The SOI slotted 2D PhC structure under consideration is started

from the 260 nm thick silicon film of the SOI wafer, and then made of

a 2D triangular lattice PhC structure of holes with lattice constant a =

380 nm. The holes are inscribed through the silicon film with radius

of r = 0.30a = 114 nm. The slotted waveguide is formed by removing

one center row of holes along - K direction and inscribing with the

rectangular slot of width d through the silicon film. The SOH slotted

PhC waveguide is obtained by first producing the SOI slotted PhC

waveguide and then filling and covering it with DDMEBT of

refractive index of 1.8 as shown in Fig. 5.14a. Figs. 5.14b, c show the

2D PhC structures of coupled five identical slotted cavities together

through the waveguide and two identical side-coupled slotted cavities

and slotted waveguide, respectively.

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ng the structural parameters of
1D, 2D PhC structures for optical bistable devices which have high
quality factor, low optical intensity and time for switching.
+ Propose and design some integrated photonic structures which
have high performance and special characteristic for bistable devices.
3. The main research contents of the thesis
+ Design and analysis the optical properties of 1D and 2D PhC
structures.
+ Optimization of the structural parameters and resonant spectra of
the grating structures to increase the quality factor and reduce the
optical intensity for switching.
+ Investigation of the bistability characteristics of the optimal
grating structures.
+ Design and simulation the narrow high-order resonance
linewidth shrinking with multiple coupled resonators in SOH slotted
2D PhCs for reduced optical switching power in bistable devices.
Differences and new ones in the research content of the thesis:
+ Currently in Vietnam, there are very few subjects and thesis
mention the PhC structures for application in optical comunication due
to the lack of fabrication equipment. This dissertation is considered as
the first in computation and simulation of optical bistable devices
using 1D and 2D PhC structures in Vietnam.
+ This dissertation uses the modern and highly accurate calculation
and simulation methods to verify the achieved theoretical results, so
the dissertation contributes to increase the professional research.
This dissertation includes five chapters:
Chapter 1. Overview
Chapter 2. Calculation and simulation methods
Chapter 3. Optimization of quality factor and resonant spectra of
grating structures
Chapter 4. Optical bistability in slab waveguide gratings
Chapter 5. Optical bistability based on interaction between sloted
cavities and waveguides in two-dimensional photonic crystals
CHAPTER 1: OVERVIEW
1.1. Photonic crystal structures
The first concept of PhCs was proposed by Yablonovitch and John
in 1987 [7]. PhCs are the periodic structures of the dielectric elements
in space. Due to the periodic of the refractive indices, the PhC
structures produce the PBGs. Depending on the geometry of the
structure, PhCs can be divided into three categories, namely one-
dimensional (1D), two-dimensional (2D) and three-dimensional (3D)
structures. The examples are shown in Figure 1.1.
Figure 1.1. 1D, 2D, and 3D PhC structures (a) 1D PhC, (b) 2D PhC, (c) 3D PhC [27].
1.2. Optical bistable devices
Two features are required for presenting the bistable behavior:
nonlinearity and feedback. Both features are available in nonlinear
optics. An optical system is shown in Figure 1.33, this system exhibits
the bistable behavior:
For small inputs (Ivào < 1 ) or large
inputs (Ivào > 2 ), each input value
has a single response (output). In
the intermediate range, 1 < Ivào <
2 , each input value corresponds
to two stable output values.
CHAPTER 2. CALCULATION AND SIMULATION METHODS
2.1. Coupled mode theory (CMT)
Using a simple LC circuit, I have given the dependence of the
voltage amplitude on time. This is the method used to calculate the
transmission and reflection spectra of the structures.
2.2. Plane wave expansion method (PWE)
In order to exploit the extraodinary properties of PhCs, the
