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Lecture 8
Chapter 3
Dispersion
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Classical theory of dispersion
Refractive index vs. wavelength
Light scattering
Huygens principle
Forward propagation
Light in bulk matter
Maxwell eq-ns in free space  EM wave speed is c 
1
 0 0
In medium, 0 and 0 in Maxwell equation must be replaced by 
and  and phase speed of EM wave in medium becomes slower:
v
1

Absolute index of refraction: n 
Relative permittivity: K E    0
Relative permeability: K B   0
c


v
 0 0
n  KE KB
For nonmagnetic transparent materials KB1: n  K E
Maxwell’s
Relation
However, n depends on frequency (dispersion) and Maxwell
equation works only for simple gases.
Light and matter
Absorption
If electron in atom is in resonance with EM
field, or in QM terms energy of photon is
suitable for electronic transition, light can be
absorbed - energy of photon converted into
higher potential energy of electron.
Elastic scattering:
electrons in atoms are ‘shaked’ by oscillating E field of light accelerated electrons re-emit EM wave at the same frequency as incident light
Light scattered elastically has the same wavelength (frequency) as incident light.
Each atom acts as a point-source of EM radiation. The resulting wave is a
superposition of initial wave and all waves created by all atoms.
Net effect: the phase velocity of the wave is slower than that in free space.
Transparent materials have no strong resonances in the visible light
range of frequencies.
Dispersion: atomic polarization
Dispersion  frequency dependence of the index of refraction n
all materials are dispersive
Let consider a simple atom in E-field:
+ and - charges separate slightly:
induced dipole moment
atomic, or ionic polarization
(nonpolar molecules/atom)
This kind of polarization is called atomic,
or ionic polarization.
Shift of charges is typically very small.
Dipole moment per unit volume is
called electric polarization, P.
For most materials P and E are proportional:
 
P
KE  1 
   0 E  P
0E
KE is not very large for non-polar materials
E
Dispersion: orientational polarization
orientational polarization
(polar molecules)
Orientational polarization:
For polar molecules (with charged ends)
polarization and KE is much greater since
molecules can reorient
H2 O
However, molecular rotation cannot occur as fast as atomic
polarization. Therefore, KE depends on frequency:
At higher frequencies KE becomes lower, and so does n
Examples
Benzene (nonpolar)
K E  2.28
K E  1.51
n  1.501
Water (polar)
K E  80.3
K E  8.96
n  1.333
Dispersion: classical theory
Classical picture.
Electron is bound to nucleus by a
‘spring’-kind of force: F   k E x
Electron may oscillate at natural
resonance frequency  0  k E me
Light wave exerts a force:
FE  qe E t   qe E0 cos t
kE
2
d
x
Equation of motion: qe E0 cos t  me 02 x  me 2
dt
Solution:
x t  
kE - elastic constant
me - electron mass
qe - electron charge
E0 - E wave amplitude
 - light angular freq.
qe me
 < 0 - x shifts in the same direction as E


E
t
02   2 
 > 0 - x shifts in the opposite direction of E
Dispersion: classical theory
N electrons per unit volume each
contribute dipole moment qex,
Electric polarization:
qe2 N me
P t   qe x t N  2
E t 
2
0   
Refraction index:
2




P
t
Nq
1
2
e

 2
n    K E  1 
 1
2 
 0 E t 
 0me   0   
For  < 0 ; n > 1, n increases with frequency
For  > 0 ; n < 1, n increases with frequency
For multiple resonances:
2


fj
Nq
2
e


fj 1
n    1 


2
2


 0me j   0 j   
j
x t  
qe me
E t 
2
2
0   
kE - elastic constant
me - electron mass
qe - electron charge
0 - electron resonance
E0 - E wave amplitude
 - light angular freq.
N - # of electrons in unit
volume
fi - fraction of oscillators
with res. freq. 0i
Dispersion
2
Nq
e
n 2    1 
 0me


fj

j   2   2 
 0j

Quantum mechanics: fi  oscillator strength
or transition probability
kE - elastic constant
me - electron mass
qe - electron charge
0 - electron resonance
E0 - E wave amplitude
 - light angular freq.
N - # of electrons in unit
volume
fi - fraction of oscillators
with res. freq. 0i
More careful treatment:
n2  1
Nqe2

2
n  2 3 0me


fj

j   2   2  i  
j
 0j
  - damping term (losses in medium)
Dispersion
n2  1
Nqe2

2
n  2 3 0me


fi

j   2   2  i  
j
 0j

2
Nq
e
n 2    1 
 0me


fi

j   2   2 
 0j

(qualitatively similar)
Transparent materials:
- do not absorb in visible range (=400-700 nm, or =(4.3-7.5)×1014 Hz)
- absorb in ultraviolet (<400 nm, or >7.5×1014 Hz)
-  < 0 and n() gradually increases with frequency,
or decreases with wavelength
normal dispersion
Refractive Index vs. Wavelength
Since resonance frequencies exist in many spectral ranges, the
refractive index varies in a complex manner.
Electronic resonances usually occur in the UV; vibrational and
rotational resonances occur in the IR; and inner-shell electronic
resonances occur in the x-ray region.
n increases with frequency, except in anomalous dispersion regions.
Dispersion
More careful treatment:


fj
n2  1
Nqe2


 2

2
2
n  2 3 0 me j  0 j    i j 
Complex index of refraction:
 - damping term (losses in medium)
A light wave in a medium
Vacuum (or air)
Medium
n=1
n=2

