The temperature dependence of the thermo-optic effects in single crystal and ceramic TGG were evaluated by using the Fizou interferometer method. The temperature dependence of the refractive index and thermal expansion are significantly improved at low temperature for both ceramics and single crystals. Our estimation using a figure of merit indicated that a TGG ceramics cooled to liquid nitrogen temperature can reduce thermal wave-front distortion by a factor of up to 4.7 with respect to that at 300 K, and can reduce thermal birefringence effects by up to a factor of 12 with respect to those at 300 K.
© 2013 Optical Society of America
Terbium gallium garnet (TGG) ceramics is a promising material for Faraday devices for the isolation, polarization control, and birefringence compensation of both high-energy and high-average-power laser systems. This material has a high Verdet constant of 36 rad/Tm  at a wavelength of 1 μm, a high thermal conductivity of 4.9 W/mK , and excellent size scalability to prevent laser damage at the high laser energies obtained by using recent ceramics technology. Ten years have passed since the first report of TGG ceramics ; now TGG ceramics is a practical solution for Faraday devices. In 2011, 45-degree Faraday rotation was demonstrated by using a TGG ceramics piece of length 20 mm and a cross-section of 5 × 5 mm with a commercial magnet system . Recently, we have demonstrated the use of TGG ceramics under laser radiation with an average power of 100 W at the wavelength of 1030 nm for estimation of the thermal birefringence effect and extinction ratio (also referred to in the literature as the isolation ratio) under high-power laser radiation .
The extinction ratio and beam quality after passing through the Faraday device are degraded significantly by the thermal birefringence and the thermal wave front distortion of using a Faraday medium under kW-level high-average-power laser radiation. The magnitude of the thermal effects greatly depends on the thermal properties of the Faraday materials. In particular, 1) the thermo-optic coefficient, (the derivative of the refractive index with respect to temperature, dn/dT), 2) the linear thermal expansion coefficient (α), and 3) the thermal conductivity (κ) are essential characteristics of the thermal effect. For 3), we have already measured the temperature dependences of κ of the TGG ceramics in the early phase of study in the TGG ceramics. The experimental data on α and dn/dT in TGG single crystal was reported at room temperature [6–8]. However, there are no data on α and dn/dT in TGG ceramics and the temperature dependence of α and dn/dT in TGG single crystals. Furthermore, there are no data about the refractive index in TGG ceramics.
The temperature dependence of the thermal expansion coefficient α and dn/dT are important properties for the design of Faraday devices for use under high-average-power operation. At cryogenic temperatures, the thermal birefringence and thermal wavefront distortion in TGG single crystal are dramatically reduced due to the improvement of the thermo-optic properties [9–12]. From this fact, the temperature dependence of α and dn/dT should be considered both in the thermal design and in estimating the performance (the isolation ratio and wavefront distortion under high-average-power operation ) of the Faraday device.
In this paper, we report the measurement of the refractive index of TGG ceramics in the wavelength range from 435.8 nm to 656.3 nm. Also, we have simultaneously measured α and dn/dT in TGG ceramics from 293 K to 64 K and in TGG single crystal from 295 K to 86 K. From these data, we characterized the temperature dependence of the thermo-optic effects of TGG ceramics by using two figures of merit (FOMs) that characterize the thermal birefringence and the thermal wavefront distortion, respectively. As a result, we have quantitatively substantiated the improvement of thermo-optic characteristics in TGG ceramics at cryogenic temperature through a comparison with the previous study of cryogenic TGG Faraday devices.
2. Experimental method
The refractive indices of the TGG ceramics were measured by the V block method (KPR-2000, Shimazu Corporation) . The refractive index of a 10 mm × 10 mm × 5 mm sample was measured at five different wavelengths. All TGG ceramics samples in this study were made by Konoshima Chemical Co., Ltd.
