Chapter 4

The Propagation of Light:

TransmissionReflectionRefraction

Macroscopic manifestations of scattering and interference occurring at the atomic level

Lecture 9

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)

Transmission of light through dense media

(Air at STP: 5×106 molecules in a cubewith a side ~500 nm)

(distance between molecules <<)

Lateral scattering: for any molecule A there is always another molecule B distance /2 in lateral direction that would emit wavelet that is exactly out of phase with respect to the one emitted by A:destructive interference

Little or no light is scattered laterally or backwards in a dense homogeneous media.

Forward scattering: constructive interferenceThe denser the substance the less light is scattered laterally

Why can’t we see a light beam?

To photograph light beams in laser labs, you need to blow some smoke into the beam…

Unless the light beam is propagating right into your eye or is scattered into it, you won’t see it. This is true for laser light and flashlights.

This is due to the facts that air is very sparse (N is relatively small), air is also not a strong scatterer, and the scattering is incoherent. This eye sees almost no light.

This eye is blinded (don’t try this at home…)

Laser show

Scattering from a crystal vs. scattering from amorphous material (e.g., glass)

A perfect crystal has perfectly regularly spaced scatterers in space.

Of course, no crystal is perfect, so there is still some scattering, but usually less than in a material with random structure, like glass.There will still be scattering from the surfaces because the air nearby is different and breaks the symmetry!

So the scattering from inside the crystal cancels out perfectly in all directions (except for the forward and a few other preferred directions).

Scattering from particles is much stronger than that from molecules.

They’re bigger, so they scatter more.

For large particles, we must first consider the fine-scale scattering from the surface microstructure and then integrate over the larger scale structure.

If the surface isn’t smooth, the scattering is incoherent.

If the surfaces are smooth, then we use Snell’s Law and angle-of-incidence-equals-angle-of-reflection.Then we add up all the waves resulting from all the input waves, taking into account their coherence, too.

Scattering: why is sky blue

1911: Einstein showed theoretically that scattered light intensity is ~4

Blue light (~480 nm) has ~1.3 times higher frequency than red light (~630 nm) and would scatter ~3 times more efficiently. Sky is blue and sun looks red at sunset.

Rayleygh scattering: scattering by particles much less than wavelength of incident light (such as atoms)

Rayleigh scattering is the elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light.

Rayleigh scattering causes the blue hue of the daytime sky and the reddening of the sun at sunset

Mie Theory

Scattering from larger spherical particles is explained by the Mie theory for arbitrary size parameter x. Mie theory also defines scattering in small size parameter, but in this case, Mie theory reduces to Rayleigh approximation.

Light scattering regimes There are many regimes of particle scattering, depending on the particle size, the light wave-length, and the refractive index. You can read an entire book on the subject:

Particle size/wavelength

Rel

ativ

e re

frac

tive

inde

x

Mie Scattering

Ray

leig

h Sc

atte

ring

Totally reflecting objectsG

eom

etric

al o

ptic

sRayleigh-Gans Scattering

Larg

e

~

1

~

0

~0 ~1 Large

Rainbow

Air

This plot considers only single scattering by spheres. Multiple scattering and scattering by non-spherical objects can get really complex!

Transmission and the index of refractionIndex of refraction n = c/v - is light propagating slower?

- But Einstein says photons can only propagate at speed c!

- An original wave propagates at speed c (red).

scattered secondary waveincident primary wave

-In the example above the scattered wave (blue) is delayed.

- The scatterer has resonance and thus cannot oscillate exactly in-phase with the incident E-field - there is phase shift between the incident beam and the forward-scattered one.-The observer in forward direction will see only the net field, i.e. the phase velocity (v) would seem to be slower (or faster) than c depending on phase shift between incident and scattered wave.But each of the waves propagates with c!!!

