All-optical switching and multistability in photonic structures
with liquid crystal defects
Andrey E. Miroshnichenko,
1,a兲
Etienne Brasselet,
2
and Yuri S. Kivshar
1
1
Nonlinear Physics Center and Center for Ultra-high bandwidth Devices for Optical Systems (CUDOS),
Research School of Physical Sciences and Engineering, Australian National University,
Canberra Australian Capital Territory 0200, Australia
2
Centre de Physique Optique Moléculaire et Hertzienne, Université Bordeaux 1, CNRS,
351 Cours de la Libération, 33405 Talence Cedex, France
共Received 30 April 2008; accepted 1 June 2008; published online 26 June 2008兲
We demonstrate that one-dimensional photonic crystals with pure nematic liquid-crystal defects can
operate as all-optical switching devices based on optical orientational nonlinearities of liquid
crystals. We show that such a periodic structure is responsible for a modulated threshold of the
optical Fréedericksz transition in the spectral domain, and this leads to all-optical switching and
light-induced multistability. This effect has no quasistatics electric field analog, and it results from
nonlinear coupling between light and a defect mode. © 2008 American Institute of Physics.
关DOI: 10.1063/1.2949076兴
The concept of photonic crystals
1
proposed two decades
ago
2,3
brought a new paradigm to achieve light propagation
control in dielectric media. In such periodic photonic struc-
tures tuning may be achieved by using materials which are
sensitive to external fields, including temperature, electric
field, or light itself. In this context, nonlinear photonic crys-
tals have retained much attention due to possible enhance-
ment of nonlinear effects.
4
Among various nonlinear optical
materials that can be implemented in actual photonic crystal
devices, liquid crystals 共LCs兲 have been recognized as an
attractive alternative material
5
due to their unique sensitivity
to external fields. Since then, many tunable photonic crystal
devices based on LC tunability have been suggested and
implemented either using complete or partial LC infiltration
into the periodic dielectric structure. In the first case, the
photonic bandgap is tuned due to refractive index changes of
the global structure,
6
while in the second case a LC-
infiltrated layer or hole generates defect modes whose fre-
quencies are controlled by local refractive index changes of
LC.
7
The case of complete infiltration is the most studied
one, and it was the first to be demonstrated; it concerns
thermal
6
and electrical
8
tunability infiltrated one-, two- and
three-dimensional photonic structures with LCs.
There exist much less studies concerning optical tuning.
One can mention the demonstration using photonic LC
fibers,
9,10
one-dimensional
11
or planar
12,13
photonic crystals
using absorbing or dye-doped LCs. In these works the reso-
nant interaction of light induces a change of the order param-
eter, phase transition or surface-mediated bulk realignment.
In contrast the nonresonant case, where well-known orienta-
tional optical nonlinearity of LC takes place,
14
has only been
explored recently.
15
In this Letter we address a general problem of all-optical
switching in periodic structures with nematic LC 共NLCs兲,
based on the optical Fréedericksz transition 共OFT兲. First, we
demonstrate a nonlinear feedback due to coupling between
light and a defect mode via optical orientational nonlineari-
ties. This feedback could be positive, resulting in all-optical
multistable switching, or negative, depending on the wave-
length detuning from a defect mode. Second, we demonstrate
that this coupling leads to different types 共first- or second-
order兲 OFT in periodic structures for the same NLC material.
While all-optical bistable switches based on photonic crys-
tals are mainly restricted to inorganic devices,
16
our results
could be envisaged as a promising opportunity for LC-
infiltrated photonic crystals technology.
The structure studied in this letter is made of ten bilayers
of SiO
2
and TiO
2
on each side of a NLC layer having a
homeotropic alignment 共molecules are perpendicular to sub-
strates兲, as shown in Fig. 1共a兲. We choose the commercially
available E7 LC which is nematic at room temperature and
exhibits a second-order OFT for a single slab under linearly
polarized excitation. The thicknesses of NLC, TiO
2
, and
SiO
2
layers are 5
m, 134 and 143 nm, respectively. We
used the dispersion law n
⬜
=1.4998+0.0067/
2
+0.0004/
4
and n
储
=1.6993+ 0.0085/
2
+0.0027/
4
共Ref. 17兲 for NLC,
n
SiO
2
⯝1.4322+ 0.0224/ 共 + 0.28兲+ 0.4005/ 共 + 0.28兲
2
and
n
TiO
2
⯝2.3147− 0.2577/ 共 + 0.11兲+ 202.4/ 共 + 0.11兲
2
, respec-
tively, where wavelength has to be taken in microns. The
resulting structure has a bandgap centered around 530 nm
that supports four defect modes, as shown in Fig. 1共b兲 where
the transmission spectra of the unreoriented structure is
presented.
