1. Introduction

Recent advance in giant micro-photonics has enabled drastic power-scaling in various photon sources based on microchip lasers [1]. Although many benefits have been realized by the miniaturization of highly bright solid-state lasers [2, 3], the excessive heat generated by densification of the excitation limits the averaged-power from miniature lasers. The temperature increase in laser gain media due to this heat causes not only thermal expansion but also influences on the optical gain and the heat capacity [4, 5]. As a result of high power pumping, lensing and birefringence induced by the severe temperature distribution in laser gain media degrades the laser performance of destructs laser gain media itself.

In order to extend the scalability in both of brightness and average power, heat management in laser cavity according to precise thermo-mechanical and -optical parameters of laser gain media is the most critical. However, there are significant discrepancies in the reported value of material parameters. Especially reported values of the linear thermal expansion coefficient, α and temperature coefficient of refractive index, dn∕dT are different more than one order. For examples, reported values of dn/dT at 1.06 μm for π-polarization in YVO₄ varies from 0.3 to 13.5 × 10⁻⁶ K⁻¹ [6, 7], and still now there seems to be no hope that this difference will converge in near future.

The variation of reported α is within a several-fold range to the utmost: for example, reported values of α in Y₃Al₅O₁₂ (YAG) varies from 2.5 to 8.2 × 10 ⁻⁶ K⁻¹ [8, 9]. Our research started from a consultation that Nd:YVO₄ crystal in the laser component with high brightness output designed for industrial application by use of the previously reported α [10]. Therefore, our report should be useful not only for scientific filed but also many real applications in industrial field urgently.

Moreover, α includes another important problem. Even though many researchers believe that YAG crystal has anisotropic thermal expansion [11, 12], it should be wrong physically. Thermal expansion can be described as a coordinate transformation per unit temperature, thus α is a second order tensor. From Onsager reciprocal relations, YAG (cubic) and vanadate (tetragonal) have one and two independent components of thermal expansion coefficient, respectively. Authors consider that the reported anisotropy in thermal expansion of YAG was contributed by experimental error, which should be re-evaluated with high accuracy. One of objectives in this work is an experimental confirmation by evaluation of dependence on crystal axes in α of YAG.

We already provided the evaluation procedure for α and dn/dT of GdVO₄ [13], however those were not proved by the evaluation of experimental errors. In this work, we tried to summarize these thermal properties of YAG, YVO₄, and GdVO₄ with high accuracy based on the careful treatment of experimental errors.

2. Methods

2.1 Thermal expansion coefficient

Thermal expansion was measured using a push-rod type dilatometer (DIL 402C, NETZSCH). Measurements were carried out under the dynamic helium atmosphere with gas flow rate was 50 ml/min, and heating rate is 4K/min within the range from 0 to 300 °C in temperature, T. The contact force of the push-rod was 0.25 N, and measured data was calibrated by an fused silica standard.

Measured samples were (111)-cut YAG single crystal (Scientific Materials Co.), (100)-cut and (001)-cut YVO₄ single crystals (ITI Electro Optics Co.), and (100)-cut and (001)-cut GdVO₄ single crystals (Shandong Newphotons Science and Technology Co., Ltd.) with the size of 8 mm in diameter and 25 mm in thickness.

2.2 Refractive index

The absolute value of refractive index, n was calculated from the angle of minimum deviation of the triangular prisms that have bases of isosceles right triangles [14]. The height of prisms and the length of long side in bases were 10 mm. We evaluated undoped and 1.0at.% Nd³⁺-doped YAG single crystal (Scientific Materials Co.), YVO₄ single crystals (ITI Electro Optics Co.), and GdVO₄ single crystals (Shandong Newphotons Science and Technology Co.,LTD). Bases of the YAG prism were parallel to (111)-plane, on the contrary, bases of vanadates were parallel to (001)-plane.

The minimum deviation angle, A_min of these prisms and prism angles, A_p were measured within the wavelength range between 420 nm and 1600 nm, where light was analyzed by the spectrometer (V-30D, Shimadzu Rika Co.). Values of A_min and A_p were detected by the CCD camera (CCD-41R, Shimadzu Rika Co.) with the resolution of 30 seconds, as shown in Fig. 1.Temperature in the environment of experimental setup was 23 °C.

 

Fig. 1 Schematic diagram of experimental setup for refractive index measurement (a) and the definition of A_p and A_min (b). The incident angle to the prism is equals to the output angle.

