Low temperature Voigt effect in the terbium gallium garnet crystal

Ali Akbar; Muhammad Waleed Khalid; Muhammad Sabieh Anwar

doi:10.1364/OE.25.030550

1. Introduction

A magnetic field induces a magnetization M inside a paramagnetic material changing its response to polarized light; qualifying as a preliminary investigation in the realm of magneto-optics [1]. The response manifests as a rotation θ and ellipticity χ imparted to the outgoing beam. The orientation of M relative to the wave vector k determines the nature, amplitude, reciprocity and order of magneto-optic activity. The strongest of these effects, and the most prominently studied, is undoubtedly Faraday rotation which is observed when M‖k. The effect is first order in M, originates from magnetic circular birefringence (MCB), and is non reciprocal, θ(−k) = −θ(k).

In paramagnetic crystals, there also exists a much smaller effect called the Voigt [2] or Cotton-Mouton effect [3] which registers when the magnetization is perpendicular to the wave vector, M ⊥ k. The phenomenon is reciprocal, θ(−k) = +θ(k) and quadratic, i.e., ∝ |M|². The Voigt effect originates from magnetic linear birefringence (MLB) and magnetic linear dichroism (MLD) which are defined shortly. Briefly these correspond to the real and imaginary values of the asymmetry in refractive indices Δn = n₁ − n₂ where n₁ and n₂ are defined for polarizations parallel and perpendicular to the magnetization. The smallness of the effect can be gauged from some typical values. For example, the asymmetry is reported at a mere (6.7 ± 0.5) × 10⁻¹⁵ for water vapor [4] at room temperature and 1 T and ≈ −5 × −4 for terbium aluminate garnet Tb₃Al₅O₁₂ at 4.2 K and 4 T [5]. Clearly sensitive techniques are required for recording these asymmetries.

For conductive and semiconducting materials, however relatively larger effects can be observed. For example we deduce a value of ≈ 3×10⁻⁴ from the data provided on the doped dilute magnetic semiconductor (Ga_0.98Mn_0.02)As [6]. For conductive media, the effective permittivity tensor $\tilde{ε}$ includes the conductivity tensor σ at optical frequency ω and is given by $\tilde{ε} = ε_{b} + i σ / ω$ . In these materials, the magneto-optic rotations are dominated by the field-dependent terms in σ rather than the non-conductive background part ε_b. This effect has been used to reveal magnetic anisotropies related to structural patterning in Co films [7], spin-orbit torques transferred across interfaces [8] and magnetization dynamics in Huesler compounds [9].

Voigt studies of non-conducting garnet crystals are less common [3, 5, 10]. In the present work, we report systematic investigations of MLB and MLD for terbium gallium garnet Tb₃Ga₅O₁₂ (TGG) at cryogenic temperatures (8–100 K) and variable dc magnetic fields. Although crystalline TGG has been extensively studied in the Faraday configuration from room temperatures all the way down to cryogenic temperatures [11, 12] but measurements in the Voigt geometry are lacking. The material’s prominence owes to some highly desirable features such as large Verdet constants, high thermal conductivity [13] and high damage threshold [14] making it into an almost archetypal material. Additionally, interesting quantum phenomena have also been detected in TGG which include the observation of the phonon Hall effect [15], excitation of magnetic resonances at THz [16] and microwave frequencies [17], and the observation of the inverse Faraday effect [18]. The present work complements previous studies and introduces an altogether new perspective for TGG crystals to be used in the unusual Voigt geometry, providing quantitative data on the magnetic field induced asymmetries in the refractive index, Δn, from which temperature dependent coefficients can also be deduced. The theoretical background is presented in Section 2 and Sections 3 and 4 describe the experimental work.

