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Abstract
Temperature and wavelength dependence of the Verdet constant of dysprosium sesquioxide ($\mathrm{{{Dy}_2}{{O}_3}}$) transparent ceramics has been measured for the temperatures ranging from cryogenic $20\,\mathrm {K}$ up to room temperature $297\,\mathrm {K}$ and for the wavelength range starting from visible $0.6\,\mathrm {{\mu }m}$ up to mid-infrared $2.3\,\mathrm {{\mu }m}$. Several absorption bands corresponding to the electronic transitions in $\mathrm{{{Dy}_2}{{O}_3}}$ have been identified and removed from the data analysis due to high absorption of the matching wavelengths within these absorption bands. The Verdet constant data has been fitted by the multi-transition model function covering the whole temperature and wavelength range. Such fitted function could be used for the detailed design of magneto-optical devices as well as for the analysis of physical properties of the material itself.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Magneto-active (MA) materials are currently used to rotate the major axis of polarization ellipse of the laser light for the distinct angle when it is placed in the external magnetic field. This specific feature, called the Faraday rotation, of the materials is used in many application areas especially for polarization control, birefringence compensation, narrow-band filtering, and optical isolation. The MA materials themselves are the subject of intensive research focused mainly to the development of the new materials and their characterization. The new materials are being developed in order to cover new spectral ranges, to have higher Verdet constant, lower absorption coefficient, better thermal properties, and scalability to large aperture components.
The coverage of the optical spectral range by the MA materials is spread from UV and visible over the near-IR to the relatively limited number of available materials for mid-IR. For UV and visible wavelengths especially fluorides like $\mathrm {CeF_3}$, $\mathrm {PrF_3}$, and $\mathrm {LiREF_3}$, in which RE substitutes one of the rare-earth elements Tb, Dy, Ho, Er, or Yb have been reported [1,2]. Probably the widest selection of MA materials is available for near-IR region, where mostly the materials containing $\mathrm {Tb^{3+}}$ ions are being used. Among these, terbium gallium garnet (TGG) [3–6] with various doping rare-earth ions [7,8], terbium aluminum garnet (TAG) [9] with various doping like Ce:TAG [10] or Ti:TAG [11], terbium scandium aluminum garnet (TSAG) [12,13] or terbium doped yttrium oxide ($\mathrm {Tb:Y_2O_3}$) [14], are the frequently studied ones. Most of these materials is also available as transparent ceramics usually allowing the production of larger aperture components compared to single crystals.
However, the majority of these near-IR crystals has a strong absorption in the mid-IR region, and therefore are not suitable for this spectral region, for which the list of potentially usable MA materials is still relatively thin. For low-power applications the ferrimagnetic (at room temperature) yttrium iron garnet (YIG) [15] can be used, since it exhibits high Verdet constant and low saturation magnetization. Promising magneto-optical (MO) characteristics in mid-IR are also expected from rare-earth ions containing fluorides like $\mathrm {CeF_3}$, $\mathrm {PrF_3}$ and $\mathrm {LiREF_4}$ (RE = Dy, Ho, or Er). However, to the best of our knowledge only the MO properties of $\mathrm {CeF_3}$ were further investigated up to $1.95\,\mathrm {{\mu }m}$ [16]. Other interesting materials having a potential to be used as MO elements for mid-IR are $\mathrm {\left (Dy_XY_{0.95-X}La_{0.05}\right )_2O_3}$ ceramics with variable doping X = 0.7, 0.8, and 0.9 reported in [17] and also $\mathrm {EuF_2}$ crystal reported in [18].
For all of these materials the Verdet constant depends on temperature and wavelength of the passing-through light and the knowledge of this dependencies is quite important for proper usage of the material in MO device. This is given by the fact, that for precise design of the device it is needed to find the most beneficial working temperature and length of MO element in combination with the magnetic field available keeping in mind the wavelength of the light using the MO device. The dependence of the Verdet constant on both, temperature and wavelength is, however, known just for TGG [19] and partially for $\mathrm {\left (Dy_XY_{0.95-X}La_{0.05}\right )_2O_3}$ ceramics [21].
