Non-linear optics Non-linear reaction of a material to an incident...

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

)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

)(~)(~ 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

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

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

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

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)

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

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:

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

Experimental results IExperimental results I

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

Experimental results IExperimental results I

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=∞

Some crucial considerationsSome crucial considerations

We have to increase the Drude scattering time τ.

There is a way to achieve this.

Experimental results IIExperimental results II

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

Experimental results IIExperimental results II

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 )

Numerical calculationsNumerical calculations Calculation @ 2.19µm

Measurement @ 2.18µm

Theoretical modelTheoretical model

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 Quantum optics

2.8.1 Short reminder2.8.2 Quantization of the electromagnetic field2.8.3 Interaction with matter2.8.4 The atom-photon bound state

2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers

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

Two-level system in vacuum/photoniccrystal

1

2

1

2

excitation

1

2

Spontaneous emissionStimulated emission

Freespace

1

2

Photoniccrystal

?

Allowed bands

Fermi‘s golden rule

• Describes the transition probability from an initial state |i> into an final state |f>.

( )fip

NiHf →∑=Γ ωπ 2

0

2

h

Transition probability per unit timedecay rate

Available polarizations Final state Initial state

Interaction Hamiltonian between electron and „vacuum“

Density of statesof final state

• Only reasonable if final states form a broad and featureless continuum ... See Quantum mechanics lectures

DOS in Photonic Crystals

• Three different regimes are present in PBGs

Den

sity

of

sta

tes

Frequency

band gap

bandedge

singularitydefect state

Purcell Factor

• Cavities can even further modify emission properties:

( ) 0P 1 Γ+=Γ F

2max

2

cav

em

effP

)(

)(

)(

E

rE

V

QF

r

ωρωρ≈

Cavity quality factor

Effective mode volumeof the cavity mode

DOS for emitter frequency

DOS for cavity frequency

Field intensity at emitter’s position

Maximum field intensity

E.M. Purcell, Phys. Rev. 69, 681(1946)

0. Introduction






Quantization of the electromagnetic field

• Quantize the free radiation field inside a photonic crystal(no real charge, no current).

• The radiation field can be composed of a vector potential and a scalar potential:

Φ∇−∂∂−=

t

AE

rr

ABrrr

×∇=

The radiation field is invariant under gaugetransformations. Since we do not have a real charge, wecan find a transformation to eliminate the scalar potential.


• Maxwell‘s equations tell us further:

[ ] 0)(0)( =∇∂∂−⇒=∇ Art

Errrrrrr

εε

• That is different from the usual obtained Coulomb gauge. But we know:

0=∇ Arr

• So, here we get:

[ ] [ ])(log0)( rAAArrrrrrrrr

εε ∇−=∇⇒=∇


• From Maxwell‘s equations we get:

At

Aµ

rrrr2

2

00

1

∂∂−=×∇×∇ ε

01

2

2

2=

∂∂−∆ Atc

r

∑ += −

λ

ωλλ

ωλλ

λλ titi eruCeruCV

A )()(1 ** rrrrr

• Expansion in Bloch functions due to photonic crystal periodicity


• Using this ansatz yields:

02

2

=

−∆ λ

ωu

c

r

• Additionally, the following equation has to be satisfied:

0=∇ λurr

rkieerurrrrr

λλλ =)(

• Therefore, try the following ansatz for the mode function:


• Now, choose the fourier coefficients in a „clever“ way:

λλ

λ εωaC

02

h→

λλ aa ˆ→ +→ λλ aa ˆ*

• As the coefficients describe the temporal evolution of a harmonic oscillation, we may replace the a by operators:

• These operators obey the same commutation relation as the one known from the HO:

'' ]ˆ,ˆ[ λλλλ δ=+aa


[ ])(*)(

0

ˆˆ2

1 trkitrki eeaeeaV

A λλ ωλλ

ωλλ

λ λωε−−+− += ∑rrrr rrh)

∫ ∑

+=

+= +

λλλλωε

2

1ˆˆ

1d

2

1 2

0

20

3rad aaB

µErH h

)))

• Now, let us collect all ingredients:

• Exercise: Derive E and B from this vector potential. With E and B we finally get the desired (and expected) result forthe free radiation operator:

0. Introduction






Interaction with matter

( ) )(2

1 2

int rAepm

Hr)))

Φ+−=

( ) 22

ˆ2

ˆˆˆˆ2

Am

epAAp

m

e ++−=

• The interaction Hamiltonian is given as the differencebetween the Hamiltonian for the electron in presence of theradiation field and in its absence.