2. Theory

A phenomenological expression for the dielectric tensor capturing magneto-optic effects for an isotropic medium magnetized in an arbitrary direction is [19]

ε_{M O} = ε_{0} (\begin{matrix} n_{o}^{2} & - i A_{1} M_{z} & i A_{1} M_{y} \\ i A_{1} M_{z} & n_{o}^{2} & - i A_{1} M_{x} \\ - i A_{1} M_{y} & i A_{1} M_{x} & n_{o}^{2} \end{matrix}) + A_{2} (\begin{matrix} M_{x}^{2} & M_{x} M_{y} & M_{x} M_{z} \\ M_{x} M_{y} & M_{y}^{2} & M_{y} M_{z} \\ M_{x} M_{z} & M_{y} M_{z} & M_{z}^{2} \end{matrix}) .

The tensor is decomposed into terms that are respectively linear and quadratic in the magnetization. The magnetization has components M_x, M_y and M_z. Furthermore n_o is the isotropic refractive index and A₁ and A₂ are frequency-dependent empirical constants. Generally A₂ ≪ A₁. We suppose that in the Voigt configuration, the magnetization is parallel to the

\hat{x}

with possibly a small residual component along

\hat{z}

which is parallel to k yielding the overall tensor,

ε_{M O} = ε_{0} (\begin{matrix} n_{o}^{2} & - i A_{1} M_{z} & 0 \\ i A_{1} M_{z} & n_{o}^{2} & - i A_{1} M_{x} \\ 0 & i A_{1} M_{x} & n_{o}^{2} \end{matrix}) + A_{2} (\begin{matrix} M_{x}^{2} & 0 & M_{x} M_{z} \\ 0 & 0 & 0 \\ M_{x} M_{z} & 0 & M_{z}^{2} \end{matrix}) .

If M_x = M_y = 0 we have the traditional Faraday effect. However, in the presence of M_x, the normal modes (polarizations of the electric field of the propagating wave) posses components along

\hat{x}

,

\hat{y}

and

\hat{z}

, turning the material into effectively biaxial. This leads to beam walk-off and E is no longer perpendicular to k even though D remains normal to k. In order to keep the problem tractable, we ignore the walk-off and ignore the x_z and y_z off-diagonal terms in the tensor yielding a phenomenological expression of a uniaxial tensor

ε_{M O} \approx ε_{0} (\begin{matrix} n_{1}^{2} & - i Q & 0 \\ i Q & n_{2}^{2} & 0 \\ 0 & 0 & n_{3}^{2} \end{matrix}),

identical to the form given in [20]. Here we define n₁ = n + Δn/2 and n₂ = n − Δn/2, n is an average refractive index and Q = A₁M_z. The experimental data show that the effective tensor given above is a plausible model for describing the combination of circular and linear dichroism and birefringence effects. Magnetic circular birefringence (MCB) and magnetic circular dichroism (MCD) relate to the off-diagonal whereas magnetic linear birefringence (MLB) and magnetic linear dichroism (MLD) relate to the diagonal terms in the tensor, Eq. (3). The latter two are the main subject of the present work.

Let’s first assume Q = 0 which which corresponds to the absence of axial magnetization, M_z = 0. The asymmetry Δn = Δn′ − iΔn″ is a complex number, its real (Δn′) and imaginary (Δn″) components are designated MLB and MLD respectively. Even though these terms show a quadratic dependence on M_x, the terms are classified linear because the normal modes of the tensor are the linear polarization states $E_{o} \hat{x}$ and $E_{o} \hat{y}$ . The wave vectors for the normal modes are given by ω/c(n ± Δn/2) where ω and c represent the angular frequency and speed of light. In order to probe Δn with maximal sensitivity, we input light that is an equal superposition of the normal modes which in Jones notation is $E_{1} = E_{o} / \sqrt{2} (\hat{x} + \hat{y})$ . After emerging from the crystal of axial length d the state becomes $E_{o} / \sqrt{2} (e^{i Δ β^{'} d / 2} \hat{x} + e^{- i Δ β^{″} d / 2} \hat{y})$ . For convenience, we have defined Δβ′ = k_oΔn′ and Δβ″ = k_oΔn″ and k_o = ω/c. The polarization angle θ (with respect to the $\hat{x}$ axis) and the ellipticity χ forms ∈ [−π/4, π/4] can be immediately extracted [21] and are given by the closed forms

\tan 2 θ = - \frac{\cos (Δ β^{'} d)}{\sinh (Δ β^{″} d)} \approx - \frac{1}{k_{o} Δ n^{″} d}

\sin 2 χ = \frac{\sin (Δ β^{'} d)}{\cosh (Δ β^{″} d)} \approx k_{o} Δ n^{'} d .

