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Field-Controllable Spin-Hall Effect of Light in Optical Crystals: A

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Field-Controllable Spin-Hall Effect of Light in Optical Crystals: A www.nature.com/scientificreports OPEN Field-controllable Spin-Hall Efect of Light in Optical Crystals: A Conoscopic Mueller Matrix Analysis Received: 31 October 2017 C. T. Samlan & Nirmal K. Viswanathan Accepted: 11 January 2018 Electric-feld applied perpendicular to the direction of propagation of paraxial beam through an Published: xx xx xxxx optical crystal dynamically modifes the spin-orbit interaction (SOI), leading to the demonstration of controllable spin-Hall efect of light (SHEL). The electro- and piezo-optic efects of the crystal modifes the radially symmetric spatial variation in the fast-axis orientation of the crystal, resulting in a complex pattern with diferent topologies due to the symmetry-breaking efect of the applied feld. This introduces spatially-varying Pancharatnam-Berry type geometric phase on to the paraxial beam of light, leading to the observation of SHEL in addition to the spin-to-vortex conversion. A wave-vector resolved conoscopic Mueller matrix measurement and analysis provides a frst glimpse of the SHEL in the biaxial crystal, identifed via the appearance of weak circular birefringence. The emergence of feld-controllable fast-axis orientation of the crystal and the resulting SHEL provides a new degree of freedom for afecting and controlling the spin and orbital angular momentum of photons to unravel the rich underlying physics of optical crystals and aid in the development of active photonic spin-Hall devices. Te collective and signifcant role played by the amplitude, phase and polarization of an electromagnetic wave in enriching the fundamental understanding of light feld and quanta (photons), is not more evident than in the spin-orbit interaction (SOI) of light – a study of the mutual infuence of the spin angular momentum (SAM) and the intrinsic and extrinsic orbital angular momentum (OAM) of light1. Te concept of SOI of light frst proposed and demonstrated by Zel′dovich et al.2, continues to be a research topic of signifcant importance, as evinced in the authoritative and comprehensive recent review articles3–5. Broadly, the SOI efect results in the spin-dependent redistribution of light intensity due to space- or wavevector-variant geometric phase. In systems with rotational symmetry, the SOI leads to spin-to-orbital angular momentum conversion (SOC) and the generation of optical vortex beams; breaking of which results in the spin-Hall Efect of light (SHEL), a spin-dependent redistribution of light intensity in the transverse plane. Of the innumerable efects and devices enabled by the SOI phenomenon, the direct manifestation of SHEL is reported in a verity of systems such as planar dielectric interface6, smoothly inhomogeneous medium2,7 and metasurfaces8, and reviewed recently in ref.9, which is of interest to us here. Te observation of SHEL is attributed to two ‘seemingly’ diferent mechanisms: the geometric phase gradient arising from spin-redirection (Rytov-Vladimirskii-Berry, RVB) phase, related to variation in the direction of propagation of a paraxial light beam due to medium inhomogeneity and the Pancharatnam-Berry (PB) phase associated with the manipulation of the state-of-polarization (SoP) of light via varying fast axis orientation8–10. Te gradient in RVB and PB phases along any linear direction respectively, in the momentum space and real space manifest themselves as SHEL in the real and momentum space and have been investigated independently in a variety of systems6–16. Te interplay between the SAM and the extrinsic OAM (trajectory) of paraxial light beam resulting in the SHEL is typically studied by invoking symmetry breaking11–16. Tough the RVB phase induced SHEL has been widely investi- gated in refection and transmission at a refractive index boundary6,11–13, the frst observation of its PB phase counterpart arising due to paraxial light propagation through a tilted anisotropic wave plate was only recently reported by us16. Accordingly, breaking of the rotation symmetry of a c-cut uniaxial crystal due to the application of electric feld across it provides an ideal system for investigating the PB phase induced controllable SHEL. Te problem in hand can be mathematically quite challenging despite the development of a new formalism based on optical singularities17. Nevertheless, the demonstration of important features of the treatment for transparent and School of Physics, University of Hyderabad, Hyderabad, 500046, India. Correspondence and requests for materials should be addressed to N.K.V. (email: [email protected]) SCIENTIFIC REPORTS | (2018) 8:2002 | DOI:10.1038/s41598-018-20402-4 1 www.nature.com/scientificreports/ Figure 1. (a) Index ellipsoid of a biaxial crystal and (b) its fast-axes orientation in the stereographic projection shows diferent topologies. chiral crystal are evident from our experimental results. Which also provides