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From these measured angles n can be calculated by

2.3 The detection of the shift of the fringes in transmission

For evaluating temperature coefficient of refractive index, dn/dT, we measured phase shifts of interference fringes in transmittance of laser materials. The transmission T₀ of thin laser material with a thickness of L is ideally modulated to T_r by multiple reflections by

 

Fig. 2 The concept of multiple reflection. E_in, r, and t are the amplitude of electric field, reflectance, and transmittance of probe light, respectively. |t|⁴ equals to T₀, and |r|² equals to R.

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Experimental setup for measurement of T_r is shown in Fig. 3. The light emitted from halogen lamp (PHL-150, Mejiro Precision Inc.) was collimated and focused to the sample that was sandwiched by temperature controlled copper plates within the range from 15 °C to 65 °C. The temperature was stabilized by a peltier device and its driver (LDT-5948, ILX Lightwave Co.) with the resolution of 0.1 °C. Transmitted light from the sample was collected into a monochrometer (TRIAX-550, HORIBA Jobin Yvon S.A.S.) and detected by an InGaAs array sensor (IGA512-1-1, HORIBA Jobin Yvon S.A.S.) with the spectral resolution of 0.12 nm. Δφ can be obtained by the least square fitting of T_r to Eqs. (3-4).

 

Fig. 3 Schematic diagram of experimental setup for evaluation of phase shift Δφ.

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2.4 Temperature coefficient of refractive index

We evaluated the modulation in T_r of (111)-, (100)-, and (110)-cut 1.0at.% Nd:YAG plates (Fujian Castech Crystals, Inc), (100)-cut 1.0at.% Nd:YVO₄ (ITI Electro Optics Co.), and (100)-cut 1.0at.% Nd:GdVO₄ (Shandong Newphotons Science and Technology Co.,LTD) plates. The thickness of YAG and YVO₄ samples were ca. 0.2 mm, and GdVO₄ sample has thickness of ca. 0.1 mm. These thicknesses were measured by a micrometer (BMD-25DM, Mitutoyo) with the resolution of 1 μm.

Temperature coefficient of refractive index dn/dT can be estimated from measured Δφ and α by

3. Results

3.1 Thermal expansion coefficient and refractive index

Figure 4 shows measured α, and calculated n are shown in Fig. 5.The difference in n of YAG between doped sample and undoped sample is within 0.0006, and markers for Nd-doped samples were situated at the same position as markers for doped samples in Fig. 5. Therefore there is a certain difference (3 times of δn) between n of 1.0at.% Nd:YAG and undoped YAG. On the contrary, differences in n of vanadates are below 0.0002, and those are lower than the evaluation error.

 

Fig. 4 Thermal expansion coefficients of YAG, YVO₄, and GdVO₄ single crystals.

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Fig. 5 Refractive indices of YAG, YVO₄, and GdVO₄ single crystals. (o) and (3) indicate ordinary and extraordinary polarization, respectively.

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3.2 Interferometric fringes in transmission

Figure 6 shows T_r of various samples depending on the temperature, T. The ratio of the modulation depth in transmission, ΔT_r is larger than 3% of transmission. We were able to evaluate φ with the estimation error of less than 0.02 rad by use of Eq. (3). The additional peaks at 1064 nm in transmittance are considered to be due to the fluorescence from Nd³⁺ excited by probe light.

 

Fig. 6 Temperature dependent T_r. (a) 1at.% Nd:YAG, (111)-cut. (b) 1.0at.% YVO₄, (100)-cut in ordinary polarization. (c) 1.0at.% GdVO₄, (100)-cut in extraordinary polarization.

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3.3 Phase shift in the fringes and temperature coefficient of refractive index

As shown in Eq. (4), the value of Δφ/φ depends not on thickness but on α, dn/dT, and ΔT. Δφ/φ of various samples at several important wavelengths are shown in Fig. 7, where 0.9 μm means 946 nm for Nd:YAG, 914 nm for Nd:YVO₄, and 912 nm for Nd:GdVO₄. Similarly 1.1 μm means 1064 nm for Nd:YAG and Nd:YVO₄, and 1063 nm for Nd:GdVO₄. Also 1.3 μm means 1319 nm for Nd:YAG, 1342 nm for Nd:YVO₄, and 1341 nm for Nd:GdVO₄. Lines in Fig. 7 are fitted by the least square method. dn/dT estimated from measured Δφ and α at room temperature are shown in Table 1. In this work room temperature means the temperature range between 15 °C and 65 °C. The high linearity in our fringe-shifts of transmittance in Fig. 7 is an evidence of high temperature-controllability in our measurements.

