集成光电子器件及设计 - 新型耦合器件与原理I：耦合模理论

第4章

新型耦合器件与原理I：耦合模理论集成光电子器件及设计2Outline

1.

Background

2.

Coupling

mode

theory

Equations

and

solutions

Codirectional

coupling;

Contradirectional

coupling;

3.

Coupling

to

excite

the

modes

in

optical

waveguides31.

Background:

mode

coupling

定义：波导中由于某种原因产生的由一种模式向另外一种模式的转

换，或多个波导组成的系统中，其中一个波导传输的模式向另外波导

的转移；

实质：模式的能量变换；

例子4光场在单根波导中的传播

理想情况：

波导没有缺陷

折射率分布均匀、规则；

沿波导保持光场形状无改变传播

实际情况：

制作波导的材料存在损耗，

光场沿传播方向振幅呈指数衰减；5方向耦合器波导中传输的导模在芯层外的倏逝场由于相互作用产生耦合，引起波导间模式功率的相互转移。Input

Section

Output

Section

1

2

34A0

A

Coupling

region

BB0sD

模式耦合6Surface

coupling:

Prism

coupler,

grating

coupler.

模式耦合

同向耦合：方向耦合器、Y分支、MZI反向耦合：Bragg

grating72.

Coupled

mode

theory

2.

1

Equations

Codirectional

coupler

(directional

coupler);

Contradirectional

coupler

(Grating);

2.

2

Coupling

to

excite

the

modes

in

optical

waveguides8

Coupled

mode

theoryThe

eigenmodes

(Ep,

and

Hp)

in

waveguide

#1,

and

#2

before

mode

coupling

satisfy

Maxwell’s

equations:

function

of

znI2nII2Waveguide

IIWaveguide

I

Refractive

index

profile

N(x,y)For

a

weakly

coupled

system,

the

field

(E,

H)

could

be

written

as

the

sum

of

eigen‐modes

in

waveguide

#1

and

#2,

i.e.,

yxn0209Maxwell

Equ.

for

the

coupling

systemnI2nII2Waveguide

IIWaveguide

Ixn020uzdAdz∇A(z)

=N

is

the

refractive

index

profile

for

the

whole

coupling

system

y1011With

,

one

has

where12The

mode

coupling

coefficient

of

a

directional

coupler.The

butt

coupling

coefficient

between

the

two

waveguides.χp<<κpq,

thus

usually

could

be

neglected.

13

The

difference

of

the

propagation

constant

between

waveguide

I

and

II:

Using

Equ.

(4‐11)

–

(4.12)

*c12exp(-j2δz)=0,Using

Equ.

(4‐12)

–

(4.11)

*c21exp(j2δz)=0,

Codirectional

coupler:

β1>0,

β2>0

Contradirectional

coupler:

β1>0,

β2<014Assume

cpq=0,

χp=0

(p,

q=1,

2).Codirectional

coupler:

β1>0,

β2>0,

the

solution

is15Initial

condition:

A(z=0),

B(z=0),

which

is

corresponding

to

the

launched

field.

16Usually,

A(z=0)=A0,

B(z=0)=0,

i.e.,

light

is

input

to

waveguide

I

only.

Power

flow

along

the

z‐direction

is

given

by

F

denotes

the

maximum

power‐

coupling

efficiency,

given

byδ=0,

F=1.0δ=2κ,

F=0.2Coupling

length,

z(m=0)Position

for

maximum

coupling

Powersplitterbychoosingthelength

L17For

the

case

when

there

is

a

loss0P

a(z)

=

P

sin2[Kz]exp(−2αz)b

0

P

(z)

=

P

cos2[Kz]exp(−2αz)when

there

is

a

loss?

•With

material

absorption

(e.g.,

metal);

•With

bending

leakage;

•With

substrate

leakage;18How

to

have

δ≠0?

δ≠0β1≠β2WG

IWG

II

β

~

the

width

w,

the

height

h,

the

indices:

n1,

n2One

of

the

parameters

for

the

two

coupled

waveguide

different

β1≠β2

for

the

fundamental

modes

in

them

δ≠0

(asymmetrical

coupler)

A

directional

coupler

might

not

work

due

to

β1≠β2

caused

by

the

fabrication

error.Different

bending

radii

β1≠β219Several

example

for

asymmetrical

couplersExample.

1:

Straight

DC

Design

parameters

for

the

optical

waveguides:

hrib=320nm,

wrib=0.95μm,

wgap=0.9μm;

Issue:

we

can

not

observe

the

coupling

in

the

fabricated

DC

structures

(the

coupling

length

varies

from

0

to

2000μm).

