Institut d'Optique d’Aquitaine, Bordeaux France · Institut d'Optique d’Aquitaine,...

Plasmonics: a few basics

Philippe Lalanne Institut d'Optique d’Aquitaine, Bordeaux – France

Laboratoire Photonique, Numérique et Nanosciences (LP2N)

Photons and nanosystems

Complex nanostructures

Cold atoms, matter waves

Biophotonic

Optics & numerics (virtual reality)

outline

Field localization (10mn)

Delocalized surface plasmons on metal surfaces

Wood anomaly

Localized plasmon

The « end » of the plasmon

optics physics chemistry

The magic confinement

photonic

plasmonic

J. Takahara et al., Proc. SPIE 5604, 158 (2004). D.K. Gramotnev and Sergey I. Bozhevolnyi, plasmonics beyond the diffraction limit, Nat. Photon. 4, 83-91 (2010).

DOS

Singularity (LDOS ng)

a/l

p/a Re(kz)

gap

0

a/l

LDOS singularity in periodic systems

Experimental recronstruction of the DOS in a photonic crystal waveguide

S. R. Huisman and al., Phys. Rev. B 86, 155154 (2012)

No singularity for real waveguide

Quenching

R. Amos and W.L. Barnes, Phys. Rev. B 55, 7249 (1997).

D

Quenching is predicted by

classical electromagnetic

theory quenching is

simply described by the

dielectric constant of the

substrate material.

x+fx/t = -(e/m) E0exp(-iwt)

Dipole momentum

p(w) = -ex(w) = e0a(w)E(w)

Dielectric constant:

.. .

0.6 1 1.4 1.8 -160

-120

-80

-40

0

40

l (µm)

Re(er)

Im(er)

electron

E

t = g-1

lp = 2pc/wp

wt-we

-=wa=eim

NeN

20

2 111r

The electron sea

tm

td

w

Dielectric thickness td (nm) G

roup Index

LDOS of MIM waveguides

Ag

AsGa

w =40 nm

w =100 nm

w =350 nm

w =

singularity

dm

d

tefft

nd

l

e

e

p

-

1

Stotal=0 M

M I

Stotal>0 M

M I

kSP

Dielectric thickness td (nm)

LDOS of MIM waveguides

Dam

pin

g (

µm

-1)

dm

md

t t

Im

d

142

e

ee a

tm

td

w

w =40 nm

w =100 nm

w =350 nm

w =

Ag

AsGa

outline

Field localization

Delocalized surface plasmons on metal

surfaces

Wood anomaly

Localized plasmon

The « end » of the plasmon

Dark-field nanoscope: G.A. Zheng et al, PNAS

107, 9043-48 (2010) .

Submicron dichroic splitter: J. Liu et al, Nat.

Comm., Nov. 2011.

Plasmonic nanofocussing for near-field

spectroscopy: S. Berweger et al., Phys. Chem.

Lett. 3, 945 (2012) .

J. Pendry et al., Science 305, 847 (2004).

R. Ulrich and M. Tacke, APL 22, 251 (1973).

"SPOOF" SPP

k//

w

Génération de plasmons avec des nanostructures

Questions:

Comment peut-on mesurer ou calculer l’efficacité de génération des

plasmons?

Comment exciter efficacement les plasmons de surface?

Comment cette efficacité varie avec les principaux paramètres?

Kuzmin et al., Opt. Lett. 32, 445 (2007). S. Ravets et al., JOSA B 26, B28 (2009).

glass

Au

q

Young slit experiment

(with a single slit illuminated)

0 -40 -30 -20 -10 0 10 20 30

glass

Au

q

q TE

Kuzmin et al., Opt. Lett. 32, 445 (2007). S. Ravets et al., JOSA B 26, B28 (2009).

How to calculate the amount of SPP

generated on the surfaces?

How to calculate the amount of SPP

generated on the surfaces?

a+ a-

)(xα)(xα2dz(z)Ez),(xH 00SP0y

-

-=

)(xα)(xα2dz(z)Hz),(xE 00SP0z

-

--=

x0

z

« Overlap integral »

Orthogonality is not

implemented with

EH* products but

with EH products

PL, J.P. Hugonin and J.C. Rodier, PRL 95, 263902 (2005)

a+ a-

x

exp[-Im(kspx)]

test

x0/l

l = 940 nm and silver

z

x

x0 x0

Il est bon de disposer de formule approcher pour mieux comprendre; ces

formules ont été établie surtout pour les fentes.

