Weak measurement of magneto-optical Goos-H&#x00E4;nchen effect

Tingting Tang; Jie Li; Li Luo; Jian Shen; Chaoyang Li; Jun Qin; Lei Bi; Junyong Hou

doi:10.1364/OE.27.017638

1. Introduction

Goos-Hänchen (GH) effect refers to a lateral shift of the reflected light beam when the total reflection occurs at the interface of two media with different refraction indices [1]. The spatial shift is of the order of the wavelength (a few micrometers in the visible or near infrared) in general and attributed to the evanescent wave that travels along the interface. Therefore, it is rather difficult to experimentally observe GH shift during a total internal reflection in optical band. Fortunately, several techniques have been developed for an experimental observation of GH effect [2–5]. Weak measurement is an important and convenient method for amplifying and measuring many weak physical phenomena [6–8] and effectively expands their applications [9,10]. It was firstly introduced for the measurement of spin Hall effect of light (SHEL) in 2008 [11,12] and for Goos-Hänchen effect in 2013 [13]. Unlike traditional measurement methods, the weak measurement realizes amplification of the tiny physical effects by introducing a concept of pre-selection and post-selection.

The enhancement and modulation of GH effect has attracted much attention of researchers for many years and it is still going on [14–17], for its potential applications in optical sensors [18–22]. In a heterostructure consisting of an electro-optic film or a magneto-electric film, both the spatial and angular GH shifts can be controlled through a variation of the direction and the magnitude of the external electric field [23–26]. It is worth mentioning that magnetic field is also a powerful tool for tunable GH shift in ferromagnetic metal, magneto-optical dielectric or graphene [27–31]. In 2014, we reported a theoretical study of the enhanced GH effect in a prism-waveguide coupling system with a MO thin film of Ce doped Y₃Fe₅O₁₂ (CeYIG) [32]. By applying opposite magnetic field across the CeYIG layer in a transverse magneto-optical Kerr effect (TMOKE) configuration, a difference in GH shift we called magneto-optical Goos-Hänchen (MOGH) shift caused by nonreciprocal phase shift (NRPS) can be observed. In this paper, we experimentally verify the theoretical prediction of MOGH effect. Using weak measurement method, we demonstrate the amplified GH and MOGH shifts.

2. Theoretical model and simulation results

Previous works have shown that magneto-optical Kerr effect can be significantly enhanced by surface plasmon resonance (SPR) in the multilayer structures contain Au-Fe or Au-Co, which is called magneto-optical surface plasmon resonance (MOSPR) [33]. When MOSPR is excited, the NRPS in the TMOKE condition caused by the magnetic field will be greatly amplified. In this way, the MOSPR-induced magnetic field-tunable singular phase retardation causes enhanced MOGH shift [18,32,34]. We consider a model of BK7 prism/Fe/Au waveguide as shown in Fig. 1(a), a p-polarized light is injected into the waveguide with an incident angle of θ, Meanwhile, a magnetic field is applied on the BK7/Fe/Au waveguide along the y-axis.

Fig. 1 (a) Schematic of the BK7/Fe/Au waveguide. (b) Reflectance curves and field (magnetic field component, H_x) distribution of the structure.

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When the magnetic field is applied along the y axis, the structure is under the Voigt geometry with TM mode as eigenmodes. The permittivity tensor of Fe can be expressed as

\hat{ε} = ε_{0} (\begin{matrix} ε_{20} & 0 & - ε_{21} \\ 0 & ε_{20} & 0 \\ - ε_{21} & 0 & ε_{20} \end{matrix}) .

Here we choose λ = 632.8 nm, d_Au = 22nm, d_Fe = 10nm, ε_prism = 1.515, n_Fe = 2.88 + 9.035i, ε₂₁ = 0.0088exp(−1.452i)∙ε₂₀ [35,36] and ε_Au = −10.4 + 1.4i [37,38].

