## Weak measurements of a large spin angular splitting of light beam on reflection at the Brewster angle |

Optics Express, Vol. 20, Issue 14, pp. 16003-16009 (2012)

http://dx.doi.org/10.1364/OE.20.016003

Acrobat PDF (1036 KB)

### Abstract

We reveal a large spin angular splitting of light beam on reflection at the Brewster angle both theoretically and experimentally. A simple weak measurements system manifesting itself for the built-in post-selection technique is proposed to explore this angular splitting. Remarkably, the directions of the spin accumulations can be switched by adjusting the initial handedness of polarization.

© 2012 OSA

## 1. Introduction

1. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. **93**(8), 083901 (2004). [CrossRef] [PubMed]

2. 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**(7), 073903 (2006). [CrossRef] [PubMed]

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

5. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. **33**(13), 1437–1439 (2008). [CrossRef] [PubMed]

6. 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**(15), 2312–2314 (2009). [CrossRef] [PubMed]

7. 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**(4), 043809 (2012). [CrossRef]

8. N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett. **36**(16), 3200–3202 (2011). [CrossRef] [PubMed]

9. Y. Qin, Y. Li, X. Feng, Y. F. Xiao, H. Yang, and Q. Gong, “Observation of the in-plane spin separation of light,” Opt. Express **19**(10), 9636–9645 (2011). [CrossRef] [PubMed]

10. C. C. Chan and T. Tamir, “Angular shift of a Gaussian beam reflected near the Brewster angle,” Opt. Lett. **10**(8), 378–380 (1985). [CrossRef] [PubMed]

12. M. Merano, N. Hermosa, A. Aiello, and J. P. Woerdman, “Demonstration of a quasi-scalar angular Goos-Hänchen effect,” Opt. Lett. **35**(21), 3562–3564 (2010). [CrossRef] [PubMed]

13. C. Leyder, M. Romanelli, J. Ph. Karr, E. Giacobino, T. C. H. Liew, M. M. Glazov, A. V. Kavokin, G. Malpuech, and A. Bramati, “Observation of the optical spin Hall effect,” Nat. Phys. **3**, 628–631 (2007). [CrossRef]

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

4. Y. Qin, Y. Li, H. He, and Q. Gong, “Measurement of spin Hall effect of reflected light,” Opt. Lett. **34**(17), 2551–2553 (2009). [CrossRef] [PubMed]

15. 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**(14), 1351–1354 (1988). [CrossRef] [PubMed]

16. 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**(4), 043806 (2011). [CrossRef]

17. N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE **7950**, 79500F (2011). [CrossRef]

18. N. Hermosa, A. Aiello, and J. P. Woerdman, “Radial mode dependence of optical beam shifts,” Opt. Lett. **37**(6), 1044–1046 (2012). [CrossRef] [PubMed]

## 2. Theoretical analysis

*z*-axis of the laboratory Cartesian frame (

*x*,

*y*,

*z*) is normal to the air-prism interface. We use the coordinate frames (

*x*,

_{i}*y*,

_{i}*z*) and (

_{i}*x*,

_{r}*y*,

_{r}*z*) to denote incidence and reflection, respectively. A left- or right-elliptically polarized light beam incidents at the air-prism interface. Here, we choose the long and short axis of the elliptical polarization beam along to the

_{r}*x*- and

_{i}*y*-axis, respectively. The elliptical polarization light beam can be decomposed into two orthogonal polarization components

_{i}*H*and

*V*. It is noted that the mechanism of the elliptical polarization beam reflection on the prism at the Brewster angel acts as a built-in post-selection in which the

*H*component is mainly cut off and is equal to the

*V*component. That is to say, this mechanism takes the role of the second polarizer in the precious weak measurements technique. After reflection, the

*H*and

*V*components overlap and induce the large spin angular splitting. Additionally, the reflection coefficient of

*H*polarization component changes its sign across the Brewster angle, which means the induced total circular polarization reverses its handedness [Fig. 1(b) and 1(c)]. In other words, the two reflected beams are ”colored” by different circular polarization. Selection of circular polarization in these beams is the post-selection procedure in the weak-measurement technique.

*H*and

*V*polarizations can be written as:

*w*

_{0}is the beam waist. The reflected angular spectrum can be obtained from Eq. (1). In the spin basis,

*Ẽ*can be obtained from the boundary conditions:

_{r}*k*= −

_{ix}*k*and

_{rx}*k*=

_{iy}*k*.

