Liquid Crystals and Light - University of Oxford · Chirality • chirality represents broken inversion symmetry • constituent molecules of liquid crystals may be chiral • observable

Liquid Crystals

and

Light

Peter Palffy-Muhoray Liquid Crystal Institute

Kent State University

3/1/2017 1

Outline

• motivation: why LCs?

• light and matter

• matter governing light

• nonlinear optics

• light governing matter

• summary

3/1/2017 2

I. Motivation

3/1/2017 3

Motivation

• Why LCs?

– interesting aspects:

1. orientationally ordered

2. responsive

3/1/2017 4

Orientation

• concept not necessary, but useful

– else have to specify position of many points

• unusual aspects:

position orientation

two particles cannot have same position

two particles can have same orientation

to change from one position to another, particle must have linear momentum

to change from one orientation to another, particle need not have angular momentum

fermion-like boson-like

3/1/2017 5

Orientation

• concept not necessary, but useful

– else have to specify position of many points

• unusual aspects:

position orientation

two particles cannot have same position

two particles can have same orientation

to change from one position to another, particle must have linear momentum

to change from one orientation to another, particle need not have angular momentum

fermion-like boson-like

3/1/2017 6

Responsivity

• LCs respond readily to stimuli

• Why?

– nematic liquid crystals:

director breaks continuous symmetry; system is

necessarily ‘SOFT’

Goldstone’s theorem:

Broken continuous symmetry → low energy excitations

→ responsive

3/1/2017 7

Goldstone modes ?

– at high T, rodlike particles are randomly oriented

– below a critical temperature, they orient so that, on the

average, they point in some direction

– this direction is arbitrary

• (Hamiltonian is invariant under rotation broken continuous symmetry no external field)

– energy of changing can only depend on

• no coupling of to external field

– energy of long wavelength excitations is vanishingly small

nn

n

0S

ˆn

n

3/1/2017 8

Goldstone modes ?

– the energy of deformation per volume goes as

– which vanishes as !

• these low-energy excitations are Goldstone modes

– they destroy long range order in 1D (Peierls) and 2D (Hohenberg-Mermin-Wagner)

– make ’soft’ materials responsive

2 2 2 2 22ˆ~ ( ) ( )q qK KA q KA

n

3/1/2017 9

II. Light

3/1/2017 10

• current of photons

• photons have:

– no rest mass

– wavelength

– frequency

– energy

– energy current density

– linear momentum

– angular momentum

(spin)

0 0m

/h

spin

E

H

h

E H

3/1/2017 11

Photons and classical fields

• classical fields are ‘wavefunctions’ of photons

• photon number density: ,

• photon current density:

• momentum conservation can explain light induced forces

and torques on reflection & absorption

2

01

2p

E

h

2

01

2p

H

h

ph

E HJ

3/1/2017 12

II. Fields and matter

3/1/2017 13

Fields and matter

• coupling is via forces on charges in matter

– electric fields create electric dipoles

linear response

– magnetic fields create magnetic dipoles

linear response

»

0 0 0 ( )r D E E P I E

0 ( )o r B H H M

0 P E

M H

1q dV

V P r

1 1

2qdV

V M r J

3/1/2017 14

Linear Response

– impulse response

– (Green’s function)

– we have, in real space

– and, in Fourier space,

• it follows that is valid in Fourier space

• key notion: response is nonlocal

/1( ) , 0 t RCg t e t

RC

0( ) ( ) (t )div t v g

0( ) ( ) ( )iV G V

D E

( , ) ( , ) ( , ) D k k E kCaveat:only if system is homogeneous(invariant under translation)

3/1/2017 15

Light Propagation

• The propagation of electromagnetic radiation is described by Maxwell’s equations:

where

and there are no free charges.

0

0

t

t

D

B

BE

DH

0

0 ( )

o r

o r

D E E P

B H H M

3/1/2017 16

The wave equation

• if the material properties do not change in time, we write

• where and are tensors.