In this work, we report on the complete temperature-wavelength dependence of dysprosium oxide $\mathrm{{{Dy}_2}{{O}_3}}$ ceramics in the temperature range from cryogenic temperature $20\,\mathrm {K}$ up to room temperature $297\,\mathrm {K}$ and spectral range starting at visible $0.6\,\mathrm {{\mu }m}$ and reaching mid-IR up to $2.3\,\mathrm {{\mu }m}$. The proper fitting function for the experimental data has also been found to allow the convenient usage of the data for calculations, MO device design, or material parameters study.
2. Experimental method
2.1 Experimental setup
The measurement technique similar to that described in [19] has been used to measure the temperature-wavelength dependence of the dysprosium sesquioxide $\mathrm{{{Dy}_2}{{O}_3}}$ based transparent ceramics Verdet constant. The experimental setup used for this measurement is shown in Fig. 1.
Fig. 1. Experimental setup used for the measurement of temperature-wavelength dependence of Verdet constant of $\mathrm {{Dy}_{2}O_3}$ ceramics. Broadband laser (spectral range $0.45-2.4\,\mathrm {{\mu }m}$), Input/Output polarizer - high contrast Glan polarizers, Adjustable HWP - achromatic half-wave plate fixed in motorized rotational stage ($0.6-2.7\,\mathrm {{\mu }m}$), Spectrometer $1 (0.2-1.1\,\mathrm {{\mu }m}$),Spectrometer $2 (0.9-2.55\,\mathrm {{\mu }m}$)
It shows the improved version of the setup used for the measurement presented in [19]. The beam from a broadband pulsed fibre laser source (NKT Photonics SuperK Compact) was linearly polarized by a high-contrast Glan polarizer (Input polarizer) and propagated through the MA sample of $\mathrm{{{Dy}_2}{{O}_3}}$ ceramics placed inside the cryostat, where it was affected by the longitudinal magnetic field of the permanent magnet. The rotation of the linear polarization plane was then measured by the detection system which consisted of an achromatic half-wave plate (Adjustable HWP) fixed in the motorized rotational stage and another high-contrast Glan polarizer (Output Polarizer) decomposing the beam onto two orthogonal linear polarizations which were detected by two spectrometers with different spectral ranges. The spectrometer in the straight direction (Spectrometer 1 - Ocean Optics HR4000CG-UV-NIR) has the spectral range from $0.19$ to $1.1\,\mathrm {{\mu }m}$ and the spectrometer in the deflected direction (Spectrometer 2 - Hamamatsu TGC11118GA) was analyzing the spectral range from $0.9$ to $2.55\,\mathrm {{\mu }m}$. It should be noted that the overall spectral range of the measurement was limited from the bottom by the HWP to $0.6\,\mathrm {{\mu }m}$ and to approximately $2.3\,\mathrm {{\mu }m}$ from the top by the decreasing intensity of the light source in this spectral region. The temperatures were varied between 20 K and 297K.
The magnetic field for the measurement was provided by the permanent magnet which was fixed on the rail allowing the removal of the magnetic field from the sample for the measurement without Faraday rotation used for the specification of the bias angle of the polarizer and the HWP. The spatial profile of the longitudinal component of the magnetic field was measured by the Gaussmeter (F.W. Bell model 9200 with probe SAB92-1802) and the effective magnetic field affecting the sample has been then obtained by the integration of this spatial profile over the length of the sample divided by the length of the sample. A very thin sample with the length $L=0.51\pm 0.01\,\mathrm {mm}$ was used for this measurement and the effective magnetic field was set for this length to be $B=1.20\pm 0.06\,\mathrm {T}$.
