A polarization-stepping method for measurement of the Faraday rotation in magneto-active materials has been modified for evaluating the Verdet constant as a function of wavelength and temperature. It has been used for a comparison of wavelength and temperature dependent Verdet constant, dominant electronic transition wavelength, and Curie-Weiss temperature in terbium gallium garnet (TGG) single crystals and TGG ceramics samples. The room-temperature values of Verdet constant varied from 36.6 to 44.2 rad/T.m at 1.064 μm wavelength and from 122.4 to 155.5 rad/T.m at 0.632 μm wavelength for all samples under investigation. These results match very well the values obtained from single wavelength measurement in the literature and show the same measurement error for single crystal and ceramics, respectively. The Curie-Weiss temperature has been evaluated for the first time in TGG ceramics. This method could be used for the detailed characterization of magneto-optical properties of the materials over a wide range of wavelengths and temperatures which is of great importance for the proper design of Faraday devices.
© 2016 Optical Society of America
Magneto-active materials possess the ability to rotate the polarization plane of incident light when they are subjected to an external magnetic field in the direction of the light propagation. This effect is abundantly used especially in laser optics for polarization control, optical isolation, birefringence compensation, or narrow-band optical filtering in gaseous media [1,2]. The design of these devices is being constantly improved in order to achieve maximal performance. It includes magnetic system design, or thermally induced depolarization compensation for the component designed for high-power laser systems. However, the main improvement is obtained with the proper selection of magneto-active materials having higher Verdet constant V, which warrants larger rotation angle, high transparency at the desired wavelength, excellent thermal properties and scalability of the components to large apertures for high power laser applications.
Currently, many new magneto-active crystals or transparent ceramics were being investigated for various wavelengths like fluorides (CeF3, PrF3, LiREF3, where RE denotes one of rare earth elements Tb, Dy, Ho, Er, or Yb) [3,4] for UV and visible spectrum wavelengths, terbium aluminum garnet ceramics (TAG) , Ce:TAG ceramics , Ti:TAG ceramics , terbium scandium aluminum garnet (TSAG) [8,9], terbium doped yttrium oxide (Tb:Y2O3) ceramics , rare earth elements doped terbium gallium garnet (RE:TGG) [11–13] and finally the most frequently used one, terbium gallium garnet (TGG) in the form of single crystal or ceramics, for near infrared region. All of these crystals exhibit great magneto-optic properties combined with high thermal conductivities and some of them can be produced also in the form of high-quality transparent ceramics greatly improving their aperture scaleability properties [14,15].
Magneto-optical properties and especially wavelength or temperature dependence of Verdet constant of TGG crystal and ceramics have been extensively studied by several authors [16–19]. Detailed characterization of magneto-optical properties is especially important for the proper design of large-aperture Faraday isolators [15,20] as well as for optimization of its performance, including operating temperature and length of magneto-active element.
In this work we report the results of Verdet constant of TGG single crytals <111> cut and TGG ceramics samples as a function of wavelength and temperature. A simple optical measurement has been developed for the precise evaluation of the Verdet constant, allowing the estimation of other important material parameters, like the electronic transition wavelength and the Curie-Weiss temperature.
2. Experimental method
2.1 Experimental setup
The experimental setup used for polarization-stepping measurement technique  has been used to obtain the spectra transmitted through the sample placed between two polarizers. The data aquisition and processing method has been, however, changed. In the  the authors captured the transmitted spectra for three angular steps and interpolated these points by cosine squared function with three parameters. Such approach is fast and simple, but can be relatively sensitive to the deviations of the measured points from cosine squared law. On the other hand, in the  the authors used same experimental setup, but captured the spectra for four angular steps. The rotational angle has been calculated for each point using known mutual inclination angle of polarizer-analyzer pair. The method described below captures the spectra for many more angular steps and uses the least-square approximation of these measured points by cosine squred law.
A schematic diagram of experimental setup used for the Verdet constant wavelength-temperature dependence of TGG ceramics measurement is shown in Fig. 1.
The beam from a broadband pulsed fiber source (NKT Photonics SuperK Compact) is linearly polarized by a high-contrast Glan polarizer and propagates through the magneto-active material sample placed inside the cryostat, where it is affected by the longitudinal magnetic field. The rotation of linear polarization is then measured by the detection system which consists of another Glan polarizer fixed in the motorized rotational stage and a diffuser, which suppresses the fiber coupling polarization dependence, and the spectrometer (Ocean Optics HR4000CG-UV-NIR). A spectral range of the measurement was limited by broadband source radiation spectrum and TGG transmission band from the bottom by 0.5 µm and by spectrometer effective range to 1.1 µm from the top. The temperature range was varied between 5 K and 300 K.
