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Optica Applicata, Vol. XLI, No. 3, 2011
Electro-optic modulation property of slow light
in coupled photonic crystal resonator arrays
D
AQUAN
YANG
*
, X
UEYING
WANG, H
UIPING
TIAN, Y
UEFENG
JI
Key Laboratory of Information Photonics and Optical Communications, Ministry of Education,
School of Information and Communication Engineering,
Beijing University of Posts & Telecommunications, No. 10, Xitucheng Road,
Haidian District, Beijing, 100876, P.R. China
*
Corresponding author: yangdq5896@163.com
A novel-designed compact, ultrafast electro-optic slow light modulator based on two-dimensional
coupled photonic crystal resonator arrays (2D CPCRAs) has been studied. The 2D CPCRAs,
realized in silicon-on-insulator (SOI) slabs with nonlinear polymer as filling material, exhibit a
single guided mode with low group velocity in all crystal directions. We realize fast electro-optic
tuning of the slow mode in a wide frequency range with low modulating voltage in this structure.
Numerical analysis shows that the frequency shift is nearly linearly increasing with the applied
voltage. In addition, for a certain frequency, changing the modulating voltage can tune the group
delay of slow mode easily.
Keywords: coupled photonic crystal resonator arrays, slow light, electro-optic polymer.
1. Introduction
Low group velocity is crucial in a variety of applications, ranging from optical delay
lines, all-optical buffers, and enhanced matter interaction [1, 2]. Although photonic
crystals can be employed to achieve low group velocities at their band edges, this is
limited to a very narrow range of wave vectors in one particular direction. Recently,
two-dimensional arrays of coupled photonic crystal resonators have been a study focus
which exhibit flat bands (which means reduced group velocities) over the entire
range of wave vectors in all crystal directions [3, 4]. This can decrease the sensitivity
of coupling and minimizes the distortion of an optical pulse propagating through
the structure.
However, for real application, the controllable all-optical devices are the most
critical components [5–7]. For example, all-optical buffers and optical storages must
be able to turn on to store and turn off to release optical data at a very rapid rate by
an external command. However, the seemingly simple function is very difficult to
754 D. Y
ANG
et al.
implement. Up to now, tunable photonic crystal devices are rarely developed due to
the lack of suitable materials with attainable changes in the refractive index being large
enough, as well as the lack of fabrication technique for those tunable materials.
Among the available materials for photonic applications, such as InP, GaAs,
silicon-on-insulator (SOI) and polymers, the polymers [8, 9] have attracted great inter-
est due to their low temperature fabrication, good mass production possibilities with
low processing cost, easy fictionalizations, and the possibility of tuning their optical
properties.
It is well known that the electro-optic effect has an ultra-high response speed of
the order of nanosecond. Recently, R
OUSSEY
et al. [10] demonstrated theoretically and
experimentally that the second-order nonlinear susceptibility can be drastically
enhanced in annealed proton exchange waveguide with photonic crystal structure
based on lithium niobate. This property opens up the possibilities for ultrafast tunable
PC devices with low power.
In this paper, we investigate a novel-designed two-dimensional coupled photonic
crystal resonator array (2D CPCRA) realized in two-dimensional photonic crystal slabs
of nonlinear polymer as substrate material, which can dynamically tune the slow light
properties and realize optical devices that could store and release optical pulses. Firstly,
the slow light transmission properties in two-dimensional arrays of coupled photonic
crystal resonators based on polymer substrate have been discussed. Secondly, we
theoretically analyze local field effect enhancement induced by the slow light
transmission and calculate the attainable refractive index changes in the structure
proposed. Finally, we research the slow light modulation by external voltage, including
frequency shifting, transmission control and storing time tuning.
2. Fabrication and photonic band gap calculation
The aim of this section is to describe the photonic crystal structure under study. We
consider polystyrene as a suitable polymer material because its refractive index
n= 1.59 is much larger than those of other polymers used in optical devices [11].
Figure 1 sketches the structure of slow light CPCRA. 2D CPCRAs studied in this
paper consist of a triangular lattice (period a) with hexagonal Si dielectric rods filled
polystyrene substrate, the refractive index of the polystyrene substrate is n (n=
= 1.59) [12]. By periodically modifying dielectric rods of a lattice photonic crystal
slab, in our design, every third lattice dielectric rod in both the x and z directions can
be removed, as shown in Fig. 1a. Figures 1b and 1c depict the unit cell and
directions of high-symmetry points,
Γ
, M, and K of the CPCRAs shown in Fig. 1a.