Les résultats sont probablement généraux.

a b

Normalization:

-incident field E=1 effective SPP cross section

-intensity incident on the slit = 1 efficiency

SPP generation by slits

describe geometrical properties -the SPP excitation peaks at a value wl/4

-for visible frequency, |a|2 reach 0.5, which means that of the power coupled out of the

slit half goes into heat

describe material properties -Immersing the sample in a dielectric enhances the SP excitation ( n2/n1)

-The SPP excitation efficiency |a|2 scales as |em(l)|-1/2

Analytical model

2/1

1

222 -el=b=a m

n

nwf

n2

n1 me

Expliquer avec Huygens pourquoi et dire que ce

resultat devrait être vrai pour bcp de géométries

Expliquer pourquoi avec l’intégrale de

recouvrement et avec les mains

PL, J.P. Hugonin and J.C. Rodier, PRL 95, 263902 (2005) & JOSAA 23, 1608 (2006).

tota

l S

P e

xci

tati

on

pro

bab

ilit

y

solid curves

(analytical model)

marks

(overlap integral)

|a|2 |b|2

Au Au

Drude model : |e| l2

exp(-z/d1)

d1=l e1/2/2p >> l

exp(-z/d2)

d2=l e-1/2/2p cte

Surface plasmon polariton

w/c

wp/c2

w/c=k/ed

Re(kSP)

)(xα)(xα2dz(z)Ez),(xH 00SP0y

-

-=

)(xα)(xα2dz(z)Hz),(xE 00SP0z

-

--=

100

101

10-1

100

(results obtained for gold)

l (µm)

|HS

P|

H. Liu et al., IEEE JSTQE 14, 1522 (2009)

Valid for all subwavelength

indentations

2/1-em

w/l

w

effic

ien

cy

(results obtained for gold)

S. B. Raghunathan et al., Opt. Express 20, 15326-15335 (2012).

Anti-symetric illumination

(never mind!)

55%

Unidirectional SPP launching with

grooves arrays

Bull eye : H. Lezec et al., Science 297, 802-804 (2002).

2 mm

2 µm

8 μm

Launcher Left

decoupler

Right

decoupler

A. Baron et al., Nano Lett. 11, 4207 (2011).

Gaussian beam (λ = 800 nm waist = 6λ)

λ

•Launching efficiency: c+ = 60%

•Contrast > 50

R(θ)

+30° -30°

-90° +90°

•Decoupling efficiency: d = 75%

•Radiation cone: < 10°

Unidirectional SPP launcher

outline

Field localization

Delocalized surface plasmons on metal surfaces

(30mn)

Wood anomaly

Localized plasmon

The « end » of the plasmon

S. Collin et al., PRL 104, 027401 (2010).

•Historique de l’anomalie de Wood

•La description plasmonique de l’anomalie

•deux types d’onde sont mises en jeux: les plasmons et les ondes

quasi-cylindriques

•Quelle est l’influence de la longueur d’onde sur le rôle de chacune

des ondes?

•Commentaire sur le spoof plasmon

“I was astounded to find that under certain conditions, the drop

from maximum illumination to minimum, a drop certainly of

from 10 to 1, occurred within a range of wavelengths not

greater than the distance between the sodium lines”.

Rapid survey of Wood’s anomalies

Discovery of the anomaly

R. W. Wood, Philos. Mag. 4, 396 (1902).

First explanation attempt by Lord Rayleigh

Rayleigh, Proc. Royal Society (London) 79, 399 (1907)

The forced resonance explanation of Fano

U. Fano, JOSA 31, 213 (1941).

k//+mK = k0

“I was astounded to find that under certain conditions, the drop

from maximum illumination to minimum, a drop certainly of

from 10 to 1, occurred within a range of wavelengths not

greater than the distance between the sodium lines”.

Rapid survey of Wood’s anomalies

Discovery of the anomaly

R. W. Wood, Philos. Mag. 4, 396 (1902).

First explanation attempt by Lord Rayleigh

Rayleigh, Proc. Royal Society (London) 79, 399 (1907)

The forced resonance explanation of Fano

U. Fano, JOSA 31, 213 (1941).

k//+mK = kSPP (>k0)

“I was astounded to find that under certain conditions, the drop

from maximum illumination to minimum, a drop certainly of

from 10 to 1, occurred within a range of wavelengths not

greater than the distance between the sodium lines”.

