1. Diffraction Light bends! Diffraction assumptions
Solution to Maxwell's Equations
The near field
Fresnel Diffraction
Some examples
The far field
Fraunhofer Diffraction
Young’s two-slit experiment
Image from Hecht, Optics.
Thanks to Prof. Rick Trebino

2. Fuzzy shadow edges make it hard to see diffraction.
Diffraction causes beams to bend around corners.
But a point source or plane wave is required to see this effect. A large source masks diffraction effects.
Rays from a point source yield, in principle, a perfect shadow of the hole, allowing diffraction ripples to be seen.
Screen
with hole
Rays from other regions of the source blur the shadow.
Example: a large source (like the sun or a light bulb) casts blurry shadows, masking the diffraction ripples.

3. Radio waves diffract around mountains, but scattering masks it.
Diffraction
Diffracted waves
Mountain
Emitter
Receiver
When the wavelength is a km long, a mountain peak is a very sharp edge!
Cloud
Tropospheric Scattering
Scattered waves
Emitter
Receiver
Mountain and cloud from:
Another effect that occurs is scattering, so diffraction’s role is not obvious.

4. Diffraction of Ocean Water Waves
Ocean waves passing through, and bending around the edges of, slits (wave breaks) in Tel Aviv, Israel:
Photo by Rick Trebino
Diffraction occurs for all waves, whatever the phenomenon.

5. Diffraction of a Wave by a Slit
Whether waves in water or electromagnetic radiation in air, passage through a slit yields a diffraction pattern that will appear more dramatic as the size of the slit approaches the wavelength of the wave.
Hecht, Optics

6. Diffraction Light does not always travel in a straight line.
Shadow of a hand
illuminated by a
Helium-Neon laser
Light does not always travel in a straight line.
It tends to bend around objects. This tendency is called diffraction.
Any wave will do this, including matter waves and acoustic waves.
Images: [
Shadow of a zinc oxide
crystal illuminated by a
electrons

7. Diffraction by an Edge x
Transmission
Even without a small slit, diffraction can be strong.
Simple propagation past an edge yields an unintuitive irradiance pattern (due to diffraction).
Laser light passing by an edge
From page 507, Hecht, Optics, 4th ed.
Electrons passing by an edge (MgO crystal)

8. Diffraction Geometry
We wish to find the light electric field after a screen with a hole in it.
This is a very general problem with far-reaching applications. x y
Incident wave-fronts
E(x,y)
This region is assumed to be much smaller than this one. x’
Aperture transmission t(x,y) y’
Observation plane
E(x’,y’) z P’ P
What is E(x’,y’) at a distance z from the plane of the aperture?

9. Diffraction Assumptions
Incident wave
Exact solutions do not exist.
The best assumptions/approximations were determined by Gustav Kirchhoff:
1) Maxwell's equations. z
2) Inside the aperture at z = 0, the field and its spatial derivative are the same as if the screen weren’t present.
3) Outside the aperture (in the shadow of the screen) at z = 0, the field and its spatial derivative are zero.
While these assumptions give the best results, they actually over-determine the problem and can be shown to yield zero field everywhere! Nevertheless, we still use them.

10. Diffraction Solution
The field in the observation plane, E(x’,y’), at a distance z from the aperture plane is given by a convolution:
Sample
Spherical wave!
where:
and:
It’s as if a spherical wave is emitted from every point in the aperture, with the strength (and phase) of the incident field at that point.

11. Huygens’ Principle
Huygens’ Principle says that every point along a wave-front emits a spherical wave that interferes with all others.
Christiaan Huygens [
Christiaan Huygens – 1695
Our solution for diffraction illustrates this idea, but is more rigorous.

12. Pattern evolution vs. propagation distance z

13. Fresnel Diffraction: Approximations
In the denominator, we can approximate r by z.
But we can’t approximate r in the exp by z because it gets multiplied by k, which is big, and where relatively small changes in kr can make a big difference! We can, however, write:
This yields:

14. Fresnel Diffraction: Approximations
Multiplying out the squares:
Factoring out the quantities independent of x and y:
This is the Fresnel integral.
It yields the light wave field at the distance z from the screen.

