Dispersion and anisotropy of thermal coefficients of the optical path (TCOP) and thermo-optic coefficients (TOCs) of Alexandrite laser crystal (Cr3+:BeAl2O4) are studied for the three principal light polarizations, E || a, E || b and E || c. Thermo-optic dispersion formulas are presented for the spectral range of 0.4-1.1 µm. All TOCs are positive and show a notable polarization-anisotropy, dna/dT = 5.9, dnb/dT = 6.9 and dnc/dT = 15.2 × 10−6 K−1 at 0.75 µm. Thermal lensing was characterized in a continuous-wave Alexandrite laser pumped at 0.532 µm and operating at 0.7509 µm (for E || b). The measured thermal lens was weak, positive and slightly astigmatic. The sensitivity factors of the thermal lens were found to be Mx = 1.74 and My = 2.38 [m−1/W].
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Alexandrite (Cr3+-doped chrysoberyl, BeAl2O4) is a well-known crystal for tunable lasers relying on vibronic coupling [1–3]. Alexandrite provides intense emission between 0.7 and 0.85 µm with a maximum at around 0.75 µm [4,5]. The corresponding stimulated-emission cross-section is relatively small, σSE = 0.7 × 10−20 cm2 (as compared to another state-of-the-art material for tunable lasers - Ti3+:α-Al2O3 or Ti:Sapphire), which is compensated by a relatively long lifetime of the upper laser level τ ~260 μs at room temperature (RT). Thus, the σSEτ product is large and the efficient and low-threshold continuous-wave (CW) laser operation of Alexandrite is possible . This crystal can be grown by a modified Czochralski method resulting in low optical losses. The Cr3+ ions in orthorombic BeAl2O4 exhibit strong polarization-anisotropy of the spectroscopic properties (the high-gain light polarization is E || b) [4,5] and linearly polarized laser output is easily achievable.
Recently, efficient multi-watt CW Alexandrite lasers with pumping in the green (e.g., at 532 nm with the use of frequency-doubled (2ω) Nd lasers)  or in the red (e.g., at ~630 nm by AlGaInP laser diodes) [7–9] were reported. Light-emitting-diode (LED) pumping of Alexandrite has been demonstrated as well . Moreover, a semiconductor saturable absorber mirror (SESAM) and Kerr-lens mode-locked (ML) Alexandrite lasers delivering 380 fs and 70 fs pulses at around 750 nm, respectively, were demonstrated [11–13]. The Alexandrite lasers have relevant applications in medicine (dermatology), space LIDAR technologies, spectroscopy  and can replace Ti:Sapphire lasers in nonlinear microscopy.
Alexandrite exhibits a combination of attractive thermal and mechanical properties, namely very high thermal conductivity (κ ~23 W/(mK) at RT), weak and almost isotropic thermal expansion (α ~6-7 × 10−6 K−1), and high optical damage threshold [3,15]. However, thermo-optical properties of Alexandrite have not been studied in detail to date.
In the present paper, we aimed to measure the thermo-optic coefficients (TOCs, dn/dT) and to characterize thermal variation of the optical path length of Alexandrite with respect to light polarization, as well as to measure the optical power of the thermal lens in a real laser element under lasing conditions.
2. Experimental techniques
2.1 Measurements of the dn/dT coefficients
Alexandrite is orthorhombic (sp. gr. Pnma) and thus optically biaxial . Its optical properties are characterized in the frame of the optical indicatrix. The optical indicatrix axes are mutually orthogonal and they coincide with the crystallographic axes a, b, c. The corresponding principal refractive indices are na, nb and nc (for polarizations E || a, E || b and E || c, respectively) with nc < na < nb. Similarly to the refractive indices, three principal TOCs exist for Alexandrite, namely dna/dT, dnb/dT and dnc/dT. Note, that no predefined relation for the corresponding TOCs is expected.