Rapid survey of Wood’s anomalies

Discovery of the anomaly

R. W. Wood, Philos. Mag. 4, 396 (1902).

First explanation attempt by Lord Rayleigh

Rayleigh, Proc. Royal Society (London) 79, 399 (1907)

The forced resonance explanation of Fano

U. Fano, JOSA 31, 213 (1941).

Modern theory of grating diffraction

Fully-vectorial numerical tools: Integral, differential methods, RCWA, …

Advanced conceptual tool: « polology » 2

0P

ZTTw-w

w-w= ~

~

w

T

The extraordinary optical transmission

T. W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolff, Nature 391, 667 (1998).

l (nm)

ratio

tran

smis

sion

e- e- e-

e-

e-

e- e- e-

e-

e-

SPP-assisted transmission?

T. W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolff, Nature 391, 667 (1998).

Debated hypothesis: The electronic character of SPP helps the coupling of the energy from the surface to the holes?

Main results from mode theory

Phenomenological polology E. Popov et al., PRB 62, 16100 (2000).

Resonance-assisted tunneling L. Martín-Moreno et al., PRL 86, 1114 (2001).

Spoof plasmon J. Pendry et al., Science 305, 847 (2004).

The Fano-type formula is very elegant as it well reproduce the

spectral lineshape with o,nly 5 real parameters. It additionnally

shows that the EOT is a resonance phenomenon.

All these analysis relies on physical ‘GLOBAL’ quantities attached to periodic ensembles; they give a good

insight into the macroscopic mechanisms responsible for the transmission, but nothing is known about the

individual plasmons that are launched inbetween the holes of the array.

2

0P

ZTTw-w

w-w= ~

~

w

T

"SPOOF" SPP

k//

resonance

resonance

More insight has been provided by Martin-Moreno who showed

that the resonance occurs at interfaces and that they boosts an

evanescent tuneling.

w Pendry et al. showed that the same resonant-assisted mechanism

occurs at low frequencies, and introduced the concept of spoof

plasmons.

SPP-assisted transmission?

If one derives a model of the EOT where only SPP are assumed to

carry the energy between adjacent hole chains and compares with

fully-vectorial computations, then one should allow us to quantify

what is really due to SPP in the EOT.

Microscopic SPP model

r t

a

b

t

b

in-plane reflection-transmission of SPP

coupling of SPP to free-space

tr-

ab

l

l

l

l=

aik

pT

SPexp2

20

2

201 1

H. Liu and P. Lalanne, Nature (London) 452, 448 (2008).

(for periodic arrays)

Actual SPP role in the EOT

H. Liu and PL, Nature (London) 452, 448 (2008).

Experimental evidence: F. van Beijnum et al., Nature 492, 411 (Dec. 2012).

0.95 1 1.05 1.1 1.15

0

0.1

0.2

a=0.68 µm

Tra

nsm

itta

nce

l/a

Normal incidence

RCWA SPP model

q2

Tra

nsm

issio

n

wavelength (nm)

750 800 850 900

0

0.05

0.1

spectra32thefromfitted

arecomplexreal

ik1T

321

SP

2

2

20

2

2012

...,q

p&p,p

a

ppq

=

tr=

tr--l

l

l

l=

exp

Normal

incidence

Measurement performed in Martin van Exter’s group (Leiden)

q=2

2a

1a

3a

q=3

1a

q = 1

q = 4

q = 2

q = 6

Direct experimental proof

q2

Tra

nsm

issio

n

wavelength (nm)

750 800 850 900

0

0.05

0.1

F. van Beijnum et al., Nature 492, 411 (Dec. 2012)

tr-

ab

l

l

l

l=

aexp

p

qikTq

SP2

20

2

2012 1

q = 1

q = 4

q = 2

q = 6

The 5 coefficients p1 (real), ab and r+t

(complex) are fitted for q = 2,3 …7

Direct experimental proof

total field

SPP field only

Quasi-CW field only

P. Lalanne, J.P. Hugonin, H.T. Liu and B. Wang, Surf. Sci. Rep. 64, 453 (2009)

l = 940 nm and silver

Quasi-cylindrical wave

Highly accurate for x < 10l

F(x) = exp(ik0x) (x/l)-m

m varies from 0.9 in the

visible to 0.5 in the far IR

102 100

10-2

100

10-4

10-6

|F(x

)| (

a.u

.)

x/l

1/x3

1/x

silver @ l=1 µm

Quasi-cylindrical wave

10-1

10-2

100

10-3

101 102 100 101 102 100

l=0.6 µm l=1 µm

l=9 µm l=3 µm

x/l x/l

|H| (a

.u.)

|H

| (a

.u.)

|H| (a

.u.)

|H

| (a

.u.)

x/l x/l

10-1

10-2

100

10-3

HSP

HCW

(x/l)-1/2

PL and J.P. Hugonin, Nature Phys. 2, 556 (2006). PL et al., Surf. Sc. Report (2009).

