Chapter 4

Scattering – Interaction with Matter Part I Fundamentals Questions? Scattering Fundamentals

Scattering can be broadly defined as the redirection of radiation from the original direction of incident , usually due to interactions with atoms, molecules, and • Reflection, refraction, diffraction, etc. are actually forms of scattering • Matter is composed of discrete electrical charges (atoms and molecules – dipoles) • Light is an oscillating EM field – excites charges, which radiate EM • These radiated EM waves are scattered waves, excited by a source external to the scatterer • The superposition of incident and scattered EM waves is what is observed Scattering Geometry

Backward scattering Forward scattering (backscattering) Light – Molecule Interaction

Simplest case: electric dipole moment oscillating in E- field

Incident plane wave Resulted total wave Types of Scattering

• – the (frequency) of the scattered light is the same as the incident light (Rayleigh and )

• – the emitted radiation has a wavelength different from that of the incident radiation (, fluorescence)

• Quasi-elastic scattering – the wavelength (frequency) of the scattered light shifts (e.g., in moving matter due to Doppler effects) More Types of Scattering • Single scattering: scattered only once • Prevails in optically thin media (τ << 1), since photons have a high probability of exiting the medium (e.g., a thin ) before being scattered again • Also favored in strongly absorbing media (ω << 1)

• Multiple scattering: prevails in optically thick, strongly scattering and non-absorbing media • Photons may be scattered hundreds of times before emerging Part II Electric Dipole Radiation – Polar and Non-Polar Molecules in Molecules E-Field Distortion of A Non-Polar Molecule

P Polar Molecule Re-Orientated in E-Field Oscillating Electric Dipole What actually happens to the fields based on a precise calculate is shown in Fig. Magnetic fields are also formed. When there is electric current, magnetic field is produced. If the current is in a straight wire circular magnetic field is generated. Its magnitude is inversely proportional to the distance from the current. Oscillating Electric Dipole Oscillating Electric Dipole Oscillating Electric Dipole The radiation E-field is fairly complicated. In far field, the E-field can be approximated as Oscillating Electric Dipole The radiation E-field is fairly complicated. In far field, the E-field can be approximated as (toroidal radiation)

2 Oscillating Electric Dipole Rayleigh Scattering Rayleigh Scattering Rayleigh Scattering Short wavelength light gets scattered more laterally in atmosphere due to the random distribution/motion of gas molecules, causing . Light Propagation through a Dennsed Medium The lateral scattering is suppressed in regularly densed medium while forward and backward scattering get enhanced. Part III

Mie Scattering Crepuscular Rays Mie and Rayleigh Scattering

Blue sky  Rayleigh scatter Strong  dependence

White clouds  Mie scatter (weak  dependence) Mie Scattering • Explains scattering around larger droplets such as Corona around the or moon, Glory and similar phenomena. Mie Scattering • Begin with Maxwell’s equations • Derive a wave equation in spherical polar coordinates • Provide boundary conditions at surface of a sphere • Solve the partial differential wave equation for dependence on r, q, f. Mie Scattering

2 4 6 2 2 4  1 m 1 m 1 m  I   r 1  2  2    2 1 3 (1 2) 4 (1 23) 

• The bracket has the form of a Bessel function of the first order and  r sin  m   Mie Scattering Mie Scattering - Fog Mie Scattering – Mie Scattering – Glare Mie and Rayleigh Scattering Part IV

Scattering Scattering from Point of View

Detector

q, f

Probability of scattered photons: Scattering Cross Section Target atoms Detector

q, f

When incident photons interact with the target atoms, each atom present an effective area to the incoming :

i.e., only the photons in this portion of area get scattered Scattering Cross Section The total area scattered all the photons where is the target atom area density, A is the total area. Then the probability of scattered photons:

Thus,

The scattering cross section  is the scattering probability divided by the surface density of target atoms. Rayleigh Scattering Cross Section Rayleigh scattering cross section,

n: index of refraction of the particle d: particle diameter Rayleigh Scattering Cross Section Rayleigh Scattering Cross Section Mie Scattering Cross Section Mie Scattering Cross Section Part V

Light in Bulk Materials Transmission and the Index of Refraction

1. The transmission of light is the on-going repetitive process of scattering and re-scattering 2. Each scattering introduce a phase shift 3. In between scattering, all photons travel with speed exactly equal to c 4. The continuous accumulation of phase shift of the transmitted wavefronts relative to the primary wave traveling in vacuum means it has a different phase velocity 5. The “v” can be less than or more than c Transmission and the Index of Refraction

Secondary wave Primary wave

f Phase Velocity For a plane wave,

If the total phase is fixed as a constant, = constant, one has Transmission and the Index of Refraction A medium can be modeled as a collection of many polarized atoms

The atom can be treated as a classic harmonic oscillator. If the displacement of the electron –e is

- + k Transmission and the Index of Refraction Applied external electric field

For the electron, one has

0 is the intrinsic frequency Transmission and the Index of Refraction

 At  = 0 electron’s displacement  becomes large, and harmonic oscillator approximation is not right. An damping term needs to be added, r

 Transmission and the Index of Refraction

The electron will performs a damped force oscillation, which introduce a phase lag/lead. Solving a Damped Oscillator

Assuming  i t E (t )  E 0 e  i t x (t )  x 0 e One has 2  it  it  it  it  m x0 e  i x0 e  kx 0 e  eE 0 e

 eE0  eE0 / m x0  2  2 2 k  m  i 0   i / m

 eE0 / m t x(t)  2 2 e 0   i / m The magnitude and phases of x(t)

eE0 1 1   / m  x  f  tan   m 2 2 2 2 2 2 (0  )  ( / m) 0   Solving a Damped Oscillator

100

50

0 Phase

-50

-100 0 2 4

0.6

0.4

0.2 Amplitude

0.0 0 2 4  Relationship between n and Oscillator The electric displacement D:

D   0E  P   0E   0E   0 (1 )E   0E

P is the (macroscopic) density of the permanent and induced electric dipole moments in the material, called the density. The n of Oscillator Model The macroscopic polarization of medium P:

Pinduced  Np  Nexnˆ Ne2 1  2 2 E m0 (0  i) N is the number of atoms per unit volume. The electric displacement D can be expressed as

D   0E  P

  0 E  Pbackground  Pinduced Ne2 1   0 E 0E  2 2 E m0 (0  i) The n of Oscillator Model 1 1 1 2 2 2 2 n  [1  (1   2 ) ] 2 1 1 1 2 2 2 2   [1  (1   2 ) ] . 2 Transmission and the Index of Refraction Transmission and the Index of Refraction