Resonances and optical constants of dielectrics:

basic light-matter interaction

Understanding the Rainbow

EEP

100

Dielectric materials: All charges are attached to specific atoms or molecules

Response to an electric field E: Microscopic displacement of charges

Relative dielectric permittivity describes how a material is polarized in response to an electric field

depends on frequency: ()

If we know the relation between P and E we can solve Maxwell’s equations

t

ttH

E

JPE

H

H

PE

0

0

0

0

1

tttc

JPE

E 02

2

02

2

22 1

leading to the wave equation:

00

1

c

In vacuum (P = J = 0):

)(0

tie rkEE

c

2

k

Exxx

ekdt

dm

dt

dm

2

2

Deriving the relation between P and E in a dielectric

Equation of motion of the electron:

ExP i

mNeNe

220

2 /

: damping coefficient for given materialk: restoring-force constant

resonance frequency:

assume E is varying harmonically, and also

mk /0 tie 0xx

xP Ne

2

2

2200

2

22 1

11

tim

Ne

cE

E

Inserting P(E) in wave equation

gives:

solution:

with complex propagation constant kz = + iα :

)(0

tzki ze EE

im

Ne

ckz 22

00

222 1

1

)(0

tzizee EESo that…

c

nkz

2

tttc

JPE

E 02

2

02

2

22 1

im

Nen

2200

22 1

1

So that we find the refractive index of the dielectric:

j jj

j

i

f

m

Nen

220

22 1

For a dielectric with multiple resonances:

Rainbow: why red outside, blue inside ?

Blue (high frequency): larger n

Red (small frequency): smaller n

Rainbow: why red outside, blue inside ?

Light scattering from small resonant particles

Metal nanoparticle plasmons

What is a plasmon?

Plasmons in the bulk oscillate at

determined by the free electron density and effective mass

Plasmons confined to surfaces that can interact with light to form propagating “surface plasmon polaritons (SPP)”

Confinement effects result in resonant SPP modes in nanoparticles

+ + +

- - -

+ - +

k

“plasma-oscillation”: density fluctuation of free electrons

0

2

m

Nedrudep

Sphere in a uniform static electric field

Bohren and Huffman (1983), p.136

particle can be considered as a dipole:

in a metal cluster placed in an electric field, the negative charges are displaced from the positive ones

electric polarizability of a sphere α

03

0 24 ERp m

m

m

304

2m

m

R

ε = ε1(ω)+i ε2(ω) = dielectric constant of the metal particle

εm = dielectric constant of the embedding mediumusually real and taken independent of frequency

minimum m 2)(

negative real dielectric constant ε1(ω)

resonant enhancement of p if

Vr r , t f

r r e i

r k

r r t ,

r k

2

r r

r k

r r 2 V

r r , t f

r r e it

y

Einc trkiinc EtrE

e, 0

x

k

y

Einc tiinc EtrE e, 0

x

k

Derivation using quasi-static approximation

zEarrr

arr

m 0221

21

lim,,

E0 m

z

r

a

Jackson (1998), p.157Bohren and Huffman (1983), p.136

Boundary conditions:

Derivation using quasi-static approximation

ar

ar

0

0

2

1

Equations:

ar

ar

0

0

2

1

E0 m

z

r

a

Equations:

Solution:

1 E0rcos m

2m

E0rcos

3m

2m

E0rcos

2 E0rcos a3 m

2m

E0

cosr2

E0rcos pcos4mr2

r p 0m

r E 0

4a3 m

2m

with:

r p 0m

r E 0e

it

Sphere in electro-magnetic field (a << ):

Jackson (1998), p.157Bohren and Huffman (1983), p.136

Derivation using quasi-static approximation

550nm

20Au

n=1.5 Ienh

Metal nanoparticles:

• Extinction = scattering + absorption• Large field enhancement near particle

At resonance, both scattering and absorption are large

albedo = scattering / extinction = sca/(abs+sca)

400 450 500 550 600 650 7000.0

0.2

0.4

0.6

0.8

Op

tica

l de

nsi

ty

Wavelength (nm)

35nm Dimer Trimer

Reosnance spectra

Extinction spectra in water

Groupings of 35nm Au NPs are obtained after surface ligand exchange (thio-PEG instead of BSPP)

Resonance tunable by dielectric environment

Ag, D=100 nm

Si3N4 (n=2.00) Si (n=3.5)

DQDQ O

H

Optics Express (2008), in press

Resonance spectra for particles on surface

σscat normalized to particle area

500 600 700 800 900 10000

2

4

6

8

10

12

14

wavelength (nm)

Qsc

at, Q

subs

30nm

10nm

30 nm

10 nm

D

Q

Appl. Phys. Lett. 93, 191113 (2008)

tot

sub

Different materials/shapes: distinct colors

Old: New:

All particles are driven by the external field and by each other

Focusing and guidance of light at nanometer length scales

(but the same principle)

Other applications of nanoparticles

Au colloids in water(M. Faraday ~1856)

(image: CALTECH)

LONGITUDINAL: restoring force reduced by coupling to neighbor Resonance shifts to lower frequency

TRANSVERSE: restoring force increased by coupling to neighbor Resonance shifts to higher frequency

An isolated sphere is symmetric, so the polarization direction does not matter.

Interaction between particles

Near field enhancement in gaps between particles: nanoscale antenna