## Abstract

The Imbert-Fedorov (IF) shift is defined as the transverse shift of barycenter of the entire beam when a circular or elliptically polarized incident beam is reflected. In this work, we examine the IF shift of Gaussian beam at the Brewster angle. Interestingly, the spin Hall effect of light takes place in the IF shift at the same time. Furthermore, this interesting phenomenon is experimentally observed using weak measurements. These findings may have useful applications in spin optics.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

When a bounded beam is reflected or refracted at a plane dielectric interface, it does not perfectly follow the geometrical-optics evolution. There usually exist four basic deviations from the geometrical-optics picture, which are in-plane (parallel to the plane of incidence) and out-of-plane (perpendicular to the plane of incidence) spatial and angular shifts, respectively. The in-plane (out-of-plane) spatial and angular shifts are known as the spatial and angular Goos-Hänchen shifts [Imbert-Fedorov (IF) shifts] [1–3]. A variant of the IF shift is the so-called Spin Hall effect of light (SHEL) [4–6]. As a result of spin-orbit coupling corresponding to geometrical Berry phase, the SHEL manifests itself as the spin-dependent shifts upon reflection or refraction. [7,8].

Recently, the SHEL has been widely researched in different physical systems such as optical physics [9–12], plasmonics [13, 14], semiconductors physics [15], metamaterial [16, 17], and even topological insulators [18,19]. Moreover, the SHEL has also drawn significant attention due to its potential application in precision metrology. For example, the SHEL has been used to measure the thickness of nanometal film [20], graphene layers [21], magneto-optical constant of Fe films [22,23], optical rotation of chiral solution [24], and the refractive index of media [25,26].

In this work, we theoretically and experimentally reveal the SHEL in IF shift in the case of a circular or elliptically polarized Gaussian beam reflected from the air-prism interface at the Brewster angle. The internal physical mechanism of this phenomenon is the spin-orbit coupling in IF shift. Particularly, the IF shift changes with the incident polarization state, while the spin splitting of the SHEL in IF shift is unchanged.

## 2. Theoretical model

Figure 1 schematically illustrates the SHEL in IF shift when the beam is reflected at the air-prism interface. The *z* axis of the Cartesian coordinate system (*x*, *y*, *z*) is perpendicular to the reflection interface at *z* = 0. The coordinate systems (*x _{i}*,

*y*,

_{i}*z*) and (

_{i}*x*,

_{r}*y*,

_{r}*z*) denote the incident and reflected beams, respectively.

_{r}Consider that the incident beam is an elliptical polarization state |*ψ _{i}*〉 = cos

*α*|

*H*〉 +

*i*sin

*α*|

*V*〉, where

*α*is the azimuth angle (the angle between the long axis of elliptical polarization beam and horizontal direction), |

*H*〉 and |

*V*〉 denote the horizontally and vertically polarized states, respectively. The total wave function is given by

*k*

_{xi},

*k*

_{yi}are the transverse wave vectors in the

*x*and

*y*directions, respectively. We assume that Φ(

*k*

_{xi},

*k*

_{yi}) is the transverse spatial distribution of Gaussian beam. Under the reflection at the air-prism interface, the initial state of system evolves to where

*θ*is the incident angle.

_{i}*r*and

_{p}*r*are the Fresnel reflection coefficients for parallel and perpendicular polarizations, respectively. And

_{s}*k*

_{0}= 2

*π*/

*λ*with

*λ*being the wavelength of light in air. According to the transversality, we can obtain

*k*

_{yr}=

*k*

_{yi}. Note that the incident angle is fixed at the Brewster angle

*θ*in our paper, so the in-plane spread of wave vectors should be taken into account [27]. The Fresnel reflection coefficient of parallel polarization can be expanded to the first-order approximation of Taylor series, where

_{B}*χ*=

*∂r*/

_{p}*∂θ*is the first-order derivative of the Fresnel reflection coefficient of parallel polarization. The total wave function evolves to

_{i}*σ*= ± denotes the left- and right-handed circular polarization, respectively.

*H*and

_{r}*V*denote the horizontally and vertically polarized components from Eq. (2), respectively. In Eqs. (6), (7), and (9),

_{r}*δ*= cot

*θ*/

_{B}*k*

_{0}, and we provide that tan

*α*≫ (

*χk*

_{xr})/(

*r*

_{s}k_{0}).

