Abstract

Potassium terbium fluoride $ {{\rm KTb}_3}{{\rm F}_{10}} $ (KTF) crystal is a promising magneto-active material for creating multi-kilowatt average-power Faraday isolators operating at the visible and near-infrared wavelengths. Nevertheless, the key material’s parameter needed for the design of any Faraday isolator—the Verdet constant, has not been comprehensively investigated yet. In this Letter, we report on measurement of the Verdet constant of the KTF crystal for wavelengths between 600 and 1500 nm and for temperatures ranging from 15 to 295 K. A suitable model for the Verdet constant as a function of wavelength and temperature has been developed and may be conveniently used for optimal design of KTF-based high-average-power Faraday isolators.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

The laser research community is constantly working on the upgrade of the existing laser systems in order to deliver new laser sources with even higher average powers, higher pulse energies, higher repetition rates, and shorter pulse durations. In addition to this effort, several laser facilities are currently providing access to the beamtime to reach a broad range of users from the different research and industrial areas. In order to fulfil this goal, it is necessary to intensify the investigations of new laser system architectures, optical components, more efficient materials, and other related technologies [1].

One of the tasks of vital importance is the development of the high-power Faraday isolators (FIs). These nonreciprocal magneto-optical devices represent indispensable components for any high-power laser system, providing the needed functionality for optical separation of amplification cascades or for protecting the system from harmful backreflections coming from the experimental part. The core of every FI is a magneto-optical element (MOE), in which the plane of polarization of polarized light is rotated by a desirable angle, which is depending on the Verdet constant (a material property), length of the MOE, and on the applied magnetic field.

The main factors hampering the successful implementation of the high-power FIs are the polarization and wavefront distortions caused by the absorption-induced thermal effects arising mainly in the MOE [2,3]. This is caused by the fact that, compared with the other optical elements, most of the magneto-active materials used for MOEs have relatively higher absorption ($ \sim {{10}^{ - 3}}\;{{\rm cm}^{ - 1}} $). The absorption gives rise to an inhomogeneous temperature distribution in the MOE, which, in turn, causes (1) nonuniformities in the polarization rotation angle, because of the temperature dependence of the Verdet constant and changes in the MOE’s length due to the thermal expansion, and (2) nonuniform changes of the polarization state, caused by the thermal-stress-induced birefringence (TSIB). Furthermore, a thermal lensing phenomenon occurs as a consequence of the thermally induced deformations and changes in the refractive index. Since none of these effects can be completely avoided, it is essential to implement effective techniques for their partial compensation or suppression.

A substantial suppression of the thermal effects may be achieved by the investigation of new magneto-active materials (crystals, ceramics, or glass) with favorable properties [3,4]. The potential of the new materials is usually assessed using the following magneto-optical figure of merit (FOM) criteria [2,3,5],

$$\!\!{\mu _{\rm Q}} = \left| {\frac{{Vk}}{{\alpha Q}}} \right|({\rm if}\;|\xi | \ge 1)\;{\rm or}\;{\mu _{{\rm Q},\xi }} = \left| {\frac{{Vk}}{{\alpha Q\xi }}} \right|({\rm if}\;|\xi | \lt 1) ,\!\!$$
$${\mu _{\rm P}} = \left| {\frac{{Vk}}{{\alpha P}}} \right|,$$
in which $ V $ is the Verdet constant, $ \alpha $ is the linear absorption coefficient, $ k $ is the thermal conductivity, $ \xi $ denotes the optical anisotropy parameter (OAP), and $ Q $ and $ P $ are the thermo-optical constants evaluating the thermally induced distortions caused by the TSIB ($ Q $) and thermal lensing ($ P $). The OAP determines the influence of the TSIB depending on the crystallographic orientation of the material. High values of the FOMs mean that the respective thermal effects are less pronounced in the material, marking the assessed material as a more advantageous choice.

