Abstract

We report on the observation of the spin separation of light in the plane of incidence when a linearly polarized beam is reflected or refracted at a planar dielectric interface. Remarkably, the in-plane spin separation reaches hundreds of nanometers, comparable with the transverse spin separation induced by the well-known spin Hall effect of light. The observation is properly explained by considering the in-plane spread of wave-vectors. This study thus offers new insights on the spinoptics and may provide a potential method to control light in optical nanodevices.

© 2011 OSA

1. Introduction

Amazing shifts of a physical light beam at a planar dielectric interface have been investigated for a long time. Among them, the in-plane Goos-Hänchen (GH) shift in the plane of incidence and the transverse Imbert-Fedorov (IF) shift perpendicular to the plane of incidence are well-known [14]. While the (linear) GH shift or the (linear) IF shift exhibits a linear positional shift of the beam center, the angular GH shift or the angular IF shift manifests the angular deviation of the beam axis, which can be considered as the shift in the wave vector space. Recently, the spin Hall effect of light (SHEL) has attracted much attention because it offer a promising tool in quantum and classical optical information processing applications [511]. As a result of an effective spin-orbit interaction, the reflected or transmitted beam slightly splits into its two spin components that acquire opposite transverse displacements perpendicular to the plane of incidence. This effect is also referred to as the optical Magnus effect of spinning particles [12,13]. SHEL has been studied at non-planar interfaces [1418] and applied to probe the spatial distributions of electron spin states in semiconductors at the nanometer scale [19,20].

By analogy with the in-plane GH shifts and the transverse IF shifts, does exist the in-plane spin separation of light (IPSSL) different from the transverse spin separation induced by the SHEL when a linearly polarized beam is reflected or refracted at the planar dielectric interface? Here we report on the observation of the IPSSL. By considering the tiny in-plane spread of wave-vectors, we theoretically explain the experimental results, which exhibit excellent agreement. This observation thus not only consummates the completeness of beam shifts but also offers a way to control the direction and magnitude of the total spin separation.

2. Experimental setup

The experimental setup shown in Fig. 1 is similar to that in Refs.8-10. Along with the standard coordinate system (x, y, z) attached to the interface (z=0) and the incidence plane (y=0), we employ the beam coordinate systems (xa, y, za), where a=I, R, T denotes incident, reflected, transmitted beams, respectively. The za axis attaches to the direction of the central wave vector.

 

Fig. 1 Experimental setup for observing the in-plane spin separation of light. The He–Ne laser generates a Gaussian beam at 632.8 nm; HWP, half-wave plate for attenuating the intensity after P1 to prevent the position-sensitive detector (PSD) from saturating; L1 and L2, lenses with 25 and 125 mm focal lengths, respectively; P1 and P2, Glan polarizers; by replacing the PSD with a CCD, the intensity profiles of the beam after P2 can be captured. The insect shows the results when the incident angle θI= 30° and the polarization angle γI=30°.

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A He–Ne laser generates a Gaussian beam at 632.8 nm, passing through a lens L1 and a Glan polarizer P1 to produce an initially linearly polarized focused beam with a polarization angle γI (the angle between x I and the central electric-field-vector). The reflected or refracted beam splits into its two spin components: the component parallel (σ=+1, right-circularly polarized, denoted by |+>) and antiparallel (σ=−1, left-circularly polarized, denoted by|>) to the central wave vector. We first focus on the reflected beam. The polarizer P2 is rotated by an angle φp2 from y in order to be crossed with the polarization direction of the central wave vector of the reflected light. Here, tanφp2=tan(γI)rsθI/rpθI, where rsθI, rpθI are the Fresnel reflection coefficients for s and p plane waves at the incident angle of θI. A position-sensitive detector (PSD) is used to measure the amplified displacement after the collimation lens L2, and this allows for calculating the original separation induced by SHEL [710] and IPSSL. When the intensity profiles are captured, a color charge-coupled device (CCD) is used instead of the PSD.

3. Results

The experimental data are presented in Fig. 2 . In addition to the y-displacement of the |+> spin component induced by the SHEL, δy|+>γI, as shown in Fig. 2(a), we observe the remarkable IPSSL along the xR-direction, δx|+>γI, as shown in Fig. 2(b). For θI =30°(or 45°), the y-displacement decreases from 88.1 nm (or 224 nm) to 0 when γI increases from 0 (|H>) to 39° (or 29°); then, the |+> spin component shifts along the −y direction and increases to 58.5 nm (or 69.5 nm) at 90° (|V>). The zero values occur at γI = tan1rpθI/rsθI. For θI = 70°, however, the y-displacement monotonously decreases from 136 nm to 50.2 nm toward –y direction. The difference results from the sign of βθI = rsθI/rpθI: when θI<θB, βθI<0; and if θI>θB, βθI>0, where θB is the Brewster angle (~57° in the experiment). The xR-displacement is always in +xR direction and reaches maximum at γI=tan1|rpθI/rsθI| that is comparable with the maximal y-displacement.

