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第7章
光學(xué)微腔:原理集成光電子器件及設(shè)計2Outline
1.
Background
2.
Optical
Micro‐cavities:
2.1.
Standing‐wave
type:
F‐P
cavity;
2.2.
Traveling‐wave:
2.2.1.
Microring
resonator;
2.2.2.
Micro‐disk
resonator.
3Standing‐wave
&
Traveling‐wave
The
Fabry–Pérot
Cavity
~
Standing‐wave
λq
2LThe
Microring
Cavity
~
Traveling‐wave42.1.
Standing‐wave:
Fabry–Pérot
CavityL1L2hPG1
G2Charles
Fabry
(1867‐1945)Alfred
Perot(1863‐1925)F‐P
resonator
(1897)567Transmission:Reflection:Transmission
(dBm)8-30-40-10-20
FSR:
~
21
nm;
Q‐value:
~2600;
Extinction
ratio:
13dB;
1.52
1.545
1.57
1.595
1.62
Wavelength
(μm)FSR
could
be
as
large
as
200nm
by
reducing
the
cavity
length
to
about
1μm.
It
is
much
larger
than
the
MRR’s
FSR.
SOI‐nanowire
F‐P
micro‐cavity
0J.
Wang,
D.
Dai,
and
S.
He.
IPRA
conference
2010,
USA.
Bragg
gratingsQ=(6.3±0.8)x101092.2
Traveling‐wave
optical
cavity/?page_id=59silica
microtoroids10Microring
resonators&
micro‐racetrack
resonators?
(0)E1'
=
k2
(0
'1)
'E2
(0
'
)exp(?
jφ2
0'1')=
eE1k2
(0
'1)
'k12
(0')=
k12
(0')
+11Model
of
a
single
ring
resonator
with
one
waveguide
General
formula
11′Method
I
2
2′l4′1′l23′(0
(
(2'
('(
(0
(
(2
('?E20)
=
k12)E10)
+k1'0)E10)?E2'
=
k12')E10)
+k1'0)E10)?
(0)?('1'
(2'(2'
('1'
(0(('('1'
(2'('('1'
(2'(2
('1'
(0(0((E10)
(0)E20)E10)E20)E10)1?
k20)k1'0)
k1'0)k20)k12')
1?
k20)k1'0)=k1'0)k20)k12')1?
k20)k1'0)=
k12)
+0k0
0
k2′1′
0
αl2'1'2'1'
φ2'1'
=
βl2'1'0βl2'1'
=
mλResonance
wavelgnth?
(0)E1'
=
k2
(0
'1)
'E2
(0
'
)E1=
k12
(0)
+k2
(0
'1)
'k12
(0')=
k12
(0')
+=?
?∏k1'2'
?
?γ
tol
exp(?
jΦtol)=
E
?
?k1'2
∏k1'2'
?
?γ
n
exp(?
jΦn)??(0
(
(2'
('(
(0
(
(2
('?E20)
=
k12)E10)
+k1'0)E10)?E2'
=
k12')E10)
+k1'0)E10)?
(0)?('1'
(2'(2'
('1'
(0(('('1'
(2'('('1'
(2'(2
('1'
(0((E20)E10)E10)
(0)E20)E10)=
k1'0)k20)k12')
1?k20)k1'0)1?k20)k1'0)
k1'0)k20)k12')
1?k20)k1'0)1′2′2′#N
1′1′
#1
1′
1
#0#n
2′
2
The
resonator’s
response
Ring
resonator
with
N
output
ports.
Through
port
2
1
Input
port
1
2output
port
#1
output
port
#N
2′
2
1output
port
#n
(0)2'1'k?
N
(n)??
n=1
?Daoxin
Dai
and
Sailing
He.
Proposal
of
a
coupled‐microring‐based
wavelength‐selective
1×N
122'
(n)2E(0)?
(n)
n?1
(m)?
m=1=
k
=
?
jkk13121′2′#01′12′2#1121′2′#N1′21#n
2′Input
portThrough
portoutput
port
#1output
port
#noutput
port
#N(
(('1'
(0
(0
(2'
(2
(0
1
1′l4′1′
The
critical
coupling
condition
2No
power
outputs
from
the
thru
port,
i.e.,
2′
E20)
/
E10)
=
0
l23′
k20)
=
k12)
/[k12)k1'0)
?
k1'0)k12')]k2′1′(0
(2'(1)
For
coupler
#0,
one
has
(0)
(0)
1'2
12'('1'k20)
=
1?k
2
k12)
=
k1'0)
=
1?k
2Finally
the
critical
coupling
condition
becomes14
Special
case
I:
all
passed
filter
(n=1)
The
critical
coupling
condition
becomes0
(0)2'1'0kαl2'1'exp(?
jφ2'1')=
e
1
1′l4′1′
2
2′l23′k2′1′and2
(0)2'1'k=
1?kα<0α>0
λPowerFSR=?
