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
IV. Matter governing light:
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
IV. Matter governing light:
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
Light propagation in helical cholesterics
• 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
Light propagation in helical cholesterics
• 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
Light propagation in helical cholesterics
•
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
Light propagation in helical cholesterics
•
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Second harmonic generation
• 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
Second harmonic generation
• 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
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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.
3/1/2017 75
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
3/1/2017 76
• 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
3/1/2017 79
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.
VI. Light governing matter
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)
Nonlinear propagation of light
• 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
Nonlinear propagation of light
• 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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VI. Light governing matter
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
2/12/2016
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:
2/12/2016 94
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?
2/12/2016 96
• 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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