NonlinearNonlinear OpticalOptical EffectsEffects in in PhcPhc and and MetamaterialsMetamaterials

Concita Sibilia

Dipartimento di Energetica

Universita’ di Roma La Sapienza

Concita.sibilia@uniroma1.it

Introduction

:Nl

OpticsNonlinear 1-D Photonic Crystals: enhancement

of

quadratic

interaction. Fields localization & Phase matchingRelevance of fields’overlap on frequency

conversion

efficiency.

Generation of Entangled two-photon state

Conclusions

Outline

Metallo-Dielectrics

Nanostructured metals

Introduction

Question:Is it possible to change the color of a monochromatic light?

Answer:Not without a laser light

output

NLO

sample

input

2. The Essence of Nonlinear Optics

When the intensity of the incident light to a material system increases the response of medium is no longer linear

Input intensity

Output

How does optical nonlinearity appear

The strength of the electric field of the light wave should be in the range of atomic fields

N

a0

e

νh20/ aeEat =

220 / mea =

esu102 7=×≈atE

Response of an optical Medium

The response of an optical medium to the incident electro magnetic field is the induced dipole moments inside the medium

μ

νh

νhνh

νh

μ μ

μ

μ

)(ˆ)( tEtP α=

Lasers made available highly coherent radiation that could be concentrated and focused to give extremely high local intensities that can reach 1018 W/cm2.Discovery of second harmonic generation by Franken et al. in 1961 is considered the beginning of the field of nonlinear optics. A rich stream of new phenomena soon followed. Nonlinear optics plays an important role in telecommunications and future computer technologies. The relatively long interaction lengths and small cross sections available in waveguides and fibers means that low energy optical pulses can achieve sufficiently high peak intensities to put in evidence non linear effects also in many transparent optical materials with weak nonlinearities.

When the strength of the applied field is comparable to the atomic field strength, the linear approximation is no longer valid. By performing a power expansion of P it is possible to consider the nonlinear terms in the polarization function:

...)()()(...)()()()( )3()2()1(32 +++=+++= tPtPtPtEtEtEtP γβα

Introduction to non linear optics

Nonlinear Polarization

• Permanent Polarization

• First order polarization:

• Second order Polarization

• Third Order Polarization

0iP

jiji EP)1(1 χ=

kjijki EEP)2(2 χ=

lkjijkli EEEP)3(3 χ=

Nonlinear Optical Interactions• The E-field of a laser beam

• 2nd order nonlinear polarization

C.C.)(~ += − tiEetE ω

)C.C.(2)(~ 22)2(*)2()2( ++= − tieEEEtP ωχχ

ωω

ω2)2(χ

2nd Order Nonlinearities • The incident optical field

• Nonlinear polarization contains the following terms

..)(~ 21 21 CCeEeEtEtiti ++= −− ωω

(OR) )(2)0(

(DFG) 2)(

(SFG) 2)(

(SHG) )2(

(SHG) )2(

*22

*11

)2(

*21

)2(21

21)2(

21

22

)2(2

21

)2(1

EEEEP

EEP

EEP

EP

EP

+=

=−

=+

=

=

χ

χωω

χωω

χω

χω

1ω

2ω)2(χ

1ω

2ω213 ωωω +=

Sum Frequency Generation

1ω3ω

2ωApplication:Tunable radiation in the UV Spectral region.

Application:Tunable radiation in the UV Spectral region.

Application:The low frequency photon, amplifies in the presence of high frequency beam . This is known as parametric amplification

Application:The low frequency photon, amplifies in the presence of high frequency beam . This is known as parametric amplification.

2ω

1ω

1ω

2ω)2(χ

2ω

1ω213 ωωω −=

Difference Frequency Generation

1ω3ω

2ω

Linear susceptibility tensor:

EP )1(0)1( χ̂ε=

For anisotropic media is a second rank tensor with 9 components (3x3 matrix).

