Experimental observation of the spin Hall effect of light on a nano-metal film via weak measurements
aa r X i v : . [ phy s i c s . op ti c s ] A p r Experimental observation of the spin Hall effect of light on a nano-metal film via weakmeasurements
Xinxing Zhou, Zhicheng Xiao, Hailu Luo, ∗ and Shuangchun Wen † Key Laboratory for Micro-/Nano-Optoelectronic Devices of Ministry of Education,College of Information Science and Engineering,Hunan University, Changsha 410082, People’s Republic of China (Dated: October 30, 2018)We theorize the spin Hall effect of light (SHEL) on a nano-metal film and demonstrate it ex-perimentally via weak measurements. A general propagation model to describe the relationshipbetween the spin-orbit coupling and the thickness of the metal film is established. It is revealedthat the spin-orbit coupling in the SHEL can be effectively modulated by adjusting the thicknessof the metal film, and the transverse displacement is sensitive to the thickness of metal film incertain range for horizontal polarization light. Importantly, a large negative transverse shift canbe observed as a consequence of the combined contribution of the ratio and the phase difference ofFresnel coefficients.
PACS numbers: 42.25.-p, 42.79.-e, 41.20.Jb
I. INTRODUCTION
The spin Hall effect of light (SHEL) manifests itself asthe split of a linearly polarized beam into left- and right-circular components when a beam propagates throughhomogeneous media. The splitting in the SHEL, gov-erned by the angular momentum conservation law [1, 2],takes place as a result of an effective spin-orbit cou-pling. The SHEL is sometimes referred to as the Imbert-Fedorov effect, which was theoretically predicted by Fe-dorov and experimentally confirmed by Imbert [3, 4].Generally the transverse shift of the SHEL is on thesubwavelength scale, and it is difficult to be directlymeasured with the conventional experimental methods.In 2008, benefiting from the weak measurement tech-nique, Hosten and Kwiat first measured the transversedisplacement of refracted light [5]. The SHEL has beenwidely investigated in different physical systems, suchas high-energy physics [6, 7], plasmonics [8–10], opticalphysics [11–16], and semiconductor physics [17, 18].The SHEL holds great potential applications, such asmanipulating electron spin states and precision metrol-ogy [5]. The SHEL itself may become a useful metrologi-cal tool for characterizing the refractive index variationsof nanostructure. Thus, the relationship between SHELand nanostructure is important, yet it is not fully un-derstood. To measure the refractive index variations atsubwavelength scale, we need to establish the relation-ship between the SHEL and the structural parametersof the nanostructure. It is well known that the SHELmanifests itself as the spin-orbit coupling. Now a ques-tion arises: What role does the structural parameters ofthe subwavelength nanostructure play in the spin-orbitcoupling?In this work, we establish a general propagation model ∗ Electronic address: [email protected] † Electronic address: [email protected] to describe the SHEL on the nano-metal film and revealthe impact of the structural parameters on the SHEL.We find that the spin-orbit coupling in the SHEL canbe effectively modulated by adjusting the thickness ofthe metal film. It should be noted that the interestingSHEL on this structure is different from that on pureglass prism [5, 15], metal bulk [19], and layered nanos-tructure [20]. The paper is organized as follows. First, weanalyze the SHEL on the nano-metal film theoretically.Our findings indicate that the transverse displacementof the SHEL is sensitive to the thickness of the metalfilm and undergoes a large negative value for horizontalpolarization. Next, we focus our attention on the exper-iment (weak measurements). Here, the sample is a BK7substrate coated with a thin Ag film. The experimentalresults are in good agreement with the theory. Finally, aconclusion is given in the fourth section.
