1. Diffraction Light bends! History of diffraction
Diffraction solution to Maxwell's Equations
Fraunhöfer Diffraction
Some examples
Young’s two-slit experiment
Prof. Rick Trebino
Georgia Tech

2. Diffraction of Ocean-Water Waves
Light does not always travel in a straight line.
It tends to bend around objects. This tendency is called diffraction.
Ocean waves passing through slits in Tel Aviv, Israel
Diffraction occurs for all waves, whatever the phenomenon.

3. 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 (decreases to) the wavelength of the wave.

4. Radio waves diffract around mountains.
When the wavelength is a km long, a mountain peak is a very sharp edge!
Another effect that occurs is scattering, so diffraction’s role is not obvious.

5. Diffraction by an Edge t(x)
Even without a small slit, diffraction can be strong.
Simple propagation past an edge yields an unintuitive intensity-fringe pattern. x
Transmission
t(x)
Laser light passing by edge
Bending
Fringes

6. Diffraction Interesting fringes in a wave that passes near an edge:
Shadow of a hand illuminated by a Helium-Neon laser.
Note the fringes around the edges.

7. Diffraction Gratings
A diffraction grating is a series of equally spaced tiny slits.

8. Why It’s Hard to See Diffraction
Diffraction tends to cause ripples at edges. But a point source is required to see this effect. A large source masks them.
Screen
with hole
Rays from a point source yield a perfect shadow of the hole. Rays from other regions blur the shadow.
Example: a large source (like the sun) casts blurry shadows, masking the diffraction ripples

9. History of Diffraction
Diffraction of light was first characterized by  Francesco Grimaldi, who also coined the term diffraction.
Isaac Newton called it inflexion of light rays. 
James Gregory (1638–1675) observed the diffraction patterns from a bird feather—the first diffraction grating to be discovered. 
Thomas Young performed a celebrated experiment in 1803 demonstrating interference from two closely spaced slits. He deduced that light must propagate as a wave. 
Augustin-Jean Fresnel did more definitive studies and calculations of diffraction in giving great support to the wave theory of light advanced by Christiaan Huygens and reinvigorated by Young, against Newton's particle theory.
Francesco Grimaldi

10. How to Profit from Diffraction…
"Amazing x-ray glasses!”
“See the bones in your hand, see through clothes!"
Really just bird feathers diffracting the image.
Wikipedia.
Marketed in the 1950s and 60s by Harold von Braunhut, who also sold “Amazing Sea Monkeys.”
Actual view through a bird feather

11. Huygens’ Principle
Huygens’ Principle says that every point along a wave-front emits a spherical wave that interferes with all others.
Christiaan Huygens – 1695
For a plane wave without an aperture, all these spherical waves still add up to a plane wave in the forward direction.
Fresnel’s solution for diffraction from an aperture illustrates this idea.

12. 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) = 0 or 1 y’
Observation plane
E(x’,y’) z P’ P
What is E(x’,y’) at a distance z from the plane of the aperture?

13. Diffraction Assumptions
Incident wave
The best assumptions were determined by Kirchhoff (in the early 19th century):
1) Maxwell's equations
2) Inside the aperture, the field is the same as if the screen were not present.
3) Outside the aperture (in the shadow of the screen), the field is zero.

14. Diffraction Solution
The field in the observation plane, E(x’,y’), at a distance z from the aperture plane is a sum of spherical waves from every point within the aperture:
Spherical wave!
where:
How to simplify this result? Because z is much larger than all the other distances, we can approximate r by z in the denominator.
But we can’t approximate r in the exp by z because it gets multiplied by k, which is big, so relatively small changes in r can make a big difference in the sines and cosines!

15. Diffraction Approximations
But, inside the exp, we can write:
This yields:

16. The Fraunhöfer Approximation
Multiplying out the squares:
Factoring out the quantities independent of x and y:
If k (x2 + y2) / 2z << 1, the quadratic terms << 1, so we can neglect them.
This means going a distance away: z >> k (x2 + y2) /2 = p (x2 + y2) /l
If (x2 + y2) < 1 mm2 and l = 1 micron, then z >> 3 m.

17. Diffraction Conventions
We’ll typically assume that a plane wave is incident on the aperture.
There’s still an exp[i(k z –w t)], but this is constant with respect to x and y.
And we’ll usually ignore the various factors in front (they don’t affect the shape of the diffracted intensity, that is, they don’t depend on x’ or y’):

18. The Fraunhöfer Diffraction Formula
We can write this result in terms of the off-axis k-vector components:
Aperture transmission function
that is: kz ky kx
kx = kx’/z and ky = ky’/z
and:
kx and ky are the off-axis k-vector components of the diffracted wave.
Diffraction is simply a Fourier transform!

19. 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,
Or:
The smaller the slit, the larger the diffraction angle and the bigger the diffraction pattern!

20. Fraunhöfer Diffraction from a Slit
Fraunhöfer Diffraction from a slit is simply the Fourier transform of a rect function, which is a sinc function. The intensity is then sinc2.
t(x) = rect(x/w)
I(x’)

21. Fraunhöfer Diffraction from a Square Aperture
Diffracted
intensity
field
The diffracted field is a sinc function in both x’ and y’ because the Fourier transform of a rect function is sinc.

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

23. 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.
Far-field intensity pattern from a large aperture

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

25. One-Slit vs. Two-Slit Patternskx

26. Interference ideas also explain extra fringes in the two-slit pattern.
The relative phase between the waves at each peak is an integral number (m) of 2p.
The relative delay is an integral number (m) of wavelengths.

27. More Views

28. Diffraction from One and Two Slits
Measured Fraunhöfer diffraction patterns
One slit
Two slits
In 1803, Thomas Young measured the above two-slit pattern, which convincingly confirmed the wave nature of light (how could particles yield such a pattern?), ending centuries of debate as to whether light was a particle or a wave.