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Lih Y. Lin

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EE 539B

Integrated Optics and Nanophotonics

1 Theory of Optical Waveguides1.1 Modes in planar waveguides

1.2 Ray-optic approach to optical waveguidetheory

Guided-wave Optoelectronics, T. Tamir, ed., Sec. 2-1, Springer Verlag.

Integrated Optics, by R. G. Hunsperger, Chapter 2, Springer Verlag.

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E-M Field in a Planar Waveguide

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

Assuming a monochromatic wave propagating in z-direction:tjzjtj eeyxet == ),()(),( ErErE

(We will deal with some Maxwells

equations, but dont be afraid of this.)0)()()( 222 =+ rErrE nk

Region I:

0),()(),( 2222

2

2

=+

yxEnkyxE

x

0),()(),( 2212

2

2=+

yxEnkyxEx

Region II:

0),()(),( 2232

2

2

=+ yxEnkyxEx

Region III:

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Modes in a Planar Waveguide

Modal solutions are sinusoidal or exponential, depending on the sign of )( 222 inkBoundary conditions:

x

yxEyxE

),(and),(

132 nn >>

must be continuous at the interface between layers.

Assuming n , lets draw possible waveguide modes:

kn3 kn2kn10x

n1

n2

n3

(The technique you learned from solving optical waveguide modes can be applied

to the design of many photonic components.)

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Guided Modes in a Planar Waveguide

Examples of guided modes in a symmetrical waveguide.

m: Mode order

Q: How to define the mode order?

Only discrete values of are allowed in a waveguide.(This is the center of the optical waveguide theory.)

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Experimental Observation of Waveguide Modes

Q1: How to choose the laser wavelength?

Q2: How to create different modes?

Q3: How to measure the modal profile?

Q4: How to tell which side is air, which side is the substrate?

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Do things in simple ways first.

Geometrical optics.

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Ray Patterns in the

Three-Layer Planar Waveguide

Remember that only discrete values of are allowed. How to solve for allowable ?Step 1:

Determine the relation between and the angle of the optical ray. Different modeshave different angles.

)sin( + hxE 22

222 nkh =+

For the m-th mode,

=

m

m

h1tan

Overall

propagation

constant

Propagation constant in z

Propagation constant in x

In the guided region,

Lower-order mode has smaller m and larger m.

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Ray Patterns for Different Modes

2

2

11

2

12 sinsin

n

n

kn

=

2

312 sin

nn

2

32

2

1 sinnn

nn

kn3

kn2

kn10 Higher-order

Lower-order

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Reflection and Refraction

2211

2211

coscos

coscos

+

=nn

nnrTE TETE rt +=1

1213 , tEErEE ==

Step 2:

Determine phase changes at the interfaces.

For TE wave:

)exp(||,)exp(|| TMTMTMTETETE jrrjrr ==2112

2112

coscos

coscos

+

=nn

nnrTM )1(

2

1TMTM r

n

nt +=For TM wave:

Phase change accompanies reflection.

Ref: Saleh and Teich, Fundamental of Photonics, Sec. 6-2.

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Total Internal Reflection for TE Wave

11

221

221

1

21

2

cos

sin

cos

sinsin

2tan

=

=

n

nncTE

2

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Total Internal Reflection for TM Wave

11

221

221

22

21

21

21

2

cos

sin

sincos

sinsin

2tan

=

=

n

nn

n

n

c

cTM

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Dispersion Equation

Transverse resonance condition:

= mhkn scf 222cos2coshknf

)(2 ,TMTEc =

)(2 ,TMTEs =

m

Dispersion equation ( vs. ):

= mhkn scf cos

Step 3:

Define transverse resonance condition.

: mode number

: 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

Solve for .

= sinfnkN fs nNn

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Graphical Solution of the Dispersion Equation

Symmetrical waveguide, s = c

Asymmetrical waveguide, s c

For fundamental mode (m = 0), there is always a solution (no cut-off) for symmetrical waveguide.

Increasing h (and/or decreasing ) will support more modes.

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Typical diagram

Cut-off

knfkns

Lower-order

Higher-order

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Numerical Solution for Dispersion Relation (I)

Define:Normalized frequency and film thickness

22sf nnkhV

22

22

sf

s

nn

nNb

Normalized guide index

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

Measure for the asymmetry

TMforTE,for22

22

4

4

22

22

sf

cs

c

f

sf

cs

nn

nn

n

na

nn

nna

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

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Numerical Solution for Dispersion Relation (II)

For TE modes, dispersion relation

b

ab

b

bmbV

+

+

+= 1

tan1

tan1 11= mhkn scf cos

m= : Mode number(Normalized) cut-off frequency:

+=

=

mVV

aV

m 0

10 tan

# of guided modes allowed:

222sf nn

hm

=

AlGaAs/GaAs/AlGaAs double heterostructure

n = 3.55/3.6/3.55

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The Goos-Hnchen Shift

=

d

dz ss

= tan)( 2/122 ss nNkz

+

=

1

tan)(

2

2

2

2

2/122

fs

ss

n

N

n

N

nNkz

For TE modes

For TM modes

The lateral ray shift indicates a penetration depth:

= tans

s zx

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Effective Waveguide Thickness

Effective thickness:

cseff xxhh ++=

Normalized effective thickness:

22sfeff nnkhH

For TE modes:

abbVH

+++=

11

Minimum H Maximum confinement

Effective waveguide thickness cannot be zero,

even for symmetrical waveguide (a = 0).

Example:

Sputtered glass, ns = 1.515, nf= 1.62,nc = 1, a = 3.9