For isotropic media

)1(χ̂

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

ˆ )1()1( χχ EP )1(0)1( χε=

Nonlinear susceptibility tensor:

zyxi EEPzyxkj

kjijki ,,;ˆ,,,

)2(0

)2( == ∑=

χε)2(χ̂ Third rank tensor (27 components)

zyxi EEEPzyxlkj

lkjijkli ,,;ˆ,,,,

)3(0

)3( == ∑=

χε

The Non linear Optical Susceptibility Tensor

[m/V]

)3(χ̂ Fourth rank tensor (81 components) [m2/V2]

Only non centrosymmetric crystals can posses a non-vanishing second order nonlinear susceptibility tensor. In a centrosymmetric crystal, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Thus a reversal of the sign of Ej and Ek must cause a reversal in the sign of Pi(2):

( ) ( )kjizyxkj

kjijkkji EEPEEEEP ,))((ˆ,)2(

,,,

)2(0

)2( −=−−=−− ∑=

χε

Possible only if all the elements of the nonlinear susceptibility tensor are zero.

The Non linear Optical Susceptibility Tensor

Second order nonlinear effects

( ),:ˆ )2(22

002

2

0002 EE

tE

tE

tE r χεμεεμσμ ∂

∂+

∂∂

+∂∂

=∇

In our one dimensional model we have:

( ),),(),(),(),( )()()(22

2321

tzEtzEtzEz

tzE ωωω ++∂∂

=∇

Performing the second order derivative and assuming that the variation of the complex field envelopes with z are small enough so that:

),()( 122

11 zEdzdkzE

dzd

ii >>Slow varying envelope approximation

SVEA

( )[ ]( )[ ]( )[ ]..)(

21),(

..)(21),(

..)(21),(

333

222

111

3)(

2)(

1)(

ccezEtzE

ccezEtzE

ccezEtzE

zktijj

zktikk

zktiii

+=

+=

+=

−

−

−

ωω

ωω

ωω

Being i,j,k the Cartesian coordinates and can each take on values x and y.

[ ]

( )

( )

( )

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡++

+⎥⎦⎤

⎢⎣⎡ ++

+⎥⎦⎤

⎢⎣⎡ +

−≅++=∇

−

−

−

zktijj

zktikk

zktiii

edz

zdEikzEk

edz

zdEikzEk

edz

zdEikzEk

tzEtzEtzEdzdE

33

22

11

321

)(2)(

)(2)(

)(2)(

21),(),(),(

333

23

222

22

111

21

)()()(2

22

ω

ω

ω

ωωω

Second order nonlinear effects

The valuation of the Laplacian within the SVEAT approximation gives:

Finally, with some algebra:

zkkki

yxkikijik

rj

r

j

zkkki

yxjijikij

rk

r

k

zkkki

yxkjkjijk

ri

r

i

eEEiEdz

dE

eEEiEdz

dE

eEEiEdz

dE

)(

,,21

)2(

3

0033

30

033

)(

,,

*31

)2(

2

002*2

20

02*2

)(

,,

*23

)2(

1

0011

10

011

321

231

123

ˆ22

ˆ22

ˆ22

−+−

=

+−−

=

−−−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛′−−=

⎟⎟⎠

⎞⎜⎜⎝

⎛′+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛′−−=

∑

∑

∑

χε

εμωεε

μσ

χε

εμωεε

μσ

χε

εμωεε

μσ

Phase Matching

ω)2(χ

ω2

•Since the optical (NLO) media are dispersive,The fundamental and the harmonic signals havedifferent propagation speeds inside the media.

•The harmonic signals generated at different points interfere destructively with each other.

•Since the optical (NLO) media are dispersive,The fundamental and the harmonic signals havedifferent propagation speeds inside the media.

