0. Introduction 1. Reminder: E-Dynamics in...

0. Introduction

1. Reminder:E-Dynamics in homogenous media and at interfaces

2. Photonic Crystals2.1 Introduction2.2 1D Photonic Crystals2.3 2D and 3D Photonic Crystals2.4 Numerical Methods2.5 Fabrication2.6 Non-linear optics and Photonic Crystals2.7 Quantumoptics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers

3. Metamaterials and Plasmonics3.1 Introduction3.2 Background3.2 Fabrication3.3 Experiments

Calculation of the band structure of a 1D Photonic Crystal

a

1ε 2ε k

E

Consider an electromagnetic wave propagating along theaxis of a 1D Photonic Crystal.

How does the dispersion relation ω(k) look like?

K. Sakoda, Optical Properties of Photonic Crystals

Calculation of the band structure of a 1D Photonic Crystal

2

2

2

22 ),(),(

)( t

txE

x

txE

x

c

∂∂=

∂∂

εWe start with the 1D wave equation:

ε -1(x) is also periodic and can be expanded in a Fourier series:

∑∞

−∞=

=m

xa

mi

m ex

π

κε

2

)(

1

The modes of a 1D Photonic Crystal are Bloch states:

ti

m

xa

mki

mkkeeEtxE ω

π−

∞

−∞=

+

∑=)

2(

),(

Calculation of the band structure of a 1D Photonic Crystal

We assume that the components with m = 0 and m = ±1 aredominant in the expansion of the inverse dielectric function:

xa

ixa

iee

x

ππ

κκκε

2

1

2

10)(

1 −

−++≈

exact

approximation

Example:

0κ

Calculation of the band structure of a 1D Photonic Crystal

We assume that the components with m = 0 and m = ±1 aredominant in the expansion of the inverse dielectric function:

Substituting ε -1(x) and E (x,t) into the wave equation, we obtain

xa

ixa

iee

x

ππ

κκκε

2

1

2

10)(

1 −

−++≈

tix

a

mki

mmk

tix

a

mki

mm

xa

ixa

i

k

k

eeE

eea

mkEeec

ωπ

ωπππ

ω

πκκκ

−

+∞

−∞=

−

+∞

−∞=

−

−

∑

∑

−=

+−

++

22

222

1

2

102 2

)1(

2

2

x∂∂

2

2

t∂∂

Calculation of the band structure of a 1D Photonic Crystal

By comparison of coefficients we have

mk

mm

Ea

mk

c

Ea

mkE

a

mk

+−=

+++

−+ +−−

2

02

2

1

2

11

2

1

2

)1(2)1(2

πκω

πκπκ

For m = 0,

++

−

−= −− 1

2

11

2

1220

2

2

0

22E

akE

ak

kc

cE

k

πκπκκω

For m = -1,

+

−

−−= −−− 0

212

2

1220

2

2

1

4

)/2(EkE

ak

akc

cE

k

κπκπκω

Calculation of the band structure of a 1D Photonic Crystal

For m = 0,

++

−

−= −− 1

2

11

2

1220

2

2

0

22E

akE

ak

kc

cE

k

πκπκκω

For m = -1,

+

−

−−= −−− 0

212

2

1220

2

2

1

4

)/2(EkE

ak

akc

cE

k

κπκπκω

Calculation of the band structure of a 1D Photonic Crystal

For m = 0,

++

−

−= −− 1

2

11

2

1220

2

2

0

22E

akE

ak

kc

cE

k

πκπκκω

For m = -1,

+

−

−−= −−− 0

212

2

1220

2

2

1

4

)/2(EkE

ak

akc

cE

k

κπκπκω

For k ≈ π/a , E0 and E-1 are dominant in the expansion.

Calculation of the band structure of a 1D Photonic Crystal

( ) 02

1

22

1022

02 =

−−− −E

akcEkck

πκκω

{ } 0)/2( 122

02

022

1 =−−+− −− EakcEkc k πκωκ

These linear equations have a nontrivial solution when the determinat of coefficients vanishes:

0)/2(

)/2(22

0222

1

221

220

2

=−−−

−−−

− akckc

akckc

k

k

πκωκπκκω

We obtain

Calculation of the band structure of a 1D Photonic Crystal

For real epsilon (κ1 = κ-1*) and |h = k - π/a| << π/a we obtain

2

2

120

10110 2

1h

ac

a

c

−

±±±≈±

κκ

κκκπκκπω

Thus, there are no modes in the interval

1010 κκπωκκπ +<<−a

c

a

c

Band edge of the dielectric band

Band edge of the air band

Calculation of the band structure of a 1D Photonic Crystal

2

2

120

10110 2

1h

ac

a

c

−

±±±≈±

κκ

κκκπκκπω

Numerical simulations:

