1 2D Bloch Wave Optics - Bilkent Universityregpot/winterschool2009/app/... · 2009. 2. 3. ·...

1© Philip Russell, MPL, Erlangen

2D Bloch Wave Optics

or the

Peculiar Properties of Light in periodic media

Philip Russell

Max-Planck Institute for the Science of Light

Erlangen, Germany

http://www.ioip.mpg.de/~russell/PC_website/index.htm


Topics

•

Peculiar Bloch waves•

nearly free photon model•

wavevector diagrams•

anatomy of a Bloch wave•

negative & positive refraction•

interference, Green’s functions•

curious focusing device•

TIR at normal incidence•

slowness diagrams & diffraction•

square lattices

•

Conclusions


“Nearly free photon”

theory

Λ

strong cumulative reflection

Bragg angle

very weak reflection from each Bragg plane


“Nearly free photon”

theory

•

assume that waves exist inside the periodic region

•

allow them to be “coupled”

by Bragg scattering:

Bragg angle

B

B

0

1

( ) ( ) exp[ ]( ) exp[ ( ) ]

E V iV i

= ⋅

+ − ⋅Kk

kr r r

r r

•

at a particular frequency, special values of Bloch wavevector yield:

•

a “magic”

combination of amplitudes that “sneaks through”

the structure without change•

these are the Bloch waves

0 ( )V r

1( )V r

y

x

2 ˆπΛy

Bloch

wavevector

Λ[a.k.a. coupled mode theory]


Solutions for a 1-D crystal

Λ

0 ( )V r

1( )V rfr

eq

ue

nc

y

wavevector

k π=

Λk π

=Λ

reminder: Gω∂

=∂

vk

free waves

dc

b

a

c

d

b

a


Topics

•










square lattices

•

Conclusions


isotropic media


Refraction & reflection

inc

ω=k

ray diagramwavevector diagram

kx

ky

medium 1medium 1

medium 2medium 2

constructionline

interfacekt

incident

reflected

refracted


Total internal reflection

ray diagramwavevector diagram

kx

ky

medium 1medium 1

medium 2medium 2

interface

constructionline

kt

incident

reflected

inc

ω=k


Topics

•










square lattices

•

Conclusions


anatomy of a Bloch wave: Real space

group velocity

Λ

2 /K π= Λ


2 /K π= Λ

K KKK K

2k1k0k1k2k

3k

anatomy of a Bloch wave: Reciprocal space


Bloch’s theorem

B

B

( , ) exp( ) exp( )

exp( () )

mm

j A jm

Pj

ωΨ = − ⋅ − ⋅

= − ⋅

∑ K r

rk

rk

r

r

function with period of lattice

Bloch wavevector

reciprocal lattice vector

2 /π= ΛK

lattice spacing

Bloch waves have multiple phase velocities & a single group velocity

/( )

( )m B

g B

v k mKφ ω

ω

= +

= ∇kv k


Bone-structure & face

http://tutorialblog.org/skull-face/

with flesh without flesh


2 /K π= Λ

1k0k1k2k

xk

yk

anatomy of a Bloch wave


Hugely enhanced design freedom ...

magnitude

& direction

of group & phase velocity can be almost independently controlled


Topics

•










square lattices

•

Conclusions


Bloch wave refraction and reflection


ray diagram

medium 1

Λ

Mono-periodic medium

constructionlines

wavevector diagram

medium 2

2

2π/Λ

1

interface

incident


Double negative refraction

wavevector diagram

group ( )ω= ∇kV k

medium 2

average average

medium 2

incident

Λ

medium 1

interface


Negative & positive refraction


medium 2

medium 1

incident

wavevector diagram

medium 2

average average

interface

Λ


kx

kyT

E c

ircl

e

TM circle

corrugated

smoothsmooth

Negative refraction (1983)

in “Confined Electrons and Photons”

(Eds

Burstein & Weisbuch) pp 585-633

Plenum Press, 1995

•

150 nm sputtered tantala

on borosilicate glass

•

~1 dB/cm losses

•

enhanced scattering in periodic region

For reprints see: www.ioip.mpg.de/~russell/PC_website/index.htm/

© Philip Russell, University of Bath


kx

kyT

E c

ircl

e

TM circle

corrugated

smoothsmooth

Double negative refraction (1983)

in “Confined Electrons and Photons”

(Eds

Burstein & Weisbuch) pp 585-633

Plenum Press, 1995

•

150 nm sputtered tantala

on borosilicate glass

•

~1 dB/cm losses

•

enhanced scattering in periodic region

© Philip Russell, University of Bath

For reprints see: www.ioip.mpg.de/~russell/PC_website/index.htm/


Topics

•










square lattices

•

Conclusions


Double negative refraction

wavevector diagram


medium 2

average average

medium 2

incident

Λ

medium 1

interfaceΔk

ray diagram


ray diagram

medium 2

medium 1

Λ

interface

Bloch wave interference

wavevector diagram

incident

2π/Δk

Λ << 2π/Δk

Phys Rev A33

(3232-3242) 1986

Δkmedium 2

average

incident


Bloch wave point-influence function

ray diagram

medium 1

Λ

medium 2

interface

incident

Phys Rev A33

(3232-3242) 1986wavevector diagram

spectralspread

medium 2

average


Bloch wave point-influence function

ray diagram

medium 1

Λ

medium 2

interface

2π/Δkmin

Δkmin

Phys Rev A33

(3232-3242) 1986wavevector diagram

spectralspread

medium 2

average


with polariser

Experimental observation

•

Pendellösung

period 50 μm

•

stop-band width 125 per mm

•

100% directional coupler is only 25 μm thick

boundary

smooth

corrugations

guide

d pen

cil be

am

300 nm50 μm


Experimental observationsPhys Rev A33

(3232-3242) 1986

no polarizer

0 -1

“0

”