 

Fig. 2 The dependence of δy|+> (a) and δx|+> (b) of the |+> spin component induced by SHEL and IPSSL on the polarization angleγI. The dots, circles and triangles are experimental data at three typical incident angles: 30°, 45° and 70°. The curves are the theoretical results. The insets show the theoretical prediction for a period from 0° to 180°.

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In the following, we theoretically demonstrate that the IPSSL originates from the tiny in-plane spread in x I direction of the wave-vector.

4. Theoretical analysis

4.1 Calculation of the transverse and in-plane spin separations at an interface

Here we derive the displacements induced by SHEL and IPSSL of the reflected beam at an interface, which are expressed in the spin basis [8,9]. Using the coordinate systems in Fig. 1, the incident beam can be considered as a wave-packet containing a distribution of wave-vectors k(I)=kI(z^I+κ(I))    centered around kIz^I, with κ(I) =(kxIx^I+kyy^)/kI =κxIx^I+κyy^, |κ(I)|<<1, and z^I is a unit vector along zI axis and kI=2π/λ with λ being the wavelength of the light in the incident medium. For the reflected beam: k(R)=kR(z^R+κ(R))    with κ(R) =(kxRx^R+kyy^)/kR=κxRx^R+κyy^, |κ(R)|<<1, and kR = kI for Snell’s Law. For an arbitrary wave-vector, k^(I)z^(z^k^(I))=k^(R)z^(z^k^(R)), connecting κ(I) and κ(R):kxI=eRkxR and kyI=kyR=ky where eR=cosθR/cosθI=1 with θR=πθI. In refraction, eT=cosθT/cosθI changes with the incident angle θI.

Because the eigenstates of the reflection and refraction are the linear p- and s-polarization states, |p(k(I,R))> and |s(k(I,R))>, we start with a horizontally and vertically polarized incident wave-packet in the s-p basis and expend everything up to first order in κ:

|H(k(I,R))>=|p(k(I,R))>cotθI,Rκy(I,R)|s(k(I,R))>   ,|V(k(I,R))   >=|s(k(I,R))   >+cotθI,Rκy(I,R)|p(k(I,R))>.

These are the states that a generic polarizer would produce, and they are accomplished with the aid of the relations: s^(k(I,R))=z^×k^(I,R)/sin(θI,R(k^(I,R))) and p^(k(I,R))=s^(k(I,R))×k^(I,R), in which 1sin(θI,R(k^(I,R)))1sinθI,RcotθI,RsinθI,RκxI,xR.

We first calculate the spin separations of the reflected beam. After reflection, s- and p-polarizations evolve as: |p(k(I))>rp|p(k(R))> and |s(k(I))>rs|s(k(R))>. We expand the Fresnel reflection coefficients rp,rs for p and s plane waves about the central wave vector (corresponding to rpθI,rsθIat the incident angle θI) [3, 7]:

rp=rpθI+rpθIkxIkI,rs=rsθI+rsθIkxIkI.

The first derivative in Eq. (2) is not considered in Refs. 7-10, though it is responsible for the angular GH effect [1,3,4,21]. The correction proportional to kxI induces the IPSSL. Putting everything together, we can obtain:

|H(k(I))>rpθI(|H(k(R))>+kxRΔH|H(k(R))>+kyδH|V(k(R))>),|V(k(I))>rsθI(|V(k(R))>+kxRΔV|V(k(R))>kyδV|H(k(R))>),
where

δH=cotθI(eRrsθI/rpθI)/kI,
δV=cotθI(eRrpθI/rsθI)/kI,
ΔH=eR(lnrp/θI)/kI=eR(rp/rpθI1)/kxI,
ΔV=eR(lnrs/θI)/kI=eR(rs/rsθI1)/kxI.

In the spin basis set |+>=12(|H>+i|V>) and |>=12(|H>i|V>), Eq. (3) are expressed by:

|H(k(I))>rpθI2[exp(ikyδH)exp(kxRΔH)|+>+exp(ikyδH)exp(kxRΔH)|>],|V(k(I))>irsθI2[exp(ikyδV)exp(kxRΔV)|+>exp(ikyδV)exp(kxRΔV)|>],
provided that kyδH,V<<1, kxRΔH,V<<1

An arbitrary input polarization can be defined as follows [79]:

|m(I)>=11+|m|2(|H>+m|V>),
where m is a complex number: m=mr+imi, and for m=i, |m(I)>=|+>; for m=i, |m(I)>=|>; |H> (m=0) and |V> (m=∞) represent the horizontal (along xI, γI=0°) and vertical polarization (along y, γI=90°), respectively. After reflection, using the Eqs. (1)-(6), the beam state |m(I)> evolves into (in the spin basis set, at the interface):

|φ>=rpθI2(1+|m|2)(1imβθI)exp[iky(δy|+>m+iΔy|+>m)]exp[ikxR(δx|+>m+iΔx|+>m)]|+>+rpθI2(1+|m|2)(1+imβθI)exp[iky(δy|>m+iΔy|>m)]exp[ikxR(δx|>m+iΔx|>m)]|>.

