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A Faraday isolator with compensation of thermally induced depolarization outside magnetic field was implemented for the first time on TGG ceramics. Stable isolation ratio of 38 dB in steady-state regime at a laser power of 300 W was demonstrated in experiment. Theoretical estimates show a feasibility of a device that would provide an isolation ratio higher than 30 dB up to laser power of 2kW.
© 2014 Optical Society of America
1. Introduction
Optical devices based on the effect of nonreciprocal rotation of polarization plane, such as Faraday rotators and isolators are widely employed in laser engineering for isolation of optical radiation, organization of multipass schemes of laser amplifiers, compensation of thermally induced birefringence in active lasers elements, and so on. The key component of such devices is a magneto-optical element placed in a constant magnetic field. When Faraday devices are used in high energy, high average power [1–3] lasers, the material of the magneto-optical element must meet stringent requirements: high magneto-optical figure-of-merit [4], large thermal shock parameter [5], and a possibility of fabricating from it large-aperture elements to prevent laser damage and reduce the arising nonlinear effects.
The technology of magneto-optical glass manufacturing enables fabricating optical elements of large aperture of tens centimeters [6], but glass has a relatively small magneto-optical figure-of-merit and thermal shock parameter. Hence, it can be used in low average power laser systems only. Magneto-optical single crystals have a figure-of-merit an order of magnitude higher than magneto-optical glasses and a large thermal shock parameter, which allows using them in laser systems with kilowatt average power. However, their growth technology limits the aperture of optical quality elements to several of centimeters, thus making application of such elements in high energy (high peak power) lasers highly problematic. An alternative way is to use magneto-optical ceramics that possesses figure-of-merit comparable with single crystals and a larger thermal shock parameter [7]. The up-to-date laser ceramics technology permits fabricating high quality optical elements of nearly any optical material with a size up to 45 cm in diameter and 1 cm thick [8]. Still more important is that this technology allows fabricating optical elements of materials that cannot be used for growing single crystals of comparable quality and size (TAG [9, 10]).
One of the widely used magneto-optical materials to date is terbium gallium garnet (TGG) thanks to its high value of Verdet constant and thermal conductivity and low absorption losses in the 500-1100 nm wavelength range. High optical quality TGG ceramics appeared relatively recently. The pioneer work in which Faraday rotation was demonstrated and the temperature dependence of the Verdet constant was measured in TGG ceramics was published in 2007 [11]. In that work a sample with a size of 5.95х5х1 mm was used and it was shown that the Verdet constants of the ceramic and single crystal samples coincide within the temperature range of 7.8 to 300 K. The first demonstration of 45 degree rotation of the radiation polarization plane at the wavelength of 1064 nm at room temperature was published in 2011 [12]. The research was done on a TGG ceramic sample having length 20 mm and cross-section 5х5 mm. It was shown that the studied ceramic sample had high optical homogeneity, the scattering losses and laser damage threshold were comparable with those in single crystal, with a potential to increase laser damage threshold by improving the technology. Yasuhara et al. [13] investigated thermally induced depolarization in a 28 mm-long ceramic sample 7 mm in diameter and showed that the absorption at the wavelength of 1030 nm in the studied sample was comparable with the absorption in the best commercially available crystals (~1.3∙10−3 1/cm). They also measured the coefficient of thermal expansion of the TGG ceramics at room temperature that proved to be only slightly different from the thermal expansion coefficient of a single crystal [14]. Thus, the present day technology of TGG ceramics enables producing elements the quality of which is comparable with the best TGG single crystals and the technological potential to fabricate large aperture elements makes TGG ceramics a promising magneto-optical material for Faraday devices used in lasers with high average and high peak power simultaneously.
At high average power of laser radiation, absorption in a magneto-optical element gives rise to thermally induced depolarization that limits application of traditional schemes of Faraday devices [4, 15]. At the present moment for room temperature traditional Faraday isolator (FI) the record isolation ratio achieved experimentally is 30 dB at average power of laser radiation of 650 W [16]. To advance to the region of higher average power with the same level of isolation ratio it is necessary to reduce the influence of thermally induced depolarization on the efficiency of FI. One of possible ways is to use Faraday devices with compensation of thermally induced depolarization [15, 17–19].