2.2 Magnetic field
One of the important parts of the experimental setup shown in Fig. 1 is a permanent magnet used as a source of external magnetic field which induces the Faraday rotation in magneto active material. The magnetic field magnitude inside the air gap of the magnet has been measured by a Gaussmeter (F.W. Bell Model 9200 with probe SAB92-1802). The longitudinal variation of the field is shown in Fig. 2.
The zero point of the longitudinal x-coordinate was put into the geometrical center of the air gap of the permanent magnet, which also corresponds to the area with the strongest magnetic field. The effective magnetic field affecting each sample of the specific length was evaluated according to mean value of magnetic field for each particular sample placed exactly in the geometrical center of the air gap. The uncertainty of the magnetic field is given by the gaussmeter measurement error as well as by the possible misplacement of the sample with respect to the exact center of the magnetic system.
In our measurements, the samples of five different lengths were used. The effective magnetic field for the longest sample with the length 7.15 mm was 1.18 ± 0.06 T, 6 mm and 4.11 mm long samples were exposed to the effective magnetic field 1.19 ± 0.06 T, and it was 1.20 ± 0.06 T for the shortest samples with the lengths 3.9 mm and 3 mm, respectively.
2.3 Angular resolution of the setup
The angular resolution of the experimental scheme has been measured with the setup with no magneto-active sample included. The measurement of the mutual angular rotation of two Glan-polarizers has been obtained with an angular step of 1 degree, which leads to 360 datapoints for each wavelength beeing fitted by squared cosine Malus law with the phase shift as a fitting parameter. As it can be seen from the results shown in Fig. 3, the angular resolution of the setup for most wavelengths between 0.5 μm and 1.1 µm is around 0.01 rad with exception of the region close to 1.1 μm, where the measurement error grows to the values around 0.04 rad. This resolution decrease is caused by the low sensitivity of the used spectrometer at these wavelengths leading to worse signal-to-noise ratio. The rotational angle measurement error comes mainly from the fitting with squared cosine function because of weak parasitic polarization-dependence of the diffusser-fiber collimator detection system causing slight deviations from squared cosine profile.
2.4 Faraday rotation angle measurement
The polarization plane rotation angle has been measured using a full-circle rotation of the analyzer. The relative intensity value has been captured every 5 degrees, leading to 73 data points for every temperature and wavelength. These data have been measured twice, once when the crystal was a subject to an external magnetic field (B coefficient) and once when it was not (N coefficient). The intensity data have been normalized to the range (0,1) and fitted with cos2(x-α) according to Malus law. The Faraday angle can be then obtained as a difference of phase shifts of these cosine-squared functions with and without magnetic field included, i.e. αB-αN. It should be noted that, because of the π-periodicity of this function, the total number of half-circle rotations have to be carefully observed. The typical example of the data obtained with 4 mm long TGG ceramics sample at the temperature of 35 K are shown in Fig. 4. The discontinuities in B-coefficient measured under magnetic field correspond to the half-circle rotations mentioned above.
The rotation angle can be then extracted from these data by simple subtraction of B and N coefficients, where it is necessary to remove the phase jumps caused by half circle rotations of the polarization plane.
2.5 Verdet constant calculation and fitting
The Verdet constant of the samples can be directly obtained from the polarization plane rotation angle α as V = α/BL, where B denotes the longitudinal magnetic field component and L is the length of the sample.
The Verdet constant of rare-earth garnets is generally a function of wavelength and temperature and consists of four different contributions [23–26]:27,28] diamagnetic contribution of the crystal lattice. In the single-oscillator model, which takes into account just one dominant electronic transition with wavelength λ0, this term can be expressed as  Vdm = -Aλ03λ2/(λ2-λ02)2 and it is caused by Zeeman splitting of ground and excited states. It should be noted that the diamagnetic term is negligible in the crystals containing paramagnetic ions like Tb3+ in TGG except at the frequencies close to resonances. Because our measurement is far from any resonance (the resonant frequency is much higher than the distance between the ground and the first excited multiplets), the diamagnetic term will be neglected for all frequencies and temperatures.
The second term, which is called the Van Vleck mixing term Vmix, comes from the overlap of wavefunctions of ground and neighbouring excited states which causes the perturbation of the amplitude elements of the electric moment by the magnetic field. In the single- oscillator model, it can be expressed as  Vmix = -Bλ02/(λ2-λ02). This term can be also considered to be temperature-independent.