•The harmonic signals generated at different points interfere destructively with each other.

frequency

Ref

ract

ive

inde

x

2ω

ω

NOTE: In order to have efficient energy transfer among the fields, a long interaction length is required, the phase mismatch should be as close as possible to zero. In other words:

k3 =k2 +k1

Second order nonlinear effects

In a more general case, the condition to be fulfilled is:

213 kkk +=

It is called phase matching condition and it can be interpreted as a momentum conservation requirement. Example: Two pump fields at frequencies w1 and w2 can generate a sum frequency (ω3 ) field. The wavevector of the generated field will fulfill:

2k1k

3k

To obtain an expression for the second harmonic power output, consideringA linearly polarized pump with non vanishing electric field component E1x we use the relation:

*3330

)2(

21

jjr EEcAreaP εε

ω

=

If we consider the SH generated linearly polarized with non vanishing field componentE3y we can write an expression for the conversion efficiency:

( );

2/)2/sin(

21 2)(

22

30

2)2(22

)(

)2(

⎥⎦⎤

⎢⎣⎡

ΔΔ′

==kL

kLAreaP

nncL

PP yxx ω

ωωω

ω

εχω

η

The conversion efficiency is linearly growing with the pump intensity. This means that the generated second harmonic intensity with respect to the pump intensity follows a quadratic law. The conversion efficiency increases with the squared length of the nonlinear medium

Optical Second harmonic Generation

Optical Second harmonic Generation

First experimental report on second harmonic generation was performed by Franken, Hill, Peters and Weinreich in 1961. (Fig: A. Yariv, Quantum electronics)

SHG can be studied as the limiting case of the three frequency interaction where two of the frequencies ω1 and ω2 are equal and ω3 =2ω1 .We assume as first approximation that the amount of power lost by the input beam is negligible so that dE1i /dz=0 and the medium is transparent at ω3 ( σ=0)

Maximum conversion efficiency is achieved for ΔkL/2=0,Δk=k2ω

-2kω

=0 is called phase matching condition and it is fulfilled as long asThe refractive indices at FF and SH frequencies are the same: n2ω

=nω

.If such condition is fulfilled, the FF and SH fields propagate with the same phase velocities

We note that the conversion efficiency is crucially determined by the sinc function:

Phase matching

In common materials the refractive index is an increasing function of the frequency thus we have nω

0.During the non linear process there is an exchange of energy between FF and SH field and the SH production is zero for any propagation length that satisfy the law:

ΔkL/2=mπ, where m is an integer.

For a mismatched interaction, the length of non linearmaterial that produce the maximum generatedsecond harmonic field can be calculated by:

( )ωωλπ

π

nnkl

kl

c

c

−=

Δ=

=Δ

2

0

4

;22

Typically in nonlinear materials Δn is of the order of 10-1to 10-2. Coherence lengths are only a few numbers of wavelenghts.

Phase matching

Undepleted pump approximation

Phase matching techniques

A widely used technique takes advantage of the natural birefringence of anisotropic crystals.Under certain circumstances it is possible to use the different refractive indices for the ordinary wave and the extraordinary wave. For example, a typical behaviour of dispersion of the refractive indices of a negative (ne

Quasi Phase matching

Phase matching techniques

Periodic modulation of the non linear coefficients tensor elements responsible for the interaction. It can be shown that the phase matching condition becomes:

Δk=2πm/ΛWhere m is an integer and Λ is the period of the nonlinearity.

EXAMPLE: If the sign of the non linear interaction is reversed at every coherence length d(z) is a periodic function of period 2lc =2π/Δk. QPM is achieved for m=1.

deff =2dbulk /π

Periodically poled LiNbO3Periodical reversal of electric field in order to induce a permanent periodically modulated electric polarization

Phase matching techniques

Considering dielectric waveguides, the modal dispersion can be used to achieve phase matching. The phase matching relation for second harmonic generation becomes:

ωω ββ~~

2 =

This condition can be easily fulfilled by considering modes for the pump and for the SH of different order or polarization.

Nevertheless the non linear process is governed by how the fields overlap: usually overlaps between fields belonging to different orders are not efficient.

( ) ( ) 2*22)2( )()(:ˆ ⎥⎦⎤

⎢⎣⎡ ⋅∝ ∫

∞+

∞−dxxExE nm

ωωχη

Phase matching techniques

Photonic CrystalsIn a photonic crystal both the linear and the nonlinear susceptibility functions can be periodically modulated. Modulation of the linear susceptibility is responsible for peculiar properties of the linear dispersion curves of these structures. Fields are characterized by a Bloch wave vector and an periodic function over the unit cell. Thus the quasi momentum conservation for second harmonic generation is:

02 =−− Gkk ωωWhere G is a reciprocal lattice vector. Usually unit cells are of the order of the wavelength or less unlike in the QPM regime where domains are inverted every coherence length.