0. Introduction

1. Reminder:E-Dynamics in homogenous media and at interfaces

2. Photonic Crystals2.1 Introduction2.2 1D Photonic Crystals

Defects2.3 2D and 3D Photonic Crystals2.4 Numerical Methods2.5 Fabrication2.6 Non-linear optics and Photonic Crystals2.7 Quantumoptics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers

3. Metamaterials and Plasmonics3.1 Introduction3.2 Background3.2 Fabrication3.3 Experiments

So far we have only discussed strictly periodic structures.

a

1ε 2ε 1ε 2ε 1ε 2ε 1ε

Question: what happens if we introduce a defect?

1ε 2ε 1ε 2ε 1ε 2ε 1ε

a

However, far away from the defect the structure should behave approximately as before.

Since the structure is no longer periodic, the modes of the “defective” Photonic Crystal are not Bloch states.

Therefore, we still can use knowledge from bandstructurecalculations whether a mode is extended or evanescent.

The periodic structures on both side of the defect act as frequency dependent mirrors.

mirror

1ε 2ε 1ε 2ε 1ε 2ε 1ε

mirror

The periodic structures on both side of the defect act as frequency dependent mirrors.

Extended modes: low reflectivity

Evanescent modes: high reflectivity

Defects may permit localized modes to exist, if the modes have frequencies inside the photonic band gap.

A localized defect mode must decay exponentially once it enters the Photonic Crystal.

Since the distance between the two “mirrors” is on the order of the wavelength of light, the modes are quantized.

(Remember the quantum-mechanical problem of a particle in a box.)

An example: “quarter wave stack” designed for λc=1240 nm

1ε 2ε

Parameters: 16 Periods

24 11 =⇒= nε

11 22 =⇒= nε

1l 2l

nml 1551 =

nml 3102 =

431011

cnmnlλ==

431022

cnmnlλ==

An example: “quarter wave stack” designed for λc=1240 nm

λ=1240 nm

2.4 Numerical Methods, T-Matrix

1ε 2ε

Next, we introduce a “λ/2 defect” in the middle of the stack.

2610 1

cdd nlnml

λ=⇒=

8 Periods8 Periods

Air defect

Numerical simulations based on T-Matrix.

2.4 Numerical Methods, T-Matrix

1ε 2ε

Next, we introduce a “λ/2 defect” in the middle of the stack.

2310 2

cdd nlnml

λ=⇒=

8 Periods8 Periods

Dielectric defect

Numerical simulations based on T-Matrix.

2.4 Numerical Methods, T-Matrix

2.4 Numerical Methods, T-Matrix

Dielectric defect: numerical simulations based on T-Matrix.

The air/dielectric defect leads to a transmission peak (T=1 !) in the middle of the photonic band gap.

The mode associated with the air/dielectric defect is localizedat the site of the defect:

• huge intensity enhancement at the position of the defect (I0 = 1 vs. IDefect≈ 65000)

• exponetial decay of the intensity inside the Photonic Crystal

Dielectric defect: numerical simulations based on T-Matrix.

2.4 Numerical Methods, T-Matrix

The spectral width of the transmission peak decreases with increasing number of periods before and after the defect.

The quality factor Q of the defect mode is defined as

Q= central frequency ω0 / full width at half maximum ∆ω

Dielectric defect: numerical simulations based on T-Matrix.

2.4 Numerical Methods, T-Matrix

The spectral width of the transmission peak decreases with increasing number of periods before and after the defect.

The quality factor Q of the defect mode is defined as

Q= central frequency ω0 / full width at half maximum ∆ω

5002.0

10 ==∆

=eV

eVQ

ωω

Example:

Intuitive picture:

Photons in defect mode bounce Q-times between both “mirrors” before leaving the structure.

Without proof:

Q ∝ (energy stored in mode)/(energy loss per cycle)

Defects play an important role in Photonic-Crystal applications since they can be used as functional elements.

In analogy: Pure Si is “boring”. By intentional doping (intoducing defects) we can fabricate functional semiconductor elements.

Defects in Photonic-Crystal can be used as …

… spectral filters (band pass filter).

… cavities for ultra small lasers.

… waveguides (line defects 2D and 3D Photonic Crystals).

… switches (if combined with nonlinear materials).

