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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 1975–1980
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Variation of polarization distribution of reflected beam caused by spin separation

Yu Jin, Zefang Wang, Yang Lv, Hao Liu, Ruifeng Liu, Pei Zhang, Hongrong Li, Hong Gao, and Fuli Li  »View Author Affiliations


Optics Express, Vol. 20, Issue 3, pp. 1975-1980 (2012)
http://dx.doi.org/10.1364/OE.20.001975


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Abstract

The variation of polarization distribution of reflected beam at specular interface and far field caused by spin separation has been studied. Due to the diffraction effect, we find a distinct difference of light polarization at the two regions. The variation of polarization distribution of reflected light provides a new method to measure the spin separation displacement caused by Spin Hall Effect of light.

© 2012 OSA

1. Introduction

In this paper, we have systematically studied the polarization property of the reflected beam and demonstrated the variation of polarization distribution from the interface to far field. The polarization distribution at interface is determined by spin separation (SHEL and IPSSL), while the polarization variation during the propagation is caused by diffraction effect. In our theory, upon the reflection at an air-glass interface, a linearly polarized incident light becomes a beam with elliptically polarized distribution. However, during the propagation after reflected, the polarization ellipticity angle gradually vanish and the major axis rotate. At far field, the reflected beam eventually evolves to linearly polarized distribution. This feature provides a feasible way to measure the displacement caused by SHEL. Our experimental results show a perfect agreement with the theoretical prediction.

2. Theoretical analysis

2.1 General description of the reflected beam

To ensure the integrity of our theory, it is necessary to review the general description of spin separation at interface [4

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

, 6

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

]. The general SHEL experimental setup and coordinate systems are shown in Fig. 1
Fig. 1 Experimental setup and coordinate systems for studying the polarization distribution. The He-Ne laser generates a fundamental Gaussian beam with the wavelength of 632.8nm.HWP, half-wave plate for adjusting the intensity after P1. L1 and L2, lenses with 100mm and 150mm focal lengths, respectively. P1 and P2, Glan polarizers. In our experiment we set the reflect interface at the back focal plane of L1 and at the front focal plane of L2. Our observation plane (CCD) is chosen at the back focal plane of L2. The refractive index of glass prism is 1.49.
. We consider the wave-packets containing distribution of wave vector k(I)and k(R)for incident beam and reflected beam, which can be expressed as: k(I)=kxIx^I+kyy^I+kIz^I and k(R)=kxRx^R+kyy^R+kRz^Rwith|kxI,R|,|kyI,R|kI,R. x^I,R,y^I,Randz^I,Rare the unit vectors along xI,R,yI,RandzI,Raxis, andkI=2π/λwith λ is the wavelength of light in the air. Since y^I,Rare always vertical direction, they can replaced by y^thereafter. We can get kI=kR from Snell’s law. For arbitrary incident wave packet, we have kxI=kxRand kyI=kyR. Considering that the incident beam is not exact planar wave, we can express Fresnel reflection coefficients, rsandrp, as:rs=rsθI+(kxI/kI)rs/θI, rp=rpθI+(kxI/kI)rp/θI, where θIis incident angle, rsθIand rpθIare s- and p-wave reflection coefficients of planar wave at incident angle of θI.

Under the action of the interface, horizontally polarized incident beam |Hand vertically polarized incident beam|V evolve as:
|H(k(I))rpθI(|H(k(R))+kxRΔH|H(k(R))+kyδH|V(k(R)),
(1a)
|V(k(I))rsθI(|V(k(R))+kxRΔV|V(k(R))kyδV|H(k(R)),
(1b)
where
δH=cotθI(1rsθI/rpθI)/kI,δV=cotθI(1rpθI/rsθI)/kI,ΔH=1(rp/rpθI1)/kxI,ΔV=1(rs/rsθI1)/kxI.
δHandδVaccount for SHEL, while ΔH and ΔVaccount for in-plane spin separation. When a horizontally polarized light with a certain intensity distribution incident on the interface, from Eq. (1a), the reflected beam then can be expressed as:
|ϕfinal(kxR,ky)=rpθIϕinitial(kxR,ky)(|H+kxRΔH|H+kyδH|V),
(2)
where, ϕinitial(kx,ky) and ϕfinal(kx,ky)are the wave functions of incident and reflected beam at interface in momentum space, respectively. The above equation can also be expressed in spin basis to get more physical meaning as described in Refs [4

