Realization of Tunable Photonic Spin Hall Effect by Tailoring the Pancharatnam-Berry Phase
Xiaohui Ling, Xinxing Zhou, Weixing Shu, Hailu Luo, Shuangchun Wen
aa r X i v : . [ phy s i c s . op ti c s ] J un Realization of Tunable Photonic Spin Hall E ff ect by Tailoring the Pancharatnam-Berry Phase Xiaohui Ling , , Xinxing Zhou , Weixing Shu , Hailu Luo , , ∗ and Shuangchun Wen , † Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province,College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060 Key Laboratory for Micro- / Nano-optoelectronic Devices of Ministry of Education,College of Physics and Microelectronic Science, Hunan University, Changsha 410082, China Department of Physics and Electronic Information Science,Hengyang Normal University, Hengyang 421002, China (Dated: September 26, 2018)
Recent developments in the field of photonic spin Halle ff ect (SHE) o ff er new opportunities for advantageousmeasurement of the optical parameters (refractive index,thickness, etc.) of nanostructures and enable spin-basedphotonics applications in the future. However, it remains achallenge to develop a tunable photonic SHE with any de-sired spin-dependent splitting for generation and manip-ulation of spin-polarized photons. Here, we demonstrateexperimentally a scheme to realize the photonic SHE tun-ably by tailoring the space-variant Pancharatnam-Berryphase (PBP). It is shown that light beams whose polariza-tion with a tunable spatial inhomogeneity can contributeto steering the space-variant PBP which creates a spin-dependent geometric phase gradient, thereby possibly re-alizing a tunable photonic SHE with any desired spin-dependent splitting. Our scheme provides a convenientmethod to manipulate the spin photon. The results can beextrapolated to other physical system with similar topo-logical origins. Spin Hall e ff ect (SHE) is a transport phenomenon in whichan electric field applied to spin particles results in a spin-dependent shift perpendicular to the electric field direction [1–3]. It plays an important role in the spintronics since it canprovide an e ff ective way for generation, manipulation, anddetection of spin-polarized electron which define the mainchallenges in spintronics applications [4–6]. Photonic SHEmanifests itself as the spin-dependent splitting of light [7–9], which can be regarded as a direct optical analogy of theSHE in electronic systems. Recently, photonic SHE has beenproved to be an advantageous metrological tool for character-izing the parameters of nanostructures [10, 11]. More impor-tantly, it o ff ers new opportunities for manipulating spin pho-tons and developing next generation of spin-controlled pho-tonic devices as counterparts of recently presented spintronicsdevices [4, 12]. Similar to those in the spintronics devices,one major issue in spin-based photonics is to develop a tun-able photonic SHE with any desired spin-dependent splittingfor generation, manipulation, and detection of spin-polarizedphotons.Photonic SHE is generally believed to be a result of topo-logical spin-orbit interaction which describes the coupling be-tween the spin (polarization) and the trajectory (orbit angu-lar momentum) of light beam propagation. The spin-orbitinteraction corresponds to two types of geometric phases:the Rytov-Vladimirskii-Berry phase associated with the evo- lution of the propagation direction of light [7–9] and thePancharatnam-Berry phase (PBP) related to the manipulationwith the polarization state of light [13, 14]. The spin-orbit in-teraction associated with the Rytov-Vladimirskii-Berry phaseis exceedingly small, and the detection of the correspondingphotonic SHE needs multiple reflections [14] or weak mea-surements [9, 15]. Recently, the strong spin-orbit interactionrelated to the PBP in metasurface, a two-dimensional electro-magnetic nanostructure, enables one to observe a giant pho-tonic SHE [16–18]. However, the PBP in the metasurfaceis fixed and unadjustable, once the nanostructure is manu-factured. Despite of recent experimental e ff orts, it remainsa challenge to realize a photonic SHE with a high tunability.In this work, we report an experimental realization oftunable photonic SHE via tailoring the space-variant PBPcontributed by a light beam with transversely inhomogeneouspolarization. We show that the space-variant polarizationis able to invoke the spin-orbit interaction even when prop-agating through a homogeneous waveplate and producesthe space-variant PBP as the inhomogeneous metasurfacedoes, with a high tunability the latter cannot achieve. Asthe beam with any desired space-variant