In this work we consider a Faraday isolator scheme with external compensation of thermally induced depolarization. The scheme was first proposed in [18]. The idea underlying this scheme was to supplement the traditional FI with a compensator that consists of two optical elements: a reciprocal polarization rotator and an additional optical element (AOE) and is placed outside FI magnetic field. The radiation depolarization arising during passage through the MOE is partially compensated during passage through the AOE. Parameters of the AOE and of the reciprocal rotator are determined by constants of the MOE and AOE materials, their inner structure (glass, single crystal, ceramics) and orientation of the crystallographic axes in the case of single crystals. The main advantage of this scheme is a possibility to upgrade traditional FIs introducing no changes in their magnetic system and MOE and to use a different, not necessarily magneto-active material [19] for AOE. AOE material may be chosen based on its availability, cost, features of material constants and their dependences, or for some other reason. Note that in this FI scheme the key point is absence of circular birefringence (Faraday rotation) rather than AOE position outside magnetic field. Therefore, if an AOE is made of a material with zero or negligible magnetic activity, it may be placed inside magnetic field and the scheme will be basically the same.
Here, we present an FI with external compensation with MOE made of TGG ceramics. Compensation of thermally induced depolarization in the FI has been achieved using both ceramic and single crystal TGG samples. The first experiments with laser power up to 300 W have been carried out. The dependence of thermally induced depolarization on laser radiation power and stability of Faraday rotation have been measured. The potential of further improvement of the chosen scheme with the use of TGG ceramics has been assessed theoretically.
2. Experimental results
2.1 Experimental setup
Schematic diagram of the experiment is shown in Fig. 1. A 300 W Yb-fiber laser operating at the wavelength of 1070 nm was a source of CW linearly polarized radiation in our experiments. The intensity distribution in the beam cross-section had Gaussian profile with a diameter of 2.6 mm by the 1/e level. Calcite wedge polarizer 6 ensured linearity of the polarization and high contrast less than 10−5 throughout the power range. Fused silica wedges 7 were used to attenuate radiation. Glan prism 8 was adjusted to minimum of the laser signal whose power Pd was measured by CCD camera 9. By turning Glan prism by 90° we measured the power of the laser signal Plaser. The integral thermally induced depolarization γ = Pd/Plaser was calculated by the ratio of the power in two positions of Glan prism.
Fig. 1 Schematic of the experiment: 1 – magneto-optical element; 2 – additional optical element; 3 – 67.5 degree quartz rotator; 4 – absorber; 5 – magnetic system; 6 – calcite wedge; 7 – fused silica wedge; 8 – Glan prism; 9 – CCD camera.
2.2. Investigation of ceramic samples
Two cylindrically shaped samples of TGG ceramics produced by Konoshima Chemical Co. (Japan) were used in the experiments. The samples had the following dimensions: sample 1 – diameter 7.04 mm, length 9.15 mm; sample 2 – diameter 7.07 mm, length 4.11 mm and the characteristic size of ceramic grains of about 1 μm. Each of them was placed into the measuring scheme in turn [Fig. 1 without elements 1, 2, 3 and 5] to measure the dependence of the integral thermally induced depolarization on the power of laser radiation. The results of the experiment are presented in Fig. 2(а) (dots). The distribution of local depolarization at low laser radiation power, the so-called “cold” depolarization, was relatively uniform [Fig. 2(с)], and the integral level of the “cold” depolarization in the ceramic samples was 10−5, which is comparable with single crystals of high optical quality.
Fig. 2 Integral thermally induced depolarization versus laser radiation power: red circles – TGG ceramics sample 1 in copper holder; red triangles – TGG ceramics sample 1 without copper holder; blue circles – TGG ceramics sample 2; green circles – TGG monocrystal (а); distribution of transmitted beam and its depolarized component at 8 W (b and c) and at 198 W (d and e) in TGG ceramic sample 1.
To provide heat sink from the optical elements they were placed in copper holders. In general, a ceramic sample with poor laser quality is usually scattered at the grain boundaries, residual pores, inclusions and defects, which results in a higher level of “cold” depolarization and, when the samples are placed in a metal heat sink, in its additional heating. The additional heating, in turn, leads to increasing the level of integral thermally induced depolarization. To confirm the additional heating by scattering, we repeated measurement of the integral thermally induced depolarization as a function of power for samples mounted in copper holders. The measured dependences coincided with those presented in Fig. 2(а) to the accuracy of the experiment, and no instability of thermally induced depolarization was observed, which indicates weak scattering of laser radiation in the studied samples.
