## Abstract

Verdet constants of beta-barium borate (β-BaB_{2}O_{4}, BBO) and lead molybdate (PbMoO_{4}, PMO) crystals are measured by comparing with that of terbium-doped glass. Measurement uncertainties caused by light intensity fluctuation, responsiveness drift of photo-detector and unstable amplifying ratio of signal processing circuits can be suppressed by comparative measurement method. Verdet constants of the BBO and PMO crystals respectively are 5.61 ± 0.17 rad.T^{−1}.m^{−1} and 51.7 ± 1.5 rad.T^{−1}.m^{−1} for light wavelength of 635nm.

© 2015 Optical Society of America

## 1. Introduction

A lot of optical crystals have both electrooptic and magnetooptic effects, such as quartz (SiO_{2}), bismuth germanate (Bi_{4}Ge_{3}O_{12}, BGO) and bismuth silicate oxide (Bi_{12}SiO_{20}, BSO) crystals, etc. Such crystals can be used for optical sensors to measure multiple variables, e. g. optical voltage, current and electric-power sensors, magnetooptic sensor with electrically adjustable sensitivity, and magnetooptic sensor based on electrooptic compensation [1–6]. Therefore, it is meaningful to find more other crystals with both electrooptic and magnetooptic effects.

In principle, all optical crystals including crystals with electrooptic effect or electrogyration effect can exhibit Faraday effects, but only some of them have considerable Verdet constants. Beta-barium borate (β-BaB_{2}O_{4}, BBO) crystal exhibits electrooptic Pockels effect, lead molybdate (PbMoO_{4}, PMO) crystal exhibits electrogyratory effects [7–9], however, up to now, magnetooptic Faraday effects and the Verdet constants of the two crystals have not yet been experimentally investigated.

Some optical methods to measure Verdet constant have been reported in literatures, e.g. the combined ac and dc measurements, using small ac magnetic field and lock-in amplifier, polarization-stepping techniques and phase-shifting interferometry, phase-sensitive low-coherence interferometry, auto-balanced photo-detection method, and analyses on Stokes parameters, etc [10–15]. The effect of light intensity fluctuation on measurement result has been removed by use of previous measurement schemes, but it is difficult to remove the effects of unstable magnetic field distribution, temperature drift of output signal, and electronic noises from photo-detector and signal processing circuits.

In this paper, Verdet constants of the BBO and PMO crystals will be experimentally investigated by use of comparative measurement method, in which effects of light intensity fluctuation, non-uniform magnetic field distribution, photo-detector and signal processing circuits on measurement result can be greatly suppressed.

## 2. Principle of comparative measurement method

Figure 1 illustrates the basic principle and experimental setup for Verdet constant measurement. Magnetooptic samples to be measured are inserted inside the solenoid coil. In the Cartesian coordinate o-*xyz*, if the transparent axis of one polarizer is oriented at an angle of 45° to the *x*-axis, and that of the other polarizer is parallel to the *x*-axis, then the normalized emerging light intensity propagating along the *z*-axis can be represented as [16]

*I*is incident light intensity, and

_{i}*k*is a coefficient related to light coupling loss, and

_{o}*θ*is the Faraday rotation angle. For a uniform magnetic flux density

*B*,

*θ*can be represented as [16]where

*V*is the Verdet constant of magnetooptic medium,

*L*is its length. If

*B*is generated by a solenoid coil with an applied current

*i*(

*t*), as shown in Fig. 1, then

*B*can be represented as [17]where 0.5≤

*k*≤1 is a coefficient related to the applied magnetic field distribution inside the coil,

_{m}*μ*

_{0}is the permeability of vacuum,

*μ*

_{r}is the relative permeability of magnetooptic medium in the coil, and

*μ*

_{r}≈1 for a non-ferromagnetic medium,

*n*is turns per unit length of the coil.

