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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides

EE529 Semiconductor Optoelectronics

Optical Waveguides

• 1. Modes in planar waveguides
• 2. Ray-optics approach
• 3. EM-wave approach
• 4. Modes in channel waveguides

“Photonic Devices,” Jia-Ming Liu, Chapter 2 “Theory of Optical Waveguides,” by H. Kogelnik, in “Guided-wave Optoelectronics,” T. Tamir, ed., Chapter 2, Springer Verlag

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 2

Why study waveguides for integrated

• ptoelectronics?

Oxide insulator Stripe electrode

Substrate

Electrode Active region where J > Jth. (Emission region) p-GaAs (Contacting layer) n-GaAs (Substrate) p-GaAs (Active layer)

Current paths

L W Cleaved reflecting surface

Elliptical laser beam

p-AlxGa1-xAs (Confining layer) n-AlxGa1-xAs (Confining layer)

1 2 3

Cleaved reflecting surface Substrate

Semiconductor laser

V(t) L iN bO

3

E O Sub str ate A B In O u t C D A B W av eg u id e E le ctro de

V(t) L iN bO

3

In E lec trod e

W av eg u id es

F ib ers A B Lo

Modulator Directional coupler Photonic integrated circuits

Light: Science and Applications (2012)

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 3

E-M Field in a Planar Waveguide

Assuming a monochromatic wave propagating in z-direction:

t j z j t j

e e y x e t

ω β − ω =

= ) , ( ) ( ) , ( E r E r E ) ( ) ( ) (

2 2 2

= + ∇ r E r r E n k

Region I:

) , ( ) ( ) , (

2 2 2 2 2 2

= β − + ∂ ∂ y x E n k y x E x

Region II:

) , ( ) ( ) , (

2 2 3 2 2 2

= β − + ∂ ∂ y x E n k y x E x

Region III:

) , ( ) ( ) , (

2 2 1 2 2 2

= β − + ∂ ∂ y x E n k y x E x

Warm-up question: What kind of structure can be a waveguide?

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 4

Modes in a Planar Waveguide

Modal solutions are sinusoidal or exponential, depending on the sign of

) (

2 2 2

β −

i

n k

Boundary conditions:

x y x E y x E ∂ ∂ ) , ( and ) , (

must be continuous at the interface between layers. Assuming

1 3 2

n n n > >

, let’s draw possible waveguide modes:

β x n1 n2 n3 kn1 kn3 kn2 (The technique you learned from solving optical waveguide modes can be applied to the design of many photonic components.)

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 5

Guided Modes in a Planar Waveguide

m: Mode order Q: How to define the mode order? Q: Can we obtain an infinite number of solutions to β with continuous values? Examples of guided modes in a symmetrical waveguide

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 6

Experimental Observation of Waveguide Modes

Q1: How to choose the laser wavelength? Q2: How to create different modes? Q3: How to tell which side is air, which side is the substrate?

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 7

Do things in simple ways first. → Geometrical optics.

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 8

Ray Patterns in the Three-Layer Planar Waveguide

) sin( γ + ∝ hx E

In the guided region,

2 2 2 2 2

n k h = + β

For the m-th mode,

        β = θ

− m m

h

1

tan

Lower-order mode has smaller θm and larger βm. Remember that only discrete values of β are allowed. How to solve for allowable β? Step 1: Determine the relation between β and the angle of the optical ray. Different modes have different angles.

Overall propagation constant →Propagation constant in z →Propagation constant in x

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 9

Ray Patterns for Different Modes

β kn1 kn3 kn2 φ2

2 1 1 2 1 2

sin sin n n kn

− −

≤ β = φ

2 3 2 2 1

sin n n n n ≤ φ ≤

2 3 1 2

sin n n

−

≥ φ

Lower-order Higher-order

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 10

Reflection at a Dielectric Interface

2 2 1 1 2 2 1 1

cos cos cos cos θ + θ θ − θ = n n n n r

TE

TE TE

r t + =1

| | exp( ), | | exp( )

TE TE TE TM TM TM

r r j r r j = ϕ = ϕ

For TE wave:

1 2 1 3

, tE E rE E = =

2 1 1 2 2 1 1 2

cos cos cos cos θ + θ θ − θ = n n n n r

TM

) 1 (

2 1 TM TM

r n n t + =

For TM wave: Step 2: Determine phase changes at the interfaces.

