Tatiana V. Teperikesperia.iesl.forth.gr/~wip/lectures/pdfs/Teperik.pdfTatiana V. Teperik LaboratoireEM2C, CNRS, Grande Voiedes Vignes 92295 Châtenay-MalabryCedex, France PDF created

Tatiana V. TeperikLaboratoire EM2C, CNRS, Grande Voie des Vignes

92295 Châtenay-Malabry Cedex, France

PDF created with pdfFactory Pro trial version www.pdffactory.com

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Outline

1. Surface plasmons. Surface plasmons, surface plasmon conditions, surface plasmon dispersion. Excitation of

surface plasmons: ATR coupler (Kretschmanns and Otto configurations), grating coupler. Propagation length of surface plasmon . Surface plasmon on thin film.

2. Localized surface plasmons.Localized surface plasmon modes in metallic particles: electrostatic solution. Localized

surface plasmon in metalic sphere, void in metal, metallic shell. Electromagnetic solution. Radiative damping. Surface plasmons and surface localized plasmons: differences and similarities

3. Plasmons in nanoporous metal structuresSurface plasmon resonance on a flat surface. Localized surface plasmon resonancein voids. Rayleigh anomalies.

4. Total light absorption by plasmonicnanostructures

Effective surface impedance model. Multi-channel model: the Breit-Wigner approximation

Summary



m b e+ =x x E&& &

Dielectric response of the metal: the Drude model

( / )( )e

e miω ω ν

= −+

Ex /e b mν =

2

( )p

e

N Nei

ωω ω ν

= = = −+

P p x E

( 1)χ ε= = −P E E

2

( ) 1( )

p

eiω

ε ωω ω ν

= −+

22

pNem

ω =

2

2( ) 1 p

e

ωε ω

ωω ν

≅ −

>>

e=p x



R.H.Ritchie, Phys. Rev. 106, 874 (1957)C.J.Powell, J.B.Swan, Phys. Rev. 118, 640 (1960)

z exp[ | | ]y zH k z−

Surface plasmons

metal

dielectric

x



Surface plasmon condition 1

Maxwell’s equations:

1

1

( ) 00

c t

c t

ε

ε

∂∇× =

∂∂

∇× = −∂

∇ ⋅ =∇ ⋅ =

H E

E H

EH

1,2i =

( ,0, )exp[ | |]exp[ ]

(0, ,0)exp[ | |]exp[ ]x z

y

i i i i

i i i

E E k z iqx i t

H k z iqx i t

ω

ω

= − −

= − −

EH

p-polarization

1 1 1 1

2 2 2 2

y x

y x

ik H Ec

ik H Ec

ωε

ωε

=

= −

22 (1)i ik q

cω

ε = −



Surface plasmon condition 2

1 2

1 2

x x

y y

E E

H H

= =

Boundary conditions:

1 21 2

1 2

1 2

0

0

y y

y y

k kH H

H Hε ε

+ = − =

1 2

1 2

0 (2)k kε ε

+ =

1 2

1 2

( )qcω ε ε

ωε ε

=+

22 (1)i ik q

cω

ε = −

1 2 0ε ε+ = nonretardedsurface plasmon

1 2k k q= =



Surface plasmon dispersion

2 2

2 2( )2

p

p

qc

ω ωωω

ω ω−

=−

p

q

cq

2

1 2

2

1

1

pωε

ωε

= −

=

/ 2pω ω=for large q

2/ 1pω ω ε= +

for large q

2 1ε ≠



Maxwell’s equations:

1

1

( ) 00

c t

c t

ε

ε

∂∇× =

∂∂

∇× = −∂

∇ ⋅ =∇ ⋅ =

H E

E H

EH

1,2i =

(0, ,0)exp[ | |] exp[ ]

( ,0, )exp[ | |] exp[ ]y

x z

i i i

i i i i

E k z iqx i t

H H k z iqx i t

ω

ω

= − −

= − −

E

H

1 2 (2)k k= −2

2 (1)i ik qcω

ε = −

contradiction

S-polarization



Excitation of surface plasmons: ATR coupler

Kretschmann’sconfiguration

Otto’sconfiguration

ω

/ 2pω

resω

qresq

pr/ sincq ε θcq

pr/cq ε

0



Excitation of surface plasmons: Grating coupler

||2 2sin ,

0,1, 2, ...

