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

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 Crystals

2.6.1 Non-linear optics2.6.2 Phase-matching2.6.3 Band-structure tuning2.6.4 All-optical switching

2.7 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

Non-linear optics

Non-linear reaction of a material to an incident EM-field

=

...)(~)(~)(~)(~ )3()2()1( +++= tPtPtPtP

)(~)(~ )1()1( tEtP χ=

)(~)(~ 2)2()2( tEtP χ=

)(~)(~ 3)3()3( tEtP χ=

(for a lossless, dispersionless medium with instanteneous reaction)

linear polarization

2nd order non-linear polarization

3rd order non-linear polarization

Polarization

)1(χ)2(χ)3(χ

linear susceptibility

2nd order suszeptibility

3rd order suszeptibility

0)2( =χ For inversion-symmetric materials

z.B. fluids, amorphous materials

0)3( ≠χ Materials with and without inversions symmetry

2

213)3( 109

Vcm−⋅≈χ

Vcm5)2( 105.1 −⋅≈χ

1)1( ≈χ

Susceptibility

Atomic fields: |E| ~ 1010 V/m

Sun light: |E| ~ 600 V/m

Laser light: |E| ~ 108 V/m

ti

nn

nerEtrE ωω −⋅=∑ ),(),(~ rrrr

n=2, lossless & dispersionless materials:

++⋅= − ..)()(~1

1 cceEtE tiωω

..)( 22 cceE ti +⋅+ − ωω

Electric field

)2(~P

)3(χ

)2(χ

)3(~P

sum-freqency generation (SFG)

difference-freqency generation (DFG)

second-harmonic generation (SHG)

third-harmonic generation (THG)

intensity dependent refractive index

Consequences in 2nd and 3rd order

)(~)(~ 2)2()2( tEtP χ=

[ ] ++⋅⋅= )()()()(2)(~2

*21

*1

)2()2( ωωωωχ EEEEtP

[ titi eEeE 21 22

21

)2( )()( ωω ωωχ −− ⋅+⋅⋅+

tieEE )(21

21)()(2 ωωωω +−⋅⋅+ +

tieEE )(21

21)()(2 ωωωω −−⋅⋅ ++

+

]..cc+

„OR“

„SHG“

„SFG“

„DFG“

Non-linear 2nd order polarization

.....)()(2...)(~ )(21

)2( 21 ++⋅⋅+= +− cceEEtP ti ωωωω

)2(χ 21 ωω +1ω

2ω 21 ωω +1ω

2ω

tunable em-radiation in the UV spectral region

„Energy-level diagram“

Sum frequency generation

.....)()(2...)(~ )(21

)2( 21 ++⋅⋅+= −− cceEEtP ti ωωωω

)2(χ 21 ωω −1ω

2ω21 ωω −

1ω 2ω

tunable em-radiation in IR region

optical parametric amplification of ω2


Difference frequency generation

[ ]4444 34444 214434421 ..2)(~ 22)2(*)2()2( cceEEEtP ti +⋅⋅+⋅⋅= − ωχχ

)2(χω ω2 ω2ω

ω

..)(~ cceEtE ti +⋅= − ωspecial: 21 ωω =


„OR“ „SHG“

Second harmonic generation

)2(~P

)3(χ

)2(χ

)3(~P

sum-freqency generation (SFG)

difference-freqency generation (DFG)

second-harmonic generation (SHG)

third-harmonic generation (THG)

intensity dependent refractive index

Consequences in 2nd and 3rd order

)(~)(~ 3)3()3( tEtP χ=

( )tEtE ωcos)(~ ⋅=same frequency ω for all waves:

444 3444 214444 34444 21)cos(

43)3cos(

41)(~ 3)3(3)3()3( tEtEtP ωχωχ ⋅⋅⋅+⋅⋅⋅=

„THG“ Non-linear polarization contribution for the incident field.