Three wave mixing

The basic equations govern the interaction of three fields due to second order nonlinearity:

kzi

r

kzi

r

kzi

r

eEEidz

dE

eEEidz

dE

eEEidzdE

Δ−

Δ

Δ−

′−=

′+=

′−=

21)2(

3

0033

*31

)2(

2

002*2

*23

)2(

1

0011

ˆ2

ˆ2

ˆ2

χε

εμω

χε

εμω

χε

εμω

Different processes can happen depending on the initial condition: ω3ω1

ω2Sum frequency generationFrequency up-conversion

Parametric amplificationDifference frequency generationStimulated parametric down conversion

Spontaneous parametric down conversionSpontaneous parametric fluorescence

Summary

Third order optical nonlinearities

Third order nonlinear polarization allow coupling of fields at different frequencies, Here we present an sketched overview of the most relevant effects that will be analyzed later:

ω1

ω3ω2

ω4

Four wave mixing

ω

3ω

Third harmonic generation

ωω

ω

ω

Optical Kerr effect

ω−ω

ω

ωs

Raman effect

−ωωs

Intensity dependence of the refractive index

The refractive index of many materials depends on the intensity of the incident optical field. If we consider an optical field of the form:

( ) 2202 )()()(,~ ωωωω EnnEn +=

[ ]xcceEt ti ˆ.)(21)(E += − ωω

Time averaged intensity is defined:

200 )(2

1 ωε EncI =

The nonlinear refractive index is defined:

For instantaneous response:

(t)E(t)E(t)Eˆ)(P )3(0χε=tNL

Intensity dependence of the refractive index

Intensity dependence of the refractive index has drastic effects both on the spatial properties of optical beams and on temporal (spectral) properties of ultrashort pulses.

Self phase modulation (SPM)Phase shift experienced by short pulse propagating through a nonlinear refractive index- spectral broadening; - Optical solitons in anomalous dispersion regimes of fibres.

Cross phase modulation (XPM)Phase shift of a field induced by a co-propagating intense field at different wavelength.

Self focusing and self defocusing of optical beamsDepending on the sign on the nonlinearity, the central (more intense) portion of aBeam experiences higher (lower) index of refraction with respect to the outer edges.An effective lens is created.

Photonic

Crystals

1D PCh

-

Band Edge

EffectsFinite size 1D photonic crystals transmission bands exhibit resonance peaks

due to the boundary conditions with the external ( homogenous ) world.In particular, resonances are sharper in proximity of the band edge.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Frequency0.00.10.20.30.40.50.60.70.80.91.0

Tran

smis

sion

Pass bands

Band gaps

0

0.2

0.4

0.6

0.8

1.0

0.56 0.57 0.58 0.59 0.60 0.61

ΔωII ΔωI

0

4

8

12

0 4 8 12

z (μm)

| Φ ω|2

No Hermitian

Eigenproblems

3

5

7

(in u

nits

of 1

/c)

tieTyixtφ

ωωω =+= )()()(

πφ mxyt ±−= )/(1tan

Dnc

Dk efft )()( ωωωφ ==

Density of modes :

dϕ/dw

= (1/D)dk/d w

DOM = dk/dw=

(1/D)(y’x-x’y)/(x^2+y^2)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Frequency

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Tran

smis

sion

Pass bands

Band gaps

:Transmission function for the structure depicted in Fig.(1). Note the frequency pass bands and band gaps.