q q n nL c L

n

π ω πθ= ± = ±

=

ω

/ 2pω

resω

qresq

/ sincq θcq

q∆

0

2q nLπ

∆ =

spsin q qcω

θ + ∆ =



Propagation length of surface plasmons

1 2

1 2

qcω ε ε

ε ε=

+

' ''q q iq= +

1/ 2

1 2

1 2

'''

qcω ε ε

ε ε

= +

3/ 2

1 2 12

1 2 1

' ''''' 2( ' )

qcω ε ε ε

ε ε ε

= +

2 ''q x−1

2 ''l

q=

Ag: 22µm λ=515 nm500µm λ=1060 nm

'' 'iqx q x iq x−

1 1 1' ''iε ε ε= + 1 1'' 'ε ε<<

∼

∼



Maxwell’s equations:1

1

( ) 00

c t

c t

ε

ε

∂∇× =

∂∂

∇× = −∂

∇ ⋅ =∇ ⋅ =

H E

E H

EH

1,2i =

2 | |2 2 e e

y

k z iqx i tH H ω− −=

p-polarization

Surface plasmons on thin film

( ,0, )

(0, ,0)x z

y

i i i

i i

E E

H

=

=

EH

even mode

odd mode

1 11 1[e e ]e

y

k z ik z iqx i tH H ω−= +

1 11 1[e e ]e

y

k z k z iqx i tH H ω− −= −2 | |

2 2 e ey

k z iqx i tH H ω− −=

1 1 1 1( ) ( ) 2 cosh[ / 2]2 2y y

d dH z H z H k d= = = − =

1 1 1 1( ) ( ) 2 sinh[ / 2]2 2y y

d dH z H z H k d= = − = − =



Surface plasmons on thin film

2 11 2

2 1

exp( ),qd k k qε εε ε

+= − = =

−m

1 / 2

1 / 2

[1 exp( )]2

[1 exp( )]2

p

p

qd

qd

ωω

ωω

+

−

= + −

= − −

2 1ε =ω

/ 2pω

q0

ω +

ω −

1 2

1 1 2

1 2

1 1 2

0tanh( / 2)

0coth( / 2)

k k d k

k k d k

ε ε

ε ε

+ =

+ =

even plasmon

odd plasmon

even plasmon

odd plasmon

221 1 /k q cε ω±= −



Surface plasmons of on Al film

/ 5pω ω=

R.B.Pettit, J.Silcox, R.Vincent, Phys. Rev. B 11 3116 (1975)H. Raether, Surface Plasmons, 1988;J M Pitarke et al., Rep. Prog. Phys. 70 1(2007).

ωp = 15 eVνe = 0.75 eV

cqa = 120Åt = 40Å ε0 = 4.

Al:

even plasmon

odd plasmon



Surbhi Lal, Stephan Link, Naomi J. HalasNature Photonics, 1 641 (2007)

Localized surface plasmons



A single plasmonic nanoshell: quasi-electrostatic limit

Laplace equation 2 0∇ Φ =2 2

2 2 2 2 21 1 1( ) sin 0

sin sinr

r r r rθ

θ θ θ θ ϕ∂ ∂ ∂Φ ∂ Φ Φ + + = ∂ ∂ ∂ ∂

General solution for a problem possessing azimuthal symmetry

( 1)

0[ ] ( , )

ll l

lm lm lml m l

A r B r Y θ ϕ∞

− +

= =−

Φ = +

( ) ( ) ( )U r P Qr

θ ϕΦ =



1 2 2 3 2

1 2 2 3 3

( ) ( ) ( )( 1)( ) ( )

l l fl ll l

ε ε ω ε ω ε ε ωε ε ω ε ω ε ε

+ + −= +

− + +

2 1 2 1/( )l lf a a h+ += +

A single plasmonic nanoshell: boundary conditions

10

( 1)2

0

( 1)3

0

( , )

[ ] ( , )

