Download Optical Electronics in Modern Communications

Summary of Lecture 18
导波条件
( pd ) 2  (hd ) 2  (n22  n12 )k02 d 2
边界条件
pd  hd tan( hd )
图解法求波导模式
波导中模式耦合的微扰理论
E y (r, t ) 
1
Am ( z ) y( m ) ( x)ei (t   m z )  c.c.

2 m
dAs(  ) i (t   m z ) dAs(  ) i (t   m z )
e

e
 c.c.
dz
dz
i  2 
(s)

P
(
r
,
t
)

y ( x ) dx
2   pert
2 t
周期波导
dA
  ab Be i 2(  ) z
dz
dB
  ba Aei 2(  ) z
dz

Coupling constant

Phase difference
§11.5 Coupled-Mode Solutions
Coupled-mode equations
As( )  A
As( )  B
dA
  ab Be i 2(  ) z
dz
dB
*
  ab
Aei 2 (  ) z
dz
Reflective wave
Incident wave
Initial conditions:
B( z  0)  B(0)
A( z  L)  0
§11.5 Coupled-Mode Solutions
Solution
A( z )eiz
B ( z )e
where
iz
i abei0 z
 B(0)
sinh[ S ( z  L)]
  sinh SL  iS cosh SL
e  i 0 z
 B(0)
  sinh SL  iS cosh SL
 { sinh[ S ( z  L)]  iS cosh[ S ( z  L)]}
S   2  ( ) 2
 |  ab |
§11.5 Coupled-Mode Solutions
Under the matching condition
  0
 ab sinh[  ( z  L)]
A( z )  B(0)

cosh L
cosh[  ( z  L)]
B( z )  B(0)
cosh L
§11.5 Coupled-Mode Solutions
z-dependent parts are exponentials with propagation constants
l
 '   0  iS   i  2  [  ( )   0 ]2

for
     0
 0  l / 
 ( )  
'
has an imaginary part
“forbidden” region
Analogous to the energy gap in semiconductors where the periodic crystal
potential causes the electron propagation constants to become complex.
For each l, there has a gap with center frequency
 ( )  ( / c)neff
 (0l )  l / 

 neff
l
2
 '
 i   


 c


 (  0 ) 2 


2
1/ 2
§11.5 Coupled-Mode Solutions
“forbidden” gap zone
( ) gap
2c

neff
and
(Im  ' ) max  
Numerical Example
n 2 ( x)  2n0 n
al(l  1 )  2 /π
β  ω με0 n0   / 
i 0 al  2
(s)
2


n
(
x
)[

(
x
)]
dx
y



4
i 0 al
2n
2 2

n
i
4


§11.6 Periodic Fibers
How to make a periodic index fiber
n( x, y, z )  n0 sin(
2
z)



2 sin 
§11.6 Periodic Fibers
Optical reflectors and filters
At frequencies near the Bragg frequency, incident mode is strongly reflected; while at
frequencies not near Bragg frequency are transmitted with no loss.
 i sinh(  2  ( ) 2 L)
A(0)
r ( ) 

B(0)   sinh(  2  ( ) 2 L)  i  2  ( ) 2 cosh(  2  ( ) 2 L)
 ( )   ( )   0 
n
0  l

Filter bandwidth
 filter
2c

neff
  0
c
neff
§11.6 Periodic Fibers
Distributed Feedback Lasers
dA
  ab Be i 2(  ) z  A
dz
dB
*
  ab
Aei 2(  ) z  B
dz
A( z )  A' ( z ) exp( z )
B( z )  B' ( z ) exp( z )
Ei ( z )  B' ( z )e[( i  ) z ]
[( i  ) z ]
Er ( z )  A' ( z )e
dA'
  ab B' e i 2(  i ) z
dz
dB'
*
  ab
A' ei 2(  i ) z
dz
e i0 z {(  i ) sinh[ S ( L  z )]  S cosh[ S ( L  z )]}
 B(0)
(  i ) sinh( SL)  S cosh( SL)
 abei z sinh[ S ( L  z )]
 B(0)
(  i ) sinh( SL)  S cosh( SL)
0
§11.6 Periodic Fibers
S 2 |  |2 (  i ) 2
Consider the condition
(  i ) sinh SL  S cosh SL
E r ( 0)
Ei ( 0)
and
Ei ( L )
Ei ( 0 )
infinite
Oscillator condition for DFB
High-gain constant
 ab sinh SL
Er (0)