The approximations above are valid because Δβ′d, Δβ″d ≪ 1. We deduce that for Q = 0 (i.e. the pure Voigt effect or the complete absence of axial magnetization) the rotation θ should exclusively be caused by MLD and the ellipticity χ by MLB.

However, a nonzero axial magnetization (M_z, Q, ≠ 0) intermixes θ and χ. We observe that the normal modes of the magneto-optic tensor are now the elliptically polarized states, $1 / \sqrt{2} {(- i (1 + \sin ξ), \cos ξ)}^{T}$ and $1 / \sqrt{2} {(\cos ξ, - i (1 + \sin ξ))}^{T}$ with the respective refractive indices $γ_{1} = {(n^{2} + \sqrt{Q^{2} + ζ^{2}})}^{1 / 2}$ and $γ_{2} = {(n^{2} - \sqrt{Q^{2} + ζ^{2}})}^{1 / 2}$ where we have defined $ζ = (n_{1}^{2} - n_{2}^{2}) / 2 \approx n Δ n$ which characterizes the asymmetry in the linear direction and tan ξ = ζ/Q which determines the mixing of the linear and circular effects. It can be shown that the input state E₁ transforms to

\frac{1}{2} \frac{E_{o}}{\sqrt{1 + \sin ξ}} (\begin{array}{l} - i e^{i θ_{1}} (1 + \sin ξ) + e^{i θ_{2}} \cos ξ \\ e^{i θ_{1}} \cos ξ + - i e^{i θ_{2}} (1 + \sin ξ) \end{array})

where θ₁ = tan⁻¹(f) + k_oγ₁d, θ₂ = tan⁻¹(1/f) + k_oγ₂d and f = (1 + sin ξ)/cos ξ. This form doesn’t itself provide much physical insight but becomes the basis for calculating the rotation and ellipticity for the intermixed linear and circular effects which become

\tan 2 θ = \frac{\cot (θ_{1} - θ_{2})}{\cos ξ}

\sin 2 χ = - \sin ξ \sin (θ_{1} - θ_{2}) .

3. Experimental procedure

The experimental arrangement is visualized in Fig. 1. A cylindrically shaped TGG crystal of width 3 mm and length 1 cm is acquired. Its long axis lies along the [111] crystallographic direction and is mounted inside a cryostat using a homemade clamping mount of copper. The vacuum chamber is fitted with non-magnetic quartz windows. A temperature controller is used to control the crystal temperature that is continuously monitored by a calibrated gallium arsenide diode. The sample is always inside a vacuum of better than 10⁻⁶ mbar. A helium-neon laser beam of wavelength 633 nm passes the optical window of the cryostat and makes normal incidence on the cross sectional face of TGG. The transmitted intensity is measured using a silicon photodetector connected to the input of a lock-in amplifier. All instruments are fully integrated into a computer controlled system. We extract the angles θ and χ from the Stokes parameters measured from the Fourier decomposition of the detected intensity signal as a quarter wave plate inserted before the final analyzer is synchronously rotated in a motorized stage. The polarimetry has been described elsewhere [22].

Fig. 1 The experimental arrangement including L = laser, O = optical chopper, P = polarizer, C= crystal, Q = quarter waveplate, A = analyzer, D = photodetector. The perceived beam path is shown in red.

