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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 21 — Oct. 21, 2013
  • pp: 25148–25158
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Complete Stokes polarimetry of magneto-optical Faraday effect in a terbium gallium garnet crystal at cryogenic temperatures

Hassaan Majeed, Amrozia Shaheen, and Muhammad Sabieh Anwar  »View Author Affiliations


Optics Express, Vol. 21, Issue 21, pp. 25148-25158 (2013)
http://dx.doi.org/10.1364/OE.21.025148


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Abstract

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 OSA

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

1. A. D. Buckingham and P. J. Stephens, “Magnetic Optical Activity,” Annu. Rev. Phys. Chem. 17, 399–432 (1966). [CrossRef]

3

3. C. A. Bennett, “Faraday Rotation,” in Principles of Physical Optics (John Wiley & Sons Inc., 2008).

]
θ=VBd
(1)
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

3. C. A. Bennett, “Faraday Rotation,” in Principles of Physical Optics (John Wiley & Sons Inc., 2008).

, 4

4. G. Fischer, “The Faraday Optical Isolator,” J. Opt. Commun. 8, 18–21 (1987).

], in optical circulators for fiber optic systems [5

5. Y. Fujii, “High-isolation polarization-independent optical circulator coupled with single-mode fibers,” J. Light-wave Technol. 9, 456–460 (1991). [CrossRef]

], in optical modulators [2

2. E. Hecht, “The Faraday Effect,” in Optics, 4th ed. (Pearson Education Inc., San Francisco, 2002).

, 6

6. R. W. Cooper and J. L. Page, “Magneto-optic light modulators,” Radio Electron. Eng. 39, 302–304 (1970). [CrossRef]

] and in current and magnetic field sensors [7

7. P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for Temperature-Dependence of the Faraday Effect by Optical Activity Temperature Shift,” IEEE Sens. J. 13, 832–837 (2013). [CrossRef]

9

9. J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7, 1468–1470 (1995). [CrossRef]

]. 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

10. J. Van Vleck and M. Hebb, “On the Paramagnetic Rotation of Tysonite,” Phys. Rev. 46, 17–32 (1934). [CrossRef]

].

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

11. J. A. Davis and R. M. Bunch, “Temperature dependence of the Faraday rotation of Hoya FR-5 glass,” Appl. Opt. 23, 633–636 (1984). [CrossRef] [PubMed]

]. 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

12. N. P. Barnes and L. B. Petway, “Variation of the Verdet constant with temperature of terbium gallium garnet,” J. Opt. Soc. Am. B 9, 1912–1915 (1992). [CrossRef]

]. 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

13. R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007). [CrossRef] [PubMed]

]. 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

14. D. Clarke and J.F. Grainger, “The description of polarized light,” in Polarized light and Optical Measurement, First ed. (Pergamon Press Ltd., Germany, 1971).

]. 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
ε=(εxxiQ0iQεxx000εzz)
(2)
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

15. R. P. Hunt, “Magneto-Optic Scattering from Thin Solid Films,” J. Appl. Phys. 38, 1652–1671 (1967). [CrossRef]

18

18. H. Hauser, R. Chabicovsky, and K. Riedling, “Magneto-Optical Methods,” in Handbook of Thin Films: Deposition and Processing, H. S. Nalwa, ed. (Academic Press, 2001).

]. 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

19. A. K. Zvezdin and V. A. Kotov, “Magnetooptical effects,” in Modern Magnetooptics and Magnetooptical Materials (Taylor & Francis Group, New York, 1997).

, 20

20. M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. S. J. Russell, “Complex Faraday Rotation in Microstructured Magneto-optical Fiber Waveguides,” Adv. Mater. 23, 2681–2688 (2011). [CrossRef] [PubMed]

]
θ=ω2cnxxQd
(3)
ψ=ω2cnxxQd
(4)
where nxx=εxx 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

19. A. K. Zvezdin and V. A. Kotov, “Magnetooptical effects,” in Modern Magnetooptics and Magnetooptical Materials (Taylor & Francis Group, New York, 1997).