Reason?

het

HCladding

wco

Core

BuffernclnconbfTop

view

Cross

section20S=1.858um,

wco1=0.95um,

wco1=0.95um

(Δw=0nm),

Lc=1250um;

neff=3.263857645725755TE21S=1.858um,

wco1=0.95um,

wco1=0.945um

(Δw=5nm),

Lc=1250um;

neff=3.263228568956343Δneff=6.29e-004

(when

Δw=5nm)δ=Δneffk0/2=0.00255μm-1κ=0.5π/Lc=0.0013μm-1TE22Example.

2

Bent

directional

coupler

(R1

≠

R2)R1R2

w1

wg

w2

R1

≠

R2If

we

choosing

different

widths,

the

bent

DC

could

be

symmetrical.TETMPBS

based

on

asymmetrical

DCDaoxin

Dai,

and

John

E

Bowers,

“Novel

ultra‐short

and

ultra‐broadband

polarization

beam

splitter

based

on

a

bent

directionalcoupler,”

Opt.

Express,

19(19):

18614‐18620

(2011)

TMTESiO2SiTE/TMwghcoSiO2w1

Siw2

SiTE~0.02~0.97R=20μm

S-bendL<10umTM~0.983<0.00124Optical

switch:

δ=0δ>>κApplication

of

using

the

asymmetrical

coupler

Control

the

state:

δ=0

or

δ

≠

0Derivation

of

coupling

coefficients

(Method

1)

Coupling

for

slab

waveguidesFor

TE

polarization,

one

hasN2‐N22≠0

in

waveguide

I

only

(|x|<a).

(|x|<a)

2526Example

1.

For

slab

waveguides

with

2a=6μm,

∆=0.3%,

v=1.5,

separation

D=4a,

the

coupling

coefficient

=0.39mm‐1,

the

coupling

length

Lc=4mm.

Using

the

eigen

value

equation,

finally

one

has

The

formula

for

calculating

the

coupling

coefficient

of

a

slab‐waveguide

coupler.Core

ICore

II++Derivation

of

coupling

coefficients

(Method

2)

Based

on

mode

interference

Ein=Eo(x)+Ee(x)odd

evenevenodd2728Derivation

of

coupling

coefficients

(Method

3)

Based

on

numerical

simulation

method:

BPMGet

the

coupling

length

from

the

light

propagation.Be

able

to

deal

with

a

complicated

case/structure.29DC

#1DC

#2arms

More

applications

of

directional

couplers

(I)Mach‐Zehnder

Interferometer

(MZI):

switcher,

modulator,

filter,

optical

sensor,

PBS,

etc.

3dB

coupler:

κl=π/430An

MZI’s

response31Connecting

an

output

port

with

one

input

port

of

an

DC.

More

applications

of

directional

couplers

(II)Ring

resonator:

switcher,

modulator,

filter,

optical

sensor,

PBS,

etc.

32More

forms

of

resonators⎪

(0)E1'

=

k2

(0

'1)

'E2

(0

'

)E1k2

(0

'1)

'k12

(0')=

k12

(0')

+33The

resonator’s

responseGeneral

formula

11′22′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)

+0

00

0

k2′1′k2'1'

=

exp(−

jφ2'1')φ2'1'

=

βl2'1'0βl2'1'

=

mλResonance

wavelgnth34The

resonator’s

response

Key

features:

FSR

(free

spectral

response).

3dB‐bandwidth,

Q

factor

=

λ/BW3dB.

Resonance

wavelengths.

⎪

(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

352'(n)2E(0)⎛

(n)

n−1

(m)⎞

m=1Power36

10.50

00.40.20.40.215401550

00.40.2

0

1530Wavelength

(nm)(a)(b)(c)(d)

1560(a)

the

through

port;

(b)

output

port

#1;

(c)

output

port

#2;

(d)

output

port

#3.

121′2′#01′12′2#1121′2′#N1′21#n

2′Input

portThrough

portoutput

port

#1output

port

#noutput

port

#NRing

resonator

with

N

output

ports,

N=3371×N

Wavelength‐selective

Power

Splitter

(By

D.

T.

Spencer,

Daoxin

Dai,

Y.

Tang,

M.

J.

R.

Heck,

and

John

E.

Bowers)38Contradirectional

coupling

in

corrugated

waveguides

(波形波导

)

Consider

a

coupler

where

the

index

is

perturbed

periodically

between

waveguide

I

and

II

(β1>0,

β2<0).

Assume

κ12(z)=κGexp(‐j(2π/Λ)z),

Λ

is

the

a

period

of

perturbation.

