1. Introduction

Recent developments in cryogenically cooled, eye-safe lasers based on Er³⁺-doped YVO₄ and GdVO₄ crystals, demonstrated record high optical-to-optical laser efficiencies and a great potential for further power scaling [1,2]. It is well understood that the thermal conductivity, the coefficient of linear thermal expansion (CTE) and the thermo-optic coefficient (dn/dT) are the key thermo-optic properties of the laser material which ultimately define its power scaling potential with preserved beam quality [3]. In the case of cryogenic solid-state laser development, the knowledge of these three parameters in the 80 – 320 K temperature range is critically important for laser design. Thermal conductivity data for orthovanadate crystals in this range are available from a few sources (e.g., [4–6]). Meanwhile, to the best of our knowledge, the data on CTE and dn/dT below room temperature has not been reported, though it is readily available in the temperature range from + 20 to + 1000 °C [4,7–10]. Based on growing interest in rare-earth doped orthovanadates, viable competitors to conventional YAG as cryogenic laser hosts, systematic measurements of their thermal and thermo-optic properties from the temperature of liquid nitrogen to a room temperature are useful in order to remedy the dearth of data.

The purpose of our work was to eliminate this data vacuum for the YVO₄ single crystal. Reported here are the measurement results for the coefficients of linear thermal expansion and thermo-optic coefficients for YVO₄ in the 80 – 320 K temperature range. The measurements were carried out using Twyman-Green interferometry and the obtained results were compared with the published room temperature data on CTE and dn/dT.

2. Experiments and results

The temperature-dependent amplitude of lattice vibrations affects the mean inter-ion bond length. As a result, linear thermal expansion and refraction indices, which define optical path length inside a laser crystal, become temperature-dependent as well. Any thermally induced change in the optical path length can be measured by interferometric methods with high precision. If two parallel faces of the crystal under investigation form an interferometer, then it can be probed with a polarized laser beam to produce an interference fringe pattern. The interval between the maxima and minima of this pattern corresponds to a round-trip optical path length change equal to half the wavelength (λ) of the probe beam and can be determined as:

The temperature-induced shift of interference fringes causes modulation of the output of the photodiode, positioned behind the slit in an opaque screen, see Fig. 1 . The fractional change in the optical path length, γ(λ,Τ), can be calculated from Eq. (1):

 

Fig. 1 Experimental set-up used for the γ(λ,T) measurements. The slit was used for selecting a single interference fringe. Three orientations (a-c) of the uniaxial YVO₄ crystal are shown with respect to the polarization of a probe beam. Inset (d) shows an arrangement for the CTE measurements in free space.

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The YVO₄ is a uniaxial crystal with tetragonal symmetry (space group D_4h), and is known to have strong anisotropy in thermal and optical properties along and perpendicular to its optical c-axis. Therefore, thermal expansion and thermo-optic coefficients have to be characterized by four parameters: α_⊥, α_||, dn_o/dT, and dn_e/dT, where subscripts || and ⊥ denote directions of thermal expansion parallel and perpendicular to the optical c-axis, respectively. For refraction indices we use the conventional subscripts - o (ordinary) and e (extraordinary) which are defined by the vector of the electric field, normal and parallel relative to the c-axis. Thus, for a uniaxial crystal, four independent measurements are required. Because the refractive indices n_o and n_e are functions of temperature and wavelength, their derivatives dn_o/dT and dn_e/dT are also temperature and wavelength dependent.

Figure 1 shows an experimental set-up for the measurements of γ(λ,T). Either a helium-neon (He-Ne) laser (633 nm) or a TEM₀₀, fiber coupled diode laser (1570 nm) was used as a probe source. Every tested crystalline sample was mounted inside the liquid nitrogen cryostat and was initially cooled down to 77 K. The temperature of the sample was then raised very slowly (~0.5° K /min) in order to maintain a uniform temperature distribution in the sample. The crystal was single-point fixed to a cold-plate in order to provide full “freedom” of its thermal expansion/contraction with changes in temperature. The temperature of the sample was monitored with a calibrated silicon diode sensor (model DT-470-SD, Lake Shore Cryotronics) attached to the top of the crystalline slab. The sensor was connected to a temperature controller (model 335, Lake Shore Cryotronics). The sinusoidal photodiode output and the temperature of the sample were recorded simultaneously.

