Mechanical effects of light spin and orbital angular momentum in liquid crystals E.Santamato Università degli Studi di Napoli Federico II Dipartimento di Scienze Fisiche Complesso di Monte S. Angelo via Cintia, 80126 Napoli Workshop on singular optics ICTP, Trieste, May 30 June 3
Lorenzo Marrucci Bruno Piccirillo Enrico Santamato Ebrahim Karimi Sergei Slussarenko Workshop on singular optics ICTP, Trieste, May 30 June 3
Outline I. Experiments on the mechanical effects induced by the light OAM in nematic liquid crystals II. A discussion on the different mechanical effects induced by the light SAM and OAM fluxes in liquid crystal Workshop on singular optics Trieste, May 30 June 3
Part I Experiments on the mechanical effects induced by the light OAM in nematic liquid crystals Workshop on singular optics Trieste, May 30 June 3
Liquid Crystals Liquid crystals have internal orientational degrees of freedom External fields (electric, magnetic and optical) may induce the LC reorientation The reorientation is described by the molecular director n Light
Liquid crystal reorientation An electrostatic field produces a torque τ E ε a 4π ( nˆ E)( nˆ E) An optical field produces a torque εa * I τ Re ( ˆ )( ˆ o ) s 8π n E n E The torque is zero when E is either parallel or perpendicular to n 3 Workshop on singular optics Trieste, May 30 June 3
The Experimental Geometry Homeotropic anchoring at the walls Normal incidence of the laser beam E
Liquid crystal are reoriented by the light polarization (SAM) (b) Linear polarization (b) Circular polarization (a) S. D. Durbin et al., Phys. Rev. Lett., 47, 1411 (1981) (b) E. Santamato et al., Phys. Rev. Lett., 57, 19 (1986) Threshold Rotation
Self-phase modulation Workshop on singular optics Trieste, May 30 June 3
Reorientation mechanism: light Angular Momentum transfer SAM transfer birefringence
What happens with unpolarized light? LC are sensitive to second-order correlations of zero-average polarization fluctuations [L. Marrucci er al., PRE, 57, 3033 (1998)] But LC are much more sensitive to the beam shape [L. Marrucci er al., Opt. Commun., 171, 131 (1999)] What s the source of this torque? Workshop on singular optics Trieste, May 30 June 3
Director azimuthal angle vs incident laser intensity 100 (deg) 80 60 40 e = 4.53 e = 1.99 e = 1.65 e = 1.35 e = 0.97 e 20 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 I/I th
A simple model for the effect A torque is generated on the upper cylindrical lens The two lenses tens to align
Reorientation mechanism: light Angular Momentum transfer SAM transfer inhomogeneity OAM transfer
The origin of the OAM torque Thin sheet of light along x
The transfer of angular momentum of light to LC LC are sensitive to the photon spin because they are birefringent and change the light polarization LC are sensitive to the photon orbital angular momentum because they are inhomogeneous and change the ray direction What happens if the light SAM and OAM compete?
Linear polarization ( = 0 ) E Standard OFT
Linear polarization ( = 45 ) E Workshop on singular optics Trieste, May 30 June 3
Linear polarization ( = 90 ) E B.Piccirillo et al., PRL, 86, 2285 (2001) At low incident power, the reorientation is in the polarization plane Above a critical power threshold nonlinear oscillations start up
Linear polarization ( = 90 ) E 90 90 80 80 70 70 60 60 (deg) 50 40 30 20 P = 350 mw (deg) 50 40 30 20 P = 450 mw 10 10 0 0 100 200 300 time (s) 0 0 100 200 300 time (s) 90 80 70 60 (deg) 50 40 30 B.Piccirillo et al., PRL, 86, 2285 (2001) 20 10 0 P = 670 mw 0 100 200 300 time (s)
Switchin on-off the polarization Low power E High power Workshop on singular optics Trieste, May 30 June 3
Circular polarization, low power Azimuthal angle The azimuthal angle increases linearly in time (precession). Ring diameter The ring diameter in the far field oscillates in time.
Circular polarization, high power n 2 x n 2 y n arctan n y x A. Vella, A. Setaro, B. Piccirillo, and E. Santamato, Phys. Rev. E, 67, 051704 (2003)
Conclusions It was experimentally shown that liquid crystals are sensitive to the light SAM and OAM When SAM and OAM are competing in a LC complex nonlinear dynamics is excited Steady states Rotating states Oscillating states Intermittency states Workshop on singular optics Trieste, May 30 June 3
Part II A discussion on the mechanical effects induced by the light SAM and OAM fluxes in liquid crystal Workshop on singular optics Trieste, May 30 June 3
The problem The main problem is connecting the forces and torques acting in the bulk of the LC sample with the fluxes of linear and angular momentum coming from outside. In particular, we are interested in the angular momentum coming from an incident light beam.
