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A new technique for measuring thermal conductivity of solids, including creation of a steady-state one-dimensional heat flow through a studied sample and measurement of temperature distribution using phase-shifting interferometry has been developed. The technique is a handy tool for studying small-size (> 1 mm) samples. The thermal conductivity of new magneto-optical materials such as Ce:TAG optical ceramics (0%, 0.05% and 0.1% doping), MgAl2O4 spinal ceramics, and Tb2O3–B2O3-GeO2 magneto-optical glass was measured.
© 2014 Optical Society of America
1. Introduction
Thermal processes in optical elements of laser system units are of great importance for developing solid-state lasers with high average power. Inhomogeneous heating of these elements gives rise to phase and polarization distortions of laser radiation and overheating of the laser active element may cause deterioration of optical characteristics. For a specified heat release, the magnitude of heating is determined by thermal conductivity of the element material [1, 2]. Hence, accurate measurement of thermal conductivity is essential for developing high-power solid-state laser systems as well as for investigating new optical materials. Thermal conductivity of many laser media may strongly depend on the level of doping, temperature, method of fabrication and may vary in different manufacturers [3]. In addition, a great number of new laser media are emerging lately the properties of which are still poorly studied (e.g., Yb:YLF and Yb:FAP uniaxial laser ceramics [4], disordered sesquioxide laser ceramics, or new magneto-optical materials [5]).
Thermal conductivity of optical materials is frequently measured by non-steady-state methods (temperature wave propagation method [6], laser flash method [7, 8]) that have such drawbacks as complicated implementation and calculation of thermal conductivity from thermal diffusivity and heat capacity, instead of direct measurement. The steady-state measurement techniques are simpler (e.g., steady-state longitudinal heat flow method), but measuring temperature distribution in the sample holders by means of several thermocouples imposes restrictions on the size of the studied samples [9]. In addition due to the big size of the heated units the measurement time is quite long. Temperature can be measured by two thermocouples only but in this case measurement accuracy is lower and additional measurement of the heat power is necessary [10]. We propose a steady-state method for thermal conductivity measurements, where temperature distribution is measured using phase-shifting interferometry [11]. This method allows studying small-size samples (characteristic size in all three dimensions is not less than 1 mm), which is a good tool for investigation of new optical materials. The measurement time of presented method is quite short (a couple of minutes). The method of thermal conductivity measurements was calibrated on samples of fused silica (analog of GE-151 in the USA), TGG crystal, and stainless steel (analog of AISI 321H in the USA). Thermal conductivity of new optical materials such as Tb2O3–B2O3-GeO2 magneto-optical glass with record content of terbium ions, Ce:TAG optical ceramics (0%, 0.05% and 0.1% doping), and MgAl2O4 spinel ceramics was measured for the first time.
2. Thermal conductivity measurement
For implementing the idea of the proposed technique a special construction unit was designed [Fig. 1(a)], that is organized as follows. Two standard transparent bodies with well known thermal conductivity are attached to two ends of the studied sample. The sample and the standard bodies are right cylinders with identical base shapes and with slightly beveled lateral surfaces. Beveling is done so that the interferometer beam path in a certain area of the standard bodies was a rectangular parallelepiped. Measurement is performed exactly in this area.
Fig. 1 Schematic of measuring thermal conductivity of solid bodies: (a) construction unit, (b) scheme of Mach-Zehnder interferometer.
The construction unit is heated by radiation through a heat spreading copper cylinder at one end and is cooled by a heatsink with running water at the other end. In this way, a steady-state one-dimensional heat flow is created in the system. Due to the small longitudinal size of the construction unit the time-to-steady-state temperature is quite short (a couple of minutes). The thermal conductivity of the studied sample is found from measurements of the power of heat flowing through the sample and the temperature jump in it. Temperature distribution inside the standard bodies is measured by the method of phase-shifting interferometry using the Mach-Zehnder interferometer [Fig. 1(b)].