In Eq. (9), the term exp[−*iσk*_{yr}*δ*] stands for the spin-orbit coupling (*δ* is the coupling strength), while the term exp[*ik*_{yr}*δ* cot *α*] stands for the IF shift (−*δ* cot *α* is the magnitude of the IF shift). Putting them together, Eq. (9) shows the spin-orbit coupling in IF shift. We call this phenomenon as the SHEL in IF shift. Interestingly, from Eq. (9), we also find that the intensity distributions of left- and right-handed circularly polarized components are the same. In such case the IF shift can be regarded as the average value of the initial spin splitting. The result of Eq. (7) also proves this point. After obtaining the initial spin splitting and IF shift, we definite the relative shift between the initial spin splitting and IF shift as the spin splitting of the SHEL in IF shift, which can be expressed as

From Eq. (10), we know that the spin splitting of the SHEL in IF shift represents a symmetric property, which is different from the asymmetric spin-dependent splitting [29]. Next, we plot Fig. 2 to clearly reveal the SHEL in IF shift. As shown in Fig. 2(a), the magnitude of IF shift decreases with the increase of azimuth angle, and equals to zero at *α* = 90°, as described by Eq. (7). Moreover, the initial spin splitting always changes with IF shift. However, as shown in Fig. 2(b), the spin splitting of the SHEL in IF shift is a fixed value, as described by Eq. (10). Note that the spin splitting of the SHEL in IF shift is different from the spin splitting of the usually SHEL where the barycenter of both incident and reflected fields are in the same position (i.e., the origin of the Cartesian coordinate system) [9]. However, in our scheme, as shown in Fig. 1, there exists the IF shift between the barycenter of incident and reflected fields.

## 3. Experimental observation

Since the spin splitting of the SHEL in IF shift is an exceedingly small value (about 66.5nm), it is extremely challenging to accurately measure it. However, weak measurements [30, 31] based on the signal enhancement technique have proven to be successful to resolve this problem. In 2008, this technique was used to observe the SHEL for the first time by Hosten and Kwiat [9]. Afterwards, this technique was widely used to estimate a variety of small physical parameters such as beam deflections [32], phase shifts [33,34], frequency shifts [35], and even temperature shifts [36].

The experimental setup is shown in Fig. 3. A Gaussian beam at 632.8 nm generated by a He-Ne laser passes through a half-wave plate for adjusting the light intensity. Then, the beam is focused by a lens (L1) and preselected by a Glan polarizer (P1) and a quarter-wave plate (QWP). And then, the beam is reflected at the air-prism interface where the SHEL in IF shift takes place. The incident angle is chosen as Brewster angle. Finally, the reflected beam is postselected by a Glan polarizer (P2) and collimated by a lens (L3). In the measurement, the transverse shift of barycenter of beam is recorded by CCD.

More concretely, the preselection of the system is achieved with an elliptical polarization state |*ψ _{i}*〉 by P1 and QWP. Then, upon reflection at the air-prism interface, the weak coupling between an observable (i.e., spin operator

*σ̂*

_{3}= |+〉〈+| − |−〉〈−|) of the system and a meter variable (i.e., transverse wave-vector component of the wave packet

*k*

_{yi}) leads to a spin splitting of the SHEL in IF shift

*δ*, which can be regarded as a consequence of the geometrical Berry phase exp[−

*ik*

_{yr}

*δσ̂*

_{3}], see Eq. (9). The evolution of the total wave function can also be expressed as

*R̂*=

*diag*(

*r*,

_{p}*r*) describes the change of polarization state of incident beam on reflection.

_{s}Finally, the postselection of the system is achieved with a linear polarization state |*ψ _{f}*〉 = cos

*β*|

*H*〉 + sin

*β*|

*V*〉 by P2, where

*β*is the postselection angle. After postselection, the small spin-dependent transverse shift is convert into a large transverse shift of the whole light field. The final wave function is given by

*z*is the free propagation distance) and |

*ψ′*〉 =

_{i}*R̂*|

*ψ*〉 is the modified preselection after reflection. Provided that

_{i}*σ̂*

_{3}

^{2}= 1, the geometrical Berry phase can be expanded to all orders with exp [−

*ik*

_{yr}

*δσ̂*

_{3}] = cos (

*k*

_{yr}

*δ)*−

*iσ̂*

_{3}sin (

*k*

_{yr}

*δ)*. The equation (12) can be rewritten as

*ω*

_{0}is the focused beam waist). Note that we consider that |

*β*| ≫ |

*δ*/

*ω*

_{0}| (i.e., the condition of the linear approximation in weak measurements [30]) in Eq. (17). The whole amplification factor from weak measurements is defined as

Figure 4(b) shows the experimental results of crossed polarization intensity profiles at different azimuth angles. These images captured by a color CCD are in agreement with the numerical simulations by calculating the intensity profile after P2, see Fig. 4(a). We see that the intensity profile of bilateral symmetry is gradually transformed into central symmetry, and finally transformed into longitudinal symmetry with the increase of azimuth angle. This is because in-plane and out-plane wave vectors (i.e., *k*_{xr} and *k*_{yr}) play different roles at different azimuth angles. When azimuth angle is small, *χ* cot *α* in Eq. (17) is large. From Eq. (3), we know that in-plane wave vector is closely related with the first-order derivative *χ*. Therefore, a large *χ* cot *α* shows that in-plane wave vector plays a leading role in the intensity distribution, which gives arise to the bilateral symmetry. With the increase of azimuth angle, *χ* cot *α* becomes small. The effect of in-plane wave vector on the intensity distribution is reduced. When *α* = 65°, both in-plane and out-plane wave vectors have the same effect on the intensity distribution, which gives arise to the central symmetry. With the further increase of azimuth angle, out-plane wave vector plays a leading role in the intensity distribution, which gives arise to the longitudinal symmetry.