Among the recently reported new materials, crystals with a negative value of the OAP, such as terbium scandium aluminum garnet (TSAG) [57], sodium terbium fluoride (NTF) [5,810], or potassium terbium fluoride (KTF) [5,1012] are attracting more and more attention. The reason is that the possession of negative OAP ensures existence of an optimal crystallographic orientation at which the polarization distortions caused by the TSIB vanish completely (provided that no external magnetic field is present) [13] or may be significantly reduced (in the presence of an external magnetic field) [5]. Such a property is extremely beneficial since it provides an additional way for decreasing the deleterious impact of the TSIB as compared with the materials with positive OAP, such as the currently most commonly used terbium gallium garnet (TGG) crystal. Provided that the material has negative OAP, an optical compensation scheme of the TSIB may be realized without the need of additional optical elements [reciprocal rotator or half-wave plate (HWP)] [7]. Moreover, the compensation schemes (with and without the additional optical elements) could be conveniently combined in order to obtain even more effective TSIB compensation, as it was recently theoretically predicted in Ref. [5].

Since the TSIB in the negative OAP crystals could be efficiently suppressed, the influence of the temperature dependence of the Verdet constant or the thermal expansion may become more decisive [5]. These contributions are not considered in the traditionally used FOM criteria [Eq. (1)] and, therefore, it has been proposed to use the so-called maximum admissible power $ {P_{{\rm max}}} $ criterion in order to evaluate the new materials performance more precisely [2,5]. The $ {P_{{\rm max}}} $ gives the maximum power at which a defined value of the isolation ratio is guaranteed considering the influence of the TSIB, temperature dependence of the Verdet constant, and thermal expansion. The isolation ratio is a key parameter of the FI evaluating the power attenuation of light propagating in the undesired direction. A 30 dB isolation ratio (attenuation to 0.1%) is usually considered as sufficient.

The promising TSAG, NTF, and KTF crystals were recently compared using the $ {P_{{\rm max}}} $ criterion in Ref. [5] assuming equal values of the applied magnetic field, identical incident beam profiles, and that the optimal crystallographic orientations are used in each of the compared crystals. In the comparison, the room temperature material properties were considered. The KTF crystal scored highest in the $ {P_{{\rm max}}} $ for a 30 dB isolation ratio ($ \sim 3\;{\rm kW} $), followed by the NTF ($ \sim 2.9\;{\rm kW} $) and the TSAG ($ \sim 1.7\;{\rm kW} $) crystals. The limit of the TGG crystal was also evaluated to be $ \sim 0.7\;{\rm kW} $. In addition to the high $ {P_{{\rm max}}} $ score, it has been very recently reported that the KTF and NTF crystals exhibit $ \sim 20 $ times and $ \sim 6 $ times lower thermal lensing effect as compared with the TGG crystal [10]. The KTF has the additional advantage of its commercial availability [14]. All of these results are marking the mentioned terbium fluorides as one of the prime candidates for utilization in the multi-kilowatt (kW) average-power class of FIs.

In this Letter, we report on comprehensive characterization of the Verdet constant of the KTF crystal as a function of wavelength (600–1500 nm) and temperature (15–295 K), which is, apart from a few values at the room temperature [11,12,15], still unknown. This lack of information is currently blocking the practical application of the KTF crystal in high-power FIs as well as further investigations of this promising material. The reported data represent an indispensable input not only for the proper design of the KTF-based MOEs’ length but also for the FI designs compensating the distortions caused by the temperature dependence of the Verdet constant [16,17]. This could be of great importance since the influence of the TSIB may be considerably lowered due to the negative value of the OAP. Another benefit is the possibility of assessment of an additional increase of $ {P_{{\rm max}}} $ by cryogenic cooling.