Here,

δy|σ>m=σδH+σmiβθI(δH+δV)+|m|2|βθI|2δV|1imσβθI|2,
Δx|σ>m=ΔH+σmiβθI(ΔH+ΔV)+|m|2|βθI|2ΔV|1imσβθI|2,
Δy|σ>m=mrβθI(δHδV)|1imσβθI|2,
δx|σ>m=σmrβθI(ΔHΔV)|1imσβθI|2,
where δy|σ> and δx|σ> are the displacements of |σ> (σ =+1 or −1) spin component, induced by SHEL and IPSSL, respectively. Equations (8) and (4) represent the interplay of the spin-orbit interaction (involving δHand δV) and the GH effect (involving ΔHand ΔV) for an arbitrary input polarization.

Equation (7) presents the analytical expression in the most general case of an arbitrarily polarized Gaussian beam. The reflected light beam experiences spin-dependent displacements not only along the y-axis (SHEL, δy|σ>m) but along the xR-axis (IPSSL, δx|σ>m). In addition, there exist the angular shifts (momentum changes, related to the terms involving Δx|σ>m, Δy|σ>m) that are not measured in our experiment. As a matter of fact, the experimental setup is only sensitive to linear displacements of the wave packets [8]. This setup converts the position displacements caused by the SHEL and IPSSL into a momentum shift, and it also converts the momentum shifts into a position displacement which is small compared to the measured displacements in the experiment.

For a linearly incident polarization, which can be described by the polarization angle γI from horizontal, Eq. (8) reduce to:

δx|σ>γI=σ2sin(2γR)(ΔHΔV),δy|σ>γI=σ(cos2γRδH+sin2γRδV),
Δx|σ>γI=cos2γRΔH+sin2γRΔV,Δy|σ>γI=12sin(2γR)(δHδV),
where cosγR=1/1+(tan(γI)rsθI/rpθI)2=1/1+tan2φP2. The angle φ P2 presents the rotation of the polarizer P2 from y in order to be crossed with the polarization direction of the central wave vector of the reflected light.

Equation (9.1) confirms the in-plane separation δx|σ>γIin xR direction in addition to the previously reported transverse separation δy|σ>γI in y direction. In Refs. 8-10, the first derivatives were neglected, rp=rpθI,rs=rsθI, ΔH,V = 0, as a result, only the SHEL was revealed. At the same time, there exist angular shift (momentum change) related terms in xR and y directions that are not explicitly dependent on the spin σ, as described by Eqs. (9.2).

Equations (9) clearly demonstrate the correlation and difference between spin separations and angular shifts for a linearly polarized incidence. The transverse spin separation in SHEL is determined by δH and δVor rsθI/rpθIwhile the angular IF shift is related to δ H-δ V. On the contrary, IPSSL is determined by ΔHV or the angle-of-incidence gradient of ln(r s/r p), while the angular GH shift is related to the ΔH and ΔV or the angle-of-incidence gradient of ln(r s) and ln(r p).

4.2 Calculation of the angular and linear GH/IF shifts

In the momentum-space representation, the reflected beam can be given by: |ψR>=Φ(kxR,ky)|kxR,ky>|ϕ>,with Φ(kxR,ky)exp[Λ2kI2(eR2kxR2+ky2)ikxR2+ky22kIzR]. From Eq. (7), we can also obtain the angular shifts, <kxRm> and <kyRm>, and zR-dependent parts of the linear shifts, <xRm> and <yRm>, i.e., the expectation values of the momentum and position of the photons:

<kxRm>=kI2ΛeR2(12kIeRθI+ΔH+|m|2|βθI|2ΔV1+|m|2βθI2),<kyRm>=kI2ΛmrβθI(δHδV)1+|m|2βθI2,
<xRm>=zRkI<kxRm>,<yRm>=zRkI<kyRm>,
where Λ=kI2w022 and w0 is the minimum waist. Equations (10) are obtained with the aid of the relations: <k>=eR<ψR|k|ψR>dkeR<ψR|ψR>dk, <r>=eR<ψR|ik|ψR>dkeR<ψR|ψR>dk. Although eR/θI=0, eT/θI0 for the refracted beam. Therefore, we keep this zero term in Eq. (10.1) for the unification of the expressions for both reflected and refracted light. Especially, for the linearly polarized incidence, Eqs. (10) reduce to:

<kxRγI>=kI2ΛeR2(12kIeRθI+Δx|σ>γI),<kyRγI>=kI2ΛΔy|σ>γI,
<xRγI>=zRkI<kxRγI>,<yRγI>=zRkI<kyRγI>.

Equations (10) and (11) are in accordance with those in Refs. 2, 4, and 7.

4.3 The intensity profile of the reflected beam after P2

The intensity profile of the reflected beam after P2 for a linearly polarized incidence can be obtained from the Eqs. (7):

Ip2γI|11+tan2γIrpθIwzR2cos2γRexp(xR2+y22wzR2)(δx|+>γIxR+δy|+>γIy)|2.
where wzR2=Λ+ikIzRkI2. Equation (12) describes a rotated double-peak intensity profile of the cross section of the light spot with a central dark fringe as shown in Fig. 3 . It can be also derived from Refs. 3, 7. The distance between the peaks is D=2kIΛ2+zR2Λ that is mainly determined by the beam waist. Therefore, this can be used to measure the beam waist.