wave blocked

with polarizer

0 -1

“-1

”

wave blocked

with polarizer

0 -1


Topics

•










square lattices

•

Conclusions


Bloch wave lens

kx

kyBragg planes

boundarieswavevector diagram near Bragg point

using negative and positive refraction


Bloch wave lens

kx

kyBragg planes




Bloch wave beam expander

•

miniature optical elements•

two-dimensional resonators•

in-plane 2D lasers•

new kinds of lenses

Electron. Lett., 20 (72-73) 1984

Bloch wave beam expander

1 mm


Topics

•










square lattices

•

Conclusions


Total internal reflection at normal incidence

slow

fast

( )g Bω= ∇kv k

“fast” mode

waveguide

evanescent

medium 2evanescent

Erice 1993


Topics

•










square lattices

•

Conclusions


Diffraction (spatial dispersion)

diffraction wanted?

•

Free-space:

diffraction is OK but rather limited & isotropic

•

Photonic crystals:

diffraction can be very strong and anisotropic with multiple beams


very fast

a bit

faster

Phase velocity is not a vector


“Slowness”

vector

•

highlights the change in refractive index with frequency and direction

( ) ( )c ω ωω

=k n

wavevector

frequencyvector refractive index


Slowness diagrams for light: TE

-1.5 -1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5 Λ

yc kω

xc kω

as the wavelength drops, the fields can discriminate better between low & high index regions

higher

lower


Λ

Slowness diagrams: Beam diffraction

air

air circle

-1.5 -1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5yc k

ω

xc kω

divergent



-1.5 -1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5

Λ

yc kω

xc kω

close to zero diffraction



-1.5 -1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5

Λ

yc kω

xc kω

convergent



-1.5 -1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5

Λ

yc kω

xc kω

zero diffraction



-1.5 -1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5

Λ



-1.5 -1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5

Λ

evanescent

evanescent



-1.5 -1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5

Λ

highly

divergent


Topics

•










square lattices

•

Conclusions


Square lattice

Brillouin zone (Bragg condition)

yk

xk

2 2 2 2x y ok k n k+ =

ok

δ

scan

angle


o 1 (cos cos )M Kx Kyε ε/ = + +

[ ]ˆ ˆ

o o0 0

( , ) exp (( ) cos ) (( sin )m n

mnm n

E x y V j k m K x k n K yδ θ θδ= =

= − + − + + ) −∑∑

2 200o o

2 210o o

2 201o o

2 211o o

0/ 2 0 / 20/ 2 / 2 000 / 2 / 20/ 2 0 / 2

s

s c

c

aq

qVk M k MVk M k MVk M k MVk M

qqk

aM

aa

⎡ ⎤ ⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎜ ⎟ ⎜ ⎟−⎢ ⎥ ⎜ ⎟ ⎜ ⎟=⎢ ⎥ ⎜ ⎟ ⎜ ⎟− −⎢ ⎥ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟−⎢ ⎥ ⎝ ⎠⎝ ⎠⎣ ⎦

Nearly-free photons in square lattice

o( )q kδ δ= − + 2 o4 ( )sinsa kπ π δ θ⎛ ⎞= − +⎜ ⎟Λ Λ⎝ ⎠

o4 ( ) cosca kπ π δ θ⎛ ⎞= − +⎜ ⎟Λ Λ⎝ ⎠

perturbation


Wavevector diagrams in square crystal

PBGlower band

edge

negative

refraction

PBG

1 2 3 4

5 6 7 8

9 10 11 12

fre

qu

en

cy

inc

rea

sin

g


upper band edge

Wavevector diagrams in square crystalPBG 13 14 15 16

17 18 19 20

21 22 23 24

fre

qu

en

cy

inc

rea

sin

g

positive

refraction


negative diffraction positive diffraction

convergent rays:

surface “nose”

points away

from direction of energy flow

divergent rays:

surface “nose”

points towards

direction of energy flow


Negative diffraction in a square-latticeZengerle, J. Mod. Opt. 34 (1589-1617) 1987

ω1frequency



2ω ω1>frequency



3 2ω ω>frequency

focusing independent of position


Reinhard

Ulrich

•

first experiments on metal-wire “meta-materials”

–

early 1970s•

first experiments on multiply-

periodic planar waveguides (“photonic crystals”) –

from 1975



Topics

•










square lattices

•

Conclusions



Conclusions

•

refractive index vector plays a crucial role in periodic media –

usually it is multi-valued•

negative & positive refraction are common in periodic media –

often at the same time•

negative diffraction is a common feature of propagation in periodic media

•

many negative effects were first reported in the early 1980s in planar periodic waveguides

•

since about 1990, periodic optical media have become known as “photonic crystals”


1 2D Bloch Wave Optics - Bilkent Universityregpot/winterschool2009/app/... · 2009. 2. 3. ·...

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