• if magnetic properties are isotropic and uniform, then

• and

with .

t

t

HE

EH

2

2t

EE

22

2( )

t

EE E

0 E

3/1/2017 17

The wave equation

• we look for solutions of the form .

• here is the wave vector,

• the speed of the wave is

• the wave equation becomes an eigenvalue equation

• with

( )

0

i te k rE E

2

0 0ˆ ˆ( ) v I kk E E

2 ˆ

k k

2v

k

0 0 k E

3/1/2017 18

The wave equation

• we have

• in vacuum,

• in general,

• refractive index:

2

0 0ˆ ˆ( ) v I kk E E

0 0

1v c

1v

r r

cn

v

02

ck

2v

k

o

n

3/1/2017 19

Dielectric tensor

• in liquid crystals, the polarizability is anisotropic

• rearrange

• and

• dielectric tensor

E

lˆ ˆ ˆ ˆ( ) ( ( ) ) p E l l E E l l

2 2 1 ˆˆ[( ) ( ) (3 )]3 3 2

p I ll I E

[ ]Q P I E

0 0

(1 ) Qr

I

1 1 1 1ˆ ˆ( )

I nnˆ ˆ( ) I nn

3/1/2017 20

IV. Matter governing light:

Light propagation in uniaxial crystals

3/1/2017 21

Solutions of the wave equation

• have

• better to consider

• index ellipsoid:

Ordinary mode:

1

0 02

1ˆ ˆ( ) rn

I kk D D

0 02

1ˆ ˆ( ) r rn

I kk E E

1

0 0 1r D D

1 1 1 1ˆ ˆ( )

I nn n

k

on en

0D0E

0H

on en

S

S E Hˆ, ,D E k n

on n

S k

( 1)r

E D

3/1/2017 22

Solutions of the wave equation

• have

• better to consider

• index ellipsoid:

Extraordinary mode:

1

0 02

1ˆ ˆ( ) rn

I kk D D

0 02

1ˆ ˆ( ) r rn

I kk E E

1

0 0 1r D D

1 1 1 1ˆ ˆ( )

I nn n

k

on en

0D0E

0H

on en

S

S E Hˆ, ,D E k n

2 2 2 2cos sin

e o

e o

n nn

n n

S k D E3/1/2017 23

Two propagating modes: ordinary & extraordinary

• have orthogonal polarizations

• rays travel in different directions

• describes light propagation in nematics

• remarkable aspect of extraordinary mode:

ray (energy, photons)

wave (phase, momentum)

• propagate in different directions !

n

k

0D

0E

0H

S

2 2

2 2 2 2

( )sin costan

cos sin

e o

e o

n n

n n

3/1/2017 24

Conoscopic pattern from nematic film

• interference between modes

Polarizer Analyzer

Microscope objective

LCE Sample

Screen

Polarizer Analyzer

Alignment direction

45°

LCE

green: experiment red: theoryp.p-m. group

3/1/2017 25

q-plates from nematic film

• spatially varying director field.

• applications in quantum computing

photoalignment

far field intensity pattern

L. Marrucci et al. 2011

3/1/2017 26


Chirality and optical activity

3/1/2017 27

Chirality

• chirality represents broken inversion symmetry

• constituent molecules of liquid crystals may be chiral

• observable consequences:

– optical activity (isotropic)

• rotation of plane of polarization

– photonic band gap with handedness (anisotropic)

• no stable propagation for some range of frequencies

• form of in isotropic chiral materials?( , ) k

3/1/2017 28

Optical activity (isotropic materials)

• expand in :

• substitute in wave equation:

– two circularly polarized modes, opposite handedness

– optical rotation:

• can write, in real space,

k1( )o i k D E

Levi-Civitapseudo-scalar

21 2( )

2 o

z

where does come from?

pseudoscalar optical rotation

t

BD E E E

3/1/2017 29

Particle shape and the pseudoscalar

• consider the chiral molecule/particle

– polarizable segments

– field only at the origin

– calculate dipole moment

– Fourier transform:

