Coupling by Periodic Corrugation 13pp

7/28/2019 Coupling by Periodic Corrugation 13pp

1/13

1

Bragg Coupling surface

Coupling caused by a periodic corrugationCoupling caused by a periodic corrugation

(laser DBR)

1

Back to the coupled modes equationsBack to the coupled modes equations(periodic corrugation)(periodic corrugation)

Propagation equation

( )( ) ( ) ( ) ( )yxEzyxyxEyx ,,,,, 202 =+

2

1Coupled modes

Phase

The source terms

==

,0

r

m

rm

( ) ( ) ( )zmkizAyx rlll

m

m

+

exp,0

2

matched

sources

Coupled

equations

Phase

matched terms

Co-

directional

( ) ( ) ( ) =l

lll ziyxEzAzyxE exp,,,The propagating waves

2

those of the propagating wavesrl

k

The coupling between and is efficient only

when the phase matching condition is satisfied

)( llA )( kkA

rlk m+=

coupling

Contra-

directional

coupling


2/13

2

Back to the coupled modes equationsBack to the coupled modes equations(periodic corrugation)(periodic corrugation)

A k

Coupled waves equations :1

Coupled modes

Phasezmzz

zrkll

l

kl

mk

=

exp,0

( ) [ ]( )== dxdyEElkzC lmkmmklrr

*

44

kllkk dxdyEE

=

rr

*

02

matched

sources

Coupled

equations

Phase

matched terms

Co-

directional

3

Phase matching condition for efficient coupling of kl AA and

0=+ rkl m

coupling

Contra-

directional

coupling

Coupled waves equationsCoupled waves equations

Consider the coupling

by a sinusoidal grating between 2 modes only

2

( )i

ziziyxn rr

2

expexp,2

=

1Coupled modes

Phase

( ) ( )

+=

+

i

eeACzkACi

z

A zkizki

r

rr

2

sin 2121111

11

21 ==r

( ) ( )

+

+=

+

i

eeACzkACi

z

A zkizki

r

rr

2sin 121222

22

matched

sources

Coupled

equations

Phase

matched terms

Co-

directional

4

Phase matching condition to be satisfied :

rr =+= 21

( Could also be : )rr == 21

coupling

Contra-

directional

coupling


3/13

3

Neglect the non phaseNeglect the non phase--matched termsmatched terms

( ) ( )( )

++=

zkizkiAi

CzkACi

z

Arrr

expexp

2sin 2

12111

1

11

1Coupled modes

Phase

the distances on which the amplitudes of the

modes change significantly are large (slowly varying envelop

approximation)

Integrate the equation on a distance small comparedto

211211 ,,

211211

1,

1,

1

CCC

L

matched

sources

Coupled

equations

Phase

matched terms

Co-

directional

5

these distances

The stronglyphase matchedterms vanish.

coupling

Contra-

directional

coupling

Neglect the non phaseNeglect the non phase--matched termsmatched terms

( )zki rACA =

1211 ( )zki reACA =

1

2122

1Coupled modes

Phase

z 1 2 z 2

( ) === dxdyEyxnECC

2

2*

102112 ,

822

matched

sources

Coupled

equations

Phase

matched terms

Co-

directional

6

zieAz

A '2

1

11

= zieA

z

A '1

2

22

=

r= 21' mismatch of the propagation constants

coupling

Contra-

directional

coupling


4/13

4

11-- codirectional couplingcodirectional coupling

r= 21' 12

2

1

1 ==

1Coupledmodes

Co-

directional

zieAz

A '2

1 = zieA

z

A '1

2 =

'*'**

* ziiAA

Conservation of energy in the coupling process

Solutions

Effect of

mismatch

Coupling

constant

Applications

7

[ ] 0222

1 =+ AAdz

d

122121 == eezz -

directional

coupling


12

2

1

1 ==

r

=21

'

1

Coupledmodes

Co-

directional

Resolution zizi eiAezA

zA '

1

'1

2

2

2

)'( =

0' 122

2

2

2

=+

+

AA

iA

zieAz

A '2

1 = zieA

z

A '1

2 =

=+

0' 122

2

2

2

AA

iA

Solutions

Effect of

mismatch

Coupling

constant

Applications

8

zz zz

( )

+

++

=

zbzaezAzi

2

2

2

2

2

'