3rd order non-linear polarization

2nd term )(~ )3( tPintensity dependent

refractive index

Innn ⋅+= 20

20

8EcnI ⋅=

π

)3(20

2

212 χπ ⋅=

cnn

00

00

2)3(

2)3(

<⇒<

>⇒>

n

n

χχ

Optical Kerr-effect

0. Introduction






2121 )( ωωωω kkkrrr

+=+ (SFG)

Energy transfer from incident waves ω2, ω1 into the created wave (ω1+ω2) is most efficient.

Microscopic explanation:

electric fields from atomic dipolmoments interfere constructively

Field enhancement along direction of emission

Phase matching (PM) condition

PM for SHG: ωω kkrr

22 =

ωωωω ⋅⋅=⋅ )(22)2( nn

normal dispersion )()2( ωω nn ≠

PM impossible!

ω

)(ωn

ω ω2

)2( ωn

)(ωn

normal dispersive material

Problem in normal bulk material

inte

nsi

ty

vph,1 vph,2 = vph,1

...

destructive interference

propagation distance

Intensity of created frequency component will never exceed a certain limit!

Frequency conversion in bulk material

PM for SHG: ωω kkrr

22 =

ωωωω ⋅⋅=⋅ )(22)2( nn

normal dispersion )()2( ωω nn ≠

PM impossible!

ω

)(ωn

ω ω2

)2( ωn

)(ωn

normal dispersive material

Problem in normal bulk material

ω

)(ωn

ω ω2

)()2(

ωω

nn =

onen

positiv uniaxial birefringent crystal

Third harmonic generationThird harmonic generation

)(t)Eχ(t)EχE(t)(χεP(t) 3(3)2(2)(1)0 K+++=

c.c.eE(t)E ti0i += ω

χ(1)ω ω

0

, χ(3)

3ω

c.c.)(eE c.c.)(e3E(t)E(t)E

ti330

ti30

3if

+++=∝

ω

ω

Frequency conversion in bulk materialFrequency conversion in bulk material

ω3=3ω1ω

ω1

n

n3

n1

Different frequency components propagate with different phase velocities.

0k3k∆k 31 ≠−=

phase mismatch

Dispersion in photonic crystalsDispersion in photonic crystals

Band structure of a 1D photonic crystal

“Effective” refractiveindex

Frequency conversion in photonic crystalsFrequency conversion in photonic crystals

ω3=3ω1 ωω1

n

n1=n3

Different frequency components propagate with same phase velocities.

0k3k∆k 31 =−=

phase matching

P. Markowicz et al., PRL 92, 083903 (2004)

ban

d g

ap

Sum frequency generationSum frequency generation

)(t)Eχ(t)EχE(t)(χεP(t) 3(3)2(2)(1)0 K+++=

ω2

χ(1)ω1 ω1

ω2

, χ(3)

2ω1+ω2

2ω2+ω1

2ω1−ω2

2ω2−ω1

3ω13ω2

0

Phase matching condition: 0kk2k∆k 321 =−−=

Frequency conversion in photonic crystalsFrequency conversion in photonic crystals

ωωseed

neff

phase matching!

band

gap

ωpump ωsignal=2ωpump-ωseed

>> 0signalseedpump kk2k∆k −−= > 0= 0< 0

ωpump

FFinite inite DDifference ifference TTime ime DDomain methodomain method

• Exact calculation of propagation of light pulses through a 1D medium.

• Solves the Maxwell equations numerically.

• Including a 2-level system via the optical Bloch equations allows the implementation of a frequency dependent refractive index in time domain calculations.

• Optical nonlinearities can easily be implemented.

MaxwellMaxwell--Bloch equationsBloch equations

xtb

yzb

xt

xzyt

PHE

EH

∂−∂−=∂

∂−=∂

00

011

1

εεεε

µ

)(1

2

21

1

3031

23

322

102

2012

1

ρρργρ

ργρρωρ

ρωρρ

−−−=∂

+−−=∂

+−=∂

TE

ET

T

xt

xt

t

h

h

Maxwell

Bloch

transition dipole moment

)()( 1 tNtPx γρ−=two-level medium

density of two-level absorbers real part of polarization

SimulationsSimulations

ε1 ε2

II) Bulk material

I) Bragg stack

ε1

?