 

Fig. 3 Cross sections and polarization distributions of a reflected Gaussian beam. (a), (b), and (c) are schematics representing a linearly polarized beam, the reflected beam for |H> incidence and the reflected beam for an arbitrary linearly polarized incident beam, respectively. The yellow arrows and ellipses indicate the polarization distributions. (d), (e) and (f) are the corresponding intensity profiles after the beams going through a crossed polarizer. The spin separation, the ellipticity and the rotation angle of the elliptical polarizations in (b) and (c) are exaggerated for a better view.

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When both IPSSL and SHEL take place, the cross sections and polarization distributions of a reflected Gaussian beam change greatly as schematically demonstrated in Fig. 3. Figure 3(a) presents the cross section of a linearly polarized Gaussian beam with a uniform polarization distribution, in which the |+> and |> components fully overlap. For the |H> (or |V>) incidence as shown in Fig. 3(b), only SHEL can be observed, i.e., the reflected beam splits only along y-axis. The polarization varies mainly in y-axis with a minor change in xR-axis due to the deflection in xR direction that is related to the Eq. (4.3). For an arbitrary linearly polarized incident beam, the polarization distribution of the reflected beam is shown in Fig. 3(c) and Fig. 1. Figures 3(d), 3(e) and 3(f) are the corresponding intensity profiles after the beams going through a crossed polarizer. For |H> incidence, the profile shows a double-peak structure with a dark fringe at the center. As shown in Fig. 3(f), the double-peak profile rotates an angle of φdf=tan1(δx|+>/δy|+>) from xR-axis when IPSSL occurs.

In the kxR-ky momentum space, the polarizations are linear with an angle of tan1[(kyδH+βθI(1+kxRΔV)tanγI)/(1+kxRΔHβθIkyδVtanγI)]from kxR-axis. For the |H> (or |V>) incidence, the angle is tan1[kyδH/(1+kxRΔH)] (or tan1[(1+kxRΔV)/δVky]), so the dark fringe would move along y-axis by slightly rotating the polarizer.

Figure 4 demonstrates the experimental results of the rotated double-peak intensity profiles at the incident angle of 30° and 70°, respectively. These images are captured by a color CCD and in excellent agreement with the numerical simulations by calculating the intensity profile of the reflected beam after P2.

 

Fig. 4 Intensity profiles of the reflected beam passing through a crossed polarizer as a function of the polarization angle γI. The incident angle θ I is 30° (a) and 70° (b), respectively. The experimental images of the intensity profiles are in excellent agreement with the numerical simulations in the upper row. The arrows indicate the total displacement of the |+> spin component.

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IPSSL does not explicitly occur for |H> or |V> input polarization, but it covertly affects spin separations. At first glance, the two cases seem the same, but they actually differ from each other. From Figs. 2 and 4, it can be clearly seen that the dark fringe rotates 180° when γI changes from |H> to |V> for θI<θB. IPSSL plays a role in these two cases, resulting in the opposite direction of the spin displacement. When θI>θB, the total spin separation rotates counterclockwise and returns clockwise when γI changes from |H> to |V>.

For the refracted beam, kxI=eTkxT and kyI=kyT=ky. The corresponding expressions for the spin separations can be obtained by replacing rs,p, rsθI,pθI, xR, zR, kxR, k(R), ψR, wzR, γR and eRwith ts,p, tsθI,pθI, xT, zT, kxT, k(T), ψT, wzT, γT and eT.

At the incident angle of 49.3°, δy|+>γI monotonously decreases from 33.3 nm to 23.3 nm and δx|+>γI grows from 0 to −8.9 nm and then returns to 0 when γI changes from |H> to |V>. The experimental data are in excellent agreement with theoretical predictions as shown in Fig. 5 . The separations are smaller than those of reflected beam because of the smaller changes of the Fresnel transmission coefficients and their first-order derivatives.

 

Fig. 5 Displacements of the |+> spin component of the refracted beam as the function of the polarization angle γI at the incident angle of 49.3°, induced by SHEL (δy|+>, triangles) and IPSSL (δx|+>, dots), respectively. The error ranges are less than 2 nm. The curves indicate the theoretical results. The inset shows the theoretical prediction for a period from 0° to 180°.

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5. Conclusion

In conclusion, we have observed a new type of spin separation i.e. the in-plane spin separation, different from the transverse spin separation induced by the well-known SHEL, when a linearly polarized Gaussian beam is reflected or refracted at a planar dielectric interface. The measurements of displacements and the intensity profiles exhibit excellent agreement with the theoretical predictions by taking into account of the tiny in-plane spread of wave-vectors. Our study broadens the comprehensive understanding of beam shifts. As a result of the simultaneous SHEL and the new IPSSL, both the magnitude and direction of the total splitting of the two spin components can be controlled by the incident angle or the polarization angle of the incident linearly polarized light beam.