Figures 5(a)–5(e) show the amplified shifts of spin splitting of the SHEL in IF shift as a function of the postselection angle. Here, the azimuth angles are selected as *α* = 30°, 45°, 60°, 75° and 90°. The black solid lines and red solid circles represent the theoretical predictions and experimental results, respectively. Note that there is no any approximate condition in Eq. (15) when the preselection and postselection are nearly orthogonal [37]. We can see that the amplified shifts first quickly increase with the postselection angle. Then, after reaching the peak value, the amplified shifts decrease gradually. From Eq. (7), we know that the IF shift is only decided by the azimuth angle. For different azimuth angles corresponding to different IF shifts, both the peak value of amplified shift and its corresponding postselection angle are different. Figure 5(f) shows the amplified shift and its corresponding spin splitting of the SHEL in IF shift as a function of azimuth angle at postselection angle *β* = 1.8°. The red solid line and the green dash line represent the theoretical predictions derived from Eqs. (15) and (10), respectively. The red hollow circles correspond to the amplified shift derived from measurement. The green solid diamonds correspond to the spin splitting in IF shift derived from the amplified shift divided by the amplification factor in Eq. (18). These experimental results are good agreement with the theoretical calculation.

In addition, although we have measured the spin splitting of the SHEL in IF using weak measurements and confirmed that the amplified shift is influenced by the IF shift, the IF shift has still not been measured in our experiment. This is because weak measurements are only helpful to the IF shift in some specially incident polarization states (e.g., the incident polarization state is a 45° linear polarization state.) [38], they are helpless to the IF shift when the incident polarization state is an elliptical polarization state in our paper. However, it is very possible to solve this issue using the standard lock-in amplifier based on heterodyne detection [39,40].

## 4. Conclusion

In conclusion, we have revealed the SHEL in IF shift when a circular or elliptically polarized Gaussian beam is reflected from the air-prism interface at the Brewster angle. The IF shift changes with the incident polarization state, while the spin splitting of the SHEL in IF is a fixed value. We found that this phenomenon is attributed to the spin-orbit coupling in IF shift. Moreover, the weak measurement method is employed to measure the spin splitting of the SHEL in IF shift. These findings not only provide a pathway for modulating the SHEL but also are useful for deep understanding of beam shifts.

## Funding

The National Key R&D Program of China (Grant No. 2017YFB0405704); The National Natural Science Foundation of China (Grant Nos. 11674234 and 11604095); The Hunan Provincial Natural Science Foundation of China (Grant No. 2017JJ3209); The Fundamental Research Funds for the Central Universities (Grant No. 2012017yjsy143).

## References

**1. **A. W. Snyder and J. D. Love, “Goos–Hänchen shift,” Appl. Opt. **15**, 236–238 (1976). [CrossRef] [PubMed]

**2. **C. Imbert, “Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam,” Phys. Rev. D **5**, 787–796 (1972). [CrossRef]

**3. **F. I. Fedorov, “To the theory of total reflection,” J. Opt. **15**, 014002 (2013). [CrossRef]

**4. **M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. **93**, 083901 (2004). [CrossRef] [PubMed]

**5. **K. Y. Bliokh and Y. P. Bliokh, “Conservation of Angular Momentum, Transverse Shift, and Spin Hall Effect in Reflection and Refraction of an Electromagnetic Wave Packet,” Phys. Rev. Lett. **96**, 073903 (2006). [CrossRef] [PubMed]

**6. **X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. **80**, 066401 (2017). [CrossRef] [PubMed]

**7. **V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A **46**, 5199–5207 (1992). [CrossRef] [PubMed]

**8. **K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interaction of light,” Nat. Photo. **9**, 796 (2015). [CrossRef]

**9. **O. Hosten and P. Kwiat, “Observation of the Spin Hall Effect of Light via Weak Measurements,” Science **319**, 787–790 (2008). [CrossRef] [PubMed]

**10. **Y. Qin, Y. Li, H. He, and Q. Gong, “Measurement of spin Hall effect of reflected light,” Opt. Lett. **34**, 2551–2553 (2009). [CrossRef] [PubMed]