calculation method is required to accurately determine the PBG. One
of the most common methods is the plane wave expansion (PWE).
This method allows for solving wave vector equations for
electromagnetic fields, calculating the eigen frequency of the PhCs. In
addition, it is also used to calculate energy diagram as well as PBG.
2.3. Finite-difference time-domain method (FDTD)
The FDTD method is one of the time domain simulation methods
based on the mesh generation. Maxwell's equations in differential
form are discrete by using the approximation method for differential
of the time and space. The finite differential equations will be solved
by software according to the leapfrog algorithm. This method aims to
provide the mathematic facilities for calculating and simulating the
Figure 1.33. Ouput versus input of the bistable device.
The dashed line represents an unstable state [85].
device characteristics using PhC structures such as: transmission
spectra, energy diagrams, and the characteristics of stability.
CHAPTER 3. OPTIMIZATION OF QUALITY FACTOR AND
RESONANT SPECTRA OF GRATING STRUCTURES
3.3. Optimization of structural parameters and resonant spectra
In this chapter, I will introduce some methods to optimizing the Q-
factors and resonant spectra of grating structures.
3.3.1. Slab waveguide grating structure combining with metallic film
Based on the study of waveguide grating structure, so that in order
to increase the Q-factor, the grating depth must be reduced, but due to
the limitation of manufacturing technology, the grating depth is not
too thin of less than 10 nm. Therefore, I have optimized the grating
structure by adding a silver (Ag) layer of thickness d (> 50 nm)
between the slab waveguide grating and the glass substrate. This thin
layer supports a strongly asymmetric resonant profile in the nonlinear
slab waveguide grating and reflects the light waves in any direction
due to its high reflectivity. These reflected waves will then be coupled
into guided-mode resonances in the grating [23].
Figure 3.14. (a) Metallic assisted guided-mode resonance structure with normally incident
light. (b) Transmission and reflection spectra for several Ag layer thicknessed d.
This results obtained with metallic assisted guided-mode resonance
(MaGMR) structure provide the enhancement Q-factor coefficients
greater than 1, therefor this structure has a higher Q-factor than grating
waveguide structure. Combining with metallic film, the Q-factor has
been enhanced.
3.3.2. Coupled grating waveguide structures
The second optimal method,
coupling two slab waveguide
gratings to obtain a higher Q-
factor and change the shape of
the resonant spectrum. Here, the
Q-factor is controlled based on
the distance between the two
slab waveguide gratings. The
schematic of two coupled
identical slab waveguide
gratings facing each other with a
gap-distance of d and horizontal shifted-alignment of s is shown in
Fig. 3.18. Each slab waveguide grating supports the Fano resoance,
where key structural parameters are defined as the guiding layer made
of chalcogenide glass (As2S3, n = 2.38) with a thickness (t) of 220 nm
on a thick glass substrate (n=1.5). The grating slit aperture (w) is
formed by a rectangular corrugation in As2S3 guided layer with the
depth and periodicity of 220 nm and 860 nm, respectively. A normally
incident plane wave with transvere electric (TE) polarization is ussed.
Figure 3.18. Sketch of coupled slab waveguide
gratings. The gap-distance d and horizontal shifted-
alignments s are tuned for exciting Fano resonances.
Figure 3.19 shows the
reflection spectra for various the
gap-distances d. With this gap-
distance 50 nm ≤d ≤ 300 nm, the
resonant wavelengths shifts
towards the short wavelenth.
The Q-factor increases as the gap-
distance d increases due to the
long distance of Fabry-Perrot
resonantor formed between two
slab waveguide gratings.
3.3.3. Multilayer dielectric grating structure
Figure 3.21. Multilayer nonlinear dielectric grating structure. The structure consists of N-pair
of bilayer As2S3/SiO2 gratings.