22

'sin

2

'cos

-

directional

coupling


5/13

5


( ) 002 =A ( ) 11 0 AA =

( )z

zeAzA 2'

2

2

2

212 2

'sin

'

+

=

Initial conditions1

Coupledmodes

Co-

directional

2

( )zi

ezizAzA 2'

2

2

2

2

2

2

112

'sin

2

'

2

'

2

'cos

+

+

+

=

( ) ( ) zPzP 22

2

2

2

2

122

'sin

'0

+

+

=

Solutions

Effect of

mismatch

Coupling

constant

Applications

9

( ) ( )

( ) ( )zPP

zzPzP

21

2

2

2

2

2

2

2

2

2

11

0

2

'sin

2

'

2

'

2

'cos0

=

+

+

++

=

2 -

directional

coupling

Effect of the mismatchEffect of the mismatch

( ) ( ) 12

'sin

12

'

10

2

2

212+

+

=

zPzP

( )2 zP

1

Coupledmodes

Co-

directional

1

1SolutionsEffect of

mismatch

Coupling

constant

Applications

02

' =

' '

10

z

25,0

2

23 2

10

10,0

-

directional

coupling

2=

2=


6/13

6

The coupling constantThe coupling constant

Superstrate cn

h

=

21

1Coupledmodes

Co-

directional

Waveguide

Substrate

fn

sn

d =

rk

22

0

22

0

11

ceffkseffk

knnknnk

dd

+

+=keffk

n

Solutions

Effect of

mismatch

Coupling

constant

Applications

11

( )( )

21

2

2

22

1

2

210 effeff

efffefff

nn

nnnn

dd

h =

-

directional

coupling

Couplage codirectionnelCouplage codirectionnelrsonnantrsonnant

Exemple 51,1=fn 50,1=sn 0,1=cn

m 6,00 =

md 4=

1

Coupledmodes

Co-

directional

m

3,149122

=

=

=

Effective indices of co-propagating TE modes

50862,1

0

11 ==

k

neff

5046,1

0

22 ==

k

neff

Resonant coupling when

=+= 0

221

Solutions

Effect of

mismatch

Coupling

constant

Applications

12

effeff 21021

1598,0 = cmmd 678,41 =Gives md 897,42 =

Full energy transfer for interaction length

cmLc 63,22

=

-

directional

coupling


7/13

7

Effect of wavelengthEffect of wavelength

Same exemple,At the critical length, in the presence of a mismatch

2

1Coupledmodes

Co-

directional

( )( ) 2

2

1

2

'21

12

sin

0

+

+=

c

c

c

LP

LP

( )

=

22' 21

0

effeff nn

Solutions

Effect of

mismatch

Coupling

constant

Applications

13

The energy transfer is very sensitive to the wavelength.Falls by 50% when

cL

8,0

0

here nm7,2

-

directional

coupling

Codirectional coupling betweenCodirectional coupling betweenguided modes and radiating modesguided modes and radiating modes

Equations valid for radiating modes, as well as for guided modes

The phase matching condition is a condition on the longitudinalpropagation constants

1

Coupledmodes

Co-

directional

=

2rk

radk

rad

cos

2cos

0

sradrad nk ==

= 2guidedrad

=

22

cos2

effs nn

Solutions

Effect of

mismatch

Coupling

constant

Applications

14

Useful input-outputcoupling device

guided

radk

00

=

cos0

seff nn

51,1=fn50,1=sn0,1=cn

m 6,00 =md 4=

50862,11 =effn

-

directional

coupling


8/13

8

Contra directional couplingContra directional coupling

112

2

1

1 ==

zieAA '

21 =

rr k== 2' 21

( ) AA =01( )LA1

1

Coupledmodes

Co-

directional

z

zieAA '

12 =

02

22

2

2

2

=

+

Az

Ai

z

A 0' 1

21

2

1

2

=

Az

Ai

z

A

( )02A( ) 02 =LA

0=Z LZ =

coupling

Contra-

directional

coupling

solutions

15

( )

+=

zzzi

ebeaezA22

2'

2

2'

22

'

2

( )

+=

zzzi

ebeaezA

22

22

2

'

1

2

'

12

'

1


( ) AA =01 ( ) 02 =LAUsing the initial conditions2

2 '

=

sand

1

Coupledmodes

Co-

directional

( ) ( )( ) ( )

sLisLs

zLsizLsseAzA

zi

sinh2

'cosh

sinh2

'cosh

0 2'

11

+

+=

( ) ( )( )zLsi

eAzAzi

'

sinh0 2

'

12

=

coupling

Contra-

directional

couplingsolutions

16

sLisLs sn2

cos +


9/13

9


=

+=

22

2021 effnk

=( )LA1

( ) znnzn c +=

2sin

1

Coupledmodes

Co-

directional

( )02A( ) 02 =LA

Reflectivity ?

