1 Optical Diffraction Theory and Its Applications on Photonic Device Design.

1

Optical Diffraction Theory and Its Applications on

Photonic Device Design

2

Diffraction

SourceFree Space

Plane Wave

Waveguideor resonator

Wave guidance for power or signal transmission

Wave manipulation for other purposes

Diffraction as a natural evolving process towards the plane wave

Direct applications: broadcasting, satellite comm., radar, etc.

3

Models for Diffraction and Scattering

Diffraction Scattering

>>D

D

D

Characteristics of the diffracted/scattered EM fields and the simulation models differ depending on the relative magnitude of the wavelength and

feature size of the object.

<<D

Quasi-Static Geometric Optics

≈D

Full Wave

Single Pole Multi Poles

Ray

4

E

H = j E J

E j H M

Generalized Maxwell’s equations in homogeneous medium

mH

2( ) ( )E E E j H M

2( ) ( )E j j E J M

2 2 1E E j J M

Generalized wave equations in homogeneous medium

2 2 1mH H j M J

Generalized Wave Equations

5

Kirchhoff’s Diffraction Formula

MJEE μjωρε

1μεω22 JMHH εjωρ

μ

1μεω m

22

rrrr,rr, δεgμωg 22 rr

rrrr,

π4

jkexpg

Scalar Green’s Function

sd

n

g

ngvdgg 22 EE

EEn

E,H

EnM s ˆHnJ s ˆ

Region 1

V

SJ,M

n̂

n

vdμεgωgμεωgvdgg 222222 EEEEE

rErrEE vdδvdgμεω22

vdμjωρε

1gvdμεωg 22 MJEE

sd

n

g

ng vdgμjω

ε

ρE

EMJE

sdn

g

ng vdgjω

μ

ρm HH

JMH ε

Note: The integral is for closed surface only!

6

Diffraction Formulas for Open Surfaces

E,HEnM s ˆ

HnJ s ˆ

S dl

ds'C

r

r'

n̂

O

Line integrals enclosing the surface must be added to amend Kirchhoff’s formula:

ρ gjωμ g dv g ds

ε n n

1g d g d

jωC C

EE J M E

E l H l

mρ gjω g dv g ds

μ n n

1g d g d

jωC C

ε

HH M J H

H l E l

7

P1(x1,y1,z1)

P2(x2,y2,z2)

Q(ξ,η,0)

O

x

y

z

R1

r' R2

r2

r1

Diffraction by Aperture

rrR 11 rrR 22

21

2111 rrr2rR rr 2

22222 rrr2rR rr

11 1

1

exp jkRE A

R

2

2

exp jkRg

4 πR

Spherical wave at Q from P1 Spherical wave at P2 from Q

8

sd

n

gE

n

EgE 1

1

11 1

1 1 21 2

ˆ ˆˆ ˆE jAgE g E ds exp jk R R ds

n n 2λ R R

2n R n R

Diffracted Field:

11 R

ˆ

R 111 1 1

1 1

exp jkRE 1ˆˆ ˆE jk An R R

n n R

22

2 2

exp jkRg 1ˆˆ ˆg jkn R 4 πR

n n R

Ignore terms of 1/R³ if R>>

Diffracted Field

22 R

ˆ

R

R1

R2n̂

o

Q

P1

P2r1

r2

r'

2

1

9

1

11 2

1 2

ˆ ˆˆ ˆj A 1E exp jkR exp jkR ds

2 λ R R

2n R n R

Spherical wave at the aperture Q from P1

Spherical wave at the point P2 from Q

Inclination factor corrected by Kirchhoff

Physical Interpretation of Kirchhoff’s Integral

Original Huygens Principle

The diffracted field is a superposition of spherical waves emanating from the wavefront of another spherical wave originating from the point source.

