Surface Plasmons. Surface plasmons: outline 1.Time-line of major discoveries 2.Surface plasmons -...

Surface Plasmons

Surface plasmons: outline1. Time-line of major discoveries2. Surface plasmons - surface mode of

electromagnetic waves on a metal surface

3. Spectroscopy of SPs in nanostructures:(a) Nanoparticles(b) Gratings, nanostructures 4. Applications: sensors, nanophotonics,

surface enhanced Raman spectroscopy (SERS)

Surface plasmons, A. Kolomenski, S. Peng, 9/24/2012

Time line

Excitation of SPs with a prism: Raether, Kretschmann

1941

1907 Rayleigh’s explanation (angle-diffraction orders)

1993-

Fano: role of surface waves, surface plasmons

1968

1991

1902 Wood anomalies: reflection on gratings (two types)

Nanoplasmonics, extraordinary transmission, etc. First biosensor on SPs

SPs allow to localize and guide EM waves!!!

1974 Surface Enhaced Raman Spectroscopy


Maxwell’s equations (SI units) in a material, differential form

density of charges

density of current

f

fJ


Wave equation

2 2( ) ( )B B B B

2

2 2 21 1

( ) ( ) ( )B

E E Bt t t tc c t

22

2 21

0B

Bc t

22

2 21

0E

Ec t

0

Double vector product rule is used a x b x c = (ac) b - (ab) c


Plane waves

0 [ ( )]B B Exp i k r t

)]([0 trkiExpEE

B i k B

E i k E

is parallel to

is parallel to B

E

Thus, we seek the solutions of the form:

From Maxwell’s equations one can see that k

B

E


Incident light

Simple system of a metal bordering a dielectric with incident plane wave

Dielectric, refractive index is dielectric permittivity

Metal (gold)

2n

2

1 r imi

Reflected light

Transmitted light


Waves at the interface

Assume that incident light is p-polarized, which means that the E-vector is parallel to the incidence plane

)]([),0,( 11111 tzkxkiExpEEE zxzx

1 1 1 1(0, ,0) [ ( )]y x zB B Exp i k x k z t

Then the vector of the magnetic field is perpendicular to the incidence plane and has the form

In medium 1, z<0, 2

1

22

12

1c

kk zx

In medium 2, z>0, 22

22

22

2c

kk zx

2 2 2 2(0, ,0) [ ( )]y x zB B Exp i k x k z t

)]([),0,( 22222 tzkxkiExpEEE zxzx

x

yz

1E

x

1xE

1zE


Boundary conditions

xx EE 21

1 1 2 2 1 2/ / , . .y y y yB B i e H H

1 1 2 2z zE E

1 1 1 2 2 2 1 1 1 2 21 2

( / ) / / ( / / ) ( / ) 0t y y tl l

EB dl B dl B dl l B B B ds ds

t

1 1 2 1 1 1 2 21 2

( ) ( ) 0z zS S S V

Eds E ds E ds S E E E dv

1 1 2 2 1 1 21 2

( ) 0ix x i

l l

BEdl E dl E dl l E E Eds ds

t

Gauss’s theorem

Stokes's theorem

Stokes's theorem

x

yz dl

0s

0s

0V Surface plasmons, A. Kolomenski, S. Peng, 9/24/2012

Relations in an E-M wave

the curl operator

ˆ ˆ ˆ

x y z

x y z

Ax y z

A A A

( ) ( ) ( )x y z xExp ikr Exp ik x ik y ik z ik Exp ikrx x

[ ]i k E i B

[ / ]i k B i E

1 1[ / ] ( ) z y

x x y z z yk B

E k B k B k B


Derivation of the dispersion equation

xx EE 21 From the other condition =>1 2

1 21 2

z zy y

k kH H

yy HH 21 One boundary condition is

Therefore we have a system of 2 homogeneous equations and a nontrivial solution is possible only if the determinant of this system is equal to 0.

011

2

2

1

1

2

2

1

10

zz

zzkkkkD

0

0

22

21

1

1

21

yz

yz

yy

Hk

Hk

HH

Assume no external currents or free charges, magnetic permeability.

1 2 0


Surface plasmon dispersion equation1 2

1 2

z zk k

)()( 212

222

222

221 k

ck

c

We square both sides

21

212

22

c

k

We introduce , wavenumber of the surface plasmon, then we obtainxkk

22 2 2 2 22 1 2 1 1 2 2

( ) ( )kc


Dispersion equation and properties of surface plasmons

We would like to have a solution which is localized to the surface, i.e. it decays with distance from on both sides from the interface.