0=Z LZ =

0 =

( ) ( )sLisLs

sLiAA

sinh2

'cosh

sinh00 12

+

=

coupling

Contra-

directional

coupling

Solutions

reflectivity

17

( )( )

sLsLs

sL

A

AR

2

2

22

222

1

2

sinh2

'cosh

sinh

0

0

+

==

22

2

'

=

s

Resonant contra directional couplingResonant contra directional coupling

02

22

021 ==

+=

effnk

=( )LA1

( ) znnzn c +=

2sin

1

Coupledmodes

Co-

directional

( ) ( )( )

( )LzL

LzAzA

sinh

cosh

cosh011

=

( )02A( ) 02 =LA

( )( )

LA

AR 2

2

1

2 tanh0

0==

0=Z LZ =

0 =coupling

Contra-

directional

couplingSolutions

Reflectivity

Resonant

18

Lcosh12 =

exemple m 550,10 = 46,1=effn nm531=

410.4 n mmL 2= %98= R

couping


10/13

10

Wavelength sensitivity of the Bragg gratingsWavelength sensitivity of the Bragg gratings

effn20

= Constructive

interferences

1

Coupledmodes

Co-

directional

( ) AA =01

( )02A

coupling

Contra-

directional

coupling

Solutions

Reflectivity

Resonant

19

Smaller the index modulationSmaller the reflectivity of each grating planeHigher the number of reflected beamHigher grating selectivity in wavelength

coup ng

selectivity

Wavelength sensitivity of the Bragg gratingsWavelength sensitivity of the Bragg gratings

( )( )

sLsLs

sL

A

AR

2

2

22

222

1

2

sinh2

'cosh

sinh

0

0

+

==

2

2

2

'

=

s

1

Coupledmodes

Co-

directional

Max when 0'=

First minimum when i.e.0sinh2 =sL 2'=

effeff nn == 20

240

0

0

n=

The smaller the index modulation

coupling

Contra-

directional

couplingSolutions

Reflectivity

Resonant

20

4

000

10.312/1

12224

+

==

+ effeff

effn

n

nnnn

the smaller The higher the grating selectivity in wavelength

couping

selectivity


11/13

11

PhotoPhoto--induced Bragg gratings filters in doped fibersinduced Bragg gratings filters in doped fibers

1

Coupledmodes

Co-

directional

coupling

Contra-

directional

coupling

Solutions

Reflectivity

Resonant

21 Doc. Highwave Optical Technologies

coup ng

Selectivity

applications

FineTrim Gain Flattening Filter

(GFF)

Flattening filtersFlattening filters

1

Coupledmodes

Co-

directional

Wavelengthrange S, C and L bands

Maximum attenuation 10 dB

Error functionTypical


12/13

12

Application of flatening filterApplication of flatening filter

1

Coupledmodes

Co-

directional

coupling

Contra-

directional

coupling

Solutions

Reflectivity

Resonant

23

coup ng

Selectivity

applications

PhotoPhoto--inscription of Bragg gratingsinscription of Bragg gratings

1

Coupledmodes

Co-

directional

coupling

Contra-

directional

couplingSolutions

Reflectivity

Resonant

24

n er er ng eamsPhase masksPoint to point writing

couping

Selectivity

Applications

technology


13/13

13

Laser diode reflectors

Distributed Back Reflectors

1

Coupledmodes

Co-

directional

Diode laser

Distributed Feed Back

coupling

Contra-

directional

coupling

Solutions

Reflectivity

Resonant

25

DFB

Provides easier longitudinal and transverse single mode operation

coup ng

Selectivity

Applications

technology

Documents

Coupling by Periodic Corrugation 13pp