Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

Guidance condition in a waveguide

2

22

0

( 1) ( 1) 1

2 2 and

(= ) ( )

y y yjk d jk d j k d

y y

z

e e emk d m kd

mk nkd

d

Metallic waveguide

,/ / ,/ / 1y yjk d jk de r e r

n2 cladding

d n1

n2 <n1

core

cladding

Dielectric waveguide (TIR)

,//,/ /Since ,jr e

,/ /2 2 2yk d m

,//2 2 1yj k d je e

,/ /Or yk d m

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

d

n2

n1

n2 <n1

core ,/ / yk d m

Numerically solve for ky (TE, TM)

12 2 2

12 2 2||

/ / 2

Remember

(sin ) : tan( ) (TE)2 cos

(sin ): tan( ) (TM)2 cos

i

i

i

i

n

nn

2 2yk 2 2

1 0( ) yn k k

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

Full analysis starting from wave equations.

n2

d n1

n2

y=0

y=d/2

y=-d/2

y

z

22

2( , , ) ( , , )EE y z t y z tt

2 0

1 0

( ) for y ; cladding2

( ) for y ; core2

dk y n k

dk y n k

Assuming , , , , j tE y z t E y z e

2 2 2 2( ) 0, where ( ) ( )E k y E k y y

What are field profiles for guided modes?

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

Consider TE Solution.

d

n2

n1

n2

y=0

y=d/2

y=-d/2

y

z

( , ) ( ) j zE y z x E y e

2 2In core, ( ) 0 k y

22 2

2

( ) ( ( ) ) ( ) 0d E y k y E ydy

=> Eigen value equation. Solve for and ( ).E y

2 2In cladding, ( ) 0k y

2 2From ( ) 0,E k y E

=> ( ) ~ sin( ) or cos( )y yE y k y k y

2 2Sign of ( ( ) ) determines the solution type.k y 1 0 2 0We know n k n k

2 2 2( ) ,yk k y 2 21 0 ( )yk n k

=> ( ) ~ exp( ) or exp(- )E y y y 2 2 2( ) ,k y 2 2

2 0( )n k

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

d

n2

n1

n2

y=0

y=d/2

y=-d/2

y

z

Solutions

: ( ) exp( ) exp( )2

| | : ( ) sin( ) cos( )2

: ( ) exp( ) exp( )2

y y

dy E y A y B y

dy E y C k y D k y

dy E y E y F y

A=0 and F=0

For easy analysis, divide the solutions into even and odd solutions

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

d

n2

n1

n2

y=0

y=d/2

y=-d/2

y

z

Even Solutions

: ( ) exp( )2

| | : ( ) cos( )2

: ( ) exp( )2

( )

y

dy E y B y

dy E y D k y

dy E y B y

E B

Odd Solutions

: ( ) exp( )2

| | : ( ) sin( )2

: ( ) exp( )2

( )

y

dy E y B y

dy E y D k y

dy E y B y

E B

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

d

n2

n1

n2

y=0

y=d/2

y=-d/2

y

z

Even Solutions

Apply boundary conditions:

: ( ) exp( )2

| | : ( ) cos( )2

: ( ) exp( )2

( )

y

dy E y B y

dy E y D k y

dy E y B y

E B

At ,2dy

( )( ) and are continuous at 2

dE y dE y ydy

(2) : tan( )(1) 2y y

dk k

exp( ) cos( ) ------- (1)2 2yd dB D k

exp( ) sin( ) ----- (2)2 2y yd dB k D k

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

d

n2

n1

n2

y=0

y=d/2

y=-d/2

y

z

Odd Solutions

: ( ) exp( )2

| | : ( ) sin( )2

: ( ) exp( )2

( )

y

dy E y B y

dy E y D k y

dy E y B y

E B

At ,2dy

Apply boundary conditions.( )( ) and are continuous at

2dE y dE y y

dy

exp( ) sin( ) ------- (1)2 2yd dB D k

(2) :(1)

exp( ) cos( ) ----- (2)2 2y yd dB k D k

cot( )2y ydk k tan( )

2 2y ydk k

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

d

n2

n1

n2

y=0

y=d/2

y=-d/2

y

zEven: tan( )

2y ydk k

For easier interpretation, do following normalization.

Let ,2 2yd dX k Y

Plot (1), (2), (3) on X-Y plane!

Odd: tan( )2 2y ydk k

Determine and that satisfy above conditions.yk

Then, tan for even (1)Y X X tan( ) for odd (2)2

Y X X

2 2 X Y 2

2 2 2 21 0 2 0[( ) ] [ ( ) ]

2d n k n k

22 2( )

2 yd k

22 2 2

0 1 2( )2d k n n

2 2 2 (3)X Y r

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

tan : even modeY X X

tan( / 2) : odd modeY X X

2ydX k

2dY

2 3

2

Observations:

22 2 2 2 2 2

0 1 2( )2dX Y k n n r

m=1

m=3

m=2

- Points where circle and tangent curvesintersect are solutions: ky, ,

With larger r, more modes exist.

- There is at least one even TE mode.

- Even, odd, even, odd …

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

E(y) profile: n1=1.5, n2=1.495, d=10m, =1m

TE1 TE2

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

E(y) profile: n1=1.5, n2=1.495, d=10m, =1m

TE1 TE2TE3

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

How much power is confined within the core

22

2

2

( )Power inside core

Total Power( )

dy

dy

y

y

E y dy

E y dy

How does change for different modes?

- Effective index: neff= /k0

d n1

n2

n2

- Confinement factor:

How does neff changes for different modes?

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

+ +( ) ~ ( )in m m

mE y a E y

n2

n1

n2

( )inE y

For , use the fact that ( )'s are orthogonal.m ma E y

Partitioning of input field into different guided modes.

( ) ( )in mE y E y dy ~ ( ) ( )n n mn

a E y E y dy 2 ( )m ma E y dy

2

( ) ( )

( )in m

mm

E y E y dya

E y dy

Dot product between Ein(y) and Em(y)Or projection of Ein(y) into basis Em(y)

(Sturm-Liouville theory)

( ) ( ) 0 if m nE y E y dy m n

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

b-V diagram for TE mode

V

b

12 2 20 1 2( )

(Normalized )V k d n n

k

d

n3

n1 core

cladding

n2

Asymmetric waveguide?

- Graphical solution

2 22 32 2

1 2

(Asymmetry factor)

n nan n

22

202 2

1 2

(Normalized )nk

bn n

For a given waveguide: V,a b from the diagram

Then, determine

- Numerical solution

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

m

Slope = c/n2

Slope = c/n1

TE1

cut-off

TE2

TE3

Schematic dispersion diagram, vs. for the slab waveguide for various TE m. modes.cut–off corresponds to V = /2. The group velocity vg at any is the slope of the vs. curve at that frequency.

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

- diagram

- For a given , phase velocity (=/) and group velocity (=d/d) can be determined.

- Group velocity determines the speed of information transfer

- Multi-mode waveguides suffers from dispersion since different modes have different group velocities

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

Homework

Lect. 5: Dielectric Waveguides

Special Topics in Photonic Devices (14/1) W.-Y. Choi

Homework: