Best Student Paper Award

Best Student Paper Award Giant Spin Hall Effect of Light in an Exotic Optical System A. Bag, S. Chandel, C. Banerjee, D. Saha, M. Pal, A. Banerjee and N. Ghosh* Indian Institute of Science Education and Research, Kolkata (Mohanpur Campus) Mohanpur, P.O. BCKV Main Office, Nadia, West Bengal 741252, India ABSTRACT We report a giant enhancement of Spin Hall ( ) shift even for normal incidence in an exotic optical system, an inhomogeneous anisotropic medium having complex spatially varying birefringent structure. The spatial variation of birefringence is obtained by changing the three dimensional orientation of liquid crystal by modulating the pixels with user-controlled greyscale value. This polarization dependent spatial variation (in a plane transverse to the direction of propagation of light) of the transmitted light beam (for incident fundamental Gaussian beam lacking any intrinsic angular momentum) through such inhomogeneous anisotropic medium was recorded using an Eigenvalue calibrated Stokes- Mueller imaging system. Giant shift was manifested as distinctly different spatial distribution of the recorded output Stokes vector elements for two orthogonal (left and right) input circular polarization states. We unravel the reason for such large enhancement of shift by performing rigorous three dimensional analysis of polarization evolution in such complex anisotropic medium. The theoretical analysis revealed that generation of large magnitude of transverse energy flow (quantified via the Poynting vector evolution inside the medium) originating from Spin Orbit Interaction ( ) in the inhomogeneous birefringent medium leads to the observation of such a large spin dependent deflection of the trajectory of light beam. Keywords: Angular momentum of light, of light, Spin Hall effect of light ( ), Birefringence, Polarization, Jones-Mueller matrices, Poynting vector 1. INTRODUCTION Light beam is associated with two forms of rotation, first being dynamical rotation of electric and magnetic field around propagation direction and second being dynamical rotation of optical wave front around beam propagation axis. These two rotations are associated with two forms of angular momentum of light, namely Spin Angular momentum ( ) and Orbital Angular momentum ( ). For paraxial light beam, is particularly related to circular polarization of light. Whereas origin of can be either intrinsic, which is a helical structure of optical wave front with an optical vortex around beam axis; or it can be extrinsic in nature, which arises from the cross product of the total momentum transported by the beam and the position of its axis relative to the origin of coordinates. It was reported that anisotropic inhomogeneous media such as liquid crystals could give rise to a previously unrecognized optical process in which the variation of occurring from the medium's birefringence gives rise to the appearance of, arising from the medium's inhomogeneity. In rotationally symmetric geometries, this process involves no net transfer of angular momentum to matter, so that the variation in the light is entirely converted in to its process was dubbed `spin-to-orbital conversion of angular momentum' [1]. of light, which refers to intrinsic coupling between the (circular polarization) and (topological phase vortex) of photons, has evoked intensive investigations in the past few years owing to fundamental interests and potential nano-optical applications. is associated with two distinct effects (a) evolution of geometric (topological) phase due to the effect of trajectory of light on polarization and (b) arising due to the effect of polarization on the trajectory itself. For refraction, reflection, or total internal reflection from inhomogeneous isotropic media, such shifts have been typically observed for oblique incidence of light beam (thus via breaking the symmetry of the system) and the observed effect have been rather small, typically of sub-wavelength magnitudes. But, very recently a huge shift for meta-materials was reported even for normal incidence [2]. Here, we observe a giant enhancement of shift for another exotic optical system spatially (radially and azimuthally) varying retardance modulated by user-controlled greyscale value in a liquid crystal system, even for normal incidence of light. In the next section we will briefly describe the experimental set-up, followed by a theoretical understanding of the optical system. Finally we will discuss the results. *email: nghosh@iiserkol.ac.in Phone: +91-9734678247 fax: +91-()33-25873 www.iiserkol.ac.in/~nghosh Nanophotonics V, edited by David L. Andrews, Jean-Michel Nunzi, Andreas Ostendorf, Proc. of SPIE Vol. 9126, 91262E 14 SPIE CCC code: 277-786X/14/$18 doi: 1117/12.51798 Proc. of SPIE Vol. 9126 91262E-1