Waveguide

I

κ(z)

Waveguide

II

z=0A(z=0)=0

z=LB(z=L)=039Phase‐matching

condition

factorThe

coupling

equation

κ12(z)=κGexp(‐j(2π/Λ)z)The

same

as

that

in

Page

154041Special

case:

Bragg

optical

waveguideIn

this

case,

waveguide

I

and

II

are

the

same,

i.e.,

β1=‐β2=kneff,

The

phase

matching

condition42Eq.

(4.50)ρL=πκGL=2αL=2κGL=2forwardbackwordforwardbackwordWavelength

dependence

of

the

transmission

Pass

band

43|φ|>

κG

Stop

bandThe

transmission

&

reflection|φ|>

κG

|φ|=044

Bragg

wavelength|φ|=0

Frequency/wavelength

dependent:κGL=2

Reflection

Transmission45(4.60)‐(4.62)T=1‐RR=tanh2(κGL)

@

the

Bragg

frequency

ωB

E.g.,

R=0.93

when

κGL=2.

Gratings

with

various

index

profile20.

A.

Inoue,

et

al.

optimization

of

fiber

Bragg

grating

for

dens

WDM

transmission

system.

IEICE

Trans.

46How

to

fabricate

a

grating?Planar

optical

waveguidesfibers

47

λ2np

sinθΛ=Two‐beam

interference

method

for

fiber

grating

双光束干涉UV

light:

krF

excimer

laser

(248nm),

SHG

Ar

laser

(244nm)

Change

the

index

of

the

Ge‐doped

fiber

core.

4849fiberTwo‐beam

interference

method

for

fiber

grating:

IPeriodically

index

profile50Two‐beam

interference

method

for

fiber

grating:

IIPlanar

optical

waveguide:

standard

micro/nano‐fabircationE‐beam

/

deep

UV

lithography:

form

patterns

on

photoresist.Dry

etching:

transfer

the

patterns

from

photoresist

to

the

dielectric

film.

5152Grating

Coupler

between

fibers

and

chips53Grating

coupler

&

PBSBOX

TE

TM

TETMFiber

core

(a)54The

coupling

system55The

application

for

grating?

Filter.

Coupler.

PBS.

Reflector

(laser).

Sensor

(stress,

temperature,

refractive

index).

Etc.

563.

Coupling

to

excite

the

modes

in

optical

waveguidesSurface

coupling:

prism

coupler,

grating

couplerTransverse

coupling:

end‐fire

coupling,

butt‐couplingIncit

be57Schematic

configuration

for

prism

couplingPrism

npθ

θ’am

αncn1βn2den

SnP

sinθ2π

λ01.

The

matching

condition:

βv

=

βP

=2.

折射率:

np>n1>n2>n03.

θ

>θc4.

Gap

width

S<λ/2.

n0Change

the

incident

angle,

light

could

be

coupled

to

different

guided‐modes.匹配液:水、甘油、二碘甲烷Coupling

in/out

with

prismsPrismPrismMatching

liquidWaveguidePhotodetectorSliding

<激光通过棱镜和薄膜之间的空气层被耦合进波导层。在耦合的某个角度，可以看到波导产生的模点。当从棱镜里面看到衍射光时，在这些耦合的角度，可以发现光强突然变弱，在光斑的中间有个垂直的黑线。通过测试所有的模点，就能够算出膜层的折射率和厚度了。为了得到这些值，膜层厚度需要足够大，至少在波导上出现两个模点。通过调整激光的直角偏振，就可以算出膜层的寻常光和非寻常光。

5859Prism

coupler

1波导损耗的测量

2薄膜及波导折射率/厚度测量

3体材料折射率的测量

4薄膜及体材料的双折射测量

5液体折射率的测量

*

Precision:

±0.0005

(even

0.0001‐0.0002)

*

Thickness:

±(0.5%+50Å)

*

Range

n:

1.0~3.35

60The

(dis)advantages☺☺☺☺效率高可以通过改变入射角激励不同的导波模式可以测量平板波导，也可以测量条形波导可以通过调整间隙实现最大耦合强度对材料要求高（折射率，吸收）入射光必须高度对准震动和温度变化会引起不稳定性61Grating

coupler在平面介质光波导上直接制作光栅利用光栅替代棱镜和间隙介质可以是正弦、三角周期性结构βv

=

β0

+(v

=

0,±1,±2,...)v2π

Λk0sinθi

=βv

无光栅时导波模传播常数光栅周期62The

(dis)advantages☺☺☺☺☺不受光波导材料折射率大小限制可以选择导波模式任一种进行激励与波导集成后，耦合效率不会因外界环境变化而变化调整光束的入射不需要很高精度可以激励宽度非常大的波导不能耦合发散光束偏振相关性•Very