Ref: Pedrotti3, “Introduction to Optics,” Sec. 23.1-23.3

→ Phase change accompanies reflection.

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 11

Phase Change on Total Internal Reflection

1 1 2 2 1 2 2 1 1 2 1 2

cos sin cos sin sin 2 tan θ − θ = θ θ − θ = φ n n n

c TE

2

1 1 2 2 1 2 2 1 2 2 2 1 2 1 2 1 2

cos sin sin cos sin sin 2 tan θ − θ = θ θ θ − θ = φ n n n n n

c c TM

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 12

Dispersion Equation

Transverse resonance condition:

π = φ − φ − θ m h kn

s c f

2 2 2 cos 2

θ cos h kn f

) ( 2

,TM TE c

φ = φ ) ( 2

,TM TE s

φ = φ m

: phase shift for the transverse passage through the film : phase shift due to total internal reflection from film/cover interface : phase shift due to total internal reflection from film/substrate interface : mode number

Dispersion equation (β vs. ω):

π = φ − φ − θ m h kn

s c f

cos

Effective guide index

θ = β ≡ sin

f

n k N

f s

n N n < <

Step 3: Define transverse resonance condition. → Solve for θ.

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 13

Graphical Solution

• f the Dispersion Equation

Symmetrical waveguide, φs = φc Asymmetrical waveguide, φs ≠ φc For a symmetrical waveguide, there is always a solution (no cutoff) for fundamental mode (m = 0). Increasing h (and/or decreasing λ) will support more modes.

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 14

Typical β – ω diagram

Cut-off β kns knf

Lower-order Higher-order

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 15

Numerical Solution for Dispersion Equation (I)

2 2 s f

n n kh V − ≡

Define: Normalized frequency and film thickness

2 2 2 2 s f s

n n n N b − − ≡

Normalized guide index

b = 0 at cut-ooff (N = ns), and approaches 1 as N → nf.

TM for TE, for

2 2 2 2 4 4 2 2 2 2 s f c s c f s f c s

n n n n n n a n n n n a − − ≡ − − ≡

Measure for the asymmetry

a = 0 for perfect symmetry (ns = nc), and a approaches infinity for strong asymmetry (ns ≠ nc, ns ~ nf).

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 16

b a b b b m b V − + + − + π = −

− −

1 tan 1 tan 1

1 1

Numerical Solution for Dispersion Equation (II)

For TE modes, dispersion relation

→ π = φ − φ − θ m h kn

s c f

cos

m = ν

: Mode number (Normalized) cut-off frequency:

π + = =

−

m V V a V

m 1

tan

# of guided modes allowed:

2 2

2

s f

n n h m − λ =

<Example> AlGaAs/GaAs/AlGaAs double heterostructure, n = 3.55/3.6/3.55. Determine a waveguide thickness supporting 0th, 0th and 1st order modes for λ = 1.55 µm..

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 17

The Goos-Hänchen Shift

s s

d z d ϕ = β

For TE modes

θ − =

−

tan ) (

2 / 1 2 2 s s

n N kz

For TM modes

        − + θ − =

−

1 tan ) (

2 2 2 2 2 / 1 2 2 f s s s

n N n N n N kz

The lateral ray shift indicates a penetration depth:

θ = tan

s s

z x

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 18

Effective Waveguide Thickness

Effective thickness:

c s eff

x x h h + + =

Normalized effective thickness:

2 2 s f eff

n n kh H − ≡ a b b V H + + + = 1 1

For TE modes:

Minimum H → Maximum confinement

Example: Sputtered glass, ns = 1.515, nf = 1.62, nc = 1, a = 3.9. Determine the minimum effective waveguide thickness.

Effective waveguide thickness cannot be zero, even for symmetrical waveguide (a = 0).

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 19

Ray-optic approach can solve for the effective index, but this is not good enough. Why?

