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Abstract
We investigated material properties of terbium aluminum garnet (TAG) ceramics and its characteristics as a material for a Faraday isolator for high-average power lasers. We measured the transmittance, absorption coefficient, and temperature dependencies of Verdet constant and thermal conductivity. Consequently, our model analysis shows that TAG ceramic isolators can be operated over 3 kW average laser power at a 30 dB isolation ratio. The power limit becomes higher if a gas cooling scheme is applied.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The Faraday isolator is one of the crucial devices in laser systems. It can prevent back reflections from optical and metal surfaces or plasma. High-average power lasers are in demand in many areas, including scientific research, aerospace applications, and various industries; hence, Faraday isolators must be operated in high-average power.
The acceptable average power of Faraday isolators is limited by the degree of depolarization [1]. The depolarization effects are mainly caused by the temperature distribution in the isolator. In other words, low absorption and high thermal conductivity are desirable for low depolarization. A high Verdet constant is also important because it contributes to a shorter isolator length and a lower heat absorption. Terbium gallium garnet (TGG Tb3Ga5O12) is widely used in the VIS–NIR region [2–7]. Isolators with TGG are demonstrated at power levels of 300 W [3], 650 W [4] and 1500 W [5]. Another candidate material is terbium aluminum garnet (TAG, Tb3Al5O12) [6], which has a larger Verdet constant than TGG [8]. An isolator based on TAG has an isolation ratio of >38 dB up to 300 W [9].
Another approach to reducing depolarization is cryogenic cooling [2,10]. This method utilizes improvements in material properties, such as thermal conductivity and Verdet constant at low temperatures. The comparison between TAG at 80 K and at 300 K, for example, indicates approximately four and three times of improvement in the Verdet constant and thermal conductivity, respectively [11].
Since a large aperture size is required for high-power laser systems, ceramic materials are preferred. We investigate herein TAG ceramic properties and its performance as an isolator and measure its transmittance and absorption coefficient at room temperature. The temperature dependencies of Verdet constant and thermal conductivity are measured between cryogenic and room temperatures. Accordingly, we calculate the acceptable average power of TAG ceramic isolators, including their temperature dependence, based on the measured material properties and analytical models.
2. Experiments and results
The key function of a Faraday isolator is the Faraday rotation determined by the Verdet constant. Understanding the other material characteristics of Faraday isolators in addition to the Verdet constant is indispensable in designing them for high-average power lasers. High transmittance is vital because isolators are transmissive optics. Accordingly, a higher thermal conductivity and a lower absorption coefficient are essential for high-average power operations. We measured the TAG ceramic properties herein. The samples were made by Konoshima Chemical Co., Ltd., Japan. The dimensions of the samples are 3 mm × 5 mm × 3.1 mm and 7.1 mm × 9.6 mm × 3.1 mm where 3.1 mm are the thickness. Figure 1 depicts the microscopic images of the sample. The grain size was estimated to be 3.7 µm through the linear intercept method [12].
Fig. 1. Microscopic image of the TAG sample taken with a confocal laser scanning microscope (Olympus OLS 1200). The sample was annealed in the atmosphere to visualize the grain boundaries.
2.1 Optical transmittance
The optical transmittance was measured with a spectrophotometer (Hitachi U-4100) at room temperature. Figure 2 illustrates the result.
Fig. 2. Optical transmittance of TAG ceramics. The solid line shows the measured transmittance with a slit of 5 nm spectral width. The dashed line shows the theoretical transmittance.
The absorption near 484 nm is considered as absorption by Tb3+ ions [13]. The transmittance around 750 nm may be affected by the spectrophotometer’s switching of the detectors. The transmittance between 500 and 1500 nm is almost identical to the theoretical transmittance calculated with the refractive indices in [8], indicating that the ceramic sample is of high quality.
2.2 Verdet constant
The sample was set in a cryostat. The temperature was changed from approximately 9 K to 300 K. The magnetic flux density at the sample position was 1.14 T. A laser beam (Nd:YAG, 1064 nm, 0.5 mW) passing through a linear polarizer was put into the sample. A half wave plate, another polarizer, and a power meter were set after the cryostat to measure the changes in the polarization angle by the TAG sample.