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4. Results and discussion

Equation (5) predicts a quadratic dependence of the ellipticity on the field. Indeed this is the observed behavior and is elucidated in Fig. 2(a). The ellipticity is measured to be ≈7° at 250 mT and 8 K. Using Eqs. (4) and (5), we can also deduce the magnetically induced asymmetries Δn enabling quantitative estimates of MLB and MLD. The estimates of MLB, Δn′, are simply proportional to χ and are therefore identified on the right axis of Fig. 2(a). These asymmetries are of the order 10⁻⁶, and increase by increasing B and lowering T. Similarly Fig. 2(b) shows the variation of the polarization angle θ. For zero field, all curves coincide at θ = 45° which is a control baseline. The overall rotations can approach ≈ 12° for fields of 250 mT at a temperature of 8 K. Nonetheless, the rotations are indeed much smaller than the conventionally observed Faraday rotation which could approach ≈ 300° under similar conditions [11]. Similar amplitudes are measured for the MLD as well as illustrated in Fig. 3. The phase sensitive measurement technique is capable of revealing these minute asymmetries whose magnitudes are in close agreement with similar paramagnetic materials [3, 5].

Fig. 2 (a) The ellipticity χ and magnetic linear birefringence Δn′ with respect to the square of magnetic field. (b) Angle θ plotted with respect to the square of magnetic field showing correspondence with the theoretical predictions (solid lines) based on intertwined linear and circular birefringence. Uncertainties in angles θ are of the order of ±0.6° and solid lines are guide to the eye.

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Fig. 3 Magnetic linear dichroism, Δn″ plotted with respect to the square of magnetic field B². Solid lines are only visual guides.

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We also notice that for θ, we obtain a quadratic dependence on B only for temperatures ≳ 30 K. These results are explicable by the intermixing of the strong circular components due to tiny misalignments of the applied field away from the Voigt, and towards the Faraday geometry. The effect is accentuated at lower temperatures and stronger magnetic fields when the Faraday rotation, even stemming from slight axial components can leave a significant impact on the observed rotation angle. Equation (7) describes the presence of concomitant circular effects. The experimental data is fit to this expression and drawn as solid lines in the figure. Suppose the field’s skew angle is θ_s away from the $\hat{x} - \hat{y}$ plane and towards the axial direction. The component B sin θ_s causes Faraday rotation due to MCB; is proportional to B, and gives rise to Q = Q_o (B sin θ_s)where Q_o = 2θ_f/(Bk_od) is a constant derived from the Faraday angles θ_f determined in other experiments [11]. The other parameter required in the analysis is ζ = nΔn = ζ_o(B cos θ_s)² which, originating from MLB and MLD, shows a quadratic dependence on B. Here n = 1.9535 is the isotropic refractive index of TGG. A nonlinear fitting procedure determines best estimates of ζ_o and the skew angles, and the former are in excellent agreement with our measurements of MLB and MLD. The data is presented in Table 1 and for a single experimental setting, provides an estimate of the skew angle: (3.0 ± 0.5)°.

Table 1. Data showing the temperatures T, asymmetries Q₀ and values of the fitted parameters ζ_o and θ_s.

View Table

The field-independent temperature dependence of the polarization angles can in fact be captured by defining the coefficients, V_χ = Δn′/B² and V_θ = Δn″/B. For TGG, these coefficients respectively approach values of 4 × 10⁻⁵ T⁻² and −1.7 × 10⁻⁶/T⁻¹ at 8 K and their complete temperature dependence can be seen in Figs. 4(a) and 4(b). These coefficients are estimated through the slopes of the χ versus B² (Fig. 2(b)) and θ versus B (Fig. 5(b)). For completeness, Fig. 5(a) depicts the variation of χ with the magnetic field B. The coefficient V_θ is predominantly caused by the strong Faraday rotation and is similar to the Verdet constant though intermixed with a minute Voigt component. The smallness of the MLB that gives rise to the coefficient V_χ, for example, can be appreciated from the guesstimate that achieving a birefringence of 1% in TGG at 8 K would require a magnetic field of strength 200 T. The inverse temperature relationship indicates proportionality with the paramagnetic nature of the magnetic susceptibility of TGG [23]. Fitting the two kinds of Voigt coefficients to the Curie-Weiss law V = C/(T − T_W) yields Curie-Weiss constant T_W estimates of −8.7 K and −7.3 K which are in excellent agreement with data acquired by magnetic susceptibility [23] or pure Faraday effect measurements [24]. Furthermore, |T_W | > TN = 0:35 K also corroborates the existence of a strongly frustrated spin system inside the TGG crystal for the temperature range [T_N, |T_W|] where T_N is the Neél temperature. We consider this being the first reported instance of the Curie-Weiss constant derived from magneto-optic measurements based on the Voigt effect. Alternatively, Fig. 6 in the supplementary information depicts the combined field and temperature dependence of the angles of the polarization ellipse at values of B. The rotation and ellipticity are relatively small at temperatures above 30 K and become pronounced only at low temperatures and strong fields. For example, no detectable rotation and ellipticity were observed at room temperature up to fields of 1 T.