– small and isotropic average

a ˆcˆ

c

b

Eˆ ˆ ˆ' c a a b

ˆ ˆ ˆ, , c a cx

y

z

00 3

0

ˆ ˆ ˆ ˆ( )( ( cos( ) i '( )sin( ))2 b

p E a a a k b a b k b

k b

20 0

20 0

{( ) i ' )}3 6

o kb

D E

00 3

0

ˆ ˆ ˆ ˆ ˆ ˆ( )( ( ) (( ' ) ( ) ( ' ) ( ))4 b

p E a a r a a b r b a a b r b

3/1/2017 30

Current Approaches

• two schools of thought:

• Landau & Lifshitz: (Casimir, Agranovich)

– spatial dispersion, Fourier space

• Drude-Born-Fedorov:

– non-locality, real space description

• two approaches have been ~reconciled, but questions

remain: (i.e. BCs for non-local interactions….)

( , ) ( , ) ( , ) D k k E k

( )

( )

D E E

B H H

V.M. Agranovich and V.L. Ginzburg, ‘Crystal Optics and Spatial Dispersion..’ (Springer, 1984)

3/1/2017 31

Optical activity (isotropic materials)

• expand in :

• substitute in wave equation:

– two circularly polarized modes, opposite handedness

– optical rotation:

– if , can ignore

• reasonable description for homogeneous, isotropic

materials

k1( )o i k D E

Levi-Civitapseudo-scalar

21 2( )

2 o

z

where does come from?

pseudoscalar optical rotation / 1

3/1/2017 32


Optical properties of helical cholesterics

3/1/2017 33

What if material is not homogeneous ?

• inhomogeneous: not invariant under translation

– in space: material properties vary with position

– in time: material properties very in time

– - if R, C vary in time, cannot use Green’s

– function method

– 0 - response is not simple convolution

• strategy: determine relation between and

solve PDE

• determination of constitutive equations can be difficult

and D E and B H

3/1/2017 34

Helical cholesteric (chiral nematic) material

• chiral molecules form helical structures

– due to symmetry; attractive and repulsive interactions

• permittivity is function of

– response is non-local

– spatially inhomogeneous

• approach:

– ignore non-locality

– work in real space

zz

3/1/2017 35

Light propagation in helical cholesterics

• here we assume

• need to solve

• construct rotating frame

• in rotating frame

• write field as

(kz )( ) i tz e E E

22

2 2( ) r

c t

EE E

ˆ ˆ ˆcos sinqzqz η x y

ˆ ˆ ˆsin cosqzqz φ x y

0

0r

ˆ ˆ(z) E E E η φ

z

η

φ

3/1/2017 36

2 2 2 20 0( ) 4 ( )np p


• then

• since and , we obtain

• where and .

22 2

2( ) 2q k E ikqE

c

22 2

2( ) 2q k E ikqE

c

2q

p

0

2 nk

2

2

3/1/2017 37

2 2 2 20 0( ) 4 ( )np p


• evaluating

• no propagation if .

2n

0

p

3

2

on en

2 0n

0

2( z )

( )i

nt

z e

E E

2 0n

3/1/2017 38

2n

0

p

3

2

onen

bandgap


•

0

2( z )

( )i

nt

z e

E E

z

η

φ

0ˆ i tE e

E φ0ˆ i tE e

E η

LCP

RCP RCP

0

2( z )

0ˆ

oin

t

E e

E φ

0

2( z )

0ˆ

ein

t

E e

E ηstanding waves at band edges

3/1/2017 39

3/1/2017 40

Twisted Nematic Cell

• how a TN cell works:

• ~cholesteric

surface alignment layer

surface alignment layer

polarization of light

follows director

field induced

alignment

0 1p

2n

0

p

3

2

onen

bandgap


•

0

2( z )

( )i

nt

z e

E E

z

η

φ

0ˆ i tE e

E φ0ˆ i tE e

E η

LCP

RCP RCP

0

2( z )

0ˆ

oin

t

E e

E φ

0

2( z )

0ˆ

ein

t

E e

E ηstanding waves at band edges

3/1/2017 41

Reflection from helical cholesterics• no propagation in the reflection band

• incident light with the ‘same handedness’ as the

helical structure is reflected

LCP RCP

Scarab beetles: Chrysina beyeri(left- and right-circular polarizedillumination)