The method used for the investigation of the Verdet constant has been already described in detail in Refs. [3,18]. The main advantage of the method is the utilization of a broadband radiation source as a probe beam, which enables a more comprehensive characterization of the Verdet constant. A simplified scheme of the experimental setup is depicted in Fig. 1. At first, the probe beam is linearly polarized by an input polarizer and then propagates through the investigated material sample, which is thermally coupled to a cryostat ledge enabling temperature control. If a magnetic field is applied on the sample, the plane of polarization of the probe beam is rotated by an angle $ \theta (\lambda ,T) = V(\lambda ,T){B_{{\rm eff}}}L $, where $ V $ is the Verdet constant of the material, $ \lambda $ is the wavelength, $ T $ is the temperature of the sample, $ {B_{{\rm eff}}} $ is an effective value of the applied magnetic field, and $ L $ is the sample length. The induced rotation angle is analyzed using the detection system consisting of an arbitrarily rotated superachromatic HWP, an output polarizer, and two spectrometers (in order to cover a broader range of detectable wavelengths). For a fixed value of $ {B_{{\rm eff}}} $ and $ T $, the detected signal on a single wavelength is proportional to a cosine-squared function, namely $ \propto \cos ^2 [ {2\phi + \theta (\lambda ,T,B) - {\phi _0}} ] $. In this function, $ \phi $ denotes the angular position of the HWP’s fast axis, and $ {\phi _0} $ refers to the initial bias angle between the HWP’s fast axis and the transmission axis of the polarizers. The spectra are gathered in a few hundreds of different angular positions of the HWP.

 

Fig. 1. 3D scheme of the experimental setup used for the characterization of the Verdet constant (on the left). An example of data, which was obtained for a fixed position of the permanent magnet and stabilized temperature of the sample, is shown on the right. Without the magnetic field, the initial phase of the individual cosine-squared functions (on each of the detectable wavelengths) is equal, as illustrated with the dashed blue line located in their maximum value. When the magnetic field is applied, the initial phases are shifted by a rotation angle $ \theta $ depending on the wavelength $ \lambda $, temperature of the sample $ T $, and the applied magnetic field $ B $.

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An example of data, which was obtained for a fixed position of the permanent magnet and stabilized temperature of the sample, is shown in Fig. 1 on the right. Without the magnetic field, the initial phase of the individual cosine-squared functions (on each of the detectable wavelengths) is equal, as illustrated with the dashed blue line located in their maximum value. However, if a nonzero magnetic field is applied, the initial phases are shifted by a nonzero rotation angle. The measurement was performed with the sample located in the maximum of the magnetic field, i.e., $ {B_{{\rm eff}}} = \int_{ - L/2}^{L/2} B(z){\rm d}z $, in which $ B(z) $ denotes the known longitudinal distribution of the magnetic field (the maximum is located at $ z = 0 $) [18]. Based on the knowledge of $ {B_{{\rm eff}}} $, $ L $, and $ \theta (\lambda ,T) $, the Verdet constant of the material as a function of wavelength and temperature may be directly calculated. In our investigation, the sample of the KTF crystal had $ 6.223 \pm 0.001\;{\rm mm} $ in length, and the corresponding effective magnetic field was equal to $ 1.188 \pm 0.028\;{\rm T} $. We have also characterized a sample of TGG crystal under the same experimental conditions, with the $ L = 5.987 \pm 0.001\;{\rm mm} $ and $ {B_{{\rm eff}}} = 1.189 \pm 0.028\;{\rm T} $.

A general model, which may be used for the description of the Verdet constant data obtained for the KTF and TGG crystals, may be written as follows [3,1820]:

$$V(\lambda ,T) = \frac{{{C_m}\lambda _0^2}}{{{\lambda ^2} - \lambda _0^2}} + \frac{{{C_p}\lambda _0^2}}{{({\lambda ^2} - \lambda _0^2)(T - {T_w})}} + \frac{{{C_g}}}{{T - {T_w}}}.$$
The model (3) describes the Verdet constant of a solid-state compound containing rare-Earth paramagnetic ions, assuming that (1) the described spectral range is sufficiently far from any resonance line, and (2) only one dominant electronic transition at wavelength $ {\lambda _0} $ is contributing to the Verdet constant. The model (3) consists of three different contributions, which are associated with the Zeeman splitting of the energy states of the material subjected to a magnetic field. The $ {C_m} $-term is the so-called mixing contribution, the $ {C_p} $-term is the paramagnetic contribution, and the $ {C_g} $-term denotes the gyromagnetic contribution. The $ {T_w} $ is the Curie–Weiss temperature. The first two contributions are connected with the electric dipole transitions, whereas the gyromagnetic contribution with the magnetic dipole transitions. A full description of these individual contributions is given in Refs. [3,1820].