In the weak birefringence approximation, when phase delay between two eigenpolarization much smaller than unity, the expression for the integral thermally induced depolarization in a ceramic element consists of two terms. The first of them is thermally induced depolarization averaged over directions of the crystallographic axes and ceramic grain lengths. The second term which depends in inverse proportion on the number of grains on the path of the beam describes the contribution of small-scale polarization perturbations caused by different directions of the crystallographic axes in the grains [20]. It was shown in [21] that the contribution of the second term does not exceed 0.5% even for 300 grains on the beam path. The number of grains on the beam path in our case is ~4000 for the thinnest sample of TGG ceramics; therefore the second term may be neglected. The first term in the absence of magnetic field is defined by
where A is the coefficient depending on the shape of the heating and probe radiation in the cross-section (when heating and measurements are done by one beam with Gaussian intensity distribution, A = 0.137); Q is thermo-optical constant equal to −17∙10−7 1/K for TGG [22]; λ is the wavelength of probe radiation equal to 1070 nm in our experiment; κ is the thermal conductivity of the studied samples (4.9 W/mK) [13]; α is the absorption coefficient; L is the length of the optical element; Plaser is the laser radiation power; p stands for the normalized power of heat release; X is the coefficient that depends on the assumptions on the probability density distribution of Euler angles determining the direction of crystallographic axes in the ceramics grains: X = (75 + 53ξ)/128 if α, β, and Φ are distributed uniformly in the intervals [-π,π], [0, π], and [-π, π], respectively [20]; X = (2 + 3ξ)/5 if α and Φ are distributed uniformly in the interval [-π,π], and angle β with probability density sin(β)/2 in the interval [0, π] [23], here ξ is the optical anisotropy parameter, for TGG ξ = 2.25 [24]. The main distinction is that, with uniform distribution of Euler angles in their intervals, the distribution of crystallographic directions in ceramic grains is not equiprobable, crystallographic directions close to the [001] orientation are more frequent. When angle β has probability density distribution sin(β)/2 in its interval, all crystallographic orientations are distributed equiprobably, hence hereinafter we will use it for describing ceramics. For a single crystal with [111] orientation X = (1 + 2ξ)/3; for all glasses ξ = 1, so X = 1 too. For a single crystal with [001] orientation, thermally induced depolarization depends on the angle θ between the crystallographic axis and the polarization of incident radiation and is specified by expression (1) with X2 = (1 + (ξ2-1)cos2(2θ)) [4, 25]. The expressions for integral thermally induced depolarization in optical element with Faraday rotation on angle φ differ from Eq. (1) by a factor of sin2(φ)/φ2 [4]. For FI, φ = 45° and the factor is equal to 8/π2.Clearly, the averaged thermally induced depolarization degree introduced by ceramics and the thermally induced depolarization degrees introduced by a single crystal with [111] orientation or by glass differ by the multiplier X only. Thus, we can say that ceramics on average introduces the same polarization distortions as a single crystal with [111] orientation with effective optical anisotropy parameter ξeff defined by
or as glass with effective parameter Qeff [26] defined byIn terms of the introduced polarization distortions, the use of ceramics in traditional FIs for some materials may be more beneficial than [111] crystals. As follows from (1), thermally induced depolarization in ceramics and in [111] single crystal differ only by parameter X, and for the γ ratio we obtain
According to (5), for all materials with ξ>1 and ξ<-11/19, the use of ceramics instead of [111] single crystal is preferable. The γ ratio is 0.91 for ξ = 2.25 corresponding to TGG. Thus, theoretically, for identical sample lengths and material absorption, γceramic is 9% less than γ[111]. For materials with −11/19<ξ<1 the situation is reversed. In a single crystal with [001] orientation, the magnitude of thermally induced depolarization depends on angle θ, but for materials with ξ in the interval from −7/3 to −1/4 the thermally induced depolarization introduced by ceramics is less than for [001] single crystal for any θ. Note that for ceramics there exists an assigned value ξ* = −2/3 at which <γ> becomes equal to zero. In ceramic materials with ξ close to ξ* the polarization distortions caused by different orientations of crystallographic axes in ceramic grains become decisive. For a TGG crystal, γ[001] is minimal at θ = π/4, and . The substitution of 1 for in Eq. (5) yields that, with other conditions being equal, thermally induced depolarization in TGG ceramics is 3 times higher than minimal thermally induced depolarization in a [001] single crystal.
