集成光電子器件及設計 - 光學微腔：原理

第7章

光學微腔：原理集成光電子器件及設計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