The dominant term in Tb3+ ions doped garnets is the paramagnetic term Vpm, arisen from different occupation of sublevels of magneto-active ion ground state in magnetic field. This term depends on temperature according to Curie-Weiss law [30,31] which is well valid for the temperatures above the magnetic state phase transition characterized by Néel temperature of TGG TN ≈0.35 K . Once again, taking into account the dominant 4f-5d transition, the paramagnetic contribution can be described as Vpm = -Cλ02/(T-Tw)(λ2-λ02). This formula describes the paramagnetic contribution to Verdet constant far from any resonance. In the case of antiferromagnetic ordering below Néel temperature like in the case of TGG, the Curie-Weiss temperature Tw is negative.
The last contribution to the Verdet constant is the only frequency-independent gyromagnetic term Vgm. This contribution comes from magnetic dipole transitions and it is important especially in infrared region where the paramagnetic contribution decreases. The gyromagnetic term is proportional to the magnetic susceptibility and therefore it depends on temperature in the same way as paramagnetic term, according to Curie-Weiss law. The gyromagnetic term is given as  Vgm = D/(T-Tw).
The Verdet constant as a function of wavelength and temperature can be obtained by the substitution of all contributions mentioned above to Eq. (1). If the diamagnetic term is neglected, the resulting form of the Verdet constant is the following [17,31]
3. Results and discussion
The experimental procedure described above allows us to simultaneously measure the Verdet constant as a function of wavelength and temperature. The measurement has been done for four TGG single crystal samples all in <111> crystallographic cut and for two TGG ceramics (abbreviated as TGGc in the graphs) samples. The samples came from different manufacturers and had different lengths. The properties of these six samples are summarized in Table 1.
The resulting Verdet constant can be shown as a surface over the (λ, T) plane as represented in Fig. 5.
The left group of the graphs shows the Verdet constant of six TGG and TGG ceramics samples, while the right-hand side group shows the relative measurement error for all measurements. It should be noted that absolute values of Verdet constant will be displayed in all figures, even if according to the convention, the Verdet constant of TGG is negative.
It should be noted here that the main contribution to the measurement error comes from an uncertainty of the magnetic field which has been specified above, while the error caused by the limited angular resolution of the measurement is at least one order of magnitude smaller for most of the data points. This is also the main reason of an error distribution which was more severe for higher temperatures and longer wavelengths, where an absolute value of the Faraday angle is smaller. In addition, since the absolute uncertainty of the magnetic field depends neither on wavelength nor sample temperature, the relative error in this region grows, because the relative error is approximatelly inversely proportional to the polarization rotation angle. The fitting parameters calculated for the data fitted by Eq. (2) are collected in Table 2.
Once the complete temperature-wavelength dependence of the Verdet constant has been obtained, one can compare the results for some typical values of temperature or wavelength. As an example, the comparison of the Verdet constant dispersion at three particular temperatures 300 K, 150 K, and 50 K is shown in Fig. 6 and the comparison of temperature dependencies at three wavelengths 532 nm, 632.8 nm, and 1064 nm are shown in Fig. 7. The relatively severe fluctuations of the values at 1064 nm wavelength were caused by the fact that there was a very sharp narrow peak at this wavelength in broadband source spectrum which considerably decreases an angular resolution of the measurement close to this wavelength.
The results obtained by this procedure were also compared with the values measured by other groups using the methods leading separately to wavelength dependence of Verdet constant at fixed temperature or temperature dependence at fixed wavelength. The most straightforward way to compare the data is the comparison of the fitting parameters of the simple curves used to fit the dependencies. The following curves are mostly used in the literature: V(λ) = Cλ/(λ2-λ02) and V(T) = CT/T, where there are two fitting parameters for wavelength dependence Cλ and λ0 and one fitting parameter for temperature dependence CT. The resulting values for our samples are represented by dots with error bars and are compared to the values taken from the literature, which are displayed by black lines, as shown in Fig. 8.
A purely wavelength or temperature dependent Verdet constant evaluated at fixed temperature or wavelength can be also deduced from Eq. (2) asFig. 9. The measurement error is represented by the colored semitransparent area around the curve.
As expected, it can be clearly seen that the Curie-Weiss temperatures Tw, which are negative for the crystals with antifferomagnetic ordering below Néel temperature like TGG, do not depend on the wavelength of the incident wave. The values of Curie-Weiss temperature reported in the literature comes from magnetic susceptibility measurement and gains similar values to our results Tw = −8.61 K [32,33] and Tw = −7 K .