Cholesteric cell with laserwritten image (unpolarizedIllumination)

3/1/2017 42

V. Nonlinear optics

3/1/2017 43

• linear optics:

• and

• nonlinear optics:

• and

Nonlinear optics

0 P E

22

02( ) ( )

t

E E E P

0 2 3( ...) P E EE EEE

22

0 0 2 32( ) ( ( ..)

t

E E E E EE EEE

nonlinear

P

3/1/2017 44

– consider motion of electron in an anharmonic potential

and a sinusoidal field

– equation of motion:

– try solution of the form

– treat as perturbations

Nonlinear optics: basics

( )U x

E

22 3

1 2 32x ... cos

xm k x k x k eE t

t

1 2 3, ,...k k kharmonic

0 1 2 3cos cos cos ...x x x t x t x t

0 2 3, , ....x x x

2 3 4

1 2 3

1 1 1( ) ..

2 3 4U x k x k x k x

3/1/2017 45

– it follows that

where is the charge of the electron, and .

– continuing gives

– the polarization is

Nonlinear optics: basics3 3

0 3 01 2 2 2 2 4

0 0

( / ) 3 ( / )

( ) ( )

e m E k e m Ex

2 2 332 21 1 1 12 2 2 2

1 0 0

/ m1 /cos cos cos3 ..

2 2( (2 ) ) ( (3 ) )

kk k mx x x t x t x t

k

P p ex

e 20 /k m

3/1/2017 46

• having the polarization, one can write explicitly

• note powers of .

Nonlinear optics: basics

0E

3/1/2017 47

– in general, the susceptibilities are written as

– where (momentum conservation)

– and (energy conservation)

Nonlinear optics: standard form

(3)

0 1 1 2 2 3 3( , ) ( , ) ( , ) ( , ) ( , )P E E E k k k k k

1 2 3 k k k k

1 2 3

3/1/2017 48

Nonlinear optics: liquid crystals

• two types of NLO effects:

– nonlinear electronic response which originates in LC

structure

• LC lasers, SHG

– linear electronic response with light induced changes in

structure

• optical Kerr effect, material response to light-induced forces and torques

3/1/2017 49

V. Nonlinear Optics

Liquid crystal lasers

3/1/2017 50

Mathematical origins

• Floquet’s theorem:

periodic coefficient in 2nd order ODE

• solution is of the form

• allowed solutions are in divided into ‘bands’ (μ real)

• no stable solutions between bands! (μ imag)

~ ( )i ze z

G. Floquet, C. R. Acad. Sci. Paris 91 ,880–882 (1880).

3/1/2017 51

Bandgaps: examples and applications

– electronics: periodic crystal structure

• electronic band gap materials: semiconductors, .....

– photonics: periodic dielectric structure

• photonic band gap materials: optical switches, lasers,..?

22 ( ( ))

2E V

m r

2 2 ( ) E r E

3/1/2017 52

Left circular Right circular

0

0.4

0.8

Fre

qu

ency (

2

c/pit

ch)

kx ( 2/pitch)

0 0.4 0.80.40.8


0

0.4

0.8

Fre

qu

ency (

2

c/pit

ch)

kx ( 2/pitch)

0 0.4 0.80.40.8

o

p

2n

0

n

n reflection

band

o

p

2n

0

n

n reflection

band

2

( )o

ni z

e z

E E

- on p en p

400 500 600 700

0.0

0.2

0.4

0.6

0.8

1.0

(nm)

Ref

lecta

nce


0

0.4

0.8

Fre

qu

ency (

2

c/pit

ch)

kx ( 2/pitch)

0 0.4 0.80.40.8


0

0.4

0.8

Fre

qu

ency (

2

c/pit

ch)

kx ( 2/pitch)

0 0.4 0.80.40.8

o

p

2n

0

n

n reflection

band

o

p

2n

0

n

n reflection

band

2

( )o

ni z

e z

E E

- on p en p


0

0.4

0.8

Fre

qu

ency (

2

c/pit

ch)

kx ( 2/pitch)