Ei (0) (  i ) sinh( SL)  S cosh( SL)
Ei ( L)
 Sei 0 L

Ei (0) (  i ) sinh( SL)  S cosh( SL)
§11.6 Periodic Fibers
Incident and reflected fileds inside an amplifying periodic waveguide
§11.7 EO Modulation and Mode Coupling
Optical modulation and switching
(1) Smaller modulation powers (see Eq. 9.5-1)
(2) Longer modulation paths (absence of diffraction)
TM  TE mode conversion
[ ppert (t )] y  rE E
( 0)
( )
x
( x)e
i (t  TM z )
E ( 0) dc field
E x( ) TM mode
Excite a TE mode
Coupled mode equations
dAm
i (  mTM   mTE ) z
 iBm e
dz
dBm
i (  mTM   mTE ) z
 iAm e
dz
d
(| Am |2  | Bm |2 )  0
dz
Initial conditions:
Bm (0)  B0
Am (0)  0
§11.7 EO Modulation and Mode Coupling
Under the matching condition

TM
m

when
Bm ( z )  B0 cos(z )
TE
m
Am ( z )  iB0 sin( z )
i
Bm ( z )  B0 e [cos( sz )  sin( sz )
s
iz
 mTM   mTE
where
Am ( z )  iB0 e
s2   2   2
iz

s
sin( sz )
2   mTM   mTE   B   A
2
Fraction of power exchanged  2
  2
§11.7 EO Modulation and Mode Coupling
Power exchange between two coupled modes
 mTM   mTE
 mTM   mTE
§11.7 EO Modulation and Mode Coupling
EO modulation in a dielectric waveguide
§11.8 Directional Coupling
Exchange of power between guided modes of adjacent waveguides is
known as directional coupling.
Applications:
Power division, modulation, switching, frequency selection, and
polarization selection
Coupled field
E y  A( z ) y( a ) ( x)ei[t (  a  M a ) z ]  B( z ) y(b) ( x)ei[t ( b  M b ) z ]
§11.8 Directional Coupling
Perturbation polarization
Ppert  eit  0{ y( a ) A( z)[nc2 ( x)  na2 ( x)]ei a z   y(b) B( z )[nc2 ( x)  nb2 ( x)]eib z }
Coupled mode equations
dA
 i ab Be i 2z
dz
dB
 i ba Aei 2z
dz
2  ( b  M b )  (  a  M a )
Initial conditions:
B(0)  B0
A(0)  0
solutions
2
2
2
2 1/ 2
Pa ( z )  P0 2
sin
[(



) z]
2
 
Pb ( z )  P0  Pa ( z )
Complete power transfer
 0
L   / 2
§11.8 Directional Coupling
For
 0
2
Fraction of power exchanged  2
  2
Identical waveguides
2h 2 pe  ps

 ( w  2 / p)( h 2  p 2 )
Channel waveguides
2h 2 pe  ps

w(h 2  p 2 )
§11.8 Directional Coupling
EO Switch
 0
The directional coupler is designed as:
So that
L   / 2
Pa (0)  0
Pa ( L)  P0
Pb (0)  P0
Pb ( L)  0
Now we apply electric field to control the phase such that:
  L  
2
So that
2
3 2
 L    L  
4
2
2
2
2
2
L 
Pa (0)  0
Pa ( L)  0
Pb (0)  P0
Pb ( L)  P0
3

2
§11.8 Directional Coupling
Multiplexing and Demultiplexing
§11.8 Directional Coupling
Multiguide directional coupler
dAn
 iAn 1  iAn 1
dz
An (0)  1
n0
An (0)  0
n0
An ( z )  (i) n J n (2z )