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 20

Guided E-M Wave in a Planar Waveguide

Cover: Film: Substrate:

) , ( ) ( ) , (

2 2 2 2 2 2 2 2

= γ − ∂ ∂ → = β − + ∂ ∂ E E x y x E k n y x E x

c c

Define:

2 2 2 2 2 2 2 2 2 2 2 2 2 2 s s s f f c c c

k n k n k n γ − = β − = κ β − = κ γ − = β − = κ ) , ( ) ( ) , (

2 2 2 2 2 2 2 2

= κ + ∂ ∂ → = β − + ∂ ∂ E E x y x E k n y x E x

f f

) , ( ) ( ) , (

2 2 2 2 2 2 2 2

= γ − ∂ ∂ → = β − + ∂ ∂ E E x y x E k n y x E x

s s

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 21

TE Modes

) (

2 2 2

β −

i

n k

Modal solutions are sinusoidal or exponential, depending on the sign of Boundary conditions:

x E E

y y

∂ ∂ and

The tangential components of E and H are continuous at the interface between layers. → continuous at the interface.

Cover: Film: Substrate:

)] ( exp[

2 2 2

h x E E E E x

c c y y c y

− γ − = → = γ − ∂ ∂

For guided modes:

) cos(

2 2 2 s f f y y f y

x E E E E x φ − κ = → = κ + ∂ ∂

) exp(

2 2 2

x E E E E x

s c y y s y

γ = → = γ − ∂ ∂

Applying boundary conditions, we obtain:

f c c f s s

κ γ = φ κ γ = φ tan , tan π = φ − φ − κ m h

c s f

→ Dispersion relation

Ec, Ef, and Es can be determined by, Optical power

dx P z H E ˆ *} Re{ 2 1 ⋅ × = ∫

Optical confinement factor

∫ ∫

∞ ∞ −

⋅ × ⋅ × = Γ dx dx

h

z H E z H E ˆ *} Re{ 2 1 ˆ *} Re{ 2 1

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 22

TM Modes

Boundary conditions:

z y

E H and

Cover: Film: Substrate: )] ( exp[

2 2 2

h x H H H H x

c c y y c y

− γ − = → = γ − ∂ ∂

) cos(

2 2 2 s f f y y f y

x H H H H x φ − κ = → = κ + ∂ ∂

) exp(

2 2 2

x H H H H x

s c y y s y

γ = → = γ − ∂ ∂

Applying boundary conditions, we obtain:

f c c f c f s s f s

n n n n κ γ         = φ κ γ         = φ

2 2

tan , tan π = φ − φ − κ m h

c s f

→ Dispersion relation continuous at the interface between the layers

x H n H

y y

∂ ∂ →

2

1 and

continuous at the interface between the layers Relation between the peak fields:

1 , 1 ) ( ) ( ) (

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

−         +         = −         +         = − = − = −

c f c s f s c c c f c s s s f s f f f

n N n N q n N n N q n q n n H n q n n H n N n H

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 23

Multilayer Matrix Theory

Focusing on TE modes first,

z y

H V E U ωµ ≡ ≡ ,

)] exp( ) exp( [ ) exp( ) exp( x j B x j A V x j B x j A U κ − κ − κ = κ + κ − =

At x = 0,

) ( ), ( V V U U ≡ ≡

      ≡               κ κ κ κ κ κ =       V U V U x x j x j x V U M ) cos( ) sin( ) sin( ) cos(

M: Characteristic matrix of the layer

        κ κ κ κ κ κ = ) cos( ) sin( ) sin( ) cos(

i i i i i i i i i i i

h h j h j h M       =      

n n

V U V U M

n

m m m m M M M M  ⋅ ⋅ =       ≡

2 1 22 21 12 11

But β, and therefore κi, is unknown …

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 24

Dispersion Relation for Multilayer Slab Waveguides

Consider guided mode. For substrate and cover,

)] exp( ) exp( [ ) exp( ) exp( x B x A j V x B x A U γ − − γ γ = γ − + γ =

In the substrate,

s s s

A j V A U γ = = ,

In the cover,

c c n c n

A j V A U γ − = = ,

Using the multilayer stack matrix theory, we obtain:

12 21 22 11

) ( m m m m j

c s c s

γ γ − = γ + γ

→ Dispersion relation for multilayer slab waveguides

<Example> Four-layer waveguides

Q: What do we know about the fields in the substrate and cover?

MathCAD programs Effective index and modal distribution in 3-layer and 4-layer waveguides.