The results obtained for the Verdet constant using the above-mentioned procedure are shown in Fig. 2. The Verdet constant data calculated at each of the detectable wavelengths (600–1500 nm) and for the characterized temperatures (15–295 K) are shown in the first row of the depicted plots. It needs to be noted that, according to the convention, the Verdet constant of both KTF and TGG crystals is negative, because the sense of the induced rotation is clockwise when the light propagates parallel to the magnetic field. The sign of the Verdet constant data depicted in Fig. 2 is opposite to its true sign. In the second row of the plots, the relative errors of the calculated Verdet constant values are presented. In the calculation of the error, we have considered the deviations of the detected signals from the cosine-squared functions, the uncertainty of the magnetic field, and the uncertainty of the sample lengths [3]. The obtained relative errors are mostly $ \lt {4}\% $ across the whole measured spectral range for all temperatures. The Verdet constant data were fit with the considered model (3); the obtained fitting parameters for each of the investigated materials are listed in Table 1. The relative deviations of the measured values from the values given by the fitted models are plotted in the last row of the plots in Fig. 2. The deviation is mostly $ \lt {10}\% $, but grows significantly when the wavelength approaches 1500 nm (up to $ \sim 30\% $). This is accounted to the increasing absorption of the $ {{\rm Tb}^{3 + }} $ ions for the wavelengths exceeding 1500 nm due to the $ ^7{{\rm F}_6} \to {{\rm F}_{0,1, \ldots }} $ electronic transitions located in the near-infrared region [15]. In order to obtain a more accurate model for the wavelengths $ \gt {1450}\;{\rm nm}$, it would be possible to include additional terms to the model (3) describing the contributions of the near-infrared transitions [21]. However, the cost of adding more fitting parameters corresponding to these transitions would be too high compared with the relatively weak impact on the overall model accuracy. Although it may provide a local improvement around the nearest vicinity of the 1500 nm wavelength, it was decided to neglect it in the current investigation.

 

Fig. 2. (a), (b) Verdet constant data. (c), (d) Relative error of the measurement. (e), (f) Relative deviation from the fitting model (3).

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Tables Icon

Table 1. Obtained Fitting Parameters for the Verdet Constant Model (3)

Some of the measured Verdet constant values for the wavelengths near the common laser wavelengths are listed in Table 2 and compared with the yet-reported measurements in the available literature. The disagreement with the reported values is mostly $ \lt {5}\% $ and may be accounted to the small sample-to-sample variations of the magneto-optical properties, and to the differences in the reported temperatures and wavelengths.

Tables Icon

Table 2. Measured Verdet Constant Values $ [ {{\rm rad}/({\rm T\cdot m)}} ] $ at the Wavelengths of 632.7 nm and 1062 nm at Room Temperature and Near the Liquid Nitrogen Temperature

As it may be easily observed from Table 2, the needed length of the KTF-based MOEs could be significantly reduced when cooled to cryogenic temperatures due to the increase of the absolute value of the Verdet constant ($ \sim 4 $ times when cooled from the room temperature to $ \sim 75\;{\rm K} $). Such a shortening would result in further suppression of the thermal effects and an increase of the material’s performance in an FI. Nevertheless, for a proper assessment of the advantage of the cryogenic cooling by the $ {P_{{\rm max}}} $ criterion, it will be needed to evaluate the yet-unknown material properties strongly depending on temperature, i.e., thermal conductivity, thermal expansion coefficient, thermo-optical constant $ Q $, and the linear absorption coefficient. This will be a topic for the future investigations of the KTF crystal.

We believe that the results presented here for the KTF’s Verdet constant as a function of wavelength and temperature can be used for the proper design and fabrication of MOEs for the multi-kW average-power class of FIs operating at the visible and near-infrared wavelengths. Since the contribution of the TSIB to the decrease of the isolation ratio may be effectively suppressed due to the negative value of the OAP, the presented results will be of great importance for a further increase of the KTF-based FIs’ performance by implementing the compensation techniques designed to minimize the contribution of the Verdet constant temperature dependence [16,17].