No Hermitian

Eigenproblems

tieTyixtφ

ωωω =+= )()()(

πφ mxyt ±−= )/(1tan

Dnc

Dk efft )()( ωωωφ ==

Vg= dw/dk=1/DOM

0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 5 1 . 0 1 . 5

ω / ω 0

|t|2

1D 1D PChPCh

--

Band Band EdgeEdge

EffectsEffects

-- GroupGroup

velocityvelocity

( ) 022

2

2

=+ ωωω εω E

cz

dzEd

The boundary conditions at the input (z=0) and output (z=L) surfaces are: EIw+Erw=Ew(0); Etw=Ew

(L)exp[-i(w/c)L)]; i(w/c)(EIw-Erw)=dEw

(0)/dz; i(w/c)Etw=(dEw

(L)/dz)exp[-i(w/c)L).

Φω

(z) =Eω

(z)/EIω

; tω

=(Etω

exp[i(ω/c)L]/EIω

;

( )

( )∫

∫

Φ

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ΦΦ−

=

∗

L

L

E

dzz

dzdz

dicV

0

2

0

2Re

ωω

ωω

ω

εω

No Hermitian

EigenproblemsThe energy velocity is defined as the ratio of the Poynting vector integrated over the volume to the average total energy stored in the same volume

1D 1D PChPCh

--

Band Band EdgeEdge

EffectsEffects

-- EnergyEnergy

velocityvelocity

( ) ( )ωω

ωgE VtV

2=

0 .4

0 .6

0 .8

1 .0

units

of c

) ;

|t|2

(a.

u)

0

0.2

0.4

0.6

0.55 0.57 0.59 0.61

N→ ∞N=20N=200

N=200N=20

ω/ω0

V E

(in

units

of

c)

G.D’Aguanno et al

-χ(2)

–

Nonlinear frequency Conversion-

SHG

ωω

2ωPnl(2ω) χ(2) E2(ω)

-Down Conversion

ω

ω

Nonlinear Interaction

2ωIn Bulk:

PM and χ(2)

Nonlinear quadratic Interaction in PhC-χ(2)

, interface , bulk (in each layer)

-P.M. (equal PHASE velocity),

For SH 2K1 = K2n(w) = n(2 w)

thanks to the geometrical dispersion-Field enhancement due to field localization

( band edge and when T=1)

-

Large conversion efficiencyM.Scalora et al 1997

Effective refractive index

πϕ mxytg ±⎟

⎠⎞

⎜⎝⎛= −1t

*Centini et al, Phys. Rev. E 60, 4891-4898 (1999)

frequency

Ref

ract

ive

inde

x 2ω

ω

Φ≡≡+= ii eeTiyxt tϕ

Dnc

DkTi effefft ˆˆlnωϕ ≡≡−=Φ

⎥⎦⎤

⎢⎣⎡ +−= )ln(

2ˆ 22 yxi

Dcn teff ϕω

Bulk Medium

ωF 2ωF

Re(

n eff)

frequency

Multilayer Structure

Phase Matching

But

Field

overlap

more important

than

P.M !!!!

The expression of the conversion efficiency (η=ΙSH /Ιpump )in the non-depleted pump regime is similar to the bulk case

22 ( , ) 2( , )

2 ( , ) ( , ) (2 , )0

8 eff pumpp s p

eff eff eff

d L Ic n n nω ω ωπ

ηε λ

+ −+ − =

( , ) (2) 2( ) *( , )2

0

1 ( ) ( ) ( )D

effd z z z dzL ω ωχ+ − + + −= Φ Φ∫

deff contains both information on PM conditions and fields overlap

*G. D’Aguanno, et al. J. Opt. Soc. Am. B 19, 2111 (2002)

Conversion Efficiency U-Pho

NL layers =λ0

/4Linear layers = λ0

/2

24

68

1030

210

60

240

90

270

120

300

150

330

180 0

n1=1; n2(ωF )=1.428; n2(2ωF )=1.616.

0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 20

0 . 2

0 . 4

0 . 6

0 . 8

1

w / w 0

T

n1=1; n2(ωF )=1.428; n2(2ωF )=1.676.