II. THEORETICAL MODEL
Figure 1 schematically illustrates the SHEL of beamreflection on a nano-metal film in Cartesian coordinatesystem. The z axis of the laboratory Cartesian frame( x, y, z ) is normal to the interface of the metal film at z = 0. The incident and reflected electric fields are pre-sented in coordinate frames ( x i , y i , z i ) and ( x r , y r , z r ),respectively. In the spin basis set, the angular spec-trum can be written as ˜ E Hi = ( ˜ E i + + ˜ E i − ) / √ E Vi = i ( ˜ E i − − ˜ E i + ) / √
2. Here, H and V representhorizontal and vertical polarizations, respectively. Thepositive and negative signs denote the left- and right-circularly polarized (spin) components, respectively.The incident monochromatic Gaussian beam can beformulated as a localized wave packet whose spectrum isarbitrarily narrow, and can be written as e E i ± = ( e ix + iσ e iy ) w √ π exp " − w ( k ix + k iy )4 , (1) FIG. 1: (Color online) Schematic of SHEL on a nano-metalfilm. A linearly polarized beam reflects on the model com-posed of air, Ag film and the BK7 glass substrate and thensplits into left- and right-circularly polarized light, respec-tively. δ + and δ − indicate the transverse shift of left- andright-circularly polarized components. Here, θ i is the inci-dent angle and the Goos-H¨anchen shift is not considered. where w is the beam waist. The polarization operator σ = ± (cid:20) e E Hr e E Vr (cid:21) = " r p k ry cot θ i ( r p + r s ) k − k ry cot θ i ( r p + r s ) k r s E Hi e E Vi (cid:21) . (2)Here, r p and r s denote Fresnel reflection coefficients forparallel and perpendicular polarizations, respectively. k is the wave number in free space.From Eqs. (1) and (2), we can obtain the expressionsof the reflected angular spectrum e E Hr = r p √ h exp(+ ik ry δ Hr ) e E r + + exp( − ik ry δ Hr ) e E r − i , (3) e E Vr = ir s √ h − exp(+ ik ry δ Vr ) e E r + + exp( − ik ry δ Vr ) e E r − i . (4)Here, δ Hr = (1 + r s /r p ) cot θ i /k , δ Vr = (1 + r p /r s ) cot θ i /k , and e E r ± can be written as e E r ± = ( e rx + iσ e ry ) w √ π exp " − w ( k rx + k ry )4 . (5)It is known that the spin-orbit coupling is the intrin-sic physical mechanism of the SHEL. We note that,in Eqs. (3) and (4), the terms exp( ± ik ry δ Hr ) and theexp( ± ik ry δ Vr ) indicate the spin-orbit coupling terms inthe case of horizontal and vertical polarizations [5]. The spin-orbit coupling terms stem from the transverse na-ture of the photon polarization: The polarizations asso-ciated with the plane-wave components undergo differentrotations in order to satisfy the transversality after reflec-tion [5]. We can find that increasing or decreasing term δ H,Vr will significantly enhance or suppress the spin-orbitcoupling effect.It is noted that the real parts of the spin-orbit couplingterms δ H,Vr denote the spatial shift of the SHEL [21].Hence, we can obtain the initial transverse displacementof the SHEL on the nano-metal film: δ H ± = ∓ λ π (cid:20) | r s || r p | cos( ϕ s − ϕ p ) (cid:21) cot θ i , (6) δ V ± = ∓ λ π (cid:20) | r p || r s | cos( ϕ p − ϕ s ) (cid:21) cot θ i , (7)where r p,s = | r p,s | exp( iϕ p,s ) and λ is wavelength of theincident beam. Calculating the reflected shifts of theSHEL requires the explicit solution of the boundary con-ditions at the interfaces. Thus, we need to know thegeneralized Fresnel reflection of the metal film, r A = R A + R ′ A exp(2 ik p ε − sin θ i d )1 + R A R ′ A exp(2 ik p ε − sin θ i d ) . (8)Here, A ∈ { p, s } , R A and R ′ A is the Fresnel reflection coef-ficients at the first interface and second interface, respec-tively. ε and d represent the permittivity and thicknessof the metal film, respectively.To obtain a clear physical picture, we plot Fig. 2 to re-veal what role the thickness of the nano-meta film playsin the spin-orbital coupling. Figure 2(a) and 2(b) showthe