Transcript Photonic Crystals

PHOTONIC STRUCTURES FROM
LC/POLYMER COMPOSITES
• Photonic crystals (general properties)
• Photonic crystal optical fibers
• Photonic quasicrystals
12 kV x5,000
1mm
• Switchable diffraction gratings from liquid crystal/polymer (LC/P) composites
• Optical amplification in periodic LC/P structures
• Switchable photonic crystals and quasicrystals from LC/P composites
• Photonic bandgap fibers from LC composites
Photonic crystals-general properties
• Photonic crystals are 1D, 2D and 3D periodic structures of dielectric media
with periodicity (lattice distance) on the scale of visible light.
• Multiple optical reflection and refraction phenomena in such structures result in
dispersion relations (k) similar to electronic band structure E(k) in crystals.
1 2
(e=je=0; m=1) + Maxwell equations 
H(r, t )  H(r)eit

E(r, t )  E(r)eit

ˆ H(r)     1 (r)  H(r)  m  2H(r)

0
 H(r)  E(r ) 
1
 1 (r )  H(r )
i
These equations have to be fulfilled taking into account the structural periodicity:
 1 (r  R)   1 (r)
Due to the periodicity monochromatic EM fields propagating in such medium have
ikr
the form Bloch waves :
Hk (r)  uk (r)e , uk (r  R)  uk (r)
with complicated dispersion relation (k). If for some interval of  all possible
solutions have nonzero imaginary part of k, waves with these  can not propagate
in the photonic crystal (PC). Such interval of  is called the photonic bandgap.
EXAMPLE OF 1D STRUCTURE
Sinusoidaly modulated refractive index:
Propagation of EM field along the z axis: E(z)=eyE(z)
by substitution:
y
   m0 0 21 ,    m0 0 2 2
z
and Maxwell equations it follows:
2E
2

(



cos
z)E  0
2
z
Solutions of this equation have a form
E ( z )  D1 A( z )emz  D2 B( z )e mz
and are known as Mathieu functions.
white regions: nonpropagating waves (m=a+ib)
shadowed regions: propagating waves (m=±ib)
Stability diagram of Mathieu functions.
BAND STRUCTURE of 1D LATTICES
sinusoidal lattice
band
gaps
z
E(z)
A(z)
Reduction to the 1st Brillouin zone.
a
2c
Periodic dielectric layers
GaAs/GaAlAs
GaAs/air
In 1D periodic dielectric structures complete photonic band-gaps exist for any value of
refractive index contrast (1/2)1. In 2D and 3D this happens only in special cases.
2D PHOTONIC CRYSTALS
=(y,z)
E
Example:
quadratic lattice of
dielectric cylinders
TE
Different behaviour for TE and TM modes!
TE modes
TE modes
TM modes
Band-gap only for TM modes
TM modes
Complete band-gap
J. D. Joannopolos, R. D. Meade, J.N. Winn, Photonic Crystals: Molding the flow of light (Princeton Uni. Press, 1995)
EXAMPLE OF A 2D PHOTONIC CRYSTAL
A-C:H = Amorphous
Hydrogenated
carbon
Holographic patterning via photoresist coating.
band-gap
regions
F. Quinonez et al., Optics Express 14, 4873 (2006).
3D PHOTONIC CRYSTALS
To obtain band-gap structures in 3D a very high refractive index contrast is required
(2/1>10) . Besides this only some lattices exhibit complete band-gaps.
One such example is Yablonovitch construction
(E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987))
Self-assembled 3D FCC photonic structure from
dielectric spheres (artificial opals)
DEFECTS IN PHOTONIC CRYSTALS
Defects in PC structures provide localization and guiding of optical waves.
g()

A cavity (defect) within the PC
structure can act as an optical
resonator!
New states appear
witin the bandgap
Such a PC acts as an extremly
narrow band otical filter
(dielectric Fabry-Perot filters)
Extended defects in 2D and 3D provide
waveguiding effects = bending of light
around sharp edges...
Photonic crystal optical fibers
PC optical fibers are a new technological chalange for improoved optical
communication lines.
Compared to standard quarz glass fibers they exhibit reduced optical loss, they can
transmit higher optical power, they exhibit lower optical nonlinearities, they can be
used also out from the conventioanl spectral region (~1.5 mm), their dispersion
properties can be tuned to the desired needs...
cladding
core
http://ab-initio.mit.edu/photons/tutorial/
Basic idea of the PC fibers:
Reflection at the surface of an
„air core“ is obtained by
surrounding it with a cladding
of the PC band-gap medium.

