1. Chapter 6 OPTICAL FIBERS AND GUIDING LAYERS
◈ The dielectric slab guide (Waveguide) ▪ Wave equation (Governing eq.):
TIR
▪ Solution:
▪ Direction separation: TE & TM

2. Transverse Electric (TE) Modes (1/3)
▪ TE field: ▪ Wave equation (previous): ▪ We can get the Eigen-value equation:
TIR
Each eigenfunction has one eigenvalue associated with it, ie, eigenfunctions and eigenvalues come in pairs .
▪ Considering :
▪ For core, we select a symmetric solution:

3. Transverse Electric (TE) Modes (2/3)
▪ To match the boundary condition, the impedance should be continuous (at the interface):
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

4. Transverse Electric (TE) Modes (3/3)
-- Even
-- Odd r
▪ Let (Normalized term), then the previous solutions are represented as:
- even case: odd case:
▪ Graphical representation
- Discrete # of the TE solutions (modes) -
- Mode depends on the radius of the circle
m=1
m=0
m=2
▪ [Ex]Higher mode  

5. Dispersion diagram for TE waves in dielectric guide
Higher mode  Less β

6. Numerical/Graphical representation
▪ Field profile of dominant mode for three different frequencies
▪ Dominant TE mode

7. Additional comprehension for waveguide
E(y) profile: n1=1.5, n2=1.495, d=10mm, l=1mm
TE1
TE2
Core x
Cladding
Even function solution
Odd function solution
TE3 →
 E or energy penetrates (leaks) at the boundary
Even function solution
 TIR backward and forward in x-direction: Standing wave case

8. Additional comprehension for waveguide
-- Even
-- Odd r
▪ Confinement factor: G
- How much power is confined within the core
- How does G change for different modes? →
 Energy penetrates (leaks) at the boundary
▪ Partitioning of input field into different guided modes.
- Discrete modes  Summation of the solutions + n2 n1

9. Supplement studying materials

10. TE1
TE2
(Haus Fig. 6.4)

11. E(y) profile: n1=1.5, n2=1.495, d=10mm, l=1mm
TE1
TE2

12. E(y) profile: n1=1.5, n2=1.495, d=10mm, l=1mm
TE1
TE2
TE3

13. How does neff changes for different modes?
- Effective index:
neff= b/k0 d n1 n2
How does neff changes for different modes?
- Confinement factor: G
How much power is confined within the core
How does G change for different modes?

14. + + Partitioning of input field into different guided modes.
(Sturm-Liouville theory)
Dot product between Ein(y) and Em(y)
Or projection of Ein(y) into basis Em(y)

15. For a given waveguide: V,a  b from the diagram
cladding
Asymmetric waveguide? d n1
core
- Numerical solution n2
- Graphical solution
b-V diagram for TE mode V b
For a given waveguide: V,a  b
from the diagram
Then, determine b
(Haus 6.11)