Funding

European Regional Development Fund (CZ.02.1.01/0.0/0.0/15_006/0000674); Japan Society for the Promotion of Science (18H01204); Horizon 2020 Framework Programme (739573); National Institute for Fusion Science (KEIN1608); Ministerstvo Školství, Mládeže a Tělovýchovy (LO1602, LM2015086).

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. C. N. Danson, C. Haefner, J. Bromage, T. Butcher, J.-C. F. Chanteloup, E. A. Chowdhury, A. Galvanauskas, L. A. Gizzi, J. Hein, D. I. Hillier, N. W. Hopps, Y. Kato, E. A. Khazanov, R. Kodama, G. Korn, R. Li, Y. Li, J. Limpert, J. Ma, C. H. Nam, D. Neely, D. Papadopoulos, R. R. Penman, L. Qian, J. J. Rocca, A. A. Shaykin, C. W. Siders, C. Spindloe, S. Szatmári, R. M. G. M. Trines, J. Zhu, P. Zhu, and J. D. Zuegel, High Power Laser Sci. Eng. 7, e54 (2019). [CrossRef]  

2. E. Khazanov, Usp. Fiz. Nauk 186, 975 (2016). [CrossRef]  

3. D. Vojna, O. Slezák, A. Lucianetti, and T. Mocek, Appl. Sci. 9, 3160 (2019). [CrossRef]  

4. J. Dai and J. Li, Scr. Mater. 155, 78 (2018). [CrossRef]  

5. I. L. Snetkov, IEEE J. Quantum Electron. 54, 1 (2018). [CrossRef]  

6. U. V. Valiev, J. B. Gruber, G. W. Burdick, I. A. Ivanov, D. Fu, V. O. Pelenovich, and N. I. Juraeva, J. Lumin. 176, 86 (2016). [CrossRef]  

7. A. V. Starobor, I. L. Snetkov, and O. V. Palashov, Opt. Lett. 43, 3774 (2018). [CrossRef]  

8. D. N. Karimov, B. P. Sobolev, I. A. Ivanov, S. I. Kanorsky, and A. V. Masalov, Crystallogr. Rep. 59, 718 (2014). [CrossRef]  

9. E. A. Mironov, O. V. Palashov, A. V. Voitovich, D. N. Karimov, and I. A. Ivanov, Opt. Lett. 40, 4919 (2015). [CrossRef]  

10. A. V. Starobor, E. A. Mironov, and O. V. Palashov, Opt. Mater. 98, 109469 (2019). [CrossRef]  

11. A. A. Jalali, E. Rogers, and K. T. Stevens, Opt. Lett. 42, 899 (2017). [CrossRef]  

12. D. E. Zelmon, G. Foundos, and K. T. Stevens, Proc. SPIE 10553, 105530E (2018). [CrossRef]  

13. I. Snetkov, A. Vyatkin, O. Palashov, and E. Khazanov, Opt. Express 20, 13357 (2012). [CrossRef]  

14. W. Schlichting, K. T. Stevens, G. Foundos, and A. Payne, Proc. SPIE 10448, 104481N (2017). [CrossRef]  

15. M. J. Weber, R. Morgret, S. Y. Leung, J. A. Griffin, D. Gabbe, and A. Linz, J. Appl. Phys. 49, 3464 (1978). [CrossRef]  

16. E. A. Mironov, A. V. Voitovich, A. V. Starobor, and O. V. Palashov, Appl. Opt. 53, 3486 (2014). [CrossRef]  

17. E. A. Mironov, A. V. Voitovich, and O. V. Palashov, Laser Phys. Lett. 13, 035001 (2016). [CrossRef]  

18. O. Slezák, R. Yasuhara, A. Lucianetti, and T. Mocek, Opt. Mater. Express 6, 3683 (2016). [CrossRef]  

19. A. K. Zvezdin and V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials, 1st ed. (Taylor & Francis Group, 1997).