_{ry}*e*sin Δ)

^{iφ}*. Here Δ represents the azimuth angle (the angle between the crystal axis of wave plate and the*

^{T}*x*-axis) and

_{i}*φ*denotes the phase difference between the two polarization components

*H*and

*V*. In the present study, we consider a elliptical polarization beam with its long and short axis along to the

*x*- and

_{i}*y*-axis. Therefore the Jones vector will be simplified to (cos Δ, +

_{i}*i*sin Δ)

*or (cos Δ, −*

^{T}*i*sin Δ)

*representing the left- or right-elliptical polarization in the case of angle*

^{T}*φ*= ±

*π*/2. And we note here that the azimuth angle Δ mentioned in the following is a tiny value allowing for a slightly elliptical polarization and its long axis along to the

*x*-axis.

_{i}*i*sin Δ)

*. Therefore, according to Eqs. (1) and (2), we can obtain the reflected angular spectrum: Here,*

^{T}*δ*= (1 +

_{ry}*r*/

_{s}*r*) cot

_{p}*θ*/

_{i}*k*

_{0}and

*η*=

*ik*+

_{ry}δ_{ry}*r*tan Δ/

_{s}*r*. At any given plane

_{p}*z*=

_{r}*const*., the transverse displacement of field centroid compared to the geometrical-optics prediction is given by where

*ξ̃*

_{r±}=

*r*cos Δ(1 +

_{p}*i*tan Δ

*k*±

_{ry}δ_{ry}*η*)

**Ẽ**

_{r}_{±}. We note that there needs a theoretical correction and the higher-order terms should be taken into account when the beam is incident near the Brewster angle [16

16. 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**(4), 043806 (2011). [CrossRef]

*r*and

_{p}*r*can be expanded as a polynomial of

_{s}*k*: Using this method, we can obtain the theoretical shift of the single circular polarization component induced by angular splitting in the case of left-elliptical polarization:

_{ix}*z*is the propagation distance. It should be noted that the reflected light beam will experience a spatial shift in the case of linear polarization and a angular displacement according to the elliptical polarization. In this work, we only consider the elliptical polarization in which the large spin angular splitting is explored.

_{r}## 3. Weak measurements system

*S*

_{3}which reveals the circular polarization state of the angular splitting [13

13. C. Leyder, M. Romanelli, J. Ph. Karr, E. Giacobino, T. C. H. Liew, M. M. Glazov, A. V. Kavokin, G. Malpuech, and A. Bramati, “Observation of the optical spin Hall effect,” Nat. Phys. **3**, 628–631 (2007). [CrossRef]

*f*= 50mm to generate beam waist

*w*

_{0}= 18.66

*μ*m and make the incident ligt beam at the Brewster angle by modulating the GLP1 along to the

*x*-axis. Then we select the incident light beam as sligltlly left-elliptical polarization by modulating the Δ=0.5° from the

_{i}*x*-axis. Limited by the large holders of the knife edge and diaphragm, the angular splitting at small propagation distance are not measured. We measure the displacements every 10mm from 150mm to 300mm [Fig. 3(b)]. The detected splitting value reaches about 1500

_{i}*μ*m at the plane of

*z*= 300mm. The solid lines represent the theoretical predictions. The experimental results are in good agreement with the theory without using parameter fit.

_{r}*S*

_{3}is introduced to describe the circular polarization state of the spin angular splitting. Here,

*S*

_{3}=+1 or −1 represents the left- or right-circular polarization. Figure 4(a) and 4(b) illustrate the theoretical polarization distribution of reflected light beam considering the different left- and right-elliptical polarization incident beam. We can clearly see that, with the incident beam elliptical polarization state changing from left (right) to right (left), the spin angular splitting will reverse the direction. As an analogy of SHE in electronic system [13

13. C. Leyder, M. Romanelli, J. Ph. Karr, E. Giacobino, T. C. H. Liew, M. M. Glazov, A. V. Kavokin, G. Malpuech, and A. Bramati, “Observation of the optical spin Hall effect,” Nat. Phys. **3**, 628–631 (2007). [CrossRef]

*S*

_{3}. The experimental setup is similar to the first experiment. A new Glan laser polarizer (GLP2) and a new quarter-wave plate (QWP2) are added behind the prism. The last Glan laser polarizer, quarter-wave plate and CCD establish a general experimental system for measuring polarization distribution. By rotating the GLP2 to two angles and holding the QWP2 along to the

*y*-axis, we can conclude the Stokes parameter

_{r}*S*

_{3}from the intensity distributions on CCD. The rotation angles are 45° and 135°, the deviations from the configuration of the

*y*-axis. The experimental results shown in Fig. 4(c) and 4(d) are in good agreement with the theoretical calculation.