The temperature dependences of α and dn/dT in ceramics and single-crystal TGG were measured by the Fizeau interferometer method [15–17]. A photograph of the TGG ceramics (Konoshima Chemical Co., Ltd.) and the single-crystal TGG (CASTECH Inc.) are shown in Fig. 1. TGG ceramics flats were diffusion-bonded to the ends of the 14.9-mm-long piece of TGG ceramics to form a Fizeau interferometer. Also, single-crystal TGG flats were also bonded to the 20.1-mm-long piece. A part of the outer surface of each of the flats is antireflection (AR)-coated at the laser wavelength to avoid the reduction of the signal-to-noise ratio along the vacuum path of the interferometer. Each of the TGG samples were placed in a vacuum chamber and attached to the cold head of the cryostat (Iwatani HE05), which was cooled to 4 K; the samples were attached with thermal conductive Ag-paste (Dotite, Fujikura Kasei Co., Ltd.) to improve the thermal contact, as shown in Fig. 2. Two Fizeau interferometers, one with a vacuum path and the other with a path containing a sample, were illuminated through the half-mirror by a He-Ne laser operating at 633 nm. The interference fringe was observed on the left side of the half-mirror. When the sample temperature was changing, this fringe pattern moved due to changes in the optical path of the interferometer. After the beam passed through a pin-hole, the intensities of the moving fringes were recorded by photo-detectors 1 and 2 as the temperature was varied while the cold head was heated. The temperature was measured by a calibrated Kp-Au thermocouple on the material surface with a nano-voltmeter (34420A, Agilent). The accuracy of the measurements is estimated to be ± 0.2 × 10−6 K−1. The temperature was varied over a range of 10 K to obtain one measurement value.
The light intensity as a function of temperature was fitted with a sinusoid to extract the physical path-length change. Note that a count of one fringe corresponds to a change in optical path length of λ/2, where λ is the wavelength of the incident light, L is the length of the interferometer path (29.8 mm for TGG ceramics and 40.2 mm for TGG single crystal for the vacuum path, 41.5 mm for TGG ceramics and 53.2 mm for TGG single crystal for the TGG medium path) and n is the refractive index along the interferometer path (1.96619 for TGG ceramics at 633 nm [as in this work] and 1.9656 for TGG single crystal at 633 nm ). α and dn/dT were determined using Eq. (1) in  with data from each of the interferometer paths.
3. Experimental results
3.1 Refractive index of TGG ceramics
The refractive indices are measured at five different wavelengths from 486.1 nm to 656.3 nm. The measured refractive indices are summarized in Table 1 and plotted in Fig. 3. The wavelength dispersion of the refractive index can be described using a single-oscillator Sellmeier equation asFig. 3, we show the fitting curve obtained by Eq. (1) with the measured refractive indices of the TGG ceramics. Then, the fitted parameter values of Ed = 9.3657 eV and E0 = 25.664 eV were obtained from our data. These values are similar to those of Ed = 9.223 eV and E0 = 25.208 for the single-crystal TGG reported in . The refractive indices in the single-crystal TGG from  are also plotted in Fig. 3 for the comparison between ceramics and a single-crystal TGG. All refractive indices data previously obtained in  were in good agreement with the fitting curve in Fig. 3 within the measured wavelength range. Note that from the experimental data and Eq. (1), we can determine the refractive index to be 1.96619 at 632.8 nm, which is the wavelength of the He-Ne laser for the measurement of the thermo-optic characteristics.
3.2 Temperature dependence of α and dn/dT
Figure 4 shows the results for the temperature dependence of α that we obtained for the TGG ceramics and single-crystal TGG samples. Our data on α for TGG single crystal is 7.3 × 10−6 K−1 at 295 K. This α value is in reasonable agreement with the previous work on the TGG crystal at room temperature, which yielded values of 9.4 × 10−6 K−1  and 6.7–7.2 × 10−6 K−1 . The α values shown in Fig. 4 for TGG ceramics are close to those of the TGG crystal. At 293 K, α for TGG ceramics is 7.1 × 10−6 K−1 . α decreased when the temperature of the TGG samples was decreased. The α of TGG single crystal at 86 K was decreased to 1.2 × 10−6, which is 6.1 times lower than the value of α at 295 K, and the α of TGG ceramics at 64 K was decreased to 3.86 × 10−7 K−1, which is 18.4 times lower than the value of α at 293 K. The temperature dependence of α in TGG ceramics corresponded to the α of a single crystal from 295 K to 86 K in our experiment.