**11. **S. Goswami, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Simultaneous weak value amplification of angular Goos–Hänchen and Imbert–Fedorov shifts in partial reflection,” Opt. Lett. **39**, 6229–6232 (2014). [CrossRef] [PubMed]

**12. **S. Goswami, S. Dhara, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Optimized weak measurements of Goos–Hänchen and Imbert–Fedorov shifts in partial reflection,” Opt. Express **24**, 6041–6051 (2016). [CrossRef] [PubMed]

**13. **Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the Spin-Based Plasmonic Effect in Nanoscale Structures,” Phys. Rev. Lett. **101**, 043903 (2008). [CrossRef] [PubMed]

**14. **Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. **109**, 013901 (2012). [CrossRef] [PubMed]

**15. **J.-M. Ménard, A. E. Mattacchione, M. Betz, and H. M. van Driel, “Imaging the spin Hall effect of light inside semiconductors via absorption,” Opt. Lett. **34**, 2312–2314 (2009). [CrossRef] [PubMed]

**16. **X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic Spin Hall Effect at Metasurfaces,” Science **339**, 1405–1407 (2013). [CrossRef] [PubMed]

**17. **W. Zhu, M. Jiang, H. Guan, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Tunable spin splitting of Laguerre–Gaussian beams in graphene metamaterials,” Photon. Res. **5**, 684–688 (2017). [CrossRef]

**18. **X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A **88**, 053840 (2013). [CrossRef]

**19. **W. J. M. Kort-Kamp, “Topological Phase Transitions in the Photonic Spin Hall Effect,” Phys. Rev. Lett. **119**, 147401 (2017). [CrossRef] [PubMed]

**20. **X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A **85**, 043809 (2012). [CrossRef]

**21. **X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. **101**, 251602 (2012). [CrossRef]

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

**23. **T. Tang, J. Li, L. Luo, P. Sun, and J. Yao, “Magneto-Optical Modulation of Photonic Spin Hall Effect of Graphene in Terahertz Region,” Adv. Opt. Mater. **6**, 1701212 (2018). [CrossRef]

**24. **X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. **41**, 4032–4035 (2016). [CrossRef] [PubMed]

**25. **L. Xie, X. Qiu, L. Luo, X. Liu, Z. Li, Z. Zhang, J. Du, and D. Wang, “Quantitative detection of the respective concentrations of chiral compounds with weak measurements,” Appl. Phys. Lett. **111**, 191106 (2017). [CrossRef]

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

**27. **H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhanced and switchable spin hall effect of light near the brewster angle on reflection,” Phys. Rev. A **84**, 043806 (2011). [CrossRef]

**28. **K. Y. Bliokh and A. Aiello, “Goos–Hänchen and Imbert–Fedorov beam shifts: an overview,” J. Opt. **15**, 014001 (2013). [CrossRef]

**29. **X. Zhou and X. Ling, “Unveiling the photonic spin Hall effect with asymmetric spin-dependent splitting,” Opt. Express **24**, 3025 (2016). [CrossRef] [PubMed]

**30. **Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin −1/2 particle can turn out to be 100,” Phys. Rev. Lett. **60**, 1351–1354 (1988). [CrossRef] [PubMed]

**31. **J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. **86**, 307–316 (2014). [CrossRef]

**32. **P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification,” Phys. Rev. Lett. **102**, 173601 (2009). [CrossRef] [PubMed]

**33. **X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase Estimation with Weak Measurement Using a White Light Source,” Phys. Rev. Lett. **111**, 033604 (2013). [CrossRef] [PubMed]

**34. **X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. **110**, 071105 (2017). [CrossRef]

**35. **D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A **82**, 063822 (2010). [CrossRef]

**36. **L. J. Salazar-Serrano, D. Barrera, W. Amaya, S. Sales, V. Pruneri, J. Capmany, and J. P. Torres, “Enhancement of the sensitivity of a temperature sensor based on fiber bragg gratings via weak value amplification,” Opt. Lett. **40**, 3962 (2015). [CrossRef] [PubMed]

**37. **X. Zhu, Y. Zhang, S. Pang, C. Qiao, Q. Liu, and S. Wu, “Quantum measurements with preselection and postselection,” Phys. Rev. A **84**, 052111 (2011). [CrossRef]

**38. **G. Jayaswal, G. Mistura, and M. Merano, “Observation of the Imbert–Fedorov effect via weak value amplification,” Opt. Lett. **39**, 2266–2269 (2014). [CrossRef] [PubMed]

**39. **H. Gilles, S. Girard, and J. Hamel, “Simple technique for measuring the Goos–Hänchen effect with polarization modulation and a position-sensitive detector,” Opt. Lett. **27**, 1421–1423 (2002). [CrossRef]

**40. **X. Yin and L. Hesselink, “Goos–Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. **89**, 261108 (2006). [CrossRef]