_{r}## 4. Conclusions

*μ*m at

*z*= 300mm. As an analogy of SHE in semiconductor microcavity [13

_{r}**3**, 628–631 (2007). [CrossRef]

## Acknowledgments

## References and links

1. | M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. |

2. | 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. |

3. | O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science |

4. | Y. Qin, Y. Li, H. He, and Q. Gong, “Measurement of spin Hall effect of reflected light,” Opt. Lett. |

5. | A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. |

6. | 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. |

7. | 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 |

8. | N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett. |

9. | Y. Qin, Y. Li, X. Feng, Y. F. Xiao, H. Yang, and Q. Gong, “Observation of the in-plane spin separation of light,” Opt. Express |

10. | C. C. Chan and T. Tamir, “Angular shift of a Gaussian beam reflected near the Brewster angle,” Opt. Lett. |

11. | M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics |

12. | M. Merano, N. Hermosa, A. Aiello, and J. P. Woerdman, “Demonstration of a quasi-scalar angular Goos-Hänchen effect,” Opt. Lett. |

13. | C. Leyder, M. Romanelli, J. Ph. Karr, E. Giacobino, T. C. H. Liew, M. M. Glazov, A. V. Kavokin, G. Malpuech, and A. Bramati, “Observation of the optical spin Hall effect,” Nat. Phys. |

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,” arXiv: 1204. 0378v2. |

15. | 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. |

16. | 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 |

17. | N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE |

18. | N. Hermosa, A. Aiello, and J. P. Woerdman, “Radial mode dependence of optical beam shifts,” Opt. Lett. |

19. | H. Luo, X. Ling, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhancing or suppressing the spin Hall effect of light in layered nanostructures,” Phys. Rev. A |

**OCIS Codes**

(240.0240) Optics at surfaces : Optics at surfaces

(260.5430) Physical optics : Polarization

(240.3695) Optics at surfaces : Linear and nonlinear light scattering from surfaces

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 16, 2012

Revised Manuscript: June 20, 2012

Manuscript Accepted: June 25, 2012

Published: June 28, 2012

**Citation**

Xinxing Zhou, Hailu Luo, and Shuangchun Wen, "Weak measurements of a large spin angular splitting of light beam on reflection at the Brewster angle," Opt. Express **20**, 16003-16009 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-16003

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### References

- M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett.93(8), 083901 (2004). [CrossRef] [PubMed]
- 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(7), 073903 (2006). [CrossRef] [PubMed]
- O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science319(5864), 787–790 (2008). [CrossRef] [PubMed]
- Y. Qin, Y. Li, H. He, and Q. Gong, “Measurement of spin Hall effect of reflected light,” Opt. Lett.34(17), 2551–2553 (2009). [CrossRef] [PubMed]
- A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett.33(13), 1437–1439 (2008). [CrossRef] [PubMed]
- 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(15), 2312–2314 (2009). [CrossRef] [PubMed]
- 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. A85(4), 043809 (2012). [CrossRef]
- N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett.36(16), 3200–3202 (2011). [CrossRef] [PubMed]
- Y. Qin, Y. Li, X. Feng, Y. F. Xiao, H. Yang, and Q. Gong, “Observation of the in-plane spin separation of light,” Opt. Express19(10), 9636–9645 (2011). [CrossRef] [PubMed]
- C. C. Chan and T. Tamir, “Angular shift of a Gaussian beam reflected near the Brewster angle,” Opt. Lett.10(8), 378–380 (1985). [CrossRef] [PubMed]
- M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics3(6), 337–340 (2009). [CrossRef]
- M. Merano, N. Hermosa, A. Aiello, and J. P. Woerdman, “Demonstration of a quasi-scalar angular Goos-Hänchen effect,” Opt. Lett.35(21), 3562–3564 (2010). [CrossRef] [PubMed]
- C. Leyder, M. Romanelli, J. Ph. Karr, E. Giacobino, T. C. H. Liew, M. M. Glazov, A. V. Kavokin, G. Malpuech, and A. Bramati, “Observation of the optical spin Hall effect,” Nat. Phys.3, 628–631 (2007). [CrossRef]
- 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,” arXiv: 1204. 0378v2.
- 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(14), 1351–1354 (1988). [CrossRef] [PubMed]
- 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. A84(4), 043806 (2011). [CrossRef]
- N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE7950, 79500F (2011). [CrossRef]
- N. Hermosa, A. Aiello, and J. P. Woerdman, “Radial mode dependence of optical beam shifts,” Opt. Lett.37(6), 1044–1046 (2012). [CrossRef] [PubMed]
- H. Luo, X. Ling, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhancing or suppressing the spin Hall effect of light in layered nanostructures,” Phys. Rev. A84(3), 033801 (2011). [CrossRef]

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