Waveguide Basics - High-Speed Circuits & Systems Laboratory

Waveguide Basics

Optical Fiber: Circular dielectric waveguide made of silica (SiO2)

y

Cladding

y

Core zrφ

Fiber axisSiO2:Ge

nn2 n1

Loss in rectangular metal waveguide

The step index optical fiber. The central region, the core, has greater refractiveindex than the outer region, the cladding. The fiber has cylindrical symmetry. Weuse the coordinates r, φ, z to represent any point in the fiber. Cladding isWhat is special about fiber?

E t l l l : 0 2dB/kuse the coordinates r, φ, z to represent any point in the fiber. Cladding isnormally much thicker than shown.

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

- Extremely low loss: 0.2dB/km- Can be very long: 100’s of km

Special Topics in Optical Comm.: Si Photonics (11/1) W.-Y. Choi

Waveguide Basics

Loss in fiber

10

5

10

OH- absorption peaks

1.0Loss(dB/km)

0.5Latticeabsorption

Rayleighscattering

1550 nm

0.05

0.11310 nm

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Wavelength (λ)


Waveguide Basics

Buffer tube: d = 1mmr

Protective polymerinc coatingCladding: d = 125 - 150 μm

Core: d = 8 - 10 μmn

n1

for single-mode fiber

n2 (For multi-mode fiber, d: several tens of μm)

The cross section of a typical single-mode fiber with a tight buffertube. (d = diameter)Optical Power Distribution

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)In Single-Mode Fiber

What confines light inside core?What confines light inside core?


Waveguide Basics

tθ 2ntE ( 1 44)=

θ θ

t

n

2

inE E ( 1 0)=

( 1.44)=

iθ rθ 1ninE rE ( 1.0)=

12 2 2cos sinnE θ θ⎡ ⎤⎣ ⎦

12 2 22sin cosn nθ θ⎡ ⎤⎣ ⎦

12 2 2

cos sin

cos sin

i ir

ii i

nErE n

θ θ

θ θ⊥

⎡ ⎤− −⎣ ⎦= =⎡ ⎤+ −⎣ ⎦

|| 12 2 22

sin cos

sin cos

i i

i i

n nr

n n

θ θ

θ θ

⎡ ⎤− −⎣ ⎦=⎡ ⎤− +⎣ ⎦

21

2 2 2 1

2cos ( )cos sin

t i

ii i

E nt nE nn

θ

θ θ⊥ = = =

⎡ ⎤+ −⎣ ⎦

|| 12 2 22

2 cos

sin cosi

i i

ntn n

θ

θ θ=

⎡ ⎤− +⎣ ⎦


i i⎣ ⎦

Waveguide Basics

tθ2 1.0n =tE

θ θ

t

inE Eiθ rθ

1 1.44n =

inE rE

1

1⎡ ⎤

What happens at 1sin ?c nθ −=

12 2 2

12 2 2

cos sin

cos sin

i ir

ii i

nErE n

θ θ

θ θ⊥

⎡ ⎤− −⎣ ⎦= =⎡ ⎤+ −⎣ ⎦

r becomes complex !( magnitude 1, phase shift)


Waveguide Basics

tθ2 1.0n =tE

θ θ

t

inE rEiθ rθ

1 1.44n =

rE

1


Waveguide Basics

d

n2

0

y=d/2

y

z3 l di l t i uid d n1

n2

y=0

y=-d/2

z3-layer dielectric waveguide

Full analysis starting from wave equations.

22

2( , , ) ( , , )EE y z t y z tμε ∂∇ =

∂ 2( , , ) ( , , )y yt

μ∂

( ) ( )Assuming , , , , j tE y z t E y z e ω= ⋅

2 2 2 2( ) 0 h ( ) ( )k k

( ) for y ; claddingdk y n k= >

2 2 2 2( ) 0, where ( ) ( )E k y E k y yμε ω∇ + = =

2 0

1 0

( ) for y ; cladding2

( ) for y ; core2

k y n k

dk y n k

= >

= <


1 0( ) y2

y

Waveguide Basics

d

n2

n1 y=0

y=d/2 y

z

Consider TE Solution.

n2

y=-d/2

( , ) ( ) j zE y z x E y e β−=2 ( )d E y 2 2

2

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

β+ − =

=> Eigen value equation Solve for and ( )E yβ

2 2 2 21 0 ( ) 0 in core => ( ) ~ sin( ) or cos( ) with ( )y y yk y E y k y k y k n kβ β− > = −

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

1 0( ) ( ) ( ) ( ) ( )y y yy y y yβ β

2 2 2 22 0( ) 0 in cladding => ( ) ~ exp( ) or exp(- ) with ( )k y E y y y n kβ α α α β− < = −


Waveguide Basics

d

n2

n1 y=0

y=d/2 y

z

n2

y=-d/2

S l tiSolutions

: ( ) exp( ) exp( )2dy E y A y B yα α> = + −

| | : ( ) sin( ) cos( )2 y ydy E y C k y D k y

d

< = +

: ( ) exp( ) exp( )2dy E y E y F yα α< − = + −

Here, A=0 and F=0.For easy analysis, divide the solutions into even and odd solutions