Number of guided modes at selected bandgap region (2D): N 
2
H

2
  L2 rcore
4
EXAMPLES OF STRUCTURES
a) Bragg reflection
fibers:cladding is made
from periodic
cylindrical layers of
two dielectric
materials.
The first example of optical fiber for
CO2 laser light. (OmniGuide, MIT)
B. Temelkuran et al., Nature 420, 650 (2002)
b) PC fiber based on 2D periodic arrangement of air channels in fused silica.
J. F. Cregan et al., Science 285, 1537 (1999)
Photonic quasicrystals
• Quasicrystals are structures with long-range positional order, but no regular periodicity
(no Bloch waves and Brillouin zone concepts, effective Brillouin zones are called (pseudo)Jones zones ).
• To desribe their structure a basic set of more than 3 lattice vectors is needed (4D, 5D...).
• They posses 5, 7, 8, 9 10, 12...fold rotational symmetry axes, which are not possible in regular
crystals.
• They can exhibit complete photonic bandgaps (actually pseudogaps!!) more readily and at
much lower refractive index contrasts than 3D photonic crystals.
Effective „Brillouin zone“ image
Hand-made icosahedral
quasicrystal for
microwave radiation
http://www.physics.princeton.edu/~steinh/quasiphoton/
W. Man et al., Nature 03977, 2005
PENROSE QASICRYSTAL STRUCTURE
R. Penrose and
R. Ammann
Penrose tiling is based on two elements:
1) Thick rombus (angles of (2/10) and (3/10)*360 deg)
2) Thin rombus (angles of (1/10) and (4/10)*360 deg)
Basic rule: no two tiles can
be touching so as to form a
single parallelogram!
Optical structures composed of such tailings can be made by appropriate drilling
and/or etching processes of dielectric films.
Example of structure
fabricated by electronic-beam litography.
M. A. Kaliteevski et al., Nanotechnology 11, 274 (2000).
PENROSE QC – band structure and density of modes
The reciprocal wavector space of QC structures in in principle full. Nevertheless,
some diffraction peaks have much higher intensity than the others. So in practice
we always „see“ only a finite number of diffraction peaks.
Optical diffraction pattern
reduced to 3 most intense
classes of peaks
Band structure calculation with 2 different
sets of RWVs
M. A. Kaliteevski et al., Nanotechnology 11, 274 (2000)
b1=(1,0)=(10000)
b2 =(cos(/5), sin(/5))=(01000)
b3 =(cos(2/5), sin(2/5))=(00100)
b4 =(cos(3/5), sin(3/5))=(00010)
b5 =(cos(4/5), sin(4/5))=(00001)
Reciprocal wave vectors of the internal „ring“.
Switchable diffraction gratings from
LC/polymer composites
Polarization optical microscopy image
of a H-PDLC transmission grating.
100 mm
grating L=1.8 mm
Optical diffraction from
a H-PDLC transmission grating
Optical grating structures from LC/P media can be obtained by:
• periodic configuration of the external electric field (patterned electrodes)
• periodic modification of the alignment layers (patterned alignment layers)
• filling of a periodic polymeric host by the liquid crystalline material
• periodically modulated photopolymerization induced phase separation process
GRATINGS BASED ON PERIODIC ELECTRODES
probe beam w0<<L
Optical microscopy, L=50 mm, L=5 mm
1
nIP   n p ( x, z )  ns ( x, z )dz 
L0
L
0,10
0,08
Field induced in-plane
birefringence.
nIP
0,06
0,04
800 V (RMS)
100 V (RMS)
0,02
0,00
0
50
100
Lateral position (mm)
150
DIFFRACTION PROPERTIES
p polarization
probe beam w0 >>L
I. Drevensek-Olenik at al., J. Appl.
Phys. 96, 6207 (2004)
I. Drevensek-Olenik
al.,Fig.
Fig.5 5
I. Drevensek-Olenik
atatal.,
Ivp
0,080,08
1,01,0
0,060,06
order
2nd2nd
order
order
3rd3rd
order
order
4th4th
order
0,040,04
Diffraction efficiency
Diffraction efficiency
0,80,8
0,60,6
0.
0,020,02
0,000,00
0 0
200200
400400
600600
1. 2. 3.
800
800
RMS
Voltage
(V)(V)
RMS
Voltage
0,40,4
0th
0thorder
order
1st
order
1st order
0,20,2
0,00,0
0 0
200
200
400
400
600
600
RMS
RMSVoltage
Voltage(V)
(V)
800
800
Diffraction efficiency:
N (U)=IN(U)/(T(U)Ivp)
FORMATION OF HOLOGRAPHIC PDLC GRATINGS
LC+polymer precursor
laser beam 1
Polymer rich regions
LC rich regions
Inhomogeneous
phase
separation
process
laser beam 2
z
1mm
Spatially inhomogeneous polymerization-induced phase separation process.
12 kV x5,000
H-PDLCs are electrically switchable
holographic media with contrast of
refractive index (1/2)1.3.
Examples of H-PDLC transmission
gratings with different values of the
grating pitch (SEM images).
L = 0.8 mm
L = 1.6 mm
L = 3.1 mm
SWITCHING PROPERTIES of H-PDLC GRATINGS
+
Bragg reflection
-
U
s
nlc  npol
p
nlc~ npol
Diffraction efficiency
efficiency
Diffraction
Uklonski izkoristek
0,5
p-polar.
s-polar.
0,4
0,3
0,2
0,1
0,0
0
5
10
15
20
E (V/mm)
Time (s)
Switching field Eth~10 V/mm, switching times  are in the ms region.
REFLECTION AND TRANSMISSION GRATINGS
n=n0+n1(r); n1  0.1
Reflection grating
Transmission grating
• switchable 1D photonic band-gap reflectors.
• selectively reflect light of specific wavelengths
• applications in reflective display devices, color filters...
• switchable deflectors for monocromatic light
• voltage controled grating structures for
spectral analysis
• applications in beam steering units, optical
interconnects...
Optical amplification in periodic
LC/polymer composite structures

