The Propagation of Light The processes of transmission, reflection, and refraction are macroscopic manifestations of scattering occurring on a submicroscopic level.

Elastic Scattering In elastic scattering, the energy of the incident photon is conserved and its propagating direction is changed by the potential of the target.

Rayleigh Scattering When a photon penetrates into a medium composed of particles whose sizes are much smaller than the wavelength of the incident photon, the scattering process is elastic and is called Rayleigh scattering. In this scattering process, the energy (and therefore the wavelength) of the incident photon is conserved and only its direction is changed. In this case, the scattering intensity is proportional to the fourth power of the reciprocal wavelength of the incident photon. The scattering of electromagnetic radiation by particles with dimensions much smaller than the wavelength of the radiation, resulting in angular separation of colors and responsible for the reddish color of sunset and the blue of the sky.

The intensity of the scattered light Example 4.1 Establish the dependence of the percentage of light scattered in Rayleigh scattering. Let is the incident amplitude, is the scattered amplitude at a distance r from the scatterer. V is the volume of the scatterer. Assume

Must be unitless, and K must has units of ( Length )2

The Transmission of Light Through Dense Media Little or no light ends up scattered laterally or backwards in a dense homogeneous medium. This makes sense from the perspective of conservation of energy– we can’t have constructive interference in every direction. Interference produces a redistribution of energy, out of the regions where it’s destructive into the regions where it’s constructive.

Constructive vs. destructive interference; Coherent vs Constructive vs. destructive interference; Coherent vs. incoherent interference Waves that combine in phase add up to relatively high irradiance. Constructive interference (coherent) = Waves that combine 180° out of phase cancel out and yield zero irradiance. Destructive interference (coherent) = Waves that combine with lots of different phases nearly cancel out and yield very low irradiance. = Incoherent addition

Scattering from molecules and small particles A plane wave impinging on a molecule or particle scatters into a spherical wave. Huygens’ Principle says that waves propagate as if each point on a wave-front emits a spherical wave (whether or not there’s a molecule or particle involved). Scattering from an individual molecule or particle is weak, but many such scatterings can add up—especially if interference is coherent and constructive.

The Transmission of Light Through Dense Media The Phases of the wavelets at P differ greatly The Transmission of Light Through Dense Media

Waves using complex amplitudes The resulting "complex amplitude" is: As written, this entire field is complex!

Complex numbers simplify optics! Adding waves of the same frequency, but different initial phase, yields a wave of the same frequency. This isn't so obvious using trigonometric functions, but it's easy with complex exponentials: where all initial phases are lumped into E1, E2, and E3.

Adding complex amplitudes When two waves add together with the same complex exponentials, we add the complex amplitudes, E0 + E0'. Constructive interference: Destructive interference: "Quadrature phase" ±90° interference: 1.0 0.2 1.2 1.0 -0.2 0.8 1.0 -0.2i 1-0.2i + + + = = = time time time Laser Absorption Slower phase velocity

Light excites atoms, which emit light that adds (or subtracts) with the input light. When light of frequency w excites an atom with resonant frequency w0: Electric field at atom Electron cloud Emitted field + = Incident light Emitted light Transmitted light On resonance (w = w0) An excited atom vibrates at the frequency of the light that excited it and re-emits the energy as light of that frequency. The crucial issue is the relative phase of the incident light and this re-emitted light. For example, if these two waves are ~180° out of phase, the beam will be attenuated. We call this absorption.

The interaction of light and matter Light excites atoms, which then emit more light. Electric field at atom Electron cloud Emitted electric field + = Incident light Emitted light Transmitted light On resonance (the light frequency is the same as that of the atom) The crucial issue is the relative phase of the incident light and this re-emitted light. If these two waves are ~180° out of phase, destructive interference occurs, and the beam will be attenuated—absorption. If they’re ~±90° out of phase: the speed of light changes—refraction.

The relative phase of emitted light with respect to the input light depends on the frequency. Electric field at atom Electron cloud Emitted field Below resonance w << w0 Weak emission.90° out of phase. On resonance w = w0 Strong emission. 180° out of phase. The emitted light is 90° phase-shifted with respect to the atom’s motion. Above resonance w >> w0 Weak emission. -90° out of phase.

Refractive index and Absorption coefficient w0 Frequency, w

Variation of the refractive index with wavelength (dispersion) causes the beautiful prismatic effects we know and love. Prism Input white beam Dispersed beam Prisms disperse white light into its various colors.

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. 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.

Light scattering regimes There are many regimes of particle scattering, depending on the particle size, the light wavelength, and the refractive index. You can read an entire book on the subject: Particle size/wavelength Refractive index Mie Scattering Rayleigh Scattering Totally reflecting objects Geometrical optics Rayleigh-Gans Scattering Large ~1 ~0 ~0 ~1 Large Air Rainbow

Mie Scattering .                                                                                                           

The mathematics of scattering If the phases aren’t random, we add the fields: Coherent Etotal = E1 + E2 + … + En I1, I2, … In are the irradiances of the various beamlets. They’re all positive real numbers and add. Ei Ej* are cross terms, which have the phase factors: exp[i(qi-qj)]. When the q’s are not random, they don’t cancel out! If the phases are random, we add the irradiances: Incoherent Itotal = I1 + I2 + … + In