'Optical Properties of Plasmonic Nanosystems'-...

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SMR: 1643/5

WINTER COLLEGE ON OPTICS ON OPTICS AND PHOTONICSIN NANOSCIENCE AND NANOTECHNOLOGY

( 7 - 18 February 2005)

"Optical Propertiesof Plasmonic Nanosystems"- II

presented by:

M. StockmanGeorgia State University

Atlanta

U.S.A.

Department of Physics and AstronomyGeorgia State UniversityAtlanta, GA 30303-3083

2/14/2005 Web: http://www.phy-astr.gsu.edu/stockmanE-mail: [email protected]

ITCP, Trieste, Italy02/10/2005 1

Theory of Nanoplasmonics 2:Optical Properties of Plasmonic

NanosystemsMark I. Stockman

Department of Physics and Astronomy, Georgia State University, Atlanta, GA

30303, USA

Support:




LECTURE 2

Nanoplasmonics of Nanosystems1. Introduction; Local fields of a sphere

2. Quasistatic eigenmodes; Theorem on Anderson localization3. Giant SERS in fractals4. Nanospheres and their aggregates; Efficient nanolens made of nanospheres5. Enhancement, depolarization and dephasing of SHG




PROBLEMS IN NANOOPTICS

Microscale

Delivery of energy to nanoscale; Adiabatical conversion of propagating EM wave to local fields

Enhancement and control of the local nanoscale fields. Enhanced near-field responses

Generation of local fields on nanoscale: SPASER




Quasistatic Approximation and EigenmodesFor a nanosystem, size is much less than the radiation wavelength. In this case, one can neglect retardation, and describe the system by quasistatic equations for electrostatic potential.

Quasistatic approximation does not imply that the excitation processes are slow. Just to the opposite, the neglectable retardation allows one to use and study ultrafast processes in nanostructures.




Example: Quasistatic fields as expansion over eigenmodes for spherical particles

j 2@l_, m_, r_, q_, j _D:= ikjja@2, l, mD 1

rl+ 1+ b@2, l, mD rly{zz SphericalHarmonicY@l, m, q, jD

j 1@l_, m_, r_, q_, j _D:= ikjja@1, l, mD 1

rl+ 1+ b@1, l, mD rly{zz SphericalHarmonicY@l, m, q, jD�. a@1, l, mD® 0

l R1+2 lHe1 - e2Le2 + lHe1 + e2LMultipolar (l,m) polarizability:

Coefficients of eigenmodes expansion:9b@2, 1, 0D® - 2$%%%%%%p3

, a@2, 1, 0D®2 ######p

3R3He1 - e2L

e1 + 2 e2, b@1, 1, 0D® - 2�!!!!!!!3 p e2

e1 + 2 e2=




-3 -2 -1 0 1 2 3x

-3

-2

-1

0

1

2

3

z

-3 -2 -1 0 1 2 3x

-3

-2

-1

0

1

2

3

z

-3 -2 -1 0 1 2 3x

-3

-2

-1

0

1

2

3

z

Quasistatic local field intensity for silver sphere

SPωω < SPωω = SPωω >




We will follow the spectral theory that allows one to separate the material and geometric properties of the system.

1. M. I. Stockman, S. V. Faleev, and D. J. Bergman, Anderson Localization vs. Delocalization of Surface Plasmons in Nanosystems: Can One State Have Both Characteristics Simultaneously?, Phys. Rev. Lett. 87, 167401-1-4 (2001).

2. M. I. Stockman, D. J. Bergman, and T. Kobayashi, Coherent Control of Nanoscale Localization of Ultrafast Optical Excitation in Nanosystems, Phys. Rev. B. 69, 054202-1-10 (2004).




Quasistatic field equations and boundary conditions

. 0),,(),,(

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

,0,0=

∂∂

=∂∂

===∂∂

∂∂

==yx LyLx

z

zyxy

zyxx

Lyxyx

ϕϕ

ϕϕϕωε rr

rr




Eigenmode problem (depending only on geometry, material independent!)

. 0),,(),,(

and , 0),,()0,,(

; 0)host( and 1)inclusion( where

,)( )()(

,0,0

2

2

=∂∂

=∂∂

===∈=∈

∂∂

=∂∂

∂∂

== yx Lyn

Lxn

znn

nnn

zyxy

zyxx

Lyxyx

s

ϕϕ

ϕϕθθ

ϕϕθ

rr

rr

rr

rr




Green’s function as an eigenmode expansion:( ) )()(

)(;, * rrrr ′

−=′ ∑ nn

n n

n

sssG ϕϕ

ωω

1

host

metal

)()(1)( parameter spectral thewhere

,)(equation an from found becan frequency complex whoseexcitation elementary are plasmons surface Physical

−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

=+

ωεωεω

γω

s

sis nnn




Local fields on nanoscale for harmonic excitation with external field at frequency ω:

( ) , ;,)()()( 3200 rdG ′′∇′−= ∫ ′ ωϕϕϕ rrrrr r

( ) , ;,),(

),(),(32

0

0

tdrdttGt

tt′′′−′∇′′

−=

∫ ′ rrr

rr

rϕ

ϕϕ

Local fields on nanoscale for short-pulse excitation with pulse field

)(0 rϕ

),(0 trϕ




Theorem on Localization of Surface Plasmons:

Any Anderson-localized (or, strongly localized) mode is dark.