For the measurements of TOCs of Alexandrite, the laser beam deviation method for a material with a linear thermal gradient was used . This method is simpler than other known techniques, e.g., minimum deviation and interferometry. The measurements were done using a 0.06 at.% Cr3+:BeAl2O4 crystal (Solix Ltd.) which was cut to a rectangular sample with dimensions of 5.58(a) × 6.22(b) × 6.87(c) mm3. All six surfaces were polished to a laser-grade quality. A set of probe lasers emitting in the spectral range of 0.4-1.1 µm was used. The probe radiation was linearly polarized. The measurements were done at 298 K. The linear temperature gradient in the sample was ~1-2 K/mm. It was determined separately for each sample orientation. The actual temperature of the hot and cold surfaces of the crystal was measured using sensitive thermocouples (type K, chromel-alumel) with a precision of 0.1 K.
The laser beam deviation method allows one to measure the so-called thermal coefficients of the optical path (TCOP), dn/dT + (n – 1)α. The precision of the TCOP measurements was 7-10% depending on the crystal cut. The n and dn/dT are determined by light polarization E and α (the linear thermal expansion coefficient) is determined by light propagation direction k. For orthorhombic Alexandrite, there are three principal α values along the a, b and c directions (αa, αb and αc, respectively). For any biaxial crystal incl. Alexandrite, a total of 6 independent TCOPs can be measured leading to 3 principal TOCs each of which is determined from two measurements .
2.2 Measurements of the thermal lens
Thermal lensing was studied in a c-cut 0.16 at.% Cr3+:BeAl2O4 (NG Synoptics) oriented for the E || b laser polarization when placed at Brewster angle. The optical power of the thermal lens D (inverse of the focal length, D = 1/f) was calculated from the measured radii of the output laser mode. For this, a ray transfer matrix formalism (ABCD law) was used and the M2 parameter of the laser beam was accounted for. The thermal lens was considered as a thin astigmatic lens located in the center of the crystal. The radii of the laser mode were measured along the horizontal (x) and vertical (y) directions using a beam profiler.
The schematic of the CW Alexandrite laser is shown in Fig. 1(a). The slab-shaped laser crystal was oriented at a Brewster angle (θB). Its dimensions were 3 × 5 × 7 mm3 (height × width × length). It was mounted in an Al-holder and passively cooled from 4 sides. A typical four-mirror laser cavity  was designed in such a way that the size of the output laser mode was sensitive to the thermal lens in the crystal. The cavity consisted of a highly-reflective (HR, at 0.75 µm) plane mirror M1, two HR concave folding mirrors R1 and R2 (radius of curvature, RoC: 100 mm), and a plane output coupler (OC) with a transmission of 5% at the laser wavelength. The mirror R1 which served as a pump mirror was coated for high transmission (HT) at 0.532 µm. The laser crystal was pumped by a CW green 2ω Nd:YVO4 laser (Finesse, Laser Quantum) emitting up to 6 W at 0.532 µm (a diffraction-limited output, TEM00 mode). The pump was focused into the Alexandrite crystal by a 150 mm lens. At the focus the pump spot diameter 2wp was ~25 µm in the vertical direction and ~44 µm in the horizontal one (due to the Brewster-angle oriented crystal). The crystal was pumped in a single-pass and about 85% of the pump was absorbed.
For the measurements of the laser beam radius, a 150 mm spherical lens placed at 34.8 cm after the OC, a cut-off filter for the green (0.532 µm) and a beam profiler were used as shown in Fig. 1(b).