The structure consits of identically layers of As2S3 and SiO2 with
thickness of t = N*(dH + dL), where N are the repetitive identical
bilayers of As2S3 and SiO2, and dH và dL are the thickness of As2S3 and
SiO2 layers, respectively. In our design, the optical thicknesses of
As2S3 and SiO2 layers are chosen to satisfy the quarter-wavelength
condition, that mean nH*dH = nL*dL = λ/4, where nH and nL are the
refractive indices of As2S3 and SiO2, respectively. In calculations, the
center wavelength λcenter = 1550 nm, dH = 162,8 nm và dL = 267,2 nm
are used. Figure 3.22 shows the transmission spectra with N = 3 pairs
Figure 3.19. Reflection spectra of the coupled slab
waveguide gratings depicted in Fig. 3.18.
of As2S3/SiO2 layers for several grating widths w from 30 nm to 150
nm. There exits two Fano resonances within the interested wavelength
regims, which are associated with the guided-mode resonances in the
long and short resonant spectra from 1460 nm to 1610 nm and from
1340 nm to 1480 nm. As it is shown, the increase of grating width w
makes the resonance shifts to the short wavelength and the Q-factor
decreases. In addition, the spectral resonances show that the side band
degrees of Fano lineshapes do not change, it even shows that the
linewidths and peaks of resonances change when the grating widths
change.
Figure 3.22. Transmission spectra of this structure depicted in Fig. 3.18 with N = 3.
We investigated and found that the Fano lineshapes were
reproducible and readily controlled via the number of layers N and the
grating width w, demonstrating the robustness of the suggested
structure. With the given grating width w of 70 nm, the resonant peaks
and Q-factors of the long and short resonances for several number of
layers N were evaluated using Fano lineshapes and plotted in Figure
3.23. When the number of layers N increase, redshifts in resonance,
higher Q-factor, and lower sidebands are obtained.
Figure 3.23. Resonant peaks and Q-factors of the structure as depicted in Fig. 3.21 for several
number of layers N.
CHAPTER 4. OPTICAL BISTABILITY IN SLAB
WAVEGUIDE GRATINGS
After optimizing the Q-factor and resonance spectra of the slab
waveguide grating structure as presented in Chapter 3, in this chapter
I will examine the bistability characteristics of optimal structures.
4.1. Optical bistability in slab waveguide grating structure
combined with metallic film
The third-order nonlinear coefficient at a working wavelength of
As2S3 is n2 = 3,12x10-18 m2/W (χ
(3) = 1,34x10-10). In order to see the
optical bistability in MaGMR, we excite the devives with an incident
CW source having a suitable working wavelength (frequency) on the
surface of the structure. In general, the relation between the working
frequency and the resonant frequency requires that [66]:
0 3 (4.1)
where, τ is a photon life time, to observe bistability. For our case of an
inverse Lorentzian shape, we choose a working wavelength at 80%
reflection, which corresponds to a frequency detuning of (ω0 - ω)τ=2
for the Lorentzian shape.
In this work, we keep the slab and Ag thickness at 380 nm and 100
nm, respectively. The grating depth δ (< 120 nm) is found close to an
optimal value. Table 4.1 shows the trends for the resonant wavelength,
the quality factor Q, and the Q-factor enhancement when the grating
depth δ changes. As the grating depth increases, the resonant
wavelength of MaGMR shifts to shorter wavelengths. It seems that the
deeper the grating depth, the more leaky the waveguide mode. The Q-
factor enhancement increases as the grating depth increases. For
example, a Q-factor enhancement of 5.56 occurs for a grating depth δ
of 120 nm.
Table 4.1. Linear and nonlinear characteristics of MaGMR gratings with a Ag thickness d =
100 nm for several grating depths.
Grating depth, δ (nm) 30 50 80 100 120
Resonant wavelength (nm) 1574,75 1560,61 1524,51 1516,81 1494,55
Q-factor 676,1 506,5 353,9 316,7 293,3
Q-factor enhancement 0,71 1,55 2,97 4,12 5,56
Reduced switching intensity 0,42 2,57 10,7 24,5 45,0
4.2. Optical bistability in coupled grating waveguide structures
Figure 4.5 shows the
calculated bistable behaviors of