The optical properties of the proposed structure are con-
sidered to be within the plane wave approximation, so that
the system depends only on coordinate z 关Fig. 1共a兲兴 and time
t. The problem is conveniently solved by using the Berreman
4⫻4 matrix approach
18
where Maxwell’s equations can be
expressed as
⌿ /
z=ik
0
D⌿, where k
0
=2
/ is the wave
vector in vacuum, D is the Berreman matrix that depends on
dielectric permittivity tensor
⑀
ij
, and ⌿ = 共E
x
,H
y
,E
y
,−H
x
兲
T
.
18
The E
z
component of the light field is obtained from the
constitutive equation
i
⑀
ij
E
j
=0. The whole structure is di-
vided into many layers with constant permittivity, which are
described by constant matrices D
n
. While the thickness of the
dielectric layers corresponds to thickness of actual materials,
the LC layer is discretized into as many sublayers as neces-
sary to ensure prescribed small variations from one sublayer
to another. The resulting matrix thus writes D = 兿D
n
. We ob-
a兲
Author to whom correspondence should be addressed. Electronic mail:
aem124@rsphysse.anu.edu.au.
APPLIED PHYSICS LETTERS 92, 253306 共2008兲
0003-6951/2008/92共25兲/253306/3/$23.00 © 2008 American Institute of Physics92, 253306-1
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
tain a light propagation problem for incident 共i兲, transmitted
共t兲, and reflected 共r兲 amplitudes that writes ⌿
t
=D共⌿
i
+⌿
r
兲.
Inside the NLC layer, the light field is coupled to the Euler–
Lagrange equations that govern the dynamics of the director
n共z,t兲 and account for dissipative, elastic, and electromag-
netic contributions.
14
We further consider the propagation of light linearly po-
larized along x-axis and assume the director to be con-
strained in the 共x,z兲 plane, which can be represented by the
reorientation angle ⌰共z ,t兲关Fig. 1共a兲兴. The problem is solved
numerically via the modal expansion procedure detailed in
Ref. 19 by searching the reorientation profile as ⌰共z,t兲
=兺
n=1
N
⌰
n
共t兲sin共
z/ L兲, where N = 10 is enough to ensure ac-
curate results. The needed Frank elastic constants for the
NLC are K
1
=11.1⫻ 10
−12
N and K
3
=16.0⫻ 10
−12
N. We in-
troduce the normalized intensity
=兩E
x
i
兩
2
/ I
lin
0
, where I
lin
0
is the
OFT threshold for a single NLC slab and linearly polarized
plane wave that corresponds to ⯝595 kW/ cm
2
, and time
=t/
NLC
, where
NLC
is the typical relaxation time,
14
NLC
⯝25 ms in our case. Note that the OFT threshold value
could be reduced by up to two order of magnitude using a
small addition of dye into the NLC.
20
We define the total
light-induced phase delay in the presence of reorientation,
⌬共
兲=k
0
兰
0
L
关n
e
共z ,
兲−n
o
兴dz, with n
o/e
being the ordinary and
effective extraordinary refractive indexes.
First, we calculate the OFT threshold
th
above which
the NLC is reoriented for pump wavelength
p
lying inside
the bandgap 关see Fig. 1共b兲兴. As was shown in Ref. 15, the
reorientation threshold is drastically reduced at a defect
wavelength with respect to the single NLC slab
th
共
d
兲Ⰶ1,
caused by a very strong light confinement at the defect mode
placed in a periodic structure. A reduction factor up to 10
3
is
obtained for the present structure. In contrast, a significant
increase of the threshold is observed when
p
is strongly
detuned from defect mode frequencies 关Fig. 1共b兲兴. Indeed,
the smaller the detuning parameter
␦
=
p
−
d
is, the better
light confinement inside the defect NLC layer, which leads to
the lower reorientation threshold.
