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Home , Diffuse reflection, Specular reflection

Lecture 8

Chapter 4 The Propagation of :

Transmission Macroscopic manifestations of and interference occurring at the atomic level Reflection Reflection

Inside the dense substance each of the scattering molecules has a pair that is /4 away and scatters backward that is out-of-phase - complete destructive interference

/4

On an interface between two substances there are many ‘unpaired’ molecules, or even if there is a pair scattering efficiency or phase will be different for different atoms or molecules Backwards scattering close to interface (~ /4) will not experience complete destructive interference: reflection

/4 Reflection

Scattering properties are reflected in in index of refraction n Reflection occurs on the surface between two materials with different n Depending on the interface type: external reflection internal reflection

ni < nt ni > nt

Example: reflection of beam falling Example: reflection of beam on (beam coming from air) - coming from under water from ~4% reflected 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 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. We’ll check the interference one direction at a time, usually far away.

This way we can approximate spherical waves by plane waves in that direction, vastly simplifying the math.

Far away, spherical wave-fronts are almost flat…

Usually, coherent constructive interference will occur in one direction, and destructive interference will occur in all others. If incoherent interference occurs, it is usually omni-directional. The mathematics of scattering The math of light scattering is analogous to that of light sources.

If the phases aren’t random, we add the fields: Coherent

Etotal = E1 + E2 + … + En

** * ItotalI12 I... I N  c Re EE 1213  EE  ... E N 1 E N 

* I1, I2, … In are the irradiances of the Ei Ej are cross terms, which have the phase various beamlets. They’re all factors: exp[i(i-j)]. When the ’s are not positive real numbers and add. random, they don’t cancel out!

If the phases are random, we add the irradiances: Incoherent

Itotal = I1 + I2 + … + In To understand scattering in a given situation, we compute phase delays.

Wave-fronts Because the phase is constant along a wave- L front, we compute the 1

phase delay from one L2 wave-front to another L3 Potential potential wave-front. wave-front L4 ii kL Scatterer

If the phase delay for all scattered waves is the same (modulo 2), then the scattering is constructive and coherent. If it varies uniformly from 0 to 2, then it’s destructive and coherent. If it’s random (perhaps due to random motion), then it’s incoherent. Coherent constructive scattering: Reflection from a smooth surface when angle of incidence equals angle of reflection A beam can only remain a plane wave if there’s a direction for which coherent constructive interference occurs.

The wave-fronts are perpendicular to the k-vectors. i r

Consider the different phase delays for different paths.

Coherent constructive interference occurs for a reflected beam if the angle of incidence = the angle of reflection: i = r. Coherent destructive scattering: Reflection from a smooth surface when the angle of incidence is not the angle of reflection

Imagine that the reflection angle is too big. The symmetry is now gone, and the phases are now all different.

 = ka sin(too big) i too big  = ka sin(i)

Potential wave front a

Coherent destructive interference occurs for a reflected beam

direction if the angle of incidence ≠ the angle of reflection: i ≠ r. Coherent scattering occurs in one (or a few) directions, with coherent destructive scattering occurring in all others.

A smooth surface scatters light coherently and constructively only in the direction whose angle of reflection equals the angle of incidence.

Looking from any other direction, you’ll see no light at all due to coherent destructive interference. Reflection: constructive interference

For constructive interference 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: E  f (k  r  t)

Wave phase along incident wavefront is the same: EA  EB  f (t) The scattered wavefront CD: points C and D must be at the same phase:  EC  f (  k  AC  t)  AC  BD ED  f (k  BD   t) phase shift on scattering atom The angle of reflection Triangles ABD and ACD: AC  BD AD  AD B  C  90

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

i = r Rays and the Law of Reflection

A 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 Reflection

1. 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 . Specular and

Smooth surface: Rough surface: diffuse reflection Example: stealth technology

Flat surfaces: do not reflect back Lecture 9

Chapter 4 The Propagation of Light:

Transmission Reflection Macroscopic manifestations of Refraction scattering and interference occurring at the atomic level 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.

Huygens Principle

Constructive interference will occur for a transmitted beam if Snell's Law is obeyed. 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:  BD  vit  AD sin i

AE  vtt  AD sint

vi sini cnt sini   ni sini  nt sint vt sint nic sint The Law of Refraction

1. (Snell’s law): ni sini  nt sint

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

The laws of refraction and reflection are reversible

Willebrord Snel van Royen (1580-1626) Magic: see over edge

See over the edge

cup no water

Apparent (virtual) image The cup seems to be shallow

Add water Spoon bending Refraction and

The wavelength changes when light enters a substance: v t v t ct ct i  t   i t ni i ntt

nii  ntt

If 0 is wavelength in vacuum (n=1): 0  nvacuum

In the media with n>1

wavelength decreases:  = 0/n speed decreases: v = c/n frequency does not change.

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

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

Skier will see red at the top of the , 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