1. Introduction

Over the past two decades generation of ultrashort laser pulses using cubic Ti³⁺:α-Al₂O₃ (Ti:Sapphire) crystal has become a well-established approach [1]. Such process strongly depends on the value of intracavity dispersion. In soliton mode-locking regime, the shortest pulses can be produced when the net intracavity dispersion is slightly negative [2]. Since laser materials typically display positive group velocity dispersion (GVD) in the near-IR spectral range, its proper compensation requires introduction of a certain amount of negative dispersion. Thus, the dispersion management relies on detailed knowledge of the dispersive properties of the laser crystal. This becomes even more critical when generation of a few-optical-cycle pulses is needed which have spectra spanning over hundreds of nm [3–5]. In this case, the spectral behavior of the dispersion also should be taken into account and plays a central role in determining the optimum design of dispersion compensating elements such as chirped mirrors [3–5]. Moreover, in the few-cycle regime, higher-order dispersive properties of materials, such as third-order dispersion (TOD), become increasingly important in pulse forming process and cannot be ignored [6].

Alongside the well-studied Ti³⁺:α-Al₂O₃ crystal, there are several other attractive candidates for generation of ultrashort laser pulses at around ~800 nm based on the Ti³⁺ and Cr³⁺ ions. Among them are the orthorhombic crystals of chrysoberyl, BeAl₂O₄ (Cr:BeAl₂O₄ – Alexandrite) and beryllium hexaaluminate, BeAl₆O₁₀. While Alexandrite is a well-known laser material [7,8], the other one was studied only recently [9]. Both crystals possess broad vibronic emission bands, they provide naturally polarized emission and have high thermal conductivity (~23 and 12 Wm⁻¹K⁻¹, respectively) [8,10]. Strong absorption in the blue-red spectral region opens the way for efficient pumping with visible laser diodes [11–13].

Recently, the first femtosecond mode-locked operation of Alexandrite laser was demonstrated producing pulses as short as 170 fs [14]. Considering a smooth wavelength tuning range of ~85 nm achieved in the continuous wave regime [15], the pulse duration can be reduced by at least one order of magnitude thus entering the few-optical-cycle regime where careful dispersion management is a must. Unfortunately, there was no detailed study of dispersive properties of BeAl₂O₄ to date and, for example, Sellmeier equations for its refractive indices are not available. In case of BeAl₆O₁₀, very limited information is available as well [10]. In addition, knowledge of Sellmeier equations is not only the necessary prerequisite for dispersive studies but it is also essential to predict refractive properties of crystals. Since both discussed crystals are optically biaxial, they exhibit internal conical refraction (CR) [16] and therefore can be used as passive (or active) conerefringent elements, e.g. in CR lasers [17,18] or beam shapers [19,20].

In the present work, we report on a detailed study of dispersive properties of Alexandrite and beryllium hexaaluminate, provide Sellmeier equations using enhanced four-parameter fitting model for both crystals and make initial analysis of their refractive properties.

2. Physical model

There are several known crystals in the xBeO–yAl₂O₃–zSiO₂ series, called in general beryllium aluminates [9,10]. In the present work, we will consider two compounds, BeAl₂O₄ (chrysoberyl, Cr:BeAl₂O₄ – Alexandrite) and BeAl₆O₁₀ (beryllium hexaaluminate). Both of them are orthorhombic (sp. gr. Pnma and Pcam, respectively). The unit cell of these crystals is characterized as a ≠ b ≠ c and α = β = γ = 90°. In particular, the lattice parameters are a = 9.404 Å, b = 5.476 Å and c = 4.727 Å (for BeAl₂O₄) a = 9.553 Å, b = 13.816 Å and c = 8.929 Å (for BeAl₆O₁₀) [10,21].

Both crystals are biaxial. Their optical properties are thus described within the frame of the optical indicatrix with the mutually orthogonal principal axes N_p, N_m and N_g (the corresponding refractive indices are n_p < n_m < n_g) [22]. These three axes are strictly linked to the crystal-physical frame, as follows: N_p = [001], N_m = [100] and N_g = [010]. The two optical axes (OA) are located in the N_p–N_g (or, equivalently, b–c) plane. The angle between the N_g-axis and each of the OA is called the optical axis angle, V_g.