This structure can be viewed as a 2D array of single-defect photonic crystal cavities
formed by removing a single Si dielectric rod. Two electrodes are placed on each
side of the waveguide. The external electrostatic field is parallel to Z axis, allowing
the largest electro-optic coefficient (
γ
33
) in polymer to be used.
Electro-optic modulation property of slow light... 755
In order to model a fully characterized photonic crystal waveguide (PCW) with
cutoff frequencies very close to the telecommunication wavelength of 1550 nm, this
waveguide was designed using a lattice spacing of a= 443 nm with Si dielectric rods
of width d= 177.2 nm.
All theoretical results presented in this paper were obtained by the two-dimensional
finite difference time domain method and the plane wave expansion method [13].
Figure 2 shows the photonic band structure and spatial mode profile for TM-like modes
of the CPCRA. A complete band gap between 0.2546 and 0.3329 (normalized
frequency, in unit
ω
a/2πc) can be observed. In the photonic bandgap, there is a single
flat guided mode over the entire range of wave vectors and in all crystal directions.
Figure 3 presents the calculated transmission spectrum. We can observe a trans-
mission peak in the middle of the bandgap. The isolated transmission peak is actually
Fig. 1. Schematic configuration of the simulated and fabricated structure in polymer substrate (a); unit
cell (b); the first Brillouin zone (c).
Electrode
Electrode
Light in
X
Y
Z
Polymer substrate
Si dielectric rods ad
MK
Γ
a
b
c
Fig. 2. Calculated band diagram of the polymer coupled photonic crystal resonator arrays (a); calculated
spatial mode profile of the CPCRA structure for TM polarization (b).
0.5
0.4
0.3
0.2
0.1
0
Γ
MK
Γ
Guided mode
Normalized frequency [
ω
a/2
π
c]
Wave vector [ka/2
π
]
E
y
K = (0.209, 0.387), m = 75,
ω
a/2
π
c = a/
λ
= 0.282277
n
eff
= 0.24808, image ratio = 0.628254 1.79538
–1.79538
4
2
0
–2
–4
Z [a]
X [a]
420–2–4
ab
756 D. Y
ANG
et al.
a thin transmission band separating two photonic bandgaps. It can be seen that
the normalized frequency of the transmission peak corresponds to the guided mode in
the band diagram obtained by plane wave expansion method. Indeed, for the fabrication
of tunable photonic crystal, it is easier to obtain a good extinction ratio by tuning
a thin peak rather than an edge of the bandgap.
As a standard definition for group velocity is the derivation of the band diagram
as follows,
(1)
In our case, since the propagation is along the
Γ
M direction, we calculate only
the group velocity for this direction as shown in Fig. 4.
We can note the existence of an almost flat horizontal band in the band gap that
corresponds to a very low group velocity. In the vicinity of band edge, the group
Fig. 3. Simulated transmission spectrum for TM-like modes.
Normalized transmission
10
–1
10
–2
10
–3
0.24 0.26 0.28 0.30 0.32
Normalized frequency [
ω
a/2
π
c]
Fig. 4. The guided mode along the propagation direction
Γ
M.
0.34
Γ
M
Normalized frequency [
ω
a/2
π
c]
Wave vector [ka/2
π
]
0.32
0.30
0.28
0.26
0.24
v
g
∂
ω
∂k
------------ c
n
ω
dnd
ω
⁄()+
-------------------------------------------c
n
g
-----------== =
Electro-optic modulation property of slow light... 757
velocity approaches zero. We have calculated an average value of the group velocity
in the structure v
g
=0.009c (c – group velocity of light in vacuum), as shown in Fig. 5.
3. Modulator
An optical wave propagating through a nonlinear electro-optic material presents
a refractive index change Δn in proportion to the electric modulating field inside
the nonlinear material based on the Pockels effect, namely, the variation depends on
the second-order susceptibility , which is expressed as [8]:
(2)
where
γ
33
is the linear electro-optic coefficient, Δn
poly
represents the extraordinary
refractive index of polystyrene, U is the applied modulating voltage, d is the distance
between electrodes.
It is well known that nonlinear effects can be greatly enhanced in systems with
slow group velocity as a result of the compression of local density of states. Nano-
structuring enhances the second-order nonlinear susceptibility of the material
compared with the bulk material. The effective susceptibility in a slow light structured
material has previously been proved to depend on the local-field factor f [14]:
(3)
where is the second-order susceptibility in the bulk polystyrene and f is the local-
-field factor. In this case, the electro-optic coefficient becomes
γ
33
f
3
. The modified
index variation can be expressed as [10]:
(4)
Fig. 5. Group velocity of the guided mode versus frequency along the propagation direction
Γ
M.