2.2 Faraday rotation angle measurement
The polarization plane rotation angle was measured using a half-circle rotation of the HWP. The relative intensity value was captured every 0.2 degree, leading to 901 data points for every temperature and wavelength. These data were measured separately in two cases: in the first case the sample was subject to the external magnetic field, while in the second case it was not. The intensity data were normalized to the range $\left \langle 0,1\right \rangle$ and fitted by the function
where $\phi _H$ is HWP azimuth angle, $\theta$ is Faraday rotation angle, and $\beta$ is output polarizer and HWP fast axis bias angle. The bias angle $\beta$ was obtained by the comparison of the fits of the data measured with and without external magnetic involved. It should be noted that, because of the $\pi$-periodicity of the function (1) the total number of half-circle polarization rotations have to be carefully observed, because it causes the discontinuities in the Faraday rotation angle. In another words, the number of half-circles reached by the rotation angle cannot be obtained from one single measurement and it is necessary to follow the trend of the rotation angle as it grows with shorter wavelengths or lower temperatures. Thanks to the large number of angular steps taken for each temperature and each wavelength the absolute measurement error of the rotation angle was well below 0.5 angular degree for all the wavelengths and temperatures under consideration. The measurement error grew just for the wavelengths close to the absorption bands, where the transmission of the sample was too low to get reliable signal. These points were, however, not taken into consideration in the analysis of the results.2.3 Verdet constant calculation and fitting
The Verdet constant of the sample can be obtained directly from the polarization plane rotation angle $\theta$ as $V=\theta /BL$, where $B$ is the longitudinal component of effective external magnetic field and $L$ denotes the length of the sample. The Verdet constant is generally, under the fixed external magnetic field, the function of wavelength of the probe beam and the temperature of the sample [20]. Using similar procedure as in the work [19] the Verdet constant can be fitted by the function of wavelength $\lambda$ and temperature $T$. Nevertheless, in the contrary to the work [21] the single electronic transition model was replaced by multi-transitions model in order to cover the whole observed spectral range by one fitting function, while e.g. the absorption band located close to $\lambda =1.2\,\mathrm {\mu m}$ changes the character of the Verdet constant evolution with the growing wavelength which makes the single transition model unsuitable to be used for the whole observed wavelength spectrum ranging from $0.6\,\mathrm {{\mu }m}$ to $2.3\,\mathrm {{\mu }m}$. The resulting form of the fitting function is the following [22,23]
The second term, which is called the Van Vleck mixing term and in Eq. (2) is represented by the terms proportional to the fitting constants $B_j$, 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 multi oscillator model, it can be expressed as [26] $-B_j\lambda _{0j}^2/\left (\lambda ^2-\lambda _{0j}^2\right )$. This term can be considered to be temperature-independent.
The third term, proportional to $C_j$ is the paramagnetic term, 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 [27] which is well valid for the temperatures above the magnetic state phase transition characterized by Néel temperature of $\mathrm{{{Dy}_2}{{O}_3}}$ $T_N\approx 1.2\,\mathrm {K}$ [28]. The paramagnetic contribution can be described as $-C_j\lambda _{0j}^2/\left (T-T_w\right )\left (\lambda ^2-\lambda _{0j}^2\right )$. In the case of antiferromagnetic ordering below Néel temperature like in the case of $\mathrm{{{Dy}_2}{{O}_3}}$, the Curie-Weiss temperature $T_w$ is negative.
The last contribution to the Verdet constant is the only frequency-independent gyromagnetic term, proportional to $D$. 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 [20] $D/\left (T-T_w\right )$.