2. EXPERIMENTAL SET-UP LASER MIRROR PINHOLE APERTURE LENS POLARISER OW PLATE OW PLATE POLARISER ü Top g:. Intl LENS APERTURE Figure 1: Schematic diagram of the experimental set-up for measuring full 4x4 Mueller matrix consisting of Laser Source, Spatial Filter,, and. (Colour online) FILTER CCD...1.1Hwu.nn...b 9 The schematic of the experimental setup for measuring the full 4 x 4 Mueller matrix is shown in the Figure 1. Mueller matrix measurement was done with proper eigenvalue calibrated experimental system, which is well practiced and previously reported [3]. laser ( = 632.8 ) was used as a source of the Gaussian beam which lacks any intrinsic. The experimental set-up mainly consist of one Polarization State Generator ( ) and Polarization State Analyzer ( ). One fixed polarizer with a computer-controlled rotatable broadband Quarter Wave Plate ( ) acts as a and four different polarization state is generated for four different orientation of. Similarly, one and a polarizer in orthogonal orientation to corresponding polarizer in makes and we analyze with those same four different orientation of. Those 16 measurements were recorded using spectrometer (ixon 3, ANDOR technology, USA). Spatial Light Modulator (, LC2, Holoeye, Germany) is between and, which projects different user controlled phase variant images. Measurement can be done over a spatial domain which will allow to investigate more optical properties like diattenuation etc. of any material. The system employed in our study is a twisted nematic liquid crystal spatial light modulator ( ), whose individual pixels are electronically addressed to form complex spatially varying birefringence effect. The spatial variation of birefringence is obtained by changing the three dimensional orientation of local liquid crystal director (tilt and twist angle of the director axis), by modulating the pixels with user-controlled spatial maps. Proc. of SPIE Vol. 9126 91262E-2

3. THEORY Liquid crystal cells of has a unique property, exploiting which we could generate a spatially varying retardance pattern [4]. This pattern rotates individual crystal cells into different three dimensional orientation depending up on the grey scale value at that pixel. These uniaxial crystals [6][7] has two axis of propagation namely, ordinary and extraordinary axis. As the crystal cells gets rotated, the extraordinary axis (or optic axis) rotates and that polarization component of light experiences different refractive index. In spatial domain, these follow the same pattern as of retardance. Inside the medium, we know that dielectric vector ( ), whose direction is rotated from that of, depending on the dielectric tensor (Ԑ) of the medium. In our laboratory set-up, there is very less effect of depolarization over propagation of light beam. Hence, we can use both Jones and Mueller formalism [8], depending on ease of applicability. Whereas intensity based 4 x 4 Mueller matrix measures all optical properties [9] of the optical medium; Jones formalism, which involves electric field vectors is mostly used for theoretical calculation involving field. Electric field is transverse to the propagation direction of light in most cases and hence two dimensional Jones formalism is mostly practiced. But to understand the origin of the shift better, we have to carefully think and use a three dimensional Jones formalism [1] as the dielectric vector ( ) rotates from and it has a component in the longitudinal direction, i.e. propagation direction of light. The Jones matrix for uniaxial retarder has one ordinary and one extraordinary axis in transverse plane, the longitudinal axis is considered as extraordinary. This makes the three dimensional Jones matrix look like ( ) = (1) ( ) Where optic axis is at an angle of with propagation direction, ( ) and are refractive indices for light parallel to ordinary and extraordinary axes. Here the angle has a spatial distribution and so has ( ). Along with this Jones matrix, rotation matrices are multiplied both side to take care of the orientation of liquid crystal cells in the laboratory frame. An inherent property of is that each layer, which are infinitesimally thin (or continuous along its thickness) is slightly rotated from its previous layer (twisted layers). This twist or rotation can be incorporated by unitary rotation matrices ( ) and ( ) where ( ) is given by ( ) = (2) 1 Inside the medium, Electric field is denoted by, which is rotated from by an angle of = tan tan. (3) After each layer of liquid crystal, this rotation is incorporated by multiplying a unitary rotation matrix ( ) = 1 (4) Proc. of SPIE Vol. 9126 91262E-3