( , )

ll

l lml m l

ll l

l l lml m l

ll

l lml m l

A r Y

B r C r Y

D r Y

θ ϕ

θ ϕ

θ ϕ

∞

= =−

∞− +

= =−

∞− +

= =−

Φ =

Φ = +

Φ =

1 21 2 1 2

2 32 3 2 3

, ,

, ,

r ar r

r a hr r

ε ε

ε ε

∂Φ ∂ΦΦ = Φ = = ∂ ∂ ∂Φ ∂ΦΦ = Φ = = + ∂ ∂



Localized surface plasmons2 2( ) 1 /pε ω ω ω= − 1 3 1ε ε= =

v1

2 1pll

ω ω+

=+

s 2 1pl

lω ω=

+

2 1 2 1/( )l lf a a h+ += +

Metallic sphereVoid in metal

2 2 1 1 1[ ] ( 1)2 1 2 4p l fl ll

ω ω± = + ± + + +

/ 2p lω ω= → ∞

0p lω ω= =surface plasmon

bulk plasmon

Metallic shell

D.W.Brandl, C.Oubre, P.Nordlander, J.Chem.Phys. 123, 024701 2005PDF created with pdfFactory Pro trial version www.pdffactory.com


s / 3pω ω=

v 2 / 3pω ω=

22 4 (1 )

4(1 ) (5 4 3 1 8 )p f

f f fω

ω±−

=− + + +m

3 3/( )f a a h= +

metallic shellvoid in metal

metallic sphere

Localized surface plasmons: Frohlich modes

a/(a+h)

ωs

ωv



A single plasmonic nanoshell: electrodynamic approach

(1) (2)

,[ ( ) ( )] ( , )lm l i lm l i lm

l mA h k r B h k r Yψ θ ϕ= +

M Eik

ψ ψ= − ∇ ×E L LM E

ji

ik

ψ ε ψµ

= ∇ × −H L L

i= − ×∇L r

2 2( ) 0jk ψ∇ + =

j j jk k ε µ=

(1,2) ( ) ( ) ( )l l lh x j x in x= ±

/k cω=

Void in metalVoid in metal

1ε

2 1ε =

(1) (1)1 2 1 2 1 1 2 1 2( )[ ( )]' ( )[ ( )]'l lh j j hρ ρ ρ ε ρ ρ ρ=

1 1 2( ) / 2 / 2kd kdρ ε ω ρ= =

M

E

ψψ

ψ

=

J.D.Jackson, Classical electrodynamics, 1975



Plasmon modes of a metallic nanoshell

T.V. Teperik, V.V. Popov, and F.J. García de Abajo, Phys. Rev. B 69, 155402 (2005).

2

2( ) 1 pe

ωε ω ω ν

ω≅ − >> hH

lδ= δ is the skin depth



Surbhi Lal, Stephan Link, Naomi J. Halas, Nature Photonics, 1 641 (2007)

.

Tunability of nanoshells.

AuSi

d=120nm



Nanoporous metal structure: technology and experiment

Pore diameter Ø ~ 500 nm (nanoscale casting technique with the electrochemical deposition of metal through a self-assembled latex template )

Technology and experiment:Department of Physics and Astronomy, Department of Chemistry, University of Southampton, United Kingdom

Jeremy J. Baumberg et al. Adv. Mat. 13 (2001), (2003); PRL 87 (2001); APL 83 (2003), Faraday Discussion (2003)



Theory:

• Instituto de Optica, Madrid, Spain• Donostia International Physics Center, San Sebastian, Spain• Institute of Radio Engineering and Electronics, (Saratov Division), RAS

Rigorous self-consistent electromagneticmultiple-scattering layer-Korringa-Kohn-Rostoker approach

N. Stefanou, V. Yannopapas, and A. Modinos, Comput. Phys. Commun. 113 49 (1998)

Nanoporous metal structure: theory



Reflection spectra of nanoporous metal surface.Localised and delocalised plasmons: anticrossing regime.

theoryexperiment

(1) (1)0 1 1 1 1 1 0 0

0 1

( )[ ( )] ' ( ) ( )[ ( )]'