The Eq. (2) can be reduced to the single variable for fixed temperature $T_f$ or fixed wavelength $\lambda _f$ as follows
3. Results and discussion
The Verdet constant of the sample of $\mathrm{{{Dy}_2}{{O}_3}}$ transparent ceramics has been measured by the above described procedure within the temperature range from $20\,\mathrm {K}$ to $297\,\mathrm {K}$ and in the wavelength ranging from $0.6\,\mathrm {{\mu }m}$ to $2.3\,\mathrm {{\mu }m}$. It should be noted that according to the convention the Verdet constant of $\mathrm{{{Dy}_2}{{O}_3}}$ is negative within almost whole studied wavelength range (with potential exception in the regions close to transition bands), because it rotates the polarization clockwise if the laser beam propagation is parallel to the magnetic field. However, it will be shown as positive in the graphs ($-V$ will be shown on y-axis) to make plots more convenient.
The optical quality of this experimental sample was not very high and that led to the lowered transmittance of the sample shown in Fig. 2 (a). The spectral regions with the transmittance $< 30\,\%$ were excluded from the data analysis due to the low strength of the detected signal leading to relatively high absolute measurement error, these areas are highlighted by the gray filled rectangles in Fig. 2 (a) and by red rectangles in Fig. 2 (b). The exception from this rule was made for the region between $1$ and $1.2\,\mathrm {{\mu }m}$ where, despite the low transmission of the sample, the signal was strong enough to provide very low measurement error as can be seen in the Fig. 2 (b). The same exception has been done for the region slightly below $1.4\,{\mu }m$ which also provided the measurement error comparable to the regions with significantly higher transmission. This was provided by the high power of the broadband source in this spectral region. On the other hand, the wavelengths above $2\,\mathrm {{\mu }m}$ suffers from the high relative error of the measurement, even if the absolute error is comparable to the rest of the wavelengths. This was caused by the very small polarization rotation angle $< 1\,\mathrm {deg}$ given by the small thickness of the sample combined with the decreasing Verdet constant for longer wavelengths, especially for higher temperatures. The region of the increased relative error can be also found in the Fig. 2 b).
Fig. 2. a) The transmittance of the $\mathrm{{{Dy}_2}{{O}_3}}$ ceramics sample. The gray rectangles are showing the spectral regions excluded from the data analysis due to weak detected signal. b) The map of the relative measurement error consisting from the measurement error of the rotation angle, magnetic field measurement uncertainty, and the sample length measurement error. The excluded spectral regions are outlined by red lines.
At first, the data have been studied at fixed temperature as a function of wavelength. The example of the obtained data for the fixed temperatures $T=20\,\mathrm {K}$ and $T=297\,\mathrm {K}$ are shown in Fig. 3. The two chosen temperatures corresponds to the minimal and maximal temperature which has been analysed in the experiment. These data were fitted by the formula (3) in order to obtain the fitting parameters $E$, $F$, $G$, and $\lambda _0$. It should be noted that the number of summands in (3) has been fixed to 3, corresponding to the two most significant absorption bands of the sample located around $\lambda _{02}\approx 1.2\,\mathrm {{\mu }m}$ and $\lambda _{03}\approx 1.6\,\mathrm {{\mu }m}$. The first summand localized at $\lambda _{01}$ corresponds to the main electronic transition responsible for the magnetic activity and is located in UV region, which is far from the spectral region under investigation. These absorption bands correspond to the electronic transitions well known from the spectroscopic measurements. The lowest $4f\rightarrow 5d$ transition (ground state $4f^9\,^6H_{15/2}$) at $\lambda _{01}$ lies outside the spectral area investigated in this work and has been measured to be $\lambda _{01}\approx 169.8\,\mathrm {nm}$ [29] for $\mathrm {Dy^{3+}}$ ions in $\mathrm {CaF_2}$ host or $\lambda _{01}\approx 185.2\,\mathrm {nm}$ [30] in $\mathrm{{{Dy}_2}{{O}_3}}$ doped borate glasses. Another transition which corresponds to the transition wavelength $\lambda _{02}$ has been identified as $^6H_{15/2}\rightarrow ^6F_{11/2}+^6H_{9/2}$ and its wavelength can be found in the literature e.g. for $\mathrm {Dy:Y_2O_3}$ ceramics [31] as $\lambda _{02}=1253\,\mathrm {nm}$ or $\lambda _{02}=1282.4\,\mathrm {nm}$ for $\mathrm {Dy^{3+}}$ ions in fluorozirconate glass [32]. The last, $\lambda _{03}$, transition corresponds to $^6H_{15/2}\rightarrow ^6H_{11/2}$ transition with the wavelength $\lambda _{03}=1667\,\mathrm {nm}$ in $\mathrm {Dy:Y_2O_3}$ ceramics [31].