Making use of Eq. (1) we plotted theoretical curves [Fig. 2(а), solid curves]. The absorption coefficient α was used as a fitting parameter. The absorption coefficient was α = 1.4∙10−3 1/cm for TGG sample 1, and α = 1.9∙10−3 1/cm for TGG sample 2. Knowing absorption coefficients and sample lengths one can calculate the ratio of the samples’ normalized power p2/p1 = 0.7 (where p1,2 is the normalized heat generation power in element 1 or 2 [see Fig. 1]).
2.3. FI with external compensation of thermally induced depolarization
A Faraday isolator with external compensation of thermally induced depolarization was made using TGG sample 1 as a MOE [see Fig. 1, element 1]. The MOE was placed in a constant-magnet system with the magnetic field magnitude of 2.5 T [Fig. 1, element 5] described in detail in [16]. Ceramic TGG sample 2 was taken for an AOE [Fig. 1, element 3]. Crystal quartz with the polarization plane angle of turn θr = 67.5° at the wavelength of 1070 nm was used as a reciprocal polarization rotator. The integral thermally induced depolarization was measured as a function of the power of laser radiation in two cases: with and without compensator (quartz rotator and AOE, elements 2 and 3 in Fig. 1, respectively). By comparing the two cases it is possible to assess efficiency of compensation of thermally induced depolarization. Results of the experiment are presented in Fig. 3(а). One can see that at maximum laser radiation power of 300W, addition of a compensator allows a 5-6-fold reduction of thermally induced depolarization, which corresponds to an increase of the isolation ratio of the device from 30 dB to 38 dB. Based on the data on absorption obtained in Sec.2.1 we calculated theoretical curves for the FI without compensator (red solid curve) and with compensator (blue solid curve). The green solid line is the theoretical curve for the compensator in which the ceramic sample has optimal parameters (p2/p1 = 0.9, θr = 67.5° [18]). In our numerical computations the ceramics was simulated as a single crystal having [111] orientation, with the corresponding substitution of the optical anisotropy parameter according to Eq. (3). The computations show that the use of the scheme with compensation and ceramic TGG sample 2 permits increasing the power by a factor of 3, from 300 W to 900 W (red and blue solid curves) retaining the isolation ratio of 30 dB, and by choosing an optimal AOE it is possible to enhance this power up to 2 kW (green solid curve).
Fig. 3 а) Integral thermally induced depolarization versus laser radiation power. Red circles – experiment and red curve – theory for FI without compensator; blue circles – experiment and blue curve – theory for FI with TGG ceramics compensator; magenta squares – experiment and magenta dotted curve – theory for FI with compensator of TGG [001] single crystal; green solid curve theoretic estimations for FI with TGG ceramics compensator with optimal parameters; green dotted curve theoretic estimations for FI with compensator of TGG [001] single crystal with optimal parameters. b) Time stability of the angle of Faraday rotation at laser radiation power of 300 W.
Stability of Faraday rotation in time was measured at a laser power of 300 W [Fig. 3(b)]. It takes ~2 minutes to reach the steady state. In the steady state at the power of 300 W, the MOE turned polarization of the laser radiation by 45.5°, which is 0.5° more than the required Faraday rotation. This drawback may be readily eliminated by removing the MOE from the region of a strong magnetic field, or by making it shorter. The marginal angle of turn may also be used to enhance radiation power without cooling by water or by Peltier elements [27].
Another experiment was aimed at compensating thermally induced depolarization in TGG ceramics by means of a TGG single crystal. Towards this end, we used a crystal TGG sample with [001] orientation 10.1 mm in diameter and 9 mm long. Results of the measurements and computations of γ as a function of laser radiation without magnetic field are presented in Fig. 2(а) (green circles and dotted line, respectively). The absorption coefficient of the crystal was α = 1.65∙10−3 1/cm, which corresponds to the ratio of normalized powers in the crystal and ceramic sample 1 p2/p1 = 1. Further, using ceramic sample 1 as a MOE and the single crystal sample as an AOE we measured the dependence of thermally induced depolarization as a function of laser radiation power [Fig. 3(а), magenta squares]. Based on the calculated ratio of normalized powers we plotted a theoretical curve for the FI with compensation using a single crystal [Fig. 3(а), magenta dotted curve] and a theoretical curve for the case of the parameters p2/p1 = 0.80, θr = 75.7°, θcrystal = 14.7° optimal for compensating thermally induced depolarization [Fig. 3(а), green dotted line].
