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```2. LORENTZ MODEL OF LIGHT MATTER INTERACTION
2.1. From microscopic to macroscopic response
 Review the main concepts in basic atom-field interactions. In particular the
Lorentz model, a pre-quantum mechanics model, and its asymptotic case for
metals, the Drude model.
 The Lorentz model explains much of classical optics via a physical picture
borrowed from mechanics. The starting point is the “mass on a spring”
description of electrons connected to nuclei. Thus, the incident electric field
induces displacement to the electron that is under the influence of a spring-like
restoring force due to the nucleus.
1
 The equation of motion for the electron can be expressed as
d 2 x(t )
dx(t )
e
2




x

E (t ),
0
2
dt
dt
m
(0.1)
  is the damping constant, 0 is the resonant frequency, e is the electronic
charge, m mass of the electron, and E the incident field.  and 0 are
characteristics of the material, the first describing the energy dissipation
property of the medium and the second the ability of the medium to store
energy.
 Since Eq. 57 is a linear differential equation, Fourier transforming both sides of
the equation gives the frequency-domain solution.
2
 Using the Fourier property of the differential operator,
dn
n

(
i

)
,
n
dt
(0.2)
 we obtain Eq. 57 in the frequency domain
e
 x( )  i x( )  0 x( )   E ( ) .
m
2
2
(0.3)
 Thus, we find the solution for the charge displacement in the frequency domain,
e
E ( )
x( )  2 m
.
2
  i  0
(0.4)
 To obtain the time domain solution, x(t), we need to Fourier-transform Eq. 60.
However, we explore further the frequency domain solution. The induced dipole
moment due to the charge displacement x() is
p( )  e  x( )
(0.5)
3
 In Eqs. 59-60 we obtained microscopic quantities, the atomic-level response.
 The macroscopic behavior of the medium is obtained from the induced
polarization P, which captures the contribution of all dipole moments within a
certain volume,
PN p .
(0.6)
 N is the volume concentration of dipoles (m-3) and the angular brackets denote
ensemble average.
 Assuming that all induced dipoles are parallel within the volume, we obtain
Ne2
E ( )
P( ) 
 2
.
2
m 0  i  
(0.7)
4
 Generally, each atom has multiple resonances or dipole-active modes, such that
Eq. 63 can be generalized to
 E ( )
Ne2
P( ) 
 2 i 2
,
m i 0i    i i
(0.8)
 The summation is over all modes, characterized by different resonant
frequencies and damping constants. The weight i is called the oscillator
strength and has the quantum mechanical meaning of a transition strength.
 For simplicity, we reverse to the single normal mode description, which captures
the origin of absorption and refraction of materials. The induced polarization
only captures the contribution of the medium itself, it excludes the vacuum
contribution.
5
 Thus,
P  0E
(0.9)
   r  1  n  1.
2
  is the dielectric susceptibility, which generally is a tensor quantity.
 However, for isotropic media, we obtain the complex scalar permittivity
 r   r '( )  i r ''( ,
Ne2
1
 r ( )  1 
 2
m 0 0   2  i
(0.10)
Ne 2

has units of frequency squared,  p 2 , and p is the plasma frequency.
m 0
6
 From Eq. 66, we readily obtain the real and imaginary parts of r,
0 2   2
 i '( )  1   p
(o 2   2 )   2 2

 r '( )   p 2
(0 2   2 ) 2   2 2
2
(0.11)
 Figure 7 illustrates the main features of  ' and  '' vs. frequency. To gain further
physical insight into Eqs. 67a-b, we discuss three different frequency regions, as
follows.
7
1.2.1 Response below the resonance,   0
 In this case, Eqs. 67a-b simplify to
p2
1
 r '( ) 1  2 
0 1  2  / 0 2
(0.12)
 p  
1
 r ''( )

0 4 1  2( / 0 ) 2
2
 Since   0 2 ,  '   '' , absorption is negligible, below the resonance the
material is transparent. Further, d  '
d
dispersion.
8
 0 , which defines a region of normal
 It can be seen that expanding the denominator in Eqs. 68a-b, we obtain
 r '( )   2 and  r ''( )   3 . In designing optics of imaging systems, the
Sellmeier equation is very efficient for describing the refractive index vs.
wavelength,
2
a

n 2 ( ) 1   2i
i   bi
 1 
i
,
ai
1  bi 2 /  2 c 
(1.68a)
2
 The summation is over several resonances, ai and bi are experimentally
determined parameters, and c is the speed of light. It can be seen that the
Sellmeier equation originates in the expression for  r '( ) in Eq. 1.68.
 As we approach resonance, this dependence becomes more complicated.
9
1.2.2 At resonance,  0
 For frequencies comparable to 0 , Eqs. 67a-b are well approximated by
 p2
 r '( ) 1 