?∏k1'2'
?
?γ
tol
exp(?
jΦtol)E1=
k12
+=
E
?
?k1'2
∏k1'2'
?
?γ
n
exp(?
jΦn)??15Special
case
II:
add‐drop
filter
(n=2)
(0)2'1'k?
N
(n)??
n=1
?2'E('1'
(2'(2
('1'
(0(0)(E20)
(0)k1'0)k20)k12')1?k20)k1'0)(0)?
(n)
n?1
(m)?
m=1
1
1′
l4′1′
(n)2
2
2′
l23′k2′1′16The
resonator’s
response
Key
features:
FSR
(free
spectral
response).
3dB‐bandwidth,
Q
factor
=
λ/BW3dB.
Resonance
wavelengths.
17Model
of
a
single
ring
resonator
with
one
waveguide
Method
IIα
is
the
loss
coefficient
of
the
ring
(zero
loss:
α
=
1).
θ
=
ωL/c,
L
=
2πr,
c
=
c0/neff,
ω
=
kc0,
k
=
2π/λThe
transmission
power
Pt1
in
the
output
waveguide,The
circulating
power
Pi2
in
the
ring
is
given
bywhere
t
=
|t|
exp
(j?t),
|t|
representing
the
coupling
losses
and
?tthe
phase
of
the
coupler.On
resonance,
(θ+?t)
=
2πm,
where
m
is
an
integer
critical
coupling:
α=|t|
1819The
spectral
response
of
an
all‐passed
filter20Model
of
a
basic
add–drop
single
ring
resonator
filterAt
resonance:Critical
coupling:21Spectral
response
of
an
add–drop
ring
resonator
filter22Some
important
parameters
FSR
(free
spectral
range):
neffL=mλn’effL=(m‐1)λ’
(neff+
Δλ
(?neff/
?
λ))L=(m‐1)(λ+Δλ)
ΔλFSR=
λ/[m
(ng/neff)]
Group
index233dB
bandwidth
(full‐width
at
half‐maximum)
|Et2|2=0.5Pt2_resonance_When
α=1,
t1=t2
(symmetrical),
one
has
Finess
Q
valueThe
intensity
enhancement
or
buildup
factor
B:
On‐resonanceLossless,
κ1=κ2
B=Qλ/(πneffL)
2425An
example
to
show
the
field
enhancement
in
the
resonator:B
~
105
Q
~1×108,
D
~
50μm,
Vm~
600
μm3Pin
=
1
mWExperimental
data1
mWPcav~
100
W,
Icav
~
2.5
GW/cm2,τ
~
100
ns,
#
of
round
trip
~
2×105.
>
100
W26Serially
Coupled
Double
Ring
Resonatorwhere
α1,2
represent
the
half
round
trip
loss
coefficients
of
ring
resonator
one
and
two
respectively.27Assuming
a
coupler
without
losses
and
symmetric
coupling
behavior,
setting
t
=
t?
andκ=?κ?,
one
hasIn
order
to
achieve
a
double
ring
resonator
filter
with
maximally
flat
response
for
the
drop
port,
one
should
make
28An
exampleR1=R2=5.08um,
n1=3.45,
n2=1.456,
k1=0.18,
k2=0.01~0.09.
k1=0.18,
k2=0.0164729Parallel
Coupled
Double
Ring
ResonatorSimplifiedRegular
model:30Finally31Parallel
Coupled
Double
Ring
Resonator
with
Coupling
Between
the
Two
Ring
ResonatorsThe
distance
Λ
between
the
rings
does
not
have
an
influence
on
the
transfer
characteristic.For
lossless
couplers
with
κ1=κ3=
κ:Chremmos
and
Uzunoglu.
PTL.
17(10):
2110‐2112
,
200532In
order
to
realize
a
maximally
flat
response
with
a
single
peak,
the
coupling
coefficients
have
to
obey
the
following
equation:The
corresponding
FWHM
is
given
by33Modeling
cascaded‐ring
resonators:
Method
IIIwhere34Numerical
simulation
for
microring
resonators:
FDTD
methodFDTD
simulation/en/fdtd/user_guide_cw_norm_ring.html/rsoft/application‐galle
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