20. K. M. Mukimov, B. Y. Sokolov, and U. V. Valiev, Phys. Status Solidi A 119, 307 (1990). [CrossRef]  

21. O. Slezák, R. Yasuhara, D. Vojna, H. Furuse, A. Lucianetti, and T. Mocek, Opt. Mater. Express 9, 2971 (2019). [CrossRef]  

22. R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, Opt. Express 15, 11255 (2007). [CrossRef]  

References

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  1. C. N. Danson, C. Haefner, J. Bromage, T. Butcher, J.-C. F. Chanteloup, E. A. Chowdhury, A. Galvanauskas, L. A. Gizzi, J. Hein, D. I. Hillier, N. W. Hopps, Y. Kato, E. A. Khazanov, R. Kodama, G. Korn, R. Li, Y. Li, J. Limpert, J. Ma, C. H. Nam, D. Neely, D. Papadopoulos, R. R. Penman, L. Qian, J. J. Rocca, A. A. Shaykin, C. W. Siders, C. Spindloe, S. Szatmári, R. M. G. M. Trines, J. Zhu, P. Zhu, and J. D. Zuegel, High Power Laser Sci. Eng. 7, e54 (2019).
    [Crossref]
  2. E. Khazanov, Usp. Fiz. Nauk 186, 975 (2016).
    [Crossref]
  3. D. Vojna, O. Slezák, A. Lucianetti, and T. Mocek, Appl. Sci. 9, 3160 (2019).
    [Crossref]
  4. J. Dai and J. Li, Scr. Mater. 155, 78 (2018).
    [Crossref]
  5. I. L. Snetkov, IEEE J. Quantum Electron. 54, 1 (2018).
    [Crossref]
  6. U. V. Valiev, J. B. Gruber, G. W. Burdick, I. A. Ivanov, D. Fu, V. O. Pelenovich, and N. I. Juraeva, J. Lumin. 176, 86 (2016).
    [Crossref]
  7. A. V. Starobor, I. L. Snetkov, and O. V. Palashov, Opt. Lett. 43, 3774 (2018).
    [Crossref]
  8. D. N. Karimov, B. P. Sobolev, I. A. Ivanov, S. I. Kanorsky, and A. V. Masalov, Crystallogr. Rep. 59, 718 (2014).
    [Crossref]
  9. E. A. Mironov, O. V. Palashov, A. V. Voitovich, D. N. Karimov, and I. A. Ivanov, Opt. Lett. 40, 4919 (2015).
    [Crossref]
  10. A. V. Starobor, E. A. Mironov, and O. V. Palashov, Opt. Mater. 98, 109469 (2019).
    [Crossref]
  11. A. A. Jalali, E. Rogers, and K. T. Stevens, Opt. Lett. 42, 899 (2017).
    [Crossref]
  12. D. E. Zelmon, G. Foundos, and K. T. Stevens, Proc. SPIE 10553, 105530E (2018).
    [Crossref]
  13. I. Snetkov, A. Vyatkin, O. Palashov, and E. Khazanov, Opt. Express 20, 13357 (2012).
    [Crossref]
  14. W. Schlichting, K. T. Stevens, G. Foundos, and A. Payne, Proc. SPIE 10448, 104481N (2017).
    [Crossref]
  15. M. J. Weber, R. Morgret, S. Y. Leung, J. A. Griffin, D. Gabbe, and A. Linz, J. Appl. Phys. 49, 3464 (1978).
    [Crossref]
  16. E. A. Mironov, A. V. Voitovich, A. V. Starobor, and O. V. Palashov, Appl. Opt. 53, 3486 (2014).
    [Crossref]
  17. E. A. Mironov, A. V. Voitovich, and O. V. Palashov, Laser Phys. Lett. 13, 035001 (2016).
    [Crossref]
  18. O. Slezák, R. Yasuhara, A. Lucianetti, and T. Mocek, Opt. Mater. Express 6, 3683 (2016).
    [Crossref]
  19. A. K. Zvezdin and V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials, 1st ed. (Taylor & Francis Group, 1997).
  20. K. M. Mukimov, B. Y. Sokolov, and U. V. Valiev, Phys. Status Solidi A 119, 307 (1990).
    [Crossref]
  21. O. Slezák, R. Yasuhara, D. Vojna, H. Furuse, A. Lucianetti, and T. Mocek, Opt. Mater. Express 9, 2971 (2019).
    [Crossref]
  22. R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, Opt. Express 15, 11255 (2007).
    [Crossref]