0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 20

0 . 2

0 . 4

0 . 6

0 . 8

1

w / w 0

T

Phase matched interaction

Non Phase matched interaction

Fields’ Overlap

* Centini et al. Opt. Letters (2004)

( , ) (2) 2( ) *( , )2

0

1 ( ) ( ) ( )D

effd z z z dzL ω ωχ+ − + + −= Φ Φ∫

22 ( , ) 2( , )

2 ( , ) ( , ) (2 , )0

8 eff pumpp s p

eff eff eff

d L Ic n n nω ω ωπ

ηε λ

+ −+ − =

For periodic finite structures, tuning the pump at the band edge resonance, PM is always achieved if the second harmonic is tuned to the second resonance*

* Centini et al Phys Rev E 60 4891 (199

Phase Matching

Huge conversion for random sequences

0.85 0.9 0.95 1 1.05 1.1 1.15 1.210-3

10-2

10-1

100

101

102

103

104

105

106

12 μm etalonof nl material

Perfectly phase matched12 micron etalon

ω/ ω

0

SH energy(a.u.)

random

Coll. with D.Wiersma (Lens)

D’Aguanno et al. “Energy Exchange Properties during SHG in finite,1-D ,PBG structures with deep gratings”

Fig.2

η+SHη-SH

(GW/cm2)

0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

RFF

TFF

η SH

+ &

ηSH

−

IFF(MW/cm2)

0

0 .0 2

0 .0 4

0 .0 6

1 4 7 1 0

)(inputFFI Fig.2

η+SHη-SH

(GW/cm2)

0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

RFF

TFF

η SH

+ &

ηSH

−

IFF(MW/cm2)

0

0 .0 2

0 .0 4

0 .0 6

1 4 7 1 0

)(inputFFI

Undepleted pumpregime

SHG Conversion SaturationSHG Conversion Saturation

FF Anom

alous Transm

ission/Reflection

N^6

0

10

20

30

0 4 8 12

b)

z(μm)

|E|2

(arb

. uni

ts)

0

0.25

0.50

0.75

1.00

-500 0 500

Δϕ=0 Δϕ=0

Δϕ=πΔϕ=π a)

z (μm)

|E|2

(arb

. uni

ts)

δφω (units of π)

Con

vers

i on

E ffic

i enc

y

Q=0.5

Fig.(6a)

0

0.005

0.010

0.015

0.020

0 0.5 1.0 1.5 2.0

Control of SHG process

0

0.2

0.4

0.6

0.8

1.0

0.5 0.7 0.9 1.1 1.3 1.5

Tran

smitt

ance

ω /ω0 Fig.1(a)

FF SH

Q=Ipump2 /Ipump1

Second Harmonic Generation for counterSecond Harmonic Generation for counter--propagating propagating pumpspumps

62 NIT ≈

Y. Dumeige et al APL 2001, JOSAB 2002

Experiments on SHG

AlGaAs/Al2 O3

Simultaneous perfect phase matching for second and third harmonic generations in ZnS/YF3 photonic crystal for visible emissions

Weixin Lu, Ping Xie, Zhao-Qing Zhang, George K. L. Wong, and Kam S. Wong

Optics Express, Vol. 14, Issue 25, pp. 12353-12358

M. Centini, et al,"Simultaneously phase-matched enhanced second and third harmonic generation," Phys. Rev. E 64, 046606 (2001)

Experiment on Type II SHG :AlGaAs/Al2 O3λp=1.55 microns

ω-p polarized

ω-s polarized

2ω-p polarized

Non collinear Type II second harmonic generation

kω kω

k2ω

Momentum Conservation

p,θ2ω p,θωs,θω

p,kωs,kωp,k2ω

p,θ2ω p,θωs,θω

p,kωs,kωp,k2ω

k y[μ

m-1

]

kx [μm-1]0 5 10

5

10

GAP

( , )sk ω⎡ +⎣( , )pk ω ⎤⎦

12

(2 , )12

pk ω= (2 , )12 pk ω=

U-Pho

Seco

nd H

arm

onic

Sig

nal [

a.u.

]

θ ω ,p [deg]

0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

20 22 24 26 28 30 32 34 36

Seco

nd H

arm

onic

Sig

nal [

a.u.