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

, 6

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

]. Then Fourier transformation gives the spatial distribution of reflected beam at the interface as:
|ψfinal(x,y)=rpθI2ψinitial(xiΔH,yδH)|++rpθI2ψinitial(xiΔH,y+δH)|=rpθI2[ψinitial(xiΔH,yδH)+ψinitial(xiΔH,y+δH)]|H+rpθIi2[ψinitial(xiΔH,yδH)ψinitial(xiΔH,y+δH)]|V,
(3)
where, ψinitial(x,y)and ψfinal(x,y)are the wave functions of incident and reflected beam at spatial space, respectively.|H=12(|++i|)and |V=12(|+i|). To obtain Eq. (3) we assume kxRΔH(V),kyδH(V)1. Following the same procedure, the description of reflected beam with vertical polarization can also be obtained, and the result is similar.

2.2 Variation of polarization distribution of the reflected beam

In this section, we only focus on the horizontally polarized incident beam and discuss its polarization properties. The result of vertically polarized is quite similar. Arbitrary polarization beam can be obtained by the linear combination of horizontal and vertical polarization. For realistic, the incident beam has a Gaussian shape as ψinitial(x,y)ex2+y2w02or ϕinitial(kx,ky)ew02(kx2+ky2)4. To better describe the polarization properties of the reflected beam, we use two parameters, the orientation angle γand the ellipticity angle χ.γdenotes the angle between the major semi-axis of the ellipse and the horizontal (x-axis) direction. χ=arccot(ε), where εis the ellipticity defined as the major-to-minor-axis ratio. Therefore γand χcan be calculated by the following equations:
tan(2γ)=2Re(EHEV*)|EH|2|EV|2,tan(2χ)=2Im(EHEV*)(|EH|2|EV|2)2+(2Re(EHEV*))2.
(4)
EHandEVare the horizontal and vertical components of the reflected beam’s wave function, respectively. Using Eqs. (3) and (4), the polarization properties of reflected beam at interface can be obtained. Figure 2(a)
Fig. 2 The results of theoretical calculation of the polarization properties at interface (a), (c) and at far field (b), (d). (a) and (b) are schematics to show the polarization property. (c) shows the calculation value ofχ(rad) at the interface and (d) shows the calculation value ofγ(rad) after 10m propagation for the reflected beam. Note that the incident beam is horizontally polarized and beam waist at the interface is 10 μm.
and 2(c) show the polarization distribution of reflected light at a certain position of the beam cross section, where (a) is the schematic plot and (c) is the calculation results. When y = 0, it is horizontally polarized. At the other position, it is elliptically polarized and the major-axis is parallel to xR-axis. Their spin directions are different between the two sides of y = 0.

After the reflect beam propagates for a long distance zR (kIw02/(2zR)2π), by using Fraunhofer diffraction theory, we can describe its property as follows:
ψ(xR,y,zR)=eikRzRiλzReikRxR2+y22zRF{ψfinal(x,y)}=eikIzRiλzReikIxR2+y22zRϕfinal(kIxRzR,kIyzR).
(5)
Fdenotes Fourier transformation. Using Eqs. (2), (4) and (5), the polarization properties of reflected beam at far field can be revealed and they are shown in Fig. 2(b) and 2(d). Unlike the results at the interface, the polarization distribution of reflected beam becomes pure linear at a certain position of the beam cross section. The polarization at y = 0 is still horizontally polarized, while the polarization at other position has rotated a small angle. And the directions of rotation are different between two sides of y = 0.

2.3 Measurement of y-displacement and intensity distribution at the back focus plane

To validate the polarization of reflected beam becomes purely linear at far field, we introduce a lens L2 with a focal length of f. We set the reflect interface at the front focal plane of L2 and the observation plane at its back focal plane. Considering the Fourier Transformation of L2, the wave function at its back focal plane can be expressed as: ψ(xR,y,zf)F{ψfinal(xR,y,0)}=ϕfinal(kIxRf,kIyf), which is similar to Eq. (5). Thus the polarization is also linear at the back focal plane, which is equivalent to far-field when L2 hasn’t been introduced. According to Eqs. (2) and (5), the polarization of the reflected beam at observation plane can be expressed by the polarization angle, θ:

tanθ=kIyfδH/(1+kIxRfΔH).
(7)

The above equation shows the polarization angle at a certain position of observation plane is determined by the position and displacements. When the incident beam is horizontally polarized, the polarization angle at xR-axis is horizontal. If we introduce a polarizer P2 with vertical polarizing direction, there will be a dark fringe at xR-axis, as show in Fig. 4(a)
Fig. 4 Intensity profiles of reflected beam at the back focal plane of lens L2. (a), (b) and (c) show the theoretical results. (d), (e) and (f) are the experimental results.
and 4(d).