polarization can beeasily generated by many methods (see Ref. [19] for a reviewand the references therein), involving the contribution of theincident polarization distribution to the space-variant PBPwill introduce a convenient degree of freedom to manipulatethe photonic SHE. RESULTSTheoretical analysis . For an inhomogeneous waveplate ormetasurface, the space-variant optical axis direction producesa spatial rotation rate and applies a space-variant PBP to alight beam that passes through it, due to the spin-orbit inter-action [12, 13]. This inhomogeneous geometric phase formsa spin-dependent geometric phase gradient and generates aspin-dependent shift ( ∆ k ) in momentum space [Fig. 1(a)]. In-deed, the inhomogeneous birefringent waveplate is not thesole way to produce the PBP. Providing the incident beamtakes a spatial rotation rate, namely exhibits an inhomoge-neous polarization, the PBP and spin-dependent momentumsplitting should still occur [Fig. 1(b)]. Recent works aboutthe spin-dependent splitting occurring at metasurfaces [16–18] can be seen as special cases where only the waveplateis spatially inhomogeneous. The scheme proposed here showthat a beam with tunable inhomogeneous polarization can also FIG. 1: Schematic illustration of the photonic SHE with multi-lobe, spin-dependent momentum splitting. (a) A collimated Gaussianbeam with homogeneous polarization (spatial rotation rate Ω β = Ω α , Ω β ,
0) passes through a homogeneous anisotropicwaveplate ( Ω α = generate the spin-dependent momentum shift even though itpasses through a homogenous waveplate.We now develop a unified theoretical model of the photonicSHE for either the polarization of light beam or the waveplateexhibits a spatial inhomogeneity. First, we consider that onlythe waveplate is inhomogeneous. It is assumed that the wave-plate is a uniaxial crystal with its optical axis direction spec-ified by a space-variant angle α ( r , ϕ ) = q ϕ + α , where q isa topological charge, ϕ = arctan( y / x ) the local azimuthal an-gle, and α the initial angle of local optical axis direction [20].Some examples of the inhomogeneous waveplate geometriesare shown in Fig. 2(a). It is the spatial version of a rotationallyhomogeneous waveplate [13]. Similar to the temporal rotationrate, the spatial rotation rate of the inhomogeneous waveplatein the polar coordinate is Ω α = d α ( r , ϕ ) / d ξ , where d ξ = r d ϕ is the rotating route with r the radius of the waveplate.Next, we consider only the polarization of the incidentbeam exhibits a spatial inhomogeneity. A linear polarization,with its electric field described by a Jones vector, is given as E in ( x , y ) = cos β sin β ! E ( x , y ) , (1)where β is the polarization angle and E ( x , y ) the complex am-plitude. Generally, β is a constant, and Eq. (1) represents ho-mogeneous linear polarizations with β = π/ x - and y -polarizations.If β is a position-dependent function, e.g., β ( r , ϕ ) = m ϕ + β ,Eq. (1) represents a beam with its polarization having cylindri-cal symmetry, i.e., the cylindrical vector beam (CVB). Here, q = 2 q = 0 q = 1 q = − q = − m = − m = − m = 2 m = 0 m = 1( b ) FIG. 2: (a) Schematic examples of the waveplate geometries for dif-ferent spatial inhomogeneity. The tangent to the lines shown indi-cates the direction of local optical axis. (b) Schematic examples ofthe polarization of CVBs for di ff erent spatial inhomogeneity. Thearrows represent the local polarization directions. m is also a topological charge and β the initial polarizationangle for ϕ =
0. Similar to the inhomogeneous anisotropicwaveplate, the inhomogeneous linear polarization has a spa-tial rotation rate Ω β = d β ( r , ϕ ) / d ξ which can be tailored bysuitably designing the polarization distribution of the CVBs[see Fig. 2(b) for examples].In the most general case, both the polarization of light andthe waveplate possess spatial rotation rates. Their relative ro-tation rate can be regarded as Ω γ = Ω α − Ω β . Analogous tothe rotational Doppler e ff ect, the PBP can be written as [13] Φ PB = − σ ± Z Ω γ d ξ = − σ ± [( q − m ) ϕ + ( α − β )] , (2)where σ + = + σ − = − − k (momentum)space, i.e., a manifestation of the photonic SHE.Unambiguously, the PBP is locally varying, and creates ageometric phase gradient which can be tunable by tailoring thewaveplate geometry and polarization distribution of the CVB.This will lead to a spin-dependent shift in k space. Under thenormal incidence, the spin-dependent shift ∆ k = ∆ k r + ∆ k ϕ can be calculated as the gradient of the PBP: ∆ k = ∇ Φ PB = − σ ± ( q − m )ˆ e ϕ , (3)where ˆ e r (ˆ e ϕ ) is the unit vector in the radial (azimuthal)direction. Here ∆ k only has the azimuthal component ˆ e ϕ ,which means that the spin-dependent splitting occurs alsoin this direction. Note that the spin-dependent splittingwill vanish if the beam polarization and inhomogeneouswaveplate have the same spatial rotation rate (i.e., q = m ).As the polarization rotation rate of the CVB can be tailored ( b ) SLM BSP1 − m HWP3HWP1 HWP2QWP1 QWP2 CCDHe-Ne Laser P2