Although the OFT threshold is independent of the sign
of
␦
for
p
⬇
d
共local parabolic approximation兲, the type of
the light-induced reorientation strongly depends on it. This
point is illustrated in Fig. 2共a兲 where the reorientation dia-
gram ⌬ as a function of
is shown for
␦
= ⫾0.25 nm around
d
=532.75 nm. A negative detuning leads to a second-order
OFT, as it is the case for the NLC slab alone, whereas a
positive detuning exhibits a first-order OFT with an relative
hysteresis width of almost 100%. To understand this behav-
ior we note that defect mode frequencies underwent a red-
shift proportional to the increase of the averaged refractive
index inside the NLC defect layer due to molecular reorien-
tation. Consequently, recalling that the threshold intensity
has a parabolic profile around defect modes
th
共
p
兲−
th
共
d
兲
⬀共
p
−
d
兲
2
, negative detuning
␦
⬍0 leads to a negative feed-
back and positive detuning
␦
⬎0 is accompanied by a posi-
500 510 520 530 540 550 560
0
50
100
150
200
250
(nm)
0
1
Transmission
λ
ρ
th
z
x
L
TiO
2
SiO
2
Θ
z
x
Laser
(
a)
(b)
NLC layer
FIG. 1. 共Color online兲共a兲 Sketch of the multilayers periodic structure with an embedded NLC defect. 共b兲 Transmission spectra of the unperturbed system 共red兲
and normalized OFT threshold 共blue兲.
FIG. 2. 共Color online兲共a兲 Reorientation diagram for positive 共blue兲 and negative 共black兲 detuning
␦
= ⫾0.25 nm around
d
=532.75 nm. Solid 共dashed兲 lines
refers to stable 共unstable兲 states. Corresponding dynamics for 共b兲
␦
⬎0 and 共c兲
␦
⬍0 of the phase delay ⌬ and pump light transmission for square-shaped
temporal excitation with
=1.
253306-2 Miroshnichenko, Brasselet, and Kivshar Appl. Phys. Lett. 92, 253306 共2008兲
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
tive feedback. All-optical bistable switching is thus achieved
by properly chosen pumping wavelength
p
⬎
d
.
The dynamics of the all-optical switching process is il-
lustrated in Fig. 2共b兲, where the unperturbed system is ex-
cited with
␦
⬎0 and
⬎
th
共
p
兲 for
⬍10 and
=0 for
⬎10. Two transmission peaks are observed, the first one at
excitation stage and the second one during relaxation pro-
cess. These peaks correspond to a transient resonance be-
tween
d
and
p
due to the displacement of the defect modes
inside the bandgap caused by reorientation. The full width at
half maximum pulse duration of the first and second peaks
are ⯝4 ⫻ 10
−4
NLC
and ⯝8 ⫻ 10
−2
NLC
, namely, 10
s and
2 ms in the present case. Initial conditions at
=0 are taken
as 共⌰
1
=10
−2
,⌰
2,...,N
=0兲, which mimics a thermal orienta-
tional fluctuation in the NLC slab. As a result, the smaller the
detuning parameter is, the smaller is the required intensity to
perform all-optical switching in periodic structures. A com-
promise has nevertheless to be found to preserve light-
induced phase delay discontinuity since it vanishes for
␦
→ 0
+
.
Multistable all-optical switching can also be achieved in
such a structure for larger intensities, caused by larger mo-
lecular reorientation resulting in larger changes of the refrac-
tive index. This leads to a possibility of many defect modes
passing the pumping wavelength
p
, and, consequently, to a
series of reduction of the pumping intensity
giving reori-
entation boosts. An example is shown in Fig. 3共a兲 for
p
=537 nm. The multistability is evidenced by several coexist-
ing stable states in the reorientation diagram. The corre-
sponding switching dynamics is shown in Fig. 3共b兲 when a
linear ramp of excitation light intensity is used. Each of six
peaks of the transmission dynamics is reminiscent of a pla-
teau in the reorientation diagram, as indicated by labels num-
bered 共1兲–共6兲, and has the same origin as those discussed
above.
In conclusions, we have predicted multistable all-optical
switching resulting from an unique nonlinear coupling be-
tween light and defect modes in a periodic dielectric struc-
ture that contain a NLC defect layer. Moreover, depending
on the pumping wavelength
p
the same NLC material may
demonstrate first-or second-order OFT in periodic structures.
