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We report the complete determination of the polarization changes caused in linearly polarized incident light due to propagation in a magneto-optically active terbium gallium garnet (TGG) single crystal, at temperatures ranging from 6.3 to 300 K. A 28-fold increase in the Verdet constant of the TGG crystal is seen as its temperature decreases to 6.3 K. In contrast with polarimetry of light emerging from a Faraday material at room temperature, polarimetry at cryogenic temperatures cannot be carried out using the conventional fixed polarizer-analyzer technique because the assumption that ellipticity is negligible becomes increasingly invalid as temperature is lowered. It is shown that complete determination of light polarization in such a case requires the determination of its Stokes parameters, otherwise inaccurate measurements will result with negative implications for practical devices.
© 2013 Optical Society of America
1. Introduction
The Faraday effect refers to the rotation of the plane of polarization of linearly polarized light when it propagates through a magneto-optically active material. Magneto-optical activity is induced in the material by applying a magnetic field parallel to the direction of light propagation. The rotation occurs because the material has different refractive indexes for the left hand circularly polarized (LHCP) and right hand circularly polarized (RHCP) propagating eigen-modes resulting in different phase velocities for these modes. The rotation, commonly referred to as Faraday rotation, is given by [1–3]
where B is the applied magnetic flux density, d is the distance travelled by light through the material parallel to the magnetic field and the constant of proportionality V is called the Verdet constant. The effect is illustrated in Fig. 1.
Fig. 1 Faraday effect: The rotation of the plane of polarization of linearly polarized light on passing through a magneto-optically active material.
Devices employing the Faraday effect have found extensive applications in optical isolators for laser systems [3, 4], in optical circulators for fiber optic systems [5], in optical modulators [2, 6] and in current and magnetic field sensors [7–9]. The principal figure of merit for a Faraday element in all of these applications is the magnitude of the Verdet constant. A large Verdet constant is desirable as it leads to large rotation per unit length and applied field. For paramagnetic materials such as the archetypal terbium gallium garnet (TGG), the Verdet constant has been measured to increase by lowering the temperature of the Faraday element, owing to the Verdet constant’s linear dependence on the material’s magnetic susceptibility [10].
This observation has prompted interest into determining the polarization characteristics of light emerging from a Faraday element as a function of temperature. Of particular interest is polarimetry at cryogenic temperatures. For example, Davis and Bunch reported a large enhancement in Faraday rotation in terbium-doped glass as the temperature was decreased to 20 K [11]. Barnes and Petway investigated both the wavelength and temperature dependence of the Verdet constant of TGG in the 266–353 K temperature range, again reporting an increased rotation at lower temperatures [12]. More recently, Yasuhara et al. measured the temperature dependence of Verdet constant in both ceramnics and single crystals, reporting a 40 times increase in the Verdet constant for the ceramics as temperature was decreased from 300 to 7.8 K [13]. In all of these cases cases the authors inferred a proportional relationship between the Verdet constant and the reciprocal of the temperature. Furthermore, in each case it was assumed that the ellipticity of light emerging from the Faraday element was negligible and the application of magnetic field solely imparted a pure rotation of the plane of linearly polarized incident light. If this assumption is indeed true, the conventional fixed polarizer-analyzer technique (Section 2.2), employed in these earlier works, can be used to quantify the Faraday rotation.
As we show in the following sections, while it may be valid to assume that the ellipticity of light emerging from a paramagnetic Faraday element is negligible at room temperature, the assumption becomes increasingly invalid as the element’s temperature is decreased. Polarimetry of light in such a case cannot be accurately carried out using a fixed polarizer-analyzer combination. In Section 2, we build up on this argument theoretically and show that accurate polarimetry in this context entails the complete measurement of the Stokes parameters [14]. In Sections 3 and 4, we outline the experimental procedure used to determine the variation of Stokes parameters of light emerging from an active TGG crystal at temperatures ranging from 6.3 to 300 K and report the results obtained. We compare (and contrast) these results with those obtained in earlier reports by naively employing the fixed polarizer-analyzer setup. Our results are significant because in order to assess the performance of a Faraday device at particular temperature, in all the aforementioned applications, accurate determination of the output polarization state is essential.