• Gauge transformation takes care of the scalar potential ...

• ... and the quadratic terms are neglected, as they describephoton-photon interaction (so-called low intensity limit), or free electron movement.

X

X


[ ] Ai

pAApAp ˆˆˆˆˆˆ,ˆ ⋅∇=⋅−⋅= h

[ ]m

pixH

ˆˆ,ˆ

atom h−=

• Exercise: Derive the commutation relation between p and A. Result: The operators p and A do not generally commutate.

• Therefore, we get (due to the chosen gauge transformation):

)log(ˆ2

ˆˆint ε∇−⋅−= A

m

iepA

m

eH

h)

• Can we get some meaning out of the p operator? Trick:

dipole moment


• Using additionally the dipole aproximation:

We end up with the following interaction Hamiltonian (longcalculation):

1≈rkierr

( )∑ −+ +=λ

ωλλ

ωλλ

λλ σσ titi eageagiH ˆˆˆˆˆ21

*12int h

∑=ij

ijijH σω ˆˆatom h

Bare atom operator

eV

dg

rh

h

r

λλ ωε

ω0

2112

2=

Coupling between atom and PC modes

Here we used:


• The complete Hamiltonian now reads:

( )∑ −+ +=λ

ωλλ

ωλλ

λλ σσ titi eageagiH ˆˆˆˆˆ21

*12int h

∑=ij

ijijH σω ˆˆatom h

intatomradˆˆˆˆ HHHH ++=

( )∑ +=λ

λλλω aaH ˆˆˆrad h

0. Introduction






The bound atom-photon state

• Model: Two-level atom in a photonic crystal with a bandstructure described by an effective massapproximation:

20 )( kkCck −+=ωω

• Choose the energy origin to coincide with the excited atomlevel:

02 =E• Eigenstates are a superposition of the atom-states and the

photonic crystal states:

tiei Ω−⊗=Ψ λ

11ˆ12atom ωh−=⇒ H

• Describe the Hamiltonian in Rotating Wave Approximation (RWA).


• Special cases: 0;2

Atom in excited state |2>

No photons inside the photonic crystal

tie 12;1 ωλ −

Atom in ground state |1>

Photon in PC mode λ

• Therefore, we use the following ansatz:

tietDtCt λω

λλ λ −∑+=Ψ ;1)(0;2)()(


• We are interested in the time evolution of the system, starting from the following initial condition: C(0)=1; D(0)=0,i.e., excited atom in empty photonic crystal.

• Therefore, derive the equations of motion for theseamplitudes (Schrödinger equation):

tietDitCitdt

di λω

λλλ λω −∑ ++=Ψ ;1))((0;2)()( &h&hh

)()(ˆ ttH Ψ=

The atom-photon bound state

';1)(ˆˆ

';1)(ˆˆ

0;2)(ˆˆ

0;2)(ˆˆ

'

'

'21

'12

21

12

λσ

λσ

σ

σ

λ

λ

ω

λλλ

ω

λ λλ

λλ

λλ

ti

ti

etDagi

etDagi

tCagi

tCagi

−

−+

+

∑∑

∑ ∑

∑

∑

+

+

+

h

h

h

h

+−= −∑ λω λω

λλ ;1)(12

tietDh

+−+∑ ';1)(ˆˆ '

