Enhanced and switchable spin Hall effect of light near the Brewster angle on reflection
Hailu Luo, Xinxing Zhou, Weixing Shu, Shuangchun Wen, Dianyuan Fan
aa r X i v : . [ phy s i c s . op ti c s ] A ug Enhanced and switchable spin Hall effect of light near the Brewster angle on reflection
Hailu Luo, Xinxing Zhou, Weixing Shu, Shuangchun Wen, ∗ and Dianyuan Fan Key Laboratory for Micro/Nano Opto-Electronic Devices of Ministry of Education,College of Information Science and Engineering,Hunan University, Changsha 410082, People’s Republic of China (Dated: October 31, 2018)We reveal an enhanced and switchable spin Hall effect (SHE) of light near Brewster angle onreflection both theoretically and experimentally. The obtained spin-dependent splitting reaches3200nm near Brewster angle, 50 times larger than the previous reported values in refraction. Wefind that the amplifying factor in week measurement is not a constant which is significantly differentfrom that in refraction. As an analogy of SHE in electronic system, a switchable spin accumulationin SHE of light is detected. We were able to switch the direction of the spin accumulations byslightly adjusting the incident angle.
PACS numbers: 42.25.-p, 42.79.-e, 41.20.Jb
I. INTRODUCTION
The spin Hall effect (SHE) of light can be regarded as adirect optical analogy of SHE in electronic system wherethe spin electrons and electric potential are replacedby spin photons and refractive index gradient, respec-tively [1–3]. Recently, the SHE of light has been inten-sively investigated in different physical systems, such ashigh-energy physics [4], plasmonics [5], optical physics [6–9], and semiconductor physics [10]. The SHE of light isgenerally believed as a result of an effective spin-orbitalinteraction, which describes the mutual influence of thespin (polarization) and trajectory of the light beam. Ingeneral, the spin-dependent splitting in these physics sys-tems is limited by a fraction of the wavelength, and there-fore it is disadvantage for potential application to nano-photonic devices.The SHE in electronic system offer an effective way tomanipulate the spin particles, and open a promising wayto potential applications in semiconductor spintronic de-vices [11–13]. The generation and manipulation of spin-polarized electrons in semiconductors define the mainchallenges of spin-based electronics [14]. In semiconduc-tor systems, the spin accumulation can be switched byaltering the directions of external magnetic field [15, 16].By rotating the polarization plane of the exciting light,the directions of spin current can be switched in a semi-conductor micro-cavity [17, 18]. Now a question arises:Whether there exists a similar phenomenon in SHE oflight? In this paper, we want to reveal an enhanced andswitchable SHE of light near Brewster angle on reflec-tion. The SHE of light has been studied in reflectionboth in theory [19–21] and in experiment [22]. However,the developed paraxial propagation model cannot be ap-plied and the experimental evidence is still absent fordescribing the SHE of light near Brewster angle.The paper is organized as follows. First, we develop ageneral propagation model to describe the SHE of light ∗ Electronic address: [email protected] near Brewster angle on reflection. Next, we attemptto reveal the enhanced SHE of light in theory and de-tect the large spin-dependent splitting in experiment viaweek measurements. The large spin-dependent splittingis found to be attributed to the large ratio between theFresnel reflection coefficients near Brewster angle. Fi-nally, we want to explore the switchable SHE of light.We demonstrate that the transverse displacements canbe tuned to either a negative, or a positive value, or evenzero, by slightly adjusting the incident angle. The under-lying secret can be interpreted from that the horizontalfield component changes its phase across the Brewsterangle. As an analogy of SHE in electronic system, thespin accumulations can be switched in the SHE of light.