Phase distribution in the interferometer arm housing the crystal is measured by the phase-shifting interferometry [11] when the heating is switched on (φhot) and switched off (φcold). An example of interference patterns without and with heating is shown at Fig. 2(a).The variation of the optical path length (L) in the interferometer arm housing the crystal when heating is switched on [Fig. 2(b)] is calculated by the formula:
The variation of the optical path length in the standard body is caused by the variation of the index of refraction with temperature (dn/dT) and by thermal expansion of the standard material (αtherm). The temperature in the standard body [Fig. 2(c)] is calculated by the formulawhere lcryst is the radiation path length in a cold standard body and n0 is the index of refraction of the standard body. The temperature in each standard body grows linearly along the longitudinal coordinate and is described by a straight line [grey dash lines in Fig. 2(d)]. Knowing thermal conductivity κstan of the standard bodies, one can calculate from their longitudinal temperature gradient (dT/dy) the power of the heat flowing through them:where S is the cross-sectional area of the standard body. The heat power in the hotter standard body is higher than in the cooler one because of heat loss to the atmosphere. Our estimates show that the magnitude of heat power in the sample is much closer to the heat power in the cooler standard body than hotter one. It’s concerned with the fact that the heat loss in the hotter body is higher because the heat transfer to the atmosphere is proportional to the temperature. Therefore, heat power in the sample (Pheat) is regarded to be equal to the heat power in the cooler standard body. For determining the temperature jump between the bases of the standard bodies adjacent to the sample, the temperature curve in the cooler standard body should be shifted along the coordinate axis by a distance separating the standard bodies (Δh), i.e. the lower grey dash line in Fig. 2(d) is shifted towards the dot-dash line. If there were no temperature jump between the standard bodies, the dot-dash temperature curve would coincide with the upper grey dash temperature curve. The shift of the curves in the area of contact between the hotter standard body and the sample is the sought temperature jump (ΔT). It consists of the temperature jump in the sample and two temperature jumps at the interface between the sample and the standard bodies (ΔTinterface). The value of ΔTinterface must be either small compared to the temperature jump in the sample or should be measured. Two measurements may be performed on samples of the same material but of different thicknesses. Thermal conductivity is calculated by the formulawhere Hsamp is the thickness of the studied sample. Note that the final result does not depend on αtherm and dn/dT of the standard bodies, and κstan is the only parameter used in the calculations. Should be added that the linear task is described above. Thermal conductivity of the sample and the standard bodies is assumed to be temperature independent. To satisfy this condition temperature jump in the construction unit shouldn’t exceed several tens of degrees. It is regulated by the heat power tuning. Thermocouple on the heat spreading copper cylinder is used for the temperature control [Fig. 1(a)]. In the opposite case the nonlinearity should be taken into account.Fig. 2 Scheme of processing experimental data: (a) interference patterns when heat is switched on and off, (b) variation of the optical path length in the interferometer arm when heat is switched on, (c) temperature variation in standard bodies when heat is switched on, (d) calculation of temperature jump between the bases of the standard bodies adjacent to the sample. Solid lines –experimental data; dash grey lines – linear approximations of temperature distribution inside the standard bodies; dot-dash line – lower dash line up-shifted by the distance separating the standard bodies.
In all experiments heating was weak and the task was considered to be linear. YAG crystals with thermal conductivity assumed to be 9.5 W/m/K [12] were used as standard bodies. YAG crystals were indium soldered to the sample. Thermal conductance of such interfaces is 20 W/cm2/K [13]. All the studied samples were shaped as right circular cylinders with slightly beveled lateral surfaces. Their dimensions are listed in Table 1.For measuring samples of definite diameter, two cylinders of the same diameter, and a height of 3 mm were fabricated from a YAG crystal. The sample temperature during the measurements was 300 K. The method was tested on samples of fused silica, TGG crystal, and stainless steel. From Table 1 it is clear that the obtained values coincide with the earlier published data to an accuracy better than 5%. The thermal conductivity of the samples made of new materials: Tb2O3–B2O3-GeO2 magneto-optical glass with record content of terbium ions, Ce:TAG optical ceramics, and MgAl2O4 spinel ceramics was measured for the first time. The results of the measurements are presented in Table 1.