By use of a photo-detector and pre-amplifying circuit, the light signal is converted into an output voltage *u _{p}*(

*t*),

*i.e.*

*s*is responsiveness of the photo-detector,

_{d}*k*is amplifying ratio of the pre-amplifier,

_{a}*u*and

_{pac}*u*respectively are the ac and dc components of

_{pdc}*u*(

_{p}*t*), and they are separated by use of a high-pass (HP) filter and a low-pass (LP) filter, as shown in Fig. 1. Finally,

*u*is divided by

_{pac}*u*using an analog divider, and the output voltage

_{pdc}*u*(

_{o}*t*) can be written as

*k*,

_{h}*k*, and

_{l}*k*respectively are amplifying ratios of high-pass filter, low-pass filter, and analog divider as shown in Fig. 1, and the approximation of sin(2

_{d}*θ*)≈2

*θ*is valid for a small angle 2

*θ*. Equation (5) shows that the effects of

*s*,

_{d}*k*,

_{a}*k*and

_{o}*I*on

_{i}*u*(

_{o}*t*) can be removed by the dividing operation. Using (2) and (3), Eq. (5) can be rewritten aswhere the coefficient

*A*

_{o}is defined as

According to Eq. (6), we can measure the root-mean-square values of *u _{o}*(

*t*) and

*i*(

*t*) by experiment,

*i.e. U*

_{o}and

*I*, and calculate out the ratio of

*U*

_{o}and

*I*to obtain a slope,

*i.e.*

*k*is the slope of linear fitting line of experimental data of

_{v}*U*

_{o}versus

*I*.

In principle, we can obtain the value of *V* from Eq. (8), since *μ _{r}*,

*k*and

_{m}*L*are given, and

*A*can be calculated by using Eq. (7). However, the measurement uncertainty is still dependent on

_{o}*k*and

_{m}*A*, including the influences of applied magnetic field distribution and signal processing circuits, etc.

_{o}In order to suppress measurement uncertainties caused by *k _{m}* and

*A*, firstly we can employ a block of magnetooptic sample with given Verdet constant to obtain its corresponding slope

_{o}*k*by experiment, and

_{v0}*k*can be represented as

_{v0}*μ*

_{r}_{0}is relative permeability of the magnetooptic sample,

*k*is a magnetic field distribution coefficient related to the sample,

_{m0}*V*and

_{0}*L*respectively are given Verdet constant and length of the sample. Similarly, the slope

_{0}*k*corresponding to a magnetooptic sample to be measured can be written aswhere

_{vx}*μ*

_{r}_{x}is relative permeability of the sample,

*k*is a corresponding magnetic field distribution coefficient,

_{mx}*V*is the Verdet constant to be measured, and

_{x}*L*is the length of the sample. Finally we can obtain

_{x}*V*from the dividing operation of Eq. (10) and Eq. (9),

_{x}*i.e.*

Equation (11) demonstrates that effect of the coefficient *A _{o}* on Verdet constant measurement can be removed by the above comparative measurement method. It is difficult to completely remove the effect of non-uniform magnetic field distribution on the measurement of

*V*unless the magnetic field distribution in the solenoid is uniform enough to make

_{x}*k*≈

_{m0}*k*. However, this unwanted effect can be suppressed by the dividing operation of

_{mx}*k*and

_{m0}*k*. A lot of magnetooptic samples to be measured are non-ferromagnetic (paramagnetic, diamagnetic, or anti-ferromagnetic), in this case,

_{mx}*μ*≈1, then Eq. (11) can be approximated as,

_{rx}## 3. Experimental result and discussion

The BBO, PMO, BGO, and quartz crystal samples to be measured are available in our laboratory, which are used for optical voltage sensors in our previous research. The BBO, PMO, and quartz crystals are uniaxial, and their sizes are 3 × 3 × 20mm^{3}, 10 × 10 × 5mm^{3}, and 6 × 6 × 31.3mm^{3} along the *x-*, *y-*, and *z-*axis, respectively. The BGO crystal is isotropic and with a size of about 12 × 3 × 35mm^{3}. All the above crystal samples and the Tb-doped glass sample are non-ferromagnetic substances [21], thus *μ _{rx}*≈

*μ*≈1 and we can use Eq. (12) to measure and calculate

_{r0}*V*. E.g. the BGO crystal is diamagnetic, and the Tb-doped glass is paramagnetic.