2. Theory
2.1. Faraday effect and magnetic circular dichroism
Magneto-optical phenomena are embodied by the presence of off-diagonal, magnetization dependent terms in the dielectric tensor of a material. For a crystal like TGG that shows uniaxial anisotropy at zero magnetization, the dielectric tensor is given by
for the longitudinal geometry shown in Fig. 1. In Eq. (2) Q = Q′ + iQ″ is the complex Voigt parameter that is proportional to the magnetization of the material and εxx and εzz are the dielectric constants for the uniaxial crystal when no magnetization (spontaneous or induced) is present [15–18]. The presence of Q′ is responsible for magnetic circular birefringence (MCR) producing different phase velocities for the two eigen-modes, resulting in rotation of the the plane of linearly polarized light, given by Eq. (1). The term Q″ on the other hand represents magnetic circular dichroism (MCD), producing different extinction coefficients for the two eigen-modes and introduces ellipticity. If the light incident on the material is linearly polarized along the x-axis, then to first order only the rotation θ and ellipticity ψ are given by [19, 20] where is the refractive index along the x and y axes at no field, ω is the light frequency and c is the speed of light in free space.For most paramagnetic Faraday elements at room temperature Q″ is negligible. This is why the conventional description of the Faraday effect in literature (as outlined in Section 1) only takes rotation into account. However, since the magnetization (and hence Q′ and Q″) of a paramagnetic material such as TGG varies inversely with temperature, ellipticity changes in incident light have to be taken into account in a Faraday effect measurement at cryogenic temperatures. It is worth clarifying, however, that Eq. (3) and (4) capture only first order effects of the magnetization on θ and ψ assuming that the magneto-optical parameters Q′ and Q″ are small valued. Therefore, these equations only capture magneto-optical effects at temperatures high enough for the small Q′ and Q″ approximation to remain valid [19, 20]. While it is intuitive to expect enhancement in both rotation and ellipticity as the temperature is lowered, the one to one correspondence between θ and Q′ and ψ and Q″ will not be true at cryogenic temperatures. In this regime, the dependence of θ and ψ on the magnetization is expected to have different and more complicated expressions.
2.2. Polarimetry using a fixed polarizer-analyzer setup
It is well known that for a Faraday element that causes only rotation (ψ is negligible) the simplest way to measure this rotation is to use two linear polarizers as shown in Fig. 2. The first polarizer P transmits x-polarized light and the second polarizer A, called the analyzer, has its transmission axis fixed at an angle of 45° with the x-axis. The intensity of light measured by the photodetector is given by Malus’s law [21],
using which the Faraday rotation θ can be obtained. In Eq. (5),I0 is the intensity of light measured when the transmission axes of P and A are parallel to each other and Imin is the intensity measured when these axes are mutually perpendicular [13]. We refer to this scheme as the fixed polarizer-analyzer setup owing to the fact that both polarizer and analyzer transmission axes are kept fixed. As stated earlier this method loses validity at temperatures below room temperature due to its inability to measure the ellipticity ψ which is now a significant component of the magneto-optical effects.Fig. 2 In polarimetry using the fixed polarizer-analyzer setup the transmission axes of polarizer P and analyzer A are oriented at 45° to each other, allowing one to determine the rotation.
It is well nigh possible to measure ψ by making a modification in the arrangement of Fig. 2, borrowing a technique commonly used for ellipticity measurements in a magneto-optic Kerr effect experiment. The technique consists of using either a variable phase retarder or a quarter wave retarder in between the Faraday element and analyzer [22]. The addition of a retarder allows the measurement of the ellipticity from the intensity signal registered on the photodetector. However, this method is only valid if both θ and ψ are small and remain decoupled. Clearly this is not the case at low temperatures.
2.3. Determination of Stokes Parameters
The state of the polarized component of light is completely described by three parameters: the angle the major axis of the ellipse makes with the horizontal reference axis (azimuth), the ratio of the semi-minor and semi-major axes (ellipticity) and the handedness of the ellipse (LHCP or RHCP). In a Faraday experiment with horizontally polarized incident light, the azimuth and ellipticity would be given by θ and |ψ| respectively whereas the handedness would be given by the sign of ψ. By convention, θ ∈ [0, π] and . Our definition of the polarization ellipse is illustrated in Fig. 3(a). When the azimuth and ellipticity are not small valued, their determination requires the measurement of the Stokes parameters of light. In terms of the amplitudes (|Ex| and |Ey|) and phases (δx and δy) of the x and y components of the electric field, these parameters are defined as [14]
It is also important to keep track of the polarized component of light . Finally, θ and ψ can be determined from the Stokes parameters by using the relations
Here η refers to the ratio of the semi-minor axis to the semi-major axis and is obtained by taking the smaller of the two roots of the equation The sign of ψ, which determines the handedness of the ellipse, is the same as that of the Stokes parameter S.Fig. 3 (a) The convention used for determining the rotation θ, ellipticity |ψ| and handedness (LHCP or RHCP) of elliptically polarized light. (b) Configuration of the retarder W and analyzer A used to measure Stokes parameters.