0 0.4 0.80.40.8


0

0.4

0.8

Fre

qu

ency (

2

c/pit

ch)

kx ( 2/pitch)

0 0.4 0.80.40.8

o

p

2n

0

n

n reflection

band

o

p

2n

0

n

n reflection

band

2

( )o

ni z

e z

E E

- on p en p

400 500 600 700

0.0

0.2

0.4

0.6

0.8

1.0

(nm)

Ref

lecta

nce

400 500 600 700400 500 600 700

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

(nm)

Ref

lecta

nce

Helical Cholesteric

• band structure:

2

p

– reflection band: band edges at and

3/1/2017 53

Lasing: feedback effects

active medium

mirrormirror

• coupled waves

• coupling:

• threshold gain: if

• standing wave & coherent emission

21ln

2th r

L kL m

( ) ( )

1

ikL L L ikL L L

n nE E e re r

L

3/1/2017 54

Lasing: cavity effects

active medium

mirrormirror

• emission rate:

• density of states:

• bare emission rate:

• emission rate:

( )o

( )dk

d

o

Fermi’s Golden Rule

( )o

3/1/2017 55

• infinite sample:

• density of states diverges

• expect ~thresholdless

lasing at band edges!

DOS in bulk cholesterics


0

0.4

0.8

Fre

qu

ency (

2

c/pit

ch)

kx ( 2/pitch)

0 0.4 0.80.40.8


0

0.4

0.8

Fre

qu

ency (

2

c/pit

ch)

kx ( 2/pitch)

0 0.4 0.80.40.8

o

p

2n

0

n

n reflection

band

o

p

2n

0

n

n reflection

band

2

( )o

ni z

e z

E E

- on p en p

400 500 600 700

0.0

0.2

0.4

0.6

0.8

1.0

(nm)

Ref

lecta

nce


0

0.4

0.8

Fre

qu

ency (

2

c/pit

ch)

kx ( 2/pitch)

0 0.4 0.80.40.8


0

0.4

0.8

Fre

qu

ency (

2

c/pit

ch)

kx ( 2/pitch)

0 0.4 0.80.40.8

o

p

2n

0

n

n reflection

band

o

p

2n

0

n

n reflection

band

2

( )o

ni z

e z

E E

- on p en p


0

0.4

0.8

Fre

qu

ency (

2

c/pit

ch)

kx ( 2/pitch)

0 0.4 0.80.40.8


0

0.4

0.8

Fre

qu

ency (

2

c/pit

ch)

kx ( 2/pitch)

0 0.4 0.80.40.8

o

p

2n

0

n

n reflection

band

o

p

2n

0

n

n reflection

band

2

( )o

ni z

e z

E E

- on p en p

400 500 600 700

0.0

0.2

0.4

0.6

0.8

1.0

(nm)

Ref

lecta

nce

400 500 600 700400 500 600 700

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

(nm)

Ref

lecta

nce

2

( )o

dk k dn

d n d

DOS

o

p

0

3/1/2017 56

Liquid crystal DFB lasers

Nd:YAG

detectorlens

plate

polarizerfiber

Nd:YAG

detector

dye-doped helical cholesteric

sample

lens

plate

polarizerfiber

polarizer

2

• Experimental setup:

25 m25 m

3/1/2017 57

Wavelength (nm)

500 600 700

Lig

ht

Inte

ns

ity

(a

.u.)

0

1

2

3

4

5

Reflection band

Laser emission em= 668nm

BLO61-E7:DCM(2%WT)

ex=355 nm.

Lasing at the low energy band edge

3/1/2017 58

Wavelength (nm)

550 600 650 700

Lig

ht

Inte

ns

ity (

a.u

.)

0

1

2

Reflection band

Laser emission em = 595nm

BLO61-E7:DCM(2%WT)

ex=355 nm.

