Reflection & RefractionReflection & Refraction

The Phase Difference1 1 10kx t

2 2 20kx t

1 2 1 10 2 20

1 2 10 20 0

( ) ( )

( ) ( ) 2

kx t kx t

xk x x

Path-length difference Inherent phase difference

If the waves are initially in-phase

00

22kxnx

21 xxnxn The optical path difference (OPD)

If 0 is constant waves are said to be coherent.

The Phase Difference

00

For constructive interference:

2 2 or 2 2

x xm m

The condition of being in phase, where crests are aligned with crests and troughs with troughs, is that

= 0, 2, 4, or any integer multiple of 2.

For identical sources, 0 = 0 rad , maximum constructive interference occurs when x = m ,

Two identical sources produce maximum constructive interference when the path-length difference is an integer number of wavelengths.

The Phase Difference

10 2

102

For destructive interference:

2 2( ) or

2 2

xm

xm

The condition of being out of phase, where crests are aligned with troughs of other, that is,

=, 3, 5 or any odd multiple of .

For identical sources, 0 = 0 rad , maximum constructive interference occurs when x = (m+ ½ ) ,

Two identical sources produce perfect destructive interference when the path-length difference is half-integer number of wavelengths.

We’ll check the interference one direction at a time, usually far away.

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.

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…

To understand scattering in a given situation, we compute phase delays.

Because the phase is constant along a wave-front, we compute the phase delay from one wave-front to another potential wave-front.

If the phase delay for all scattered waves is the same (modulo 2), then the scattering is constructive and coherent. If it varies continuously from 0 to 2, then it’s destructive and coherent.

If it’s random (perhaps due to random motion), then it’s incoherent.

i ik L

Wave-fronts

L4

L2

L3

L1

Scatterer

Potentialwave-front

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.

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 .

Coherent constructive interference occurs for a reflected beam if the angle of incidence = the angle of reflection: i = r.

i r

The wave-fronts are perpendicular to the k-vectors.

Consider the different phase

delays for different paths.

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.

Coherent destructive interference occurs for a reflected beam direction if the angle of incidence ≠ the angle of reflection: i ≠ r.

i too big

Potential wave front

a

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

Coherent scattering usually 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.

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.

Coherent scattering typically occurs in only one or a few directions; incoherent scattering occurs in all directions.

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…)

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.

ttii nn sinsin Snell’s Law

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.

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 perhaps a few other preferred directions).

Scattering from large particles

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.

Diffraction GratingsIf light impinges on a periodic array of grooves, scattering ideas tell us what happens. There will be constructive interference if the delay between adjacent beamlets is an integral number of

wavelengths .

sin( ) sin( )m ia m

where m is any integer.

A grating can have solutions for zero, one, or many values of m, or “orders.”

Remember that m and the dif-fracted angle can be negative, too.

Path difference AB – CD =

mAB = a sin(m)

CD = a sin(i)

Diffraction orders

Because the diffraction angle depends on , different wavelengths are separated in the +1 (and -1) orders.

No wavelength dependence in zero order.

The longer the wavelength, the larger its diffraction angle in nonzero orders.

Diffraction angle, m

Zeroth order

First order

Minus first order

It’s helpful to know the variation of the diffracted angle vs. wavelength.

Differentiating the grating equation, with respect to wavelength:

Diffraction-grating dispersion

cos( ) mm

da m

d

sin( ) sin( )m ia m

cos( )m

m

d m

d a

Rearranging:

[i is a constant]

Thus, to separate different colors maximally, make a small, work in high order (make m large), and use a diffraction angle near 90 degrees.

Gratings typically have an order of magnitude more dispersion than prisms.

Wavelength-dependent incoherent molecular scat-tering: Why the sky is blue

Air molecules scatter light, and the scattering is proportional to 4.

Shorter-wavelength light is scattered out of the beam, leaving longer-wavelength light behind, so the sun appears yellow.

In space, the sun is white, and the sky is black.

Light from the sun

Air

Sunsets involve longer path lengths and hence more scattering.

As you know, the sun and clouds can appear red.

Edvard Munch’s “The Scream” was also affected by the eruption of Krakatoa, which poured ash into the sky worldwide. Munch Museum/Munch

Ellingsen Group/VBK, Vienna

Earth

Sunset ray

Noon ray

Atmosphere

Note the cool sunset.