?

ωpump

ωseed

ε1>ε2 , χ(3)>χ(3) , 80 periods

ωpump

ωseed

χ(3)1

χ(3)2

1 2

χ(3)1

ResultsResults

ωseed

ωpump

Input spectrum

ω (eV/ħ)

inte

nsity

Parameters fora TiO2/Ta2O5material system:

dTiO2=110nm,

dTa2O5=120.5nm

εb,TiO2=3.978,

εb, Ta2O5=3.833

Eg, TiO2=4.425eV,

Eg, Ta2O5=4.457eV

dcv,TiO2=72.82eA,

dcv, Ta2O5=46.97eA

N=1024m-3

ResultsResults

ωseed

ωpump

Spectrum after bulk material

ω (eV/ħ)

inte

nsity

zoom x30

ωsignal=2ωpump-ωseed

Creation of a newfrequency component!

ResultsResults

Spectrum after Bragg stack

ωseed

ωpump

Gain of theseed pulse.

ω (eV/ħ)

inte

nsity ωsignal=2ωpump-ωseed

Creation of the newfrequency component

much more efficient than in bulk material!zoom x30

ResultsResults

Spectrum after Bragg stack

phase matching!

ω (eV/ħ)

inte

nsi

ty

Comparison photonic crystal Comparison photonic crystal –– bulk materialbulk material

ban

d g

ap

Enhancement of frequency conversion efficiency of the processωsignal=2ωpump-ωseed

by more than two orders of magnitude.ω

sign

al: (

Inte

nsity

Bra

gg/In

tens

ityB

ulk) x 105

Influence of disorderInfluence of disorder

ε1 ε2

What happens if the layer thickness is not perfectly λ/4?

Perfect λ/4 stack

“frozen phonon”disorder

ε1 ε2

Comparison photonic crystal Comparison photonic crystal –– bulk materialbulk material

ban

d g

ap

Enhancement of frequency conversion efficiency of the processωsignal=2ωpump-ωseed

by more than two orders of magnitude.ω

sign

al: (

Inte

nsity

Bra

gg/In

tens

ityB

ulk) x 105

x 75

Frequencyconversion isvery robustagainstimperfections!

“Frozen phonon”disorder of 30%(!)

Perfect λ/4 stack

inte

nsity

inte

nsity

FDTD-calculations: Proof of principle

A. Zakery et al., J. Non-Cryst. Solids 330, 1 (2003)

104 layers

FDTD: Influence of frequency and disorder

2.5

2.6

2.7

2.8

effe

ctiv

e in

dex

O. Toader et al., Phys. Rev. E 70, 046605 (2004)

I3~L2 sinc2(2∆k/L) with ∆k=2k1-k2-k3

ћω (eV)

Sample fabrication and characterization

ω2 ω1 ω3

As2S3: 125 nmAs2Se3: 106 nm

104 layers

Experimental setup

•two OPA •pulse duration ωi=80-150 fs•repetition rate 1 kHz

Theory

Experimental results and theoryχ(3)-process ω3=2ω1-ω2

I3~L2 sinc2(2∆k/L) with ∆k=2k1-k2-k3

Experiment

2.5

2.6

2.7

2.8

Effe

ctiv

e in

dex

ω1 = 0.894 eV/ћ, ω2 = 0.536 eV/ћ

ћω (eV)

104 layers

152 layers

0. Introduction






Tuning of position and width of the PBG!

