Kauranen Erice2012 Part2 19072012 · – resonant surface modes exist • Proper approach –...

TAMPERE UNIVERSITY OF TECHNOLOGYDepartment of Physics

Nonlinear optics with metals

Martti KauranenDepartment of Physics

Tampere University of Technology

Finland


Outline

• Part II: Second-Order Response of Nanoscale Metals– higher-multipole radiation– dipole limit in effective response– nanodimers with nanogaps– local-field effects

Nanophotonics Summer School, Erice 19.7.2012 2


Second-order nonlinear optics

• Second-order polarization

• Vector quantities E P

• Tensorial response


(2) 2 (2) 2 2( ) ( ) i tP t E t E e

second-harmonic generation

( ) i tE t E e

(2) (2)

,:i j kijk

j kP E E P χ EE

x

y

z

Ex

Ey

(2) 2y yxx xP E


Symmetry issues

• Spatial symmetry– symmetry operations– interdependent tensor components

• Inversion

• Surfaces and thin films– centrosymmetry broken– probes based on SHG and SFG


(2) (2) (2)0ijk ijk lmnC

mirrorplane

r r(2) 2 (2) 2: ( ) : P χ E χ E P

E E P P

(2) 0χ


Multipole interactions

• Hamiltonian

• Second-order response

• Magnetic and quadrupole effects– different symmetry properties


EQBmE :H

EEχM :2mee

EEχBEχEEχP eeQeemeee ::2

EEχQ :2Qee

weak

e (m)

m (e)e

eem

electric-dipole-forbidden effects can occur

TAMPERE UNIVERSITY OF TECHNOLOGYDepartment of Physics Nanophotonics Summer School, Erice 19.7.2012 6

Multipolar?

• Light-matter interaction Hamiltonian

• Scattering (Heinz, Dadap, Brevet, …)

– dipolar interaction– retardation across particles

• Multipolar structures (Zyss, …)

– octupolar molecules

EQBmE :H

multipolar susceptibilities

Mie-type multipolar radiation patterns

eeeχeemχ

meeχ

eeQχQeeχ


Metal nanoparticles

• Plasmon resonances– collective oscillations of

conduction electrons

• Resonances depend on– size and shape– mutual ordering and coupling– dielectric environment

• Nanoscale variations

– local fields– ”hot spots”

– material properties– strong gradients

tieEE 0

enhanced nonlinearresponse

multipole effects

localfield


Second-order response

• Symmetry rule– noncentrosymmetric structures needed

• Normal incidence– avoid coupling with traditional

surface nonlinearity– sample must appear noncentrosymmetric

• Basic shapes– L-shaped nanoparticles– T-shaped nanodimers with a nanogap

• Typical sample dimensions– period 400-500 nm– gold thickness 20 nm



Theoretical descriptions

• Traditional susceptibility

• Effective medium approach?– sub-wavelength structure– resonant surface modes exist

• Proper approach– coupling of radiation fields to modes– local material properties– local nonlinear sources

,(2 ) ( ) ( )i ijk j k

j kP E E

nanoscale variations

excitation depends on experimental details

Dadap et al., PRL 83, 4045 (1999)JOSAB 21, 1328 (2004)


Nonlinear response tensor

• Definition [JOptA 8, S278 (2006)]

– macroscopic input-output fields (”scattering matrix”)

• Disadvantage– specific to experiment, not the sample itself

• Advantages– avoids nanoscopic difficulties– directly measurable quantities– all multipoles implicit– electric-dipole selection rules– equivalent to effective-medium susceptibility

,(2 ) ( ) ( )i ijk j k

j kE A E E

y

x


Tensor analysis

• Fundamental beam [JOptA 8, S278 (2006)]

– QWP modulation of polarization

• Polarized SHG signals

• Fit coefficients

2 2(2 )i i x i y i x yE f E g E h E E

x y xxx yxxf A A

relative complexvalues of Aijk

0 90 180 270 360QWP Angle [degrees]

0

1

2

3

4

SH In

tens

ity [a

rb. u

nits

]

Ex

x

y

Ex

EyQWP

1060 nm170 fs


Early results

• Linear spectra[JOptA 7, S110 (2005); APL 86, 183109 (2005)]

– axis shifts and dispersion of axes– optical activity

• Second-harmonic response[Opt. Exp. 12, 5418 (2004); 14, 950 (2006)]

– ”forbidden signals”– circular-difference effects– chiral symmetry breaking due to defects– varying levels of equivalent signals

• Nonlinear microscopy of nanodots[New J. Phys. 10,013001 (2008)]

– inhomogeneous tensorial SHG and THG

x yAB

Y

X


Multipole effects

• Multipole sources– electric dipoles– magnetic dipoles– electric quadrupoles

• Higher multipoles– magnetic dipoles and electric

quadrupoles cannot be separated?– can be separated from electric

dipoles

opposite interferencein transmission and

reflectionB

p

m

E

E

E

E

E

EQ

B

B

B

BB


Multipole experiment

• Fundamental beam– modulate polarization

with quarter-wave plate– angle of incidence

very small (~1°)

• Second-harmonic signals– s-polarized detection– compare reflected and

transmitted lineshapes


0 45 90 135 180

0.0

0.5

1.0

QWP Angle [deg.]