Shown in Figs. 1a–1c are three different orientations of the uniaxial YVO₄ crystal with respect to the light polarization and the propagation direction of the probe beam. For every chosen wavelength of the probe beam, these orientations allow us to obtain three independent functions γ = f(T) containing different combinations of two CTEs and two thermo-optic coefficients:

In order to find all four aforementioned components${\alpha }_{\left|\right|}$,${\alpha }_{\perp }$,$\frac{1}{n_o}\cdot \frac{d{n}_o}{dT}$ and $\frac{1}{n_e}\cdot \frac{d{n}_e}{dT}$, either α_|| or α_⊥ has to be known in advance or measured independently. With the first probe beam, it is necessary to measure all three coefficients of γ₁, γ₂ and γ₃. While using the second beam (with a different wavelength), it is sufficient to measure only two fractional coefficients because CTEs α_|| and α_⊥ are wavelength-independent.

As was already mentioned, there is no available data on the linear thermal expansion of YVO₄ below room temperature. Therefore, in order to determine CTE α_⊥(T) in this range, we built a free space interferometer by carefully mounting two small fused silica wedge-prisms on top of the YVO₄ crystal, see Fig. 1, inset d. This interferometer was probed with the He-Ne beam at 633 nm in a manner similar to that described above.

Figure 2a shows the measured temperature behavior of α_⊥(T). Figures 2b–2d show fractional changes (γ₁(T), γ₂(T) and γ₃(T)) in the optical path length, for the three different crystal orientations with respect to the probe beam propagation direction and the probe beam polarization (shown in Figs. 1a, 1b, 1c). The measurements were performed on a 20.1 mm long, 3.07 mm thick, sample provided by Photop Technologies, Inc.

 

Fig. 2 (a) Temperature dependence of the coefficient of linear thermal expansion α_⊥ of the YVO₄ crystal, as was measured using a free space interferometer shown in Fig. 1d. The maximum observed scatter of data points is ± 10% (b,c) - Fractional change in optical path length γ₁ and γ₂ versus temperature of the YVO₄ crystal measured with 633 nm and 1570 nm probe beams (d) - fractional change in the optical path length γ₃ measured with 633 nm wavelength.

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In all our measurements, crystalline samples were initially cooled down to approximately 77.7 – 77.8 K and then their temperatures were slowly brought up to approximately 320 −325K. For each recorded temperature point in the above range, there was a corresponding point of the intensity of the fringe pattern. The resulting “sinusoidal” function had a variable period, which shortens with the rising temperature. However, it is an extremely rare event, when the minima (or maxima) of the fringe pattern coincide with the lowest temperature in the available temperature range. In addition, it was difficult to identify a peak (or a valley) of the fringe pattern for temperatures below ~100K, especially for very “slow” changes in thermal expansion. Therefore, in Fig. 2, for fractional changes γ, we presented our experimental data from the temperature of 80K and up, see Figs. 2b–2d; but for the CTE α_⊥ - only from 100K.

The fractional changes γ₁(T), γ₂(T) and γ₃(T), measured with the same wavelength of the probe beam, and the CTE α_⊥(T) can be fit to a second order polynomial respectively:

Using the expressions (3)–(6), the temperature dependence of the CTE α_|| can be defined as:

Figure 3 depicts temperature dependencies of CTEs α_⊥ and α_|| plotted according to expressions (6) and (7). We can compare the measured values of α_⊥ and α_|| with published data [4,7–9]. As it was mentioned before, a comparison can only be done for room temperature values. There is a noticeable spread in the CTE data obtained by different authors, especially for the α_⊥. Indeed, α_⊥ values were reported in the range of (1.69 – 4.43)∙10⁻⁶ K⁻¹ while α_|| values were reported in the range of (8.19 – 11.4)∙10⁻⁶ K⁻¹. Our measurements of α_⊥ = 2.2·10⁻⁶ K⁻¹ (expansion normal to the c-axis) and α_║ = 8.4·10⁻⁶ K⁻¹ (expansion parallel to the c-axis) are falling well within the published range and are in the best agreement with the data for YVO₄ at 300 K published in [8].

 

Fig. 3 Temperature dependencies of the linear thermal expansion coefficients of the YVO₄ crystal in the direction normal to (α_⊥) and parallel to (α_║) the c-axis. The dependence α_⊥(T) is the same as linear extrapolation of the data in Fig. 2a.