Waveguide Basics

d

n2

n1 y=0

y=d/2 y

z

n2

y=-d/2

E S l tiEven Solutions Apply boundary conditions:( )( ) and are continuous at

2dE y dE y y

d= ±

( ) ( )d E B 2

exp( ) cos( ) ------- (1)2 2y

dyd dB D kα− =

: ( ) exp( )2

| | : ( ) cos( )2 y

dy E y B y

dy E y D k y

α> = −

< =

exp( ) sin( ) ----- (2)2 2

(2)

y yd dB k D k

d

α α− − = −2

: ( ) exp( )2

( )

dy E y B y

E B

α< − =

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

dk kα==> =( )E B=


Waveguide Basics

d

n2

n1 y=0

y=d/2 y

z

n2

y=-d/2

Odd Solutions Apply boundary conditions.( )( ) and are continuous at dE y dE y y = ±d ( ) d e co uous

2

exp( ) sin( ) ------- (1)2 2y

y ydy

d dB D kα− =

: ( ) exp( )2

| | : ( ) sin( )2 y

dy E y B y

dy E y D k y

α> = −

< = 2 2

exp( ) cos( ) ----- (2)2 2

(2)

y yd dB k D k

d d

α α

π

− − =

| | ( ) ( )2

: ( ) exp( )2

yy y y

dy E y B yα< − = −

(2) cot( ) tan( )(1) 2 2 2y y y y

d dk k k k πα==> = − = −( )E B= −


Waveguide Basics

d

n2

n1 y=0

y=d/2 y

z

n2

y=-d/2

What do these mean?

Even: tan( )2

Odd: tan( )

y ydk k

dk k

α

πα

=

=What do these mean? Odd: tan( )2 2

Determine and that satisfy above conditions.

y y

y

k k

k

α

α

= −

For graphical analysis do following normalizationFor graphical analysis, do following normalization.

Let ,2 2

Then tan for even

yd dX k Y

Y X X

α= =

=

2 2

Then, tan for even

tan( ) for odd2

Y X X

Y X X

d d

π=

= −

⎛ ⎞ ⎛ ⎞

Plot these on X-Y plane.

2 2 2 2 2 2 2 21 0 2 0

22 2 2 2

0 1 2

But ( ) [( ) ( ) ]2 2

( )

yd dX Y k n k n k

d k n n r

α β β⎛ ⎞ ⎛ ⎞+ = + = − + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= − =⎜ ⎟⎝ ⎠


0 1 2( )2⎜ ⎟

⎝ ⎠

Waveguide Basics

tan : even modeY X X=

tan( / 2) : odd modeY X X π= −

2dY α= ⋅

tan( / 2) : odd modeY X X π= −2

2 2 2 2 2 20 1 2( )

2dX Y k n n r⎛ ⎞+ = − =⎜ ⎟

⎝ ⎠m=1

2

Observations:- Points where circle and tangent curvesm=3

m=2

2ydX k=

2π π 3

2π

intersect are solution points

for ky and α (TE mode).− With larger r (larger d, smaller λ, larger

2 2) d i tn12-n2

2), more modes exist.- There is at least one even mode.- Even, odd, even, odd …


Waveguide Basics

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

TE1 TETE1 TE2TE3


Waveguide Basics

Wave is not entirely confined within core: Confinement factor

dy=2

2

2

( )Power inside core

y

dy

E y dy=−

Γ = =

∫

2Total Power( )

y

y

E y dy=∞

=−∞

Γ = =

∫

For higher modes, how does Γ change?


Waveguide Basics

ωSlope = c/n2

Sl /Slope = c/n1

TE2

TE3

ωcut-off

TE2

βm

TE1

Schematic dispersion diagram, ω vs. β for the slab waveguide for various TE m. modes.ωcut–off corresponds to V = π/2. The group velocity vg at any ω is the slope of the ω vs. βGroup velocities are different for different modes => modal dispersioncut off p g p y g y p βcurve at that frequency.

© 1999 S O K O l i (P i H ll)

Group velocities are different for different modes > modal dispersionNeed a single-mode waveguide in order to avoid signal distortion.

How do you design a single mode waveguide?


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

Waveguide Basics

b-V diagram for TE mode

d

n3

n1 core

cladding

12 2 2( )k d

co e

n2

b

2 2 20 1 2( )

(Normalized )V k d n n

k= −

b22

202 2

1 2

nkb

n n

β⎛ ⎞ −⎜ ⎟⎝ ⎠=

−1 2

2 2

(Normalized )n n

β

V

2 22 32 2

1 2

(Asymmetry factor)

n nan n

−=

−


V

Waveguide Basics

Issues for practical waveguidesp g

- Precise control of dimension and refractive index - Low loss at desired λ- Mass production possible- Integration desirable (Integrated Optics)- Electrical control of refractive index (Electro-Optic effect)

Materials used for waveguidesMaterials used for waveguides

- Silica Optical fiber- Semiconductors: GaAlAs, InGaAsP, Si/SiO22

- Dielectric materials: LiNbO3 with Ti doping


Waveguide Basics

LiNbO3 waveguide

V(t)

d

Thin buffer layerCoplanar strip electrodes

Polarized

Ea LiNbO3

dPolarizedinputlight

L

Cross-sectionEO Substratez

y

x WaveguideLiNbO 3

Integrated tranverse Pockels cell phase modulator in which a waveguide is diffusedinto an electro-optic (EO) substrate. Coplanar strip electrodes apply a transversefield Ea through the waveguide. The substrate is an x-cut LiNbO3 and typically thereis a thin dielectric buffer layer (e.g. ~200 nm thick SiO2) between the surfaceelectrodes and the substrate to separate the electrodes away from the waveguide.?1999 S.O. Kasap, Optoelectronics (Prentice Hall)


p, p ( )

Waveguide Basics

Si/SiO waveguide


Waveguide Basics

Problem 1


Waveguide Basics

Problem 2


Waveguide Basics

Problem 3


Waveguide Basics - High-Speed Circuits & Systems Laboratory

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