(result for silver)

HSP 1/|em|1/2

HCW cte

Frequency dependence (important)

2/32 d

m

pe

e

(perfect metal)

q (°)

l=0.6 µm

l=1 µm

l=3 µm

l=10 µm

PC

S. Ravets et al., JOSA B 26, B28 (2009).

Fa

r-fi

eld

in

ten

sit

y (

a.u

.)

SPP mainly

Quasi-CW only

Quasi-CW & SPP

Quasi-CW mainly

J. Pendry et al., Science 305, 847 (2004).

"SPOOF" SPP

k//

w

Spoof=coherent interaction of holes

with quasi-cylindrical waves

Field localization

Delocalized surface plasmons on metal surfaces

Wood anomaly (60 mn)

Localized plasmon

The « end » of the plasmon

outline

Metallic resonance

nanoantenna

sensing

Solar cell

Frequency conversion

How efficiently can you excite it?

-from the near field? Purcell?

-from far field?

What is the resonance mode?

-how to define it « properly »?

-What is the mode volume?

-What are the limiting quantities for Q?

Application to sensing

-analytical formula of the resonance shift

Metallic resonance

+ +

+

+ +

- -

-

- -

E

bh

bhVee

e-e=a

23

J.D. Jackson, Classical Electrodynamics

plane wave E0

he

be

V << l3

02 =ee bh

Resonance is achieved for a

single fixed wavelength, such

that 02 =ee bh

Why tiny metallic NP resonate?

52

Cross-section shrinks to zero, neff of the plamonic mode diverges, and L shrings!

(The Fabry-Perot electric-dipole resonance mode scales down (no cutoff) )

IMIM MIM IMI

L

td

m

d

d

efft

npe

el-= 0

d

d

m

eff

tRe

nReL

e

ep-=

l-

2

20 /

J. Yang et al., Opt. Express 20, 16880-16891 (2012)

"Ultrasmall metal-insulator-metal nanoresonators: impact of slow-wave effects on the quality factor"

Resonance is achieved for any

wavelength, just by scaling

down dimensions.

Why tiny metallic NP resonate?

Q factor at deep sub-λ scale

Quasi-static limit : Q factor of a localized plasmon resonance is

determined only by εmetal.

εω

ωQε

metal0S

metal

Re( )

=2Im( )

F. Wang et al., Phys. Rev. Lett. 97, 206806 (2006).

J. Yang et al., Opt. Express 20, 16880-16891 (2012)

"Ultrasmall metal-insulator-metal nanoresonators: impact of slow-wave effects on the quality factor"

(Ag: Q ~ 70 @ λ=600nm)

Far-field excitation

Near-field

excitation

Excitation of metal resonance

mmmm

mmm

i

i

ErH

HE

HE

~)~,(~~

~~~

ω~2Imω~ReQω~frequencycomplexaforsourcewithout

equationssMaxwell'ofsolutionsare~

,~

modesnormalquasiThe

m

mm

wew-=

mw=

=

-

0

What is a metallic resonance?

w rrkk as~

expcomplex~

complex~ i

wwew-w

w-=wa rErErrr minc

mm

~,,d~, 3

Excitation coefficient a

1dif 3 =ww-ww rHμHEεE~~~~

Energy of a dispersive

material? yes but only when

absorption is small. No

energy consideration in the

derivation.

Very easy!

Only a single hypothesis : material is reciprocal

rErErE m

m

minc

~,, waw=w

Far-field excitation

Near-field

excitation The scattered fiedl is expanded in

the QNM basis

Complex coordinate

transform (PML)

X = a(1+im) x

Y = a (1+im) y

Z = a (1+im) z

Analytical

continuation in the

complex plane with

PMLs

Rkrk im1iexpiexp ~~

remove the divergence problem for

suitable m’s by transforming the

exponentially diverging field into an

exponentially damped field

0

The normalization issue

Complex coordinate

transform (PML)

X = a(1+im) x

Y = a (1+im) y

Z = a (1+im) z

is an invariant under

space coordinate transforms

is invariant too and can be calculated with

any PML, by computing the integral in real

space and in the PML.

w rEεE 3d

rEεE 3dV ww=~~

m

First (?) time the field in the PML is explicitly considered to evaluate a physical quantity.

The normalization issue

Open-source software for resonance

calculation

0

4

8

12

50 nm

1w~

2w~3w~

Re(w)

Im(w)

4w~

2w~ 3w~ 4w~1w~

Freeware implemented with COMSOL multiphysics can be downloaded at

www.lp2n.institutoptique.fr

Q. Bai,et al., Opt. Express 21, 27371 (2013).

Coll. Mathias Perrin/LOMA

0.7 0.8 0.9 1 1.1 1.2 0

0.02

0.04

0.06

0.08

Wavelength (µm)

0.7 0.8 0.9 1 1.1 1.2 0

100

200

300

400

Wavelength (µm)

Purcell factor Extinction cross section (µm2)

wwew-w

w-=wa rErErrr minc

3

m

~,,d~,

Q. Bai,et al., Opt. Express 21, 27371 (2013).

Cro

ss S

ecti

on

700 750 800 850 900 950 0

1

2

3

4

5

Wavelength (nm)

rrErEr 3d

perturb