the perfect alignment coupled
slab wavelength gratings for the
gap-distance d of 50 nm, 100 nm,
170 nm, and 300 nm. Bistable
behaviors are clearly observed.
In each bistable curve, the
incident intensity for
switching can be estimated as
the input intensity for which the reflection increases abruptly in the
dotted solid curve. The estimated switching intensities are 1427,1
MW/cm2; 104,1 MW/cm2; 16,2 MW/cm2; và 2,2 MW/cm2;
Figure 4.5. Bistability curves of the coupled
gratings for various gap-distances d of 50 nm,
100 nm, 170 nm, and 300 nm, respectively.
corresponding to the quality factors: Q = 2104, 2543, 3759, và 8522;
and asym metric factor q = 1,609; 1,110; 0,835; và 0,655. In contrast
to the Lorentzian resonance, these Fano-based results do not follow
the 1/Q2 dependence rule of the switching intensity. While the Q-
factors increase gradually, the switching intensities dramatically
decrease due to a reduction of asymmetric factor q. The Q-factor
increases 4.0 times but the switching intensity decreases 648.7 times.
4.3. Optical bistability in multilayer dielectric grating structure
Figure 4.9 shows the dependence of transmission (ratio between
the transmitted and incident intensities) on the incident intensity of the
optical switching/bistability for the long (Fig. 4.9a,b) and short (Fig.
4.9c,d) resonances. For the long resonance, the operating wavelengths
are chosen at resonant dip and 10% of transmission as shown in the
insets of Fig. 4.9a,b. Figure 4.9a,b with the operating wavelength at
10% of transmission, blue (arrows pointing up/right) and red curves
(arrows pointing left/down), it shows that the bistability behaviors and
the switching intensities are 0.50 MW∕cm2 and 1178.56 MW/cm2 for
grating widths w =30 nm and 150 nm, respectively. Whereas with the
operating wavelengths at the resonant dips, the bistability behaviors
have not occurred (black curves, on left) even the switching points at
0.04 MW/cm2 and 50.35 MW/cm2 of input intensities and high
contrasts are observed for grating widths w = 30 nm and 150 nm,
respectively. When the operating wavelength moves away from the
resonant dip, the switching intensity is higher and the bistability region
is broader. This is attributed to the wavelength detuning,
which implies a broader detuning bandwidth and, thus, a higher
resonance shift amount is required to change the state. For the short
resonance, the operating wavelength is chosen at 1/e transmission as
shown in the insets of Figure 4.9c,d.
Figure 4.9. Optical switching/bistability behaviors in the nonlinear-pair-grating layers for
grating widths w of 30 nm (a and c) and 150 nm (b and d) with operating wavelengths in the
long (a and b) and short (c and d) resonances.
Figure 4.10 shows the
calculated switching intensity
for various Q-factors. The
fitting equation and the line
for the switching intensity are
also noted. It is clearly seen
that the switching intensities
decrease roughly as 1∕Q2.4 and
1/Q2.3 for bistability and
switching behaviors,
Figure 4.10. Optical incident intensity for the
switching of optical switching/bistability devices
based on 3-pair-grating layers for various Q-factors.
respectively. It is well known that the switching intensities of an
established Lorentzian lineshape optical bistable device in photonic
crystal slabs or slab waveguide gratings scale as 1/Q2, where Q =
λo/Δλ, λo and Δλ are the resonant wavelength and full-width at half-
maximum, respectively. This implies that the switching intensity
based on Fano resonances decrease faster than that of the Lorentzian
lineshapes. If the nonlinear characteristics of the Fano resonances are
similar to that of a Lorentzian lineshape, the normalized switching
intensity should be proportional to the 1/(Δλ)2.
CHAPTER 5. OPTICAL BISTABILITY BASED ON
INTERACTION BETWEEN SLOTED CAVITIES AND
WAVEGUIDES IN TWO-DIMENSIONAL PHOTONIC
CRYSTALS
5.1. Photonic devices and two-dimensional photonic crystal
structure using silicon photonic material
Silicon-on-insulator has become the foundation of silicon photonic
materials due to a number of advantages [128,129]: (i) promote the
