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
Thanks to Prof. Rick Trebino
Georgia Tech

2. Coherent vs. Incoherent Interference
Waves that combine in phase add up to relatively high irradiance.
Coherent constructive interference =
Waves that combine 180° out of phase cancel out and yield zero irradiance.
Coherent destructive interference =
Waves that combine with many different random phases nearly cancel out and yield low—but not zero—irradiance. =
Incoherent interference

3. Interfering Many Waves with the Same w and k: 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. Incoherence Mathematics: Adding N Waves
Coherent constructive addition: Assume all N individual irradiances, polarizations, frequencies, propagation directions, and phases are equal.
Then:
Incoherent addition: Now assume the phases are random.
m = m’
m ≠ m’ 1
Individual irradiances N
≈ 0
Coherent constructive irradiance: I N2.
Incoherent irradiance: I N.
Coherent destructive irradiance: I = 0.

5. Adding Different Fields with Random Phases
If there are many different-irradiance fields, each with a random phase, qm:
Cross terms (m ≠ m’)
I1, I2, … IN are the irradiances of the various beamlets (m = m’). They’re all positive real numbers and they add.
Em Em’* have the phase factors: exp[i(qm- qm’)]. When the q’s are random, this sum isn’t 0, but it’s small compared to the sum of the irradiances.
All the relative phases
I ≈ I1 + I2 + … + IN Re Im
I1+I2+…+IN
We could call this effect Phase Incoherence.

6. Incandescent light (hot filament) Temperature = 2500-3000K
Hot electrons jump between many very closely spaced levels (solid metal). Produce all colors. Mostly infrared at temp of normal filament.
>90% is worthless Infrared radiation (IR = longer than ~700nm) λ IR
~10% of energy is
useful visible light P

7. How do fields with random phase arise?
Anything that’s so complicated that we don’t know the details, physicists call random.
Mole-cules
The positions of emitters can be complex, e.g., molecules in a light bulb.
The source that excites emitters can also be very complicated.
Collisions of emitters com- plicate the matter even more.
Basically, everything that emits light is random, unless we impose order upon it, for example, as in a laser.

8. Temporal Incoherence: Adding Many Different Colors Interfering in Time
Frequency
Short pulse
Locked phases
Time
Intensity vs. time
Random phases
Light bulb
Coherent constructive interference
tc = 1/Dn
Coherent destructive interference
Incoherent interference
tc = 1/Dn

9. Coherent Interference in Space of Many Beams from a Perfect Lens
How tightly can we focus a beam?
Consider the rays in pairs of symmetrically propagating directions and add up all the fields at the focus, yielding fringes with a spacing of l/(2sinq), where q is the ray angle relative to the axis.
Note that the smallest fringe spacing possible occurs for counter- propagating rays (q = 90°). And it’s l/2.

10. Spatial coherence length: xc = 1/Dkx
Coherent Interference in Space of Many Beams from a Perfect Lens: l/2 Limit
We added up all the spatial fringes from crossing beams at the focus of an infinitely large perfect lens and found that, at best, we could focus a beam to a diameter of ~l/2 (= 2p/Dkx):
Irradiance E
Spatial coherence length: xc = 1/Dkx
where Dkx = 2k = 2(2p/l) = the total range of kx’s (the min is 0, and the max is 2k)
xc = 2p/Dkx
= 1/Dkx x
The interference is coherent and constructive at x = 0, yielding a minimal-sized beam there of width 2p/Dkx = 1/Dkx.
And it is coherent and destructive elsewhere.
Analogous to the coherence time!

11. Incoherent Interference of Beams in Space from a Lousy Lens or a Light Bulb
Now, what if all the spatial fringes are randomly out of phase with each other?
The fringes now shift in phase by random amounts: E
Irradiance
xc = 1/Dkx x
Analogous to the temporal case, the fluctuations are on a length scale similar to the perfect in-phase case, but are much weaker and there are many more spikes. The beam is spatially incoherent.
Also analogously, the coherence length is: ~ 1/Dkx, where Dkx is the range of off-axis k’s (for this beam and also the perfect focus).