, 20

20. M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. S. J. Russell, “Complex Faraday Rotation in Microstructured Magneto-optical Fiber Waveguides,” Adv. Mater. 23, 2681–2688 (2011). [CrossRef] [PubMed]

]. 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

21. E. Hecht, “Malus’s Law,” in Optics, 4th Edition ed. (Pearson Education Inc., San Francisco, 2002), pp. 332–333.

],
I=I0cos2(θπ4)+Imin
(5)
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

13. R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007). [CrossRef] [PubMed]

]. 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

22. S. Polisetty, J. Scheffler, S. Sahoo, Y. Wang, T. Mukherjee, X. He, and C. Binek, “Optimization of magneto-optical Kerr setup: Analyzing experimental assemblies using Jones matrix formalism,” Rev. Sci. Instrum. 79, 055107–055106 (2008). [CrossRef] [PubMed]

]. 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 ψ[π4,π4]. 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

14. D. Clarke and J.F. Grainger, “The description of polarized light,” in Polarized light and Optical Measurement, First ed. (Pergamon Press Ltd., Germany, 1971).

]
I=|Ex|2+|Ey|2M=|Ex|2|Ey|2C=2|Ex||Ey|cos(δyδx),andS=2|Ex||Ey|sin(δyδx).
(6)

It is also important to keep track of the polarized component of light Ip=(M2+C2+S2)12. Finally, θ and ψ can be determined from the Stokes parameters by using the relations
θ=12arctan(CM),and|ψ|=arctan(η).
(7)
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
2η1+η2=|S|Ip.
(8)
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

23. H. G. Berry, G. Gabrielse, and A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. 16, 3200–3205 (1977). [CrossRef] [PubMed]

]
IT(α,β,δr)=12[I+(M2cos2α+C2sin2α)(1+cosδr)]+S2sinδrsin(2α2β)+14[(Mcos2αCsin2α)cos4β+(Msin2α+Ccos2α)sin4β](1cosδr)
(9)
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

23. H. G. Berry, G. Gabrielse, and A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. 16, 3200–3205 (1977). [CrossRef] [PubMed]

]. 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

23. H. G. Berry, G. Gabrielse, and A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. 16, 3200–3205 (1977). [CrossRef] [PubMed]

]
I=C01+cosδr1cosδr[C4cos(4α+4β0)+S4sin(4α+4β0)],M=21cosδr[C4cos(2α+4β0)+S4sin(2α+4β0)],C=21cosδr[S4cos(2α+4β0)C4sin(2α+4β0)],andS=S2sinδrcos(2α+4β0)
(10)
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

23. H. G. Berry, G. Gabrielse, and A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. 16, 3200–3205 (1977). [CrossRef] [PubMed]

]. 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

11. J. A. Davis and R. M. Bunch, “Temperature dependence of the Faraday rotation of Hoya FR-5 glass,” Appl. Opt. 23, 633–636 (1984). [CrossRef] [PubMed]

, 13

13. R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007). [CrossRef] [PubMed]

]. 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

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

26. A. V. Starobor, D. S. Zheleznov, O. V. Palashov, and E. A. Khazanov, “Magnetoactive media for cryogenic Faraday isolators,” J. Opt. Soc. Am. B 28, 1409–1415 (2011). [CrossRef]

].

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.

References and links

1.

A. D. Buckingham and P. J. Stephens, “Magnetic Optical Activity,” Annu. Rev. Phys. Chem. 17, 399–432 (1966). [CrossRef]

2.

E. Hecht, “The Faraday Effect,” in Optics, 4th ed. (Pearson Education Inc., San Francisco, 2002).

3.

C. A. Bennett, “Faraday Rotation,” in Principles of Physical Optics (John Wiley & Sons Inc., 2008).

4.

G. Fischer, “The Faraday Optical Isolator,” J. Opt. Commun. 8, 18–21 (1987).

5.

Y. Fujii, “High-isolation polarization-independent optical circulator coupled with single-mode fibers,” J. Light-wave Technol. 9, 456–460 (1991). [CrossRef]

6.

R. W. Cooper and J. L. Page, “Magneto-optic light modulators,” Radio Electron. Eng. 39, 302–304 (1970). [CrossRef]

7.

P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for Temperature-Dependence of the Faraday Effect by Optical Activity Temperature Shift,” IEEE Sens. J. 13, 832–837 (2013). [CrossRef]

8.