initial transverse shifts of the SHEL with differentfilm thickness. In the case of horizontal polarization, wefind that the transverse displacement is extremely sensi-tive to the thickness when it is less than about 10nm. Wefind that this interesting phenomenon is attributed to thelarge variations of | r s | / | r p | [Fig. 2(c)]. However, as forvertical polarizations, the transverse shift is insensitiveto the thickness because of small variations of | r p | / | r s | [Fig. 2(d)]. It should be noted that, from Eqs. (3) and (6),the term of | r s | / | r p | plays a dominant role in spin-orbitcoupling. Hence, we can enhance or suppress the SHELeffectively by modulating the thickness of the metal film.Similar effect can also be observed in a layered nanos-tructures, in which the transverse displacement changesperiodically with the air gap increasing or decreasing dueto the optical Fabry-Perot resonance [20].In the case of horizontal polarization, the transverseshift experiences large negative value [Fig. 2(a)], whichis different from the SHEL on a metal bulk [19]. FromEqs. (6) and (7), we can find that, for a fixed incident an-gle, negative shifts entail the combined contributions ofthe large ratio of Fresnel coefficients ( | r s | / | r p | or | r p | / | r s | )and phase difference induced negative cos( ϕ s − ϕ p ) orcos( ϕ p − ϕ s ) [Fig. 2(e) and 2(f)] which are due to thematerial properties of the metal film. We conclude that θ i (degrees) δ + V ( n m ) d=2 nmd=5 nmd=10 nmd=30 nmd=60 nm (b) θ i (degrees) | r p | / | r s | (d) θ i (degrees) δ + H ( n m ) d=2 nmd=5 nmd=10 nmd=30 nmd=60 nm (a) θ i (degrees) | r s | / | r p | (c) θ i (degrees) c o s ( ϕ s - ϕ p ) (e) θ i (degrees) c o s ( ϕ p - ϕ s ) (f) FIG. 2: (Color online) Role of the thickness of nano-meta filmin the SHEL. (a) and (b) represent the transverse displace-ments of the SHEL on the thin Ag film under the condition ofhorizontal and vertical polarization. We choose the thicknessof the thin metal film from 2 to 60nm. (c) and (d) show thevalue of | r s | / | r p | and | r p | / | r s | . (e) and (f) denote the value ofcos( ϕ s − ϕ p ) and cos( ϕ p − ϕ s ) for the different thickness. Here,the permittivity of Ag is chosen as ε = −
18 + 0 . i and therefractive index of the BK7 substrate is chosen as n = 1 . . large negative transverse displacement only exists in thecase of horizontal polarization while always is positive un-der the condition of vertical polarizations. It is indicatedthat by rotating the polarization of incident light beam,we are able to switch the direction of the spin accumu-lation [22] effectively. Similar phenomena also occur inelectronic system. Here, the spin accumulation can beswitched by altering the directions of an external mag-netic field [23–25]. By rotating the polarization plane ofthe exciting light, the directions of spin current can beswitched in a semiconductor microcavity due to the spinHall effect [26, 27]. III. EXPERIMENTAL OBSERVATION
To detect the tiny transverse shifts, we use the sig-nal enhancement technique known as the weak measure-ments [28, 29]. Note that the weak measurements hasattracted a lot of attention and holds great promise forprecision metrology [30–35]. The theoretical analysis ofthe SHEL on nano-metal film has yielded two major re-sults: sensitive SHEL in extremely thin metal film and