SH In

tens

ity

TransmissionReflection

clear differencesobserved

[PRL 98, 167403 (2007)]

p-pol s-polQWP

PMTPMT

s-pol

1060 nm170 fs

2

2 sample

TAMPERE UNIVERSITY OF TECHNOLOGYDepartment of Physics CFN Summer School, Bad Herrenalb 29.8.2010 15

Results of tensor analysis

• Symmetric and antisymmetric parts

• Result [PRL 98, 167403 (2007)]

sxxx xxx

s asxyy xyy xyy

s asxxy xxy xxy

A AA A A

A A A

allowed resonant

allowed nonresonant

forbidden (chiral)

transmission reflection |As| |Aas|

xxxxyyxxy

10.66 - 0.58i0.51 - 0.13i

10.37 - 0.67i0.37 - 0.26i

10.810.48

10.150.10

higher multipole amplitude ~20%

assume dipolar

x

y


Phenomenological model

• Full tensor analysis [Opt. Express 16, 17196 (2008)]

– ”forbidden” signals dominate and have strong multipole part

– chiral symmetry breaking

• Role of surface defects– non-equivalent defects at symmetrically

opposite sites– local dipolar sources retarded along the

direction of observation

/ 2 / 21 2 1 2 2 1(2 ) ( ) / 2ika ikae e ika E p p p p p p

effectivedipole effective

quadrupole

22

z

a


Effective multipole tensors

• Effective dipolar and magnetic tensors[New J. Phys. 13, 023025 (2011)]


metal trans. + + +

metal refl. + + -

substrate trans. + - +

substrate refl. + - -

eeexxxA mee

xxyA eyxxmeA

x

y

xy

y y

multipolar components about50% of dipolar components

eee eem meexxx xxx xxy yxxA A A A

0 45 90 135 1800.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

SH

inte

nsity

QWP rotation angle (degrees)


New samples

• Significantly improved quality[Opt. Express 18, 16601(2010)]

– narrow extinction peaks– high-order resonances observed– stronger SHG signals

• Four equivalent signals– all lineshapes overlap


effective dipolelimit reached

0 45 90 135 1800.0

0.2

0.4

0.6

0.8

1.0


Nor

mal

ized

SH

inte

nsity

path open for tailorablenonlinear metamaterials


Enhancement in nanogaps

gap-dependent FWM[Danckwerts and Novotny,

PRL 98, 026104 (2007)]

bowtie antenna[Fromm et al.,NanoLett. 4, 957 (2004)]

resonant antenna[Mühlschlegel et al.,

Science 308, 1607 (2005)]

self-similar spheresfor SHG[Li et al., PRB 72,153401 (2005)]

coupled dimers[Atay et al., NanoLett. 4, 1627 (2004)]


Designer dimers for SHG

• Symmetry rule– noncentrosymmetric structures needed

• Nanodimers [Nano Lett. 7, 1251 (2007)]

– T shape– noncentrosymmetric– vary gap between the bars– resonant with 1060 nm laser

• Expected result– only Y polarization enhanced– smallest gap leads to largest

enhancement

X

Y


Gap dependence of SHG

• SHG signals allowed by symmetry– pure polarization combinations for

normal incidence

,YXX YYYA A

X

Y

0 10 20 30 40

0

10

20

30

40

50

60

SHG

Inte

nsity

(arb

. uni

ts)

Gap (nm)

YXX YYY

nontrivial gap dependence

small gaplow response

YYY

YXX

1060 nm170 fs


Calculated local-field distributions

• Fundamental field– plasmon resonance with

the dimer– strong local-field effects– polarization conversion

• Second-harmonic field– off-resonant– weak local-field effects

local field contains new polarization

components Y incident, Y local

2 nm 23 nm

X incident, Y local

2 nm 23 nm

500 nm


Origin of SHG

• Local-field distribution– hot spots near the boundary of the dimer

• Surface nonlinearity– dominated by local component– integrate response around dimer perimeter

• Gap region– formally noncentrosymmetric– responses from top and

bottom tend to cancel nn X

Y

nnn

nnn

n

nnn

nnn

n

nnn

parts with opposite normal tend to cancel

asymmetric field distribution required

[Nano Lett. 7, 1251 (2007)]


Chiral symmetry breaking

• Slanted bar orientations– reflection symmetry broken

• Circular-difference response

samples are chiral

reflection plane

2 LHC RHC

LHC RHC

I ICDRI I

0 45 90 135 180

0,0

0,2

0,4

0,6

0,8

1,0

Nor

mal

ized

SH

G In

tens

ity

QWP Angle0 10 20 30 40

0

20

40

60

80

100

120

CD

Res

pons

e (%

)Gap (nm)


0 10 20 30 400

20

40

60

80

100

120C

D R

espo

nse

(%)

Gap (nm)


Local-field calculations


Part II: Conclusions

• Multipole effects– higher multipoles arise from surface defects

• Dipole limit reached– improved sample quality– multipole effects suppressed– prerequisite for nonlinear metamaterials

• Nanodimers– complicated gap dependence of SHG– symmetry and polarization of local fields


22

0 45 90 135 1800.0

0.2

0.4

0.6

0.8

1.0 M-T M-R S-T S-R

Nor

mal

ized

SH

inte

nsity


Kauranen Erice2012 Part2 19072012 · – resonant surface modes exist • Proper approach –...

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