Difference between SAM and OAM in liquid crystals SAM s OAM l Workshop on singular optics Trieste, May 30 June 3
Hydrodynamics of liquid crystals Force density on the fluid p l s f div ˆ r f divlˆ ω τ divsˆ ω Linear momentum density flux (stress) Orbital angular momentum density flux Intrinsic angular momentum density flux Torque density on n l-s coupling torque p v l s div( Lˆ Sˆ) divjˆ Linear momentum density l s r p I( n n) Orbital angular momentum density Intrinsic angular momentum density
Definitions (div Tˆ ) L x T skewsymmetric part of ˆ
The elastic free energy 1 2 2 Fe [ k1 div n k2 ( n rot n ) k3 ( n rot n ) ] 2 Splay Twist Bend h Fe div ˆ ( r) n n F e ( n ) molecular field field equations Workshop on singular optics Trieste, May 30 June 3
The electromagnetic free energy F B em H 1 ( B 16 * ic rote H D * E) Monochromatic field at frequency ω Average over the optical period Nonmagnetic optical medium Field equations 2 rotrote k 0 ˆ E
Electromagnetic force and torque in the medium f τ em em em n n 1 E 16 em F * 1 ( D 16 E * E c.c. ) on the fluid on the director n The presence of the optical field do not change the viscous forces and torques, because no source of entropy is associated with light.
The splitting of SAM and OAM The stress tensor is not unique We may add a divergence free tensor to Lˆ Ŝ and change and so to leave the hydrodynamics equations invariant For example, we may choose so to have S ˆ 0 ˆ ˆ The splitting J ˆ L ˆ S ˆ is may mathematically be dictated by ambiguous. physics?
The choice Ŝ = 0 (spinless gauge) Sˆ Lˆ l s 0 Jˆ divjˆ ω ω ˆ em = Maxwell s e.m. tensor D.Forster et al., Phys. Rev. Lett., 26, 1016, (1971) [Harvard group]. s T 0 ω 0 ˆ ˆ Dynamical constraints not always satisfied True if the inertia I is neglected (overdamped motion of n) True in the limit of isotropic fluids
Stress tensors and SAM fluxes Workshop on singular optics Trieste, May 30 June 3
The limit of the isotropic medium (vacuum) One elastic constant (k 1 = k 2 = k 3 ) No internal torque ( = 0) Symmetric elastic stress tensor No optical birefringence (D = 0 E) Symmetric electromagnetic stress tensor Workshop on singular optics Trieste, May 30 June 3
Properties of the three groups of stress tensors Block I (Harvard group) Elastic and e.m. stress tensors are symmetric Unrecognizable SAM flux The total elastic torque is identically zero (no LC inertia) em em The e.m. flux densities L and S are not divergence free in the isotropic limit Block II (Ericksen) Elastic and e.m. stress tensors are not symmetric even in the isotropic limit SAM and OAM can be formally recognized The e.m. flux densities are not divergence free in the isotropic limit Block III L em and S em Elastic and e.m. stress tensors become symmetric in the isotropic limit SAM and OAM can be formally recognized em em The e.m. flux densities L and S are divergence free in the isotropic limit Workshop on singular optics Trieste, May 30 June 3
Example: the rotating dipole radiation Block III 3 S ˆem ˆ em ik0 S u d L L d ( *) V u V 6 p p J.H.Crichton and P. Martson, J. Diff. Eqs., 4, 37 (2000) Block II 3 1 1 2 1 e Lz ik0 (1 e )ln ( *) 2 3 z 4e 8e 1 e p p 3 e = surface ellipticity ik0 Sz ( p p*) Lz 3 S. M. Barnett, J. Opt. B: Quantum Semiclassiical Opt. 4, S7 (2002) Workshop on singular optics Trieste, May 30 June 3
Conclusions The equation governing the electro-optic hydrodynamic of LC has been entirely worked out Explicit expression of energy, SAM and OAM flux densities have been calculated in different gauges It was proved that one gauge exist where a) The e.m. stress tensor becomes symmetric in the zero birefringence limit b) The e.m. SAM and OAM flux density become independently conserved quantities (divergence free) in the zero birefringence limit c) The LC stress tensor becomes symmetric in the zero elastic anisotropy limit d) The LC SAM and OAM flux density become independently conserved quantities (divergence free) in the zero in the zero elastic anisotropy limit e) The SAM and OAM LC degrees of freedom become decoupled and couple separately each other and are coupled, respectively, with the SAM and OAM fluxes of the e.m. field Workshop on singular optics Trieste, May 30 June 3
Thanks for your attention