Corollary: It is impossible that all surface-plasmon modes of any system are Anderson-localized (or, strongly localized).




Geometry of Random Planar Composite

0 5 10 15 20Li10 - 14

10 - 11

10 - 8

10 - 5

10 - 2

fi

Distribution of Eigenmodes over their Oscillator Strength fiand Localization Radius Li

Phys. Rev. Lett. 87, 167401 (2001).

0 10 20 30y0

10

20

30

z




Local Field Intensities for Four Eigenmodes Representative of Each Class of Eigenmodes

si=0.1995 , Li=2.1

30

y 30 z

0.1

0.3

Ei2

30

y

si=0.2, Li=11.2

30y 30 z

0.01

0.03

Ei2

30y

si=0.201 , Li=9.6

30

y 30 z

0.

0.01

Ei2

30

y

si=0.2015 , Li=1.

30

y 30 z

0

0.1

0.2

0.3

Ei2

30

y

Localized Luminous Delocalized Luminous

Delocalized Dark Localized Dark

07.0=f 02.0=f

910~ −f 910~ −f

si=0.1995 , Li=2.1

30y 30 z

0.

0.0005

0.001

Ei2

30y




0. 0.5 1.

4

8

12

16

20

s

L 3D Space 8‰32‰32

0. 0.5 1.

4

8

12

16

20

s

L 3D Space 4‰32‰32

0. 0.5 1.

4

8

12

16

20

s

L 2D Space 32‰32

0. 0.5 1.

2

4

6

8

10

s

L 3D Space 16‰16‰16

0. 0.5 1.

2

4

6

8

10

s

L 3D Space 4‰16‰16

0. 0.5 1.

2

4

6

8

10

s

L 2D Space 16‰16

L

Distribution of Eigenmodes over their Localization Length and Spectral Parameter




Giant Surface Enhanced Raman Scattering

M. I. Stockman, V. M. Shalaev, M. Moskovits, R. Botet, and T. F. George, Enhanced Raman Scattering by Fractal Clusters: Scale Invariant Theory,Phys. Rev. B 46(5), 2821-2830 (1992).




Single-molecule Surface-Enhanced Raman Scattering:

•K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. Dasari, and M. S. Feld, Single Molecule Detection Using Surface-Enhanced Raman Scattering (SERS), Phys. Rev. Lett. 78, 1667-1670 (1997).

•S. M. Nie and S. R. Emery, Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering, Science 275, 1102-1106 (1997).

•Z. J. Wang, S. L. Pan, T. D. Krauss, H. Du, and L. J. Rothberg, The Structural Basis for Giant Enhancement Enabling Single-Molecule Raman Scattering, Proc. Natl. Acad. Sci. USA 100, 8638-8643 (2003).

SERS Enhanced by factor of 1310~




Enhancement of Optical Responses in Fractals

Self-similar fractal geometry

Eiz

yx

0

100

200

t= 39.01 fs

Local optical fields in fractal cluster [MIS, Phys. Rev. Lett. 84, 1011 (2000) ].

9

4SERS

10~~ EG




SERS enhanced by a factor 1012 - 1014




Theory

Experiment

Comparison of theoretical predictions and experimental data for the SERS enhancement coefficient from silver colloid clusters.




K. Li, M. I. Stockman, and D. J. Bergman, Self-Similar Chain of Metal Nanospheres as an Efficient Nanolens, Phys. Rev. Lett. 91, 227402 (2003).

Efficient Self-Similar Nanolens of Nanospheres

Silver Nanospheres




++

+++++

--

--

--

-

Optical Electric Field

-

Underlying physics of local field enhancement in efficient nanolens: Cascade enhancement

Giant local fields in the minimum gap:

Nanoscale localization of optical energy




Local Fields for Silver 3-Sphere Nanolens

d=0.3R

w=3.37 eV

100

300

500

ÈEÈw=3.25 eV

0

500

1000

1500

ÈEÈ0

w=0.77 eV

0

5

10

15

ÈEÈ0

K. Li, M. I. Stockman, and D. J. Bergman, Self-Similar Chain of Metal Nanospheres as an Efficient Nanolens, Phys. Rev. Lett. 91, 227402 (2003).

Giant Local Field Enhancement in Nanolens




FDTD computations. C. Oubre and P. Nordlander (Private Communication).

d= 0.3 R




R. Hillenbrand and F. Keilmann, Optical oscillation modes of plasmon particles observed in direct space by phase-contrast near-field microscopy, Appl. Phys. B 73, 239–243 (2001)




CONCLUSIONS

• A self-similar chain of metal nanospheres makes an efficient nanolens focusing energy of optical field, concentrating it a nanoscale gap between the smallest nanosphere

• The optical field in the nanofocus is enhanced by more then three orders of magnitude

• A molecule adsorbed in this nanofocus will exhibit Raman scattering enhanced by a factor on order or greater than 1013.




