16.711 Lecture 4 Optical fibers

Last Lecture

• Fiber modes• Fiber Losses• Dispersion in single-mode fibers• Dispersion induced limitations• Dispersion management• The Graded index fibers

16.711 Lecture 4 Mode-coupling

Today

• Mode–coupling theory• Directional coupler• Coupling between guided and un-guided modes• Coupling from waveguide to free space• Mode coupling by scattering• Coupling to radiating modes at waveguide bands

16.711 Lecture 4 Mode-coupling

• Why knowledge of mode coupling is important?

Mode–coupling theory

1. DFB Lasers/DBR Lasers2. VCSEL structure3. March-Zehnder interferometer4. Direction coupler5. Optimized device structures

16.711 Lecture 4 Mode-coupling

• Recall mode equations

Mode–coupling theory

TE Mode:

,0 yzx HEE

Notice:

• Mode profile E(x, y) doesn’t change with z if no index perturbation.• No coupling between different modes if no index perturbation.

16.711 Lecture 4 Mode-coupling

• index perturbations

Mode–coupling theory

),(...)(2)( 220

20

20

2 znnnznnnxn

,t

BE

,

t

DH

,)(2

2

0 t

DE

,)(

2

2

02

t

DEE

,02

2

02

t

DE

,)()( 200 EznEzPED

,0)(

2

222 E

c

znE

• Generalized propagation equation

16.711 Lecture 4 Mode-coupling

Mode–coupling equation

m m

zimym

zimmym

mm ezAxEznkedz

zdAixE )()()(

)(2)( 22

0

,)()( )( m

tzimym

mezAxEE

,0)(

2

222 E

c

znE

• perturbation theory, TE mode

nm

zimymyn

nnyn

n

n nmezAdxxExEi

nkzAdxxE

i

nk

dz

zdA )(*22

02

220 )()()(

2)()(

2

)(

nm

zimnmnn

n mnezAizAiCdz

zdA )()()(

dxxEnk

C ynn

n

222

0 )(2

,)()(2

*22

0 dxxExEi

nkymyn

nmn

16.711 Lecture 4 Mode-coupling

• discussion of the mode-coupling equation

nm

zimymyn

nnyn

n

n nmezAdxxExEi

nkzAdxxE

i

nk

dz

zdA )(*22

02

220 )()()(

2)()(

2

)(

1. if no index perturbation, A(z) is an constant, goes back to the normal equation.

2. Cn measures how much the index perturbation changes the propagation constant n.

3. If index perturbation doesn’t vary with z, coupling between the mode depends on the .

m

zimnmnn

n mnezAizAiCdz

zdA )()()(

,0)()(

zAiCdz

zdAnn

n ,)( ziCn

nezA ,)()()( )(0

zCiymnym

nneAxEzAxEE

mn

,)()( zjCnn

nezEzA ,)()(

m

zimnm

n mnezEdz

zdE ],[ nnmmmn CC

16.711 Lecture 4 Mode-coupling

• directional coupler

,)()(

m

zimnm

n mnezEdz

zdE ],[ nnmmmn CC

),()(

2211 zEdz

zdE

),()(

1122 zEdz

zdE

,*1221

,0)()(

2

2

2

1 dz

zEd

dz

zEd

16.711 Lecture 4 Mode-coupling

• directional coupler --- solution of the differential equation set

),sin()0()cos()0()( 211 zzjEzzEzE

),cos()0()sin()0()( 212 zzEzzjEzE

(1) E2(z=0) =0:

),(cos)()( 20

2

11 zIzEzI

),(sin)()( 20

2

22 zIzEzI ,

4

L

• Output field power is half of the exciting waves.

• the field of the unexcited waveguide is delayed by /2 with respect to the exciting wave.

16.711 Lecture 4 Mode-coupling

• directional coupler --- Applications

(1) Signal switch

),(cos)()( 20

2

11 zIzEzI

(2) WDM coupler

16.711 Lecture 4 Mode-coupling

,sin22

1)( 2010

2

20

2

10

2

1 EEEELE

(2)

,0)(2 zI

,4

L

if

• Application: Y modulator

,)0( 11 EzE ,)0( 22jeEzE

,sin22

1)( 2010

2

20

2

10

2

2 EEEELE

,1sin,2010 EE

16.711 Lecture 4 Mode-coupling

• Phase-mismatched directional coupler

,21 ,)(

)(221

1 zjezEdz

zdE

,)()(

1122 zjezEdz

zdE

,0)()()(

1

212

12

zEdz

zdEj

dz

zEd

• General solution E10 =1, E20=0:

16.711 Lecture 4 Mode-coupling

• Distributed Bragg Reflective (DBR) Structure

),cos(),( Kznzxn

,2

K

Mode–coupling equation

nm

zimnmnn

n mnezEizEiCdz

zdE )()()(

dxxEnk

C ynn

n

222

0 )(2

,)()(2

*22

0 dxxExEi

nkymyn

nmn

,)()(2

)(2

220

2220 dxxEnee

kdxxEn

kC yn

jKzjKz

nyn

nn

),cos(2)(),( 222 Kznnnnzxn

,)()()(2

*20 dxxExEee

i

kymyn

jKzjKz

nmn

16.711 Lecture 4 Mode-coupling

• Distributed Bragg Reflective (DBR) Structure

Small variations

),)(()( )2()2(

21 KiKi eeziGEdz

zdE

,021 CC

),)(()( )2()2(

12 KiKi eeziGEdz

zdE

Phase matching condition:

;02 K

16.711 Lecture 4 Mode-coupling

• General solution

Phase matching condition:

;02 K

16.711 Lecture 4 Mode-coupling

• Example

General Phase matching condition:Grating assistant coupling:

16.711 Lecture 4 Mode-coupling

• Phase mismatched Bragg Mirror

Phase mismatching is related to wavelength, wavelength sensitive mirror.

16.711 Lecture 4 Mode-coupling

• Coupling between guided and unguided modes

,)()(

2211 zjezEdz

zdE

(a) prism coupling

(b) grating assisted coupling

(c) end-fired coupling

16.711 Lecture 4 Mode-coupling

• end-fired coupling

,)()()()(0

0

H N

qqmqbmbc fDFTfDFTdxxfxf

• Coupling efficiency:

• fm(x) is the outgoing guided mode profile, and fg(x) is the incoming waveguide excited mode profile.

,cos 2210 effnnn

16.711 Lecture 4 Mode-coupling

• Coupling form waveguide to Free-space

,|))((|)( ))((arg(( qxfDFTkzjN

Nqq exfDFTxE

• Model coupling by scattering:

16.711 Lecture 4 Mode-coupling

• Coupling to radiating modes at waveguide bends

,)(eff

g n

cCv

),2/

()(R

WR

n

cAv

effg

),()2/

()(0

claddingvn

c

R

WR

n

cAv g

effg

• Example:

mmn

nWR eff 21

2

1

0