3. Results and discussion
3.1 Thermo-optic coefficients
At first, we measured the six principal TCOPs. Their dispersion is illustrated in Fig. 2(a-c). All TCOPs are positive in the whole studied spectral range. Thus, positive (focusing) thermal lens is expected for various possible orientations of the Alexandrite laser crystals. The TCOP values show a polarization-anisotropy which is especially clear for the a-cut and b-cut crystals. In order to calculate the TOCs, i.e., dn/dT = TCOP – (n – 1)α, we used the literature data on the refractive index (calculated from the Sellmeier equations reported in ) and on the linear thermal expansion coefficients (αa = 5.9, αb = 6.1, αc = 6.7 × 10−6 K−1 [3,14]). The results are shown in Fig. 2(d). All three principal TOCs for Alexandrite are positive and show a notable anisotropy. For the whole studied spectral range, dnc/dT > dnb/dT > dna/dT. The TOCs determined from the measurements for different crystal cut were in good agreement with each other, as indicated by the error bars in Fig. 2(d).
To calculate the TCOPs and dn/dT at the particular laser wavelength, the dispersion of the dn/dT was modeled taking into account (i) volumetric thermal expansion (expressed by the αvol = αa + αb + αc coefficient) and (ii) temperature dependence of the electronic bandgap Eg (expressed by a temperature derivative, dEg/dT) :
Here, i = a, b, c, λ is the light wavelength; λg [μm] = 1.2398/Eg [eV], n(λ) is the Sellmeier equation, n∞ is the refractive index in the long-wavelength infrared limit, see Ref . The dn/dT value can be represented as a sum of two terms related to (i) and (ii) effects, (dn/dT)α + (dn/dT)g, which have negative and positive values, respectively. The experimental data in Fig. 2(d) were modeled with Eq. (1) with Eg and dEg/dT as free parameters leading to the thermo-optic dispersion curves. The best-fit parameters, depending on the light polarization, are in the range of 5.7-6.3 eV and −1.4 – 2.7 × 10−4 eV/K, respectively. For Alexandrite, the density-functional theory predicts a direct bandgap of 6.45 eV  while the UV absorption edge is located at about 9 eV . According to the Eq. (1), the positive dn/dT coefficients of Alexandrite are related to the weak thermal expansion, so that the contribution of the (dn/dT)g term is dominant.
The thermo-optic dispersion formulas can be also represented in a simplified form :Table 1.
Using the derived thermo-optic dispersion formulas, we calculated TOCs at 0.75 µm as dna/dT = 5.9, dnb/dT = 6.9 and dnc/dT = 15.2 × 10−6 K−1. The anisotropy of the dn/dT values is much stronger than that of the refractive indices, na = 1.737, nb = 1.742, nc = 1.735 at 0.75 µm . The values of the dna/dT and dnb/dT are lower than 9.4 and 8.3 × 10−6 K−1, respectively, previously measured at 1150 nm . There is no previous data on the dnc/dT. Furthermore, we calculated the dispersion curves for the TCOP values, TCOP(λ) = dn/dT(λ) + (n(λ) – 1)α, see Fig. 2(a-c). The six principal TCOPs at 0.75 µm are listed in Table 2. In particular, for a c-cut crystal and light polarization E || b (orientation studied in Section 3.2), TCOP = 11.9 × 10−6 K−1.
3.2 Thermal lensing
We started with characterization of the CW laser regime. The Alexandrite laser, Fig. 1(a), operated at 0.7509 µm (fractional quantum defect for the pump and laser photons, ηq = 1 – λp/λL = 29.2%). The laser output was linearly polarized (E || b). The maximum output power reached 1.11 W with a slope efficiency of 26.8% (with respect to the absorbed pump power Pabs). The optical-to-optical efficiency with respect to the incident pump power (Pinc) was ~18%.
In Fig. 3, we show the spatial profiles of the output mode from the Alexandrite laser corresponding to various levels of output power. Close to the laser threshold, the beam profile was nearly circular. At higher pump powers, the beam was distorted and became elliptic with its major semiaxis being parallel to the horizontal direction x. This indicated the action of an astigmatic thermal lens. To proceed with the ABCD modeling of thermal lens, the M2 parameters of the laser beam were measured along the x and y directions using a standard ISO procedure (with a focusing lens, see Fig. 1(b)).