The data on the principal refractive indices of BeAl₂O₄ and BeAl₆O₁₀ in the spectral range covering visible and near-IR were taken from [8] and [10], respectively. The following four-parameter equation for the Sellmeier fit was used:

Using the obtained dispersion curves, we calculated dispersion of the optical axis angle V_g [16]:

Beryllium aluminates when doped with Cr³⁺ ions show very broad emission band in the red and near-IR spectral region spanning from ~0.65 up to 0.8 μm (for Alexandrite) or even up to ~1 μm (for BeAl₆O₁₀) [9] which is related to the ⁴T₂ → ⁴A₂ transition of the Cr³⁺ ions. For the Ti³⁺ ions (²E → ²T₂ transition) in these crystals, the emission ranges are 0.65 – 1.05 μm and 0.62 – 1.15 μm, respectively [9]. This makes the Cr³⁺- and Ti³⁺-doped BeAl₂O₄ and BeAl₆O₁₀ crystals very attractive for generation of ultrashort laser pulses. The information about dispersion characteristics of both laser hosts is thus crucial for proper design of mode-locked oscillators. The important parameters for dispersion compensation are the group velocity dispersion (GVD) and third-order dispersion (TOD) which can be calculated as [23]:

For the description of refraction in dielectric crystals, a useful parameter is the molar polarizability α_m. It can be determined from the Clausius–Mossotti relation. Strictly speaking, the latter can be applied only for homogeneous and isotropic dielectric. However, for particular classes of anisotropic materials (including orthorhombic crystals) for which optical indicatrix frame is linked to the crystal-physical one, it can be generalized for the principal αⁱ_m values along the optical indicatrix axes [24]:

3. Results and discussion

Doping of the considered crystals with laser-active Cr³⁺ and Ti³⁺ ions results in the emission in the near-IR spectral region. For Alexandrite (Cr:BeAl₂O₄), the typical emission wavelength is 755 nm. For Cr:BeAl₆O₁₀, it is ~830 nm. As Alexandrite and BeAl₆O₁₀ crystals are optically biaxial, they exhibit internal conical refraction and if cut along one of the OA they can be used as passive or active conerefringent elements for visible and near-IR light. Thus, we discuss dispersive and refractive properties of these materials in the spectral range from ~0.25 to ~1 μm. Both BeAl₂O₄ and BeAl₆O₁₀ have relatively large optical bandgaps, E_g ~5.5 eV and thus they are transparent up to λ_g ~0.23 μm [10,21].

Data on the principal refractive indices n_i as well as best-fitting Sellmeier curves for Alexandrite and BeAl₆O₁₀ crystals at room temperature are shown in Fig. 1. As one can see, the proposed Sellmeier fits describe perfectly the dispersion of the refractive indices (within an error of Δn ~10⁻⁵...10⁻⁴). The corresponding set of Sellmeier coefficients A_i, B_i, C_i and D_i is compiled in Table 1. Due to the availability of n_i for Alexandrite crystal, the derived Sellmeier equations are valid up to at least 2.5 μm. For Alexandrite, n_p = 1.734(5), n_m = 1.736(4) and n_g = 1.741(9) at the wavelength of 755 nm. For BeAl₆O₁₀, n_p = 1.731(2), n_m = 1.733(0) and n_g = 1.737(5) at 830 nm. Thus, the refractive indices of BeAl₆O₁₀ are slightly lower than those of Alexandrite. No change of the attribution of the optical indicatrix axes with respect to the crystal-physical frame is observed in the visible and near-IR. It should be noted that dispersion of BeAl₆O₁₀ was previously described with a simple three-term Sellmeier equation [10] which is less precise in the near-IR spectral range.

 

Fig. 1 Dispersion of the principal refractive indices n_i (i = p, m, g) for Alexandrite (a) and BeAl₆O₁₀ (b) crystals. Symbols: experimental data adopted from [8,10], curves: Sellmeier fits in accordance with Eq. (1), insets: relative size of the unit-cell and orientation of optical indicatrix axes (N_p, N_m and N_g) with respect to the crystallographic ones.

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Table 1. Sellmeier Coefficients in Eq. (1) for BeAl₂O₄ and BeAl₆O₁₀ Crystals

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In Fig. 2(a), we have analyzed dispersion of the optical axis angle V_g for both crystals which is relevant for their potential CR-based applications. For this, Eq. (2) has been used. For Alexandrite V_g = 30°31′ at 755 nm and for BeAl₆O₁₀ V_g = 33°05′ at 830 nm. For Alexandrite, V_g is minimized in the visible and it further increases in the near-IR. For BeAl₆O₁₀, V_g decreases monotonously with the wavelength. For rays with any polarization propagating along the OA in the considered crystals, the refractive index will be the same, n_OA = n_m. The results on the full angle of the cone of CR, 2A_CR, are shown in Fig. 2(b). For Alexandrite 2A_CR = 12'44” (3.70 mrad) at 755 nm and for BeAl₆O₁₀ 2A_CR = 11'23” (3.31 mrad) at 830 nm.

 

Fig. 2 (a) Positive biaxial Alexandrite and BeAl₆O₁₀ crystals: (a) dispersion of the optical axis angle V_g, as calculated with Eq. (2); inset: orientation of the optical axes with respect to the optical indicatrix and crystal-physical frames; (b) dispersion of the full angle of the cone of CR, 2A_CR, as calculated with Eq. (3), inset: scheme explaining definition of this angle.

