集成光電子器件及設(shè)計(jì)-2平面介質(zhì)光波導(dǎo)和耦合模理論

第二章

平面介質(zhì)光波導(dǎo)和耦合模理論集成光電子器件及設(shè)計(jì)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

定義：波導(dǎo)中由于某種原因產(chǎn)生的由一種模式向另外一種模式的轉(zhuǎn)

換，或多個(gè)波導(dǎo)組成的系統(tǒng)中，其中一個(gè)波導(dǎo)傳輸?shù)哪Ｊ较蛄硗獠▽?dǎo)

的轉(zhuǎn)移；

實(shí)質(zhì)：模式的能量變換；

例子4光場(chǎng)在單根波導(dǎo)中的傳播

理想情況：

波導(dǎo)沒有缺陷

折射率分布均勻、規(guī)則；

沿波導(dǎo)保持光場(chǎng)形狀無改變傳播

實(shí)際情況：

制作波導(dǎo)的材料存在損耗，

光場(chǎng)沿傳播方向振幅呈指數(shù)衰減；5方向耦合器波導(dǎo)中傳輸?shù)膶?dǎo)模在芯層外的倏逝場(chǎng)由于相互作用產(chǎn)生耦合，引起波導(dǎo)間模式功率的相互轉(zhuǎn)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:

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.,

function

of

znI2nII2Waveguide

IIWaveguide

Iyxn0209Maxwell

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≠β2β

~

the

width

w,

the

height

h,

the

indices:

n1,

n2WG

IWG

IIOne

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

BuffernclnconbfCross

sectionTop

view

20S=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'

=

k2'1'

0()E2'

(0)k2'1'

=

exp(?

jφ2'1')φ2'1'

=

βl2'1'E1k2'1'

0()k12'

(0)=

k12'

(0)

+βl2'1'

=

mλ33The

resonator’s

responseGeneral

formula

11′22′l4′1′l23′k2′1′(0

(

1'2'

((

(0

(

(2

(?E20)

=

k12)E10)

+k1'0)E1'0)?E2'

=

k12')E10)

+k0()E1'0)?

(0)?2'1'

1'2'1'2'

2'1'

(0(('2'1'

1'2'(2'1'

1'2'(2

2'1'

(0(0((E1'0)

(0)E20)E10)E20)E10)1?

k0()k0()

k0()k0()k12')

1?

k0()k0()=k1'0)k0()k12')1?

k0()k0()=

k12)

+0

0

0

00Resonance

wavelgnth34The

resonator’s

response

Key

features:

FSR

(free

spectral

response).

3dB‐bandwidth,

Q

factor

=

λ/BW3dB.

Resonance

wavelengths.

?

(0)E1'

=

k2'1'

0()E2'

(0)E1=

k12

(0)

+k2'1'

0()k12'

(0)=

k12'

(0)

+=?

?∏k1'2'

?

?γ

tol

exp(?

jΦtol)=

E

?

?k1'2

∏k1'2'

?

?γ

n

exp(?

jΦn)??(0

(

1'2'

((

(0

(

(2

(?E20)

=

k12)E10)

+k1'0)E1'0)?E2'

=

k12')E10)

+k0()E1'0)?

(0)?2'1'

1'2'1'2'

2'1'

(0(('2'1'

1'2'(2'1'

1'2'(2

2'1'

(0((E20)E10)E1'0)

(0)E20)E10)=

k1'0)k0()k12')

1?

k0()k0()1?

k0()k0()

k0()k0()k12')

1?

k0()k0()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

(波形波導(dǎo)

)

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

154041Bragg

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|=044κGL=2

Reflection

Transmission

Bragg

wavelength|φ|=0

Frequency/wavelength

dependent:45(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

雙光束干涉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

couplingαPrism

npθdena

θ’mncβn0n1n2SnP

sinθ2π

λ01.

The

matching

condition:

βv

=

βP

=2.

折射率:

np>n1>n2>n03.

θ

>θc4.

Gap

width

S<λ/2.