In the commonly used Grüneisen theory, the linear expansion coefficient of an isotropic solid is given by19]. We can express the temperature dependence of CV(T) within the temperature range of our experiment as shown in Eq. (3),Eqs. (2) and (3), the linear thermal expansion can be expressed as a function of temperature by using two constants, A and B. The thermal coefficients of K(T), γG, and ρ(T) are relatively small , so we approximate those components as constants.Eq. (4) and found that A = 1.804 K and B = 457.3 for the TGG ceramics and A = 2.1276 K and B = 489.51 for the single-crystal TGG. From the fitting curves of our experiment, we can also confirm that the temperature dependence of the coefficient α for TGG ceramics is close to the value of the TGG single crystal.
Figure 5 shows the results for the temperature dependence of dn/dT. This dn/dT for single-crystal TGG of 1.79 × 10−5 K−1 at 295 K is in reasonable agreement with the previous work on the single-crystal TGG at room temperature of 2.0 × 10−5 K−1  and 1.8–2.1 × 10−5 K−1 . The dn/dT value of TGG single crystal at 86 K is decreased to 5.9 × 10−6 K−1, which is 6 times lower than that at 295 K. The dn/dT value of TGG ceramics at 293 K is 1.7 × 10−5 K−1. This is a slightly lower value than that of the single crystal. This value of TGG ceramics at 64 K also decreased to 3 × 10−6 K−1, which is 5.7 times lower than that at 293 K.
There is no simple relationship to represent the temperature dependence of the dn/dT data such as the temperature dependence of thermal expansion . Therefore, we used a fit to a third-order polynomial (shown in as Eq. (5)) to interpolate the dn/dT values within this temperature range in this study.
The room-temperature and cryogenic values of α and dn/dT in TGG ceramics and single crystal are summarized in Table 2. In the near future, we will focus on studying the mechanism that causes the difference between the crystal and ceramics by using previously proposed theoretical models that described the temperature dependence of dn/dT in detail to study the physics of the change in refractive index with temperature in TGG materials.
4. Effect of thermo-optic effect reduction under cryogenic temperature
For the estimation of the temperature dependence of the thermo-optic effects in TGG materials with a rod geometry, we used two parameters based on the figure of merit for thermal birefringence and for thermal lensing in laser material, proposed by T. Y. Fan . One is the figure of merit for thermal lensing FOMDrod, and the other is the figure of merit for thermal birefringence FOMBrod. The thermal lens effect is mainly determined by dn/dT in a rod geometry. Therefore, Eq. (6) is defined without the contribution of the thermal expansion. Then,Eq. (6) and Eq. (7) with the experimental results for α and dn/dT for ceramics and single-crystal TGG. We used the temperature dependence of κ for the TGG ceramics from our previous measurement in  and that of the single-crystal TGG from the sample R156 in  over the temperature range of 300 K to 80 K.
Figure 6 shows the temperature dependence of FOMDrod and FOMBrod normalized to the values at 300 K. From the results, the normalized FOMDrod for the crystal indicates that the amount of wavefront distortion at 77 K is 5.4 times lower than that at 300 K. This means that the thermal lens effect in single-crystal TGG decreases to 1/5.4 at 77 K compared with that at 300 K. In TGG single crystal, a decreasing of the thermal lens power at low temperature was measured in the previous experiment . This work reports that thermal lens power decreased 3.6-fold at 77 K. Our simple calculation by using the FOM is consistent with that previous work. Also, they reported the reduction of the thermal birefringence effect in single-crystal TGG at low temperature. At 86 K, their measurements showed that that the thermal birefringence effect was decreased by a factor of 8 with respect to that at 300 K. In our estimation, the value of FOMBrod indicates that the amount of thermal birefringence effect at 86 K and 77 K are 11.8 times and 18 times lower than that at 300 K, respectively. The previously reported result in  for this enhancement effect for thermal birefringence at low temperature is 32% smaller than our current result.
