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

Iztok Bajc

Adviser: Prof. dr. Slobodan Žumer

Fakulteta za matematiko in fiziko

Univerza v Ljubljani

Slovenija

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

Motivations, classical and new applications

Motivations

• Interesting and fast evolving field.• Liquid crystals well represented field in Slovenia.

Why to approach this thematic?

M. Ravnik, S. Žumer, Soft Matter, 2009.

• One of the priorities of the EU project (Hierarchy)

in which I’m involved.

M. Humar, M. Ravnik, S. Pajk, I. Muševič,

Nature Photonics, 2009.

• New potential applications:

• Metamaterials.

• Microcavities - microresonators.

• Requirement of very effective modeling codes.

Challenge to find the right approaches.

(Hot topics!)

• LCD (Liquid Crystal Displays).

Classical applications of liquid crystals

Liquid crystals

have unique

optical properties.

• Eye protecting filters for welding helmets (Balder)

• Polarizing glasses for 3D vision

New potential applications: metamaterials, microresonators

• Solid state metamaterials:

• Photonic crystals:

• Soft metamaterials?

Nematic droplet.

Figures: I. Muševič, CLC Ljubljana Conference, 2010.

Whispering Gallery Modes (WGM ) in a microresonator.

Nematic Liquid Crystals

Nematic liquid crystals

E

• (The same happens, if temperature is lowered)

• Electric or magnetic field can change their phase form isotropic liquid to partially ordered mesophase.

• Molecules are rodlike.

• Tend to align in a preferred direction.

• Liquid crystals are a liquid, oily material.

• They flow like a liquid...

• ... but can be partially ordered - like crystals.

Description of nematic liquid crystals

• Basic quantities

)(rn

Director )(rS

Scalar order

parameter

Quantifies the degree of order of the

orientation:

-1/2 ideal biaxial liquid

0 isotropic liquid

1 ideally aligned liquid

(all molecules parallel)

12

1 S

1n

Points in preferenced orientation.

Alternative description with Q-tensor field

221123

2eeee

PInn

SQ

• traceless:

its largest eigenvector and its corrispondent eigenvalue.n

S

New quantity: tensor order parameter :)(rQ

Q 0332211 QQQ 221133 QQQ

Only 5 independent

components of Q are required.

• symmetric:Qjiij QQ

2211

2322

131211

QQ

QQ

QQQ

Q

Free-energy functional

2ijijkijkijijij

k

ij

k

ijbulk )QC(Q

4

1 QQ BQ

3

1 QAQ

2

1

x

Q

x

Q L

2

1 f

dVQQfdVQQfQFborder

surf

bulk

bulk ),(),()(

• Director and order nematic structure follow from minimizing the Landau-de Gennes functional:

2)0(ijijsurf )Q-(QW

2

1 f

Elastic energy

Surface energy

Thermodynamic energy

L – elastic constants

A, B, C – material constants

W – surface energy

Colloidal particles in nematic

• We get disclination lines (topological defects) around the particles:

• Inclusion of colloidal particles in a thin sheet of nematic LC.

Colloidal structures

-crystals in nematic.

Strong attractive forces

between particles.

Inclusion of colloidal particles

Structures of colloidal particles in nematic

1D structures

2D structures - crystals

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).

Large 3D structures:

Modeling Requirements

Actual finite difference code in C is:

• Robust and effective for smaller or periodic systems.

• But uses uniform grid (uniform resolution).

Computations until now

A job needs 2h to converge.

You double the resolution

Then it will run for 2 days.

Example:

New modeling requirements

Moving objects (due to

nematic elastic forces).

Mesh adaptivity in 3D, preferably

with anisotropic metric.

Parallel processing

(computer clusters).

Meshes by Cécile Dobrzynski, Institut de Mathématiques de Bordeaux.

Newton iteration of tensor fields

If function (of one variable):

0)(' xf

)('')('

1 k

k

xfxf

kk xx )(')('' kkk QFQQF

kkk QQQ 1

0)(')( QFQF

First variation of functional:

( - test functions)

Newton iteration:Newton iteration:

Finite Element Method (FEM)

Advantages: – Mesh can be locally refined less mesh point needed.– Around each point we have an interpolating function (spline).

Test calculations in 3D:One colloidal particle

• Central section of 3D simulation box mesh

• Mesh points: 17 000; Tetrahedra: 100 000

• Mesh generation’s time: 5 sec (TetGen)

2 microns

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

2 microns

• Central section of the order parameter field S.

• In green: sections of Saturn ring defect.

Topological defect

2 microns

Test calculations in 3D : More particles

Future work: external fields for photonic systems

Electric field on a nematic droplet

)(Q

A large field E change Q.

Iteration needed

Also changes.

By tuning electric field

Figures: I. Muševič, CLC Ljubljana Conference, 2010.

0E

we switch between optical modes.

Electromagnetic waves – linear/nonlinear optics

• Detail dimensions comparable with wavelength.

Ray optics not adequate.2 microns

• Nematic is a lossy medium.

• Also nonhomegeneously anisotropic.

Birefringence

• Full system description needed (diffraction,...).

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

Basis:

Computational electromagnetics

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.

Frequency domain eigenproblems

0

)(2

1

H

Hc

Hr

)(k

• Seeking for eigenmodes.

• Aim: band structure of photonic crystals.

• Periodic boundary conditions.

1)

Eigenequation

• Reduces to a matrix eigenproblem:

BxAx 2

Pictures from site of Steve Johnoson (MIT).(+ condition)

Frequency domain responses

• Seeking for stationary state.

• Aims: absorption & transmittivity.

• At fixed frequency .?

Ec

iJEtc

rH

Hc

iHtc

E

1)(

1

+ Absorbing Boundary Conditions (ABC).

• Reduces to a matrix linear system:

bAx

2)

Time-domain propagation

Time evolution of electromagnetic waves.

Start with FDTD (Finite Difference Time Domain) numerical method:

1. Ready code freely available.

2. Easily supports nonlinear optical effects.

3. Gain feeling and experience for smaller systems.

Next: possibility of passing to FEM will be considered.

3)

?Micro-optical elements?

Micro-waveguides??

? ?

Work has been finansed by EU:

Hierarchy Project, Marie-Curie Actions

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