributions to the overall polarizabilities of the crystal lattices, except for Nd³⁺ ions in stoichiometric compounds. The dependences between polarizability volumes and stimulated-emission cross-sections exhibit different points for Nd³⁺ ions doped in Y₃Al₅O₁₂, YAlO₃, and LiYF₄ compared to the other mentioned Ln³⁺ ions in the same lattices. It is expected that the polarization effects highlighted in this work would promote further studies: ab initio calculations on polarizabilities of lanthanide laser crystals and on the special position of Nd³⁺ ions in regard to the emission cross-sections in dielectric crystals.