FIG. 3: (Color online) Experimental setup: Sample, a BK7glass substrate coated with Ag film. L1 and L2, lenses witheffective focal length 50mm and 250mm, respectively. HWP,half-wave plate (for adjusting the intensity). P1 and P2,Glan Laser polarizers. CCD, charge-coupled device (CoherentLaserCam HR). The light source is a 17mW linearly polar-ized He-Ne laser at 632 . large negative beam shift of horizontal polarized inci-dent beam. However, we inevitably face a major obsta-cle that prevents us from experimentally corroboratingthe first claim because fabricating Ag film thinner than10nm would unavoidably involve large technical errors.Nonetheless, we still attempt to verify the validity of ourtheory by measuring the SHEL in large film thickness. Inthis section, we choose the BK7 glass substrate coated Agfilm as our sample (with three different thickness 10nm,30nm and 60nm.Our experimental setup shown in Fig. 3 is similar tothat in Refs [5, 15]. A Gauss beam generated by a He-Ne laser firstly impinges onto the HWP which is used tocontrol the light intensity to prevent the charge-coupleddevice (CCD) from saturation. And then, the light beampasses through a short focal length lens (L1) and a po-larizer (P1) to produce an initially linearly polarized fo-cused beam. When the beam reaches the sample inter-face, the SHEL takes place. The sample is a BK7 glasssubstrate coated with a thin Ag film whose permittiv-ity is ε = −
18 + 0 . i at 632 . ◦ ± ∆. Inour weak measurements experiment, we choose the angle∆ = 0 . ◦ . Then we use L2 to collimate the beam andmake the beam shifts insensitive to the distance betweenL2 and the CCD. Finally, we use a CCD to measure theamplified shift after L2. It should be mentioned that theamplified factor A w is not a constant, which verifies thesimilar result of our previous work [22].We measure the displacements of the SHEL on thenano-metal film every 5 ◦ from 30 ◦ to 85 ◦ in the case ofhorizontal and vertical polarization, respectively. Lim- FIG. 4: (Color online) The amplified displacements of hor-izontal polarized (left column) and vertical polarized (rightcolumn) beam reflection on Ag film with different thicknesses:[(a),(b)] 10nm and 12nm, [(c),(d)] 30nm, and [(e),(f)] 60nm. A w represents the amplified factor of the weak measurements.The lines indicate the theoretical value and the dots show theexperimental results (the error ranges are less than 10 µ m).The insets show the measured field distributions from theCCD. ited by the large holders of the lens, polarizers and He-Ne laser, displacements at small incident angles were notmeasured. Figure 4 plots the amplified displacement inboth theoretically and experimentally. In the case of hor-izontal polarization, the shift first experiences a negativevalue and then increases with the incident angle. Af-ter reaching the peak value in the incident angle about75 ◦ , the shift decreases rapidly. For three different thick-nesses, the negative shifts are vary. With the thicknessincreasing, the range of negative shift decreases. In thecase of vertical polarization, the shift first increases withthe incident angle and also decreases rapidly after the peak value. But, there exists no negative values com-pared with the horizontal polarization.It should be noted that the experimental results are ingood agreement with the theoretical ones when the filmthicknesses are 30nm and 60nm [Fig. 4(c)-(f)]. However,we observe a small deviation when the thickness is 10nm[Fig. 4(a) and 4(b)]. Note that the thickness of the nano-meta film has an error in the range of ± IV. CONCLUSIONS
In conclusion, we have observed the SHEL on a nano-metal film experimentally via weak measurements. Wehave found that the spin-orbit coupling effect can be ef-fectively manipulated by adjusting the thickness of themetal film. Our findings indicate that the transverse dis-placement is sensitive to the thickness of the metal filmin certain range. Hence, altering the metal film thick-ness will enhance or suppress the SHEL significantly. Asan analogy of spin Hall effect in an electronic system, weare able to switch the directions of the spin accumulationin SHEL effectively by rotating the polarization of inci-dent light beam. These findings provide a pathway formodulating the SHEL and thereby open the possibilityof developing nanophotonic applications.