_{x}Experimental setup is shown in Fig. 1, where LD is a collimating diode laser source with light wavelength of *λ*_{0} = 635 ± 0.1 nm, light intensity of ~1mW, and diameter of ~1mm, the extinction ratio of each prism polarizer (P1, P2) is greater than 10^{5}, ac voltage regulator with a rated power of 3kVA is used as the ac current supply, a resistor with resistance of 10Ω and rated power of 2kW is cascaded with the ac voltage regulator and a solenoid coil to generate magnetic field. The total turns of the coil are 266, and *n*≈6045 turns/m, the length of its axis is ~44.0mm, and the inner diameter is ~20.5mm. The modulated light is transmitted to photo-detector (PD) through a piece of multimode plastic optical fiber (POF). According to Eq. (1), the angle *θ* can be demodulated only from the emerging light intensity *I _{ox}*. However, the light is a linearly polarized light, and we sometimes need to adjust its azimuth angle during measurement, e.g. when the quartz crystal exhibits an optical activity of 18.7° × 31.3≈585° for light wavelength of 635nm. This azimuth variation will lead to extra measurement uncertainty if the light is directly coupled into the photo-detector via free space, because the responsiveness of silicon-based photo-detector is sensitive to incident light polarization state. The application of the POF in Fig. 1 can greatly suppress the above measurement uncertainty since the multimode fiber can depolarize the light.

At middle point of the coil axis, *i. e. z* = 22mm, magnetic flux density *B* has been measured several times as a function of applied current *I* by use of a Gauss-meter (Model: Lakeshore 421). Typical experimental data in the current range of 4A and their linear fitting line are shown in Fig. 2, and the corresponding equation is *B*≈(77.3 ± 0.3)*I*−0.16(Gs).

In addition, magnetic flux density distribution *B*(*z*) along the *z*-axis has also been measured by using the above Gauss-meter. Typical experimental data of *B*(*z*) versus position *z* are shown in Fig. 3 for an applied current *I* = 1.88A, and the corresponding equation of polynomial fitting curve is

*z*<44mm, the position z is calibrated by a ruler with minimum resolution of 1mm, the maximum value of

*B*(

*z*) is

*B*

_{max}(22)≈145.3 ± 1.7Gs at

*z*≈22mm.

_{m}If the middle point of magnetooptic sample with a length of *L _{x}* is located at

*z*≈22mm, then the coefficient

_{m}*k*can be approximately evaluated by using the following formula,

_{mx}Verdet constant of the Tb-doped glass is given as *V*_{0} = −95.7rad⋅T^{−1}⋅m^{−1} for light wavelength *λ*_{0} = 632.8nm (from He-Ne laser) at room temperature of 20°C (manufactured by Xi'an Aofa Optoelectronics Technology Inc. www.xaot.com). It is a cylinder sample with a diameter of *ϕ* = 6mm and a length of *L*_{0} = 30mm. According to Eq. (14), and *L _{x}* =

*L*

_{0}= 30mm for the Tb-doped glass sample, we can get

*k*

_{m}_{0}≈0.9458 ± 0.0161. Here, measurement uncertainty of

*k*

_{m}_{0}is mainly caused by measurement errors of

*B*and

*z*, and assuming that the maximum position shift of crystal in the middle point of the axis is within 21mm<

*z*<23mm. We have performed the measurement of

_{m}*U*

_{o}versus

*I*several times. Typical experimental data together with their linear fitting line are shown in Fig. 4, and the corresponding slope of the fitting line is

*k*

_{v}_{0}≈400.9 ± 1.7(mV/A). Average temperature in the laboratory during doing experiments was about 20°C (From March 17 to April 11, 2011).

Similarly, typical experimental data of *U*_{o} versus *I* and their linear fitting line for the BBO crystal are also shown in Fig. 4, the slope is *k _{v}*

_{1}≈16.2 ± 0.2(mV/A).

*L*=

_{x}*L*

_{1}= 20 ± 0.01mm for the BBO crystal sample, then we can calculate out

*k*

_{m}_{1}≈0.9776 ± 0.0171 according to Eq. (14). Verdet constant of the BBO crystal

*V*

_{1}can be calculated according to Eq. (12). If we omit the uncertainties of

*μ*and

_{rx}*V*, then relative error of

_{0}*V*

_{1}can be calculated by [18]

*V*

_{1}≈5.61(1 ±

*δ*

_{1})≈5.61 ± 0.17 rad⋅T

^{−1}⋅m

^{−1}.