Absorption depth = 1/
k0
nk0
n
E ( z , t )  E0 (0) exp[i (k0 z   t )]
Wavelength decreases
E0 (0) exp[( / 2) z ] exp[i (nk0 z   t )]
The speed of light, the wavelength (and k), and the amplitude change,
but the frequency, , doesn’t change.
Absorption Coefficient and the Irradiance
The irradiance is proportional to the (average) square of the field.
Since E(z)
 exp(-z/2), the irradiance is then:
I(z) = I(0) exp(-z)
Beer-Lambert law
where I(0) is the irradiance at z = 0, and I(z) is the irradiance at z.
Thus, due to absorption, a beam’s irradiance exponentially
decreases as it propagates through a medium.
The 1/e distance, 1/, is a rough measure of the distance light can
propagate into a medium (the penetration depth).
Refractive index and Absorption coefficient

Absorption
coefficient
n–1
Refractive
index
0
0
0
Ne 2
 /2

2 0 c0 me (0   ) 2  ( / 2) 2
Frequency, 
0  
Ne 2
n 1 
4 0 me (0   ) 2  ( / 2) 2
Lecture 8
Chapter 4
The Propagation of Light:
Transmission
Reflection
Refraction
Macroscopic manifestations of
scattering occurring on atomic level
Reminder: Light and matter
Absorption
If the electron in atom is in resonance with EM
field, or in QM terms energy of photon is
suitable for electronic transition, light can be
absorbed - energy of photon converted into
higher potential energy of electron.
Elastic scattering:
electrons in atoms are ‘shaked’ by oscillating E field of light accelerated electrons re-emit EM wave at the same frequency as incident light
Light scattered elastically has the same wavelength (frequency) as incident light.
Each atom acts as a point-source of EM radiation. The resulting wave is a
superposition of initial wave and all waves created by all atoms.
Net effect: the phase velocity of the wave is slower than that in free space.
Light Scattering
Molecule
When light encounters
matter, matter not only reemits light in the forward
direction (leading to
absorption and refractive
index), but it also re-emits
light in all other directions.
Light source
This is called scattering.
Light scattering is everywhere.
All molecules scatter light. Surfaces scatter light. Scattering
causes milk and clouds to be white and water to be blue. It is
the basis of nearly all optical phenomena.
Scattering can be coherent or incoherent.
Scattered spherical waves often
combine to form plane waves.
A plane wave impinging on a surface (that is, lots of very
small closely spaced scatterers!) will produce a reflected
plane wave because all the spherical wavelets interfere
constructively along a flat surface.
Huygens’s Principle
Wavefront becomes distorted.
Can we predict what would be its shape?
Huygens’s Principle (1690):
Every point on a propagating wavefront serves as the source of
spherical secondary wavelet of the same frequency propagating at
the same speed.
The wavefront at some later time is the envelope of these wavelets
(interference).
plane wave
spherical wave
Constructive vs. destructive interference;
Coherent vs. incoherent interference
Waves that combine
in phase add up to
relatively high irradiance.
Waves that combine 180°
out of phase cancel out
and yield zero irradiance.
Waves that combine with
lots of different phases
nearly cancel out and
yield very low irradiance.
=
Constructive
interference
(coherent)
=
Destructive
interference
(coherent)
=
Incoherent
addition
Interfering many waves: in phase, out of
phase, or with random phase…
Im
Re
If we plot the
complex
amplitudes:
Waves adding exactly
out of phase, adding to
zero (coherent
destructive addition)
Waves adding exactly
in phase (coherent
constructive addition)
Waves adding with
random phase,
partially canceling
(incoherent
addition)
Forward propagation.
At point P the scattered waves are more or
less in-phase:
constructive interference of wavelets
scattered in forward direction.
Note: the scattered (reradiated) field is 1800 out
of phase with the incident beam
True for low and high density substance
Scattering and interference: low density matter
(distance between molecules >>)
light
molecules
(Upper atmosphere)
no steady interference,
random phases
Random, widely spaced scatterers emit wavelets that are essentially
independent of each another in all directions except forward.
Laterally scattered light has no interference pattern.
Comparison on-axis vs. off-axis light scattering
Forward (on-axis) light
scattering: scattered wavelets
have nonrandom (equal!)
relative phases in the forward
direction.
Off-axis light scattering: scattered
wavelets have random relative
phases in the direction of interest
due to the often random placement of molecular scatterers.
Forward scattering is coherent—
even if the scatterers are randomly
arranged in space.
Path lengths are equal.
Off-axis scattering is incoherent
when the scatterers are randomly
arranged in space.
Path lengths are random.