1. Coherence, Incoherence, and Light Scattering
Coherence vs. incoherence
Coherence in light sources
Light bulbs vs. lasers
Coherence in light scattering
Molecules scatter spherical waves
Spherical waves can add up to plane waves
Reflected and diffracted beams at surfaces
Why the sky and swimming pools are blue
Prof. Rick Trebino
Georgia Tech

2. 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 many different random phases nearly cancel out and yield very low irradiance.
Incoherent addition

3. Interfering many waves: in phase, out of phase, or with random phase…
Waves adding exactly in phase (coherent constructive addition) Re Im
If we plot the complex amplitudes:
Waves adding with random phase, partially canceling (incoherent addition)
Waves adding exactly out of phase, adding to zero (coherent destructive addition)

4. The relative phases are the key.
Recall that the irradiance of the sum of two waves is:
E1 and E2 are complex amplitudes. ~ Re Im A qi
If we write the amplitudes in terms of their intensities, Ii, and absolute phases, qi, Im
The intensity can be anywhere between
q1 – q2
and Re I
depending on the value of q1 – q2

5. Adding many fields with random phases
If there are many fields, each with a random phase, qi:
I1, I2, … In are the irradiances of the various beamlets. They’re all positive real numbers and they add.
Ei Ej* are cross terms, which have the phase factors: exp[i(qi-qj)]. When the q’s are random, they cancel out!
All the relative phases
Itotal = I1 + I2 + … + In Re Im
I1+I2+…+IN
The intensities simply add!
Two 20W light bulbs yield 40W.

6. Light bulbs Light from a light bulb is very complicated!
2. It’s not a point source, so we must add waves with many different directions. It has a small spatial coherence area.
1. It has many colors (it’s white), so we must add waves of many values of w (and hence k-magnitudes). It has a short coherence time.
Light bulbs
3. Even along one direction, many different molecules are emitting light with random relative phases (the effect we just considered).
Light from a light bulb is very complicated!

7. Light from a light bulb is incoherent.
When many light waves add with random phases, we say the light is incoherent, and the light wave total irradiance is just the sum of the individual irradiances.
Itotal = I1 + I2 + … + In
Other characteristics of incoherent light:
1. It’s relatively weak.
2. It’s omni-directional.
3. Its irradiance is proportional to the number of emitters.

8. Coherent vs. Incoherent Light
Laser
Coherent light:
1. It’s strong.
2. It’s uni-directional.
3. Total irradiance N2 or 0.
4. Total irradiance is the mag-square of the sum of individual fields.
Incoherent light:
1. It’s relatively weak.
2. It’s omni-directional.
3. Total irradiance N.
4. Total irradiance is the sum of individual irradiances.
Etotal = E1 + E2 + … + En
Itotal = I1 + I2 + … + In

9. Light Scattering Molecule
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 source
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.

10. Note that k and r are not vectors here!
Spherical waves
A spherical wave is also a solution to Maxwell's equations and is a good model for the light scattered by a molecule.
Note that k and r are not vectors here!
where k is a scalar, and
r is the radial magnitude.
A spherical wave has spherical wave-fronts.
Unlike a plane wave, whose amplitude remains constant as it
propagates, a spherical wave weakens. Its irradiance goes as 1/r2.

11. 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.

12. 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.

13. 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
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

14. To understand scattering in a given situation, we compute phase delays.
Wave-fronts
Because the phase is constant along a wave-front, we compute the phase delay from one wave-front to another potential wave-front. L1 L2 L3
Potential wave-front L4
Scatterer
If the phase delay for all scattered waves is the same (modulo 2p), then the scattering is constructive and coherent. If it varies uniformly from 0 to 2p, then it’s destructive and coherent.
If it’s random (perhaps due to random motion), then it’s incoherent.

15. 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.
Consider the different phase delays for different paths. qr qi
Incident wave-front
Potential outgoing wave-front
Coherent constructive interference occurs for a reflected beam if the angle of incidence = the angle of reflection: qi = qr.

16. 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.
f = ka sin(qi)
f = ka sin(qtoo big) qi
qtoo big
Potential wave front a
Coherent destructive interference occurs for a reflected beam direction if the angle of incidence ≠ the angle of reflection: qi ≠ qr.