0][ 1 zz zikExp

0][ 2 zz zikExp

This is possible, if

0, 112

22

2

1 qiqkc

k z

0, 222

12

2

2 qiqkc

k z

1 1 1[ ] [ ( )] [ ] 0zzExp ik z Exp i iq z Exp q z

Indeed, then we have waves localized near the interface

2 1 1[ ] [ ( )] [ ] 0zzExp ik z Exp i iq z Exp q z


Dispersion equation analysis

1 2

1 2

z zk k

1 2

1 21 2

, 0q q

and q q

This is only possible, if 1 20 0or

21

212

22

c

k

If we look again at the dispersion equation

,k must be real (propagating wave!), then withnegative, we see that the condition for surface waves to exist is

1 20 0or

1 20 0 ( )and dielectric

1 2 1 20, . .i e Surface plasmons, A. Kolomenski, S. Peng, 9/24/2012

Relation of Plasmonics to SOME other fields

Metamaterials

Plasmonics

NanotechnologyOptics

Biotechnology

SERS High harmonics generator coherent control imaging

Electronics

Opto-electronics

molecular interactions nano-sensors proteomics

nanostructuresnanophotonicsnanoantennas

The Growth of the Field of Surface Plasmons

PIETER G. KIK and MARK L. BRONGERSMA SURFACE PLASMON NANOPHOTONICS, (2007)

illustrated by the number of scientific articles published annually containing the phrase “surface plasmon” in either the title or abstract

Surface plasmons (or surface plasmon polaritons), Part 2: outline

1. Why SP named so?2. Excitation of SPs: with a prism or a

grating3. Spectroscopy of SPs in nanostructures:(a) Nanoparticles(b) Gratings, nanostructures 4. Applications: sensors, nanophotonics,

surface enhanced Raman spectroscopy (SERS)

Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012

Dielectric constant of a metal, Drude model2

02

00 0 2

0 0

20 0

0 2 21

, ~ ( )

~ ( )

e

e

r

Ni

i e e

d xm eE E E exp i t

dteE

then x x exp i t xm

D E P E

eE Ne EP ex Nex

m m

Consequently, 2 2

2 200

1 1 ,where pr p

ee

Ne Ne

mm

plasmon frequency

For free electrons!


Remarks to Drude’s formula

2

2 *0

,where pb p

e

Ne

m

Bound electrons should be taken into account, then 1-> ,bwhich takes into account the contribution of bound electrons.

Also the mass of electron should be replaced with the effective mass of electron in the metal, .*

em

For << p:

Influence of attenuationtieE

dt

dxm

dt

xdm e02

2

2 2

2 3' 1 , "p p

Plasmons correspond to , these are eigen (free) oscillations of the electronic plasma.

0


Electrons oscillating in the SP field

metal

dielectric

Interface

There is a longitudinal component in the electric field of SP, because E-M field is coupled to oscillations of the electronic density (plasmonic oscillations).

This is why tp exite SPs one needs a p-polarization of the incident light.


Graphing dispersion equation of SPs

,k

2

1 2 *0

,where pb p

e

Ne

m

1/21 2

1 2ck

(

Light line: ck

For excitation of SPs we need to slow down light!


Surface plasmon excitation: Coupling of light to SPs with a prism

metal film (n1)

prism (n0)

sample (n2)

incident laser beam

reflected beam

SPW evanescentwave

0: critical angle Optical arrangement used to

excite the surface-plasmon wave based on the Kretschmann-Raether configuration where a thin metal film is sandwiched between the prism and the sample.

Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012E. Kretschmann, Z. Phys. 241, 313-324 (1971).

SPR curves for different wavelengthsSPR curves for different wavelengthsGold film (d=47nm) contacting water

50 60 70 80 900.0

0.2

0.4

0.6

0.8

1.0 =1230 nm

=633 nm

=490 nmREFL

ECTI

ON

CO

EFFI

CIEN

T

INCIDENCE ANGLE (deg)Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012

Momentum conservation

)(sin2

spkn

Energy conservation

SPLight

Conditions for the Surface Plasmon Resonance (SPR): phase matching!!!

k

ck

SP

)sin/( 2 ck

ck / 2

0

ksp

p

Resonance excitation with a prism


Surface Plasmon Part 3

Graphing dispersion equation of SPs2

1 2

2

*0

,

where

pb

pe

Ne

m

1/2 1/22

2221/2 22

1 22 2

1 22 22 2

,

pp b

b

p pb b

ck ck kc

,k

(

Light line: 2/ck

For excitation of SPs we need to slow down light!