( ) / 2 / 2l lh j j h

d c d c

ρ ρ ρ ε ω ρ ρ ρ

ρ ω ε ω ρ ω

=

= =

surface plasmons2

2 ( )1 ( )pqq

cω ε ω

ε ω = +

||pq pq= +q k g|a|=|b|=505 nm, d = 500 nm φ = 0O, p-polarization || sin /k cω θ=

void plasmons

T.V.Teperik et al., Optics Express 14,1965 (2006). pq p q= +g A B



Field snapshots of plasmon modesexcited on nanoporous metal surface

bonding state

anti-bonding state

T.V.Teperik et al., Optics Express14,1965 (2006).

localized plasmon

surface plasmon

||extE

||extE

||extE ||

extE



Resonant diffraction on the nanoprous metal surface:Rayleigh anomalies versus localized plasmons

theoryexperiment

|a|=|b|=515 nm, h=5 nm, d = 500 nmφ = 0O p-polarization

/pqq cω=

surface plasmons(1) (1)

0 1 1 1 1 1 0 0( )[ ( )]' ( ) ( )[ ( )]'l lh j j hρ ρ ρ ε ω ρ ρ ρ= void plasmons

T.V. Teperik et al., Optics Express 14, 11964 (2006)

2 2

2 22p

pq pp

qc c

ω ωω ωω ω

ω ω

−= ≅ <<

−grazing photons

||pq pq= +q k g



Total light absorption by nanoporous metal surface

theoryexperiment

|a|=|b|=505 nm d = 500 nm,normal incidencet=d+h is the nanoporous film thickness



1. Effective surface impedance model2. Multi-channel model: the Breit-Wigner approximation



Drude model:

2

2

Ohm

H

ee

e

ee

mRe N

mLe N

νδ

δ

=

=

2

( )( )

ee

e

e Nm i

σ ων ω

=−

01

e e ee

Z R i L Zωσ δ

= = − ≠

0e e eR Z L R

Planar surface of metal⇒

0ZStrong reflection

0l l ll



Inverted plasmonic nanostructures

l e l eN N R R<< <<1/l lL Cω ω= −

0l lZ R Z Totalabsorption

eff 00

eff 0

Z Zr ZZ Z



Equivalent RLC circuits of plasmonic nanostructures

2 ( , )0

F| | s vl lC f b ε =

( ) ( ),s vb a d b d= − =

2 ( , )

Hl s v

l e

mLe Nδ

= ∆

( . )

2 ( . )

2 Ohms vl

l s vl e

mRe N

νδ

= = ∆

( ) ( ) ( ) ( )/s s s sl l l lZ R i L i Cω ω= − +

( ) ( )( )

2 ( ) ( ) ( ) ( )1

v vv l l

l v v v vl l l l

R i LZL C i R C

ωω ω

−=

− −



2( ) ( ) 2( )eff 2 ( ) ( ) ( )

[ ]2 [ ]

v vlv lv v v

l e l l

mZ i

e N iβ ω

δ ω ω ν= −

∆ − −

Effective surface impedance

( ) ( ) ( )eff 2 2 ( )( )

2 [ ]s s sl lss

l el

i mZ ie N

ω ω νδβ

= − −∆

lattice of spheres

lattice of voids

,l l lω ω ν ω≈ <<

( , ) ( , )

1l s v s v

l lL Cω = frequency of

the l-th plasmon mode

( , )s vlν dissipative damping of

the l-th plasmon mode

Lorentzianapproximation

( , ) 2| | 1s vlβ < coupling coefficient

( )

( )

0 at 0

seff

l lveff

ZZ

ν ω ω = = = → ∞

series resonanceparallel resonance



Light absorption by a resonant surface

( , ) ( , ) 2 ( , ) 2

( , ) ( , ) 2 ( , ) 2

( ) ( )*( ) ( )

s v s v s vl l l

s v s v s vl l l

R rr γ ν ω ωγ ν ω ω

− + −= ≈

+ + −

( , ) ( , )