In the BK7/Fe/Au waveguide, the reflection coefficients can be calculated by magneto-optical transfer matrix method (MOTMM) [39]. For a plane wave propagating in the magnetic anisotropy film with representation as $E = E_{0} \exp [i (ω t - γ \cdot r)],$ , where the wave vector is $γ = \frac{ω}{c} (\hat{x} N_{x} + \hat{z} N_{z}),$ N_x and N_z are effective refractive indices, four vertical components of wave vector $\frac{ω}{c} N_{z j} (j = 1, 2, 3, 4)$ can be obtained by solving the Maxwell’s equations. In the Fe film, the elements of the dynamic matrix(D^(Fe)) are

\begin{array}{l} D_{11} = i ε_{21} N_{x}^{} N_{z 1}^{}, \\ D_{13} = i ε_{21} N_{x}^{} N_{z 2}^{}, \\ D_{21} = i ε_{21} N_{x}^{} N_{z 1}^{2}, \\ D_{23} = i ε_{21} N_{x}^{} N_{z 2}^{2}, \end{array}

\begin{array}{l} D_{31} = (ε_{20} - N_{x}^{2}) (ε_{20} - N_{x}^{2} - N_{z 1}^{2}) + ε_{21}^{2}, \\ D_{33} = (ε_{20} - N_{x}^{2}) (ε_{20} - N_{x}^{2} - N_{z 2}^{2}) + ε_{21}^{2}, \\ D_{41} = N_{z 3} [ε_{20} (ε_{20} - N_{x}^{2} - N_{z 1}^{2}) + ε_{21}^{2}], \\ D_{43} = N_{z 4} [ε_{20} (ε_{20} - N_{x}^{2} - N_{z 2}^{2}) + ε_{21}^{2}] . \end{array}

Then the dynamic matrix D of Fe can be written as

D^{(F e)} = [\begin{matrix} D_{11} & - D_{11} & D_{13} & - D_{13} \\ D_{21} & D_{21} & D_{23} & D_{23} \\ D_{31} & D_{31} & D_{33} & D_{33} \\ D_{41} & - D_{41} & D_{43} & - D_{43} \end{matrix}] .

In isotropic layers, $N_{z j}^{} = N_{z 0}^{} (j = 1, 2, 3, 4),$ in which $N_{z 0}^{} = N_{}^{2} - N_{x}^{2},$ $N_{}^{2} = μ ε_{20}^{},$ so the dynamic matrix can be simplified as

D^{} = [\begin{matrix} 1 & 1 & 0 & 0 \\ \frac{N_{z 0}^{}}{μ} & - \frac{N_{z 0}^{}}{μ} & 0 & 0 \\ 0 & 0 & \frac{N_{z 0}^{}}{N^{}} & \frac{N_{z 0}^{}}{N^{}} \\ 0 & 0 & - \frac{N_{}^{}}{μ^{}} & \frac{N_{}^{}}{μ^{}} \end{matrix}] .

In addition, the propagation matrix of isotropic layers and magnetic layer is

P^{(n)} = [\begin{matrix} e^{i (ω / c) N_{z 1}^{(n)} d^{(n)}} & 0 & 0 & 0 \\ 0 & e^{i (ω / c) N_{z 2}^{(n)} d^{(n)}} & 0 & 0 \\ 0 & 0 & e^{i (ω / c) N_{z 3}^{(n)} d^{(n)}} & 0 \\ 0 & 0 & 0 & e^{i (ω / c) N_{z 4}^{(n)} d^{(n)}} \end{matrix}],

where d⁽ⁿ⁾ is the thickness of nth layer. Then we can get a matrix Q which connects the electric field amplitudes of cover and substrate layer by combining matrix P and matrix D as follows

Q = D^{{(1)}^{- 1}} D^{(2)} P^{(2)} D^{(2)}^{- 1} D^{(3)} .