Fig. 4 Temperature dependence of the Voigt coefficients (a) V_χ and (b) V_θ. The solid lines are fits to hyperbolic curves showing an inverse temperature dependence.

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Fig. 5 The polarization angles (a) χ and (b) θ determined from Stokes polarimetry are plotted as function of the magnetic field B whereas in the main text, we choose to plot these angles as functions of B².

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Fig. 6 Temperature dependence of (a) angle of the polarization ellipse θ and (b) ellipticity χ. Solid lines are only visual guides. Data is shown for various settings of the magnetic field.

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5. Conclusion

We have conducted a complete characterization for magneto-optical effects for TGG in the Voigt geometry. With the help of a Fourier decomposition of the signal and phase-sensitive detection it is possible to detect minute asymmetries in the refractive indices. In particular, we show that ideally MLB imparts ellipticity to the emergent light while MLD rotates the polarization ellipse. The simple decoupling of rotation and ellipticity, however, breaks down in the presence of circular magneto-optic effects. This intertwining can be tackled from a theoretical perspective and our experimental data is systematically analyzed and seen to corroborate theoretical predictions. We can perform temperature dependent measurements of the polarization ellipse, helping us to quantify coefficients for ellipticity and rotation in the Voit geometry, determining the Curie-Weiss constant as well as the direction of a magnetic field misaligned from the nominal.

Applying uniform axial fields as in the Faraday configuration, may sometimes become restrictive with magnet pole pieces intervening the optical beam path. Furthermore, in the area of integrated magneto-photonic circuits, single-sided magnetic geometries, or on-chip surface coils are preferred. The Voigt configuration may become useful in these scenarios and especially when reciprocal and non-reciprocal components are integrated for building optical circulators or polarization preserving isolators. But at the same time, we should be wary that the ellipticity is large and comparable to rotations, which limits the usefulness of this geometry for practical applications [25]. Several other useful ideas, however, have been surveyed in this article. The recent study determining the Néel vector in an antiferromagnet through MLD could reinvigorate the practical utilization of the Voigt effect [26].

6. Supplementary information

6.1. Method for calculation uncertainties in the FFT data, Stokes parameters and angles of the polarization ellipse

We outline the method employed to quantify uncertainties from intensity of light, collected at the photodetector after emerging from some birefringent element, in our case the TGG crystal. Following the experimental scheme visualized in Fig. 1, the polarization state of transmitted intensity of light can be completely characterized by Stokes parameters (I, M, C, S) [22]. The transmitted intensity I_T is given by [21, 22]

\begin{array}{l} I_{T} (α, β, δ) = \frac{1}{2} [1 + (\frac{M}{2} \cos 2 α + \frac{C}{2} \sin 2 α) (1 + \cos δ)] + \frac{1}{2} [\sin δ \sin (2 α - β)] \\ = \frac{1}{4} [(M \cos 2 α - C \sin 2 α) \cos 4 β + (M \sin 2 α + C \cos 2 α) \sin 4 β] (1 - \cos δ) \end{array}

where α is the analyzer angle, β is the retarder angle of QWP and δ is the retardation. By virtue of the discrete FFT, the uncertainty propagates from the intensity I_T into each Fourier coefficient, and hence into the Stokes parameters and derived quantities, including θ and χ. The uncertainties in the Fourier coefficients labeled C_k and S_k are (using the notation Δx for the uncertainty in x)