( , ) ( , ) 2 ( , ) 2

41( ) ( )

s v s vl l

s v s v s vl l l

A R γ νγ ν ω ω

= − ≈+ + −

Reflectance Absorbance

0 at

1 l l l

RA

γ ν ω ω=

= ==

( ) 2 2 ( )( ) 0 | |

2

s ss l l e

lZ e N

mβ δ

γ∆

=radiative damping of the l-th plasmon modeof a metallic sphere

eff 00

eff 0

120 [Ohm]Z Zr ZZ Z

π−

= =+

( ) 2 ( ) 2( )

2 ( )0

| | [ ]2

v vv l l

l vl e

mZ e N

β ωγ

δ=

∆

radiative damping of the l-th plasmon modeof a void in metal

l l

0eff

T. Teperik, V. Popov, and F. Garcıa de Abajo, J. Opt. A: Pure Appl. Opt. 9, S458 (2007)



Total light absorption by a lattice of voids in silver

( ) ( )v vl lγ ν<<

weak coupling

⇓weak absorption

( ) ( )v vl lγ ν>>

strong coupling

⇓re-radiation

⇓weak absorption

t=25 nm (dash-dotted curve)15 nm (solid curve)8 nm (dashed curve)

total lightabsorption

( ) ( )s sl lγ ν=

0effZ Z=

T. V. Teperik, V. V. Popov, and F. J. García de Abajo, Phys. Rev. B 71, 085408 (2005)



Total light absorption by a lattice of gold spheres

( ) ( )s sl lγ ν<<

weak coupling

⇓weak absorption

( ) ( )s sl lγ ν>>

strong coupling

⇓re-radiation

⇓weak absorption

total lightabsorption

0effZ Z=

t=210 nm (dash-dotted curve)170 nm (solid curve)70 nm (dashed curve)

( ) ( )s sl lγ ν=

T. Teperik, V. Popov, and F. Garcıa de Abajo, J. Opt. A: Pure Appl. Opt. 9 S458 (2007)



Excitation of a plasmon resonance at the nanostructuredmetallic surface ensures matching between the surface impedance and the free-space impedance and, hence,

enables obtaining the total absorption of light on a high-conductivity metal surface

Effective surface impedance model



partially disordered silver films• O. Hunderi and H. P. Myers, J. Phys. F: Metal Phys. 3, 683 (1973)

metal diffraction grating and double-period metal gratings• M. C. Hutley and D. Maystre, Optics Communications 19, 431 (1976)• D. Maystre and R. Petit, Optics Communications 17, 196 (1976)• W.-C. Tan, J. R. Sambles, and T. W. Preist, Phys. Rev. B 61, 13177 (1999)•E.Popov and L.Tsonev, Surface Science Letters 271, L378 (1992)

doped silicon lamellar grating• F. Marquier, M. Laroche, R. Carminati, J.-J. Greffet, Journal of Heat Transfer 129, 11 (2007)•J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, Y. Chen, Nature 416, 61 (2002)

semiconductor and metal-semiconductor-metal nanostructures• S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, Appl. Phys. Lett. 85, 194 (2004)•T.V. Teperik, F.J. García de Abajo, V.V. Popov, and M.S. Shur, Appl. Phys. Lett. 90 251910 (2007).

multiplayer of metallic nanoparticles and nanopores in metal• T. V. Teperik, V. V. Popov, and F. J. García de Abajo, Phys. Rev. B 71, 085408 (2005)• T. Teperik, V. Popov, and F. Garcıa de Abajo, J. Opt. A: Pure Appl. Opt. 0, 0 (2007)• S.Kachan, O. Stenzel, and A. Ponyavina, Appl. Phys. B 84, 281 (2006)

overdense plasma slab (in the microwave frequency range)•Y. P. Bliokh, J. Felsteiner, and Y. Z. Slutsker, Phys. Rev. Lett. 95, 165003 (2005).