...)(

1 2

1

2

10 +⋅+⋅+=−

−axi

axi

eexn

ππ

κκκ

e.g. 1D PhC:

PBG

2π

2π−

1010 κκπωκκπ +<<−ac

ac

↑n ↓

↑↓n

PBG

PBGwith Innn ⋅+= 20

Bandstructure tuning

• 2D PhC (air holes; non-linear Kerr-dielectric)

• pump-beam ωp optical Kerr-effect

• signal-beam ωs close to PBG

Tunable superprism effect

pump „on“

shift of PBS

optical Kerr-effect

strong variation of vg

ωs close to PBG

strong variation of propagation direction

photonic band structure (PBS)

Influence of pump frequency

ωp closer to PBG

vg reduced

enhanced optical Kerr-effect

enhanced shift of PBG with samepump-power

Influence of pump frequency

photonic band structure (PBS)

iso-frequency surfaces (IFS)

pump „off“ pump „on“

pump „on“

Shift of PBG

optical Kerr-effect

Tunability of propagation direction

Change of IFS

IFS⊥gvr

0. Introduction






Switching schemeSwitching scheme

Refractive index change shifts stop bandProbe beam can be switched

Nonlinear optics and allNonlinear optics and all--optical switchingoptical switching

Optically induced refractive index changeOptically induced refractive index change

Optical Kerr effect Free carrier generation

ε=n

∆n<0∆n>0

mass electron effective densitycarrier free

0for constant dielectric

*

emN

N =∞ε

Nonlinear polarization:

)(t)Eχ (t)EχE(t)(χεP(t)

)(

)()(

K+++=

33

221

0

Refractive index: n=n0+n2I

n2~χ (3)

I light intensity

*

0

22

)(e

p

p

m

Ne

εω

ωω

εωε =

−= ∞ with

Drude model:

Refractive index:


PumpPump--probe mechanismprobe mechanism

pump beam

probe beam

nonlinearPhotonicCrystal

detector

The pump beam changes optical properties of the nonlinear medium in such a way that the transmittance of the probe beam is modified.


Experimental SetupExperimental Setup

• pulse duration τp=80-150 fs• repetition rate ν=1 kHz

• Iprobe:Ipump=1:30• w0,probe=16 µm,

w0,pump=100 µm

Experimental setupExperimental setup

probe beam

pump beam

optical delay line

silicon inverse opal

Spatial and temporal overlapSpatial and temporal overlap

Robust and easymethod to findthe spatial and temporal overlapfor arbitraryfrequency combinations


Temporal resolutionTemporal resolution

-600 -400 -200 0 200 400 600

0

1

2

3

4

5

6fr

eque

ncy

mix

ing

sign

al [

a.u.

]

probe delay [fs]

Crosscorrelation ωprobe+2ωpump (yellow point)

FWHM=170fs


MeasurementMeasurement

Experimental results IExperimental results I

0.85

0.90

0.95

1.00

1.05

rela

tive

tran

smitt

ance

0 2 4 6probe delay (ps)

λprobe=2.39µm

0 2 4 6

0.5

0.6

0.7

0.8

0.9

1.0

rela

tive

tran

smitt

ance

probe delay (ps)

λprobe=1.90µm

∆T/T=49%


Same behavior regardless of the spectral position of the probe beam:Large induced absorption


HypothesisHypothesis

Indications that the measurements of the nonlinear optical response can be explained by free carrier generation:

• Nonlinear optical response is huge• Exponential decay• Quadratic dependence of ∆T/T from

pump intensityFree carrier generation by two-photon absorption

Quadratic fit

Some crucial considerationsSome crucial considerations

/T

Reference experimentReference experiment

Sample: A thin unstructured silicon film made in the same CVD chamber with same parameters (as silicon Photonic Crystals)

A thin unstructured silicon film shows the same nonlinear optical behavior as a high quality silicon Photonic Crystal.


DrudeDrude modelmodel

( )

+

++

−= ∞ 2322*0

2

)/1(

)/1(

/1

1)(

τωωτ

τωεεωε i

m

Ne

e

τ =0.5fs for hydrogenized amorphous (CVD-)silicon

Imaginary part of ε is dominant.