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 25

Multilayer Matrix Theory for TM Modes

12 2 2 21 2 22 2 11

) ( m n n m n m n m j

c s c s c c s s

γ γ − = γ + γ −

z y

E V H U , ωε ≡ ≡ )] exp( ) exp( [ ) exp( ) exp(

2

x j B x j A n V x j B x j A U κ − κ − κ − = κ + κ − =

Therefore,

      κ − → κ →

2

n TM TE

Dispersion relation:

            κ κ κ − κ κ − κ = ) cos( ) sin( ) sin( ) cos(

2 2 i i i i i i i i i i i i i

h h n j h n j h M

Characteristic matrix of the i-th layer:

)] exp( ) exp( [ ) exp( ) exp( x j B x j A V x j B x j A U κ − κ − κ = κ + κ − =

TE Modes: (except the phase terms)

12 21 22 11

) ( m m m m j

c s c s

γ γ − = γ + γ

        κ κ κ κ κ κ = ) cos( ) sin( ) sin( ) cos(

i i i i i i i i i i i

h h j h j h M

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 26

What is Next?

What we learned cannot be practical waveguides.

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 27

Channel Waveguide Structures

Q: Can you think of more?

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 28

The Method of Field Shadows (I)

Ignore the fields and refractive indices in the shaded field shadow regions. → Results in separable index profiles. Works well as long as the fields are well confined in the high index (nf) region of the waveguide. → Not applicable at cutoff.

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 29

The Method of Field Shadows (II)

) ( ) ( ) , ( y Y x X y x E =

2 2 2 y x

β + β = β

2 2 2 y x

N N N + =

2 2 2 2 s f y s f x

n n kw V n n kh V − = − =

Assuming a buried channel waveguide structure. Obtain Nx and Ny, therefore N, by using the dispersion relation chart and

2 2 2 2 2 2 2 2 2 2

/ 2 / 2

x s f x f s y s f y f s

N n n b n n N n n b n n − + = − − + = −

Or instead of solving for Nx and Ny, we can use

1

2 2 2 2

− + = − − =

y x s f s

b b n n n N b

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 30

Method of Field Shadows Exercise

Ti:LiNbO3 buried channel waveguide for λ = 0.8 µm

nf = 2.234 ns = 2.214

Determine effective index and modal distribution

2µm 1µm

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 31

The Effective-Index Method

(1) Determine the normalized thickness of the channel and lateral guides.

2 2 2 2

,

s f l s f f

n n kl V n n kh V − = − = (2) Use the dispersion relation chart to determine the normalized guide indices bf and bl. Determine the corresponding effective indices.

) (

2 2 , 2 2 , s f l f s l f

n n b n N − + =

(3) Determine the normalized width.

2 2 l f eq

N N kw V − =

Then determine the normalized guide index beq using the dispersion relation chart. (4) The effective index of the waveguide can be determined from

) (

2 2 2 2 2 2 2 2 l f eq l l f l eq

N N b N N N N N N b − + = → − − =

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 32

Effective-Index Method Exercise

Ti:LiNbO3 rib waveguide for λ = 0.8 µm, 2.234

f

n = , 2.214

s

n = , 1

c

n = , 1.8 h = µm, 1 l = µm, 2 w = µm. Determine the effective index.

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 33

Effective Index Parameters for Channel Waveguides

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 34

Numerical Comparisons Between Different Methods

Effective index method provides good approximation even near cut-off.

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Waveguides 35

To Complete the Story …

A: Use the matrix method from Multilayer Stack Theory to determine Nf and Nl, then continue on Step (3) in the Effective Index Method. Q1: What about waveguides which have more than 3 layers in each region?

Al0.35Ga0.65As, ns=3.4 Al0.25Ga0.75As, n1=3.45 0.5 µm x = 0 Air, nc=1 0.5 µm 0.3 µm GaAs, n2=3.6 x y y = 0

Q2: What about 2-D modal profile in the waveguide?

, , , 2 2 2 , , ,

• r

,

• r

,

• r

2 2 2 ,

• r
• r

2 2 2 2

exp( ) exp( ) ( ) ( ) exp( ) exp( ) ( ) ( ) ( ) ( )

x i x i x i x i i f l y f l y f l y f l y f l f l x y i

E A j x B j x kn kN E C j y D j y kN kN kn kN = − κ + κ κ = − = − κ + κ κ = − κ + κ = −