Change

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

<激光通過棱鏡和薄膜之間的空氣層被耦合進(jìn)波導(dǎo)層。在耦合的某個(gè)角度，可以看到波導(dǎo)產(chǎn)生的模點(diǎn)。當(dāng)從棱鏡里面看到衍射光時(shí)，在這些耦合的角度，可以發(fā)現(xiàn)光強(qiáng)突然變?nèi)?，在光斑的中間有個(gè)垂直的黑線。通過測(cè)試所有的模點(diǎn)，就能夠算出膜層的折射率和厚度了。為了得到這些值，膜層厚度需要足夠大，至少在波導(dǎo)上出現(xiàn)兩個(gè)模點(diǎn)。通過調(diào)整激光的直角偏振，就可以算出膜層的尋常光和非尋常光。

5859Prism

coupler

1波導(dǎo)損耗的測(cè)量

2薄膜及波導(dǎo)折射率/厚度測(cè)量

3體材料折射率的測(cè)量

4薄膜及體材料的雙折射測(cè)量

5液體折射率的測(cè)量

*

Precision:

±0.0005

(even

0.0001‐0.0002)

*

Thickness:

±(0.5%+50?)

*

Range

n:

1.0~3.35

60The

(dis)advantages????效率高可以通過改變?nèi)肷浣羌?lì)不同的導(dǎo)波模式可以測(cè)量平板波導(dǎo)，也可以測(cè)量條形波導(dǎo)可以通過調(diào)整間隙實(shí)現(xiàn)最大耦合強(qiáng)度對(duì)材料要求高（折射率，吸收）入射光必須高度對(duì)準(zhǔn)震動(dòng)和溫度變化會(huì)引起不穩(wěn)定性61Grating

coupler在平面介質(zhì)光波導(dǎo)上直接制作光柵利用光柵替代棱鏡和間隙介質(zhì)可以是正弦、三角周期性結(jié)構(gòu)βv

=

β0

+(v

=

0,±1,±2,...)v2π

Λk0sinθi

=βv

無光柵時(shí)導(dǎo)波模傳播常數(shù)光柵周期62The

(dis)advantages?????不受光波導(dǎo)材料折射率大小限制可以選擇導(dǎo)波模式任一種進(jìn)行激勵(lì)與波導(dǎo)集成后，耦合效率不會(huì)因外界環(huán)境變化而變化調(diào)整光束的入射不需要很高精度可以激勵(lì)寬度非常大的波導(dǎo)不能耦合發(fā)散光束偏振相關(guān)性?Very

useful

for

the

coupling

between

silicon

nanowire

and

fiber;?Useful

for

wafer‐scale

test.

63Transverse

Coupling

(橫向耦合

)光纖‐平面光波導(dǎo)平面光波導(dǎo)‐平面光波導(dǎo)半導(dǎo)體激光器‐平面光波導(dǎo)

聚焦耦合（end‐fire）

對(duì)接耦合（butt‐coupling)64The

facet

should

be

polished

for

higher

coupling

efficiency.Gaussian

beam

(W)Gaussian

beam

(w)End‐fire

(聚焦耦合方法)

Facet

of

the

optical

waveguideDevices

under

test

(DUT)Light

sourcePolarizerMultimode

fiber65The

setup

for

end‐fire

coupling

Splitter

Camera

(monitoring

the

mode

profile)OSA66The

setup

for

the

end‐fire

coupling

system1234568767Butt‐coupling

systemVertical

direction

(μm)Vertical

direction

(μm)68Coupling

coefficient標(biāo)準(zhǔn)光纖和波導(dǎo)在端面耦合時(shí)模式失配損耗是插入損耗的主要因素；利用光纖和波導(dǎo)模場(chǎng)的重疊積分可以得到兩者耦合時(shí)的損耗；改變波導(dǎo)的幾何尺寸，從而改變波導(dǎo)的模場(chǎng)分布，可以使波導(dǎo)的模場(chǎng)和光纖的模場(chǎng)達(dá)到較好的耦合。=∫∫Er(x,y)

?Enor*y)dxdyf

(x,f

f

(x,

∫∫Er(x,y)

?Enor*y)dx