strengths of the technology of electronic components which have been
perfect on crystalline silicon, (ii) the material cost is relatively cheap,
durable in operation and proactively sizing the components down to a
few tens of nanometers and (iii) the high refractive index difference
between crystalline silicon and silicon oxide, is very effective in
propagating of light. Silicon photonic material promises to fabricate
photonic integrated circuits (PICs) on the same large SOI plate.
Integrating the materials on a large SOI plate is essential, for example
minimizing the effect of the free carrier in the optical sensors. The SOI
plate fabrication technique is compatible with the complementary
metal oxide semiconductor (CMOS) technology, thus achieving high
accuracy.
5.2. Slot waveguide and cavity
To reduce the calculation time
without reducing the accuracy of
the simulation results, I use the
method to estimate the effective
refractive index of the PhCs plate
to bring the structure of the PhCs
plate to 2D PhCs structure.
The parameters of SOH
(Silicon organic hybrid -
SOH) plate are given as follows: refractive index of silicon plate nSi =
3.48; silicon thickness d = 220 nm and refractive index of organic
material DDMEBT nDDMEBT = 1.8, I found that the effective refractive
index of SOH plate is n = 2.9812.
Figure 5.8a shows the structure of a slot waveguide with a narrow
width d = 50 nm with the following structural parameters: the effective
refractive index of the SOH n = 2.9812, the lattice constant a = 380
nm, the air hole radius r = 0.3a, and the refractive index of the organic
material DDMEBT nDDMEBT = 1.8, filled in the holes.
Figure 5.11a shows a cavity with the slot width at the center d = 50
nm, the central of slot length L. The slot width gradually increases at
equal intervals of 10 nm/a until the wall width is reached prevent
electromagnetic waves d = 120 nm. Figure 5.11b shows the
distribution of the electric field inside the cavity with a slot length L =
Figure 5.5. Relationship diagram between the
transmission coefficient and the effective
refractive index of the structure.
1a. With this
resonator
structure, I
obtained the
quality factor Q
= 2403. Similar
in Figure 5.11c
and Figure 5.11d
is the distribution
of the electric
field inside the
cavity with slot
length L = 3a and
L = 5a
corresponds to the
quality factor Q =
6161 and Q =
9163.
5.3. Interaction between resonator and slot waveguide
5.3.1.1 Theoretical model
Figure 5.12 shows a schematic diagram of n identical resonators
through the bus waveguide.
Figure 5.12. Schematic of coupled n identical resonators through the bus waveguide.
Figure 5.8. (a) Slot waveguide channel with narrow width d,
(b) Energy diagram of the waveguide channel,
(c) Distribution of electric field within the waveguide channel.
Figure 5.11. (a) The slot cavity, (b, c, d) is the electric field
distribution within the cavity with slot length L = 1a, 3a and 5a,
respectively.
It is assumed that each resonator has a resonant frequency ωo and
large enough Q factor so that the direct coupling between resonators
is negligible and the resonators interact through the bus waveguide.
1/τ is the decay rate into the bus waveguide from each resonator. s+1
and s-1 are the amplitudes of the incoming and the outgoing waves in
the first resonator; s+n and s-n are defined similar for the n resonator.
The temporal changes of the mode amplitudes of the resonators a1,
a2,,an are derived:
1
0 1 2 1
0 1 1
0 1
1
[ ( ) ]
( 2 )
1
[ ( ) ]
i
i i i
n
n n n
da
j a a k s
dt
da
j a a a
dt
da
j a a k s
dt
với 1 < i < n
where 2je
is the coupling coefficient from the waveguide to
the resonator. δ and µ represent the shifted resonant frequency and
direct coupling coefficient, respectively, which are given by δ = cotφ/τ
and µ = -jcscφ/τ. φ is the phase shift between two adjacent resonators
through the bus waveguide. |a|2 and |s|2 are the energy stored in the
resonator and the wave power, respectively. The transmission spectra
of the coupling n identical resonators can be calculated in the
frequency domain with s+n = 0 can be given by:
22