 

Fig. 7 Relation between Δφ/φ and temperature in YAG, YVO₄, and GdVO₄ single crystals. (a) Δφ/φ at 0.9 μm. (b) Δφ/φ at 1.1 μm. (c) Δφ/φ at 1.3 μm 1.0at.%. (ord) and (ext) indicate ordinary and extraordinary polarization, respectively.

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Table 1. α and dn/dT of YAG, YVO₄, and GdVO₄ single crystals at room temperature

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4. Discussions

4.1 Sellmeier equations for YAG, YVO₄, and GdVO₄ single crystals

In order to quantize the relationship between n and λ, we can use Sellmeier equations. We can fit n of YAG crystals to Sellmeier equations given by

Table 2. Parameters in sellmeier equation for YAG, YVO₄, and GdVO₄ single crystals

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4.2 Thermal expansion coefficient along various planes in YAG single crystal

The correct measurement of α should provide a certain answer to the conflict between Onsager reciprocal relations and the common sense of anisotropic α of YAG in the solid-state laser researchers [11,12]. Authors consider that the reported anisotropy in thermal expansion of YAG should be re-evaluated with high accuracy by use of our evaluation procedure.

Similarly to Eq. 54), we can estimate α by Δφ/φ and dn/dT by

Table 3. α of YAG crystal estimated from Δφ/φ and dn/dT

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Figure 8 shows clearly that the dependence of Δφ/φ on crystal direction is quite small. This is the reason why the difference in calculated α along various orientations.

 

Fig. 8 Temperature changes of Δφ/φ in Nd:YAG plates. Lines are fitted by the least square method.

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5. Summary

We evaluated thermo-mechanical and -optical properties of YAG, YVO₄, and GdVO₄ by with less than 2% of evaluation error, and it was also revealed that thermal expansion of YAG is independent on the crystal orientation against conventional knowledge. We tabulated thermal parameters of YAG, YVO₄, and GdVO₄, and it will be the reliable data table for thermal design and heat management not only of microchip lasers but also of a plenty of solid-state lasers.

Appendix-1 Evaluation errors for α and dn/dT

Three runs of measurements by the dilatometer brought no difference larger than 1.0 nm, and the evaluation error of measured thermal expansion was 10⁻⁹ m /10K / 0.025m = 4×10⁻⁹/K. Thus the experimental error in α is below 1% due to the resolution of the dilatometer.

30 seconds of resolution of our spectrometer was enough large to bring no difference in reading of the value of angles, thus it was considered to be δA.

0.02 rad of δφ is the maximum of errors during least square fitting. Since small δλ [5] and δn, δλ/λ and δn/n can be ignored.

Major part of evaluation error is due to Δφ and ΔT, and those depend on the effective data interval. Under the assumption that dn/dT is constant within the measured temperature range, ΔT become 50°C.

and 5Parameters for error calculation in Eq. (6) and calculated errors are summarized in Table 4, respectively.

Table 4. Parameters for error evaluation

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Table 5. Experimental errors in α and dn/dT of YAG, YVO₄, and GdVO₄ single crystals

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The superiority of the evaluation method for dn/dT by the measurement of the temperature shift in the fringes on transmission is high reproducibility. In 1991, authors demonstrated the frequency tuning of Nd:YVO₄ microchip laser [15], where the ratio of the oscillation frequency and the frequency tuning range, Δν/ν is the same as the ratio of the optical path of a microchip and it’s temperature deviation: this is definitively equal to Δφ/φ in this work. Figure 9 comparatively shows the Δν/ν evaluated in 1991 and the simulated value from Δφ/φ in this work (because temperature range in these two experiments were different). The accordance with these two lines in Fig. 9 directly proves the small reproducible error of evaluation methods that use fringes in transmissions due to multiple-reflections inside samples. From our error evaluations, we can conclude that α and dn/dT reported in this work is highly accurate compared with other traditional works.

 

Fig. 9 Temperature dependence of Δν/ν in Nd:YVO₄ microchip laser compared with calculated shift based on the value in this work.

