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

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

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

E V iV i

+ − ⋅Kk

kr 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

2 ˆπΛy

wavevector

Λ[a.k.a. coupled mode theory]

Solutions for a 1-D crystal

0 ( )V r

1( )V rfr

wavevector

Λk π

reminder: Gω∂

free waves

Topics

square lattices

Conclusions

isotropic media

Refraction & reflection

ray diagramwavevector diagram

medium 1medium 1

medium 2medium 2

constructionline

interfacekt

incident

reflected

refracted

Total internal reflection

ray diagramwavevector diagram

medium 1medium 1

medium 2medium 2

interface

constructionline

incident

reflected

Topics

square lattices

Conclusions

anatomy of a Bloch wave: Real space

group velocity

2 /K π= Λ

K KKK K

2k1k0k1k2k

anatomy of a Bloch wave: Reciprocal space

Bloch’s theorem

( , ) exp( ) exp( )

exp( () )

j A jm

ωΨ = − ⋅ − ⋅

= − ⋅

∑ K 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

v k mKφ ω

= ∇kv k

Bone-structure & face

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

with flesh without flesh

2 /K π= Λ

1k0k1k2k

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

Mono-periodic medium

constructionlines

wavevector diagram

medium 2

2π/Λ

interface

incident

Double negative refraction

wavevector diagram

group ( )ω= ∇kV k

medium 2

average average

medium 2

incident

medium 1

interface

Negative & positive refraction

medium 2

medium 1

incident

wavevector diagram

medium 2

average average

interface

TM circle

corrugated

smoothsmooth

Negative refraction (1983)

in “Confined Electrons and Photons”

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/

TM circle

corrugated

smoothsmooth

Double negative refraction (1983)

in “Confined Electrons and Photons”

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/

Topics

square lattices

Conclusions

Double negative refraction

wavevector diagram

medium 2

average average

medium 2

incident

medium 1

interfaceΔk

ray diagram

medium 2

medium 1

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

medium 2

interface

incident

Phys Rev A33

(3232-3242) 1986wavevector diagram

spectralspread

medium 2

average

Bloch wave point-influence function

ray diagram

medium 1

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

cil be

300 nm50 μm

Experimental observationsPhys Rev A33

(3232-3242) 1986

no polarizer

wave blocked

with polarizer

wave blocked

with polarizer

Topics

square lattices

Conclusions

Bloch wave lens

kyBragg planes

boundarieswavevector diagram near Bragg point

using negative and positive refraction

Bloch wave lens

kyBragg planes

Bloch wave lens

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

Topics

square lattices

Conclusions

Total internal reflection at normal incidence

( )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

faster

Phase velocity is not a vector

“Slowness”

vector

highlights the change in refractive index with frequency and direction

( ) ( )c ω ωω

wavevector

frequencyvector refractive index

Slowness diagrams for light: TE

-1.5 -1 -0.5 0.5 1 1.5

1.5 Λ

yc kω

xc kω

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

higher

Slowness diagrams: Beam diffraction

air circle

-1.5 -1 -0.5 0.5 1 1.5

1.5yc k

xc kω

divergent

-1.5 -1 -0.5 0.5 1 1.5

yc kω

xc kω

close to zero diffraction

-1.5 -1 -0.5 0.5 1 1.5

yc kω

xc kω

convergent

-1.5 -1 -0.5 0.5 1 1.5

yc kω

xc kω

zero diffraction

-1.5 -1 -0.5 0.5 1 1.5

evanescent

-1.5 -1 -0.5 0.5 1 1.5

highly

divergent

Topics

square lattices

Conclusions

Square lattice

Brillouin zone (Bragg condition)

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

o 1 (cos cos )M Kx Kyε ε/ = + +

[ ]ˆ ˆ

o o0 0

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

qVk M k MVk M k MVk M k MVk M

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

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

negative

refraction

1 2 3 4

5 6 7 8

9 10 11 12

upper band edge

Wavevector diagrams in square crystalPBG 13 14 15 16

17 18 19 20

21 22 23 24

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

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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