15. Diffraction Conventions
We’ll typically assume that a plane wave is incident on the aperture.
It still has an exp[i(kz–wt)], but it’s constant with respect to x and y. The exp(ikz) is already in our result below.
And we’ll usually ignore the various factors in front:

16. Fresnel Diffraction from a Slit: Example
This irradiance vs. position immediately after a slit illuminated by a laser:
Irradiance
Slit x'
From page 504, Hecht Optics, 4th ed.
Incident plane wave

17. Fresnel Diffraction from a Slit (cont’d)
The irradiance vs. position somewhat farther away from the slit:
Far from the slit z
Close to the slit
Slit
I(x’,z)
Incident plane wave
From page 447, Hecht, Optics, 4th ed.

18. The Spot of Arago
If a beam encounters a stop (which blocks the beam center), it develops a hole, which fills in as it propagates and diffracts:
Also called Poisson’s spot. It was Poisson (who adhered to the particle-theory of light) who pointed out the “contradiction” of this spot and concluded that it showed why the wave theory of light should be abandoned. Arago then did the experiment, which confirmed the existence of the spot. But G.F. Maraldi had actually observed the spot almost a century earlier.
This center-spot irradiance can be quite high and can do some damage!

19. The Spot of Arago
If a beam encounters a stop (which blocks the beam center), it develops a hole, which fills in as it propagates and diffracts:
Interestingly, the hole fills in from the center first! x
Stop
Beam after some distance
Input beam with hole x'
Also called Poisson’s spot. It was Poisson (who adhered to the particle-theory of light) who pointed out the “contradiction” of this spot and concluded that it showed why the wave theory of light should be abandoned. Arago then did the experiment, which confirmed the existence of the spot. But G.F. Maraldi had actually observed the spot almost a century earlier.
This center-spot irradiance can be quite high and can do some damage!

20. The Spot of Arago
If a beam encounters a stop (which blocks the beam center), it develops a hole, which fills in as it propagates and diffracts:
Also called Poisson’s spot. It was Poisson (who adhered to the particle-theory of light) who pointed out the “contradiction” of this spot and concluded that it showed why the wave theory of light should be abandoned. Arago then did the experiment, which confirmed the existence of the spot. But G.F. Maraldi had actually observed the spot almost a century earlier.
The images show simulated Arago spots in the shadow of a disc of varying diameter (4 mm, 2 mm, 1 mm – left to right) at a distance of 1 m from the disc. The point source has a wavelength of 633 nm (e.g. He-Ne Laser) and is located 1 m from the disc. The image width corresponds to 16 mm.

21. Fresnel Diffraction from an Array of Slits: The Talbot Effect
One of the few Fresnel diffraction problems that can be solved analytically is an array of slits.
The beam pattern alternates between two different fringe patterns.
ZT = 2d2/l
Screen with array of slits
Diffraction patterns
Figure from Zhou, et al, OPN, 15, 11, p. 46 (2004)

22. The Talbot Carpet What goes on in between the solvable planes?
The beam propagates in this direction.
wikipedia
The slits are here.

23. Diffraction Approximated
C(x)
S(x) x
These integrals come up:
Such effects can be modeled by measuring the distance along a Cornu Spiral.
For a cool Java applet that computes Fresnel diffraction patterns, try
But the Fresnel diffraction regime is too complex, so we must look elsewhere for most useful diffraction effects.

24. Fraunhofer Diffraction: The Far Field
Recall the Fresnel diffraction result:
Let D be the size of the aperture: D2 ≥ x2 + y2.
When kD2/2z << 1, the quadratic terms << 1, so we can neglect them.
This condition means going a distance away: z >> kD2/2 = pD2/l
If D = 1mm and l = 1mm, then z >> 3m. In this case:
Notice the factors kx’/z and ky’/z, which multiply x and y in the exp.