2

2 22 1/22

, 0; . 12( )

p p pb m b m

b

k when then For we have

m

Approximation of small losses Approximation of small losses

Rk k k

i i rad

p p i i rad

1

4 1 22

1 22

( )

[ ( )] ( )

k np p ( / )2 0 t h e m e t a l f i l m i s i n f i n i t e l y t h i c k

n p r r r r [ / ( ) ] / 1 2 1 21 2

k p d e s c r i b e s t h e c o r r e c t i o n d u e t o f i n i t e t h i c k n e s s

i 1 2, i n t e r n a l l o s s e s i n t h e f i l m a n d i n t h e m e d i u m

r a d r a d i a t i v e l o s s


A. Kolomenski et al., Applied Optics, Vol. 48, 5683-5691 (2009)

The influence of the thickness of the gold film on the properties of SPs

40 42 44 46 48 500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Ref

lect

ivity

Incidence angle (deg)

10 nm 20 nm 50 nm 80 nm 120 nm

(a)

20 40 60 80 100 120 1400

20

40

60

80

100

120

140

160

180 (b) =633 nm =805 nm

Atte

nuat

ion

leng

th (m

)

Film thickness (nm)

(a) SP resonance curves at 633 nm for different film thicknesses.(b) The dependence of the attenuation length on the film thickness for 633 nm and

805 nm. The dielectric constants published by Palik are used.

Gold

Glass

Air

-1res00 )cos( kLsp


Examples: changes in the flow cell, bio-molecular binding reactions

B

0 10 20 30 40 50 60250

300

350

400

450

500

550B

A

HRP

BB

NHS/EDC

SPR

angl

e (p

ixel

s)Time (min)

Example: binding of monoclonal antibody to horseradish peroxidase protein

0.30

0.35

0.40

0.45

0.50

70.50 70.75 71.00 71.25 71.50

INCIDENCE ANGLE (deg)

C=0.82%C=0%

0.64 deg

Applied this sensing technique to myofibers and tubulin molecule.

A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, Applied Optics 36, 6539-6547 (1997).

Sensitivity and detection limit(relationships between different quantities)

angular resolution -4deg=2 RU

changes of the refractive index n-6

average thickness of the protein layer d=0.03 Å

surface concentration d=3 pg/mm2

with mprotein=24 Da surface concentration of molecules ns=1010 cm-2

A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, Applied Optics 36, 6539-6547 (1997).

600 800 1000 1200 1400 1600 1800 2000 2200 24001

10

100

1000

exact, from [1]

approximate, from [1]

exact, from [2]

exact, from [3]Atte

nuat

ion

leng

th (

m)

Wavelength (nm)

Au

600 800 1000 1200 1400 1600 1800 2000 2200 24001

10

100

1000

exact, from [1]

approximate, from [1]

exact, from [2]

exact, from [3]Atte

nuat

ion

leng

th (

m)

Wavelength (nm)

Ag

1. American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, 1972), p. 105.2. U. Schröder, Surf. Sci. 102, 118-130 (1981).3. Handbook of Optical Constants of Solids, E.D. Palik, ed. (Academic1985).

Attenuation lengths of SPs for gold and silver films in contact with air, calculated for a broad spectral range

Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012A. Kolomenski et al., Applied Optics, Vol. 48, 5683-5691 (2009)

SPs:

dielectric2 ZZ

EE

1 metal

)|kexp(|~ 1z z

~ exp( | | ) k2z z

21)Re(

01frequency;plasmon

,2

2

1 :electrons free ofion Approximat

2121

222with wavegPropagatin

p<

p

cxk

Condition of existence:

•Spatially localized to the surface E-M wave•Oscillations of the electronic density. •Have E -longitudinal component•Are excited with p-polarized light and the local field can significantly exceed the field in the exciting beam.

ep m

ne

0

2

Summary of surface plasmons

b


42 43 44 45 460

10

20

30

40

50

60

70

80

90

100

110

633nm 633nm with

1,eff.

805nm 805nm with

1,eff.

|t 012(

)|2

Angle (deg.)

Dependence of the near field intensity enhancement factor on the back side of the gold film vs. the angle for two wavelengths 633 nm and 805 nm

Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012A. Kolomenski et al., Applied Optics, Vol. 48, 5683-5691 (2009)

SP resonance: coupling with a grating (conservation of momentum)

ki

θ

ki sin(θ)kg

kSP

kSP = ki sin(θ) - kg

ki θ

ki sin(θ) kg

kSP

kSP = ki sin(θ) + kg

+1 order coupling -1 order coupling

grating


Conditions for the resonance excitation of SPsLight line

( / )c n k

0 frequency of the source

required additional momentum

Light line, suited for resonance excitation

,k

SPs are slower than light, and therefore for the same frequency their momentum is larger.To enable the resonance excitation additional momentum must be provided.