Part 3( a ) Part 1 Part 2 BS + m M FIG. 3: (a) Experimental apparatus for generating the CVBs andmeasuring the tunable photonic SHE. Part 1: A collimated and hori-zontal polarized TEM beam from a He-Ne laser (632.8 nm, 21mW)followed by a polarizer (P1) is converted into a vortex-bearing LG , m beam using a reflective phase-only spatial light modulator (SLM)(Holoeye Pluto-Vis, Germany). Using a modified Mach-Zender in-terferometer comprised of two beam splitters (BS) and one mirror(M), a LG , − m beam is produced by an odd number of reflections.A half-wave plate (HWP1) with 45 ◦ inclined to horizontal directionconverts the horizontal polarization to vertical one and a quarter-wave plate (QWP1) gives the two beams orthogonal circular polar-izations. Their collinear superposition generates a CVB on the equa-tor of the higher-order Poincar´e sphere. In order to determine theconcrete point on the equator, two cascaded half-wave plates (HWP2and HWP3) are employed [19, 27]. Part 2: The CVB passes throughthe homogeneous birefringent waveplate (HBW). The photonic SHEcan be detected by Part 3 (a quarter-wave plate followed by a polar-izer and a CCD camera) which is a typical setup for measuring theStokes parameters S . The inset: An example of the phase picturedisplayed on SLM. (b) Higher-order Poincar´e spheres. Left panelfor the case of m = m = −
1. The equatorrepresents the linearly polarized CVBs, the poles circularly polar-ized vortex beams, and intermediate points between the poles andequator elliptically polarized CVBs. The solid circles with arrowsin the north / south poles indicate locally circular polarizations, whilethe dashed ones represent the direction of vortex. arbitrarily via suitably designing the polarization distributionof the CVB, it can serve as an independent and convenientdegree of freedom for manipulating the PBP and photonicSHE. Experimental results.
To demonstrate the role of the inho-mogeneous polarization of incident beam in the tunable pho-tonic SHE, a CVB with its polarization exhibiting tunablespatial rotation rate is generated experimentally and passesthrough a homogeneous waveplate [Fig. 3(a)]. It is a typicalexample of the Fig. 1(b). Part 1 of the experiment setup canproduce a CVB on the equator of the higher-order Poincar´esphere where the polarization distribution possesses a cylin-drical symmetry [25, 26] [Fig. 3(b)]. The CVB on the equa-tor is generated by interfering the two beams from a modified
FIG. 4: Tunable photonic SHE for the CVBs exhibiting di ff erent po-larization inhomogeneity m under the condition of q = α = β = Mach-Zender interferometer. Two cascaded half-wave plates(HWP2 and HWP3) are employed to determine the polariza-tion distribution of the linearly polarized CVB. By rotating theoptical axis directions of the two half-wave plates, we can ob-tain any desired polarization on the equator of the higher-orderPonicar´e spheres [19, 27]. The CVB passes through a homo-geneous birefringent waveplate (part 2, HBW) and the Stokesparameter S is measured by the part 3. Modulating the phasepicture displayed on the SLM, we can acquire a CVB withany desired polarization distribution, and thereby introduce atunable geometric phase gradient even if the light beam prop-agates through a homogeneous anisotropic waveplate.The spin-orbit interaction between the CVB and the HBWbrings about a tailorable PBP and creates a geometric phasegradient in the azimuthal direction ˆ e ϕ [see Eqs. (2) and (3)].This phase gradient can result in a k -space spin-dependentsplitting also in this direction. And then the real-space split-ting manifested as multi-lobe patterns can be directly mea-sured in the far field (Fig. 4), which is attributed to the coher-ent superposition contributed by the local k -space shift in thenear field. For m = ±