Figure 3 depicts the result. At room temperature, our TAG ceramics had a Verdet constant of 52.2 rad/T/m, which is >40% higher than that of the TGG single crystal and ceramics (36.2 and 36.4 rad/T/m, respectively, at 1053 nm [14]). This value is almost the same as that of the single crystal TAG (52.4 rad/T/m at 1064 nm [8]). We compared the temperature dependence of our sample with ceramics produced in SIOM [15]. While the value of this work is slightly higher than the other work, basically the trend is almost the same.
Fig. 3. Temperature dependencies of the Verdet constant of the TAG ceramics (circles). The dashed line shows $V = a \cdot {T^{ - 1}}$ ($a = 16187\; [{\textrm{K} \cdot \textrm{rad}/\textrm{T}/\textrm{m}} ]$: fitting parameter) fitted with the data between 50 and 300 K. The characteristics of ceramics in [15] are shown in crosses.
The Verdet constant generally became higher as the temperature became lower in accordance with the ${\textrm{T}^{ - 1}}$ fitting. The constant deviated from ${\textrm{T}^{ - 1}}$ fitting at very low temperatures (ca. < 20 K). One possible reason for this is that the cryocooler had a limited cooling capacity at very low temperatures, and heating from the environment made the temperature at the laser position higher than that measured by a thermocouple, which was not attached to the sample itself.
2.3 Thermal conductivity
The thermal conductivity of the TAG ceramics was measured at various temperatures. We put a chip resistor as a heater on the bottom of the TAG sample. Two thermocouples were attached to the sample with a distance of d and installed in a cryostat. The temperature was controlled in a range of approximately 4–300 K. The thermal conductivity $\mathrm{\kappa }$ is given by $\kappa = {\dot{Q}_{\textrm{heat}}}d/({\mathrm{\Delta }TS} )$, where ${\dot{Q}_{\textrm{heat}}}$ is the heat load; $\mathrm{\Delta }T$ is the temperature difference of the two thermocouples; and S is the sample’s cross-section.
Figure 4 shows the result. The thermal conductivity at room temperature was 6.1 W/m/K, which is comparable to that in a prior report (6.5 W/m/K [16]). At lower temperatures, the value became larger and reached a peak of 50.3 W/m/K at 20 K. This is a significant advantage compared with the TGG ceramics, whose maximum value was 6.95 W/m/K at 60 K [17]. The difference between single crystal [18] and ceramics is small if the temperature is approximately 70 K or higher. The difference becomes larger at lower temperatures, where the thermal conductivity of ceramics peaks out.
Fig. 4. Thermal conductivity of the TAG ceramics (circles). The dashed line shows the spline interpolation curve. The characteristics of single crystal are shown in triangles.
2.4 Absorption coefficient
The absorption coefficient of the TAG ceramics was assessed at room temperature. A 400 W beam from a fiber laser (1070 nm) was injected into the TAG sample. The sample temperature was then measured with a thermography camera (256 px × 320 px, 60 fps). The heat from laser absorption is given by ${Q_{\textrm{heat}}} = {L_s}\int\!\!\!\int \mathrm{\Delta }TC\rho \textrm{d}x\textrm{d}y$, provided that no heat is dissipated from the sample to the outside. Here, ${L_s}$ is the sample length; $\Delta T$ is the temperature rise at $({x,\; y} )$ measured with the thermography camera; C is the heat capacity ($0.446\; \textrm{J}/({\textrm{g} \cdot \textrm{K}} )$ from [19]); and $\rho $ is the density ($6.06\; \textrm{g}/\textrm{c}{\textrm{m}^3}$ from [20]). The absorption coefficient is given by $\alpha ={-} \frac{1}{{{L_s}}}\ln \left[ {\left( {{I_0} - \frac{{\textrm{d}{Q_{\textrm{heat}}}}}{{\textrm{d}t}}} \right)/{I_0}} \right]$, where ${I_0}$ is the laser power. We can assume that the heat dissipation is low if we measure them within a short period from the laser incident.
The result for the TAG ceramics was $2.1 \times {10^{ - 3}}\; \textrm{c}{\textrm{m}^{ - \textrm{1}}}$. which is higher than the previously estimated values of $0.9-1.5 \times {10^{ - 3}}\; \textrm{c}{\textrm{m}^{ - \textrm{1}}}$ [9]. This result may be attributed to the difference in the samples or measurement method.