Acknowledgements

The authors acknowledge financial support from the National Natural Science Foundation of China under Grant Nos. 10821062, 11023003 and 11074013, and the National Basic Research Program of China under Grant Nos. 2006CB921601 and 2007CB307001.

References and links

1. 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 3(6), 337–340 (2009). [CrossRef]  

2. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010). [CrossRef]  

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

4. K. Y. Bliokh, I. V. Shadrivov, and Y. S. Kivshar, “Goos-Hanchen and Imbert-Fedorov shifts of polarized vortex beams,” Opt. Lett. 34(3), 389–391 (2009). [CrossRef]   [PubMed]  

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

6. 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]  

7. 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]  

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

9. 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]  

10. Y. Qin, Y. Li, X. B. Feng, Z. P. Liu, H. Y. He, Y. F. Xiao, and Q. H. Gong, “Spin Hall effect of reflected light at the air-uniaxial crystal interface,” Opt. Express 18(16), 16832–16839 (2010). [CrossRef]   [PubMed]  

11. H. L. Luo, S. C. Wen, W. X. Shu, Z. X. Tang, Y. H. Zou, and D. Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80(4), 043810 (2009). [CrossRef]  

12. A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45(11), 8204–8208 (1992). [CrossRef]   [PubMed]  

13. A. Bérard and H. Mohrbach, “Spin Hall effect and Berry phase of spinning particles,” Phys. Lett. A 352(3), 190–195 (2006). [CrossRef]  

14. A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009). [CrossRef]   [PubMed]  

15. K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2(12), 748–753 (2008). [CrossRef]  

16. K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008). [CrossRef]   [PubMed]  

17. K. Y. Bliokh and A. S. Desyatnikov, “Spin and orbital Hall effects for diffracting optical beams in gradient-index media,” Phys. Rev. A 79(1), 011807 (2009). [CrossRef]  

18. D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009). [CrossRef]   [PubMed]  

19. J. M. Menard, 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]  

20. J. M. Ménard, A. E. Mattacchione, H. M. van Driel, C. Hautmann, and M. Betz, “Ultrafast optical imaging of the spin Hall effect of light in semiconductors,” Phys. Rev. B 82(4), 045303 (2010). [CrossRef]  

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References

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  1. 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 3(6), 337–340 (2009).
    [CrossRef]
  2. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
    [CrossRef]
  3. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hanchen and Imbert-Fedorov shifts,” Opt. Lett. 33(13), 1437–1439 (2008).
    [CrossRef] [PubMed]
  4. K. Y. Bliokh, I. V. Shadrivov, and Y. S. Kivshar, “Goos-Hanchen and Imbert-Fedorov shifts of polarized vortex beams,” Opt. Lett. 34(3), 389–391 (2009).
    [CrossRef] [PubMed]
  5. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
    [CrossRef] [PubMed]
  6. 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]
  7. 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]
  8. O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008).
    [CrossRef] [PubMed]
  9. 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]
  10. Y. Qin, Y. Li, X. B. Feng, Z. P. Liu, H. Y. He, Y. F. Xiao, and Q. H. Gong, “Spin Hall effect of reflected light at the air-uniaxial crystal interface,” Opt. Express 18(16), 16832–16839 (2010).
    [CrossRef] [PubMed]
  11. H. L. Luo, S. C. Wen, W. X. Shu, Z. X. Tang, Y. H. Zou, and D. Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80(4), 043810 (2009).
    [CrossRef]
  12. A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45(11), 8204–8208 (1992).
    [CrossRef] [PubMed]
  13. A. Bérard and H. Mohrbach, “Spin Hall effect and Berry phase of spinning particles,” Phys. Lett. A 352(3), 190–195 (2006).
    [CrossRef]
  14. A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
    [CrossRef] [PubMed]
  15. K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2(12), 748–753 (2008).
    [CrossRef]
  16. K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
    [CrossRef] [PubMed]
  17. K. Y. Bliokh and A. S. Desyatnikov, “Spin and orbital Hall effects for diffracting optical beams in gradient-index media,” Phys. Rev. A 79(1), 011807 (2009).
    [CrossRef]
  18. D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009).
    [CrossRef] [PubMed]
  19. J. M. Menard, 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]
  20. J. M. Ménard, A. E. Mattacchione, H. M. van Driel, C. Hautmann, and M. Betz, “Ultrafast optical imaging of the spin Hall effect of light in semiconductors,” Phys. Rev. B 82(4), 045303 (2010).
    [CrossRef]
  21. K. Artmann, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
    [CrossRef]

2010

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[CrossRef]

J. M. Ménard, A. E. Mattacchione, H. M. van Driel, C. Hautmann, and M. Betz, “Ultrafast optical imaging of the spin Hall effect of light in semiconductors,” Phys. Rev. B 82(4), 045303 (2010).
[CrossRef]

Y. Qin, Y. Li, X. B. Feng, Z. P. Liu, H. Y. He, Y. F. Xiao, and Q. H. Gong, “Spin Hall effect of reflected light at the air-uniaxial crystal interface,” Opt. Express 18(16), 16832–16839 (2010).
[CrossRef] [PubMed]

