Understanding the Rainbow

Dielectric materials:
All charges are attached to specific atoms or molecules
Response to an electric field E :
Microscopic displacement of charges

P
 
0


E

Relative dielectric permittivity
   
1


E
0
 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
 
E

 
H

1

0

0
 
P



H

E

 
0
 
0

E
 t


H
 t

P
 t

J



 leading to the wave equation :
 2
E

1 c 2
 2
E
 t 2
 
0
 2
P
 t 2
 
0

J
 t
In vacuum (P = J = 0):
E k


E
2

0
 e

 i
 c
( k
 r
  t ) c

1

0

0

Deriving the relation between P and E in a dielectric
Equation of motion of the electron:
P
 
Ne x m d
2 x dt
2
 m
 d x
 k x dt
  e E
 : damping coefficient for given material
k: restoring-force constant resonance frequency:

0
 k / m assume E is varying harmonically, and also x
 x
0 e
 i
 t

P
 
Ne x


0
2
Ne
2
 
2
/ m
 i

E

Inserting P(E) in wave equation 
2 E

1 c
2
 2
E
 t
2
 
0
 2
P
 t
2
 
0

J
 t gives:
 2
E

1 c
2

1

Ne
2 m

0



0
2 
1

2  i



  2
E
 t
2 solution:
E

E
0 e i ( k z z
  t ) with complex propagation constant k z
=

+ i α : k z
2 


  c 

 2




1

Ne
2 m

0




0
2  
1
2  i








E

E
0 e
  z e i (
 z
  t ) k z

2



 n c
So that…

So that we find the refractive index of the dielectric: n
2  



1

Ne
2 m

0




0
2  
1
2  i








For a dielectric with multiple resonances: n
2 
1

Ne
2 m

0
 j

2 j
 
2 f j
 i
 j


Rainbow: why red outside, blue inside ?

Rainbow: why red outside, blue inside ?
Red (small frequency): smaller n
Blue (high frequency): larger n

What is a plasmon?
“plasma-oscillation”: density fluctuation of free electrons
+ + +
k
+ +
Plasmons in the bulk oscillate at
 p drude 
Ne
2 m

0 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

Sphere in a uniform static electric field
 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
 p

4

0
R
3



 
2
 m m
 m

E
0 electric polarizability of a sphere α
 
4

0
R
3
 
2
 m m resonant enhancement of p if

(

)

2
 m
 minimum
 negative real dielectric constant ε
1
(ω)
Bohren and Huffman (1983), p.136
ε = ε
1
(ω)+i ε
2
(ω) = dielectric constant of the metal particle
ε m
= dielectric constant of the embedding medium usually real and taken independent of frequency


 x x
Derivation using quasi-static approximation
V
  r

 f
  e
  i
 r k  r
  t

, r k  r
 2 r k 
2


  V
   f
  e  i  t
E inc
 k y

E inc


E
0 e i
  k 

  t
 k
E inc y

E inc

 E
0 e
 i  t

Derivation using quasi-static approximation
Equations:

1

2


0
0

 r r

 a a


E
0
 m
 a q r z
Boundary conditions:

1
 
2
 r  a

, 
 
1
 r
  m
 
 r
2
 r  a

, r lim
 

2
  E
0 z
Jackson (1998), p.157
Bohren and Huffman (1983), p.136


Derivation using quasi-static approximation
Equations:

1

2


0
0

 r r

 a a


E
0
 m
 a q r z

Solution:

1
  E
0 r cos q 


  
  2  m m

2 with:
  E
0 r
 r cos

0
 m q  a 3


 r
E
0
  
  2  m m
  4  a 3


  
  2  m m




E
0 r cos q  


3  m
  2  m


E
0 cos r 2 q
  E
Sphere in electromagnetic field (a <<  ): r
 
0
 m
 r
E
0 e  i  t


E
0 r cos q
0 r cos q 
4 p cos
 q m r 2
Jackson (1998), p.157
Bohren and Huffman (1983), p.136


Metal nanoparticles:
• Extinction = scattering + absorption
• Large field enhancement near particle n=1.5
I enh
20
Au
550nm
At resonance, both scattering and absorption are large albedo = scattering / extinction = s sca
/( s abs
+ s sca
)

Reosnance spectra
Groupings of 35nm Au NPs are obtained after surface ligand exchange (thio-PEG instead of BSPP)
0.8
35nm
Dimer
Trimer
0.6
0.4
0.2
0.0
400 450 500 550 600 650
Wavelength (nm)
Extinction spectra in water
700

Resonance tunable by dielectric environment
Ag, D=100 nm
Si
3
N
4
(n=2.00) Si (n=3.5)
Q
D O Q
D
H
Optics Express (2008), in press

Resonance spectra for particles on surface
14
12
10
8
6
4
2
0
500
σ scat normalized to particle area
Q
30 nm
600
30nm sub tot
D
10 nm 10nm
700 800 wavelength (nm)
900 1000
Appl. Phys. Lett. 93 , 191113 (2008)

Other applications of nanoparticles
Old: New:
(but the same principle)
Different materials/shapes: distinct colors
Au colloids in water
(M. Faraday ~1856)
All particles are driven by the external field and by each other
Focusing and guidance of light at nanometer length scales
(image: CALTECH)

Interaction between particles
An isolated sphere is symmetric, so the polarization direction does not matter.
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
Near field enhancement in gaps between particles: nanoscale antenna