Hence, for each point in spatial domain, effective Jones matrix of the optical system will depend on different parameters and can be generalized as = ( ) ( ) ( ) (5) For multiple layers of, different terms of couples with each other and gives rise to circular diattenuation which is manifested by shift. Though for simple calculation, this theory is developed by approximating light beam as a plane wave. For proper quantitative analysis, full three dimensional propagation of a Gaussian light beam is necessary and it is underway. Ideally, the light beam is Gaussian in nature, which can be thought as a sum of many plane waves; but for simplicity in calculation we started with single plane wave and evaluated the shift originating from this particular retardance variation. As a plane wave has only vector i.e. the propagation direction is assumed to be in direction of laboratory frame. We assume that light is not bending in its trajectory and hence, all the spatial points are equivalent in context to direction of propagation. We have analytically calculated the Electric field vector, magnetic field vector and the Poynting vector for the transmitted light. Beam shift in the transverse direction (to propagation) is a direct consequence of transverse component of Poynting vector i.e. and. From another view point one can say, if light propagates in one particular direction, energy flow is in that direction only, but if due to some modification in the system some part of energy is flowing in the transverse direction (to propagation), that is associated with shift of beam in that transverse direction [11][12][13]. Now, transverse component of Poynting vector arises due to the presence of longitudinal component of Electric and Magnetic field ( and ) which justifies our use of the three dimensional Jones matrix. We have calculated the components of Poynting vector, separately for left and right circularly polarized light ( or ) which comes out to be, for an input (1 ± ) ( or ) = 2 ±, = 2 ±, = 2 + 1 (6a) (6b) (6c) Where is birefringence. It is clear from the above equations that component of the Poynting vector along propagation i.e. -direction is same for both and. Whereas, it is different and opposite in nature for the transverse component of it. This clearly gives an idea that flow of energy if is opposite for and which is manifested in the SH shift [14]. 4. RESULTS AND DISCUSSION Experimental set-up allowed us to record the full (4 x 4) Mueller matrix over a spatial domain of the beam. And For any particular initial polarization of incident light we can gather information of the polarization of the light after it is transmitted through the optical system ( with azimuthal retardance pattern). Here, for the recorded Mueller matrix, if transmitted light for incident light with and is observed we see the output stokes vector are like in Figure 2, Proc. of SPIE Vol. 9126 91262E-4

Figure 2: Stokes vector of transmitted light beam where incident polarization of light was and. First element of the Stokes vector indicates a deformation of beam intensity profile in the spatial domain. (Colour online) It is clear from the above picture of Stokes vector that after getting transmitted through, initial circular beam profile gets modified and for the intensity profile shifts to the upper part of the circle and lower part has almost zero intensity; similarly for the intensity profile shiftss to the lower part of the circle and upper part has almost zero intensity. A closer look in the first element of stokes vector i.e. the intensity profile of the beam is here in Figure 3. RCP nn KY 2 LCP Ky 1.8 1.6 1.4 1.2 im-1 1.8 6.6.4.2 nn - - -Ky Figure 3: A closer look to the first element of the Stokes vector i.e. the total intensity profile of the transmitted beam (incident light beam polarization was and ) ) (Colour online) From the pixel readings above we can calculate the shift, which is around 3 5 μ in size whereas generally we see SH shift in sub-wavelength order. Also, this shift is observed for normally incident light beam. In spatial domain flow of energy for and is plotted in Figure 4a & 4b which clearly explains the above. Proc. of SPIE Vol. 9126 91262E-5

