lecture8 ch3-4 dispersion scattering.ppt
TRANSCRIPT
Chapter 3
Dispersion
Lecture 8
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 is00
1
c
In medium, 0 and 0 in Maxwell equation must be replaced by and and phase speed of EM wave in medium becomes slower:
1
v
Absolute index of refraction:00
vcn
Relative permittivity:
0
0
B
E
KK
Relative permeability: BE KKn
EKn Maxwell’sRelationFor nonmagnetic transparent materials KB1:
However, n depends on frequency (dispersion) and Maxwell equation works only for simple gases.
Light and matter
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.
AbsorptionIf 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.
Transparent materials have no strong resonances in the visible light range of frequencies.
Dispersion: atomic polarizationDispersion frequency dependence of the index of refraction n
all materials are dispersive
Let consider a simple atom in E-field:
E+ and - charges separate slightly:induced dipole moment
Dipole moment per unit volume is called electric polarization, P.
PE
0
For most materials P and E are proportional:
atomic, or ionic polarization(nonpolar molecules/atom)
This kind of polarization is called atomic, or ionic polarization.Shift of charges is typically very small.
EPKE0
1
KE is not very large for non-polar materials
Dispersion: orientational polarization
Orientational polarization:For polar molecules (with charged ends) polarization and KE is much greater since molecules can reorient
orientational polarization(polar molecules)
H2O
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
ExamplesBenzene (nonpolar)
501.151.1
28.2
nK
K
E
E
Water (polar)
333.196.8
3.80
nK
K
E
E
Dispersion: classical theoryClassical picture.Electron is bound to nucleus by a ‘spring’-kind of force: xkF E
Electron may oscillate at natural resonance frequency eE mk0
Light wave exerts a force: tEqtEqF eeE cos0
kE - elastic constantme - electron massqe - electron chargeE0 - E wave amplitude - light angular freq.
Equation of motion:2
2200 cos
dtxdmxmtEq eee
kE
Solution:
tEmqtx ee22
0 < 0 - x shifts in the same direction as E
> 0 - x shifts in the opposite direction of E
Dispersion: classical theory
kE - elastic constantme - electron massqe - electron charge0 - electron resonanceE0 - E wave amplitude - light angular freq.N - # of electrons in unit
volumefi - fraction of oscillators
with res. freq. 0i
tEmqtx ee22
0
N electrons per unit volume each contribute dipole moment qex,Electric polarization:
tEmNqNtxqtP eee 22
0
2
2200
2
0
2 111
e
eE m
NqtE
tPKn
Refraction index:
For multiple resonances:
j j
j
e
e fm
Nqn 2200
22 1
1
jjf
For < 0 ; n > 1, n increases with frequencyFor > 0 ; n < 1, n increases with frequency
Dispersion
j j
j
e
e fm
Nqn 2200
22 1
kE - elastic constantme - electron massqe - electron charge0 - electron resonanceE0 - E wave amplitude - light angular freq.N - # of electrons in unit
volumefi - fraction of oscillators
with res. freq. 0i
Quantum mechanics: fi oscillator strengthor transition probability
More careful treatment:
j jj
j
e
e
if
mNq
nn
2200
2
2
2
321
- damping term (losses in medium)
Dispersion
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
j jj
i
e
e
if
mNq
nn
2200
2
2
2
321
j j
i
e
e fm
Nqn 2200
22 1
(qualitatively similar)
normal dispersion
Refractive Index vs. WavelengthSince resonance frequencies exist in many spectral ranges, the refractive index varies in a complex manner.
Electronic resonances usually occur in the UV; vibrational androtational resonances occur in the IR; and inner-shell electronicresonances occur in the x-ray region.
n increases with frequency, except in anomalous dispersion regions.
Dispersion
More careful treatment:
2 2
2 2 20 0
12 3
je
je j j
fn Nqn m i
- damping term (losses in medium)
Complex index of refraction:
A light wave in a medium
The speed of light, the wavelength (and k), and the amplitude change, but the frequency, , doesn’t change.
n = 1 n = 2
k0 nk0
Vacuum (or air) Medium
Absorption depth = 1/
nWavelength decreases
00 exp[( / 2) ](0) exp[ ( )]E iz k tn z 0 0( , ) (0) exp[ ( )]E z t E i k z t
The irradiance is proportional to the (average) square of the field.
Since E(z) exp(-z/2), the irradiance is then:
Absorption Coefficient and the Irradiance
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).
I(z) = I(0) exp(-z) Beer-Lambert law
Refractive index and Absorption coefficient
2 20
2 2 2 20 0 0 0 0
/ 2 12 ( ) ( / 2) 4 ( ) ( / 2)e e
Ne Nenc m m
0
Absorption coefficient
Refractive index
0
n–1
Frequency, 0
Chapter 4
The Propagation of Light:
TransmissionReflectionRefraction
Macroscopic manifestations of scattering occurring on atomic level
Lecture 8
Reminder: Light and matter
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.
AbsorptionIf 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.
Light Scattering
When light encounters matter, matter not only re-emits light in the forward direction (leading to absorption and refractive index), but it also re-emits light in all other directions.
This is called scattering.
Light scattering is everywhere.
Scattering can be coherent or incoherent.
Light source
Molecule
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.
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 phasesnearly cancel out and yield very low irradiance.
=
=
=
Constructive interference(coherent)
Destructive interference(coherent)
Incoherentaddition
Interfering many waves: in phase, out of phase, or with random phase…
Waves adding exactly in phase (coherent constructive addition)
Waves adding with random phase, partially canceling (incoherent addition)
If we plot the complex amplitudes:
Re
Im
Waves adding exactly out of phase, adding to zero (coherent destructive 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
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.
no steady interference, random phases
moleculeslight(distance between molecules >>)
(Upper atmosphere)
Comparison on-axis vs. off-axis light scattering
Off-axis light scattering: scattered wavelets have random relative
phases in the direction of interest due to the often random place-
ment of molecular scatterers.
Forward scattering is coherent—even if the scatterers are randomly arranged in space.
Path lengths are equal.
Off-axis scattering is incoherentwhen the scatterers are randomly arranged in space.
Path lengths are random.
Forward (on-axis) light scattering: scattered wavelets have nonrandom (equal!) relative phases in the forward direction.