Finally, the reflection coefficients in BK7/Fe/Au multilayer structure can be calculated as

\begin{array}{l} r_{s s} = \frac{Q_{21} Q_{33} - Q_{23} Q_{31}}{Q_{11} Q_{33} - Q_{13} Q_{31}}, \\ r_{p s} = \frac{Q_{41} Q_{33} - Q_{43} Q_{31}}{Q_{11} Q_{33} - Q_{13} Q_{31}}, \\ r_{s p} = \frac{Q_{11} Q_{23} - Q_{21} Q_{13}}{Q_{11} Q_{33} - Q_{13} Q_{31}}, \\ r_{p p} = \frac{Q_{11} Q_{43} - Q_{41} Q_{13}}{Q_{11} Q_{33} - Q_{13} Q_{31}} . \end{array}

As shown in Fig. 1(b), the reflection curve has no absorption peak for s-polarized incident light. While for p-polarized incident light there is an absorption peak near 46° which is the angle to excite SPR. For further understanding we plot the field distribution (magnetic field component, H_x) for p-polarized incident light at the absorption peak conditions as the inset of Fig. 1(b). when θ = 46.5°, most of the incident light is coupled into the BK7/Fe/Au waveguide and most of the power is concentrated at the interface of gold and air. Thus, the field distribution is rather sensitive to the refractive index variation of air layer. If the air layer is substituted with different sample, we can get a highly sensitive waveguide to refractive index variation.

According to stationary phase method, the GH shift of the BK7/Fe/Au waveguide can be expressed as

L_{GH} = \frac{1}{k_{0} n_{1}} \cdot \frac{d φ}{d θ},

here

φ

is the phase shift of reflected light which can be calculated by MOTMM.

For magnetic field applied along the + y or –y directions, the propagation constant can be expressed as

β_{0}^{y +} = β_{0} + Δ β, β_{0}^{y -} = β_{0} - Δ β,

where

Δ β

is the NRPS caused by the MO material. By reversing the magnetic field along + y or –y directions, both

ε_{21}

and

Δ β

show reversed sign, and the MOGH effect can be observed. Meanwhile the nonreciprocal phase shift leads to a MOGH shift as

L_{MOGH} = L_{GH} (+ H) - L_{GH} (- H) = \frac{1}{k_{0} n_{1}} \cdot (\frac{d φ^{y+}}{d θ} - \frac{d φ^{y-}}{d θ}),

where

φ_{}^{y +}

and

φ_{}^{y -}

denote the phase shifts for magnetic field applied along + y or –y directions, respectively.

We give the GH and MOGH shifts as shown in Figs. 2(a) and 2(b), respectively. Here we choose the same parameters as that in Fig. 1(b). The GH shift peak corresponding to the absorption peak of Fig. 1(b) achieves 12.5 λ. This means when the incident light is mostly coupled into the waveguide, the GH shift of the reflected light achieves the maximum value. When the thicknesses of Fe and Au change, the GH shift peaks appear at different incident angles. This is because different thicknesses of Fe and Au lead to SPR excitation at different incident angles. When SPR is excited, most incident light is coupled into the waveguide. In this case, the GH shift is enlarged. Therefore, the GH shift peaks always appear at the angle which SPR is excited.

Fig. 2 Simulation results of GH shift (a) and MOGH shift (b) for different thickness of Fe and Au.

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Meanwhile in order to get the maximum MOGH shift, we give MOGH shifts for different layer thickness of BK7/Fe/Au waveguide as shown in Fig. 2(b). Here we define the TMOKE as [(R( + H)-R(-H))/(R( + H) + R(-H))] which is also given in the inset of Fig. 2(b). As Fig. 2(b) shows, the MOGH shift peaks are corresponding to TMOKE peaks. This is can be explained as that MOGH shift originates from the NRPS of Fe layer which is also shown as TMOKE value. When the thicknesses of Fe and Au are 10 nm and 22 nm respectively, the maximum MOGH shift achieves 250 nm.

However, we notice the MOGH shift is still too small to be applied in refractive index sensing. Therefore, we make use of weak measurement method to amplify the GH and MOGH shifts in experiment. Based on the description of standard wave optics, the calculation formula of relative displacement after weak measurement amplification can be written as [40]

S_{G H} = \frac{2 [{| r_{H} |}^{2} (z ρ - z_{r} χ) + {| r_{V} |}^{2} (z σ - z_{r} τ) + ξ | r_{H} | | r_{V} |]}{2 k_{0} z_{r} ({| r_{H} |}^{2} + {| r_{V} |}^{2}) + {| r_{H} |}^{2} (χ^{2} + ρ^{2}) + {| r_{V} |}^{2} (σ^{2} + τ^{2}) - ς},

where

ξ = \cos (\pm 2 Δ) [z_{r} (χ + τ) - z (ρ + σ)] - \sin (2 Δ) [z_{r} (σ - ρ) + z (τ - χ)],