We have also measured Verdet constants of the PMO, quartz (SiO_{2}) and BGO crystal samples for *λ*_{0} = 635nm by using the above comparative measurement method, related experimental results are listed in Table 1, where *V*_{PMO}≈51.7 ± 1.5rad⋅T^{−1}⋅m^{−1} for the PMO crystal. Verdet constants of the BBO, PMO, and quartz crystals were measured along their principal dielectric axes, *i.e.* the *z-* or *c*-axes. *V*_{BGO}≈32.2 ± 0.81rad⋅T^{−1}⋅m^{−1} for the BGO crystal, which approximates to 30.8rad⋅T^{−1}⋅m^{−1} in literature [19]. Similarly, *V*_{sio2}≈4.88 ± 0.14rad⋅T^{−1}⋅m^{−1} for quartz crystal, which also approximates to *V*_{sio2} = 0.017' Gs^{−1}⋅cm^{−1}≈4.945 rad⋅T^{−1}⋅m^{−1} for *λ*_{0} = 589.3nm reported in [20]. The differentials between the literature readings for the above two crystals and the readings from above measurement results are mainly caused by different crystal samples and different light wavelengths and spectrums of lasers, etc.

According to Eq. (11), effects of interrogation light intensity fluctuation, responsiveness drift of photo-detector, and unstable amplifying ratio of signal processing circuits on measurement result can be removed, and effect of non-uniform magnetic field distribution can also be decreased by the dividing operation of *k _{m}*

_{0}and

*k*. Measurement uncertainties are mainly induced by the two coefficients

_{mx}*k*

_{m}_{0}and

*k*, and the two slopes

_{mx}*k*and

_{v0}*k*. During the measurement, locations of different crystal samples in the solenoid coil should be identical. Otherwise, the location errors will lead to additional measurement uncertainties of

_{vx}*k*

_{m}_{0}and

*k*. Assuming that location errors of the crystal samples along the axis are less than 2mm, then the related uncertainties are very small and have been included in the above calculation of measurement uncertainties.

_{mx}In addition, the difference of interrogation light wavelengths and ambient temperature drift of crystal samples will lead to measurement uncertainties which need to be further evaluated. The interrogation light wavelengths respectively are *λ*_{0} = 632.8nm for the given Verdet constant of the Tb-doped glass as reference and *λ*_{0} = 635 ± 0.1nm for the samples to be measured. This error can be removed by using a given Verdet constant of the Tb-doped glass corresponding to *λ*_{0} = 635nm as reference, however this parameter is not available at present. Average room temperature was 20°C, but a small ambient temperature drift was inevitable during experiments due to electrocaloric effect of the coil under certain applied current. This measurement uncertainty can be suppressed by using a coil with small electrocaloric effect, such as a coil wrapped by use of enameled copper wire with larger diameter.

Further research may focus on the above-mentioned measurement uncertainty analyses and calculation, measurement of the dispersion property of Verdet constants for the BBO and PMO crystals, and their temperature dependences.

Since the BBO and PMO crystals have considerable magnetooptic Faraday effects, in principle, they can be used for optical magnetic field and current-related sensors, e.g. the single BBO crystal can also be used for the simultaneous measurement of current and voltage, optical electric-power sensor, and magneto-optic sensor with electrically adjustable sensitivity or electrooptic compensation, etc. like as the optical sensors reported in [1–6].

## 4. Conclusion

By comparing with the given Verdet constant of a Tb-doped glass, we have measured Verdet constants of the BBO and PMO crystal samples. For light wavelength of 635nm, Verdet constant of the BBO crystal is 5.61 ± 0.17 rad⋅T^{−1}⋅m^{−1} and that of the PMO crystal is 51.7 ± 1.5 rad⋅T^{−1}⋅m^{−1}. The effects of light intensity fluctuation, responsiveness variation of photo-detector, and unstable amplifying ratio of signal processing circuits on measurement result can be suppressed by the comparative measurement method. Magnetooptic effects in the BBO and PMO crystals have potential applications to novel optical magnetic field and current-related sensors.

## Acknowledgments

Authors would like to sincerely thank two anonymous reviewers and Prof. Yanbiao Liao of Tsinghua University for their helpful and constructive comments and suggestions, and Prof. Mingrong Zhang of the Beijing Glass Research Institute, Dr. Xuan Zhang of the Bright Crystals Technology, Inc. Prof. Rong Zeng of Tsinghua University, and Dr. Yanshun Zhang of Beihang University, for their helps in experiments.

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