B. Taheri, A.F. Munoz, P. Palffy-Muhoray, R. Twieg,Mol.Cryst. Liq. Cryst. . 358, 73, 2001

Lasing at the high energy band edge

3/1/2017 59

Mechanically tunable cholesteric elastomer laser

Wavelength (nm)

540 550 560 570 580 590 600 610

Lig

ht

Inte

ns

ity (

a.u

.)

0

1

2

3

ex =532nm, 35ps

Si O

CH3

H n

O C

O

O OCH3

O C

O

O

CH3

CH3

CH3

CH3

CH3

O

O

O

9

9

9

H. Finkelmann, S-T. Kim, A. Munoz, P. Palffy-Muhoray and B. Taheri, Adv. Mat. 13, 1069 (2001)

tunable rubber laser!

3/1/2017 60

– increasing E-field decreases pitch

Electrically tunable heliconical cholesteric laser

J. Xiang, F. Minkowski, D.A. Paterson, J.M.D. Storey, C.T. Imrie, O.D. Lavrentovich, P. P-M, PNAS 113, 12925 (2016)

3/1/2017 61

V. Nonlinear Optics

Second Harmonic Generation

3/1/2017 62

Second harmonic generation by LCs

• seen by electron cannot be even; need

• inversion symmetry must be broken:

– DC electric field

• electric field induced second harmonic: EFISH

– surface normal

• surface second harmonic generation

– more generally

• variety of ways

( )U x 2 0k

N

E

3/1/2017 63

Second harmonic generation

• in liquid crystals, , and

– splay

– bend

– order parameter gradients

• break the symmetry.

• SHG can probe director field/order parameter structure

ˆ ˆ( ) I nn

ˆ ˆ( )n n

ˆ ˆ( ) n n

Q

flexo-electricity

order-electricity

3/1/2017 64


• periodic director field

• SHG can probe order parameter field gradients

phase matching conditionprobes structure

micrograph of sample

K.M. Hsia, T. Kosa and P.P-M, SPIE Vol. 3015, 32 (1997)

E7, optically buffed40 μm thick, 25 μm strips

3/1/2017 65


• parabolic focal conic defects in nematic twist-bend phase

• SHG can unravel details of defect structure

S. A. Pardaev, S. M. Shamid, M. G. Tamba,.C. Welch, G. H. Mehl, J. T. Gleeson, D. W. Allender, J. V. Selinger, B. Ellman, A. Jakli, S. Sprunt. Soft Matter, 12, 4472 (2016)

3/1/2017 66

VI. Light governing matter

3/1/2017 67

Light induced forces and torques

• Minkowski stress tensor:

– force:

– torque density:

• Einstein-Laub torque density:

1( )

2T ED HB E D H B I

T d F A

τ D E B H

23ˆ ˆ ˆ( )

4T d

τ r r r

3/1/2017 68

VI. Light governing matter:

Light induced forces

3/1/2017 69

Light induced forces

• light exerts outward force on liquid surface

• reflected light pushes down

• reflection coeff.

• transmitted light pushes up

• net pressure

• effect small; similar for water and nematics .

0

2r

I hP R

h

0 0

(1 )( )t

I nh hP R

h

2

2

( 1)

( 1)

nR

n

2 ( 1)I

P nc

N.G.C. Astrath, L.C. Malacarne, M.L. Baesso, G.V.B. Lukasievicz, S.E. Bialkowski, Nature Communications 5, 4363 (2014)

10H nm

3/1/2017 70

Ashkin, A. & Dziedzic, J. M. Radiation pressure on a free liquid surface. Phys. Rev. Lett. 30, 139–142 (1973).

VI. Light governing matter:

Light induced torques

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Optical Freedericksz transition

• Freedericksz transition:

– configurational transition in nematic cell due to or field

– threshold field:

– net torque on plates is zero.

• Question: can light create such a transition?

B.Ya. Zel’dovich, N.F. Pilipetskii, A.V. Sukhov, N.V. Tabiryan, Pis’ma Eksp. Teor. Fiz. 31, 287 (1980)

E H

0E0E

1

0

c

KE

d

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V. K. Freedericksz and V. Zolina, Trans. Faraday Soc. 29, 919 (1933).