Optimal parameters for compensation of thermally induced depolarization in magneto-optic ceramics are different for ceramic and single crystal optical elements. If both elements are made of the same ceramic material, the compensation is equivalent to the case of the external compensation considered in the work [18], when elements 1 and 2 [Fig. 1] are made of the same single crystal with [111] orientation. The main difference is arising of small-scale polarization distortions [28] that may be neglected, as was mentioned above, because of a great number of grains on the beam path and because these distortions do not influence the magnitude of optimal parameters. If both elements are made of the same material but one element is ceramic and the other monocrystalline with [001] orientation, the compensation is equivalent to the one considered in [19], when element 1 is made of a single crystal with [111] orientation of one material, and element 2 is made of a single crystal with [001] orientation of another material. The magnitude of optimal parameters depends on the fact that the ceramics behaves like the [111] single crystal of another material with ξeff, and that a single crystal with [001] orientation is used as an AOE. To conclude, we present optimal parameters for the case of compensation of thermally induced depolarization in a TGG ceramic MOE using a TGG [111] single crystal: p2/p1 = 0.86, θr = 67.5° and for compensation of thermally induced depolarization in a TGG single crystal MOE with [111] orientation using a TGG [001] single crystal: p2/p1 = 0.83, θr = 75.7°, θcrystal = 14.7°.
3. Conclusion
Two samples of TGG ceramics made by Konoshima Chemical Co. have been investigated. The samples had high optical quality, optical homogeneity and low absorption (~1.4∙10−3 1/cm) comparable with high-quality TGG single crystal samples.
With the use of the scheme with external compensation, we demonstrated for the first time in experiment compensation of thermally induced depolarization in the FI with a ceramic magneto-optical element by means of ceramic and single crystal samples. We constructed and tested a Faraday isolator with external compensation of thermally induced depolarization based entirely of TGG ceramics. This scheme allows to upgrade traditional Faraday isolators introducing no changes in their magnetic system and magneto-optical element and to use a different, not necessarily magneto-active material [19] for additional optical element. The use of a compensator at a power of 300 W enabled increasing the isolation ratio from 30 dB to 38 dB. The theoretical estimates show that a Faraday isolator with external compensation based on such ceramics will provide an isolation ratio of 30 dB at a 2 kilowatt average power. Taking into consideration that the present day ceramics technology permits producing large-aperture elements, TGG ceramics is a highly promising material for Faraday devices used in lasers with high energy and at the same time high average power.
Acknowledgment
This work was supported by mega-grant of Government of the Russian Federation No. 14.B25.31.0024.
References and links
1. C. D. Orth, S. A. Payne, and W. F. Krupke, “A diode pumped solid state laser driver for inertial fusion energy,” Nucl. Fusion 36(1), 75–116 (1996). [CrossRef]
2. R. Yasuhara, T. Kawashima, T. Sekine, T. Kurita, T. Ikegawa, O. Matsumoto, M. Miyamoto, H. Kan, H. Yoshida, J. Kawanaka, M. Nakatsuka, N. Miyanaga, Y. Izawa, and T. Kanabe, “213 W average power of 2.4 GW pulsed thermally controlled Nd:glass zigzag slab laser with a stimulated Brillouin scattering mirror,” Opt. Lett. 33(15), 1711–1713 (2008). [CrossRef] [PubMed]
3. J. Chanteloup and D. Albach, “Current status on high average power and energy diode pumped solid state lasers,” IEEE Photonics J. 3(2), 245–248 (2011). [CrossRef]
4. E. A. Khazanov, O. V. Kulagin, S. Yoshida, D. Tanner, and D. Reitze, “Investigation of self-induced depolarization of laser radiation in terbium gallium garnet,” IEEE J. Quantum Electron. 35(8), 1116–1122 (1999). [CrossRef]
5. W. Koechner, Solid-State Laser Engineering (Springer, 1999).
6. N. Miyanaga, H. Azechi, K. A. Tanaka, T. Kanabe, T. Jitsuno, J. Kawanaka, Y. Fujimoto, R. Kodama, H. Shiraga, K. Knodo, K. Tsubakimoto, H. Habara, J. Lu, G. Xu, N. Morio, S. Matsuo, E. Miyaji, Y. Kawakami, Y. Izawa, and K. Mima, “10-kJ PW laser for the FIREX-I program,” J. Phys. IV France 133, 81–87 (2005).
7. K. Ueda, J.-F. Bisson, H. Yagi, K. Takaichi, A. Shirakawa, T. Yanagitani, and A. A. Kaminskii, “Scalable ceramic lasers,” Laser Phys. 15, 927–938 (2005).