1, both the splitting patterns have fourlobes with 2-fold rotational ( C ) symmetry. The only di ff er-ence is their anti-phase distribution of the S pattern due totheir just opposite PBP. The splitting pattern for m = C ) and 6-fold ( C ) rotational sym-metry. Actually, the lobe number is equal to 4 | m | . As ϕ isthe azimuthal angle taking values in the range of 0 to 2 π , Φ PB falls in the range of 0 to 4 m π for σ + and 0 to − m π for σ − ,respectively. In the spin-dependent splitting pattern, σ + and σ − just has a phase di ff erence of π , and they respectively re-peat again for each 2 π of Φ PB . Therefore, the spin-dependentsplitting manifests as a pattern of alternative σ + and σ − withrotational symmetry, and the total number of the lobes is 4 | m | ,which can be switched by modulating the topological charge( m ) of the CVB, i.e., changing the phase picture shown on theSLM. Note that if a spin-dependent splitting in the radial di-rection ˆ e r is desired, a CVB with polarization inhomogeneity(and the corresponding PBP gradient) in this direction mustbe created.Now let the beam polarization evolve along the equator of FIG. 5: Revolving the spin-dependent splitting of the photonic SHEby evolving the polarization of incident CVB along the equator ofhigher-order Poincar´e sphere ( m = the higher-order Poincar´e sphere where the polarization hasthe same spatial rotation rate but di ff erent initial polarizationangles. When the beams propagating through a homogeneousanisotropic waveplate, the PBP could result in a photonicSHE with rotatable splitting patterns. Figure 5 shows theexperimental results of the spin-dependent splitting under theevolution of polarization for m =
1. Due to the homomor-phism between the physical SU(2) space of the light beamand the topological SO(3) space of the higher-order Poincar´esphere, the change of beam polarization of 180 ◦ correspondsto the rotation of 360 ◦ in the equator of the sphere. Thesplitting lobes also revolve 360 ◦ and return to the originalstate. This, in turn, indicates the topological characteristic ofthe photonic SHE. DISCUSSION
The inhomogeneous metasurface which can produce a de-sired transverse phase gradient has been employed to shapethe wavefront and manipulate the out-of-plane reflection andrefraction of light based on the generalized Snell’s law [28].Therefore, the metasurface can serve as a powerful tool fordesigning optical components like the inhomogeneous wave-plates with locally controllable polarization and geometricphase. The major issues, such as high losses, cost-ine ff ectivefabrications, and unadjustable structures, hamper the develop-ment of metasurface technology [29]. But no such di ffi cultyexists when a beam with inhomogeneous polarization is em-ployed to generate the geometric phase. It can produce thesame geometric phase gradient as the inhomogeneous meta-surface does, in particular with a high tunability because abeam with any desired polarization distribution can be con-veniently achieved by modulating the phase picture displayedon the SLM (Fig. 3). So, the CVB o ff ers a convenient degreeof freedom to manipulate the PBP and photonic SHE.In summary, tunable photonic SHE has been realizedexperimentally via tailoring the space-variant PBP producedby a light beam with transversely inhomogeneous polariza-tion. By suitably designing the polarization inhomogeneity,a tunable photonic SHE with any desired spin-dependentsplitting (rotatable multi-lobe patterns) is realizable. Thoughthe incident beam with spatial-variant polarization was constructed by a particular method in the experiment, sucha scheme has a general meaning in that inhomogeneous po-larization distributions provide a robust method to modulatethe space-variant PBP, thereby a new way to controllablymanipulate the photonic SHE and spin photon. The resultedSHE is large enough for direct measurements, in contrastwith the indirect technology using the weak measurementand can be viewed as a photonic version of the famousStern-Gerlach experiment. Of particular interest is that ourresults can be generalized to other physical system due to thesimilar topological origins, such as vortex-bearing electronbeam [30, 31]. METHODSExperimental measurements of the Stokes parameters S . A quarter-wave plate followed by a polarizer and a CCDcamera [see the part 3 of the Fig. 3(a)] is a typical setup tomeasure the Stokes Parameter S pixel by pixel. By suitablysetting the optical axis direction of the quarter-wave plateand the transmission axis of the polarizer, we can obtain theintensities of the left- and right-handed circular polarizationcomponents, respectively. And after a series of calculations,we finally can obtain the S for each pixel. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] Murakami, S., Nagaosa, N., & Zhang, S. C. Dissipationlessquantum spin current at room temperature.
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