Normalized frequency [
ω
a/2
π
c]
0.015
Group velocity V
g
[c]
0.284
0.010
0.005
00.285 0.286 0.287 0.288 0.289
χ
2〈〉
Δn
poly
1
2
-------–n
poly
3
γ
33
U
d
---------
=
χ
2〈〉
χ
PC
2〈〉
f
3
χ
bulk
2〈〉
=
χ
bulk
2〈〉
Δn
poly
1
2
-------–n
poly
3
f
3
U
d
---------
=
758 D. Y
ANG
et al.
The local field inside the photonic crystal structure can be calculated following
the same calculation as in Ref. [9]. The local field factor is calculated with the group
velocity inside the bulk polystyrene substrate , and the group velocity
in photonic crystal structure . The local field factor f in PCW can be calculated as,
(5)
Thus, Eq. (4) becomes
(6)
Considering CPCRA discussed in the above section, with the reduction of group
velocity, the local field factor increases sharply. Taking the average of the group
velocity of the guided mode, the result is 0.009c, considering
γ
33
= 80 pm/V [8],
n
poly
= 1.59, the distance between the two electrodes d=12.6a=5.58μm.
Substituting = 0.009c into Eq. (5), we obtain the local-field factor f to be
6.177. Thus variation of the refractive index versus the applied voltage is as shown
in Fig. 6.
Due to the slow light of the guided mode, the value of Δn decreases sharply. When
the modulating voltage is 10 V, Δn reaches –0.17. The result shows the significance
of the slow light in the enhancing electro-optic effect.
Considering that the position of photonic bandgap (PBG) and guided mode depend
directly on the value of the refractive index of substrate material, a significant shift of
the guided slow mode has been obtained with external voltage variation. Taking into
account f= 8.36 and U= 0 V (corresponding to the situation without an applied
voltage), 5 V and 10 V, the guided mode shifts to higher frequency, as Fig. 7 shows.
The normalized frequencies of the guided mode are 0.2878, 0.2968, 0.3063,
v
g
bulk
cn
poly
⁄=
v
g
PC
fv
g
bulk
v
g
PC
-----------------=
Δn1
2
-------–n
poly
3
γ
33
f
3
U
d
---------
=
v
g
PC
Fig. 6. Variation of the refractive index for an applied voltage obtained from Eq. (4).
0
–0.02
–0.04
–0.06
–0.08
–0.10
–0.12
–0.14
–0.16
–0.1802 4 6810
Refractive index change
Δ
n
Applied voltage U [V ]
Electro-optic modulation property of slow light... 759
respectively, corresponding to the guided mode cutoff wavelength shift by approxi-
mately 0, 46.7, and 93 nm. By tuning the voltage more exactly, a more refined mode
shift can be obtained. It is concluded that the wavelength shift is nearly linearly
increased with the applied modulating voltage increasing. Modulation sensitivity is
about 9.34 nm/V. The flexible dynamic tuning of the slow mode can meet the require-
ments for the use of optical buffer in all-optical network in principle.
Shifts of the photonic crystal band and guided mode caused by refractive index
variation can be applied to tune light transmission in the photonic crystal structures
by changing the voltage dynamically and externally with low power. The external
controlled guided mode shift can be conveniently utilized for slow mode selection and
slow light cutting off or turning on. For a multiple wavelength system, by tuning
the external voltage exactly, one can select the slow mode which should be delayed or
stored.
In order to provide a clear understanding of the modulation property of the structure
proposed, the light propagation of continuous wave (1550 nm) transmitted within this
CPCRA at both 0 V and 5 V has been respectively simulated. The simulation results
by finite-difference time-domain (FDTD) are shown in Fig. 8. For example, at 0 V,
the light with wavelength 1550 nm is on the guided mode which can propagate in
the coupled resonator optical waveguides in PCs (Fig. 8a) [15]. When applying
a voltage of 5 V, it cannot transmit along the coupled resonator optical waveguides
any more (Fig. 8b). Because the device goes into strong cutoff at this wavelength
(1550 nm), this causes the light to be reflected back out of the CPCRA at the input.
The simulation results show the effective control of the propagation of light with
the changing of applied voltage distinctly. So, for a fixed single frequency, by tuning
the voltage to change the guided mode frequency, one can control the turning on or
turning off slow mode. This is critical technology in external, dynamical and
controllable optical delay lines, all-optical buffers and storages.