ς = 2 | r_{H} | | r_{V} | [\cos (\pm 2 Δ) (2 k_{0} z_{r} + ρ σ + χ τ) + \sin (\pm 2 Δ) (ρ τ - χ σ)],

ρ = Re (\partial \ln r_{H} / \partial θ_{i}),

σ = Re (\partial \ln r_{V} / \partial θ_{i}),

χ = Im (\partial \ln r_{H} / \partial θ_{i})

and

τ = Im (\partial \ln r_{V} / \partial θ_{i}) .

The MOGH shift after amplification can be obtained as

S_{M O G H} = S_{G H} (+ H) - S_{G H} (- H) .

3. Experiment results

3.1. Experiment platform setup

The optical path of weak measurement is shown in Fig. 3 [41,42]. The light source is a He-Ne laser with a wavelength of 632.8 nm which is focused using a lens with a focal length of 100 mm. The output light is polarized by a Glan laser polarizer of P1. A BK7 prism with a shape of equilateral right triangle is used as a coupling device, and the sample is fixed to the bevel of the prism by refractive index matching liquid. A magnetic field which is sufficient to saturate the magnetization of the iron is applied on the sample by an electromagnet. In the post-selection section, we make use of a quarter-wave plate for phase compensation of the s- and p-polarized components. A half-wave plate is then used to adjust the light intensity. After that a polarizer of P2 is used for post-selection, and a lens with focal length of 250 mm is used for beam collimation. Finally, the data are detected by a CCD camera. In the experiment, we first adjust the optical axis of P1 to be 45 degrees, and then set the polarizer of P2 to be orthogonal to P1. After that, we rotate P2 by a small angle of $Δ$ which is called as amplification angle. Actually, the measured GH shift is the difference between GH shifts of s-polarized and p-polarized components reflected from the sample.

Fig. 3 Experiment platform of the weak measurement for GH shift.

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3.2. Sample fabrication and characterization

Based on the above theoretical analysis and simulation results, we choose the iron thickness as 10nm and the gold thickness as 22nm. In our experiment, the Fe and Au films are both deposited on a BK7 substrate by the magnetron sputtering method. We first place the BK7 substrate in acetone solution and clean it with ultrasonic for 10 minutes. After that, the substrate is quickly transferred into the isopropanol solution for further cleaning and rinsed by deionized water for 3 minutes. After the preliminary processes, we transfer the BK7 substrate into the sputtering chamber to deposit Fe and Au films on it. The basic pressure is about 5′10⁻⁴ Pa. For Fe film deposition, we use the direct current (DC) source with a power of 50 W and a pressure of 0.5 Pa in argon. Then we deposit Au on the sample by use of radio frequency (RF) source with a power of 100 W with the same pressure and atmosphere. The thickness of each layer is measured by the cross-section measurement using scanning electron microscope (SEM). The measured results show that the thickness of Fe and Au films are 10 nm and 22 nm, respectively. The sample parameters can be regarded to be consistent with design. In order to measure the crystallinity of the BK7/Fe/Au multilayer films, we give the results of X-ray diffraction (XRD) as shown in Fig. 4. Both Au and Fe are highly crystallized. It should be noticed that the wave packet around 25 degrees is caused by the amorphous of BK7 substrate.

Fig. 4 The crystallinity of the BK7/Fe/Au multilayer films by use of XRD.

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3.3. Measurement results

In the measurement of GH and MOGH shifts, the incident angle was chosen to be 46.8 degrees, and the amplification angle was changed from −3 degrees to 3 degrees with a step size of 0.2 degrees. These theoretical results of GH shift distributions are verified by experiment results. In Fig. 5(a) the detectable maximum relevant displacement is approximately 700 microns. The light intensity distributions received by CCD camera at an amplification angle of 1.2 degrees and −1.2 degrees are given as insets of Fig. 5(a). The intensity at −1.2 degrees is not average, so the deviation is obvious in the range of −1 to −3 degrees. The uneven spot may be caused by SPR excited by the p-polarized component of the reflected beam [37]. In addition, although the GH shift is enhanced near the SPR angle, the multilayer structure results in poor amplification due to the inherent characteristics of the weak measurement. In addition, Fig. 5(b) shows the corresponding MOGH shift with a maximum value of 120 microns. It can be found the results of theory and experiment are basically consistent. Compare with GH shift, MOGH shift exhibits a much higher sensitivity to the refractive index of the waveguide [32]. Thus, MOGH effect may have potential applications in refractive index detection.