17. 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.

18. 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.

19. Why can’t we see a light 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…)
To photograph light beams in laser labs, you need to blow some smoke into the beam…

20. 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.

21. On-axis vs. off-axis light scattering
Forward (on-axis) light scattering: scattered wavelets have nonrandom (equal!) relative phases in the forward direction.
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.
Randomly spaced scatterers in a plane
Incident wave
Incident wave
Forward scattering is coherent—
even if the scatterers are randomly
arranged in space.
Path lengths are equal.
Off-axis scattering is incoherent
when the scatterers are randomly
arranged in space.
Path lengths are random.

22. Scattering from a crystal vs. scattering from amorphous material (e. g
Scattering from a crystal vs. scattering from amorphous material (e.g., glass)
A perfect crystal has perfectly regularly spaced scatterers in space.
So the scattering from inside the crystal cancels out perfectly in all directions (except for the forward and perhaps a few other preferred directions).
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!

23. 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.

24. Light scattering regimes
There are many regimes of particle scattering, depend-ing on the particle size, the light wave-length, and the refractive index. You can read an entire book on the subject:
Particle size/wavelength
~ ~ Large
Air
Rayleigh-Gans Scattering
Geometrical optics
Relative refractive index
Large ~1
Mie Scattering
Rayleigh Scattering
Rainbow
Totally reflecting objects
This plot considers only single scattering by spheres. Multiple scattering and scattering by non-spherical objects can get really complex!

25. Diffraction Gratings a qm qm qi a qi
Scattering ideas explain what happens when light impinges on a periodic array of grooves. Constructive interference occurs if the delay between adjacent beamlets is an integral number, m, of wavelengths. a
Scatterer A D C B
Potential diffracted wave-front
Incident wave-front qi qm qm a
Path difference: AB – CD = ml qi
where m is any integer.
A grating has solutions of zero, one, or many values of m, or orders.
Remember that m and qm can be negative, too.
AB = a sin(qm)
CD = a sin(qi)
Scatterer

26. Diffraction orders
Because the diffraction angle depends on l, different wavelengths are separated in the nonzero orders.
Diffraction angle, qm(l)
Zeroth order
First order
Minus first order
Incidence angle, qi
No wavelength dependence occurs in zero order.
The longer the wavelength, the larger its deflection in each nonzero order.

27. Diffraction-grating dispersion
Because diffraction gratings are used to separate colors, it’s helpful to know the variation of the diffracted angle vs. wavelength.
Differentiating the grating equation,
with respect to wavelength:
[qi is constant]
Rearranging:
Gratings typically have an order of magnitude more dispersion than prisms.
Thus, to separate different colors maximally, make a small, work in high order (make m large), and use a diffraction angle near 90 degrees.

28. Any surface or medium with periodically varying a or n is a diffraction grating.
Gratings can work in reflection (r) or transmission (t).
Transmission gratings can be amplitude (a) or phase (n) gratings.

29. Real diffraction gratings
White light diffracted by a real grating.
m = 0
m =1
m = 2
m = -1
Diffracted white light
(multiple orders)
(diffracted white light)
(gratings)
(cd)
The dots on a CD are equally spaced (although some are missing, of course), so it acts like a diffraction grating.
Diffraction gratings

30. World’s largest diffraction grating
Lawrence Livermore National Lab

31. Wavelength-dependent incoherent molecular scat-tering: Why the sky is blue.
Light from the sun
Air
Air molecules scatter light, and the scattering is proportional to w4.
Sun picture courtesy of the Consortium for Optics and Imaging Education
Shorter-wavelength light is scattered out of the beam, leaving longer-wavelength light behind, so the sun appears yellow.
In space, there’s no scattering, so the sun is white, and the sky is black.

32. Why is some ice blue?
Picture taken by Rick Trebino in the Antarctic Sound, 2006.
High pressure (over time) squeezes the air bubbles out, leaving molecular scattering as the main source of scattering.

33. Sunsets involve longer path lengths and hence more scattering.
Note the cool sunset.
Noon ray
Sunset ray
Atmosphere
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