]

θ ω ,p [deg]

0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

20 22 24 26 28 30 32 34 360

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

20 22 24 26 28 30 32 34 3620 22 24 26 28 30 32 34 36

P spectrum

S spectrum

P spectrum around SH

Ref

lect

ion

wavelength

1.51 μm

1.51 μm

755 nm

1.51 μm

1.51 μm

755 nm

1.51 μm

1.51 μm

755 nm

1.51 μm

1.51 μm

755 nm

Angular measurements:

PM interaction: FF fields tuned at the band edge resonance, SH tuned at the second resonance near the band edge

Slight Phase mismatch but the SH is tuned at the band edge resonance (higher field localization)

Fields at the Pass BandsNo Overlap inside the Structure

Almost no energy inside the photonic crystal

27 degsk ,ω

pk ,2ωp,ωθ pk ,ω

*Bosco et al APL 05

U-Pho

Nonlinearities in metals• In metals bulk production of second harmonic

radiation is usually inhibited due to symmetry reasons and only an electric quadrupole and a magnetic dipole terms may be active

• These reasons do not exhist at surface due to the lack of symmetry in a sheet region of the order of a few Å’s thickness between vacuum and the metal. In this case the discontinuity of the electric field is important

• Therefore, although the production of radiation has a very low efficiency (~10-10) it is interesting to examine what happens

the term E x B originating from the Lorentz force acts on the current produced at the first order by B (it is nonlinear in B). It is a bulk current in the metal and does not irradiate except if there is a boundary.

The other term is a surface electric quadrupole term. It is present in nonlinear reflection from a metal and originates from the discontinuity of the normal component of E at the interface which generates nonlinear currents. The surface currents are a tangential component and a bulk one directed along z.

The SH polarization from the perpendicular and parallel surface currents can be written as

(2 ) ( ) ( ) ( )surfacez z zaω ∝ ω ω ωP E E (2 ) ( ) ( ) ( )surfacex x zbω ∝ ω ω ωP E E

Dependence on fundamental beam polarization direction

• A typical apparatus is shown in the figure

• Since the surface nonlinear sources are in the simplest theory proportional to (∇·E)E, then incident fields polarized only in the plane of the surface (normal to the plane of incidence) do not contribute to the harmonic signal.

• However the bulk terms proportional to E×H lead to harmonic generation for all incidence polarizations

• Neglecting bulk terms, a [ cos(φ)

]4 dependence on the polarization is predicted, where φ

is the electric field angle

relative to the plane of incidence.

• Deviations from this relation are indicative of contributions from the bulk term, and in particular the ratio

• M = ISH (φ=90°) / ISH (φ=0°)

• yields information about the relative strengths of the surface versus bulk terms

• F.Brown et al., PRL 14 (1965) 1029• F.Brown and R.E.Parks, PRL 16 (1966) 507• N. Bloembergen et al., PR 174 (1968) 813

Dependence on fundamental beam polarization direction

High Pass band filter

800 nm150 fs1 kHz

Neutral density filters

A

PM

VA

Sample

Dichroic filters

BS

Prism

L2

L1

800nm

400nm

LaserSystem

HP-F

λ/2

α

A

PM

VA

Sample

Dichroic filters

BS

Prism

L2

L1

800nm

400nm

LaserSystem

HP-F

λ/2

α

PM

VA

Sample

Dichroic filters

BS

Prism

L2

L1

800nm

400nm

LaserSystem

HP-F

λ/2

α

p (TM)

s (TE)

φ

φ=0°

φ=90°Experimental Setup

Nonlinearities in metals

p (TM)

s (TE)

φ

φ=0°

φ=90°

When the fundamental wave is s polarized, only the bulk contribution is present

0153045607590

105120135150

-100 -50 0 50 100

Sample6 Ag/Ta2O5 Noise ≈ 1 mVSample5 Ag/Ta2O5

s polarized (TE)

p polarized (TM)

φ= angle between E and the incindent plane [ ° ]

SH

Sig

nal [

mV

]

Metallo-dielectric multilayer SH production in reflection

0153045607590

105120135150165180

-100 -50 0 50 100Noise ≈ 1mV

Fitting curve: Smax⋅ cos(φ)