band gap
k
N(E2)
N(E1)
• Near band edges cg=d/dk 0, therefore effective refractive
index c0/cg= ng , which results in high reflectivity R at
surfaces.
• Consequently an optical beam makes many roundtrips in the
medium before it escapes out from the structure!
• This effect is the basis of Distributed Feed-Back (DFB) laser
systems.
=(E2-E1)/h
z
G= G0/(1+ P/Ps)
Gain coefficiet
Optical dP=2GPL-LP
power
Gain Loss (L(1-R))
• If in such a structure there exist an atomic
transition with inverted population of
energy levels (N(E2)/N(E1))>1, optical
amplification via stimulated emission will
occur.
• Due to large reflectivity R at sample
surfaces optical gain per round-trip can
become stronger than optical losses per
round trip, therefore lasing takes place.
LASING FROM dye-doped H-PDLC GRATINGS
To achieve stimulated emission, specific dyes are added to the LC/polymer mixture.
The dye molecules are brought to excited state via usual absorption of a VIS or two
photon absorption of an IR pump beam. Lasing builds up out from the fluorescence
spectrum. noradiative decay
absorption
lasing
The output wavelenght  of such a laser
depends on the type of the dye and the
grating period L of the H-PDLC.
Y. J. Liu et al., Appl. Phys. Lett. 88, 061107 (2006)
SWITCHABLE LASING
V.K. S. Hsiao et al.,
Optics Express 13, 3787 (2005)
Laser emission power can be modulated by voltage applied to the H-PDLC grating.
Voltage reduces the width of the band-gap and the effective refractive index ng.
Switchable photonic crystals and
quasicrystals from LC/P composites
• By appropriate orientaion of 4 or more curing laser beams 3D periodic
interference patterns can be formed within the H-PDLC structures.
• In photopolymerization induced phase separation process LC material
concentrates in the intensity minima.
• The specific feature of these PCs is that their optical properties can be
modulated by low voltage external fields.
• The main problem is relatively low contrast of the refractive index
• The feld-induced
changes are affected by
anisotropy of the droplet
shapes.
example of a 2D
intereference
pattern
FCC H-PDLC PHOTONIC CRYSTAL
Optical interference of 4 above described
coherent laser beams produces intensity
profile with FCC lattice in the 3D space.
p
External voltage modifies the widths as well
as the positions (with respect to k and ) of
the band-gaps.
M. Escuti et al., Opt. Lett. 28, 522 (2003).
PHOTONIC QUASICRYSTALS
Five beam interference pattern
E j  Aj e j e