One would expect the same behavior for the 4f-5d electronic transition wavelength λ0. However it seems from the graph that there is some temperature dependence of this parameter. This effect is probably caused by the fact, that our model has been approximated by taking into account just one dominant transition wavelength. Note that H. A. Kramers  derived from quantum-mechanical considerations that the Verdet constant is proportional to the sum of the contributions Cij/(ν2-νij2), where ν is the frequency of the incident ligh, where Cij denotes the probability of the corresponding electronic transition, and νij stands for the transition frequency between corresponding electonic states. The rest of the parameters is following the dependencies 1/(T-Tw) and 1/(λ2-λ02) according to the expectation made for E, F, G, and H parameters under the Eq. (2).
The Verdet constant of terbium gallium garnet single crystal and ceramics has been measured as a function of both wavelength and temperature by one optical measurement process. The data have been taken for six samples in total to validate the reliability of the measurement procedure. The Verdet constant has been investigated in the (λ,T) region covering 0.5 – 1.1 μm in wavelength and 5 – 300 K in temperature. The obtained set of data allowed to fit the data with the approximate wavelength-temperature dependence function represented by Eq. (2) and deduce some material parameters of TGG like 4f-5d electronic transition wavelength or Curie-Weiss temperature. It has been found, that the angular resolution of the experimental setup in this case was around 0.01 rad in most of the region under investigation.
To the best of our knowledge, the Curie Weiss-temperature has been measured for the first time for TGG ceramics samples. It has been found that all the material parameters under investigation had similar values for single crystal or ceramics with small sample-to-sample differences.
This method has been found to be very versatile, providing a lot of information from a single measurement realized with a relatively simple experimental setup. Thus, we consider the method presented here a valuable tool for the detailed characterization of new magneto-active materials which are being developed worldwide, and for the design of future high-power Faraday isolators.
State budget of the Czech Republic (projects LO1602 and LM2015086); Czech Academy of Sciences (Grant No. JSPS-16-13), by the JSPS KAKENHI (26709072 and the 15K13386); the Matching Planner Program from Japan Science and Technology Agency; National Institute for Fusion Science (Grant No. ULHH036).
References and links
2. W. Kiefer, R. Löw, J. Wrachtrup, and I. Gerhardt, “Na-Faraday filter: The optimum point,” Sci. Rep. 4, 6552 (2014). [CrossRef]
3. V. Vasyliev, E. G. Víllora, M. Nakamura, Y. Sugahara, and K. Shimamura, “UV-visible Faraday rotators based on rare-earth fluoride single crystals: LiREF4 (RE = Tb, Dy, Ho, Er, and Yb), PrF3 and CeF3,” Opt. Express 20(13), 14460–14470 (2012). [CrossRef] [PubMed]
6. D. Zheleznov, A. Starobor, O. Palashov, H. Lin, and S. Zhou, “Improving characteristics of Faraday isolators based on TAG ceramics by cerium doping,” Opt. Lett. 39(7), 2183–2186 (2014). [CrossRef] [PubMed]
7. 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), 191–196 (2016). [CrossRef]
8. A. Yoshikawa, Y. Kagamitani, D. A. Pawlak, H. Sato, H. Machida, and T. Fukuda, “Czochralski growth of Tb3Sc2Al3O12 single crystal for Faraday rotator,” Mater. Res. Bull. 37(1), 1–10 (2002). [CrossRef]
9. I. L. Snetkov, R. Yasuhara, A. V. Starobor, E. A. Mironov, and O. V. Palashov, “Thermo-Optical and Magneto-Optical Characteristics of Terbium Scandium Aluminum Garnet Crystals,” IEEE J. Quantum Electron. 51(7), 7000307 (2015). [CrossRef]
10. I. L. Snetkov, D. A. Permin, S. S. Balabanov, and O. V. Palashov, “Wavelength dependence of Verdet constant of Tb3+:Y2O3 ceramics,” Appl. Phys. Lett. 108(16), 161905 (2016). [CrossRef]