Acknowledgments
One of the authors (X. Z.) thanks Dr. Y. Qin andDr. N. Hermosa for helpful discussions. We are sincerelygrateful to the anonymous referee, whose comments haveled to a significant improvement on our paper. This re-search was supported by the National Natural ScienceFoundation of China (Grants Nos. 61025024, 11074068). [1] M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev.Lett. , 083901 (2004).[2] K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. ,073903 (2006).[3] F. I. Fedorov, Dokl. Akad. Nauk SSSR , 465 (1955).[4] C. Imbert, Phys. Rev. D , 787 (1972).[5] O. Hosten and P. Kwiat, Science , 787 (2008).[6] P. Gosselin, A. B´erard, and H. Mohrbach, Phys. Rev. D , 084035 (2007).[7] C. A. Dartora, G. G. Cabrera, K. Z. Nobrega, V. F. Mon-tagner, M. H. K. Matielli, F. K. R. de Campos, and H. T. S. Filho, Phys. Rev. A , 012110 (2011).[8] Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, Phys.Rev. Lett. , 043903 (2008).[9] K. Y. Bliokh , Y. Gorodetski, V. Kleiner, and E. Hasman,Phys. Rev. Lett. , 030404 (2008).[10] L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M.Planken, and H. P. Urbach, Phys. Rev. Lett. , 083903(2010).[11] K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, NaturePhoton. , 748 (2008).[12] A. Aiello and J. P. Woerdman, Opt. Lett. , 1437 (2008).[13] D. Haefner, S. Sukhov, and A. Dogariu, Phys. Rev. Lett. , 123903 (2009).[14] O. G. Rodr´ıguez-Herrera, D. Lara, K. Y. Bliokh, E.A. Ostrovskaya, and C. Dainty, Phys. Rev. Lett. ,253601 (2010).[15] Y. Qin, Y. Li, H. He, and Q. Gong, Opt. Lett. , 2551(2009).[16] H. Luo, S. Wen, W. Shu, Z. Tang, Y. Zou, and D. Fan,Phys. Rev. A , 043810 (2009).[17] J.-M. M´enard, A. E. Mattacchione, M. Betz, and H. M.van Driel, Opt. Lett. , 2312 (2009).[18] J.-M. M´enard, A. E. Mattacchione, H. M. van Driel,C. Hautmann, and M. Betz, Phys. Rev. B , 045303(2010).[19] N. Hermosa, A. M. Nugrowati, A. Aiello and J. P. Wo-erdman, Opt. Lett. , 3200 (2011).[20] H. Luo, X. Ling, X Zhou, W. Shu, S. Wen, and D. Fan,Phys. Rev. A , 033801 (2011).[21] A. Aiello, M. Merano, and J. P. Woerdman, Phys. Rev.A , 061801 (2009).[22] H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, Phys.Rev. A , 043806 (2011).[23] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung-wirth, and A. H. MacDonald, Phys. Rev. Lett. , 126603(2004).[24] T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S.Maekawa, Phys. Rev. Lett. , 156601 (2007). [25] G. Mih´aly, M. Csontos, S. Bord´acs, I. K´ezsmarki, T. Wo-jtowicz, X. Liu, B. Jank´o, and J. K. Furdyna, Phys. Rev.Lett. , 107201 (2008).[26] A. Kavokin, G. Malpuech, and M. Glazov, Phys. Rev.Lett. , 136601 (2005).[27] C. Leyder, M. Romanelli, J. Ph. Karr, E. Giacobino, T.C. H. Liew, M. M. Glazov, A. V. Kavokin, G. Malpuech,and A. Bramati, Nature Phys. , 628 (2007).[28] Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev.Lett. , 1351 (1988).[29] N. W. M. Ritchie, J. G. Story, and R. G. Hulet, Phys.Rev. Lett. , 1107 (1991).[30] G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph,and H. M. Wiseman, Phys. Rev. Lett. , 220405 (2005).[31] P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C.Howell, Phys. Rev. Lett. , 173601 (2009).[32] N. Brunner, and C. Simon, Phys. Rev. Lett. , 010405(2010).[33] S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P.Mirin, L. K. Shalm, and A. M. Steinberg, Science ,1170 (2011).[34] A. Feizpour, X. X. Xing, and A. M. Steinberg, Phys. Rev.Lett. , 133603 (2011).[35] O. Zilberberg, A. Romito, and Y. Gefen, Phys. Rev. Lett.106