2019 (4)

D. Vojna, O. Slezák, A. Lucianetti, and T. Mocek, Appl. Sci. 9, 3160 (2019).
[Crossref]

C. N. Danson, C. Haefner, J. Bromage, T. Butcher, J.-C. F. Chanteloup, E. A. Chowdhury, A. Galvanauskas, L. A. Gizzi, J. Hein, D. I. Hillier, N. W. Hopps, Y. Kato, E. A. Khazanov, R. Kodama, G. Korn, R. Li, Y. Li, J. Limpert, J. Ma, C. H. Nam, D. Neely, D. Papadopoulos, R. R. Penman, L. Qian, J. J. Rocca, A. A. Shaykin, C. W. Siders, C. Spindloe, S. Szatmári, R. M. G. M. Trines, J. Zhu, P. Zhu, and J. D. Zuegel, High Power Laser Sci. Eng. 7, e54 (2019).
[Crossref]

A. V. Starobor, E. A. Mironov, and O. V. Palashov, Opt. Mater. 98, 109469 (2019).
[Crossref]

O. Slezák, R. Yasuhara, D. Vojna, H. Furuse, A. Lucianetti, and T. Mocek, Opt. Mater. Express 9, 2971 (2019).
[Crossref]

2018 (4)

D. E. Zelmon, G. Foundos, and K. T. Stevens, Proc. SPIE 10553, 105530E (2018).
[Crossref]

A. V. Starobor, I. L. Snetkov, and O. V. Palashov, Opt. Lett. 43, 3774 (2018).
[Crossref]

J. Dai and J. Li, Scr. Mater. 155, 78 (2018).
[Crossref]

I. L. Snetkov, IEEE J. Quantum Electron. 54, 1 (2018).
[Crossref]

2017 (2)

A. A. Jalali, E. Rogers, and K. T. Stevens, Opt. Lett. 42, 899 (2017).
[Crossref]

W. Schlichting, K. T. Stevens, G. Foundos, and A. Payne, Proc. SPIE 10448, 104481N (2017).
[Crossref]

2016 (4)

U. V. Valiev, J. B. Gruber, G. W. Burdick, I. A. Ivanov, D. Fu, V. O. Pelenovich, and N. I. Juraeva, J. Lumin. 176, 86 (2016).
[Crossref]

E. Khazanov, Usp. Fiz. Nauk 186, 975 (2016).
[Crossref]

E. A. Mironov, A. V. Voitovich, and O. V. Palashov, Laser Phys. Lett. 13, 035001 (2016).
[Crossref]

O. Slezák, R. Yasuhara, A. Lucianetti, and T. Mocek, Opt. Mater. Express 6, 3683 (2016).
[Crossref]

2015 (1)

2014 (2)

E. A. Mironov, A. V. Voitovich, A. V. Starobor, and O. V. Palashov, Appl. Opt. 53, 3486 (2014).
[Crossref]

D. N. Karimov, B. P. Sobolev, I. A. Ivanov, S. I. Kanorsky, and A. V. Masalov, Crystallogr. Rep. 59, 718 (2014).
[Crossref]

2012 (1)

2007 (1)

1990 (1)

K. M. Mukimov, B. Y. Sokolov, and U. V. Valiev, Phys. Status Solidi A 119, 307 (1990).
[Crossref]

1978 (1)

M. J. Weber, R. Morgret, S. Y. Leung, J. A. Griffin, D. Gabbe, and A. Linz, J. Appl. Phys. 49, 3464 (1978).
[Crossref]

Bromage, J.