R1

R2n̂

o

Q

P1

P2r1

r2

r'

2

1

For backward wave:

0ˆˆˆˆ 1 RnRn 2

For forward wave:

cosθ2ˆˆˆˆ 1 RnRn 2

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Free-space field at P2 from P1

1 20 1

1 2

exp jk r rE A

r r

Diffraction Coefficient

2 1 1 2 1 20

E jˆ ˆ ˆ ˆD exp jk R R r r ds

E 2 λF n r n r

21 r

1

r

1

F

1

11 1

1 1 21 2

ˆ ˆˆ ˆE jAgE g E ds exp jk R R ds

n n 2λ R R

2n R n R

11 2

1 2

ˆ ˆ ˆ ˆjAE exp jk R R ds

2 λ r r

2 1n r n r

Diffraction Coefficient

F: Focus length

11

Different Approximations for Kirchhoff’s Integral

2i

iii

2ii

iii

ˆr2

1ˆr

rr

ˆ21rR

rrrrrr

rrrr

R1

R2n̂

o

Q

P1

P2

r1

r2

r'

Wavefront curvature correction

Plane wavefront correction

Spherical wave centered at Pi

i=1,2

Fresnel Diffraction

Fraunhofer Diffraction

Geometrical Optics

2iii i

i i i

exp jkrexp jkR kˆ ˆexp jk exp j

R r 2 r

r r r r r r

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Expression of Diffraction Coefficient

2

22

1

21

212121 r2

ˆ

r2

ˆˆˆrrRR

rrrrrrrrrrr

2 1 1 2 1 2

jˆ ˆ ˆ ˆD exp jk R R r r ds

2λF

n r n r

2 1ˆ ˆ ˆ ˆ

D j exp jkf η,ξ ds2λF

n r n r

2222

1111

zˆyˆxˆ

zˆyˆxˆ

yyxr

zyxr

ξˆηˆ yxr

1

111 r

ξyηxˆ

rr

2

222

1

21122

212121 r2

ξnηm

r2

ξnηmξη

r

1

r

1

2

1ξnnηmmξη,f

2

222 r

ξyηxˆ

rr

ξη,frrRR 2121

1

11 r

xm

1

11 r

yn

2

22 r

yn

2

22 r

xm

13

Fresnel Diffraction

R1

R2

o

QP1

P2

r1

r2

r'

x2

z2

z1

x1 n̂

2

1

21 ˆˆˆ rzr

1122 ˆˆcosθcosθˆˆ rnrn 2211 sinθmsinθm

0yy 21

22

22 ξθcosηF2

1ξη,f

21 r

1

r

1

F

1

2 2

1 1

2 2 22 2jcosθ η ξ cos θ

D exp jk d dλF 2F

η

14

2 1ˆ ˆ ˆ ˆ

D j exp jkf η,ξ ds2 λF

n r n r

ξnnηmmξη,f 2121

2

2

1

1x r

x

r

xkk

2

2

1

1y r

y

r

ykk

2

222

1

21122

212121 r2

ξnηm

r2

ξnηmξη

r

1

r

1

2

1ξnnηmmξη,f

Fraunhofer Diffraction

Neglect the higher order terms in η and ξ

2 1x y

ˆ ˆ ˆ ˆD j exp j k η k ξ d d

2λFη ξ

n r n r

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Fraunhofer Diffraction

• P1 on axis, 2 small ( < 30°)• Uniform illumination

over aperture

1ˆˆ 1rn 2cosθˆˆ 2rn

2

2x y

Aperture

1 cos θD j exp j k η k ξ d d

2λFη

2

2x r

xkk

2

2y r

ykk

Equivalent to the stationary

phase approximation

16

Fresnel and Fraunhofer Diffraction

• In Fraunhofer diffraction, both incident and diffracted waves may be considered to be plane (i.e. both S and P are a large distance away)

• If either S or P are close enough that wavefront curvature is not negligible, then we have Fresnel diffraction

P

S

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Fraunhofer Diffraction

For non-uniform illumination at the aperture, and by restoring the free

space field factor, we find: 2 2 2

020

0 0 0 02 2 2 2 2 20 0

( , , ) ( , )j x y z zje

E x y z z DE Ax y z x y z

0 022 2 2 2

2 0

1 cos( )cos( )

2

z z

r x y z

2 2 2

2 0F r x y z where

0 0

0 02 2 2 2 2 20 0

2 ( )0 0

,( , ) ( , , 0) |j x y

x y

x y z x y z

A E x y z e dxdy

The coordinates in the new system are all normalized by the wavelength, i.e.,

( , , ) ( , , ) /x y z x y z

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Plane Wave Propagation in Free Space

In homogeneous medium, the plane wave solution to Maxwell’s equations

can be expressed as: 0

2( )( ) ( ) 2 ( )