Δ C_{k} = \frac{2}{N} \frac{1}{1 + δ_{k, 0} + δ_{k, L}} \sqrt{{(Δ I_{T_{1}} \cos (ω_{k} β_{1}))}^{2} + {(Δ I_{T_{2}} \cos (ω_{k} β_{2}))}^{2} + \dots + {(Δ I_{T_{N}} \cos (ω_{k} β_{N}))}^{2}}

Δ S_{k} = \frac{2}{N} \frac{1}{1 + δ_{k, 0} + δ_{k, L}} \sqrt{{(Δ I_{T_{1}} \sin (ω_{k} β_{1}))}^{2} + {(Δ I_{T_{2}} \sin (ω_{k} β_{2}))}^{2} + \dots + {(Δ I_{T_{N}} \sin (ω_{k} β_{N}))}^{2}}

where β_i = (i − 1)Δβ and

ω_{k} = \frac{2 π k}{N Δ β}

, and Δβ is the step size of the rotating QWP. We are particularly interested in the coefficients (C₀, C₂, C₄, S₂, S₂).

Subsequently, the uncertainties in the Stokes parameters I, M, C, S are given by

Δ M = \frac{2}{1 - \cos δ} \sqrt{{(C_{4} \cos (2 α + 4 β_{o}) Δ C_{4})}^{2} + {(S_{4} \cos (2 α + 4 β_{o}) Δ S_{4})}^{2}}

Δ C = \frac{2}{1 - \cos δ} \sqrt{{(C_{4} \cos (2 α + 4 β_{o}) Δ C_{4})}^{2} + {(S_{4} \cos (2 α + 4 β_{o}) Δ S_{4})}^{2}}

Δ S = \sqrt{{(\frac{1}{\sin δ \cos (2 α + 4 β_{o})} Δ S_{4})}^{2}}

Δ I = \sqrt{{(Δ C_{o})}^{2} + {(\frac{1 + \cos δ}{1 - \cos δ} \sqrt{{(C_{4} \cos (2 α + 4 β_{o}) Δ C_{4})}^{2} + {(S_{4} \cos (2 α + 4 β_{o}) Δ S_{4})}^{2}})}^{2}} .

Finally, the uncertainties in rotation θ and ellipticity χ of the light are determined as,

Δ θ = \sqrt{{(\frac{- 0.5 C}{C^{2} + M^{2}})}^{2} + {(\frac{0.5 M}{C^{2} + M^{2}})}^{2}}

Δ χ = \frac{1}{(C^{2} + M^{2} + S^{2})} \sqrt{{(0.5 \sqrt{C^{2} + M^{2}} Δ S)}^{2} + {(\frac{0.5 M S}{\sqrt{C^{2} + M^{2}}} Δ M)}^{2} + {(\frac{- 0.5 C S}{\sqrt{C^{2} + M^{2}}} Δ C)}^{2}}

Δ \tan (2 θ) = \frac{C}{M} \sqrt{{(\frac{Δ C}{C})}^{2} + {(\frac{Δ M}{M})}^{2}} .

6.2. Supplementary experimental data

Funding

This study was supported by the Higher Education Commission Pakistan (NRPU 2028) and the LUMS Faculty Initiative Fund (FIF-319).

Acknowledgments

We thank Amrozia Shaheen for providing help in polarimetry measurements and Hafiz Rizwan for constructing housing for the cryostat used in this work.

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T (K)	Q_o (×10⁻⁴ rad T⁻¹)	ζ_o (×10⁻⁶ T⁻²)	θ_s (°)
8	5.6	7.8	3.4
10	4.3	7.3	3.7
20	2.6	4.7	3.6
40	1.4	1.7	2.5
80	0.6	0.6	2.5

Low temperature Voigt effect in the terbium gallium garnet crystal

Abstract

1. Introduction

2. Theory

3. Experimental procedure

4. Results and discussion

5. Conclusion

6. Supplementary information

6.1. Method for calculation uncertainties in the FFT data, Stokes parameters and angles of the polarization ellipse

Funding

Acknowledgments

References and links

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