11. Z. Chen, Y. Hang, L. Yang, J. Wang, X. Wang, J. Hong, P. Zhang, C. Shi, and Y. Wang, “Fabrication and characterization of cerium-doped terbium gallium garnet with high magneto-optical properties,” Opt. Lett. 40(5), 820–822 (2015). [CrossRef] [PubMed]
12. Z. Chen, Y. Hang, X. Wang, and J. Hong, “Fabrication and characterization of TGG crystals containing paramagnetic rare-earth ions,” Solid State Commun. 241, 38–42 (2016). [CrossRef]
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] [PubMed]
15. R. Yasuhara, I. Snetkov, A. Starobor, and O. Palashov, “Terbium gallium garnet ceramic-based Faraday isolator with compensation of thermally induced depolarization for high-energy pulsed lasers with kilowatt average power,” Appl. Phys. Lett. 105(24), 241104 (2014). [CrossRef]
16. N. P. Barnes and L. B. Petway, “Variation of the Verdet constant with temperature of terbium gallium garnet,” J. Opt. Soc. Am. B 9(10), 1912–1915 (1992). [CrossRef]
17. K. M. Mukimov, B. Yu. Sokolov, and U. V. Valiev, “The Faraday Effect of Rare-Earth Ions in Garnets,” Phys. Status Solidi 119(1), 307–315 (1990). [CrossRef]
18. 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]
19. O. Slezak, R. Yasuhara, A. Lucianetti, and T. Mocek, “Wavelength dependence of magneto-optic properties of terbium gallium garnet ceramics,” Opt. Express 23(10), 13641–13647 (2015). [CrossRef]
20. O. V. Palashov, D. S. Zheleznov, A. V. Voitovich, V. V. Zelenogorsky, E. E. Kamenetsky, E. A. Khazanov, R. M. Martin, K. L. Dooley, L. Williams, A. Lucianetti, V. Questschke, G. Mueller, D. H. Reitze, D. B. Tanner, E. Genin, B. Canuel, and J. Marque, “High-vacuum compatible high-power Faraday isolators for gravitational-wave interferometers,” J. Opt. Soc. Am. B 29(7), 1784–1792 (2012). [CrossRef]
22. D. N. Karimov, B. P. Sobolev, I. A. Ivanov, S. I. Kanorsky, and A. V. Masalov, “Growth and Magneto-Optical Properties of Na0.37Tb0.63F2.26 Cubic Single Crystal,” Crystallogr. Rep. 59(5), 718–723 (2014). [CrossRef]
23. R. Serber, “The theory of the Faraday effect in molecules,” Phys. Rev. 41(4), 489–506 (1932). [CrossRef]
24. J. H. Van Vleck and M. H. Hebb, “On the Paramagnetic Rotation of Tysonite,” Phys. Rev. 46(1), 17–32 (1934). [CrossRef]
25. A. D. Buckingham and P. J. Stephens, “Magnetic optical activity,” Annu. Rev. Phys. Chem. 17(1), 399–432 (1966). [CrossRef]
26. G. S. Krinchik and M. V. Chetkin, “Transparent ferromagnets,” Usp. Fiziol. Nauk 98(5), 3–25 (1969). [CrossRef]
27. M. J. Weber, “Faraday Rotator Materials for Laser Systems,”, Proc. SPIE: Laser and Nonlinear Optical Materials, 0681, 75 (1987) [CrossRef]
28. Y. J. Yu and R. K. Osborn, “Effect of nonlinear Refractive Index on Faraday Rotation,” Phys. Rev. A 15(6), 2404–2409 (1977). [CrossRef]
29. A. K. Zvezdin and V. A. Kotov, eds., Modern Magnetooptics and Magnetooptical Materials (Condensed Matter Physics), 1st ed. (CRC Press, 1997)
30. C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, Inc. 2004).
31. U. V. Valiev, J. B. Gruber, and G. W. Burdick, Magnetooptical Spectroscopy of the Rare-Earth Compounds: Deveopment and Application (Scientific Research Publishing, 2012).
32. U. Löw, S. Zherlitsyn, K. Araki, M. Akatsu, Y. Nemoto, T. Goto, U. Zeitler, and B. Lüthi, “Magneto-elastic effects in Tb3Ga5O12,” J. Phys. Soc. Jpn. 83(4), 044603 (2014). [CrossRef]
33. K. Kamazawa, D. Louca, R. Morinaga, T. J. Sato, Q. Huang, J. R. D. Copley, and Y. Qiu, “Filed-induced antiferromagnetism and competition in the metamagnetic state of terbium gallium garnet,” Phys. Rev. B 78(6), 064412 (2008). [CrossRef]
34. U. Löw, S. Zherlitsyn, M. Ozerov, U. Schaufuss, V. Kataev, B. Wolf, and B. Lüthi, “Magnetization, Magnetic Susceptibility and ESR in Tb3Ga5O12,” Eur. Phys. J. B 86(3), 87 (2013). [CrossRef]
35. H. A. Kramers, “Théorie générale de la rotation paramagnétique dans les cristaux,” Proc. Koninkl. Akad. Wet. 33, 959–972 (1930).