Fig. 7. Modulated band diagram of photonic crystal waveguide. The three lines correspond to the different
applied voltages of 0 V (solid line), 5 V (dotted line), and 10 V (dash-dotted line).
0.36
0.34
0.32
0.30
0.28
0.26
Γ
MK
Γ
U = 10 V
Normalized frequency [
ω
a/2
π
c]
Wave vector [ka/2
π
]
0.24
U = 5 V
U = 0 V
760 D. Y
ANG
et al.
Figure 9 illustrates the group velocity of the guided mode with three different
applied voltages. It shows that the variation of modulating voltage controls the trans-
mission of the guided mode. In addition, for an identical frequency of guided mode
under transmission (1550 nm, as the dash-dotted vertical line shows), the group
velocity will decrease greatly with the increase of the applied voltages. This type of
behavior clearly provides the capability of group velocity modulation at determinate
frequency by changing the modulating voltage dynamically and externally, which
corresponds to the tuning of the storing time of light for real application.
Fig. 8. Simulations of light propagation through the CPCRA with different applied voltages. T
1
, T
2
and
T
3
represent different times throughout the light propagation. U=0V (a), U=5V (b).
U = 5 V
U = 0 V
U = 5 V U = 5 V
U = 0 V U = 0 V
1.0
–1.0
1.0
–1.0
T
1
T
2
T
3
T
1
T
2
T
3
a
b
Fig. 9. The variation of group velocity of guided mode versus normalized frequency. The three lines
correspond to the different applied voltages of 0 V (solid line), 0.3 V (dotted line), 0.5 V (dashed line).
0.015
0.010
0.005
01530 1540 1550 1560
U = 0.5 V
U = 0.3 V
U = 0 V
Group velocity V
g
[c]
Wavelength [nm]
Electro-optic modulation property of slow light... 761
Generally, the storage time T
s
of a buffer is defined as,
(7)
where L is the length of CPCRA and v
g
is the group velocity of light transmission.
Equation (7) shows that the storage time is direct ratio to the length of delay line, but
inverse ratio to the group velocity. We just make the hypothesis that the length of
CPCRA discussed in this paper is 7.97 μm, and the group delay can be determined as
Fig. 10 shows. In the vicinity of band edge, the group delay approaches 200 ps.
We calculated the group delay when different modulating voltages are applied to
the structure corresponding to the group velocity tuning in Fig. 9, as shown in Fig. 11.
For the wavelength of 1550 nm, we can see a group delay of the guided mode to
T
s
Lv
g
⁄=
Fig. 10. Calculated group delay of the guided mode versus the wavelength along the propa-gation
direction
Γ
M.
200
1535 1540 1550 1560
Group delay [ps]
Wavelength [nm]
160
120
80
40
01545 1555
Fig. 11. The variation of group delay of guided mode as applied voltage increased. The three lines
correspond to the different applied voltages of 0 V (solid line), 0.3 V (dotted line), 0.5 V (dashed line).
14
1549 1549.5 1550 1551
Group delay [ps]
Wavelength [nm]
12
10
8
6
21550.5
U = 0.5 V
U = 0.3 V
U = 0 V
4
762 D. Y
ANG
et al.
increase sharply as the modulating voltage increases. When modulating voltage is 0 V,
0.3 V and 0.5 V, the group delay is about 3.1 ps, 4.3 ps and 7.6 ps, respectively. So,
for a certain frequency of guided mode under transmission, the flexible tuning of slow
mode can not only be utilized for slow light selection, but also to manipulate the storing
time of slow light in optical delay lines, all-optical buffers or optical storages.
4. Conclusions
We have designed a novel ultrafast electro-optic slow light modulator based on 2D
coupled photonic crystal resonator arrays (CPCRAs). The structure supports flat
guided mode in all crystal directions, which refers to an ultra low group velocity of
10
–3
c. Employing the polymer as substrate material and local field effect induced by
the slow light transmission gives rise to the fast electro-optic tuning of the slow mode
in a wide frequency range with low modulating voltage in this structure. The wave-
length shift modulation sensitivity is about 9.34 nm/V. The flexible tuning of slow
mode can not only be utilized for slow light frequency selection, but also to manipulate
the storing and releasing of slow light pulses with a given frequency in optical delay
lines, all-optical buffers or optical storages. The study presented here can be extended
to 3D coupled resonator arrays, as well as to other types of resonators, including those
not based on photonic crystals.
Acknowledgements – This research was supported in part by NSFC (No. 60932004 ), National 973 Program
(No. 2011CB302702), National 863, P.R. China.
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