.4 8 1 15 3 -.1 -.2 -.3 -.4 8 1 1 1 1 5 1 15.3.2 -.2 -.3 -.4 Ñ N L4 Ó N Ó IN L 8.2 5.5 1 1 -.6 1-1 -5 -.2 5 1 15 -.25 Figure 4a: Flow of Poynting vector (component wise) for transmitted beam (incident polarization is and ). is same for both and ; but and are different and opposite. (Colour online) 9 Ó 8 8 8 Ó 8 8 8 1 8 D7.6 1.5 1.4 1.3 1.2 5 1 15 o Figure 4b: Flow of Poynting vector (component wise) for transmitted beam (incident polarization is and ). is same for both and ; but and are different and opposite. (Colour online) This plot is generated through simulation in Mat lab. thickness is taken to be 1 with number of layers = 1, which practically approaches infinite limit, consequently each layer thickness becomes infinitesimally small, which can be considered as continuous limit. 5. CONCLUSION The theoretical analysis revealed that generation of large magnitude of transverse energy flow (quantified via the Poynting vector evolution inside the medium) originating from in the inhomogeneous birefringent medium leads to the observation of such a large spin dependent deflection of the trajectory of light beam. This theoretical results are in Proc. of SPIE Vol. 9126 91262E-6

good agreement with experimental results too. A proper Gaussian wave propagation is underway and that will exactly reveal the quantitative information about the system. REFERENCES [1] Marrucci L., Manzo C. and Paparo D., Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media, Phys. Rev. Lett. 96, 16395 (6) [2] Yin X., Ye Z., Rho J., Wang Y. and Zhang X., Photonic Spin Hall Effect at Metasurfaces, Science 339, 15 (13) [3] Purwar H., Soni J., Lakhotia H., Chandel S., Banerjee C. and Ghosh N., Development and eigenvalue calibration of an automated spectral Mueller matrix system for biomedical polarimetry, Proc. SPIE 823, 19 (12) [4] Hermerschmidt A., Quiram S., Kallmeyer F. and Eichler H. J., Determination of the Jones matrix of an LC cell and derivation of the physical parameters of the LC molecules, SPIE 6587-46 (7) [5] Bliokh K. Y. and Aiello A., Goos Hanchen and Imbert Fedorov beam shifts: an overview, J. Opt. 15, 11 (13) [6] Ghatak A., [Optics], 4 th Ed., Tata McGraw-Hill Publishing Company Ltd., New Delhi, 22.1-22.33 (9) [7] Born M. and Wolf E., [Principle of Optics], 7 th (Expanded) Ed., Cambridge University Press, Cambridge (5) [8] Simon B. N., Simon S., Gori F., Santarsiero M., Borghi R., Mukunda N. and Simon R., Nonquantum Entanglement Resolves a Basic Issue in Polarization Optics, Phys. Rev. Lett. 14, 2391 (1) [9] Kravtsov Y. A., Bieg B. and Bliokh K. Y., Stokes-vector evolution in a weakly anisotropic inhomogeneous medium, J. Opt. Soc. Am. 24, 1 (7) [1] Yu F. H. and Kwok H. S., Explicit representation of extended Jones matrix for oblique light propagation through a crystalline slab, J. Opt. Soc. Am. A-16, 11 (1999) [11] Bekshaev A., Improved theory for the polarization-dependent transverse shift of a paraxial light beam in free space, Ukr. J. Phys. Opt. 12-1, 1 (11) [12] Aiello A., Lindlein N., Marquardt C. and Leuchs G., Transverse Angular Momentum and Geometric Spin Hall Effect of Light, Phys. Rev. Lett. 13, 11 (9) [13] Kravtsov Y. A., Berczynski P., Bieg B., Bliokh K. Y. and Czyz Z. H., Polarization Evolution in Weakly Anisotropic Media: Quasi-Isotropic Approximation (QIA) of Geometrical Optics Method and Its Recent Generalizations, PIERS Proc. 85-89 (7) [14] Chipman R. A., Polarimetry, Chap. 22 in [Handbook of Optics], 2 nd Ed., McGraw-Hill, New York, Vol. 2, pp. 22.1 22.37 (1994). [15] Soni J., Ghosh S., Mansha S., Kumar A., Dutta Gupta S., Banerjee A. and Ghosh N., Enhancing spin-orbit interaction of light by plasmonic nanostructures, Opt. Lett. 38-1 (13) [16] Nastishin Y. A. and Nastyshyn S. Y., Explicit representation of extended Jones matrix for oblique light propagation through a crystalline slab, Ukr. J. Phys. Opt. 12-4, 191 (11) Proc. of SPIE Vol. 9126 91262E-7