*
21 2
( )
1/
n
n
i
i
s k k
T f
s j f
(5.11)
With
1
1 1
1/
1
/
n
n
n
i
i
i i
a
f
a j
a
f
a j f
(5.12)
is the frequency detuned from the resonant frequency, = - o.
Eq. (5.11) gives us the
transmission spectra as
show in Fig. 5.13. As
shown in Fig. 513a, for φ
= π/2, the transmission
spectra are symmetric
and their center resonant
frequencies remain
stationary with nearly
flattop at unity and a little
fluctuation. With
deviation of the phase
shift /2, the
symmetry of
transmission spectra will be broken and the depth of valley will
increase with an increase of the detuning frequency. Another
characteristic is that the transmission spectrum shifted to a lower
frequency (or higher frequency) with increase (or decrease) of the
phase shift . Fig. 5.13d shows the fifth-order filter for several phase
shifts . As can be seen, the different resonant peaks located on both
sides of the central resonant frequency depending on the phase shift
larger or smaller than /2. The linewidth and depth of valley of the
Figure 5.13. Theoretical transmission spectra of the structure
depicted in Fig. 5.12 for several numbers of resonators by
using the CMT with: (a) = π/2, (b) = π/3, (c) = 2π/3 và
(d) The transmission spectra of the fifth-order filters for
several phase-shifts
resonances far from the center resonance tend to become narrower and
deeper, respectively. The induced resonance linewidth shrinking of the
right- (or left-) most peak with multiple coupled resonator for phase
shift /2 shows better than that with phase shift = /2. For
switching/bistability applications, the choice of phase shift = /2 is
easy comparison among transmission spectra for different number of
resonators n by maintaining the same center frequency, even phase
shift = /2 is not the best choice in terms of switching power.
5.3.1.2 Simulation results
The SOI slotted 2D PhC structure under consideration is started
from the 260 nm thick silicon film of the SOI wafer, and then made of
a 2D triangular lattice PhC structure of holes with lattice constant a =
380 nm. The holes are inscribed through the silicon film with radius
of r = 0.30a = 114 nm. The slotted waveguide is formed by removing
one center row of holes along - K direction and inscribing with the
rectangular slot of width d through the silicon film. The SOH slotted
PhC waveguide is obtained by first producing the SOI slotted PhC
waveguide and then filling and covering it with DDMEBT of
refractive index of 1.8 as shown in Fig. 5.14a. Figs. 5.14b, c show the
2D PhC structures of coupled five identical slotted cavities together
through the waveguide and two identical side-coupled slotted cavities
and slotted waveguide, respectively. Each cavity is formed by
gradually changing the slotted width from 50 nm at the center to 120
nm of both sides as shown in Fig. 5.13d. The increasing step of slotted
width of cavity is 10 nm for each periodicity, whereas the slotted
waveguide widths at the input and output ports are kept at 50 nm.
Figure 5.14. (a) Sketch of SOH slotted PhC waveguide. (b) and (c) are the designs of the fifth-
order filter and two side-coupled resonators and waveguide, respectively. The details of the one
resonator in a slotted 2D-PhCs are shown in (d).
Fig. 5.15a shows the transmittance characteristics for n from 1 to 5
by using the FDTD method. The center wavelength which is the
resonant peak of the single resonator at λ1 = 1555.28 nm and it is used
for optical communication. The Q-factor of single resonator is
estimated at 4462. The transmission spectra of the higher-order filters
(n > 1) have n resonant lineshapes and the linewidths of the rightmost
resonant peak in the transmission spectra, which are used for bistable
switching operations, are estimated. Since their full-width at half-
maximum (FWHM) cannot be defined as seen in Fig. 5.15a (note that
the first dip on the left side of the right-most peak does not go below
50%), so that the linewidth and Q-factor can be estimated as fitting the
right-most resonance to the Fano lineshape as follows [160]:
2
2
( )
( )
1
q
R F
(5

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