3. Discussion
3.1 Origin of depolarization
Understanding the origin of depolarization is important in analyzing the maximum allowable power of Faraday isolators. The depolarization arises from the piezooptic effect, inhomogeneity of the Faraday rotation angle, and reflection from the surfaces.
The first one is caused by mechanical, residual, or thermal stress. In our analysis, we assumed that the first and second factors were negligible. With regard to the second effect, the Faraday rotation angle $\mathrm{\theta }$ is written by $\theta = VBL$, where V is the Verdet constant; B is the magnetic flux density; and L is the isolator length. V is a temperature-dependent material constant; thus, both the material inhomogeneity and the temperature nonuniformity can vary V. We assumed that V is affected only by the temperature, and that B and L are constant over the aperture. The third effect can deteriorate the polarization because two-times-reflected light is 90° polarized against the non-reflected light. The effect is small if the surface reflection is smaller than 0.2% [21]. Overall, we considered the two origins: piezooptic effect and distribution of the Verdet constant. Both were caused by the thermal effect. We define the depolarization caused by the former effect as ${\gamma _P}$. and the latter as ${\gamma _V}$. If the both two ${\gamma _P}$ and ${\gamma _V}$ are small, the depolarization is given by $\gamma = {\gamma _P} + {\gamma _V}$ as an approximation.
Depolarization arises from the temperature distribution; therefore, the cooling scheme should be considered. A number of studies [1,9,22–24] have treated these isolators as “rod type” isolators, in which the shape is cylindcal, and the cylinder surface is cooled. The large Verdet constant of TAG, especially in low temperatures, shortens the isolator length. For instance, the length at 1 T and 77 K is ∼3.7 mm. Such short length or thinness raises a question of whether it is deemed as a rod. As [21] discussed, another cooling scheme is possible, in which the isolator is similar to a thin disk and cooled by gas flow. We investigated cases for “large thin disk type” isolators, where the beam radius is large enough compared to the thickness, and the disk radius is large enough compared to the beam radius. A similar cooling concept is applied to cryogenically cooled Yb:YAG laser systems [25,26].
The depolarization models (${\gamma _P}$ and ${\gamma _V}$) are expressed in [1] (rod type isolators) and [21] (large thin disk type isolators). Table 1 summarizes the equations, where p is the normalized power; $L,\; h$ is the isolator length (or thickness); $\lambda $ is the wavelength; Q is the thermo-optic coefficient; $\alpha $ is the absorption coefficient; ${P_0}$ is the laser power; $\kappa $ is the thermal conductivity; $\xi $ is the parameter for isotropy; and ${r_0}$ is the beam radius. The constant A is determined by the beam shape. In the case of a Gaussian beam, the figures are ${A_{\textrm{rod}}} \approx 0.137$, ${\textrm{A}^{\prime}_{\textrm{rod}}} \approx 0.268$ [1], ${\textrm{A}_{\textrm{disk}}} \approx 5.07 \times {10^{ - 3}}$ (calculated from an equation in [21]), $\textrm{A}{^{\prime}_{\textrm{disk}}} \approx {16^2} \times 3.6 \times {10^{ - 5}}$ [21]. The definition of X for ceramic materials is addressed in [27]. $\mathrm{\nu }$ denotes the Poisson’s ratio of the material. Here we used the value of YAG ceramics [28].
Table 1. Formulae for the isolator depolarization.
Our measurement results were used in the analysis, while Q and $\xi $ at room temperature were regarded as the same as those of the TGG due to the lack of available data. We calculated $({1/V\; \textrm{d}V/\textrm{d}T} )$ from the ${\textrm{T}^{ - 1}}$ fitting curve because the differential $\textrm{d}V/\textrm{d}T$ would be discontinuous if the measured data or the linear interpolation were used. As to Q, we considered the temperature dependence of the linear expansion $({{\alpha_L} = 1/L\; \textrm{d}L/\textrm{d}T} )$ because it widely changes as a function of the temperature [29]. We assumed that the linear expansion of TAG changes similar to that of TGG in [29]. The other terms were assumed constant. Taken together, Q at temperature T is provided with $Q(T )= {Q_{\textrm{RT}}} \times {\alpha _L}(T )\div {\alpha _{L,\; \textrm{RT}}}$, where subscript RT denotes the value at room temperature. Our model predicts that Q at 80 K is ∼7.6 times and Q at 86 K is ∼5.9 times smaller than that at 300 K. That is similar to the result in [30] or [7].