Fig. 5 Comparison between theoretical and experimental results of GH (a) and MOGH (b) shift for different amplification angle.

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It is worth mentioning that the GH shift and MOGH shift of p-polarized light in the MOSPR structure are significantly enhanced by the weak measurement method. However, if a s-polarized beam is incident into the BK7/Fe/Au waveguide, the corresponding GH and MOGH shifts are rather small. In our experiment, the GH shift is determined by the difference between GH shifts of p- and s-polarized components. We can conclude that when the incident angle or amplification angle are different, GH shift for the p-polarized beam can be significantly changed while that for s-polarized beam remains almost unchanged. Therefore, this phenomenon is more conducive to realize precision measurement of GH and MOGH effect in a SPR waveguide.

4. Conclusion

In this paper, we study the theoretical and experimental results of GH and MOGH effects of the BK7/Fe/Au waveguide in which SPR is excited and enhances both GH and MOGH effects. By using weak measurement method, the GH and MOGH shifts are further enlarged. The maximum MOGH shift of the proposed BK7/Fe/Au waveguide achieves 120 μm when optimum thicknesses are chosen. When SPR is excited, most of the power is concentrated at the interface of gold and air. As MOGH effect exhibits a higher sensitivity to the refractive index of sample than GH shift, it can be applied in refractive index detection. The proven MOGH effect solves several major problems related to the application of GH effect and paves the way for more sensitive practical devices in the future. In addition, it may lead to advances in the optical field manipulation at metasurfaces.

Funding

Sichuan Science and Technology Program (2019JDJQ0003); Open Project Program of State Key Laboratory of Marine Resource Utilization in South China Sea (2019010); Special Project of Science and Technology Strategic Cooperation between Nanchong City and Southwest Petroleum University (NC17SY4012); National Natural Science Foundation of China (NSFC) (61505016, 51522204).

References

1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. 436, 333–346 (1947). [CrossRef]

2. L. Chen, Z. Cao, F. Ou, H. Li, Q. Shen, and H. Qiao, “Observation of large positive and negative lateral shifts of a reflected beam from symmetrical metal-cladding waveguides,” Opt. Lett. 32(11), 1432–1434 (2007). [CrossRef] [PubMed]

3. K. V. Sreekanth, Q. L. Ouyang, S. Han, K.-T. Yong, and R. Singh, “Giant enhancement in Goos-Hänchen shift at the singular phase of a nanophotonic cavity,” Appl. Phys. Lett. 112(16), 161109 (2018). [CrossRef]

4. Y. Hirai, K. Matsunaga, Y. Neo, T. Matsumoto, and M. Tomita, “Observation of Goos-Hänchen shift in plasmon-induced transparency,” Appl. Phys. Lett. 112(5), 051101 (2018). [CrossRef]

5. O. J. S. Santana and L. E. E. de Araujo, “Direct measurement of the composite Goos-Hänchen shift of an optical beam,” Opt. Lett. 43(16), 4037–4040 (2018). [CrossRef] [PubMed]

6. S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332(6034), 1170–1173 (2011). [CrossRef] [PubMed]

7. J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474(7350), 188–191 (2011). [CrossRef] [PubMed]

8. G. Strübi and C. Bruder, “Measuring ultrasmall time delays of light by joint weak measurements,” Phys. Rev. Lett. 110(8), 083605 (2013). [CrossRef] [PubMed]

9. J. X. Zhou, H. L. Qian, C. F. Chen, J. X. Zhao, G. R. Li, Q. Y. Wu, H. L. Luo, S. C. Wen, and Z. W. Liu, “Optical edge detection based on high-efficiency dielectric metasurface,” Proc. Natl. Acad. Sci. U.S.A. 116(22), 11137–11140 (2019).