Reflection

ReflectionInside the dense substance each of the scattering molecules has a pair that is /2 away and scatters backward wave that is out-of-phase - complete destructive interference

/2

On an interface between two substances there are many ‘unpaired’ molecules, or even if there is a pair scattering efficiency the phase will be different fordifferent atoms or molecules

/2

Backwards scattering close to interface (~ /4) will not experience complete destructive interference: reflection

ni > ntni < nt

ReflectionScattering properties are reflected in the index of refraction n

Reflection occurs on the surface between two materials with different n

external reflection internal reflectionDepending on the interface type:

Example: reflection of beam falling on glass (beam coming from air) -~4% reflected

Example: reflection of beam coming from under water from water-air interface

Reflection: microscopic view

When an incident plane wave front strikes the surface at some angle it does not reach all the atoms along the surface simultaneously.

Each consequent atom will scatter at slightly different phase, while the spherical wave created by previous atom had a chance to move away some distance.

The resulting reflected wave front created as a superposition of all scattered wavelets will emerge also at an angle to the surface.

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.

Reflection: constructive interference

For constructive interference the spherical waves created by the atoms on the surface must arrive in-phase.

Let us consider two atoms on the surface.

Wave function depends only on xt-t: )( trkfE

Wave phase along incident wavefront is the same: )( tfEE BA

The scattered wavefront CD: points C and D must be at the same phase:

)(

)(

tBDkfE

tACkfE

D

C

phase shift on scattering atom

BDAC

The angle of reflection

90

CBADAD

BDAC

Triangles ABD and ACD:

The Law of Reflection (1st part):The angle-of-incidence equals the angle-of-reflection

i = r

Rays and the Law of ReflectionA ray is a line drawn in space along the direction of flow of radiant energy.Rays are straight and they are perpendicular to the wavefront

Conventionally talk about rays instead of wavefronts

The Law of Reflection1. The angle-of-incidence equals the angle-of-reflection (i = r)2. The incident ray, the perpendicular to the surface and the

reflected ray all lie in a plane (plane-of-incidence)

Incoherent scattering: reflection from a rough surface

No matter which direction we look at it, each scattered wave from a rough surface has a different phase. So scattering is incoherent, and we’ll see weak light in all directions.

This is why rough surfaces look different from smooth surfaces and mirrors.

Specular and diffuse reflection

Smooth surface: specular reflection

Rough surface: diffuse reflection

Example: stealth technology

Flat surfaces: microwaves do not reflect back

Chapter 4

The Propagation of Light:

TransmissionReflectionRefraction

Macroscopic manifestations of scattering and interference occurring at the atomic level

Lecture 9

What about light that scatters on transmission through a surface?

•Again, a beam can remain a plane wave if there is a direction for which constructive interference occurs.

Constructive interference will occur for a transmitted beam if Snell's Law is obeyed.

Huygens Principle

Refraction

Incident beams are bent when they enter a substance with different index of refraction - refraction.

The phase difference between wavefronts AB and ED must be the same - it must take the same time for the wavefront to cover distance BD and AE:

tt

ii

ADtAE

ADtBD

sin

sin

v

v

t

i

t

i

sinsin

vv

t

i

i

t

cncn

sinsin

ttii nn sinsin

1. (Snell’s law):

2. The incident, reflected and refracted rays all lie in the plane of incidence

ttii nn sinsin

The Law of Refraction

Willebrord Snel van Royen (1580-1626)

The laws of refraction and reflection are reversible

Magic: see over edge

See over the edge

Apparent (virtual) imageThe cup seems to be shallow

cupno water

Add water

Spoon bending

Refraction and wavelength

The wavelength changes when light enters a substance:

ti

vv

tt ti

ti ti ntc

ntc

ti ti nn

If 0 is wavelength in vacuum (n=1): vacuumn0

In the media with n>1 wavelength decreases: = 0/nspeed decreases: v = c/nfrequency does not change.

Dispersion

Speed of light in matter depends on frequency (or wavelength)Refraction index depends on wavelengthAmount of bending depends on wavelength

Rainbows: white light is separated into colors due to dispersion in water droplets

Rainbows

Skier will see red at the top of the rainbow, and blue at bottom. Rainbows are one of the most beautiful examples of dispersion in nature.

Secondary rainbow

In the secondary rainbow pattern is reversed

There are 2 total internal reflections in water droplets that form the second rainbow