0 0 0 0x y z

j x y zj k x k y k z jk x y z j x y zE e E e E e E e

2 21 By assuming the plane wave is propagating along z, we have:

2 ( )( , ) ( , , 0) j x yA E x y z e dxdy

For an arbitrary wave propagating along z, upon its passing through an aperture at z=0, we can expand the wave in terms of the plane waves:

2 200 2 12( , ) ( , ) ( , ) j zj zA A e A e

2 ( )( , , 0) ( , ) j x yE x y z A e d d

Upon its further propagating from z=0 to z=z0

2 200 2 12 2 ( ) 2 ( )

0( , , ) [ ( , ) ] [ ( , ) ]j zj z j x y j x yE x y z z A e e d d A e e d d

Once the distribution of a propagating wave is known at a specific location

z=0, we will be able to compute its distribution at any location z.

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Plane Wave Expression

0 0

0 02 2 2 2 2 20 0

2 ( )0 0

,( , ) ( , , 0) |j x y

x y

x y z x y z

A E x y z e dxdy

At far field, i.e., z0 is a large number, we can introduce the stationary-phase

approximation: 2

0 0 00

0 0 0 0

1[ ( ) ''( )( ) ] ( )( ) 2

0 0{ '( ) 0} { '( ) 0} 0

2( ) ( ) ( )

''( )

jA g x g x x x jAg xjAg x

x g x x g x

jI f x e dx f x e dx f x e

Ag x

We obtain the same Fraunhofer diffraction formula, i.e., the far field angular distribution is the Fourier transform of the field distribution at the aperture.

to obtain2 2 2

020

0 0 0 02 2 2 2 2 20 0

( , , ) ( , )j x y z zje

E x y z z DE Ax y z x y z

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Applications

Aperture:phaser or diaphragm

θ

0( , , ) ( , ) [ ( , , 0)] :E x y z z A F E x y z D

( , , 0) :E x y z d

Low spatial frequency region region

High spatial frequency region region

Far field (Fraunhofer diffraction) condition: 0dD z

(all normalized by the wavelength)

0z

To ensure this condition without going too far: insert a Fresnel lens to bring the Fourier transform closer to f – focal length of the lens

21

Functional Photonic Component Design with Slab-waveguide: Building Block

Substrate

Normal slab-waveguide: low index region

Slab-waveguide with thicker cladding: high index region

Fabrication: standard lithography + etching

Vertically confined by the slab-waveguide

Propagation

2D diffraction

Aperture: phaser

22

Wavelength Multiplexer/Demultiplexer

Input ridge waveguide Output ridge

waveguides

W1, W2 W1

W2

Slab-waveguide low index region

Slab-waveguide high index region

Input lens for beam expanding

Output lens for beam refocusing

Blazed diffractional grating

23

Optical Switch

Output ridge waveguides

Input ridge waveguide Focused beam

with no bias

Slab-waveguide low index region

Slab-waveguide high index region

Input lens for beam expanding

Output lens for beam refocusing

Biased wedge for beam steering

Focused beam under bias

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Beam Splitter

Output ridge waveguides

Input ridge waveguide

Slab-waveguide low index region

Slab-waveguide high index region

Input lens for beam expanding

Output lens for beam refocusing

1st phase plate DC filter plateFourier transform lens

25

128 Channel Beam Splitter: Design

y

A

L0

L1L2

L5W

z

x

L3G

L4

F

S00

S10

S20

S21

S01

S11

p

Q1 Q2

L4L5

x

z

26

128 Channel Beam Splitter: Simulation Result

27

Polarization Beam Combiner/Splitter

28

Form Birefringence

δ t

Ex

Ey

kz

hl

[ ( ) ] / [ (1 ) ]x h x l x h l xD E t E t f f E

[ / ( ) / ] / [ / (1 ) / ]y y h y l h l yE D t D t f f D

(1 )h lf f

/ [ (1 ) ]h l l hf f

29

Polarization Beam Combiner/SplitterPolarization Beam Combiner/Splitter

• X-pol

• loss:0.633 dB

• Crosstalk35.424dB

• Y-pol

• loss:0.66 dB

• Crosstalk36.029dB

30

Time Lens

31

Input/Output Simulation Result