10. T. F. Zhu, Y. J. Lou, Y. H. Zhou, J. H. Zhang, J. Y. Huang, Y. Li, H. L. Luo, S. C. Wen, S. Y. Zhu, Q. H. Gong, M. Qiu, and Z. C. Ruan, “Generalized spatial differentiation from the spin Hall effect of light and Its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019). [CrossRef]

11. O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008). [CrossRef] [PubMed]

12. X. X. Zhou, Z. Xiao, H. L. Luo, and S. C. Wen, “Experimental observation of the spin Hall effect of light on a nano-metal film via weak measurements,” Phys. Rev. A 85(4), 043809 (2012). [CrossRef]

13. G. Jayaswal, G. Mistura, and M. Merano, “Weak measurement of the Goos-Hänchen shift,” Opt. Lett. 38(8), 1232–1234 (2013). [CrossRef] [PubMed]

14. Y. Q. Kang, Y. J. Xiang, and C. Y. Luo, “Tunable enhanced Goos–Hänchen shift of light beam reflected from graphene-based hyperbolic metamaterials,” Appl. Phys. B 124(6), 115 (2018). [CrossRef]

15. Y. P. Wong, Y. Miao, J. Skarda, and O. Solgaard, “Large negative and positive optical Goos-Hänchen shift in photonic crystals,” Opt. Lett. 43(12), 2803–2806 (2018). [CrossRef] [PubMed]

16. X. B. Jiao, Z. Qiao, W. Q. Gao, and S. H. Shen, “Tunable Goos–Hänchen and Imbert–Fedorov shifts,” Opt. Commun. 436, 239–242 (2019). [CrossRef]

17. G. Z. Ye, W. S. Zhang, and H. L. Luo, “Goos-Hänchen and Imbert-Fedorov effects in Weyl semimetals,” Phys. Rev. A (Coll. Park) 99(2), 023807 (2019). [CrossRef]

18. X. B. Yin and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89(26), 261108 (2006). [CrossRef]

19. T. T. Tang, C. Y. Li, L. Luo, Y. F. Zhang, and J. Li, “Goos–Hänchen effect in semiconductor metamaterial waveguide and its application as a biosensor,” Appl. Phys. B 122, 167 (2016). [CrossRef]

20. X. Wang, C. Yin, J. Sun, H. Li, Y. Wang, M. Ran, and Z. Cao, “High-sensitivity temperature sensor using the ultrahigh order mode-enhanced Goos-Hänchen effect,” Opt. Express 21(11), 13380–13385 (2013). [CrossRef] [PubMed]

21. K. V. Sreekanth, Q. L. Ouyang, S. Sreejith, S. Zeng, W. Lishu, E. Ilker, W. Dong, M. ElKabbash, Y. Ting, C. T. Lim, M. Hinczewski, G. Strangi, K.-T. Yong, R. E. Simpson, and R. Singh, “Phase-change-material-based low-loss visible-frequency hyperbolic metamaterials for ultrasensitive label-free biosensing,” Adv. Opt. Mater. 1900081 (Epub 12 April 2019).

22. T. Shui, W. X. Yang, Q. Y. Zhang, X. Liu, and L. Li, “Squeezing-induced giant Goos-Hänchen shift and hypersensitized displacement sensor in a two-level atomic system,” Phys. Rev. A (Coll. Park) 99(1), 013806 (2019). [CrossRef]

23. Y. S. Dadoenkova, F. F. L. Bentivegna, N. N. Dadoenkova, R. V. Petrov, I. L. Lyubchanskii, and M. I. Bichurin, “Controlling the Goos-Hänchen shift with external electric and magnetic fields in an electro-optic/magneto-electric heterostructure,” J. Appl. Phys. 119(20), 203101 (2016). [CrossRef]

24. Y. S. Dadoenkova, F. F. L. Bentivegna, V. V. Svetukhin, A. V. Zhukov, R. V. Petrov, and M. I. Bichurin, “Controlling optical beam shifts upon reflection from a magneto-electric liquid-crystal-based system for applications to chemical vapor sensing,” Appl. Phys. B 123(4), 107–109 (2017).