Fig. 3. The dispersion of the Verdet constant at fixed temperature. a) The result for the lowest temperature under investigation $T=20\,\mathrm {K}$ and b) The result for the highest temperature under investigation $T=297\,\mathrm {K}$. The data excluded from the analysis are marked by black crosses while the data used for fitting are represented by blue circles. The fitted function given by formula (3) is given by the solid line and the transmittance showing the low-signal regions is represented by the dashed line. The relative deviation of the fitted function from the data $\Delta V$ is also shown for each temperature.
It should be noted, that also another transitions e.g. $^6H_{15/2}\rightarrow ^6F_{5/2}$ at $\approx 800\,\mathrm {nm}$ or $^6H_{15/2}\rightarrow ^6F_{7/2}$ at $\approx 900\,\mathrm {nm}$ can be identified from the measured data, however, due to low cross-section of these transitions (about 30-60 times lower than the transition at $\lambda _{02}=1253\,\mathrm {nm}$ [31]), their influence is restricted just to the very close neighborhood of the transition wavelengths, where these form a very narrow peaks visible in the Verdet constant spectrum. Therefore, the influence of these transitions to the rest of the spectrum is negligible and they were not included in the fitting function.
Since the $E_1$-term in (3) is inversely proportional to $\left (\lambda ^2-\lambda _0^2\right )^2$ its contribution in the studied region is small compared to the $F_1$-term and therefore this term can be omitted in this case.
A similar procedure has been used to analyze the data as solely temperature-dependent under fixed wavelength. Under such condition the data should follow the formula (4). The results for fixed wavelengths $\lambda _1=0.63\,\mathrm {{\mu }m}$, $\lambda _2=1.0\,\mathrm {{\mu }m}$, and $\lambda _3=2.0\,\mathrm {{\mu }m}$ together with fitted functions are shown in the Fig. 4.The relative deviation of the fitted function from the measured data ${\Delta }V$ is shown in the Fig. 4 as well, and according to the measurement error map it grows with the increasing wavelength, being below $5\,\%$ for $0.63\,\mathrm {{\mu }m}$, below $10\,\%$ at $1.0\,\mathrm {{\mu }m}$, and reaching the values around $40\,\%$ at some points for $2.0\,\mathrm {{\mu }m}$. This effect is, once again, given by the decreasing rotation angle of the thin sample and, even with the similar absolute error, its relative magnitude grows quickly as the rotation angle drops with the increasing wavelength and temperature.
Fig. 4. The Verdet constant of $\mathrm{{{Dy}_2}{{O}_3}}$ ceramics as a function of temperature at three fixed wavelengths $\lambda _1=0.63\,\mathrm {{\mu }m}$, $\lambda _2=1.0\,\mathrm {{\mu }m}$, and $\lambda _3=2.0\,\mathrm {{\mu }m}$. The measured data are marked by the crosses, while the solid lines represent the fitted functions according to the Eq. (4). The relative deviation of the fitted functions from the data $\Delta V$ is also shown in the bottom graph.
