Download Geon Lim

Chapter 6 OPTICAL FIBERS AND GUIDING LAYERS
◈ The dielectric slab guide (Waveguide)
▪ Wave equation (Governing eq.):
 2 E  x, z , t 
 E  x, z, t   
t 2
xd
TIR
x
x0
2
x  d
z
-1-
 , 0
 i , 0
 , 0
( i   )
▪ Solution:
E  x,z, t   E  x, z  e jt
(k 2   2 0 )
 E  x, z   k  x  E  x, z   0
2
2
  i 0
k  x  
 0
for x  d
for x  d
▪ Direction separation: TE & TM
Advanced Optoelectronics (13/2)
Geon Lim
Transverse Electric (TE) Modes (1/3)
▪ TE field:
xd
ˆ y e j  z
E  x, z   yE
2 E  x, z   k  x  E  x, z   0
2
▪ We can get the Eigen-value equation:
d 2 Ey  x 
dx
2


 k  x    2 Ey  x   0
2


▪ Considering sign k  x 2   2 :
k  x   2  0
2
for x  d  core 
k  x    2  0 for x  d  cladding 
2
▪ For core, we select a symmetric solution:
 A cos k x x

E y  x    Be x x
 Be x x

TIR
x
x0
▪ Wave equation (previous):
x d
xd
x  d
Advanced Optoelectronics (13/2)
x  d
-2-
z
 , 0
 i , 0
 , 0
( i   )
Each eigenfunction f j  x  has one eigenvalue  j
associated with it, ie, eigenfunctions and eigenvalues
come in pairs  f j  x  ,  j  .
 j
A sin k x x x  d


0

 j
H z  x   
Be x x x  d
 0
 j
Be x x x  d

j E y  z 
 0
Hz 
0 x
 2   x2   2 0
 2  k x2   2 0 i
Geon Lim
Transverse Electric (TE) Modes (2/3)
-3-
▪ To match the boundary condition, the impedance should be continuous (at the interface):
Ey
Hx
continuity
tan  k x d  
x
(even solution case)
kx
 

tan  k x d    x (odd solution case)
2  kx


 x / k x moves toward the origin
and intersections are lost
▪ All higher-order modes (m>0) have a cutoff
 Waves are not guided below a certain critical frequency
Advanced Optoelectronics (13/2)
Geon Lim
Transverse Electric (TE) Modes (3/3)
-4-
▪ Let X  k x d Y   x d (Normalized term), then the previous solutions are represented as:
- even case: Y  X tan X
- odd case: Y  X tan  X   / 2 
-- Even
Y   xd
X 2  Y 2  d 2  k x2   x2   d 2 2 0   i     r 2
▪ Graphical representation
- Discrete # of the TE solutions (modes)
- k x , x    E y  x 
- Mode depends on the radius of the circle r 2  d 2 2 0  i   
r
-- Odd
m=1
m=0
m=2
X  kxd
m1
▪ [Ex]Higher mode  k x      


m2   m1 
Advanced Optoelectronics (13/2)
Geon Lim
Dispersion diagram for TE waves in dielectric guide-5 2  k x2   2 0 i
Higher mode  Less β
Advanced Optoelectronics (13/2)
Geon Lim
Numerical/Graphical representation
▪ Field profile of dominant mode for three different
frequencies
Advanced Optoelectronics (13/2)
-6-
▪ Dominant TE mode
Geon Lim
Additional comprehension for waveguide
-7-
E(y) profile: n1=1.5, n2=1.495, d=10m, =1m
TE1
TE2
Core
x
Even function solution
x 
Odd function solution
Cladding
x 
TE3
m  → x 
Even function solution
 E or energy penetrates (leaks) at
the boundary
 TIR backward and forward in x-direction: Standing wave case
Advanced Optoelectronics (13/2)
Geon Lim
Additional comprehension for waveguide
▪ Confinement factor: 
 How much power is confined within the core
y
d
2

Power inside core


Total Power
-8-
- How does  change for different modes?
-- Even
-- Odd
Y   xd
2
E ( y ) dy
d
2
y 
y 

m  → x 
m
2
E ( y ) dy
y 
r
x 
▪ Partitioning of input field into different guided modes.
 Energy penetrates (leaks) at
the boundary
→ 
X  kxd
- Discrete modes  Summation of the solutions
n2
Ein ( y)
n1
+
+
Ein ( y ) ~  am Em ( y )
m
n2
Advanced Optoelectronics (13/2)
Geon Lim