20
p

20
2
 r ''( )
  0
   0 
1 


/
2


2
(0.13)
 /2
   0 
1 


/
2


2
 Under these conditions, the absorption is significant,   0   , and the
absorption line has a characteristic shape, Lorentzian line. This shape has a
central frequency 0 and a full width half maximum of  .
10
 While 0 has a clear physical significance of the frequency at which the system
“resonates”, or absorbs strongly, the meaning of  is somewhat more subtle.
 The damping constant  represents the average frequency at which electrons
collide with atoms, which generates loss of energy. Thus,   1/  col , with  col
the average time between collisions.
 Finally, around resonance, d  '
d
 0 , which defines anomalous dispersion.
11
1.2.3 Above the resonance,   0
 Well above the resonance, the following equations apply:
2
 r '( ) 1   p  4
   2 2

 r ''( ) 1   p 2  4
   2 2
2
(0.14)
 The absorption becomes less significant, as expected in a frequency range away
from the resonance. The dispersion is normal again, d  '
d
 0.
 This Lorentz oscillatory model provides great insight into the classical lightmatter interaction. In the following section, we will investigate the particular
situation of metals, when the charge moves freely within the material.
12
1.3 Drude model of light-metals interaction
 The optical properties of metals were first introduced by Drude in the context of
conductivity. In highly conductive materials, the restoring force in Eq. 57,
m0 2 x , vanishes, establishing that the charge can move freely. Under these
conditions, we obtain Drude’s model, in which Eqs. 67a-b reduce to (0  0)
 p2
 r '( )  1  2
  2
(0.15)
p

 2
   2
2
 r ''( ) 
 Typically   1/  col   , the frequency of collisions is much lower than that
of optical frequencies.
13
 In this high frequency limit,  r '( ) 1 
p2
2
and  r ''( )
 p 2
3
. From the
Fresnel equations, we derive the reflectivity coefficient. For normal incidence,
the intensity-based reflectivity is
n 1
R( ) 
,
n 1
2
(0.16)
 n is the (complex) refractive index. Since n   r , Eq. 72 becomes
R ( ) 
 r ' i r ''  1
 r ' i r ''  1
2
(0.17)
14
 Figure 7 illustrates the frequency dependence of  ' ,  '' , n’, n’’ and R for various
values of  p and  .
r’’
r’
R()
p
Figure 7. Frequency dependence of dielectric permeability and reflectivity around the plasma frequency ( p =10, =1) .
15
 At    p ,  r '( ) vanishes. In this case, the real part of the refractive index, n’,
can also vanish. This implies that the wavelength in the material is infinite,
  0 / n   .
 To gain a physical understanding of the plasma frequency  p , consider a thin
film of metal.
x{ + + + + + + + + + + + + + + + +
E
P
- - - - - - - - - - - - - - - -
Figure 9. Exiting surface plasmon resonance.
 The applied electric field induces a polarization P   0  E , with    r  1.
16
 Tuning the frequency of the incident field to the plasma frequency,  r  0 ,
  1, and P   0E . The induced polarization is the total charge times the
displacement per unit volume,
P   N  e x
(0.18)
 Therefore, the electric field is
E

P
0
Ne
0
(0.19)
 x
 If we construct the electric force due to the charge displacement, F  eE , we
obtain
F 
Ne2
 x
0
 ke   x.
(0.20)
17
 In Eq. 76, we define ke as the “spring” constant of the restoring force. By
definition, the system is characterized by a resonant frequency,  p  ke / m .
This is the plasma frequency associated with the thin film,
Ne2
.
p 
m 0

(0.21)
From Maxwell’s equations, we have that   H 
D
E D
. At plasma
 0

t
t t
frequency, D   0 E , and the magnetic field vanishes. This indicates that there
is no bulk propagation of electromagnetic field.
.
18
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