8. J. Lu, M. Prabhu, J. Xu, K. Ueda, H. Yagi, T. Yanagitani, and A. A. Kaminskii, “Highly efficient 2% Nd:yttrium aluminum garnet ceramic laser,” Appl. Phys. Lett. 77(23), 3707–3709 (2000). [CrossRef]
9. 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]
10. C. Chen, S. Zhou, H. Lin, and Q. Yi, “Fabrication and performance optimization of the magneto-optical (Tb1−xRx)3Al5O12 (R = Y, Ce) transparent ceramics,” Appl. Phys. Lett. 101(13), 131908 (2012). [CrossRef]
11. 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] [PubMed]
12. H. Yoshida, K. Tsubakimoto, Y. Fujimoto, K. Mikami, H. Fujita, N. Miyanaga, H. Nozawa, H. Yagi, T. Yanagitani, Y. Nagata, and H. Kinoshita, “Optical properties and Faraday effect of ceramic terbium gallium garnet for a room temperature Faraday rotator,” Opt. Express 19(16), 15181–15187 (2011). [CrossRef] [PubMed]
13. R. Yasuhara and H. Furuse, “Thermally induced depolarization in TGG ceramics,” Opt. Lett. 38(10), 1751–1753 (2013). [CrossRef] [PubMed]
14. 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]
15. E. A. Khazanov, “Compensation of thermally induced polarization distortions in Faraday isolators,” Quantum Electron. 29(1), 59–64 (1999). [CrossRef]
16. 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]
17. E. A. Khazanov, A. A. Anastasiyev, N. F. Andreev, A. Voytovich, and O. V. Palashov, “Compensation of birefringence in active elements with a novel Faraday mirror operating at high average power,” Appl. Opt. 41(15), 2947–2954 (2002). [CrossRef] [PubMed]
18. I. L. Snetkov, I. Mukhin, O. Palashov, and E. A. Khazanov, “Compensation of thermally induced depolarization in Faraday isolators for high average power lasers,” Opt. Express 19(7), 6366–6376 (2011). [CrossRef] [PubMed]
19. I. L. Snetkov and O. V. Palashov, “Compensation of thermal effects in Faraday isolator for high average power lasers,” Appl. Phys. B 109(2), 239–247 (2012). [CrossRef]
20. M. A. Kagan and E. A. Khazanov, “Compensation for thermally induced birefringence in polycrystalline ceramic active elements,” Quantum Electron. 33(10), 876–882 (2003). [CrossRef]
21. E. A. Khazanov, “Thermally induced birefringence in Nd:YAG ceramics,” Opt. Lett. 27(9), 716–718 (2002). [CrossRef] [PubMed]
22. E. Khazanov, “Faraday isolators for high average power lasers,” in Advances in Solid State Lasers Development and Applications (INTECH, 2010), Chap. 3.
23. 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]
24. E. Khazanov, N. Andreev, O. Palashov, A. Poteomkin, A. Sergeev, O. Mehl, and D. H. Reitze, “Effect of terbium gallium garnet crystal orientation on the isolation ratio of a Faraday isolator at high average power,” Appl. Opt. 41(3), 483–492 (2002). [CrossRef] [PubMed]
25. W. Koechner and D. K. Rice, “Birefringence of YAG:Nd laser rods as a function of growth direction,” J. Opt. Soc. Am. 61(6), 758–766 (1971). [CrossRef]
26. I. L. Snetkov, D. E. Silin, O. V. Palashov, E. A. Khazanov, H. Yagi, T. Yanagitani, H. Yoneda, A. Shirakawa, K.-i. Ueda, and A. A. Kaminskii, “Study of the thermo-optical constants of Yb doped Y2O3, Lu2O3 and Sc2O3 ceramic materials,” Opt. Express 21(18), 21254–21263 (2013). [CrossRef] [PubMed]
27. O. V. Palashov, I. B. Ievlev, E. A. Perevezentsev, E. V. Katin, and E. A. Khazanov, “Cooling and thermal stabilisation of Faraday rotators in the temperature range 300 – 200 K using Peltier elements,” Quantum Electron. 41(9), 858–861 (2011). [CrossRef]
28. M. A. Kagan and E. A. Khazanov, “Thermally induced birefringence in Faraday devices made from terbium gallium garnet-polycrystalline ceramics,” Appl. Opt. 43(32), 6030–6039 (2004). [CrossRef] [PubMed]