answer to the non-conservation of total angular momentum (TAM) of the electromagnetic feld propagating at an angle with reference to the symmetry-axis of an optical crystal18,19. Standard experimental techniques of Stokes polarimetry and quantum weak measurement6,16 not readily suitable for measuring the weak SHEL due to the complex PB phase gradient across the beam cross-section due to spatially-varying complex fast axis orientation of the crystal. We develop and demonstrate a wavevector resolved conoscopic Mueller matrix analysis (CMMA) to map the complex topo- logical pattern of the crystal fast-axis orientation and its transformation and the resulting appearance of SHEL by measuring the circular birefringence as its manifestation. Te efect observed varies across the interaction region, and is enhanced due to the applied electric feld, which demonstrates the feld-controllable nature of the SHEL. Te results are of signifcance due to emerging interest in the control and tunability of SHEL via diferent mechanisms15,20–22. To experimentally investigate the feld-controlled SHEL, we use Potassium Dihydrogen Phosphate (KDP) crystal, an important class of hydrogen-bonded ferroelectric crystal and one of the most widely investigated opto- electronic material for its linear and nonlinear optical effects and their applications23–25. For KDP crystal, non-centrosymmetric, with tetragonal structure (point group 42m) and 4-fold inversion symmetry are not the only possibility26,27. Being an electro-optic as well as a piezo-optic medium, the radially symmetric fast-axis ori- entation of the uniaxial structure gets modifed considerably with the application of external electric feld in transverse direction to the propagation of light. Te non-vanishing electro-optic coefcient (r41), piezo-electric constant (d14) and strain-optic constant (p55) of the KDP crystal all together contribute to the changes in the fast axis direction as well as the principal refractive index values26–32. Te corresponding transformation in the index ellipsoid of the KDP crystal from uniaxial tetragonal to biaxial orthorhombic and to biaxial monoclinic can be expressed as, nx 00 no 00 00n = 00n y o 00nz 00ne no 00 ⇒ 00nn+ 32E η oox 32 00nnee− Exη np01/ dE o 55 14 x ⇒ + 32 00nnooExη 1/ pdEn0 − nE32η 55 14 xxee (1) 2 2 2 rn41 o ne where, η = 2 2 and the applied electric feld is along the x-axis (Ex). Te of-diagonal elements in the last 2 nno − e matrix can be assumed to be negligible for lower values of Ex. Tus, under the infuence of external feld, the fast axis orientation of the KDP crystal gets modifed signifcantly from radial symmetry as shown in Fig. 1(b). Te corresponding ferroelectric-to-paraelectric phase transition leading to low-dimensional symmetry, including a centrosymmetric and biaxial orthorhombic (Fdd2) and monoclinic (2/m) crystal structures and a build-up of crystal structure asymmetry with reference to the optic-axis have been reported28–32. The partially broken SCIENTIFIC REPORTS | (2018) 8:2002 | DOI:10.1038/s41598-018-20402-4 2 www.nature.com/scientificreports/ Figure 2. Schematic illustration of the analogy between (a) spatial walk-of of o-mode and e-modes due to linear birefringence and (b) SHEL due to induced circular birefringence. Optic-axis of the crystal is indicated by purple color arrow. rotational symmetry is evident from the scheme of fast-axis orientation in the momentum space (via stereo- graphic projection) given in Fig. 1(b), which exhibit diferent topologies, marked in diferent colors at diferent regions. Tis feature of the feld-induced transformation of the crystal structure is found to be highly desirable for the demonstration of simultaneously present SOI and controllable observation of SHEL. Te complex topology of the fast-axis orientation (μ) perceived by the spread of wavevectors in a paraxial optical beam, as it propagates through a biaxial crystal is given by33 1 − sin2φθcos μ = 1 tan 2 2 Δ−sincθφos2cosθ (2) where θ is the angle between the wavevector and the c-axis of the KDP crystal and φ is the azimuthal coordinate of the stereographic projection plane (Fig. 1(b)). It is important to note here the electric feld Ex dependency of μ 22 2222 through the asymmetric term Δ, given by Δ= nnzy()−−nnxx/(nnyz). To understand the SOI of light in optical crystals, consider circularly polarized (CP) paraxial light beam prop- agating along Z-axis, making an angle θc with the optic-axes of the biaxial crystal (Fig. 1(a)). Diferent local topol- ogies in the fast-axis orientation, shown in Fig. 1(b), leads to diferent manifestation of the SOI. To exemplify this, we consider the plane wave component σ± in the paraxial beam, which makes an angle θ with the z-axis. Te fnal state, afer propagation through the crystal, can be expressed in the right- (R), σ+ and lef- (L), σ− CP basis as34 δ δ ±i2μ Eiout = cos σ± + sin σ e 2 2 (3) where δ is the phase retardation experienced by the linear polarization component of the plane wave due to linear birefringence of the crystal. Te second term in eqn.
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