Although we measured the absorption coefficient at room temperature, it may change in lower temperatures. In the case of TGG crystal, the absorption coefficient becomes two times larger with temperature change from 300 K to 80 K [30]. This change significantly affects the amount of depolarization. We assumed that the absorption coefficient becomes twofold with temperature change from 300 K to 80 K as in [30]. The coefficient at other temperatures is interpolated or extrapolated linearly.
3.2 Rod type isolators
We analyzed the maximum acceptable power of rod type isolators assuming that the criterion for the isolation ratio was 30 dB. Figure 5(a) depicts the depolarization under several conditions. Here, we supposed two magnetic flux densities: 1 T was close to our measurement setup, while 2.5 T was taken after [9]. The power limit was hundreds of watts at room temperature and over 3 kW at 77 K. This limit is almost half of the value predicted by [31]. This may be attributed to that our sample has higher absorption coefficient than others as described in Section 2.4. Panel (b) illustrates the temperature dependence of limits. The allowable power became higher at low temperatures, but peaked around 50 K, which was not revealed by [31]. We must consider the effect of ${\gamma _V}$ in low temperatures [24]. The contribution ratio of ${\gamma _P}$ and ${\gamma _V}$ to the total depolarization is shown in the fill patterns. Although ${\gamma _V}$ was negligible [1] at room temperature, ${\gamma _V}$ became a key factor below approximately 90 K because T became lower, ${\gamma _P}$ shrunk with the decrease of $|{1/L\; \textrm{d}L/\textrm{d}T} |$ in Q but the term $|{1/V\; \textrm{d}V/\textrm{d}T} |$ in ${\gamma _V}$ monotonically increased. Note that $|{1/V\; \textrm{d}V/\textrm{d}T} |= 1/T$ assuming that V was inversely proportional to T.
Fig. 5. (a) Depolarization in the rod type isolators. (b) Temperature dependence of the maximum acceptable power (lines) and the ratio of origin of depolarization (fill patterns).
Another potential pitfall of cryogenic cooling is the magnetic field effect. A stronger magnetic field makes the isolator length shorter and normally provides a higher power limit. However, that effect disappears in temperatures around the maximum performance. At 50 K for example, the limitation power at 1 T is approximately 3.3 kW and almost identical to the limitation power at 2.5 T. This effect is interpreted by the dominance of ${\gamma _V}$, which is insensitive to the isolator length.
3.3 Large thin disk type isolators
We investigated the depolarization in large thin disk type isolators. The ratios of depolarization for the thin disk type isolators to rod type ones (i.e., ${\gamma _{P\textrm{, disk}}}/{\gamma _{P\textrm{, rod}}}$ and ${\gamma _{V\textrm{, disk}}}/{\gamma _{V\textrm{, rod}}}$) were almost the same and approximately $0.021{({h/{r_0}} )^4}$, suggesting that the power limitation of the thin disk type isolators would be enhanced by orders of magnitude. In the following discussion, we assume ${r_0} = 30\; \textrm{mm}$ because a large aperture would be required in high-power laser systems.
Figure 6 illustrates the depolarization in several cases. The asterisks represent ${r_0}/h > 3$, where the error caused by the approximation is less than 4% [21]. This isolator type produced considerably lower depolarizations than rod type ones. Even at 150 K, which is the typical temperature of DiPOLE [25], it is smaller than the rod type equivalent at 77 K. The laser absorption by the isolator at 150 K is as small as 0.25%; hence, it would be within the capacity of cryostats for high-average power lasers.
We would like to point out that the maximum power may be confined by other effects, including the nonuniformity of the material parameters or the magnetic field, beam shape distortion, mechanical and residual stress, imperfections in the thermal modeling, and non-linear effects. Although our calculation predicts the isolation ratio for a 10-kW laser at 1 T, 150 K is better than 30 dB, temperature dependencies of material properties other than the thermal conductivity and the Verdet constant are required to estimate the power limit of TAG more accurately.
4. Conclusions
We measured the TAG ceramic properties and investigated its performance as a magneto-optical material for the isolator. The material was transparent almost to the theoretical limit, and the absorption was as low as $2.1 \times {10^{ - 3}}\; \textrm{c}{\textrm{m}^{ - \textrm{1}}}$. We examined the temperature dependences of the Verdet constant and the thermal conductivity. These TAG ceramic properties were superior to those of TGG, especially at cryogenic temperatures.