The last step is the construction of the fitted function of temperature-wavelength dependent data. The complete set of data has been fitted by the formula (2). The initial guesses of 13 fitting parameters have been estimated from the above obtained parameters of single-variable fits. The relations between $A,B,C,D$ and $E,F,G,H,I$ parameters are given as $E_j\left (T\right )=\frac {A_j}{T-T_w},$ $F_j\left (T\right )=B_j+\frac {C_j}{T-T_w},$ $G\left (T\right )=\frac {D}{T-T_w},$ and $H\left (\lambda \right )=\mathop{\sum}\limits_j\left [A_j\frac {\lambda _{0j}^3\lambda ^2}{\left (\lambda ^2-\lambda _{0j}^2\right )^2}+C_j\frac {\lambda _{0j^2}}{\lambda ^2-\lambda _{0j}^2}\right ]-D,$ $I\left (\lambda \right )=\mathop{\sum}\limits_jB_j\frac {\lambda _{0j}^2}{\lambda ^2-\lambda _{0j}^2}.$
As an example, let us consider the fitting parameters $E_2\left (T\right )$ and $F_2\left (T\right )$ corresponding to the transition close to $1.2\,\mathrm {{\mu }m}$. The results for these parameters are shown in the Fig. 5 for all temperatures. These data were fitted by the appropriate functions in order to obtain the estimations of the constant parameters $A_2$, $B_2$, $C_2$, and $T_w$. The standard algorithms for nonlinear least-square fitting have been used for this purpose. In these two cases the fitted functions $E_2\left (T\right )=0.064/\left (T+21.1\right )$ and $F_2\left (T\right )=-5.2+4356/\left (T+11.3\right )$ have been obtained.
Fig. 5. Example of the parameters obtained from the 1-D fitting of the $\lambda$-dependent data at fixed temperature.
These initial guesses of $A,B,C,D,\lambda _0,$ and $T_w$ parameters were used for the fitting of the data by nonlinear least-square method using the trust-region optimization algorithm to obtain the fitted function of wavelength and temperature. The bisquare robust weight function has been used for the suppression of the influence of the points very close to the transitions which were not considered in the function (2) e.g. the points close to $\lambda =0.8\,\mathrm {{\mu }m}$ or $\lambda =0.9\,\mathrm {{\mu }m}$, which were not excluded from the analysis by the transmission criterion. These points do not follow the dependence prescribed by the function (2).
The resulting data for the wavelength range $0.6-2.3\,\mathrm {{\mu }m}$ and temperature range $20-297\,\mathrm {K}$ together with the fitted function (2) is shown in the Fig. 6.The data obtained by the spectrometer 1 (S1) ranging from $0.6-0.94\,\mathrm {{\mu }m}$ is marked by the black squares, the data measured by the spectrometer 2 (S2) ranging from $0.94-2.3\,\mathrm {{\mu }m}$ is marked by the black circles, and the data which was omitted because of the signal insufficiency (LS) caused by the high absorption of the sample, according to the criteria mentioned in the paragraph about the wavelength-only dependent data, are represented by the red crosses. The fitted function (FF) is shown as a colored mesh. The fitting parameters including their 95% confidence bounds are collected in the Table 1.
Fig. 6. The temperature-wavelength dependence of $\mathrm{{{Dy}_2}{{O}_3}}$ ceramics Verdet constant. The measured data are highlighted by the markers: squares for the data measured by spectrometer (S1) in the spectral range $0.6-0.94\,\mathrm {{\mu }m}$, circles for the data captured by the spectrometer (S2) in the spectral range $0.94-2.3\,\mathrm {{\mu }m}$, and red crosses (LS) for the data which were omitted due to signal insufficiency in the corresponding spectral range. The mesh plot represents the fitted function (FF).
Table 1. Fitted parameters obtained from the 2-D fitting.
The relative deviation of the fitted function from the measured data is shown in the Fig. 7. Once again, the spectral regions with high absorption are highlighted by the red rectangles. By the direct comparison of the Fig. 7 with the map of measurement error Fig. 2 b) it can be observed that the areas of the high deviation from the fitted function corresponds to the areas with increased measurement error caused mostly by the small rotation angle. Another band of high relative deviation can be observed around the wavelength of $1.4\,\mathrm {{\mu }m}$ which is the around the wavelength, where the Verdet constant crosses zero, thus pushing the relative error to the fast growth around this wavelength.