The absolute values of FOMDrod and FOMBrod for ceramics and single crystal are almost the same at room temperature. However, a difference between FOMDrod and FOMBrod for ceramics and single crystal appears at low temperature due to the difference in κ between ceramics and a crystal. The value of FOMDrod for the ceramics indicates that the amounts of wavefront distortion at 77 K is 4.7 times lower than that at 300 K. The value of FOMBrod for the ceramics indicates that the amount of thermal birefringence at 86 K and 77 K are 7.9 times and 12 times lower than that at 300 K. The normalized FOMDrod and FOMBrod in TGG ceramics and single crystal are summarized in Table 3. These values for the ceramics are slightly lower than the corresponding values for the single crystal; however, a dramatic reduction of the thermal effects in TGG ceramics is expected at cryogenic temperature just as in TGG single crystal. Based on this reasoning, TGG ceramics can be used for Faraday devices for high-average-power and high-energy laser systems [22–24]. These devices should be produced in a large size in order to form a large aperture, to avoid laser-induced damage, and to provide thermal resistance under high-power-laser operation. In addition, a large ceramics is cost effective because many pieces of TGG mediums can be obtained from one batch of the ceramics process with same quality (high quality). As the result, a Faraday device made from TGG ceramics operating at low temperature is one candidate for such an application.
In this study, we have measured the refractive index of TGG ceramics in the wavelength range from 435.8 nm to 656.3 nm. Also, we have simultaneously measured α and dn/dT in TGG ceramics from 293 K to 64 K and in TGG single crystal from 295 K to 86 K. From these data, we characterized the temperature dependence of the thermo-optic effects of TGG ceramics by using the figures of merit FOMDrod and FOMBrod.
From the result, the refractive index of TGG ceramics at 632.8 nm is 1.96619. This is near the refractive index value of single-crystal TGG.
α for TGG single crystal is 7.3 × 10−6 K−1 at 295 K. The value of α for TGG ceramics is close to the corresponding value for the TGG crystal. At 293 K, α for the TGG ceramics is 7.1 × 10−6 K−1. The α of TGG single crystal at 86 K was decreased to 1.2 × 10−6, which is 6.1 times lower than the value of α at 295 K, and α of TGG ceramics at 64 K was decreased to 3.86 × 10−7 K−1, which is 18.4 times lower than the value of α at 293 K.
The dn/dT value of TGG single crystal at 86 K was decreased to 5.9 × 10−6 K−1, which is 6 times lower than that at 295 K. The dn/dT value of TGG ceramics was 1.7 × 10−5 K−1 at 293 K. This is a slightly lower value than that of the single crystal. The dn/dT of TGG ceramics at 64 K was also decreased, to 3 × 10−6 K−1, which is 5.7 times lower than that at 295 K.
The estimation of the thermo-optic effect by using FOMDrod and FOMBrod indicate the dramatic reduction of thermal effects in TGG ceramics. At liquid nitrogen temperature (77 K), thermal birefringence effects decrease to 12 times lower than those at 300 K, and thermal lens effects are 4.7 times lower than those at 300 K. These data will assist the development of Faraday devices that can be used under high-average-power operation.
This work was supported by JSPS KAKENHI Grant No. 23760813 and Grant No. 25289341, and was performed with the support and under the auspices of the National Institute for Fusion Science (Grant No. UJHH002, NIFS Collaboration Research program of NIFS11KLEH021) and was also performed as a part of a joint research project of the Institute of Laser Engineering, Osaka University (under contract subject “2013B2-02”).
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