Optical Freedericksz transition

• polarized light can cause transition if intensity is above

threshold

• interesting to consider momentum transport

3/1/2017 73

laserray

table

_hp =

B.Ya. Zel’dovich, N.F. Pilipetskii, A.V. Sukhov, N.V. Tabiryan, Pis’ma Eksp. Teor. Fiz. 31, 287 (1980)

Light induced torques in nematics

laserray

table

_hp =

n

k

D

E E x H

index ellipsoid periscope

torque

angular momentum transfer:

torque results fromchange in extrinsicangular momentumof light.

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opt τ D E

0opt el τ τ

Optical Freedericksz transition in nematics is well understood.

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Optical torque on nematic+dye: Janossy effect

1

addition of 1% of anthroquinone dye

reduction of threshold intensity by ~ 100!

laser

- dye

1100

without dye: 0

with dye: ?? 0

el opt

vol vol

el opt

vol el

Torque on nematic causing elastic deformationCANNOT come from light! Source of torque?? (A. Saupe)

1, I. Janossy, A.DD. Lloyd and B.S. Wherret, Mol. Cryst.Liq. Cryst. 179, 1, 1990.I. Janossy, Phys. Rev. E 49, 2957, 1994.

diversion

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• light causes the director to reorient, against a restoring

elastic torque, essentially without the transfer of angular

momentum.

• light causes rotation without exerting a torque!

the Puzzle: diversion

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Landauer’s Blowtorch*:

* R. Landauer, J. Stat. Phys. 53, 233 (1988)

particles in periodic potential

& temperature field

particles near sides with negative

slope are more likely to be

excited & diffuse over barrier

net current flows to the left!

heat source drives current

without transfer of momentum.

x

T(x)

x

U(x)

diversion

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Landauer’s Blowtorch*

* R. Landauer, J. Stat. Phys. 53, 233 (1988)

dye in nematic potential

& temperature field

particles near sides with negative

slope are more likely to be

excited & diffuse over barrier

net current flows to the left!

laser drives orientational current

without transfer of momentum.

θ

I(θ)

θ

U(θ)

laser

diversion

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Orientational ratchet

• dye & nematic molecular motor

• dye is rotor in field of nematic

• torque on nematic is proportional to orientational dye current

• shear carries angular momentum to cell walls

• large optical nonlinearity

• light provides energy but not momentum to drive molecular motor

T. Kosa, Weinan E and P. Palffy-Muhoray, Int. J. Eng. Sci. 38, 1077 2000M. Kreuzer, L, Marrucci and D. Paparo, J.Nonlin.Opt.Phys. 9, 157, 2000.

diversion

Light induced torques in nematics

laserray

table

_hp =

n

k

D

E E x H

index ellipsoid periscope

torque

angular momentum transfer:

torque results fromchange in extrinsicangular momentumof light.

3/1/2017 80

opt τ D E

0opt el τ τ

Optical Freedericksz transition in nematics is well understood.


Nonlinear propagation

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Nonlinear propagation of light

• simple & beautiful experiment:

– polarized light traverses a capillary with nematic

• alignment is parallel to walls

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E. Braun, L. P. Faucheu, A . Libchaber, D. W. McLaughlin, D. J. Muraki, M. J. Shelley, Europhys. Lett. 23, 239 (1993)


• observe filamentation and undulation of self-focused

laser beams

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20.84 /kW cm

21.87 /kW cm

23.36 /kW cm


• simple model

– inner equation: models filamented structure

– outer problem: describes large-scale beam-nematic

interaction

• describes self-focusing,

self-trapping of light and beam undulation

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Solid liquid crystals and

Light driven motors

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Liquid Crystal Elastomers

• anisotropic rubbers

with combined features of

LIQUID CRYSTALS

&

ELASTIC SOLIDS

proposed by de Gennes

produced by Finkelmann

P.G. de Gennes, 1932-2007

2/12/2016

P.G. de Gennes, C.R. Seances Acad.Sci.218, 725 (1975)

H. Finkelmann, H.J. Kock, G. Rehage, Makromol. Chem., Rapid Commun. 2, 317 (1981)

H. Finkelmann

86

• free energy density*:

• sum of free energies of liquid crystal + elastic solid,

plus new coupling term

• effect of strain on LC order is same as external field

• effect of LC order on strain is same as external stress

Liquid Crystal Elastomer

*P.G. de Gennes, C.R. Seances Acad.Sci.218, 725 (1975)

Qe

2 21 1..

