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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.

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The equation of motion for the electron can be expressed as 2

202

( ) ( ) ( ),d x t dx t ex E tdt 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.

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Using the Fourier property of the differential operator,

( ) ,n

nn

d idt

(0.2)

we obtain Eq. 57 in the frequency domain

2 20( ) ( ) ( ) ( )ex i x x E

m . (0.3)

Thus, we find the solution for the charge displacement in the frequency domain,

2 20

( )( )

e Emx

i

. (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

( ) ( )e p x (0.5)

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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,

.NP 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 2

2 20

( )( ) .Ne EPm i

(0.7)

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Generally, each atom has multiple resonances or dipole-active modes, such that

Eq. 63 can be generalized to 2

2 20

( )( ) ,i

i i i

ENePm 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.

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Thus,

0

21 1.r n

P E (0.9)

is the dielectric susceptibility, which generally is a tensor quantity.

However, for isotropic media, we obtain the complex scalar permittivity

'( ) ''( ,r r ri 2

2 20 0

1( ) 1rNem i

(0.10)

2

0

Nem

has units of frequency squared, 2p , and p is the plasma frequency.

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From Eq. 66, we readily obtain the real and imaginary parts of r, 2 2

2 02 2 2 2

22 2 2 2 2

0

'( ) 1( )

'( )( )

i po

r p

(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.

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1.2.1 Response below the resonance, 0

In this case, Eqs. 67a-b simplify to

2

220 0

2

4 20 0

1'( ) 11 2 /

1''( )1 2( / )

pr

pr

(0.12)

Since 20 , ' '' , absorption is negligible, below the resonance the

material is transparent. Further, ' 0dd

, which defines a region of normal

dispersion.

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It can be seen that expanding the denominator in Eqs. 68a-b, we obtain 2'( )r and 3''( )r . In designing optics of imaging systems, the

Sellmeier equation is very efficient for describing the refractive index vs.

wavelength,

22

2

22

( ) 1

11 / 2

i

i i

i

i i

anb

ab c

, (1.68a)

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.

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1.2.2 At resonance, 0

For frequencies comparable to 0 , Eqs. 67a-b are well approximated by 2

02

0 0

2

20 0

'( ) 12

1/ 2

/ 2''( )2

1/ 2

pr

pr

(0.13)

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 .

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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, ' 0dd

, which defines anomalous dispersion.

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1.2.3 Above the resonance, 0

Well above the resonance, the following equations apply: 2

24 2 2

24 2 2

'( ) 1

''( ) 1

r p

r p

(0.14)

The absorption becomes less significant, as expected in a frequency range away

from the resonance. The dispersion is normal again, ' 0dd

.

This Lorentz oscillatory model provides great insight into the classical light-

matter interaction. In the following section, we will investigate the particular

situation of metals, when the charge moves freely within the material.

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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, 2

0m 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) 2

2 2

2

2 2

'( ) 1

''( )

pr

pr

(0.15)

Typically 1/ col , the frequency of collisions is much lower than that

of optical frequencies.

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In this high frequency limit, 2

2'( ) 1 pr

and 2

3''( ) pr

. From the

Fresnel equations, we derive the reflectivity coefficient. For normal incidence,

the intensity-based reflectivity is 21( ) ,

1nRn

(0.16)

n is the (complex) refractive index. Since rn , Eq. 72 becomes 2

' '' 1( )

' '' 1r r

r r

iR

i

(0.17)

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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) .

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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. + + + + + + + + + + + + + + + +

- - - - - - - - - - - - - - - -

E P

x{

Figure 9. Exiting surface plasmon resonance.

The applied electric field induces a polarization 0 P E , with 1r .

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Tuning the frequency of the incident field to the plasma frequency, 0r ,

1 , and 0 P E . The induced polarization is the total charge times the

displacement per unit volume,

P N e x (0.18)

Therefore, the electric field is

0

0

PE

Ne x

(0.19)

If we construct the electric force due to the charge displacement, F eE , we

obtain 2

0

.e

NeF x

k x

(0.20)

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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 ek m .

This is the plasma frequency associated with the thin film, 2

0p

Nem

. (0.21)

From Maxwell’s equations, we have that 0D E Dt t t

H . At plasma

frequency, 0D E , and the magnetic field vanishes. This indicates that there

is no bulk propagation of electromagnetic field. .