2009

K. Y. Bliokh, I. V. Shadrivov, and Y. S. Kivshar, “Goos-Hanchen and Imbert-Fedorov shifts of polarized vortex beams,” Opt. Lett. 34(3), 389–391 (2009).
[CrossRef] [PubMed]

J. M. Menard, 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]

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]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[CrossRef] [PubMed]

K. Y. Bliokh and A. S. Desyatnikov, “Spin and orbital Hall effects for diffracting optical beams in gradient-index media,” Phys. Rev. A 79(1), 011807 (2009).
[CrossRef]

D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009).
[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. Photonics 3(6), 337–340 (2009).
[CrossRef]

H. L. Luo, S. C. Wen, W. X. Shu, Z. X. Tang, Y. H. Zou, and D. Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80(4), 043810 (2009).
[CrossRef]

2008

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

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2(12), 748–753 (2008).
[CrossRef]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[CrossRef] [PubMed]

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

2007

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]

2006

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]

A. Bérard and H. Mohrbach, “Spin Hall effect and Berry phase of spinning particles,” Phys. Lett. A 352(3), 190–195 (2006).
[CrossRef]

2004

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

1992

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45(11), 8204–8208 (1992).
[CrossRef] [PubMed]

1948

K. Artmann, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[CrossRef]

Aiello, A.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[CrossRef]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[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. Photonics 3(6), 337–340 (2009).
[CrossRef]

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

Artmann, K.

K. Artmann, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[CrossRef]

Bérard, A.

A. Bérard and H. Mohrbach, “Spin Hall effect and Berry phase of spinning particles,” Phys. Lett. A 352(3), 190–195 (2006).
[CrossRef]

Betz, M.

J. M. Ménard, A. E. Mattacchione, H. M. van Driel, C. Hautmann, and M. Betz, “Ultrafast optical imaging of the spin Hall effect of light in semiconductors,” Phys. Rev. B 82(4), 045303 (2010).
[CrossRef]

J. M. Menard, 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]

Bliokh, K. Y.

K. Y. Bliokh, I. V. Shadrivov, and Y. S. Kivshar, “Goos-Hanchen and Imbert-Fedorov shifts of polarized vortex beams,” Opt. Lett. 34(3), 389–391 (2009).
[CrossRef] [PubMed]

K. Y. Bliokh and A. S. Desyatnikov, “Spin and orbital Hall effects for diffracting optical beams in gradient-index media,” Phys. Rev. A 79(1), 011807 (2009).
[CrossRef]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[CrossRef] [PubMed]

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2(12), 748–753 (2008).
[CrossRef]

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]

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]

Bliokh, Y. P.

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]

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]

Desyatnikov, A. S.

K. Y. Bliokh and A. S. Desyatnikov, “Spin and orbital Hall effects for diffracting optical beams in gradient-index media,” Phys. Rev. A 79(1), 011807 (2009).
[CrossRef]

Dogariu, A.

D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009).
[CrossRef] [PubMed]

Dooghin, A. V.

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45(11), 8204–8208 (1992).
[CrossRef] [PubMed]

Fan, D. Y.

H. L. Luo, S. C. Wen, W. X. Shu, Z. X. Tang, Y. H. Zou, and D. Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80(4), 043810 (2009).
[CrossRef]

Feng, X. B.

Gong, Q. H.

Gorodetski, Y.

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[CrossRef] [PubMed]

Haefner, D.

D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009).
[CrossRef] [PubMed]

Hasman, E.

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2(12), 748–753 (2008).
[CrossRef]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[CrossRef] [PubMed]

Hautmann, C.

J. M. Ménard, A. E. Mattacchione, H. M. van Driel, C. Hautmann, and M. Betz, “Ultrafast optical imaging of the spin Hall effect of light in semiconductors,” Phys. Rev. B 82(4), 045303 (2010).
[CrossRef]

He, H. Y.

Hermosa, N.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[CrossRef]

Hosten, O.

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

Kivshar, Y. S.

Kleiner, V.

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2(12), 748–753 (2008).
[CrossRef]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[CrossRef] [PubMed]

Kundikova, N. D.

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45(11), 8204–8208 (1992).
[CrossRef] [PubMed]

Kwiat, P.

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

Leuchs, G.

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[CrossRef] [PubMed]

Li, Y.

Liberman, V. S.

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45(11), 8204–8208 (1992).
[CrossRef] [PubMed]

Lindlein, N.

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[CrossRef] [PubMed]

Liu, Z. P.

Luo, H. L.

H. L. Luo, S. C. Wen, W. X. Shu, Z. X. Tang, Y. H. Zou, and D. Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80(4), 043810 (2009).
[CrossRef]

Marquardt, C.

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[CrossRef] [PubMed]

Mattacchione, A. E.

J. M. Ménard, A. E. Mattacchione, H. M. van Driel, C. Hautmann, and M. Betz, “Ultrafast optical imaging of the spin Hall effect of light in semiconductors,” Phys. Rev. B 82(4), 045303 (2010).
[CrossRef]

J. M. Menard, 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]

Menard, J. M.