1'

22 2

extA Y QE eQe σEQ eF

2/12/2016 87

director

director

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Appearance of nematic LCE samples

50mm x 5mm x .3mm

birefringent sample between

crossed polarizers

88

Experimental Results (Ikeda et al. )

Yanlei Yu,Makoto Nakano, Tomiki

Ikeda, Nature 425, 125 (2003)

timescale: 10 s

LC + diacrylate network+ functionalizedazo-chromophore

Yanlei Yu, Makoto Nakano, Tomiki Ikeda, Nature 425, 125 (2003)

2/12/2016 89

Photoinduced Bending

2/12/2016

sample: nematic elastomer EC4OCH3

+ 0.1% dissolved

Disperse Orange 1 azo dye5mm

300 m

Response time: 70ms

M. Chamacho-Lopez, H. Finkelmann, P. Palffy-Muhoray, M. Shelley, Nature Mat. 3, 307, (2004)

90

Swimming away from the light

2/12/2016

• floating nematic LCE sample illuminated from above

Laser beam

container

waterElastomer

M. Chamacho-Lopez, H. Finkelmann, P. Palffy-Muhoray, M. Shelley, Nature Mat. 3, 307, (2004)

91

Rotary motor: light driven

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M. Yamada, M. Kondo, J. Mamiya, Y. Yu, M. Kinoshita, C. Barrett, T. Ikeda, Angew. Chem. 47, 4986 (2008)

Rotary motor: light driven

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M. Yamada, M. Kondo, J. Mamiya, Y. Yu, M. Kinoshita, C. Barrett, T. Ikeda, Angew. Chem. 47, 4986 (2008)

key aspect: light changes degree of orientational order Q

Model

– bend energy density:

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2 2

1 1/

2b

cl EI

R R

1 2

1

3 4

2 3 4

22 2

2

1

22 2 2

2

2 1

1ˆ( ) ( ( ))

1 1ˆ( ) ( ( ) ( ) ,

o

o

s s

s s

s s s

s s s

c ds c dsR s s

c ds c ds c dsR s s R

R Rz

R Rz

1s

3s

4s

2s

xsos

2

31

4

2R1R

stiffness natural curvaturefn. of positionfn. of position

constraints

Numerical solution: energy minimization

2/12/2016 95

local spontaneous curvaturedue to light (heat, solvent,...)

s

X. Zheng, P.P-M. unpublished

How does rotation come about?

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• away from the small wheel,

– longer lever arm on one side

• near small wheel

– greater point torque on one side

• curvature is the same, but have

spontaneous curvature on one side

Angular momentum current

• before irradiation

– F and F are forces provided by supports

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F F

TT

TT

Angular momentum current

• on UV irradiation

– F, -F, f and -f are forces provided by supports

– angular momentum flows in from outside to start

• light provides energy but not momentum to drive macroscopic motor

2/12/2016 98

F F

T T

TT

UV

ff

VI

Summary

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Summary

• LCs are SOFT

– director deformations are Goldstone modes

• LC anisotropy can guide light

• controlling director field enables light control

– LCDs, tunable lenses, gratings, etc.

• material response is non-local

– chirality

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Summary

• periodic structures are photonic bandgaps

– cholesteric lasers, mirrors, polarizers

• NLO response is HUGE due to coupling to soft modes

– times conventional materials (but slow)

• mechanisms via momentum or energy transfer

The softness and anisotropy of liquid crystals

enriches light matter interactions.

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Liquid Crystals and Light - University of Oxford · Chirality • chirality represents broken inversion symmetry • constituent molecules of liquid crystals may be chiral • observable