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.