~~~,~~

.

wew-=w

w~Re

wd ~Imd

Application to sensing

J. Yang et al., (in preparation)

w~Re w~Im

A. Curto et al., Science 329, 930 (2010).

The quantum-dot luminescence is totally governed by the antenna

- radiation diagram

- Purcell factor

Yagi-Uda antenna

Nano-antenna

Nano-antenna

G.M. Akselrod et al., Nat. Photon. DOI:

10.1038/NPHOTON.2014.228

“Probing the mechanisms of large Purcell enhancement in

plasmonic nanoantennas”

Purcell

Radiation

efficiency

20

220

20

2

20

0 )(4 w-ww

w

w

w=

Q

with4

3

M

3

2 V

Q

n

l

p=

Classical Lorentzian shape

F

F

Classical Purcell formula

e

e=

2

32

M

)()(max

d)()(V

rEr

rrEr

~

~Only valid for large Q

(error scales as 1/Q as

Q)

w-wd=n

nDOS ~

w

DO

S

modal-expansion of the LDOS

R.K. Chang and A.J. Campillo, Optical processes in microcavities, (World Scientific, 1996).

w-wd=n

nDOS ~

2

n

n

nLDOS rErr~~, ew-wd=w

w

DO

S

R.K. Chang and A.J. Campillo, Optical processes in microcavities, (World Scientific, 1996).

1dV

32

nnn =e= rrErEE~~

,~

modal-expansion of the LDOS

w-wd=n

nDOS ~

2

n

n2n

2n

n 1LDOS rErr

~

~, e

gw-wp

g=w

w

DO

S

R.K. Chang and A.J. Campillo, Optical processes in microcavities, (World Scientific, 1996).

2

n

n

nLDOS rErr~~, ew-wd=w

1dV

32

nnn =e= rrErEE~~

,~

modal-expansion of the LDOS

with4

3

M

3

2

l

p=

V

QRe

nF

Revisiting the Purcell formula

20

20

3

M

)r(2

dV

uE

rEεE

e

ww=

~

n

~~

w

w-w

w-ww

w

w

w=

M

M

0

0

20

220

20

2

20

0 Re

Im21

)(4 V

VQ

QF

Derivation based on reciprocity arguments, see C. Sauvan et al., PRL 110, 237401 (2013) & Q.

Bai et al., Opt. Express 21, 27371 (2013).

Circle: Green-tensor calculation (decay in all modes)

Blue line: revised Purcell formula (with a single mode)

Non-Lorentzian response with

metallic resonance

00010

i73V3

1,

l--=

00010i34V

3

2,

l=

the contribution of a quasi-normal mode to the total

power radiated by a source may be detrimental (it may

reduce the decay rate), even when the frequencies of the

source and the mode are matched.

Multi-resonance case

85 nm

145 nm

45 nm

Au

80 nm

20 nm

outline

Field localization

Delocalized surface plasmons on metal surfaces

Wood anomaly

Localized plasmon (1H20)

The « end » of the plasmon

0 2 4 6 8

Heat diffusion –

acoustic relaxation

e- - phonon

relaxation

e- abs

delay (ps)

ΔR

/R (

10

3 )

t e-e

= 1

00

fs

te-ph = 1 ps

0

0.5

1.5

1

pump

probe

S.D. Brorson, J.G. Fujimoto and E.P. Ippen, PRL 59 (1987).

The plasmon decay and then what?

SPP7 2015: first apparition of hot

electrons in the main topic list of the

SPP conference series

First review paper appeared in Jan. 2015

Hot electrons=hot topic

Many applications

Plasmon induced hot carriers

M.W. Knight et al., Science 332,702 (2011).

-photodetectors with spectral responses

circumventing band gap limitations

-chemical catalysis close to metal surfaces

Même quand il meurt, le plasmon renait de ses cendres.

Le plasmon est éternel, « offrer donc un plasmon »

The end