T. Yoshino, S.-i. Torihata, M. Yokota, and N. Tsukada, “Faraday-Effect Optical Current Sensor with a Garnet Film /Ring Core in a Transverse Configuration,” Appl. Opt. 42, 1769–1772 (2003). [CrossRef] [PubMed]

9.

J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7, 1468–1470 (1995). [CrossRef]

10.

J. Van Vleck and M. Hebb, “On the Paramagnetic Rotation of Tysonite,” Phys. Rev. 46, 17–32 (1934). [CrossRef]

11.

J. A. Davis and R. M. Bunch, “Temperature dependence of the Faraday rotation of Hoya FR-5 glass,” Appl. Opt. 23, 633–636 (1984). [CrossRef] [PubMed]

12.

N. P. Barnes and L. B. Petway, “Variation of the Verdet constant with temperature of terbium gallium garnet,” J. Opt. Soc. Am. B 9, 1912–1915 (1992). [CrossRef]

13.

R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15, 11255–11261 (2007). [CrossRef] [PubMed]

14.

D. Clarke and J.F. Grainger, “The description of polarized light,” in Polarized light and Optical Measurement, First ed. (Pergamon Press Ltd., Germany, 1971).

15.

R. P. Hunt, “Magneto-Optic Scattering from Thin Solid Films,” J. Appl. Phys. 38, 1652–1671 (1967). [CrossRef]

16.

C. C. Robinson, “Electromagnetic Theory of the Kerr and the Faraday Effects for Oblique Incidence,” J. Opt. Soc. Am. 54, 1220–1222 (1964). [CrossRef]

17.

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Fundamental magneto-optics,” J. Appl. Phys. 68, 4203–4207 (1990). [CrossRef]

18.

H. Hauser, R. Chabicovsky, and K. Riedling, “Magneto-Optical Methods,” in Handbook of Thin Films: Deposition and Processing, H. S. Nalwa, ed. (Academic Press, 2001).

19.

A. K. Zvezdin and V. A. Kotov, “Magnetooptical effects,” in Modern Magnetooptics and Magnetooptical Materials (Taylor & Francis Group, New York, 1997).

20.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. S. J. Russell, “Complex Faraday Rotation in Microstructured Magneto-optical Fiber Waveguides,” Adv. Mater. 23, 2681–2688 (2011). [CrossRef] [PubMed]

21.

E. Hecht, “Malus’s Law,” in Optics, 4th Edition ed. (Pearson Education Inc., San Francisco, 2002), pp. 332–333.

22.

S. Polisetty, J. Scheffler, S. Sahoo, Y. Wang, T. Mukherjee, X. He, and C. Binek, “Optimization of magneto-optical Kerr setup: Analyzing experimental assemblies using Jones matrix formalism,” Rev. Sci. Instrum. 79, 055107–055106 (2008). [CrossRef] [PubMed]

23.

H. G. Berry, G. Gabrielse, and A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. 16, 3200–3205 (1977). [CrossRef] [PubMed]

24.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2011). [CrossRef]

25.

J. C. Martinez, M. B. A. Jalil, and S. G. Tan, “Giant Faraday and Kerr rotation with strained graphene,” Opt. Lett. 37, 3237–3239 (2012). [CrossRef] [PubMed]

26.

A. V. Starobor, D. S. Zheleznov, O. V. Palashov, and E. A. Khazanov, “Magnetoactive media for cryogenic Faraday isolators,” J. Opt. Soc. Am. B 28, 1409–1415 (2011). [CrossRef]

OCIS Codes
(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry
(120.6810) Instrumentation, measurement, and metrology : Thermal effects
(230.2240) Optical devices : Faraday effect
(230.3810) Optical devices : Magneto-optic systems

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: July 26, 2013
Revised Manuscript: September 7, 2013
Manuscript Accepted: September 13, 2013
Published: October 15, 2013

Citation
Hassaan Majeed, Amrozia Shaheen, and Muhammad Sabieh Anwar, "Complete Stokes polarimetry of magneto-optical Faraday effect in a terbium gallium garnet crystal at cryogenic temperatures," Opt. Express 21, 25148-25158 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-25148


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References

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