An example of evaluation of the beam quality factors, M2x,y, for the Alexandrite laser operating at maximum output power is shown in Fig. 4(a). The beam quality factors were M2x = 1.85 and M2y = 1.47. The measured M2x,y parameters plotted vs. Pabs are presented in Fig. 4(b). The beam quality was lower (M2 is higher) in the horizontal direction. With the increase of the pump power, both M2x,y parameters tend to increase.
The results on the calculated optical power of the thermal lens Dx,y are shown in Fig. 4(c). As expected from the corresponding positive TCOP value for a c-cut crystal and E || b (see Section 3.1), thermal lens is positive for both directions (also referred as principal meridional planes). The optical power increases linearly with the absorbed pump power. This dependence is typically expressed by the so-called sensitivity factors, Mx,y = dDx,y/dPabs. In our case, Mx = 1.74 and My = 2.38 m−1/W. The error in the determination of M-factors was 0.05 m−1/W. The difference in the M-factors is expressed by the astigmatism degree, S/M = |My – Mx|/My = 27% (S/M = 0% for a spherical lens and 100% for a cylindrical one). Note that the difference of M-factors can be partially attributed to the different radii of the pump beam in the Brewster-angle oriented laser crystal. The radius of the pump beam is larger in the horizontal direction, so weaker thermo-optic aberrations are expected. However, one cannot describe the two directions separately, as the temperature field is formed in a bulk crystal.
For a longitudinally laser-pumped active element, the sensitivity factor of the thermal lens can be theoretically calculated as Mx,y = ηhΔx,y/(2Spκ) , where ηh ≈ηq is the fractional heat loading approximated as a quantum defect in the case of Alexandrite, Δx,y is the “generalized” TOC (representing a joint action of the thermo-optic, photo-elastic and thermal expansion effects), Sp = π〈wp〉2/cos(θB) is the effective pump spot area accounting for the Brewster-angle orientation of the laser crystal and the divergence of the pump beam, 〈wp〉 ~70 µm is the root-mean square radius of the pump beam averaged through the whole length of the laser crystal. According to this formula, we estimated Δx,y = 2SpκMx,y/ηh, resulting in Δx = 10.1 and Δy = 13.8 × 10−6 K−1 values which are in good agreement with the TCOP value determined above in Section 3.1.
To conclude, we have studied dispersion and anisotropy of the dn/dT coefficients and TCOPs of Alexandrite laser crystal. All three principal dn/dT are positive (due to the dominant effect of temperature variation of the bandgap over the weak thermal expansion) and they exhibit a notable polarization-anisotropy, dnc/dT > dnb/dT > dna/dT. For the high-gain laser polarization (E || b), dn/dT has an intermediate value of 6.9 × 10−6 K−1 at 0.75 µm. Positive dn/dT underlies positive (focusing) thermal lens of Alexandrite lasers.
This was experimentally proven using a green-laser-pumped c-cut Alexandrite crystal emitting at 0.7509 µm with the E || b polarization. The sensitivity factors of the thermal lens under highly focused pump were as weak as Mx = 1.74 and My = 2.38 m−1/W and the astigmatism degree for a Brewster-oriented laser element was only 27%. The optical power of the thermal lens was varying linearly with the absorbed pump power. Positive thermal lens in Alexandrite makes it suitable for microchip-type lasers.
We believe that a detailed knowledge of the thermo-optical properties of Alexandrite crystal will help in designing laser cavities of high-power CW and ML oscillators based on the Kerr lensing, SESAMs, graphene or their combination [20,21].
Natural Sciences and Engineering Research Council of Canada; the Canadian Foundation for Innovation and the University of Manitoba; Government of the Russian Federation (Grant 074-U01) through ITMO Post-Doctoral Fellowship scheme.
The authors acknowledge funding from the Natural Sciences and Engineering Research Council of Canada, the Canadian Foundation for Innovation and the University of Manitoba. P.L. acknowledges financial support from the Government of the Russian Federation (Grant 074-U01) through ITMO Post-Doctoral Fellowship scheme.
References and links
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