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Resonances and optical
constants of dielectrics:
basic light-matter interaction
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   0   1E
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
E  
1
0
P
H  0
E P
  H  0

J
t t
H
  E  0
t
leading to the wave equation:

2
2

1

E

P
J
2


E






0
0
2
2
2
c

t

t
t


In vacuum (P = J = 0):
E  E 0 e  i (k r t )
k 
2



c
c
1
 0 0
Deriving the relation between P and E in a dielectric
Equation of motion of the electron:
P   Nex
d 2x
dx
m 2  m
 kx  eE
dt
dt
: damping coefficient for given material
k: restoring-force constant
resonance frequency:
0  k / m
assume E is varying harmonically, and also

Ne2 / m
P   Nex  2
E
2
 0    i
x  x 0 e it
Inserting P(E) in wave equation
1  2E
 2P
J
 E  2 2  0 2  0
c t
t
t
2
gives:
2

1
Ne
2 E  2 1 
c  m 0

   2E
1
  2   2  i   t 2
 0

solution:
E  E0ei ( k z z t )
with complex propagation constant kz =  + iα :
 
2
kz   
c
2
E  E0e
 Ne 2
1 
 m
0



1
 2
 
2
 0    i  
z i ( z t )
e
kz 
2

So that…

n
c
So that we find the refractive index of the dielectric:
 Ne 2
n  1 
 m 0
2


1
 2
 
2
 0    i  
For a dielectric with multiple
resonances:
2
Ne
n2  1 
m 0

j
fj
2
j
  2  i j
Rainbow: why red outside, blue inside ?
Rainbow: why red outside, blue inside ?
Red (small frequency): smaller n
Blue (high frequency): larger n
Light scattering from small
resonant particles
Metal nanoparticle plasmons
What is a plasmon?
“plasma-oscillation”: density fluctuation of free electrons
+
+ +
Ne2
Plasmons in the bulk oscillate at  p

m 0
determined by the free electron density and
effective mass
drude
-
-
-
k
+
-
+
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


3   m
p  40 R
 m E0
  2 m
electric polarizability of a sphere α
resonant enhancement of p if
 ( )  2 m  minimum
 negative real dielectric constant ε1(ω)
Bohren and Huffman (1983), p.136
  4 0 R 3
  m
  2 m
ε = ε1(ω)+i ε2(ω) =
dielectric constant of
the metal particle
εm = dielectric constant
of the embedding medium
usually real and taken
independent of frequency
Derivation using quasi-static approximation
r
r
r ik rr t  r 2
V r ,t   f r e
, k 

r r
r
r
r it
r    k  r  2  V r ,t   f r e
Einc

k
 
 i kr t 
E inc r ,t   E 0 e
y
x
Einc
k
 
 it
Einc r ,t   E 0 e
y
x
Derivation using quasi-static approximation
Equations:
1  0
 2  0
E0
r  a 
r  a 
m

r
q
z
a
Boundary conditions:
1   2
r  a ,  1   m  2
r
r
r  a ,
lim  2  E 0 z
r 
Jackson (1998), p.157
Bohren and Huffman (1983), p.136
Derivation using quasi-static approximation
Equations:
1  0
 2  0
E0
r  a 
r  a 
m

r
q
z
a
Solution:
    m 
 3 m 
1  E 0 r cos q  
E 0 r cosq  
E 0 r cos q
  2 m 
  2 m 

 cos q
pcos q
3  m
2  E 0 r cos q  a 
E 0 2  E 0 r cos q 
2


2

r
4

r

m 
m
r
r
p
  0 m E 0
with:
Sphere in electromagnetic field (a << ):





r it
m
r
  4 a 3 

p   0 m E 0e


2



m
Jackson (1998), p.157
Bohren and Huffman (1983), p.136
Metal nanoparticles:
• Extinction = scattering + absorption
• Large field enhancement near particle
n=1.5
Ienh
20
Au
550nm
At resonance, both scattering and absorption are large
albedo = scattering / extinction = ssca/(sabs+ssca)
Reosnance spectra
Groupings of 35nm Au NPs
are obtained after surface
ligand exchange (thio-PEG
instead of BSPP)
Optical density
0.8
35nm
Dimer
Trimer
0.6
0.4
0.2
0.0
400
450
500
550
600
650
700
Wavelength (nm)
Extinction spectra in water
Resonance tunable by dielectric environment
Ag, D=100 nm
Si3N4 (n=2.00)
Q
D
O
Si (n=3.5)
Q
D
H
Optics Express (2008), in press
Resonance spectra for particles on surface
σscat normalized to particle area
14
12
Q
Qscat, Qsubs
10
30 nm
30nm
tot
8
sub
6
D
4
2
0
500
10 nm10nm
600
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
All particles are
driven by the
external field
and by each
other
Au colloids in water
(M. Faraday ~1856)
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