This explains the large overall induced absorption.

time scattering Drude mass electron effective for constant dielectric

densitycarrier free with

τ*

emNε

N0=∞


We have to increase the Drude scattering time τ.

There is a way to achieve this.

Excursion aExcursion a--Si:HSi:H

electron trap

defect states

hole trap

defect states

mobility edge

mobility edge

localized bandstates

delocalized bandstates

ener

gy

ener

gy

x density of states

Si

SiSi

SiSi

Si

SiSi

Si“danglingbond”

Si

SiSi

SiH

crystalline silicon amorphous silicon amorphous & hydrogenized silicon


Amorphous silicon Amorphous silicon nanocrystallinenanocrystalline siliconsilicon

a-Si:Htempering at 600°C

and 10-5mbarfor 24 hours

nc-Si

Oven

It is easy to convert an a-Si:H sample into the nanocrystalline state.


Reference experiment againReference experiment again

Sample: A thin unstructured silicon film made in the same CVD chamber with same parameters (as silicon Photonic Crystals)

Nonlinear optical behavior has changed completely!

a-Si:H

nc-Si

Explanation:- Re(ε) smaller- Re(n) smaller- Less surface reflection- More transmittance

Dispersive response


Tempering the silicon inverse opalsTempering the silicon inverse opals

1. Silicon inverse opals survive tempering.2. The linear optical properties change:

stopband shifts towards shorter wavelength as na-Si:H=3.95 and nnc-Si=3.48.

before tempering

after tempering


MeasurementMeasurement

Experimental results IIExperimental results II

rela

tive

tran

smitt

ance

probe delay (ps)

λprobe=1.80µm

0 5 10 15 200.20.30.40.50.60.70.80.91.01.1

rela

tive

tran

smitt

ance

probe delay (ps)0 5 10 15 20

0.81.01.21.41.61.82.02.22.4

λprobe=2.32µm


Strong dispersive response,band edge shift of about100nm, probe transmittancechange > 130%


TransferTransfer--matrix calculations in matrix calculations in ““scalarscalar--wavewave--approximationapproximation””

Transfer-matrix: 1D model, calculates the transmittance of

1D heterostructures

„scalar-wave-approximation“: replaces the 3D dielectric constant

ε(r) of an inverse opal by an effective 1D

frequency dependent dielectric constant in one

propagation direction

Theoretical modelTheoretical model

TransferTransfer--matrix calculations in matrix calculations in ““scalarscalar--wavewave--approximationapproximation””

N=0

N=1020cm-3, τ=0.5fs

N=1020cm-3, τ=10fs( )

+

++

−= ∞ 2322*0

2

)/1(

)/1(

/1

1)(

τωωτ

τωεεωε i

m

Ne

e


1.0 1.5 2.0 2.5 3.0 3.50.0

0.2

0.4

0.6

0.8

1.0

tran

smitt

ance

wavelength (µm) tran

smitt

ance

probe delay (ps)

Numerical calculationsNumerical calculations Calculation @ 1.83µm

Measurement @ 1.80µm

tran

smitt

ance

1E20 1E19

0.3

0.4

0.5

0.6

0.7

0.8

free carrier density N (cm-3)

0 5 10 15 20

0.20.30.40.50.60.70.8


1.0 1.5 2.0 2.5 3.0 3.50.0

0.2

0.4

0.6

0.8

1.0

tran

smitt

ance

wavelength (µm)0.3

0.4

0.5

0.6

0.7

tran

smitt

ance

0 5 10 15 20probe delay (ps)

0.3

0.4

0.5

0.6

tran

smitt

ance

-31E20 1E19free carrier density N (cm )




1.0 1.5 2.0 2.5 3.0 3.50.0

0.2

0.4

0.6

0.8

1.0

tran

smitt

ance

wavelength (µm) tran

smitt

ance

0 5 10 15 20probe delay (ps)



tran

smitt

ance

-31E20 1E19free carrier density N (cm )

0.65

0.70

0.75

0.80

0.55

0.60

0.65

0.70

0.75


Documents

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