The Stokes parameters of any polarization state of light can be measured by passing the light through a phase retarder W followed by a linear polarizer (analyzer) A and measuring the transmitted intensity (Fig. 3(b)). For a beam of light with Stokes parameters (I, M, C, S) this transmitted intensity is determined by using Mueller’s calculus and is given by [23]
where β and α are defined in Fig. 3(b) and δr is the phase shift caused by the retarder between electric field components along its fast and slow axes.In order to determine (I, M, C, S), we have adopted the method outlined by Berry et al. which involves the measurement of the variation in transmitted intensity IT as the retarder (β) is rotated for fixed α and δr[23]. Since Eq. (9) is a sum of even sine and cosine harmonics, IT, measured as a function of β, forms a sum of Fourier components. In our experiment, we determined the Fourier sine and cosine coefficients of the measured IT versus β data and calculated the Stokes parameters from these coefficients by using the expressions [23]
where C0, C2, C4 and S2, S4 are the Fourier cosine and sine coefficients respectively, with the subscript referring to the order of the harmonic and β0 is the initial angle between the fast axis of the retarder W and the x-axis.3. Experimental procedure
Figure 4 schematically illustrates the experimental setup. The light source was an intensity stabilized 785 nm laser diode (L785P100, Thorlabs). The transmission axis of the linear polarizer P was fixed along the x-axis. The TGG crystal (Castech Inc.) was cylindrical in shape with a length of 1 cm and diameter of 3 mm. The laser beam was incident on the circular cross-section of the cylinder along the designated optical axis (z axis) of the uniaxial crystal. The crystal was mounted inside the vacuum shroud of the cryostat (Janis Research) using a devised clamping assembly made out of 99.9% pure copper. Thermal grease (Apiezon) was applied between the sample holder and the cold-head of the cryostat in order to maximize thermal conductivity. Fused quartz windows on the sides of the vacuum shroud allowed laser beam access to the crystal inside. A computer interfaced temperature controller (Model 331, LakeShore) was used to vary the temperature from 6.3–300 K using a 25 W resistive heater while the temperature of crystal was monitored using a Gallium Arsenide diode temperature sensor (TG-120-CU-4L, LakeShore).
Fig. 4 The experimental arrangement including P = Polarizer, W = Retarder, A = Analyzer, PD = Photodetector.
An electromagnet (3470, GMW Associates) was used as the magnetic field source. The laser light passed through holes axially bored in the poles of the electromagnet. The electromagnet’s power supply (DLM60-10, Sorensen) as well as the temperature monitoring and control system were fully integrated using LABView allowing automated scanning of the magnetic field and temperature for rapid turnkey measurements. The magnetic field B was controlled by varying the current passing through the electromagnet coils. The calibration constant relating current and magnetic field was pre-determined using a gauss meter (410-SCT, LakeShore).
A 780 nm quarter wave plate (WPMQ05M-780, Thorlabs) was used as the retarder W which was mounted in a precision motorized rotation stage (PRM1/MZ8E, Thorlabs) allowing the angular displacement β to be varied with a resolution better than 0.1°. The retardation δr at 785 nm was pre-determined as follows: IT vs β was measured for light with known Stokes parameters, the Fourier coefficients for the data were determined and δr was calculated using Eq. (9) [23]. The transmission axis of the analyzer A was fixed at α = 0. The transmitted intensity IT of the laser, after passing through all the aforementioned components, was measured by a photodetector (DET 36A, Thorlabs) configured in the photoconductive configuration. The photodetector was connected to the input of a lock-in amplifier (SR830, Stanford Research Systems) allowing phase sensitive detection. The laser diode intensity was modulated by applying a 102 kHz alternating signal. The amplifier output was directly read into the computer.
4. Results and discussion
Figure 5 shows the values of θ and ψ determined from the measured Stokes parameters using Eq. (7) and (8), plotted against the applied field B in the temperature range of 6.3–300 K. The rotation θ shows a linear relationship with magnetic field B for the entire temperature range, following Eq. (1) and (3). The slope of θ vs B curve determines the Verdet constant V whose temperature variation is shown in Fig. 6. An approximately linear relationship between V and 1/T can be seen in the inset of Fig. 6. This directly follows from the Curie law dependence of magnetization on temperature and has been reported in earlier studies [11, 13]. In order to confirm that the source of this magneto-optical signal was indeed the TGG crystal, we also obtained θ vs B and ψ vs B curves, at each temperature, for the case where no TGG crystal was present in the cryostat of our experimental setup (blank reading) and the resulting magneto-optical signal was found to be negligible.