Table 1. Results of measuring thermal conductivity in solid bodies
The uncertainty of the technique was assessed comprehensively. The interference measurement error leads to the < 1% uncertainty in the Pheat and ΔT determination. It is concerned with the accuracy of measuring optical path length in the interferometer arm and by accuracy of determining the sample boundaries in the interference pattern. The use of the method of phase retrieval by one interference pattern instead of phase-shifting interferometry will reduce measurement accuracy. The inaccuracy of the data of interface thermal conductance between the standard bodies and the sample leads to the < 1% uncertainty in the ΔTinterface determination [13]. It increases in the studies of samples with high thermal conductivity and is the major factor limiting the measured thermal conductivity from above. For accurate measurement the following condition should be satisfied: κ/Hsamp << 2ΔKinter, where ΔKinter - inaccuracy of interfaces thermal conductance data. With appropriate choice of parameters it is possible to investigate samples with thermal conductivity up to 100 W/m/K. The use instead of indium solder of interface materials having lower thermal conductivity (for example, different types of heat conducting pastes) will reduce accuracy of the experiment. Numerical analysis of heat processes in the system demonstrated that stress associated with the difference of thermal expansion coefficients arise at the interface between the standard bodies and the sample. This affects heat expansion of the standard bodies in the contact area. The corresponding measurement uncertainty in the Pheat and ΔT determination was < 3%. This effect increases with increasing difference of the thermal expansion coefficients of the standard bodies and the sample and with increasing Young’s modulus of the interface material. For its impact on the final result not to exceed the mentioned uncertainty, the interferogram of the boundary region of the standard body (the distance from the sample < 0.5 mm) is not used in the calculations. This effect may be avoided if the standard bodies have the base size larger than that of the sample, but in this case measurement error due to heat flow inhomogeneity is added. The uncertainty in the Pheat determination due to heat loss to the environment < 5% was the main factor limiting accuracy of the experiment. Heat loss leads to the heat power underestimation and two-dimensional temperature distribution formation. This effect is manifested stronger in samples with low thermal conductivity and is the major factor limiting the measured thermal conductivity from below. For measurement the following condition should be satisfied: Kenv*Senv << Kconst*S, where Kenv ~10−3 W/cm2/K – heat transfer coefficient to the environment [17], Kconst - heat transfer coefficient through the construction unit, Senv – the area of the construction unit contact with the environment. With a proper choice of parameters, samples with thermal conductivity over 0.5 W/m/K may be investigated. Heat loss may be reduced by increasing sample diameter or by placing the system into a vacuum chamber. As a result, the total inaccuracy of the technique at suitable parameters of the experiment doesn’t exceed 10% and may be improved further.
3. Conclusion
New technique for measuring thermal conductivity of solid bodies has been developed. It may be used for materials with thermal conductivity in the 0.5 to 100 W/m/K range. The technique allows studying samples shaped as right cylinders with characteristic dimensions from 1 to 20 mm. The technique allows short measurement time, works indoors and is simple both technically and in terms of calculations. It has no basic restrictions for measuring thermal conductivity at different temperatures; it is expected that a setup for measurements in the 80 K – 400 K temperature range will be designed in the near future. The proposed method was successfully used to measure thermal conductivity of new magneto-optical materials: Tb2O3–B2O3-GeO2 magneto-optical glass with record content of terbium ions, Ce:TAG optical ceramics, and MgAl2O4 spinel ceramics. We hope that, by virtue of its simplicity and accuracy, the considered technique will be broadly used in the works concerned with development of new optical materials needed for creation of solid-state laser systems.
Acknowledgments
This work was supported by the mega-grant of the Government of the Russian Federation No. 14.B25.31.0024 executed at the Institute of Applied Physics RAS.