We also discussed the performance of TAG isolators based on model analyses. The power limit for 30 dB isolation in the rod type isolators was over 3 kW at 77 K. The acceptable power peaked out at around 50 K. Moreover, intensifying the magnetic field played almost no role around this temperature.
The depolarizations in another type of cooling scheme isolators were also analyzed. The depolarization of a large thin disk type at 150 K was smaller than that of a rod type one at 77 K. Although the gas cooling system is required for this type, the heat generation would be far smaller than the cooling capacity for cryogenic laser systems. All temperature-dependent properties of TAG ceramic are not studied yet, however, its characteristics such as high thermal conductivity suggest that the TAG ceramics material is potentially applicable to isolators in high-average power laser systems.
Funding
KAKENHI (18H01899); JST-Mirai (JPMJMI17A1).
Disclosures
KH: Mitsubishi Heavy Industries, Ltd. (E), TY: Konoshima Chemical Co., Ltd. (E).
References
1. E. A. Khazanov, O. V. Kulagin, S. Yoshida, D. B. Tanner, and D. H. Reitze, “Investigation of self-induced depolarization of laser radiation in terbium gallium garnet,” IEEE J. Quantum Electron. 35(8), 1116–1122 (1999). [CrossRef]
2. D. Vojna, O. Slezák, A. Lucianetti, and T. Mocek, “Verdet constant of magneto-active materials developed for high-power Faraday devices,” Appl. Sci. 9(15), 3160 (2019). [CrossRef]
3. I. L. Snetkov, R. Yasuhara, A. V. Starobor, and O. V. Palashov, “TGG ceramics based Faraday isolator with external compensation of thermally induced depolarization,” Opt. Express 22(4), 4144–4151 (2014). [CrossRef]
4. E. A. Mironov, I. L. Snetkov, A. V. Voitovich, and O. V. Palashov, “Permanent-magnet Faraday isolator with the field intensity of 25 kOe,” Quantum Electron. 43(8), 740–743 (2013). [CrossRef]
5. I. L. Snetkov, A. V. Voitovich, O. V. Palashov, and E. A. Khazanov, “Review of Faraday isolators for kilowatt average power lasers,” IEEE J. Quantum Electron. 50(6), 434–443 (2014). [CrossRef]
6. J. Dai and J. Li, “Promising magneto-optical ceramics for high power Faraday isolators,” Scr. Mater. 155, 78–84 (2018). [CrossRef]
7. D. S. Zheleznov, A. V. Voitovich, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Considerable reduction of thermooptical distortions in Faraday isolators cooled to 77 K,” Quantum Electron. 36(4), 383–388 (2006). [CrossRef]
8. M. Geho, T. Sekijima, and T. Fujii, “Growth of terbium aluminum garnet (Tb3Al5O12; TAG) single crystals by the hybrid laser floating zone machine,” J. Cryst. Growth 267(1-2), 188–193 (2004). [CrossRef]
9. D. Zheleznov, A. Starobor, O. Palashov, C. Chen, and S. Zhou, “High-power Faraday isolators based on TAG ceramics,” Opt. Express 22(3), 2578–2583 (2014). [CrossRef]
10. D. S. Zheleznov, V. V. Zelenogorskii, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Cryogenic Faraday isolator,” Quantum Electron. 40(3), 276–281 (2010). [CrossRef]
11. S. Tokita, M. Nishio, H. Uehara, T. Yanagitani, K. Fujioka, J. Kawanaka, and R. Yasuhara, “Terbium aluminium garnet ceramics for ultrahigh power laser isolators,” in 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC), 2019.
