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Nematic colloids for photonic systems (with schemes for complex structures)

Nematic colloids for photonic systems (with schemes for complex structures). Iztok Bajc Adviser: Prof. dr. Slobodan Žumer. Fa k ult eta za matematiko in fiziko Univer za v Ljubljan i Slovenija. Outline. Motivations, classical and new applications Nematic liquid crystals

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Nematic colloids for photonic systems (with schemes for complex structures)

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  1. Nematic colloids for photonic systems(with schemes for complex structures) Iztok Bajc Adviser: Prof. dr. Slobodan Žumer Fakultetazamatematikoinfiziko Univerzav Ljubljani Slovenija

  2. Outline • Motivations, classical and new applications • Nematic liquid crystals • Colloidal particles in nematic • Modeling requirements for large 3D systems • Test calculations (3D) • Future work: external fields for photonic systems

  3. Motivations, classical and new applications

  4. Motivations Why to approach this thematic? • Interesting and fast evolving field. • Liquid crystals well represented field in Slovenia. • New potential applications: M. Ravnik, S. Žumer, Soft Matter, 2009. • Metamaterials. • Microcavities - microresonators. (Hot topics!) • One of the priorities of the EU project (Hierarchy) in which I’m involved. • Requirement of very effective modeling codes. Challenge to find the right approaches. M. Humar, M. Ravnik, S. Pajk, I. Muševič, Nature Photonics, 2009.

  5. Classical applications of liquid crystals • LCD (Liquid Crystal Displays). • Eye protecting filters for welding helmets (Balder) Liquid crystals have unique optical properties. • Polarizing glasses for 3D vision

  6. New potential applications: metamaterials, microresonators • Photonic crystals: Nematic droplet. Whispering Gallery Modes (WGM ) in a microresonator. • Solid state metamaterials: • Soft metamaterials? Figures: I. Muševič, CLC Ljubljana Conference, 2010.

  7. Nematic Liquid Crystals

  8. Nematic liquid crystals • Liquid crystals are a liquid, oily material. • They flow like a liquid... • ... but can be partially ordered - like crystals. • Molecules are rodlike. • Tend to align in a preferreddirection. • Electric or magneticfield can change their phaseform isotropic liquid to partially ordered mesophase. • (The same happens, if temperatureis lowered)

  9. Description ofnematic liquid crystals • Basicquantities Scalar order parameter Director Quantifies the degree of order of the orientation: -1/2idealbiaxialliquid 0isotropicliquid 1ideally alignedliquid (all molecules parallel) Points in preferenced orientation.

  10. Alternative descriptionwith Q-tensor field New quantity: tensor order parameter: its largest eigenvector and its corrispondent eigenvalue. • traceless: • symmetric: Only5 independent components of Q are required.

  11. Free-energy functional • Director and order nematicstructure follow from minimizing the Landau-de Gennes functional: Elastic energy Thermodynamic energy L – elastic constants A, B, C – material constants W – surface energy Surface energy

  12. Colloidal particlesin nematic

  13. Inclusion of colloidal particles • Inclusion of colloidal particles in a thin sheet of nematic LC. • We get disclination lines (topological defects) around the particles: Strongattractive forces betweenparticles. Colloidal structures -crystals in nematic.

  14. Structures of colloidalparticles in nematic 1D structures 3D structures 2D structures - crystals

  15. Large 3D structures: 12- and 10- cluster in 90° twisted nematic cell. Experiments by U. Tkalec, 2010 (to be published). 3×3×3 dipolar crystal in homeotropically oriented nematic. Experiment by Andriy Nych, 2010 (to be published).

  16. Modeling Requirements

  17. Computations until now Actualfinite difference code in C is: • Robust and effective for smaller or periodic systems. • But uses uniform grid (uniform resolution). Example: A job needs 2h to converge. You double the resolution Then it will run for 2 days.

  18. New modeling requirements Mesh adaptivity in 3D, preferably with anisotropic metric. Moving objects (due to nematic elastic forces). Parallelprocessing (computer clusters). Meshes by Cécile Dobrzynski, Institut de Mathématiques de Bordeaux.

  19. Newton iteration of tensor fields Finite Element Method (FEM) Advantages: • Mesh can be locally refined less mesh point needed. • Around each point we have an interpolating function (spline). First variation of functional: If function (of one variable): Newtoniteration: Newtoniteration: ( - test functions)

  20. Test calculations in 3D:One colloidal particle

  21. 2 microns • Central section of 3D simulation box mesh • Mesh points: 17 000; Tetrahedra: 100 000 • Mesh generation’s time: 5 sec (TetGen)

  22. 2 microns • Central section: director field n (in green). • Newton’s method took 19 iterations (total time: 54 min).

  23. Topological defect 2 microns • Central section of the order parameter fieldS. • In green: sections of Saturn ringdefect.

  24. Test calculations in 3D : More particles

  25. Future work: external fieldsfor photonic systems

  26. Electric field on a nematic droplet By tuning electric field we switch between optical modes. A large field E change Q. Also changes. Iteration needed Figures: I. Muševič, CLC Ljubljana Conference, 2010.

  27. Electromagnetic waves – linear/nonlinear optics • Detail dimensions comparable with wavelength. 2 microns Ray optics not adequate. • Full systemdescription needed (diffraction,...). • Nematic is a lossy medium. • Also nonhomegeneously anisotropic. Birefringence

  28. Computational electromagnetics Basis: Numerical solution of Maxwell equations Computational photonics Mature field for homogeneous medium and periodic structures (e.g. photonic crystals). But young for nonhomegenously anysotropic media ! Computational soft photonics

  29. Computational approaches Book Joannopoulos et alt., Photonic Crystals, points out three cathegories of problems: 1) Frequency-domain eigenproblems 2) Frequency-domain response 3) Time-domain propagation [1] Joannopoulos et alt., Photonic Crystals, Molding the flow of Light, 2nd ed,Princeton University Press, 2008.

  30. 1) Frequency domain eigenproblems • Seeking for eigenmodes. • Aim: band structure of photonic crystals. Eigenequation Pictures from site of Steve Johnoson (MIT). (+ condition) • Periodic boundary conditions. • Reduces to a matrix eigenproblem:

  31. 2) Frequency domain responses • Seeking for stationary state. • Aims: absorption & transmittivity. • At fixed frequency . ? + Absorbing Boundary Conditions (ABC). • Reduces to a matrix linear system:

  32. 3) Time-domain propagation Timeevolution of electromagnetic waves. ? ? Micro-waveguides? ? ? Micro-optical elements? Start with FDTD (Finite Difference Time Domain) numerical method: • Ready code freely available. • Easily supports nonlinear optical effects. • Gain feeling and experience for smaller systems. Next: possibility of passing to FEM will be considered.

  33. Acknowledgments: • Slobodan Žumer (adviser) • Miha Ravnik, Rudolf Peierls Centre for Theoretical Physics, Univerza v Oxfordu, in FMF-UL. • Frédéric Hecht, Laboratoire Jacques-Louis Lyon, UPMC, Paris 6. • Daniel Svenšek • Igor Muševič • Miha Škarabot • Martin Čopič • Uroš Tkalec Work has been finansed by EU: Hierarchy Project, Marie-Curie Actions

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