4

Fitting curve: Smax⋅ cos(φ)

4- a ⋅ cos(φ)2 ⋅ sin(φ)2 + Smin⋅ sin(φ)4

s polarized (TE)p polarized (TM)

φ= angle between E and the incindent plane [ ° ]

SH

Sig

nal [

mV

]

0

10

20

30

40

50

200 400 600 800 1000

Wavelength [ nm ]

Tran

smitt

ance

[ %

]

Ag (20 nm)

Ta2 O5 (124 nm)

BK7 glass

The surface contribution is TM when the fundamental is TM(P), if the fundamental is TE there is no discontinuity of the normal component of E and the SH vanishes.

The bulk is always present, but only at the interface.

If fundamental is TM surface and bulk are present; if fundamental is TE only bulk is present.

600nm

0

2x10-12

4x10-12

6x10-12

8x10-12

-50 -25 0 25 50

pump ⊥ to the ripplespump = to the ripples

incidence angle (deg)

SH

con

vers

ion

effic

ienc

y

gold nanowires 5nm

S.H. signal

Sample AGold 5 nm/glass

ω 2ω

Second

Harmonic

Generation from nanostructured metals

Ti: Sapphireλ=800 nmλ nm SHG

λ=400 nmλ=400 nmλ=800 nmλ=800 nm

λ=400 nm+ λ=800 nm

λ=400 nm+ λ=800 nm

Dichroic filter

PM

Second Harmonic Generation Set-up (Reflection)

Laser p polarized

Ti:SapphireLambda=800nmPulse duration =130fsRepetition rate=1kHzAverage Power=78mW

Spot diameter on the sample=1.6mm

SHG

λ=400 nmλ=400 nm

λ=800 nmλ=800 nm

λ=400 nm+ λ=800 nm

λ=400 nm+ λ=800 nm

Dichroic filter

Ti: Sapphireλ=800 nmλ nm

PM

Second Harmonic Generation Set-up (Transmission)

pω )2(χiω

sω

Low efficiencySizeSpatial and frequency filtering needed

BULK MATERIAL

ENTANGLED PHOTON GENERATION

STATE of the ART: 10^6 twin photons/s at low temperature in a

Photonic Crystal Fiber

Optics Express, Vol. 13, Issue 2, pp. 534-544

1+1 D Ph Crystal

is ωω ,

si ωω ,

SizeHigh brightness per modeNarrow linewidthGuided entangled photon source

ENTANGLED PHOTON GENERATION In PhC

SizeHigh brightness per modeNarrow linewidth

1D Ph Crystal

pω

pω

( ) ( ) )()()()( )()( zzAzzAE nnnnn−−++ Θ+Θ=

Conclusions

Structures, suitable for degenerate and nondegenerate

generations of entangled photon pairs, were suggested using GaN/ALN layers

and AlGaAs/Al2O3.

We showed that it is possible to enhance or suppress the nonlinear effect by properly choosing the parameters of the structure.

Despite of what happens in bulk materials, the key feature for high efficiency of the nonlinear process is localization and overlap of

interacting fields in nonlinear layers.

- INTERlink Project -Miur.Italy-Phoremost NoE. EU

COST P11-ESF

ERO-US

Slide Number 1Slide Number 2Introduction2. The Essence of Nonlinear Optics How does optical nonlinearity appear Response of an optical MediumSlide Number 7Nonlinear PolarizationNonlinear Optical Interactions2nd Order Nonlinearities Sum Frequency GenerationSlide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Phase Matching Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 311D PCh - Band Edge EffectsSlide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45Second Harmonic Generation for counter-propagating pumps�Slide Number 47Slide Number 48Slide Number 49Slide Number 50Slide Number 51Slide Number 52Slide Number 53Nonlinearities in metals Slide Number 55Dependence on fundamental beam polarization direction Dependence on fundamental beam polarization direction Slide Number 58Slide Number 59Slide Number 60Slide Number 61Slide Number 62Slide Number 63Slide Number 64Slide Number 65Slide Number 66