Ménard, J. M.

J. M. Ménard, A. E. Mattacchione, H. M. van Driel, C. Hautmann, and M. Betz, “Ultrafast optical imaging of the spin Hall effect of light in semiconductors,” Phys. Rev. B 82(4), 045303 (2010).
[CrossRef]

Merano, M.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[CrossRef]

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 3(6), 337–340 (2009).
[CrossRef]

Mohrbach, H.

A. Bérard and H. Mohrbach, “Spin Hall effect and Berry phase of spinning particles,” Phys. Lett. A 352(3), 190–195 (2006).
[CrossRef]

Murakami, S.

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

Nagaosa, N.

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

Niv, A.

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2(12), 748–753 (2008).
[CrossRef]

Onoda, M.

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

Qin, Y.

Shadrivov, I. V.

Shu, W. X.

H. L. Luo, S. C. Wen, W. X. Shu, Z. X. Tang, Y. H. Zou, and D. Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80(4), 043810 (2009).
[CrossRef]

Sukhov, S.

D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009).
[CrossRef] [PubMed]

Tang, Z. X.

H. L. Luo, S. C. Wen, W. X. Shu, Z. X. Tang, Y. H. Zou, and D. Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80(4), 043810 (2009).
[CrossRef]

van Driel, H. M.

J. M. Ménard, A. E. Mattacchione, H. M. van Driel, C. Hautmann, and M. Betz, “Ultrafast optical imaging of the spin Hall effect of light in semiconductors,” Phys. Rev. B 82(4), 045303 (2010).
[CrossRef]

J. M. Menard, 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]

van Exter, M. P.

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 3(6), 337–340 (2009).
[CrossRef]

Wen, S. C.

H. L. Luo, S. C. Wen, W. X. Shu, Z. X. Tang, Y. H. Zou, and D. Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80(4), 043810 (2009).
[CrossRef]

Woerdman, J. P.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[CrossRef]

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 3(6), 337–340 (2009).
[CrossRef]

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

Xiao, Y. F.

Zel’dovich, B. Y.

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45(11), 8204–8208 (1992).
[CrossRef] [PubMed]

Zou, Y. H.

H. L. Luo, S. C. Wen, W. X. Shu, Z. X. Tang, Y. H. Zou, and D. Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80(4), 043810 (2009).
[CrossRef]

Ann. Phys.

K. Artmann, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[CrossRef]

Nat. Photonics

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2(12), 748–753 (2008).
[CrossRef]

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 3(6), 337–340 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

A. Bérard and H. Mohrbach, “Spin Hall effect and Berry phase of spinning particles,” Phys. Lett. A 352(3), 190–195 (2006).
[CrossRef]

Phys. Rev. A

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[CrossRef]

H. L. Luo, S. C. Wen, W. X. Shu, Z. X. Tang, Y. H. Zou, and D. Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80(4), 043810 (2009).
[CrossRef]

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45(11), 8204–8208 (1992).
[CrossRef] [PubMed]

K. Y. Bliokh and A. S. Desyatnikov, “Spin and orbital Hall effects for diffracting optical beams in gradient-index media,” Phys. Rev. A 79(1), 011807 (2009).
[CrossRef]

Phys. Rev. B

J. M. Ménard, A. E. Mattacchione, H. M. van Driel, C. Hautmann, and M. Betz, “Ultrafast optical imaging of the spin Hall effect of light in semiconductors,” Phys. Rev. B 82(4), 045303 (2010).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

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]

Phys. Rev. Lett.

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]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[CrossRef] [PubMed]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[CrossRef] [PubMed]

D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009).
[CrossRef] [PubMed]

Science

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

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Figures (5)

Fig. 1
Fig. 1

Experimental setup for observing the in-plane spin separation of light. The He–Ne laser generates a Gaussian beam at 632.8 nm; HWP, half-wave plate for attenuating the intensity after P1 to prevent the position-sensitive detector (PSD) from saturating; L1 and L2, lenses with 25 and 125 mm focal lengths, respectively; P1 and P2, Glan polarizers; by replacing the PSD with a CCD, the intensity profiles of the beam after P2 can be captured. The insect shows the results when the incident angle θI = 30° and the polarization angle γI =30°.

Fig. 2
Fig. 2

The dependence of δ y | + > (a) and δ x | + > (b) of the | + > spin component induced by SHEL and IPSSL on the polarization angle γ I . The dots, circles and triangles are experimental data at three typical incident angles: 30°, 45° and 70°. The curves are the theoretical results. The insets show the theoretical prediction for a period from 0° to 180°.

Fig. 3
Fig. 3

Cross sections and polarization distributions of a reflected Gaussian beam. (a), (b), and (c) are schematics representing a linearly polarized beam, the reflected beam for |H> incidence and the reflected beam for an arbitrary linearly polarized incident beam, respectively. The yellow arrows and ellipses indicate the polarization distributions. (d), (e) and (f) are the corresponding intensity profiles after the beams going through a crossed polarizer. The spin separation, the ellipticity and the rotation angle of the elliptical polarizations in (b) and (c) are exaggerated for a better view.

