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