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Optically biaxial crystals are classified as positive and negative [22]. If the intermediate principal refractive index (n_m) is closer to the minimum one (n_p) than to the maximum one (n_g), the crystal is classified as positive. In our case OAs are located closer to the N_g-axis, i.e. V_g < 45°. In other words, the acute angle 2V_g between the two OAs is bisected by the N_g-axis. Both Alexandrite and BeAl₆O₁₀ crystals are thus positive biaxial.

Results on the GVD and TOD are shown in Fig. 3. For this, Eq. (4) has been used. Their anisotropy for the principal light polarizations is very weak. Due to the anisotropy of absorption and stimulated emission cross-sections, the polarization of interest for Alexandrite is E || b (b = N_g) and it is E || a (a = N_m) for Cr³⁺ doped BeAl₆O₁₀. GVD for both crystals is positive and relatively small in the visible and near-IR spectral regions. For Alexandrite, GVD = 60.7 fs²/mm (at 755 nm, for E || b) and for BeAl₆O₁₀ it is slightly lower, 53.1 fs²/mm (at 830 nm, E || a). These values can be compared to GVD of 56.6 fs²/mm for Ti:α-Al₂O₃ at 800 nm [23]. For both studied crystals, GVD decreases monotonously with the wavelength which is typical for dielectric crystals. TOD is also positive. It reaches minimum in the red spectral region and further increases in the near-IR. For Alexandrite, TOD = 39.5 fs³/mm and for BeAl₆O₁₀ it is larger, 45.2 fs³/mm (for the respective light wavelengths and polarizations). These values are also similar to 41.4 fs³/mm of TOD at 800 nm for Ti:α-Al₂O₃ [23].

 

Fig. 3 (a,b) Group velocity dispersion (GVD) and (c,d) third-order dispersion (TOD) versus light wavelength for Alexandrite (BeAl₂O₄) (a,c) and BeAl₆O₁₀ (b,d) crystals, as calculated with Eq. (4).

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Finally, the analysis of anisotropy of polarizability is needed for further description of temperature variation of the refractive index expressed by the dn/dT coefficient. In order to determine molar polarizability of Alexandrite and BeAl₆O₁₀ crystals, we have used Eq. (5) with the following material parameters: M = 126.97 and 330.89 g/mol, ρ = 3.70 and 3.74 g/cm³, respectively. The results are shown in Fig. 4.

 

Fig. 4 Dispersion of molecular polarizability α_m for Alexandrite (a) and BeAl₆O₁₀ (b) crystals, as calculated with Eq. (5).

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The anisotropy of the principal αⁱ_m values is weak which justifies the use of the “generalized” Clausius-Mossotti relation. Molecular polarizability decreases monotonously with the wavelength. At ~1 μm, polarization-averaged value is ‹α_m› = 5.44 ų for Alexandrite and ‹α_m› = 14.0 ų for BeAl₆O₁₀. This value can be taken as a long-wavelength one due to the weak dispersion of α_m in the near-IR. Big difference in ‹α_m› for two crystals is mainly attributed to the different number of O^2- ions in one formula unit. It is known that ionic polarizability of O^2- depends strongly on the compound and can vary as 0.5–3.2 ų [25]. The actual value of α(O^2-) can be estimated by subtracting ionic polarizabilities of cations which are less sensitive to the crystal structure from the molecular one. The assumption of simple additivity for polarizabilities is reasonable for many crystals [25]. In our case, α(Be²⁺) = 0.04 ų and α(Al²⁺) = 0.067 ų, so ‹α(O^2-)› = 1.32 ų for BeAl₂O₄ and 1.36 ų for BeAl₆O₁₀. These two values are in close agreement with each other and they are also very close to the value for beryllium oxide BeO, α(O^2-) = 1.29 ų [25]. Thus, the additivity approximation can explain the variation of polarizability in beryllium-containing simple and complex oxides.

4. Conclusion

We presented a detailed comparative analysis of dispersive properties for orthorhombic beryllium aluminates, namely chrysoberyl, BeAl₂O₄ (Cr:BeAl₂O₄ - Alexandrite) and BeAl₆O₁₀ (beryllium hexaaluminate), focusing on the parameters relevant for applications of these Cr³⁺-and Ti³⁺-doped crystals in ultrafast mode-locked oscillators. Sellmeier equations were derived for both materials based on the previously reported refractive indices. GVD and TOD values were calculated, yielding very similar and reasonably low values for both hosts in the red spectral region, ~60 fs²/mm and ~40 fs³/mm, respectively. Position of the optical axes and conicity angles were also predicted for these biaxial crystals which is of practical importance for their applications as CR-elements in lasers and laser beam shapers. Further work on BeAl₂O₄ and BeAl₆O₁₀ crystals will be focused on description of their thermo-optic properties. As a prerequisite for this, molecular polarizability of both compounds has been calculated.