Fig. 4
Fig. 4

Intensity profiles of the reflected beam passing through a crossed polarizer as a function of the polarization angle γ I . The incident angle θ I is 30° (a) and 70° (b), respectively. The experimental images of the intensity profiles are in excellent agreement with the numerical simulations in the upper row. The arrows indicate the total displacement of the | + > spin component.

Fig. 5
Fig. 5

Displacements of the | + > spin component of the refracted beam as the function of the polarization angle γ I at the incident angle of 49.3°, induced by SHEL ( δ y | + > , triangles) and IPSSL ( δ x | + > , dots), respectively. The error ranges are less than 2 nm. The curves indicate the theoretical results. The inset shows the theoretical prediction for a period from 0° to 180°.

Equations (21)

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| H ( k ( I , R ) ) > = | p ( k ( I , R ) ) > cot θ I , R κ y ( I , R ) | s ( k ( I , R ) ) >     , | V ( k ( I , R ) )     > = | s ( k ( I , R ) )     > + cot θ I , R κ y ( I , R ) | p ( k ( I , R ) ) > .
r p = r p θ I + r p θ I k x I k I , r s = r s θ I + r s θ I k x I k I .
| H ( k ( I ) ) > r p θ I ( | H ( k ( R ) ) > + k x R Δ H | H ( k ( R ) ) > + k y δ H | V ( k ( R ) ) > ) , | V ( k ( I ) ) > r s θ I ( | V ( k ( R ) ) > + k x R Δ V | V ( k ( R ) ) > k y δ V | H ( k ( R ) ) > ) ,
δ H = cot θ I ( e R r s θ I / r p θ I ) / k I ,
δ V = cot θ I ( e R r p θ I / r s θ I ) / k I ,
Δ H = e R ( ln r p / θ I ) / k I = e R ( r p / r p θ I 1 ) / k x I ,
Δ V = e R ( ln r s / θ I ) / k I = e R ( r s / r s θ I 1 ) / k x I .
| H ( k ( I ) ) > r p θ I 2 [ exp ( i k y δ H ) exp ( k x R Δ H ) | + > + exp ( i k y δ H ) exp ( k x R Δ H ) | > ] , | V ( k ( I ) ) > i r s θ I 2 [ exp ( i k y δ V ) exp ( k x R Δ V ) | + > exp ( i k y δ V ) exp ( k x R Δ V ) | > ] ,
| m ( I ) > = 1 1 + | m | 2 ( | H > + m | V > ) ,
| φ > = r p θ I 2 ( 1 + | m | 2 ) ( 1 i m β θ I ) exp [ i k y ( δ y | + > m + i Δ y | + > m ) ] exp [ i k x R ( δ x | + > m + i Δ x | + > m ) ] | + > + r p θ I 2 ( 1 + | m | 2 ) ( 1 + i m β θ I ) exp [ i k y ( δ y | > m + i Δ y | > m ) ] exp [ i k x R ( δ x | > m + i Δ x | > m ) ] | > .
δ y | σ > m = σ δ H + σ m i β θ I ( δ H + δ V ) + | m | 2 | β θ I | 2 δ V | 1 i m σ β θ I | 2 ,
Δ x | σ > m = Δ H + σ m i β θ I ( Δ H + Δ V ) + | m | 2 | β θ I | 2 Δ V | 1 i m σ β θ I | 2 ,
Δ y | σ > m = m r β θ I ( δ H δ V ) | 1 i m σ β θ I | 2 ,
δ x | σ > m = σ m r β θ I ( Δ H Δ V ) | 1 i m σ β θ I | 2 ,
δ x | σ > γ I = σ 2 sin ( 2 γ R ) ( Δ H Δ V ) , δ y | σ > γ I = σ ( cos 2 γ R δ H + sin 2 γ R δ V ) ,
Δ x | σ > γ I = cos 2 γ R Δ H + sin 2 γ R Δ V , Δ y | σ > γ I = 1 2 sin ( 2 γ R ) ( δ H δ V ) ,
< k x R m > = k I 2 Λ e R 2 ( 1 2 k I e R θ I + Δ H + | m | 2 | β θ I | 2 Δ V 1 + | m | 2 β θ I 2 ) , < k y R m > = k I 2 Λ m r β θ I ( δ H δ V ) 1 + | m | 2 β θ I 2 ,
< x R m > = z R k I < k x R m > , < y R m > = z R k I < k y R m > ,
< k x R γ I > = k I 2 Λ e R 2 ( 1 2 k I e R θ I + Δ x | σ > γ I ) , < k y R γ I > = k I 2 Λ Δ y | σ > γ I ,
< x R γ I > = z R k I < k x R γ I > , < y R γ I > = z R k I < k y R γ I > .
I p 2 γ I | 1 1 + tan 2 γ I r p θ I w z R 2 cos 2 γ R exp ( x R 2 + y 2 2 w z R 2 ) ( δ x | + > γ I x R + δ y | + > γ I y ) | 2 .

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