SP dispersion curve

The crossing of the SP curve and the light line means resonance excitation for desired frequency 0


Conditions for the resonance excitation of SPs

Conditions for the resonance excitation of SPs:a photon is converted into a surface plasmon.

General laws must be observed:

(1)Energy conservation,

(2) Momentum conservation,

( / 2 )(2 )light light SP light SPh h h h

, ,x light SP x light SPhk hk k k

xk

zk

zk is changing xk is not changing

SP xk k


λ

θ

Schematic of experiment on spectroscopy of SP modes in nanostructures :transmission measurements in the far field

Charge Coupled Device(CCD)

Sample(nanostructure)

Laserbeam

GGratingating

This setup maps intensity distributionover angle and wavelength and thus reveals SP modes that affect transmission.


Study of the Interaction of 7 fs Rainbow Laser Pulses with Gold Nanostructure Grating: Coupling to Surface Plasmons

Wavelength (nm)

Intensity

Angle ofIncidence

650

800

-5°

0°

5°

Transmission dependence

The valley area (x-structure) the laser light is efficiently converted into SPs, about 80% .

10 µm

0 µm

5 µm

10 µm0 µm 5 µm

0.00 nm

58.00 nm

AFM image of the nanostructure:

A. Kolomenskii et al., Optics Express, 19, 6587-6598 (2011). Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012

Mie theory and dipole approximation

)()]2)([

)(9)(

222/3

idr

imext V

c

For small nanoparticles (R<<, or roughly 2R< /10): dipole

approximation

where V is the particle volume, frequency light, εm and are the dielectric functions of the surrounding medium and the particle material.When is small or varies slowly, the resonance takes place at

)(2

2

2

1 ,0)2()(

p

rdr d

p

21max

0)()()( iir

t=0 t=T/2

Electronic cluster

Ionic clusterElectric field

Light Electronic plasmaoscillations

=>


Extinction spectra of Ag n-particles in solution

350 375 400 425 450 4750.0

0.2

0.4

0.6

0.8

1.0

1.2

Ag 27 nm particles Ag 48 nm particles

Ext

inct

ion

(a.

u.)

Wavelength, nm

The oscillations of a n-particle, induced by a pump pulse, modulate (displace) the plasmon absorption band. For efficient detection the probe wavelength was selected at the steeper portion of the slope of this band.


S. N. Jerebtsov et al. Phys. Rev. B Vol. 76, 184301 (2007).

Bowtie nano-antenna and measured intensity enhancement

Intensity enhancement vs wavelength

3D finite difference time domain (FDTD) simulations

Fabricated by Electron Beam Lithography(EBL) bowtie antennas. Indium tin oxide substrate. Gap was varied, thickness 20 nm.

Kino et al. In: Surface Plasmon nanophotonics, p.125 (2007).


Experimental setup for study of “hot spots” for SERS Raman signals from individual Ag n-particles

Raman microscope with sensitive CCD cameras for imaging the sample in scattering and using Raman signal. Notch filters were used to suppress the excitation light. Low concentration of n-particles needed to separate individual particles.

Futamata et al. Vibrational Spectroscopy 35, 121-129 (2004).

Raman spectroscopyPhoton scattering on molecules

Elastic orRayleigh scattering

Inelastic orRaman scattering

h

h h(-) h(+)

Stocks Anti-Stocks

Raman scattering increases when hproduces electronic transition


Surface Enhanced Raman Spectroscopy (SERS) of DNA bases

Spectra of individual n-particles

Stongest enhancement ~1010 from pairs of particles with axis parallel to polarization

Characteristic stretching modes in heterocycles suited for DNA sequencing :adenine 718 and 893 cm-1;guanine 641cm-1; cytosine 791 cm-1; thymine 616, 743 and 807 cm-1.

Time evolution (whole scale 1 s) demonstratesRaman peaks and blinking effect, known for single molecule detection.

Futamata et al. Vibrational Spectroscopy 35, 121-129 (2004).

Surface Plasmons. Surface plasmons: outline 1.Time-line of major discoveries 2.Surface plasmons -...

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Transcript of Surface Plasmons. Surface plasmons: outline 1.Time-line of major discoveries 2.Surface plasmons -...