12. Light Bulbs Light from a light bulb is incoherent in several ways.
1. It emits light in all directions (it’s omnidirectional).
Phase incoherent
2. For a given direction and wavelength, many different molecules are emitting light with random relative phases.
Temporally incoherent
3. It emits many colors (it’s white), so we must add waves of many values of w (and k-magnitudes). So it has a short coherence time.
4. It’s not a point source, so we must add waves with many different origins. So it has a small spatial coherence area.
Spatially incoherent

13. What does light-bulb light really look like?
Poor temporal coherence means that the field amplitude varies on a time scale of 1/Dn (~ a few fs), and poor spatial coherence means that it varies on a length scale of 1/Dk (typically ~ l <1mm).
E (x,y,t)2 (not temporally or spatially averaged)
No one has ever measured light-bulb light with such temporal or spatial resolution.
A simulated snapshot of monochromatic, spatially incoherent light with high spatial resolution
A simulated movie of white light with high spatial and temporal resolution
B&W image:
Color image:
Note that both of these images are actually of something quite different, but they have precisely the same behavior as the incoherent light they’re used to illustrate here. x y
The natural temporal and spatial averages in measurements wash all this structure out. Time averaging is even built into the irradiance definition.

14. Coherent vs. Incoherent Light
Laser
Coherent light:
1. It’s intense (or dark).
2. It’s unidirectional.
3. Total irradiance N2 or 0 (depending on the direction).
4. Total irradiance is the mag-square of the sum of individual fields.
Incoherent light:
1. It’s relatively weak.
2. It’s omnidirectional.
3. Total irradiance N (independent of the direction).
4. Total irradiance is the sum of individual irradiances.
E = E1 + E2 + … + EN
I ≈ I1 + I2 + … + IN

15. Light Scattering Particle, bubble, droplet, or 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. It’s how we see things. Scattering causes milk and clouds to be white and water to be blue. It’s the basis of nearly all optical phenomena.
Like light itself, scattering can also be coherent or incoherent, depending on the relative phases.

16. Electromagnetic waves http://phet.colorado.edu
Radio Waves.jar

17. Note that k and r are not vectors here.
Spherical Waves
Recall that a spherical wave is also a solution to Maxwell's equations.
Note that k and r are not vectors here.
where k is a scalar, and
r is the radial coordinate.
A good approximation of the light scattered from a small particle is a spherical wave.
A spherical wave has spherical wave-fronts.
As a spherical wave propagates a large distance away from its source, its wave-fronts approach planes.

18. 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.
This will vastly simplify the math.

19. Scattered spherical waves often combine to form plane waves anyway.
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.

20. The Mathematics of Scattering
The math of light scattering is the same as that of light sources.
If the phases aren’t random, we add the fields:
Coherent
E = E1 + E2 + … + EN
I1, I2, … IN are the irradiances of the various beamlets. They’re all positive real numbers and add.
Em Em’* are cross terms, which have the phase factors: exp[i(qm-qm’)]. The inter-ference can be constructive or destructive.
If the phases are random, we add the irradiances:
Incoherent
When the q’s are random, the cross terms are negligible.
I ≈ I1 + I2 + … + IN

21. To understand scattering in a given situation, compute phase delays.
Wave-fronts
Because the phase is, by definition, 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 delays for all scattered waves are the same (modulo 2p), then the scattering is constructive and coherent. If they vary uniformly from 0 to 2p, then it’s destructive and coherent.
If it’s random (or just very complicated), then it’s incoherent.

22. Coherent Constructive Scattering: Reflection from a Smooth Surface When qi = qr
A plane wave can only scatter into 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
Equal paths mean that coherent constructive interference occurs for a reflected beam if the angle of incidence = the angle of reflection: qi = qr.

23. Coherent Destructive Scattering: Reflection from a Smooth Surface When qi ≠ qr
Imagine that the reflection angle is too big. The symmetry is now gone, and the phases are now all different (and nonrandom).
f = ka sin(qi)
f = ka sin(qtoo big)
The phases perfectly cancel: qi
qtoo big
Potential wave front a
Unequal path lengths mean that coherent destructive interference occurs for a reflected beam direction. This is because the angle of incidence ≠ the angle of reflection: qi ≠ qr.

24. Diffraction Gratings a qm qm qi a qi
Scattering ideas explain what happens when light impinges on a periodic array of scatterers.
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.
AB = a sin(qm)
CD = a sin(qi)
Scatterer
A grating has solutions for one or many values of m, or orders.
Remember that m and qm can be negative, too.