25. qx = kx /k = x’/z and qy = ky /k = y’/z
Angles in Diffraction
We can also think in terms of angles and off-axis k-vectors. y’
(x’ y’)
qx ≈ x’/z = kx /k
E(x,y) y
qy ≈ y’/z = ky /k x x’ z kz ky kx
E(x’,y’)
Incident wave-fronts
Observation plane
qx = kx /k = x’/z and qy = ky /k = y’/z
kx ≈ kx’/z and ky ≈ ky’/z

26. The Fraunhofer Diffraction Formulakz ky kx
We can write this result in terms of the off-axis k-vector components:
kx = kx’/z and ky = ky’/z
The Fraunhofer diffraction integral becomes:
E(x,y) = constant if a plane wave
Aperture transmission function
A two-dimensional Fourier transform!
that is:

27. The Uncertainty Principle in Diffraction!
kx = kx’/z
Because the diffraction pattern is the Fourier transform of the slit, there’s an uncertainty principle between the slit width and diffraction pattern width!
If the input field is a plane wave and Dx is the slit width and Dkx is the proportional to the beam angular width after the screen,
The quantum-mechanical uncertainty principle for photons
Or:
The smaller the slit, the larger the diffraction angle and the bigger the diffraction pattern!
And the longer the wavelength, the bigger the diffraction pattern.

28. Fraunhofer Diffraction from a Slit
Fraunhofer diffraction from a slit is simply the Fourier transform of a rect function, which is a sinc function. The irradiance is then sinc2.
t(x) = rect(x/w)
I(x’)
Fraunhofer Irradiance [

29. Fraunhofer Diffraction from a Square Aperture
Diffracted
irradiance
field
The diffracted field is a sinc function in both x’ and y’ because the Fourier transform of a rect function is sinc.
Hecht, Optics.

30. Diffraction from a Circular Aperture
Diffracted Irradiance
A circular aperture yields a diffracted Airy Pattern,
which looks a lot like a sinc function, but actually involves a Bessel function.
Diffracted field
Hecht, Optics.

31. Diffraction from Small and Large Circular Apertures
Far-field intensity pattern from a small aperture
Recall the Scale Theorem!
This is the Uncertainty Principle for diffraction.
Hecht, Optics.
Far-field intensity pattern from a large aperture

32. Fraunhofer Diffraction from Two Slitsx a -a w
t(x)
t(x) = rect[(x+a)/w] + rect[(x- a)/w]
kx’/z
I(x’)

33. Convolution

34. Diffraction from One- and Two-Slit Screens
In 1803, Thomas Young performed his historic two-slit measurement, which proved definitively that light is a wave.
Fraunhofer diffraction patterns
One slit
Two slits
Experimental result
Thomas Young  ( )
This pattern is too complex to occur by chance.

35. Diffraction from Multiple Slits
Slit Pattern
Diffraction
Pattern
Infinitely many equally spaced slits (a Shah function!) yields a far-field pattern that’s the Fourier transform, that is, the Shah function. 1 2 ∞
Hecht

36. Two Slits and Spatial Coherence
If the spatial coherence length is less than the slit separation, then the relative phase of the light transmitted through each slit will vary randomly, washing out the fine-scale fringes, and a one-slit pattern (the incoherent sum of the irradiances from two one-slit patterns) will be observed.
Max
Fraunhofer diffraction patterns
Good spatial coherence
Poor spatial coherence

37. Young’s Two-Slit Experiment and Photons
Use a beam so weak that only one photon is detected at a time.
Now the photon can only pass through only one slit at a time, yielding a one-slit pattern (independent of whichever slit it passes through). x'
Possible diffraction patterns
Each photon passes through only one slit (it doesn’t matter which).
Two-slit pattern, which should not occur.

38. Dimming the Light Incident on Two Slits
Dimming the light in a two-slit experiment can yield single photons at the screen.
Perform the experiment many times, detecting a single photon at a time. And plot the resulting detected photon locations on the wall far away for all the measurements.
Here’s what, in fact, happens:
The two-slit pattern emerges!
Each individual photon goes through both slits! [
Even individual photons behave like waves.

39. Photons don’t interfere with each other. They interfere with themselves.
This is a famous statement by the great 20th-century physicist, Paul Dirac.
It means that photons act like waves at the screen and so interfere to produce the two-slit pattern.
But they act like particles when we detect them.
Photo of Dirac:
Paul Dirac ( )
The same is true of all wave/particles.
It turns out that the act of learning which slit the wave/particle passes through transforms a two-slit pattern into a one-slit one.
When you’re looking, everything is a particle; when you’re not, it’s a wave.