Total light absorption in plasmonic nanostructures



1. The light transmission through the entire structure is forbidden

2. The total light absorption effect relies on the excitation of intrinsic resonance in the structure

3. The total light absorption effect requires specific conditions of effective coupling of light with resonant excitations in the system

Total light absorption in plasmonic nanostructures



1 111 exp( ) exp( )ik z S ik zk ⊥ ⊥

⊥

= − +B e 2 2c k kω ⊥= +

1111 11 11

0

exp(2 ) exp(2 )/ 2

iMS i ii

δ δω ω

Γ= −

− + Γ

211 1S j+ =

2

11

0 / 2M j

iω ωΓ

=− + Γ

1112

M =

22 0

11 2 20

( )( ) / 4

R S ω ωω ω

−= =

− + Γ

211 11 0 11Re ( ) / 2M iM Mω ωΓ = − + Γ

00 atR ω ω= =

Multi-channel model: the Breit-Wigner approximation

L. Landau and E. Lifshitz (Butterworth-Heinemann, Oxford, 1996).A. G. Borisov, F. J. Garcıa de Abajo, S. V. Shabanov, Phys. Rev. B 71, 075408 (2005).N. A. Gippius, S. G. Tikhodeev, T. Ishihara, Phys. Rev. B 72, 045138 (2005).



If the trapped electromagnetic mode (resonance) is such that

(i) there is only specular reflection from the nanostructured metal surface with no diffracted beams;

(ii) there is no polarization conversion, and

(iii) the radiative decay of the resonance is equal to its dissipative decay, then the whole energy from the incident light will be transformed into the losses in metal

Total light absorption conditions

22 0

11 2 20

( )( ) / 4

R S ω ωω ω

−= =

− + Γ 0

0at

1 1RA R

ω ω=

= = − =

T.V.Teperik, F.J.Garcıa de Abajo, A.G.Borisov, M.Abdelsalam, P.N.Bartlett, Y.Sugawara, J.J.Baumberg,Nature Photonics, 2008



Absorption spectra of nanoporous metal surface.

(1) (1)0 1 1 1 1 1 0 0

0 1

( )[ ( )] ' ( ) ( )[ ( )]'

( ) / 2 / 2l lh j j h

d c d c

ρ ρ ρ ε ω ρ ρ ρ

ρ ω ε ω ρ ω

=

= =

surface plasmons2

2 ( )1 ( )pqq

cω ε ω

ε ω = +

||pq pq= +q k g|a|=|b|=505 nm, d = 500 nm, φ = 0O, p-polarization || sin /k cω θ=

void plasmons

t=1.08d t=1.024d

theoryexperiment

T.V.Teperik et al., Optics Express 14,1965 (2006).PDF created with pdfFactory Pro trial version www.pdffactory.com


Omnidirectional total light absorption

Angle of incidenceis 20 deg

Azimuthal angle is 0 deg

T.V.Teperik, F.J.Garcıa de Abajo, A.G.Borisov, M.Abdelsalam, P.N.Bartlett, Y.Sugawara, J.J.Baumberg,Nature Photonics, 2008



Plasmonics is the rapidly emerging field that is concerned primarily with the manipulation of light at the nanoscale, based on the exploiting the both localized

and propagating surface plasmons. Owing to considerable advances made in nanotechnology a large variety of structures can be synthesized with controllable

size and narrow size distribution. Modern elaborated theory allows us to describe their unique plasmonic

properties. It is believed that plasmonic components can be successfully used for technologically important

applications such as sensing and plasmonic guiding.



Collaborators:

Javier F. García de AbajoInstituto de Optica, Madrid, Spain

Vyacheslav V. PopovInstitute of Radio Engineering and Electronics (Saratov Branch),

Russian Academy of Sciences, Saratov, Russia

Andrei G. BorisovLaboratoire des Collisions Atomiques et Moléculaires, UMR 8625

CNRS-Université Paris-Sud, 91405 Orsay Cedex, France

Tim A. Kelf, Yoshihiro Sugawara, Jeremy J. BaumbergM. Abdelsalam, P. N. Bartlett

University of SouthamptonSouthampton, SO17 1BJ, United Kingdom

Jean-Jacques GreffetLaboratoire EM2C, CNRS, Grande Voie des Vignes

92295 Châtenay-Malabry Cedex, France





Documents

Tatiana V. Teperikesperia.iesl.forth.gr/~wip/lectures/pdfs/Teperik.pdfTatiana V. Teperik LaboratoireEM2C, CNRS, Grande Voiedes Vignes 92295 Châtenay-MalabryCedex, France PDF created