II. GENERAL PROPAGATION MODEL
We first develop a general propagation model to de-scribe the SHE of light near the Brewster angle on re-flection. The z axis of the laboratory Cartesian frame( x, y, z ) is normal to the air-prism interface. We use thecoordinate frames ( x i , y i , z i ) and ( x r , y r , z r ) to denote in-cident and reflection, respectively [Fig. 1(a)]. In the spinbasis set, the incident angular spectrum can be writtenas: ˜ E Hi = 1 √ E i + + ˜ E i − ) , (1)˜ E Vi = 1 √ i ( ˜ E i − − ˜ E i + ) . (2)Here, H and V represent horizontal and vertical po-larizations, respectively. ˜ E i + = ( e ix + i e iy ) ˜ E i / √ E i − = ( e ix − i e iy ) ˜ E i / √ E i = w √ π exp " − w ( k ix + k iy )4 , (3) FIG. 1: (color online) (a) Plane-wave components in fourquadrant acquire different polarization rotations upon reflec-tion to satisfy transversality. The polarizations associatedwith the angular spectrum components in incidence (b) expe-rience different rotations in reflection (c). where w is the beam waist. The complex amplitude forthe reflected beam can be conveniently expressed as E r ( x r , y r , z r ) = Z dk rx dk ry ˜ E r ( k rx , k ry ) × exp[ i ( k rx x r + k ry y r + k rz z r )] , (4)where k rz = q k r − ( k rx + k ry ) and ˜ E r ( k rx , k ry ) is thereflected angular spectrum.The reflected angular spectrum is related to the bound-ary distribution of the electric field by means of the re-lation [2] (cid:20) ˜ E Hr ˜ E Vr (cid:21) = " r p k ry ( r p + r s ) cot θ i k − k ry ( r p + r s ) cot θ i k r s ˜ E Hi ˜ E Vi (cid:21) , (5)where r p and r s denote the Fresnel reflection coefficientsfor parallel and perpendicular polarizations, respectively.By making use of Taylor series expansion based on thearbitrary angular spectrum component, r p and r s can beexpanded as a polynomial of k ix : r p,s ( k ix ) = r p,s ( k ix = 0) + k ix (cid:20) ∂r p,s ( k ix ) ∂k ix (cid:21) k ix =0 + N X j =2 k Nix j ! " ∂ j r p,s ( k ix ) ∂k jix k ix =0 . (6)The reflection coefficient changes its sign across theBrewster angle, which means the electric field reversesits directions [Fig. 1(b) and 1(c)]. The polarizations as-sociated with the angular spectrum components experi-ence different rotations in order to satisfy the boundarycondition after reflection.In the spin basis set, the reflected angular spectrumcan be written as:˜ E Hr = 1 √ E r + + ˜ E r − ) , (7) ˜ E Vr = 1 √ i ( ˜ E r − − ˜ E r + ) . (8)We consider the incident Gaussian beam with H po-larization. From the boundary condition, we obtain k rx = − k ix and k ry = k iy . In fact, after the incident an-gular spectrum is known, Eq. (4) together with Eqs. (3)-(8) provides the general expression of the reflected field: E Hr ± = r p ( e rx ± i e ry ) √ πw z R z R + iz r exp (cid:20) − k x r + y r z R + iz r (cid:21) × (cid:20) r p − ixz R + iz r ∂r p ∂θ i ± yz R + iz r ( r p + r s ) ± ixy ( z R + iz r ) (cid:18) ∂r p ∂θ i + ∂r s ∂θ i (cid:19) (cid:21) exp( ik r z r ) , (9)where z R = k w / III. SPIN HALL EFFECT OF LIGHT
We now determine the sin-dependent splitting of fieldcentroid. At any given plane z a = const., the trans-verse displacement of field centroid compared to thegeometrical-optics prediction is given by δ H ± = R R y r I H ± ( x r , y r , z r )d x r d y r R R I H ± ( x r , y r , z r )d x r d y r . (10)The intensity distribution of beam is closely linked to thelongitudinal momentum currents I ( x r , y r , z r ) ∝ p r · e rz .The time-averaged linear momentum density associatedwith the electromagnetic field can be shown to be p r ∝ Re[ E r × H ∗ r ], where the magnetic field can be obtainedby H r = − ik − r ∇ × E r .To detect the displacements, we use the signal en-hancement technique [3] known from weak measure-ments [23, 24]. In principle, this enhancement mecha-nism of this setup can be perfectly presented in a classi-cal description [20]. Figure 2 illustrates the experimen-tal setup. A Gaussian beam generated by a He-Ne laserpasses through a short focal length lens (Lens1) and a po-larizer (GLP1) to produce an initially polarized focusedbeam. When the beam impinges onto the prism interface,the SHE of light generates. The prism was mounted to arotation stage which allows for precise control of the in-cident angle θ i . The incident beam is preselected in the H polarization state ( α = 0) by GLP1, and then postse-lected ( β = π/ V = sin ∆ e rx + cos ∆ e ry . (11)In our measurement, we chose ∆ = 2 ± . ◦ . Note thatthe interesting cross polarization effect can be observedas ∆ = 0 [25]. As the reflected beam of light splits by sev-eral wavelengths, the intensity distribution on the prism FIG. 2: (Color online) (a) Experimental setup for charac-terizing the SHE of light in reflection near Brewster angle.Prism with refractive index n = 1 .