'' λω λω

λλλλλλ

tietDaah

Act like a delta functions


0;2)(

;1)(

tietDgi

tCgi

λωλ

λλ

λλ λ

−∑

∑ +

h

h

+−= −∑ λω λω

λλ ;1)(12

tietDh

+−∑ λω λω

λλ ;1)(12

tietDh

tietDitCi λω

λλλ λω −∑ ++ ;1))((0;2)( &h&h


• Comparison of coefficients yields equations of motion:

tietDgtCt

λω

λλλ

−∑=∂∂

)()(

tietCgtDt

λωλλ )()( =

∂∂

• Formal integration finally yields the time evolution of theexcited state:

'd )'()( '

0

tetCgtD tit

λωλλ ∫=

∫ −=∂∂ t

ttCttGtCt 0

'd )'( )'()(

λ;1

0;2


• Here the Greens function (or the so-called memory kernel) describes the effect of the electromagnetic environment on the atomic system:

∑ −−−Θ=−λ

ωλ

λ )'(2 )'()'( ttiegttttG

• The proper evaluation of this Greens function is beyond thescope of this lecture. If interested, see the paper by N. Vats, K. Busch, „Theory of fluorescence in photonic crystals“.

• All interesting details are hidden in the coupling constantand in the summation over the different modes.

For in-depth reading: Phys. Rev. A 65, 043808 (2002)


Temporal evolution of the excited state population for an initially excited two-level atom near an anisotropic band-edge for various values of the detuning δ of the atomic frequency from the band-edge frequency.

0. Introduction


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 Quantum optics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers


Motivation

“I call any geometrical figure, orgroup of points, chiral, and say that ithas chirality, if its image in a plane mirror, ideally realized, cannot bebrought to coincide with itself.“

Baltimore Lectures, given in 1884 by Lord Kelvin.

“On the maintenance of vibrations byforces of double frequency, and on thepropagation of waves through a mediumendowed with a periodic structure”

Philosophical Magazine, 1887 by Lord Rayleigh.

Chirality

• An object is called chiral, if its mirror image cannot be overlaid with the original via rotation.

• A chiral object and its mirror image are called enantiomorphs or enantiomers.

• Example: Your hands.

Motivation & Basics

• Examples:

Aminoacid

ThalidomidDNA (RH)

Lime

www.google.com

Motivation & Basics

)( kztixxx epEE −= ωrr

)( kztiyyy epEE −= ωrr

( ) )( kztiyyxx epEpEE −+= ωrrr

∆==⋅ iiijji aeEpp ;δrr

Choose two orthogonal linear polarizations as base:

Adjusting the phase difference allows for any polarization state:

Motivation & Basics

• Examples:

E

y

Ex

ELIN

Ex

ERCP

Ex

Ex

polarization of light

linear polarized circular polarized

Ey Ey

Chirality <–> Optical Activity

• The term optical activity derives from the interaction of chiral materials with polarized light:

An optically active material rotates the plane of polarization of a beam of plane polarized light in a counterclockwise or clockwise direction, depending on the handedness of the material.

• This property was first observed by Jean-BaptisteBiot in 1815.

Lakhtakia, A. (ed.) (1990). Selected Papers on Natural Optical Activity (SPIE Milestone Volume 15)

Motivation & Basics

• Interaction of polarization with chiral matter• Circular birefringence (optical activity)

•hht

L - aevo(LH)

But how to distinguish between L- and D-versions of chiral objects (e.g. 2-Butanol C4H10O ) ?

D - extro (RH)

ChemSketch Freeware

Motivation & Basics


•t

•t

polarization plane turns right !

RCPLCP nn >

Optical activity = Circular dichroism

• Circular dichroic materials exhibit different refractive indices for left- or right-circular polarized light.

• This leads to a rotation of linear polarized light, with the rotation angle

( )RL nnd −=0λ

πβ

Motivation & Basics

• Interaction of polarization with (chiral) matter• Circular birefringence (optical activity)

Example: NaClO3 turns 3,1°/mm (λ0= 589nm and 20°C)

)(0

RCPLCP nnd −=

λπβ

• natural

• induced (e.g. Faraday-effect)

Example: H2O turns 2°/cm (B=1T, λ0= 589nm and 20°C)

BdTV ),(ωβ =

Motivation & Basics


• Circular dichroism

•t•t

• But remember the linear effects !

only 50% RCP

is transmitted

S- and R- Isomers of Bromchlorf-lourmethan (Wikipedia)

Beetles of the Plusiotisfamily:Plusiotis batesi shows selective reflectivity under circularly polarized light, Plusiotis resplendens does not.

