chapter 6 optical fibers and guiding layers
DESCRIPTION
Chapter 6 OPTICAL FIBERS AND GUIDING LAYERS. ◈ The dielectric slab guide (Waveguide) ▪ Wave equation (Governing eq.):. TIR. ▪ Solution:. ▪ Direction separation: TE & TM. Transverse Electric (TE) Modes (1/3). ▪ TE field: ▪ Wave equation (previous): - PowerPoint PPT PresentationTRANSCRIPT
Advanced Optoelectronics (13/2) Geon Lim
▪ Solution:
Chapter 6 OPTICAL FIBERS AND GUIDING LAYERS
◈ The dielectric slab guide (Waveguide)▪ Wave equation (Governing eq.):
x d
0x
x d
x
z
0,
0,i
0, TIR
( )i
22
2
, ,, ,
E x z tE x z t
t
,z, t , j tE x E x z e
2 20( )k
22 , , 0E x z k x E x z
0
0
for
for i x d
k xx d
▪ Direction separation: TE & TM
-1-
Advanced Optoelectronics (13/2) Geon Lim
▪ TE field:
▪ Wave equation (previous):
▪ We can get the Eigen-value equation:
Transverse Electric (TE) Modes (1/3)
ˆ, j zyE x z yE e
22 , , 0E x z k x E x z
x d
0x
x d
x
z
0,
0,i
0, TIR
( )i
2
2 22 0y
y
d E xk x E x
dx Each eigenfunction has one eigenvalue
associated with it, ie, eigenfunctions and eigenvalues come in pairs .
jf x j
,j jf x ▪ Considering : 2 2sign k x
2 2
2 2
0 for core
0 for cladding
k x x d
k x x d
▪ For core, we select a symmetric solution:
cos
x
x
x
xy
x
A k x x d
E x Be x dx dBe
0
0
0
sin
x
x
x
xz
x
j A k x x d
jH x Be x d
j Be x d
0
yz
E zjHx
2 2 20x
2 2 20x ik
-2-
Advanced Optoelectronics (13/2) Geon Lim
Transverse Electric (TE) Modes (2/3)
▪ To match the boundary condition, the impedance should be continuous (at the interface):
continuityy
x
EH
tan (even solution case)xx
x
k dk
tan (odd solution case)2
xx
x
k dk
/x xk moves toward the originand intersections are lost
▪ All higher-order modes (m>0) have a cutoff Waves are not guided below a certain critical frequency
-3-
Advanced Optoelectronics (13/2) Geon Lim
▪ Let (Normalized term), then the previous solutions are represented as: - even case: - odd case:
Transverse Electric (TE) Modes (3/3)
▪ [Ex]Higher mode xk
1m
2 1m m
xX k d xY dtanY X X tan / 2Y X X
2 2 2 2 2 2 2 20x x iX Y d k d r
xX k d
xY d -- Even-- Odd
rm=0
m=1
m=2
▪ Graphical representation - Discrete # of the TE solutions (modes)- - Mode depends on the radius of the circle
2 2 20 ir d
,x x yk E x
-4-
Advanced Optoelectronics (13/2) Geon Lim
Dispersion diagram for TE waves in dielectric guide
2 2 20x ik
Higher mode Less β
-5-
Advanced Optoelectronics (13/2) Geon Lim
Numerical/Graphical representation
▪ Field profile of dominant mode for three different frequencies
▪ Dominant TE mode
-6-
Advanced Optoelectronics (13/2) Geon Lim
Additional comprehension for waveguide
E(y) profile: n1=1.5, n2=1.495, d=10m, =1m
TE1 TE2
TE3 Even function solution Odd function solution
Even function solution
TIR backward and forward in x-direction: Standing wave case
x
m → x
E or energy penetrates (leaks) at the boundary
x
x
Core
Cladding
-7-
Advanced Optoelectronics (13/2) Geon Lim
Additional comprehension for waveguide
22
2
2
( )Power inside core
Total Power( )
dy
dy
y
y
E y dy
E y dy
- How does change for different modes? ▪ Confinement factor: How much power is confined within the core
x
+ +( ) ~ ( )in m m
m
E y a E yn2
n1
n2
( )inE y
- Discrete modes Summation of the solutions
▪ Partitioning of input field into different guided modes.xX k d
xY d -- Even-- Odd
rm
m → x
Energy penetrates (leaks) at the boundary
→
-8-