chapter 6 optical fibers and guiding layers

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-


▪ 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-



▪ 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-


▪ Let (Normalized term), then the previous solutions are represented as: - even case: - odd case:


▪ [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-


Dispersion diagram for TE waves in dielectric guide

2 2 20x ik

Higher mode Less β

-5-


Numerical/Graphical representation

▪ Field profile of dominant mode for three different frequencies

▪ Dominant TE mode

-6-


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-


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-

chapter 6 optical fibers and guiding layers

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