C. N. Danson, C. Haefner, J. Bromage, T. Butcher, J.-C. F. Chanteloup, E. A. Chowdhury, A. Galvanauskas, L. A. Gizzi, J. Hein, D. I. Hillier, N. W. Hopps, Y. Kato, E. A. Khazanov, R. Kodama, G. Korn, R. Li, Y. Li, J. Limpert, J. Ma, C. H. Nam, D. Neely, D. Papadopoulos, R. R. Penman, L. Qian, J. J. Rocca, A. A. Shaykin, C. W. Siders, C. Spindloe, S. Szatmári, R. M. G. M. Trines, J. Zhu, P. Zhu, and J. D. Zuegel, High Power Laser Sci. Eng. 7, e54 (2019).
[Crossref]

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C. N. Danson, C. Haefner, J. Bromage, T. Butcher, J.-C. F. Chanteloup, E. A. Chowdhury, A. Galvanauskas, L. A. Gizzi, J. Hein, D. I. Hillier, N. W. Hopps, Y. Kato, E. A. Khazanov, R. Kodama, G. Korn, R. Li, Y. Li, J. Limpert, J. Ma, C. H. Nam, D. Neely, D. Papadopoulos, R. R. Penman, L. Qian, J. J. Rocca, A. A. Shaykin, C. W. Siders, C. Spindloe, S. Szatmári, R. M. G. M. Trines, J. Zhu, P. Zhu, and J. D. Zuegel, High Power Laser Sci. Eng. 7, e54 (2019).
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C. N. Danson, C. Haefner, J. Bromage, T. Butcher, J.-C. F. Chanteloup, E. A. Chowdhury, A. Galvanauskas, L. A. Gizzi, J. Hein, D. I. Hillier, N. W. Hopps, Y. Kato, E. A. Khazanov, R. Kodama, G. Korn, R. Li, Y. Li, J. Limpert, J. Ma, C. H. Nam, D. Neely, D. Papadopoulos, R. R. Penman, L. Qian, J. J. Rocca, A. A. Shaykin, C. W. Siders, C. Spindloe, S. Szatmári, R. M. G. M. Trines, J. Zhu, P. Zhu, and J. D. Zuegel, High Power Laser Sci. Eng. 7, e54 (2019).
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Appl. Opt. (1)

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[Crossref]

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I. L. Snetkov, IEEE J. Quantum Electron. 54, 1 (2018).
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J. Lumin. (1)

U. V. Valiev, J. B. Gruber, G. W. Burdick, I. A. Ivanov, D. Fu, V. O. Pelenovich, and N. I. Juraeva, J. Lumin. 176, 86 (2016).
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Opt. Express (2)

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Opt. Mater. Express (2)

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K. M. Mukimov, B. Y. Sokolov, and U. V. Valiev, Phys. Status Solidi A 119, 307 (1990).
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Proc. SPIE (2)

D. E. Zelmon, G. Foundos, and K. T. Stevens, Proc. SPIE 10553, 105530E (2018).
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Figures (2)

Fig. 1.
Fig. 1. 3D scheme of the experimental setup used for the characterization of the Verdet constant (on the left). An example of data, which was obtained for a fixed position of the permanent magnet and stabilized temperature of the sample, is shown on the right. Without the magnetic field, the initial phase of the individual cosine-squared functions (on each of the detectable wavelengths) is equal, as illustrated with the dashed blue line located in their maximum value. When the magnetic field is applied, the initial phases are shifted by a rotation angle $ \theta $ depending on the wavelength $ \lambda $ , temperature of the sample $ T $ , and the applied magnetic field $ B $ .
Fig. 2.
Fig. 2. (a), (b) Verdet constant data. (c), (d) Relative error of the measurement. (e), (f) Relative deviation from the fitting model (3).

Tables (2)

Tables Icon

Table 1. Obtained Fitting Parameters for the Verdet Constant Model (3)

Tables Icon

Table 2. Measured Verdet Constant Values [ r a d / ( T m ) ] at the Wavelengths of 632.7 nm and 1062 nm at Room Temperature and Near the Liquid Nitrogen Temperature

Equations (3)

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μ Q = | V k α Q | ( i f | ξ | 1 ) o r μ Q , ξ = | V k α Q ξ | ( i f | ξ | < 1 ) ,
μ P = | V k α P | ,
V ( λ , T ) = C m λ 0 2 λ 2 λ 0 2 + C p λ 0 2 ( λ 2 λ 0 2 ) ( T T w ) + C g T T w .

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