12. H. Abrams, “Grain size measurement by the intercept method,” Metallography 4(1), 59–78 (1971). [CrossRef]
13. H. Furuse, R. Yasuhara, K. Hiraga, and S. Zhou, “High Verdet constant of Ti-doped terbium aluminum garnet (TAG) ceramics,” Opt. Mater. Express 6(1), 191 (2016). [CrossRef]
14. R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15(18), 11255–11261 (2007). [CrossRef]
15. A. Starobor, C. Chen, D. Zheleznov, O. Palashov, R. Yasuhara, and S. Zhou, “Study of the properties and prospects of Ce:TAG and TGG magnetooptical ceramics for optical isolators for lasers with high average power,” Opt. Mater. Express 4(10), 2127–2132 (2014). [CrossRef]
16. H. Lin, S. Zhou, and H. Teng, “Synthesis of Tb3Al5O12 (TAG) transparent ceramics for potential magneto-optical applications,” Opt. Mater. 33(11), 1833–1836 (2011). [CrossRef]
17. R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Measurement of magneto-optical property and thermal conductivity on TGG ceramic for Faraday material of high-peak and high average power laser,” Rev. Laser Eng. 35(12), 806–810 (2007). [CrossRef]
18. G. A. Slack and D. W. Oliver, “Thermal conductivity of garnets and phonon scattering by rare-earth ions,” Phys. Rev. B 4(2), 592–609 (1971). [CrossRef]
19. L. T. Denisova, Y. F. Kargin, and V. M. Denisov, “Heat capacity of rare-earth cuprates, orthovanadates, and aluminum garnets, gallium garnets, and iron garnets,” Phys. Solid State 57(8), 1699–1703 (2015). [CrossRef]
20. W. DeSorbo, “Magneto-optical properties of terbium aluminum garnet at liquid-helium temperatures,” Phys. Rev. 158(3), 839–842 (1967). [CrossRef]
21. I. B. Mukhin and E. A. Khazanov, “Use of thin discs in Faraday isolators for high-average-power lasers,” Quantum Electron. 34(10), 973–978 (2004). [CrossRef]
22. A. V. Starobor, D. S. Zheleznov, O. V. Palashov, and E. A. Khazanov, “Magnetoactive media for cryogenic Faraday isolators,” J. Opt. Soc. Am. B 28(6), 1409–1415 (2011). [CrossRef]
23. R. Yasuhara and H. Furuse, “Thermally induced depolarization in TGG ceramics,” Opt. Lett. 38(10), 1751–1753 (2013). [CrossRef]
24. A. V. Starobor, D. S. Zheleznov, and O. V. Palashov, “The temperature dependence of the thermo-optical properties of TAG optical ceramics,” Laser Phys. 26(2), 025801 (2016). [CrossRef]
25. S. Banerjee, P. D. Mason, K. Ertel, P. J. Phillips, M. De Vido, O. Chekhlov, M. Divoky, J. Pilar, J. Smith, T. Butcher, A. Lintern, S. Tomlinson, W. Shaikh, C. Hooker, A. Lucianetti, C. Hernandez-Gomez, T. Mocek, C. Edwards, and J. L. Collier,“100 J-level nanosecond pulsed diode pumped solid state laser,” Opt. Lett.41(9), 2089–2092 (2016). [CrossRef]
26. M. Kurata, T. Sekine, Y. Hatano, Y. Muramatsu, T. Morita, Y. Kabeya, T. Iguchi, T. Kurita, Y. Takeuchi, K. Kawai, Y. Tamaoki, Y. Kato, S. Tokita, and J. Kawanaka, “Development of a 100 J class cryogenically cooled multi-disk Yb:YAG ceramics laser,” in Laser Congress 2019 (ASSL, LAC, LS&C), OSA Technical Digest (Optical Society of America, 2019), paper ATu5A.3.
27. A. G. Vyatkin and E. A. Khazanov, “Thermally induced scattering of radiation in laser ceramics with arbitrary grain size,” J. Opt. Soc. Am. B 29(12), 3307–3316 (2012). [CrossRef]
28. Konoshima Chemical Co., Ltd., “Transparent Polycrystalline YAG ceramics,” http://konoshima.co.jp/eng/ceramics/YAG_HP.pdf.
29. R. Yasuhara, H. Nozawa, T. Yanagitani, S. Motokoshi, and J. Kawanaka, “Temperature dependence of thermo-optic effects of single-crystal and ceramic TGG,” Opt. Express 21(25), 31443–31452 (2013). [CrossRef]
30. D. S. Zheleznov, A. V. Starobor, O. V. Palashov, and E. A. Khazanov, “Cryogenic Faraday isolator with a disk-shaped magneto-optical element,” J. Opt. Soc. Am. B 29(4), 786–792 (2012). [CrossRef]
31. A. Starobor and O. Palashov, “The temperature dependence of thermooptical properties of magnetooptical TAG ceramics doped with silicon and titanium,” Opt. Mater. 78, 15–20 (2018). [CrossRef]
