Overview
Introduction – Recently, waveguide-coupled
-ring,
-disk, and
Aim – To study the properties of polygonal
-sphere resonators are
-pillar cavities (triangle, parallelogram and
proposed and demonstrated as potential optical
hexagon) by geometrical analysis, k-space
channel add-drop filters for dense wavelength
calculation and simulation.
division
multiplexing
communications.
The
(DWDM)
waveguide-coupled
-ring and -disk cavities are of interests due
Objectives – to find the similarities of the modes
of
equilateral
triangle,
60°
equilateral
to their wavelength selectivity and compact
parallelogram and regular hexagon polygonal
sizes (of ~10 m lateral dimensions and of
-pillar cavities, and approximate the modes
≈ m height) and high-Q resonances for
respectively.
high-density integration in photonic integrated
chips.
Light can be partially confined by nearly total
internal reflection (TIR) at the
-pillar
resonator sidewall. The -pillar cavity can be
evanescently
side-coupled
or
vertically
coupled with waveguides. The main drawback
of the laterally coupled circular
-ring and
-disk cavities is the short interaction length
between the curved cavity sidewall and the
straight
waveguide
sidewall.
The
short
interaction length imposes a sub-micrometer
Figure 1. Wave Pattern in equilateral triangle
gap spacing for evanescent coupling. In order
(3.10 m sidewall length, 3.45 refractive index,
to improve the lateral coupling length, channel
1.55 m wavelength, TM mode)
add/drop filters based on race-track shaped
resonators and polygonal
have been recently proposed.
-pillar resonators
Results – Geometrical analysis & K-space calculation
Software was developed for simulating the ray
The equations of ray trajectory of equilateral
trajectory inside polygonal -cavities.
triangle were proved,
cosθ i
xi +1 = −
sin
θ i +1 =
π
3
π
6
+ u iθ i
xi −
1 + ui
1 − ui
L +
L
2
2
ui − θ i
[
]
[
]
ui = 1
when
xi >
L
1 − 3 tan θ i
2
ui = -1
when
xi <
L
1 − 3 tan θ i
2
ui is undefined when xi =
Figure 2. Leosoft RayTracer 1.0
For regular hexagon, side length = L,
π (m1 + m2 )
triangle is closed-loop and wavefront-matched
for any input angle which is k-space angle of
3L
3πm3
ky =
3L
κ = tan −1
]
My theorem “the ray trajectory of the equilateral
K-space solution
kx =
[
L
1 − 3 tan θ i
2
, m1 , m2 , m3 ∈ Z
equilateral triangle” was proved completely. My
theorem could be extended to 60° equilateral
3 (m1 + m2 )
3m3
parallelogram and regular hexagon.
For 60° equilateral parallelogram, length = L,
2π
(2m1 + m2 )
3L
2 3πm2
ky =
3L
kx =
κ = tan −1
kx
= tan −1
ky
,
m1 , m2 ∈ Z
Figure 3. Input Angle = tan −1
4 3
≈ 44.704655698°
3
Figure 4. Input angle = tan −1
3
≈ 40.893394649°
2
3 (2m1 + m2 )
3m2
For equilateral triangle, length = L,
2π
(2m1 + m2 )
3L
, m1 , m2 ∈ Z
2 3πm2
ky =
3L
kx =
κ = tan −1
3 (2m1 + m2 )
3m2
Results – Simulation*
tri310_wg020_ag015_n345.wmn.csv
1
0.9
0.8
Normalized Intensity
0.7
0.6
Series1
Series2
0.5
0.4
0.3
0.2
0.1
0
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
Wavelength ( m)
par310_wg020_ag015_n345.wmn.csv
1
0.9
0.8
Normalized Intensity
0.7
0.6
Series1
Series2
0.5
0.4
0.3
0.2
0.1
0
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
Wavelength ( m)
hex310_wg020_ag015_345.wmn.csv
1
0.9
0.8
Normalized Intensity
0.7
0.6
Series1
Series2
0.5
0.4
0.3
0.2
0.1
0
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
Wavelength ( m)
tri310_wg020_ag015_n345_Y.wmn.csv
1
0.9
0.8
Normalized Intensity
0.7
0.6
Series1
Series2
Series3
0.5
0.4
0.3
0.2
0.1
0
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
Wavelength ( m)
*The interaction length of the sidewalls was 3.10µm. Refractive index was 3.45. Waveguide width was 0.2µm (single-mode). Air-gap width was 0.15µm.
Series 1 was the spectrum of the throughput port of polygon µ-cavity. Series 2 & 3 were the spectrums of the drop port. TE mode.