25. Y. S. Dadoenkova, F. F. L. Bentivegna, N. N. Dadoenkova, I. L. Lyubchanskii, and Y. P. Lee, “Influence of misfit strain on the Goos–Hänchen shift upon reflection from a magnetic film on a nonmagnetic substrate,” J. Opt. Soc. Am. B 33(3), 393–404 (2016). [CrossRef]

26. Y. S. Dadoenkova, N. N. Dadoenkova, J. W. Klos, M. Krawczyk, and I. L. Lyubchanskii, “Goos-Hänchen effect in light transmission through biperiodic photonic-magnonic crystals,” Phys. Rev. A (Coll. Park) 96(4), 043804 (2017). [CrossRef]

27. I. J. Singh and V. P. Nayyar, “Lateral displacement of a light beam at a ferrite interface,” J. Appl. Phys. 69(11), 7820–7824 (1991). [CrossRef]

28. S. B. Borisov, N. N. Dadoenkova, I. Lyubchanskii, and M. I. Lyubchanskii, “Guse-Hanchen effect for the light reflected from the interface formed by bigyrotropic and nongyrotropic media,” Opt. Spectrosc. 85(2), 225–231 (1998).

29. X. Zeng, M. Al-Amri, and M. S. Zubairy, “Tunable Goos-Hänchen shift from graphene ribbon array,” Opt. Express 25(20), 23579–23588 (2017). [CrossRef] [PubMed]

30. W. J. Wu, S. Z. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A (Coll. Park) 96(4), 043814 (2017). [CrossRef]

31. T. T. Tang, J. Li, M. Zhu, L. Luo, J. Yao, N. Li, and P. Zhang, “Realization of tunable Goos-Hanchen effect with magneto-optical effect in graphene,” Carbon 135, 29–34 (2018). [CrossRef]

32. T. Tang, J. Qin, J. Xie, L. Deng, and L. Bi, “Magneto-optical Goos-Hänchen effect in a prism-waveguide coupling structure,” Opt. Express 22(22), 27042–27055 (2014). [CrossRef] [PubMed]

33. G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013). [CrossRef]

34. N. Goswami, A. Kar, and A. Saha, “Long range surface plasmon resonance enhanced electro-optically tunable Goos–Hänchen shift and Imbert–Fedorov shift in ZnSe prism,” Opt. Commun. 330, 169–174 (2014). [CrossRef]

35. X. D. Qiu, X. X. Zhou, D. J. Hu, J. L. Du, F. H. Gao, Z. Y. Zhang, and H. L. Luo, “Determination of magneto-optical constant of Fe films with weak measurements,” Appl. Phys. Lett. 105(13), 131111 (2014). [CrossRef]

36. P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9(12), 5056–5070 (1974). [CrossRef]

37. X. Zhou, X. Ling, Z. Zhang, H. Luo, and S. Wen, “Observation of spin Hall effect in photon tunneling via weak measurements,” Sci. Rep. 4(1), 7388 (2015). [CrossRef] [PubMed]

38. E. D. Palik, Handbook of optical constants of solids (Academic, 1998).

39. J. Li, T. Tang, L. Luo, N. Li, and P. Zhang, “Spin Hall effect of reflected light in dielectric magneto-optical thin film with a double-negative metamaterial substrate,” Opt. Express 25(16), 19117–19128 (2017). [CrossRef] [PubMed]

40. S. Z. Chen, C. Q. Mi, L. Cai, M. X. Liu, H. L. Luo, and S. C. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017). [CrossRef]

41. X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018). [CrossRef] [PubMed]

42. X. X. Zhou, Z. C. Xiao, H. L. Luo, and S. C. Wen, “Experimental observation of the spin Hall effect of light on a nano-metal film via weak measurements,” Phys. Rev. A 85(4), 043809 (2012). [CrossRef]

Weak measurement of magneto-optical Goos-Hänchen effect

Abstract

1. Introduction

2. Theoretical model and simulation results

3. Experiment results

3.1. Experiment platform setup

3.2. Sample fabrication and characterization

3.3. Measurement results

4. Conclusion

Funding

References

Cited By

Figures (5)

Equations (13)

Optics Express