Chirality in nature

Motivation & Basics

• Interaction of polarization with chiral matter

So far:

Chiral objects are much smaller than thewavelength they are observed with !

Question:

What will happen if we reach feature sizeswhich are comparable to the wavelength of light ?

Chirality <–> Optical Activity

• Artificial composite materials displaying the analog of optical activity but in the microwave regime were introduced by J.C. Bose in 1898, and gained considerable attention from the mid-1980s.

Bose, J. C. (1898). "On the rotation of plane of polarisation of electric waves by a twisted structure, Proc. R. Soc. Lond. (Vol. 63, pp. 146-152)

Ernest L. Eliel and Samuel H. Wilen (1994). The Sterochemistry of Organic Compounds, Wiley-Interscience.

Polarization stop bands

• Can we fabricate artificial structures, which allow for the propagation of one circular polarized component, but not for the other?

• Can we construct an optical diode, which allows one wavelength of light to propagate in forward but not in backward direction?

• Can we construct an optical isolator, which prevents backreflected light from entering a laser?

Example I: Optical diode with chiral cholestericliquid crystals (CLC)

CLC are chiral structures with a pitch P. These structures show selective reflectivity for circularly polarized light.

If combined with a half-wave plate, this heterostructure acts like an optical diode.

Optical diode with chiral cholesteric liquid crystals (CLC)

• Band diagrams for the single CLCs (LHS).

• Band diagram for a LCP and RCP heterostructure.


• LHS: simulated transmittance spectra for single CLC for different wavelength.

• RHS: simulated transmittance for the heterostructure: blue forward, red backward propagation.


• Experimental transmittance for the heterostructure: blue forward, red backward propagation, for light with the same (solid lines) or opposite handedness (broken lines).

Nature Mater. 4, 383 (2005)

Example II: Chiral photonic crystals

5 µm 5 µm

1.5 µm

Properties of 3D Chiral Photonic Crystals

“Spiral three-dimensionalphotonic-band-gap structure“

A. Chutinan & S. NodaPhys. Rev. B 57 (1998)

• Photonic band gap material

“Proposed Square Spiral Microfabrication Architecture for Large

Three-Dimensional Photonic Band Gap Crystals“

O. Toader & S. JohnScience 292 (2001)

Left and right handed spirals

Different behavior for left and right circularlypolarized light.

M. Thiel et al., submitted (2006)

Experiment: Polarization stop bands


Theory: Polarization stop bands


An intuitive explanation ...

A „poor-man‘s“ optical isolator• 1D-3D heterostructure: first measurements

A Compact Thin-film polarizer

M. Thiel et al., submitted

• The concept of the polarizer

quarter-wave plate

quarter-wave plate

chiral photonic crystal

A Compact Thin-film polarizer• Scattering matrix calculations

M. Thiel et al., submitted

rx = 0.22 µm

d = 0.78 µmh = 3.70 µm

a = 1.30 µm

A Compact Thin-film polarizer• 1D-3D-1D heterostructure

Another design: twisted woodpiles


Left-handed structure Right-handed structure

S-Matrix simulations


• Twisteds woodpiles are much easier to fabricate.

• Polarization stopbands reach into the NIR and even the visible region.


• Experimental verification.


Example III: Glancing angle deposition

• Different Alq3 samples prepared via GLAD:

• (a), (b) five turns with different evaporation angle

• (c) 19 turns

• (d) 40 turns

Appl. Phys. Lett. 88, 251106 (2006)

Properties of GLAD samples

• (a) Selective transmittance of circularly polarized light spectra for samples deposited at values of 75°, 76°, 80°, and 85°.

• (b) The peak selective transmittance of circularly polarized light and peak degree of circularly polarization to the photoluminescent output of the films tend to be higher for samples fabricated at higher deposition angles.

Appl. Phys. Lett. 88, 251106 (2006)

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