Δθ=kIΔyfδHorΔy=ΔθδHkIf.
(8)

Thus the spin separation displacement (δH) of SHEL can be obtained by knowing ΔyandΔθ. We can also use the polarization property to obtain the intensity distribution at the observation plane (back focal plane of L2). It can be expressed as:

I(xR,y,f)|ew2(kI2xR2+kI2y2)/(4f2)[δHkIycos(Δθ)/f+(1+ΔHkIxR/f)sin(Δθ)]|2.
(9)

Thus the intensity profiles of reflected beam at back focal plane with different rotating angleΔθ of P2 can be obtained, and the theoretical results are shown in Fig. 4(a)4(c).

3. Experiment

Above theoretical analysis shows SHEL has great influence on the polarization distribution of reflected beam. According to Eqs. (7) and (8), an amplified factor of kIy/f converts δH into variation of polarization angle. Thus when we rotate P2, a noticeable shift of dark fringe caused by a tiny displacement of SHEL can be observed. By measuring dark fringe shift at certain rotation angle of P2, SHEL displacement can be obtained. To versify our theoretical prediction, we accomplish an experiment to measure Δyversus Δθ. The experimental setup is shown in Fig. 1. The light with 632.8nm generated by He-Ne laser passes through HWP and P1, and becomes horizontally polarized. When the beam reflects from the interface, it suffers from polarization evolution as discussed above. We use a charge-coupled device (CCD) to record different intensity profiles when we rotate the P2. The results are shown in Fig. 4(d)4(f). Clearly, the dark fringe moves as Δθchanges. We then measure the shift of dark fringe along the y-axis, Δy, by processing the pictures we get from CCD. The experimental data are showing in Fig. 5
Fig. 5 Relation betweenΔy(the distance dark fringe move along y-axis) andΔθ(the angle rotate away from vertical direction) for horizontally polarized incident beam. The circles, triangles and dots are experimental data at three incident angles: 30, 45 and 66°. The solid lines are theoretical results. The displacements of SHEL, δH, are calculated of 89.5, 231.5 and −213.3nm at three incident angles 30, 45 and 66°, respectively.
. The solid lines are the theoretical calculation based on Eq. (8) by using our experimental parameters, where δHat different incident angles are calculated from Eq. (1). The experimental data and theory agree perfectly. Note that at certain incident angle the displacement of SHEL is fixed and it determines the slope of the line.

4. Conclusion

The variation of polarization distribution of reflected beam at the interface and far field caused by spin separation has been studied. We find a distinct difference of light polarization between the two regions due to the diffraction effect. The polarization evolution of light also provides a new method to measure the spin separation displacement caused by Spin Hall Effect of light. Our experimental results exhibit good agreement with the theoretical prediction.

Acknowledgments

We thank Dr. Dong Wei for very useful discussion on this work. We acknowledge financial support from the National Natural Science Foundation of China (NSFC) under grants 11074198, 11004158, 11174233, and Special Prophase Project on the National Basic Research Program of China under grants 2011CB311807.

References and links

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.

K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(6), 066609 (2007). [CrossRef] [PubMed]

4.

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.

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

6.

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

7.

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]

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: December 5, 2011
Revised Manuscript: January 4, 2012
Manuscript Accepted: January 5, 2012
Published: January 13, 2012

Citation
Yu Jin, Zefang Wang, Yang Lv, Hao Liu, Ruifeng Liu, Pei Zhang, Hongrong Li, Hong Gao, and Fuli Li, "Variation of polarization distribution of reflected beam caused by spin separation," Opt. Express 20, 1975-1980 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-1975


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References

  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. K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(6), 066609 (2007). [CrossRef] [PubMed]
  4. O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science319(5864), 787–790 (2008). [CrossRef] [PubMed]
  5. Y. Qin, Y. Li, H. Y. He, and Q. H. Gong, “Measurement of spin Hall effect of reflected light,” Opt. Lett.34(17), 2551–2553 (2009). [CrossRef] [PubMed]
  6. Y. Qin, Y. Li, X. B. Feng, Y. F. Xiao, H. Yang, and Q. H. Gong, “Observation of the in-plane spin separation of light,” Opt. Express19(10), 9636–9645 (2011). [CrossRef] [PubMed]
  7. 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]

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