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It should be noted that at cryogenic temperatures the measured values of α_⊥ and α_|| are only 5.3% and 28% of the corresponding CTE values measured around 300 K. Their ratio, which reflects the degree of optical anisotropy, becomes even larger with temperature reduction.

A similar fitting procedure for γ₁(T) and γ₂(T) dependencies measured with the 1570 nm probe beam (see Figs. 2b, 2c) in the 80 - 320 K range yields:

After subtracting Eq. (6) from Eqs. (3), (4), (8) and (9) and subtracting the expression (7) from expression (5), the refractive indices n_o and n_e were derived by integrating simple first order differential equations. The initial values of these indices at room temperature were taken from [10]: $n_o^{}(633nm)=1.99279$,$n_e^{}(633nm)=2.21533$,$n_o^{}(1570nm)=1.94426$and $n_e^{}(1570nm)=2.14809$. Then, using the expressions (1)–(7) and the corresponding refractive indices of the YVO₄, the thermo-optic coefficients $\frac{dn}{dT}$ can be calculated.

Figures 4 and 5 show the behavior of refractive indices (n_o, n_e) and thermo-optic coefficients ($\frac{d{n}_o^{}}{dT}$, $\frac{d{n}_e^{}}{dT}$) for the 633 nm and 1570 nm wavelengths We believe that these data are reported for the first time. However, for the room temperature and above, the thermo-optic coefficients were measured at the 633 nm wavelength in [10]: $\frac{d{n}_o^{}}{dT}~17.4\cdot {10}^{-6}{K}^{-1}$, $\frac{d{n}_e^{}}{dT}~12.5\cdot {10}^{-6}{K}^{-1}$ and can be compared with our room temperature data. It appears that our measurements yielded slightly lower values: $\frac{d{n}_o^{}}{dT}(633nm)=15.6\cdot {10}^{-6}{K}^{-1}$ and $\frac{d{n}_e^{}}{dT}(633nm)=9.5\cdot {10}^{-6}{K}^{-1}$.

 

Fig. 4 Temperature dependencies of YVO₄ refractive indices at 633 nm and 1570 nm. a. ordinary indices n_o. b. extra-ordinary indices ne.

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Fig. 5 Temperature dependencies of thermo-optic coefficients of YVO₄ crystal according to Eqs. (10)–(13) for the 633 and 1570 nm wavelengths.

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The obtained temperature dependencies can be approximated by the expressions:

It can be seen from Fig. 5 that the thermo-optic coefficients of YVO₄ are considerably lower at cryogenic temperature compared to their room temperature values (3.5 – 5 times). There is also a noticeable anisotropy in the thermo-optic coefficients at room temperature: $\frac{d{n}_o^{}}{dT}$ >> $\frac{d{n}_e^{}}{dT}$. At cryogenic temperatures, anisotropy remains strong at 1570 nm, but becomes significantly smaller at 633 nm. A comparison of thermo-optic coefficients of the YVO₄ crystal at different wavelengths (633 nm and 1570 nm), but at the same temperature, shows that these coefficients are always higher at 1570 nm. At the same time, at cryogenic temperatures, thermo-optic coefficients of YVO₄ are very small for both wavelengths.

3. Conclusions

We report the measurement results for coefficients of linear thermal expansion and thermo-optic coefficients of a uniaxial YVO₄ crystal over a wide range of temperatures from room- down to the liquid nitrogen temperature. To the best of our knowledge, CTE and dn/dT values for this important laser host, in the 80 - 320 K temperature range, are reported for the first time. The CTE data were obtained for two directions - parallel (α_║) and perpendicular (α_⊥) to the optical axis of the crystal. The corresponding polynomial expressions for the observed temperature dependencies of CTE were derived. Thermo-optic coefficients dn/dT were measured for two wavelengths, 633 nm and 1570 nm, for ordinary and extraordinary beams. The CTE values at cryo temperatures (~80K) α_⊥ ~0.11∙10⁻⁶ K⁻¹ and α_║ ~2.43∙10⁻⁶ K⁻¹ are approximately 5.3% and 28% of those at room temperature. CTE of YVO₄ in the direction perpendicular to the optical axis is one of the lowest among known laser hosts, offering significant advantage in power scaling with preserved beam quality compared to conventional YAG.