Fig. 5 The angles θ and ψ as a function of applied field B for the 6.3–300 K temperature range. For visual clarity the data for the 6.3–20 K and 40–300 K temperature ranges have been shown in separate plots. For the same reason the data for the 40–300 K temperature range are shown in 40 K increments, even though the raw data in this range were taken at 20 K increments.
Fig. 6 The Verdet constant V plotted against temperature. The inset shows a plot of V against 1/T on logarithmic axes, highlighting a linear relationship between them.
There are two significant points of interest in the ellipticity data shown in Fig. 5(c) and 5(d). First, it important to note that as temperature lowers, the magnitude of ψ becomes significant in comparison with θ and accurate polarimetry using the conventional fixed polarizer-analyzer setup (Section 2.2) is rendered invalid. Second, while for T ≥ 40 K, ψ varies linearly as B, there is significant non-linearity in the ψ vs B data for T < 40 K. This is a departure from the linear relationship between ψ and magnetization expressed by Eq. (4) and indicates the dependence of ellipticity on higher order terms of magnetization at low temperatures. Further work is required to understand the exact nature and origin of this non-linearity.
For comparison purposes, we also measured the Faraday rotation θ using Eq. (5), employing the simplistic fixed polarizer-analyzer configuration illustrated in Fig. 2. Figure 7 shows the data obtained using this method. In the 6.3–20 K temperature range, θ shows a non-linear dependence on B. This non-linearity is only visible if the the applied magnetic field B at a particular temperature is large enough and disappears for T ≥ 40 K. We can hence conclude that the ellipticity signal is intertwined and mixed into the θ measured using this naive approach. This is due to the fact that in this experimental configuration there is no way to separate the ellipticity from the rotation, resulting in significant interference of the former in the latter.
Fig. 7 Data obtained using the fixed polarizer-analyzer configuration in the temperature ranges of (a) 6.3–20 K (b) 40–300 K
This interference is accentuated at low temperatures when ellipticity becomes significant in comparison with rotation. Comparison of Fig. 5 and 7 emphasizes the importance of factoring in ellipticity effects for accurate magneto-optical characterization at low temperatures.
5. Conclusion
We have reported the complete Stokes polarimetry of light emerging from a TGG single crystal in the 6.3–300 K temperature range. The results show that at low temperatures, in addition to enhanced rotation, a significant ellipticity is also imparted by the magneto-optical crystal. We compared the results obtained through Stokes polarimetry with those obtained by using the conventional fixed polarizer-analyzer configuration employed in previous studies. The comparison indicates that in order to completely determine the output polarization characteristics of light in a Faraday experiment at cryogenic temperatures, the fixed polarizer-analyzer configuration can lead to inaccurate and sometimes misleading results and complete Stokes polarimetry is required. Our work is significant because in all the applications of the Faraday effect, a complete determination of the polarization state of output light is essential for device design. For example, in Faraday effect based optical isolators, the isolation ratio is a key figure of merit and worsens with increase in the ellipticity caused by the Faraday element. Therefore, any advantages of lowering the temperature of the isolator (higher Verdet constant) can be offset by a deterioration in the device’s isolation ratio. This means that careful quantification of both rotation and ellipticity is required for successful isolator design. Another relevant example is that of Faraday effect based optical modulators wherein an increase in ellipticity of light can lead to a loss in modulation depth. When operating such a device at low temperatures one, therefore, needs to quantify the ellipticity in order to calculate the correct modulation depth. Another interesting example is that of using graphene as a Faraday element. Since very large angular Faraday rotations are expected when using graphene, it is important to perform a complete tomography of the output light by determining its Stokes parameters [24, 25].
Needless to say, there is still some ambiguity about the exact origin of the ellipticity we have measured. For T ≥ 40 K, the primary origin of the ellipticity appears to be first-order MCD in the Faraday element, given by Eq. (4). At T < 40 K, however, there is significant non-linearly in the ψ vs B data indicating the dependence of ψ on higher order magnetization terms. Other effects such as thermal linear birefringence may also contribute to the various compounding effects contributing to the ellipticity of the output light [26].
Acknowledgments
This work was funded by the Higher Education Commission (HEC) of Pakistan under the NRPU project 2028. We would like to thank Hafiz Rizwan for help in building the cryostat’s mounts, and Rabiya Salman and Aysha Aftab for initial room temperature measurements.
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