515 (BK7 at 632 . . α and β with x r . interface is nearly unchanged. After the second polarizerGLP2, the two splitting components interfere, and pro-duce a field redistribution whose centroid is significantlyamplified. We use a CCD to measure the amplified dis-placement after a long focal length lens (Lens2).The week measurement of SHE of light is schemati-cally shown in Fig. 3(a). The theoretical transverse dis-placements given in Eq. (10) show that the two oppositespin components would have opposite tendency versus θ i [Fig. 3(b)]. It indicates that the SHE of light can begreatly enhanced near the Brewster angle. We obtain thevalue of spin-dependent splitting 3200nm at θ i = 56 ◦ and50 times larger than the previous reported values of re-fraction [3]. The relevant amplitude of the reflected fieldat the plane of z r can be obtained as V · E Hr . The ampli-fied displacement of field centroid δ w at the CCD is muchlarger than the original displacement | δ H ± | . Calculation ofthe centroid of the distribution of V · E Hr yields the ampli-fying factor A w = δ w /δ H + . Our experimental results forthe amplified displacement δ w versus the incident angle θ i are reported in Fig. 3(c). We measure the displacementsevery 0 . ◦ from 52 ◦ to 60 ◦ . The measure values allowfor calculating the original displacement caused by SHEof light. The solid lines represent the theoretical predic-tions. It should be noted that the amplifying factor inweek measurements is always the same in refraction [3].However, it presents a valley near Brewster angle on re-flection [Fig. 3(d)]. The experimental results are in goodagreement with the theory without using parameter fit.From Eq. (9), we know that the transverse displace-ments are related to the ratio between the Fresnel trans-mission coefficients r p and r s . The reflection coefficient ofhorizontal polarization r p vanishes at exactly the Brew-ster angle, and changes its sign across the angle. Hence, FIG. 3: (Color online) (a) Presection and postselection of po-larization give rise to an interference in the CCD, shifting itto its final centroid position proportional to A w = δ w /δ H + . (b)Theoretical spin-dependent transverse splitting of spin com-ponents at the prism interface. (c) Theoretical and experi-mental results for amplifying displacements δ w . Insets showthe measured field distribution. (d) Theoretical and experi-mental results for amplifying factor A w in the week measure-ment. Inset presents a full view. the large spin-dependent splitting in SHE of light is at-tributed to the large ratio of r s /r p near the Brewsterangle. On the contrary, a small ratio of r s /r p wouldgreatly suppress the SHE of light. It should be men-tioned that a large value of ∂r p /∂θ i near Brewster anglewill lead to large Goos-Hanchen shifts [26] and angularshifts [27]. It should be noted that the horizontal com-ponent of electric field alters its phase, however the ver-tical component does not. As a result, the phase differ-ence arg[ r s ] − arg[ r p ] experiences a variation π , and thespin accumulation would reverse its directions accord-ingly. Due to the reversed spin-dependent splitting, thespin accumulation can be switched by slightly adjustingthe incident angle.The SHE of light may open new opportunities for ma-nipulating photon spin and developing new generation ofall-optical devices as counterpart of recently presentedspintronics devices [3, 14]. It should be mentioned thatthe spatial separation of the spin components is verysmall in the refraction. Hence, it is disadvantage forpotential application to nano-photonic devices. In re-fraction [3] and photon tunneling [28] the reversed spinaccumulation requires the reversed refractive index gra-dient. As shown in above, the transverse displacementscan be tuned to either a negative, or a positive value, oreven zero, by just adjusting the incident angle. Hence,our scheme provide more flexibility for switching the di-rection of the spin accumulations. These interesting phe-nomena open a promising way to some potential appli-cations in spin-based nano-photonic devices. Because ofthe close similarity of Brewster angle in optical physics,condensed matter [29], and plasmonics [30], by properlyfacilitating the reflection near Brewster angle, the SHEmay be effectively modulated in these physical systems. IV. CONCLUSIONS
In conclusion, we have revealed an enhanced andswitchable spin-dependent splitting near Brewster an-gle on reflection. The detected spin-dependent splitting reaches 3200nm near Brewster angle, and 50 times largerthan the previous reported values in refraction. We havefound that the amplifying factor is not a constant whichis significantly different from that in the refraction case.The enhanced spin-dependent splitting is found to be at-tributed to the large ratio between the Fresnel reflectioncoefficients near Brewster angle. As an analogy of SHE inelectronic system, the switchable SHE of light has beendetected, which can be interpreted from the inversion ofhorizontal electric field vector across the Brewster angle.We were able to switch the directions of the spin accu-mulation, by slightly adjusting the incident angle nearBrewster angle. These findings provide a novel pathwayfor modulating the SHE of light, and thereby open thepossibility for developing new nano-photonic devices.
Acknowledgments
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