25. Diffraction Orders
Because the diffraction angle depends on l, different wavelengths are separated in the nonzero orders.
No wavelength dependence occurs in zero order.
There, light is simply transmitted.
Or, in reflection, angle of incidence = angle of reflection.
Diffraction angle, q1(l)
Zeroth order
First order
Minus first order
Incidence angle, qi
The longer the wavelength, the larger its deflection in each nonzero order.

26. 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.

27. 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.

28. 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

29. World’s Largest Diffraction Grating
Lawrence Livermore National Lab

30. 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 random phase.
So scattering is incoherent, and we’ll see weak light in all directions.
This is why rough surfaces aren’t shiny and look different from smooth surfaces and mirrors.
It’s also why we can see most things from any angle.

31. Scattering from a Real-World Surface
Why does this happen?
4GIFS.com; Image from reddit.com: [

32. 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.

33. 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—important for x-ray crystallography).
Coherent destructive interference
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!

34. Scattering Off Macroscopic Particles
Fresnel’s equations for reflection and transmission and Snell’s Law apply to light propagating through particles, like water droplets in a cloud.
Light source
Water droplets
Light scattering in a cloud
An interesting exception is the rainbow. Why can it be colorful? The answer is that it is due to a very thin cloud, and only single scattering occurs. In most clouds, multiple scattering occurs, as in the figure, and the scattered light is white. Any rainbow-like separation of colors is destroyed by multiple scattering in a thick cloud.
Reflectivities are high (> ~4% per surface), so much light is scattered.
And there is little variation with respect to wavelength, so a white input beam scatters as white.

35. Macroscopic-Particle vs. Molecular Scattering
Particles, droplets, bubbles, snowflakes, etc. (>>l), are relatively big, so they scatter strongly.
Dispersive effects in them are weak, so this scattering is usually white (clouds, snow, milk).
But when a medium is uniform and lacks such large particles, molecular scattering can be seen.
Although there are many more of them, molecules are much weaker scatterers, so their contribution is only noticeable when larger particle scattering is absent.
Or it may not be visible at all.

36. Rayleigh Scattering: Visible Light in Air
Consider the forced oscillator with incident-light frequency w << w0 (w0 = the resonance frequency) and with the emitted-light propagation direction different from that of the incident beam. y x z
The inhomogeneous wave equation:
where:
Constant
Now the scattered-light initial condition is Ex = 0 (it doesn’t add to the incident light), as was true for absorption/refraction. But we can see:
The emitted E-field frequency is w, and its amplitude will be
So the emitted irradiance will be

37. Scattered Intensity vs. Wavelength

38. Wavelength-Dependent Incoherent Molecular Scat-tering: Why the Sky is Blue
Light from the sun
Air
Air molecules Rayleigh-scatter light, and the scattering intensity 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.

39. Sunsets/rises involve longer path lengths and hence more scattering.
Sunset ray
Noon ray
Atmosphere
Due to the greater loss of blue light, the sun and nearby clouds can appear orange or even red.
Image of earth from Warren Rogers Modern Physics lectures
Schematic from:
Moonrise (actually the supermoon of Nov. 14, 2016) by Rick Trebino from Tybee Island, GA.
Moonrises/sets are similar. Note that the bottom edge of the moon is redder than the top due to the greater length of atmosphere its rays pass through.

40. Small particles also Rayleigh scatter.
The eruption of Krakatoa affected Edvard Munch’s The Scream.
It poured gas and microscopic particles (more scattering) and into the sky worldwide.
Note the awesome sunset.
Munch Museum/Munch Ellingsen Group/VBK, Vienna

41. Why is some ice blue?
Picture taken by Rick Trebino in the Antarctic Sound, 2006.
High pressure (over time) squeezes out the air bubbles (which, when present, are much stronger and wavelength-independent scatterers), leaving molecular scattering (w4) as the main source of scattering.

42. 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 flashlight beams.
This is due to the facts that air is very sparse (N is relatively small), air molecules are also not strong scatterers, and the scattering (called Rayleigh 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…

43. 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!

44. Single vs. Multiple Scattering
Water-droplet scattering is responsible for both clouds and rainbows.
So why are clouds white and rainbows colorful?
To see a rainbow, a typical photon can be scattered at most once.
Once this light is scattered again, the angular dependence of the colors, required for a rainbow, is lost.
Clouds are much thicker and so multiply scatter light and are white.
Notice that rainbows only occur when the sky is relatively clear, and a relatively thin region of rain exists to generate the rainbow.
Picture: