2 Laser measurement of the Verdet constant temperature dependence of Ho₂O₃

A simplified scheme of the experimental setup used for the laser measurement of the Verdet constant temperature dependence is depicted in Figure 1. During the experiment a laser probe beam ( $\unicode[STIX]{x1D706}=1.064~\unicode[STIX]{x03BC}\text{m}$ ) was linearly polarized by an input Glan polarizer and propagated through a measured material sample, which was placed inside a cryostat enabling temperature control. An external magnetic field was applied on the sample, inducing Faraday rotation of the propagating probe beam. With the polarization plane rotated, the beam exited the sample and was further processed by a detection system. The detection system consisted of an output Glan polarizer (the analyzer), which can be arbitrarily rotated around the optical axis, and a power meter.

Figure 2. An example of measured data gathered for one temperature step (at 300 K).

The corresponding transmitted laser power was detected in a large number (a few hundreds) of analyzer angular positions with a $0.1^{\circ }$ angular step. The measurement was performed for several temperatures ranging from 305 down to 15 K. For each temperature $T_{i}$ , the measurement was taken twice, with and without an external magnetic field applied. An example of measured data gathered for one temperature step is depicted in Figure 2. According to the Malus law, the power data gathered in each individual measurement was fitted with the following cosine squared function

(1) $$\begin{eqnarray}P(\unicode[STIX]{x1D6FC},T_{i})=A_{1}(T_{i})\cos ^{2}[\unicode[STIX]{x1D6FC}+A_{2}(T_{i})]+A_{3}(T_{i}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ denotes the analyzer angular position, $T_{i}$ is the temperature, $P(\unicode[STIX]{x1D6FC},T_{i})$ refers to the recorded power data, and $A_{1-3}(T_{i})$ are the fitting parameters. The polarization plane rotation angle $\unicode[STIX]{x1D703}(T_{i})$ was directly calculated as a difference between the initial phase shifts $\unicode[STIX]{x1D703}(T_{i})=A_{2N}(T_{i})-A_{2B}(T_{i})$ obtained for the cases with the magnetic field – $A_{2B}(T_{i})$ – and without it – $A_{2N}(T_{i})$ .

The Verdet constant $V(T_{i})$ was obtained from the rotational angle $\unicode[STIX]{x1D703}(T_{i})$ using the following expression

(2) $$\begin{eqnarray}V(T_{i})=\frac{\unicode[STIX]{x1D703}(T_{i})}{\int _{-L/2}^{L/2}B(l)\,\text{d}l}=\frac{\unicode[STIX]{x1D703}(T_{i})}{B_{\text{eff}}L},\end{eqnarray}$$

in which $B(l)$ refers to the known longitudinal distribution of applied magnetic field^{[Reference Slezak, Yasuhara, Lucianetti and Mocek17]} and $L$ is the sample length. $B_{\text{eff}}$ denotes the effective magnetic field magnitude which was obtained from the distribution by numeric integration over the length of the sample. Although an effort was made to place the sample exactly in the maximum of $B(l)$ of the permanent magnet, an uncertainty of $B_{\text{eff}}$ , given by the magnetic field measurement error as well as by an unintended misplacement of the sample, needs to be considered. The respective values derived for the uncertainty are listed in Table 1. The experimental data obtained for Verdet constant temperature dependence was further fitted with a simplified model function^{[Reference Slezak, Yasuhara, Lucianetti and Mocek17]}

(3) $$\begin{eqnarray}V(T)=\frac{G}{T-T_{c}}+H,\end{eqnarray}$$

where $G$ , $H$ and $T_{c}$ represent fitting parameters.

Table 1. Material parameters and the fitted parameters.

Apart from Ho₂O₃ ceramics sample, a TGG ceramic sample from Konoshima Chemical, Co., Ltd. was also measured using the above described experimental method. TGG ceramics represent a valid comparative reference of our measurement since it is nowadays the most widely used magneto-optical material for wavelengths around $1~\unicode[STIX]{x03BC}\text{m}$ . The experimental results obtained for both samples (Ho₂O₃ and TGG) are depicted in Figure 3, along with the fitting curves and other comparative measurements. The material parameters and the obtained fitting parameters are listed in Table 1. The relative error of the measurement was 4.17% for the Ho₂O₃ ceramics sample, and 4.18% for the TGG ceramics sample, only weakly changing across the whole measured temperature range.

Figure 3. Experimental results for laser measurement of the Verdet constant temperature dependence of Ho₂O₃ ceramics.

Our experimental results obtained for the TGG sample are in good agreement with the previously reported measurements^{[Reference Yasuhara, Tokita, Kawanaka, Kawashima, Kan, Yagi, Nozawa, Yanagitani, Fujimoto, Yoshida and Nakatsuka18]}, as well as with our own unpublished laser measurements of the same TGG ceramics sample. The only exception in agreement can be observed at the lowest temperatures. However, it should be noted that in Ref. [Reference Yasuhara, Tokita, Kawanaka, Kawashima, Kan, Yagi, Nozawa, Yanagitani, Fujimoto, Yoshida and Nakatsuka18] the laser wavelength was $1.053~\unicode[STIX]{x03BC}\text{m}$ , and this difference can account for the slight discrepancy in the Verdet constant at the lowest temperatures. The other possible cause of the disagreement between the new and the unpublished measurements can be accounted to a larger value of $\text{d}V/\text{d}T$ at lower temperatures, causing a larger absolute error of the Verdet constant, as a consequence of temperature fluctuations during the measurement. A slightly different mutual position of the sample and the magnet between the compared measurements also greatly contributes to the discrepancy at low temperatures. However, this issue may be overcome in the future via more mechanically sophisticated magnet alignment, which is currently under investigation.

The Verdet constant temperature dependence of Ho₂O₃ at $1.064~\unicode[STIX]{x03BC}\text{m}$ is yet unknown and therefore no comparison can be made. Nevertheless, considering the good agreement of the TGG results, the presented results for Ho₂O₃ seem to be valid.

3 Verdet constant dispersion of Ho₂O₃ ceramics-based magneto-optical materials

Recent publications in the field of magneto-optical materials research showed that doping of a magneto-optical material by RE elements can boost the Faraday rotation effect in the host material^{[Reference Zheleznov, Starobor, Palashov, Lin and Zhou22–Reference Chen, Yang, Wang and Yin24]}. The Verdet constant dispersion of Ho₂O₃ doped with Tb³⁺ or Ce³⁺ ions was measured in order to investigate the impact of doping on the Faraday effect in Ho₂O₃.

A simplified scheme of the experimental setup used for measurement of the Verdet constant dispersion is depicted in Figure 4. The experimental setup, as well as the experimental procedure, was similar to the laser measurement of temperature dependence of the Verdet constant, described in Section 2. There were a few differences which will be briefly explained in the next paragraph. The full description of the Verdet constant dispersion measurement is given in Ref. [Reference Vojna, Yasuhara, Slezak, Muzik, Lucianetti and Mocek15].

Figure 4. Simplified scheme of the experimental setup used for measurement of the Verdet constant dispersion.

Instead of a laser, a broadband radiation source (NKT Photonics SuperK Compact) was used for the Verdet constant dispersion measurement. The samples under investigation were no longer kept in the cryostat, but fixed in a mechanical holder outside it. The detection system needed to be modified as well, now consisting of the rotating output Glan polarizer (the analyzer), a silver-coated diffusive reflector, an integration sphere, and a spectrometer. The diffusive reflector helped to suppress the polarization dependence of the fiber coupling, through which the captured signal was transmitted to spectrometer (Ocean Optics HR4000CG-UV-NIR). The effective spectral range of the measurement was 0.5– $1~\unicode[STIX]{x03BC}\text{m}$ . The transmitted intensity spectra were captured using a full-circle rotation of the analyzer, with a $2^{\circ }$ angular step, giving a total number of 181 angular points for each wavelength detected with the spectrometer. The intensity data was, again, gathered for two distinctive cases – with and without an external magnetic field. The measured data for each wavelength $\unicode[STIX]{x1D706}_{i}$ was fitted with the following cosine squared function

(4) $$\begin{eqnarray}I_{n}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}_{i})=\cos ^{2}[\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FC}_{0}(\unicode[STIX]{x1D706}_{i})],\end{eqnarray}$$

where $I_{n}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}_{i})$ denotes the measured intensity data normalized to unity, $\unicode[STIX]{x1D6FC}$ is the analyzer position, and $\unicode[STIX]{x1D6FC}_{0}(\unicode[STIX]{x1D706}_{i})$ represents the fitting parameter, the initial phase shift. The polarization plane rotation angle $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D706}_{i})$ was directly calculated as the difference between the initial phase shifts $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D706}_{i})=\unicode[STIX]{x1D6FC}_{0N}(\unicode[STIX]{x1D706}_{i})-\unicode[STIX]{x1D6FC}_{0B}(\unicode[STIX]{x1D706}_{i})$ obtained for the cases when the external magnetic field was applied - $\unicode[STIX]{x1D6FC}_{0B}(\unicode[STIX]{x1D706}_{i})$ - and when it was not - $\unicode[STIX]{x1D6FC}_{0N}(\unicode[STIX]{x1D706}_{i})$ . The Verdet constant dispersion $V(\unicode[STIX]{x1D706}_{i})$ was obtained from the rotational angle $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D706}_{i})$ using an expression analog to the expression (2), namely $V(\unicode[STIX]{x1D706}_{i})=\unicode[STIX]{x1D703}(\unicode[STIX]{x1D706}_{i})/(B_{\text{eff}}L)$ . The measurement was performed at room temperature.

Table 2. Material parameters for the samples under investigation for the Verdet constant dispersion measurement.

Figure 5. The experimental results obtained for the Verdet constant dispersion of Ho₂O₃ ceramics-based magneto-optical materials.

Specific parameters of the samples under investigation are listed in Table 2. The experimental results obtained for the Verdet constant dispersion $V(\unicode[STIX]{x1D706}_{i})$ are depicted in Figure 5. Figure 5 further contains values of the Verdet constant obtained by the laser measurement. The relative error of the measurement was around 4% in the whole spectral range of the measurement. At this point, it is worth to remark that the Ho₂O₃ material possesses several electronic transitions within the measured spectral range^{[Reference Furuse and Yasuhara27]}. A proper model function appropriately describing the obtained Verdet constant dispersion is currently under investigation; therefore, only the raw data are presented in this paper, which, for the purposes of a preliminary doping impact investigation, is satisfactory. Examining the experimental results we may conclude that doping of Ho₂O₃ by RE elements has no significant impact on the Verdet constant. However, it needs to be taken into account that: (a) the doping level of the investigated samples was relatively low, (b) the investigated samples were relatively shorter, thereby inducing only a small Faraday rotation of the probe beam, and finally (c) the optical quality of the provided material samples was limited. The in-line transmittance of the material samples was not investigated within the scope of this work; however, as it was mentioned in the previous work^{[Reference Furuse and Yasuhara27]}, the in-line transmittance of the nondoped Ho₂O₃ ceramics sample at $1~\unicode[STIX]{x03BC}\text{m}$ wavelength is ${\sim}60\%$ . The optical quality of the doped samples was comparable to the nondoped sample from the previous work.

After a significant improvement of provided Ho₂O₃ samples’ optical quality, longer samples may be produced and then a more comprehensive characterization of RE elements-doped Ho₂O₃ samples can be made.