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Appendix-2 Comparison to past reports on α and dn/dT

We summarize the comparison between this work to past reports on α and dn/dT in Table 6. α of YAG in this work is similar to [16], while dn/dT is almost the same as the maximum of previously reported value [17]. α and dn/dT of YVO₄ in this work are close to [18] and [19], respectively. In the case of GdVO₄, α in this work is proximate to [20] and dn/dT is larger than any other reported value.

Table 6. Maximum and minimum values of previously reported α and dn/dT

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Appendix-3 Relations between thermal effect and thermal parameters

As well as thermal conductivity κ [4], α and dn/dT of laser gain media are quite important thermal parameters for describing thermal effects in the laser cavity. For example, thermal lensing is severely depends on these parameters. Lens-effects inside a gain medium are caused by three kinds of changes of optical path: thermal expansion, refractive index change induced by thermal strain, and temperature dependent refractive index. The curvature of thermal lens is the summation of curvatures formed by these optical path changes. Optical path changes caused by thermal expansion and refractive index change induced by thermal strain are proportional to α, while optical path changes caused by temperature dependent refractive index are proportional to dn/dT. The focal length of thermal lens is the inverse of amount of these curvatures, and proportional to thermal conductivity. In the case of end-pumped solid-state rod lasers, the focal length of thermal lens f can be expressed by

(13)$$f=\frac{\pi w_p{}^2\kappa }{{\eta }_a{\eta }_{h}P}{\left[\frac{\alpha r\left(n-1\right)}{l}+\alpha n^{3}C+\frac{1}{2}\frac{dn}{dT}\right]}^{-1},$$

where w_p, η_a, η_h, P, r, l, and C are the radius of pump beam, the pump absorption efficiency, the thermal load, the induced pump power, the radius and length of rod, and photo-elastic coefficient for desired polarizations, respectively [29].

On the contrary, the evaluation of α and dn/dT from the pump-induced therml lensing of laser cavities requires the precise evaluations of mode-matching and thermal load which are dependent on the pump intensity. Therefore, in this case it should be important for the estimation of experimental errors to clarify the uncertainty against not only w_p, η_a, η_h, P, r, l, and C but also measured focal lens including focusing aberrations and astigmatism.

Appendix-4 Derivation of Eq. (2)

The experimental error of the refractive index estimated from the minimum deviation angle is given by Eq. (2), which can be derived by the differentiation of Eq. (1). Differentiating by A_p gives

(14)$$\begin{array}{c}\frac{\partial n}{\partial A_p}=\frac{1}{2}\cos \left(\frac{A_{\min }+A_p}{2}\right)/\sin \frac{A_p}{2}-\frac{1}{2}\cos \frac{A_p}{2}\sin \left(\frac{A_{\min }+A_p}{2}\right)/{\sin }^2\frac{A_p}{2}\\ =\frac{\sqrt{1-n^2{\sin }^2\frac{A_p}{2}}-n\cos \frac{A_p}{2}}{2\sin \frac{A_p}{2}},\end{array}$$

and differentiating by A_min gives

(15)$$\frac{\partial n}{\partial A_{\min }}=\frac{1}{2}\cos \left(\frac{A_{\min }+A_p}{2}\right)/\sin \frac{A_p}{2}=\frac{\sqrt{1-n^2{\sin }^2\frac{A_p}{2}}}{2\sin \frac{A_p}{2}}.$$

Therefore, we can estimate experimental error δn by

(16)$$\begin{array}{c}\delta n=\sqrt{{\left(\frac{\partial n}{\partial A_p}\right)}^2\delta A_p{}^2+{\left(\frac{\partial n}{\partial A_{\min }}\right)}^2\delta A_{\min }{}^2}=\sqrt{{\left(\frac{\partial n}{\partial A_p}\right)}^2+{\left(\frac{\partial n}{\partial A_{\min }}\right)}^2}\delta A\\ =\frac{\sqrt{2+n^2-3n^2{\sin }^2\frac{A_p}{2}-2n\cos \frac{A_p}{2}\sqrt{1-n^2{\sin }^2\frac{A_p}{2}}}}{2\sin \frac{A_p}{2}}\delta A.\end{array}$$

Acknowledgement

Authors thank to Prof. G. Aka for his kind help in refractive index measurement, Netzsch-Gerätebau GmbH for measurements of thermal expansion. This work was partially supported by Genesis Research Institute, Inc., and by the Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

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