Fig. 7. The relative deviation of the fitted function (2) from the experimental data. The areas of lowered signal are highlighted by the red rectangles. The areas with increased relative deviation corresponds to the wavelengths and temperatures with the Faraday rotation angle comparable to the measurement precision, usually around or less than 1 angular degree.
The transmission profile of the $\mathrm{{{Dy}_2}{{O}_3}}$ ceramics implies its usability mainly in the spectral region between $1.8\,\mathrm {{\mu }m}$ and $2.3\,\mathrm {{\mu }m}$. The Verdet constant values for some of the selected laser wavelength mainly from the mentioned region with He:Ne laser as a reference are collected in Table 2.
Table 2. List of the Verdet constant values for the chosen wavelengths.
4. Conclusions
The Verdet constant of the dysprosium sesquioxide $\mathrm{{{Dy}_2}{{O}_3}}$ based transparent ceramics has been measured as a function of both wavelength and temperature. The data have been obtained in the spectral region covering the range from $0.6\,\mathrm {{\mu }m}$ up to $2.3\,\mathrm {{\mu }m}$ for various temperatures within the range from cryogenic $20\,\mathrm {K}$ up to room temperature $297\,\mathrm {K}$. The obtained set of data allowed to fit the data with the approximate temperature-wavelength dependent function represented by Eq. (2) and deduce some material parameters of $\mathrm{{{Dy}_2}{{O}_3}}$ like the wavelength of the lowest $4f\rightarrow 5d$ transition $\lambda _{01}=184.0\,\mathrm {nm}$, which is in very good agreement with the values which can be found in the literature e.g. $\lambda _{01}=185.2\,\mathrm {nm}$ for $\mathrm{{{Dy}_2}{{O}_3}}$ doped borate glasses [30]. Another significant electronic transitions $^6H_{15/2}\rightarrow {}^6F_{11/2}+^6H_{9/2}$ and $^6H_{15/2}\rightarrow {}^6H_{11/2}$ have been localized at $\lambda _{02}=1250.5\,\mathrm {nm}$ and $\lambda _{03}=1606.2\,\mathrm {nm}$, respectively. The Curie-Weiss temperature has been also obtained from the measurement as $T_w=-18.1\,\mathrm {K}$. Once again the obtained value is very close to the values which has been obtained from the magnetic susceptibility measurements and are usually referred to be $\approx -17\,\mathrm {K}$ [28,33]
To the best of our knowledge, the multi-transition model has been used to fit the temperature-wavelength dependent Verdet constant data for the first time and can provide a formula for the evaluation of the Verdet constant for any combination of input wavelength and temperature. The formula seems to be valid on the whole wavelength range from $0.6\,\mathrm {{\mu }m}$ up to $2.3\,\mathrm {{\mu }m}$ with exception of the bands of high absorption of $\mathrm{{{Dy}_2}{{O}_3}}$ located in $0.785\,\mathrm {{\mu }m}$ – $0.805\,\mathrm {{\mu }m}$, $0.87\,\mathrm {{\mu }m}$ – $0.92\,\mathrm {{\mu }m}$, $1.19\,\mathrm {{\mu }m}$ – $1.32\,\mathrm {{\mu }m}$, and $1.57\,\mathrm {{\mu }m}$ – $1.69\,\mathrm {{\mu }m}$.
Funding
European Regional Development Fund (ERDF) and the state budget of the Czech Republic project HiLASE CoE (CZ.02.1.01/0.0/0.0/15_006/0000674); Japan Society for the Promotion of Science (JSPS) KAKENHI (18H01204); Horizon 2020 Framework Programme (H2020) (739573); National Institute for Fusion Science (NIFS) (KEIN1608); Ministerstvo Školství, Mládeže a Tělovýchovy (MŠMT) of the Czech Republic Programmes NPU I Project (LO1602); Large Research Infrastructure Project (No. LM2015086).
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