References and links
1. I. I. Kuznetsov, I. B. Mukhin, D. E. Silin, A. G. Vyatkin, O. L. Vadimova, and O. V. Palashov, “Thermal effects in end-pumped Yb:YAG thin-disk and Yb:YAG/YAG composite active element,” IEEE J. Quantum Electron. 50(3), 133–140 (2014). [CrossRef]
2. V. Sazegari, M. R. Milani, and A. K. Jafari, “Structural and optical behavior due to thermal effects in end-pumped Yb:YAG disk lasers,” Appl. Opt. 49(36), 6910–6916 (2010). [CrossRef] [PubMed]
3. I. B. Mukhin, O. V. Palashov, E. A. Khazanov, A. G. Vyatkin, and E. A. Perevezentsev, “Laser and thermal characteristics of Yb: YAG crystals in the 80 — 300 K temperature range,” Quantum Electron. 41(11), 1045–1050 (2011). [CrossRef]
4. Y. Sato, J. Akiyama, and T. Taira, “Orientation control of micro-domains in anisotropic laser ceramics,” Opt. Mater. Express 3(6), 829–841 (2013). [CrossRef]
5. V. Vasyliev, P. Molina, M. Nakamura, E. G. Víllora, and K. Shimamura, “Magneto-optical properties of Tb0.81Ca0.19F2.81 and Tb0.76Sr0.24F2.76 single crystals,” Opt. Mater. 33(11), 1710–1714 (2011). [CrossRef]
6. J. Morikawa and T. Hashimoto, “Analysis of high-order harmonics of temperature wave for Fourier transform thermal analysis,” Jpn. J. Appl. Phys. 37(Part 2, No. 12A), L1484–L1487 (1998). [CrossRef]
7. W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity,” J. Appl. Phys. 32(9), 1679–1684 (1961). [CrossRef]
8. Y. Sato and T. Taira, “The studies of thermal conductivity in GdVO4, YVO4, and Y3Al5O12 measured by quasi-one-dimensional flash method,” Opt. Express 14(22), 10528–10536 (2006). [CrossRef] [PubMed]
9. ASTM E 1225, “Standard test method for thermal conductivity of solids using the guarded-comparative-longitudinal heat flow technique,” http://www.astm.org/Standards/E1225.htm.
10. R. Yasuhara, H. Furuse, A. Iwamoto, J. Kawanaka, and T. Yanagitani, “Evaluation of thermo-optic characteristics of cryogenically cooled Yb:YAG ceramics,” Opt. Express 20(28), 29531–29539 (2012). [CrossRef] [PubMed]
11. K. Creath, “Phase-measurement interferometry techniques,” Progress in Optics 26, 349–393 (1989). [CrossRef]
12. B. Wang, H. Jiang, X. Jia, Q. Zhang, D. Sun, and S. Yin, “Thermal conductivity of doped YAG and GGG laser crystal,” Front. Optoelectron. China 1(1-2), 138–141 (2008). [CrossRef]
13. I. Kuznetsov, I. Mukhin, O. Vadimova, E. Perevezentsev, and O. Palashov, “Comparison of thermal effects in Yb:YAG disk laser head at room and cryogenic temperature conditions,” in Advanced Solid-State Lasers Congress, OSA Technical Digest (online) 2013), paper AM4A.33. [CrossRef]
14. Thermalinfo, “Table 1. Thermal conductivity of steel and cast iron,” http://thermalinfo.ru/publ/tverdye_veshhestva/metally_i_splavy/teploprovodnost_teploemkost_stalej_i_chuguna/7-1-0-8.
15. CrystalTechno, “Material Fused Silica (KV),” http://www.crystaltechno.com/FS_visible_en.htm.
16. I. Ivanov, A. Bulkanov, E. Khazanov, I. Mukhin, O. Palashov, V. Tsvetkov, and P. Popov, “Terbium gallium garnet for high average power Faraday isolators: modern aspects of growing and characterization,” in CLEO/Europe and EQEC 2009 Conference Digest, 2009), paper CE_P12.
17. A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28(4), 1057–1069 (1992). [CrossRef]