…

0. Introduction

1. Reminder:E-Dynamics in homogenous media and at interfaces

2. Photonic Crystals2.1 Introduction2.2 1D Photonic Crystals2.3 2D and 3D Photonic Crystals

2.3.1 Band structures2.3.2 Refraction Law for Photonic Crystals2.3.3 Defects2.3.4 CROW

2.4 Numerical Methods2.5 Fabrication2.6 Non-linear optics and Photonic Crystals2.7 Quantumoptics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers

3. Metamaterials and Plasmonics3.1 Introduction3.2 Background3.2 Fabrication3.3 Experiments

Example - 2D square lattice of dielectric rods:

For 2D-Photonic Crystals and in-plane propagation (kz=0)

, , and do not depend on z.)(rvε)(rE

vr)(rHvr

∂∂−=

∂∂−

∂∂

∂∂−

∂∂

∂∂−

∂∂

z

y

x

xy

zx

yz

H

H

H

t

Ey

Ex

Ex

Ez

Ez

Ey

0µ

∂∂=

∂∂−

∂∂

∂∂−

∂∂

∂∂−

∂∂

z

y

x

xy

zx

yz

E

E

E

tr

Hy

Hx

Hx

Hz

Hz

Hy

)(0

rεε

For 2D-Photonic Crystals and in-plane propagation,

Maxwell’s equations …

… decouple into two sets of equations ({Ez,Hx,Hy} and {Ex,Ey,Hz}).

),(1

),()(

12

2

22

2

2

2

trEtc

trEyxr zz

rrr ∂

∂=

∂∂+

∂∂

ε

Taking the curl once more, we obtain ...

),(1

),()(

1

)(

12

2

2trH

tctrH

yryxrx zz

rrrr ∂

∂=

∂∂

∂∂+

∂∂

∂∂

εε

... wave equations for the components Ez and Hz.

The remaining field components can be deduced from Ez ,

Hz and Maxwell‘s equations.

Example - 2D square lattice of dielectric rods: TM-Polarization

Example - 2D square lattice of dielectric rods: TE-Polarization

How does the band structure look like?

„Empty lattice“:

xK

21 εε ≈yK

1. Brillouin zone

Γ

X

M

xKa/π+a/π−

a/π−

a/π+

yK

Band structure of an empty 2D square-lattice for TE polarization

What is the origin of this band?

xKa/π+a/π−

21 εε ≈

a/π−

a/π+

yK

Band structure of an empty 2D square-lattice for TE polarization

xKa/π+a/π−

21 εε ≈

a/π−

a/π+

yK

Band structure of an empty 2D square-lattice for TE polarization

xKa/π+a/π−

21 εε ≈

a/π−

a/π+

yK

Band structure of an empty 2D square-lattice for TE polarization

)0,( xkk =v

∆ :kx

ky

XΓ ∆∆

∆∆`̀Γ`

a

π−

a

π

a

π

X`

a

π−

εω

kcv

=⇒

kx

ky

XΓ ∆∆

∆∆`̀Γ`

a

π−

a

π

a

π

X`

a

π−

=

akk x

π2,

v`∆ :

ε

π

εω

22 2

+

==⇒a

kckc xv

2D square-lattice of GaAs rods

r

a

aGaAs ( ε = 11.56 )

Air ( ε = 1 )

Band structure of an 2D square-lattice of GaAs rods, r/a=0.1

TE polarization

MIT Photonic-Bands (http://ab-initio.mit.edu/mpb/)

Small value chosen for didactic reasons!

MIT Photonic-Bands (http://ab-initio.mit.edu/mpb/)

Band structure of an 2D square-lattice of GaAs rods, r/a=0.1

TM polarization

2D Band Gap!

J.D. Joannopouls, Photonic Crystals

Gap map for 2D square-lattice of dielectric rods (Si)

2D triangular-lattice of air cylinders in Si

a

Si ( ε = 11.9 )

Air ( ε = 1 )

2D triangular-lattice of air cylinders in Si

yk

xk

Irreducible Brillouin zone

Γ

KM

MIT Photonic-Bands (http://ab-initio.mit.edu/mpb/)

Band structure of an 2D triangular-lattice of air cylinders in Si, r/a=0.44

complete2D Band Gap!

TM polarization

TE polarization

J.D. Joannopouls, Photonic Crystals

Gap map for 2D triangular-lattice of air cylinders in Si

2D theory is fine, but we live in a 3D world!

•There are no band gaps for propagation in z-direction.

•Even for in plane propagation, we require a large aspectratio (height/period) in order to meet experimental constraints (beam diameter).

•Scattering losses in the 3rd dimension are responsible forlow transmittance in experiments with 2D PhotonicCrystals.

Some general problems with 2D Photonic Crystals:

•index guiding for the 3rd dimension=> Photonic Crystal Slabs

Strategies to overcome these problems:

•3D Photonic Crystals: 4 examples