i ( k jx x  k jy y  k jz z  j )
j=1..5
I ( x, y )   A2j  2  A j Al (e j el ) cos (k jx  klx ) x  (k jy  kly ) y   j   l )
j

j ,l  j
calculation for
• TM polarized beams
• wavelength = 532 nm,
• angle of incidence with
respect to the xy plane  = 30o.
• Ai = A, i = 0, i = 1..5
H-PDLC PHOTONIC QUASICRYSTAL
Expected refractive index profile n(x,y) for synchronized phases
Ai = A, i = 0, i = 1..5
200
100
0
-100
Calculated Fraunhoffer
diffraction pattern.
-200
-200
-100
0
square=10x10 mm2
100
200
H-PDLC PHOTONIC QUASICRYSTAL
Expected refractive index profile: n(x,y)
Ai=A, i=1..5, 1 =0.55, 2 =5.99, 3 =1.89, 4 =1.90, 5 =1.82
asynchronized phases
(usual case!!)
Fraunhoffer diffraction
square=10x10 mm2
EXAMPLE OF QUASICRYSTAL STRUCTURE
SEM-image of broken and
etched H-PDLC morphology.
The structure is very irregular.
Volume fraction of the phase separated LC seems
to be very low (small isolated droplets)
Images of diffraction pattern observed on a
far field screen: a) E=0, b) E=100 V/mm.
a)
b)
IRREGULARITY OF DIFFRACTION EFFICIENCY
diffraction efficiency: n,N= In,N/Iin
nomenclature
-4
3rd order
2nd order
Diffraction efficiency (10 )
30
1st order
4th order
Sample region
3
1
4
2
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9
10
Relative diffraction efficiency
Diffraction peak
1.3
2.peak
5.peak
6.peak
8.peak
10.peak
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0
100
200
300
400
1st diffraction order
Irregularities are correlated over the entire sample
region.

They result not from local sample imperfections,
but from the imperfect holographic writing process.
RMS Voltage [V]
Electric field induced effect are very dispersed.
Various peaks within the 1st
diffraction order
Photonic bandgap fibers from LC composites
If usual air core in the PC fibers is filled with the LC, fiber properties can be tuned
by temperature or external electric and magnetic fields. This property is important
for waveguide modulators, couplers, filters, sensors and similar elements...
T. T. Larsen at el., Optics Express 11, 2589 (2003)
PC fiber filled with cholesteric LC.
Transmitted light at different temperatures.
CONCLUSIONS
Photosensitive polymer/LC composites provide a simple way to make
switchable PC structures
All the structures reported during the last years are typically very imperfect.
Extended work is needed in order to optimize appropriate material
compositions, curing parameters and switching properties.
What about other (besides the electrooptic effect) nonlinear optical
properties of such structures?
These are for the moment completely unexplored and will very probably be the
topic of many near future PhD theses....