DEPOLARIZATION AND DEPHASING IN SECOND HARMONIC GENERATION IN RANDOM METAL

NANOSTRUCTURED SYSTEMS

Mark I. Stockman, David J. Bergman, Cristelle Anceau, Sophie Brasselet, and Joseph Zyss, Enhanced Second Harmonic Generation By Metal Surfaces with Nanoscale Roughness: Nanoscale Dephasing, Depolarization, and Correlations, Phys. Rev. Lett. 92, 057402-1-4 (2004).




Experimental evidence of strong depolarization, dephasing, and giant fluctuations of SHG:C. Anceau, S. Brasselet, J. Zyss, and P. Gadenne, Local second harmonic generation enhancement on gold nanostructures probed by 2-photon microscopy, Opt. Lett. 28, 713 (2003).

Department of Physics and Astronomy Georgia State University Atlanta, GA 30303-3083


ITCP, Trieste, Italy 02/10/2005 29

S. I. Bozhevolnyi, J. Beermann, and V. Coello, Phys. Rev. Lett. 19, 197403 (2003)




rrrr

∂∂

=)( )()( 2(2))2( θχ EPNL

The total SH Polarization is found as:

rdG rNL

host′′⋅⎥⎦

⎤⎢⎣⎡ ′

′∂∂

= ∫ 3)2()2( )2 ;,( )(4)( ωεπϕ rrrP

rr

⎥⎦

⎤⎢⎣

⎡−=

∇−=

)2()(1)2 ,( where

),(4

1-)2 ,()()(

host

)2()2(NL

)2(total

ωθεωε

ϕπωε

s

PP

rr

rrrr

The SH potential is calculated as:




Geometry of the system SH Polarization

20

)2(

)2( )(

E

rP

χ

0 10 20 30x0

10

20

30

z

0

)(

E

rE

Local linear field

102030x

102030 z

0

100

200

102030x

10 20 30

10

20

30

10 20 30x

10

20

30

z102030

x102030 z

0

20

40

102030x




Im Pz(2)Re Pz

(2)

Re Px(2) Im Px

(2)

Depolarization of SH




Dephasing of SH

Polarized component Depolarized component

500ReP z

H2L

ImPzH2L

- 500 500ReP x

H2L

- 500

ImPxH2L




Correlation Functions for Different Fill Fractions p (Color Coded):

P = 0.35, 0.50, 0.60, 0.75, 0.95

0 2 4 6 8r0.

0.5

1.

CHrL

0 2 4 6 8r0.

0.5

1.

Cphase HrL

0 2 4 6 8r0.

0.5

1.

Cpolariz HrLH.J.Simon




Giant fluctuations of SHG local fields (new result)

Giant fluctuations of local optical fields introduced:M.I.Stockman, L.N.Pandey, L.S.Muratov, and T.F.George, Giant Fluctuations of Local Optical Fields in Fractal Clusters, Phys. Rev. Lett. 72(15), 2486-2489 (1994). The universal index –1.5 for fundamental and –2.5 for SH

10 - 210 - 1 10 102 103ISH

10 - 6

10 - 4

10 - 2

PHISHL- 0.773087 - 2.39585 x




CONCLUSIONS•Second harmonic polarization P(2)(r) in a nonlinear random planar composite is a highly singular function concentrated in hot spots. P(2)(r) is significantly different from local fields E(r).

•SHP P(2)(r) is both dephased and depolarized at the nanoscale. SHG from a random composite on any microscopic scale is incoherent, i.e., hyper-Raleigh scattering

•The local intensity of SHG undergoes giant (non-Gaussian) fluctuations with scaling distribution function whose index is not universal




Ivan A. Larkin and Mark I. Stockman, Imperfect Perfect Lens, Phys. Nano Lett. 5, 339 – 343 (2005)

Extreme Nanoplasmonics: Fundamental limit on the minimum scale of the energy concentration/spatial resolution




At small distances a, interactions transfer large momentum a

p h~

or wave vector a

k 1~

For large enough k, interacting electron system exhibits strong spatial dispersion and Landau damping . These effects are described by Klimontovich-Silin-Lindhart (RPA) formula

scm 10~~ 8

Fvak

≤ωω




Pendry’s Perfect Lens in near field

Metal slab

a

bObject2a-b

Ideal image

λ<<ba,It is required that the relative dielectric permittivity 1−=ε




Estimate of the ultimate spatial resolution of the “perfect lens”

where Q~10-100 is the surface plasmon resonance quality factor




Imaging

Image of a washer as produced by a silver slab of 5 nm thickness in GaAs host: (a) without spatial dispersion, (b) with spatial dispersion. Photon energy: 2.2 eV.

10 20

0.5

1.

Image of a point charge produced by a 5 nm silver slab in GaAs environment, photon energy: 2.2 eV

r




CONCLUSIONS

1. At small distances (on order of a few nanometers), the conventional local electrodynamics is no more applicable.

2. Nonlocality of permittivity and Landau damping become important.

3. These effects limit the spatial resolution of the “perfect lens” to no better than 5 nm

4. Other resonant, plasmonic effects are also limited by these effects and die out at the scale of a few nanometers.

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