It is important to note that there is no quasistatics electric
field analog to the optical case studied here because the
electro-optical tunability explored earlier
21
is not accompa-
nied by a coupling between excitation field and defect
modes. Additional functionalities are expected when nonpla-
nar molecular light-induced reordering takes place.
19
This work was supported by Discovery and Center of
Excellence projects of the Australian Research Council.
1
See, e.g., J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic
Crystals: Molding the Flow of Light 共Princeton University Press,
Princeton, NY, 1995兲, and references therein.
2
E. Yablonovitch, Phys. Rev. Lett. 58, 2059 共1987兲.
3
S. John, Phys. Rev. Lett. 58, 2486 共1987兲.
4
M. Soljačić and J. D. Joannopoulos, Nat. Mater. 3,211共2004兲.
5
K. Busch and S. John, Phys. Rev. Lett. 83, 967 共1999兲.
6
K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, and M. Ozaki,
Appl. Phys. Lett. 75, 932 共1999兲.
7
Y.-K. Ha, Y.-C. Yang, J.-E. Kim, H. Y. Parka, C.-S. Kee, H. Lim, and J.-C.
Lee, Appl. Phys. Lett. 79,15共2001兲.
8
D. Kang, J. E. Maclennan, N. A. Clark, A. A. Zakhidov, and R. H.
Baughman, Phys. Rev. Lett. 86, 4052 共2001兲.
9
T. T. Alkeskojld, L. A. Bjarklev, D. S. Hermann, A. Anawati, J. Broeng, J.
Li, and S. T. Wu, Opt. Express 12,5857共2004兲.
10
J. Tuominen, H. J. Hoffrén, and H. Ludvigsen, J. Eur. Opt. Soc. Rapid
Publ. 2, 07016 共2007兲.
11
H. Yoshida, C. H. Lee, Y. Miura, A. Fujii, and M. Ozaki, Appl. Phys. Lett.
90, 071107 共2007兲.
12
B. Maune, J. Witzens, T. Baehr-Jones, M. Kolodrubetz, H. Atwater, A.
Scherer, R. Hagen, and Y. Qiu, Opt. Express 13, 4699 共2005兲.
13
P. El-Kallassi, R. Ferrini, L. Zuppiroli, N. Le Thomas, R. Houdré, A.
Berrier, S. Anand, and A. Talneau, J. Opt. Soc. Am. B 24,2165共2007兲.
14
N. V. Tabiryan, A. V. Sukhov, and B. Ya. Zel’dovich, Mol. Cryst. Liq.
Cryst. 136,1共1986兲.
15
A. E. Miroshnichenko, I. Pinkevych, and Y. S. Kivshar, Opt. Express 14,
2839 共2006兲.
16
M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe,
Opt. Express 13, 2678 共2005兲.
17
J. Li, S.-T. Wu, S. Brugioni, R. Meucci and S. Faetti, J. Appl. Phys. 97,
073501 共2005兲.
18
D. W. Berreman, J. Opt. Soc. Am. 62, 502 共1972兲.
19
E. Brasselet, T. V. Galstian, L. J. Dubé, D. O. Krimer, and L. Kramer, J.
Opt. Soc. Am. B 22, 1671 共2005兲.
20
L. Marrucci, D. Paparo, P. Maddalena, E. Massera, E. Prudnikova, and E.
Santamato, J. Chem. Phys. 107, 9783 共1997兲.
21
R. Ozaki, H. Moritake, K. Yoshino, and M. Ozaki, J. Appl. Phys. 101,
033503 共2007兲.
0 100 200
0
0.5
1
1.
5
ρ
∆ /2π
0 50 100 150
0
100
200
300
τ
ρ
0
1
Tr
a
n
s
mi
ss
i
o
n
150 151
0
1
(1)
(2)
(3)
(1)
(2)
(3)
(4)
(5)
(6)
(4)
(5)
(6)
(a)
(b)
FIG. 3. 共Color online兲共a兲 Multistable reorientation diagram for positive
detuning at
p
=537 nm. Solid 共dashed兲 lines refers to stable 共unstable兲
states. 共b兲 Corresponding dynamics of pump light transmission for triangle-
shaped temporal excitation.
253306-3 Miroshnichenko, Brasselet, and Kivshar Appl. Phys. Lett. 92, 253306 共2008兲
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
