1. Chapter 13: Fresnel Diffraction Chapter 13: Fresnel Diffraction

2. Diffraction
Geometrical optics… I
…light can’t turn a corner.

3. Francesco Maria Grimaldi
Diffraction
Physical optics…
Francesco Maria Grimaldi
( ) I
…actually, it can.

4. Huygens-Fresnel principle
every point on a wavefront may be regarded as a secondary source of spherical wavelets
The propagated wave follows the periphery of the wavelets.
Huygens, just add the wavelets considering interference!

5. Huygens-Fresnel principle
If one perturbs a plane wavefront, the Huygens wavelets will no longer constructively interfere at all points in space. Adding the wavelets by physical optics explains why light can turn corners and create fringes around images of objects.

6. Obliquity factor (oblique  obscure?)
- wavelets propagate isotropically—in forward and reverse directions
- to use the Huygens approach, modify amplitude of wavefront as a function of q:

7. Calculating the diffracted wave
spherical waves:
Fresnel-Kirchhoff diffraction integral:
phase shift
obliquity factor
In general, not an easy task. Lucky for you, Fresnel made it simpler.

8. Diffraction dudes: Fresnel and Fraunhofer
Contemporaries, but not collaborators (nor competitors).

9. Fresnel vs. Fraunhofer diffractionP
Fraunhofer:
both incident and diffracted waves may be considered to be planar (i.e. both S and P are far from the aperture)
Fresnel:
occurs when either S or P are close enough to the aperture that wavefront curvature is not negligible

10. Fresnel diffraction from an edge

11. Fresnel diffraction from a slit
irradiance vs. position, just after a slit illuminated by a laser x1
Irradiance

12. From Fresnel to Fraunhofer diffraction
Incident plane wave

13. Fresnel vs. Fraunhofer diffraction
view from source:
view from point of interest:
near field 
where d represents p or q (=distance from source or point to aperture)
A is aperture area

14. Fresnel number, F Fresnel diffraction occurs when:
source:
Fresnel diffraction occurs when:
point:
Fraunhofer diffraction occurs when:
where h = aperture or slit size
l = wavelength
d = distance from the aperture (p or q)

15. Fresnel diffraction from an array of slits:
the Talbot effect
ZT = 2d2/l
Screen with array of slits
Diffraction patterns
one of the few Fresnel diffraction problems that can be solved analytically
beam pattern alternates between two different fringe patterns

16. Rolling out the optical carpet: Talbot effect

17. Getting into the zone
Fresnel’s approach to diffraction from circular aperatures
zone spacing = l/2:
r1 = r0 + l/2
r2 = r0 + l
r3 = r0 + 3l/2
rn = r0 + nl/2 …
these are called the Fresnel zones

18. Stay in the zone
as we draw a phasor diagram where each zone is subdivided into 15 subzones 5 5½ 4 3 2 1
half-period zones
obliquity factor shortens successive phasors
circles do not close, but spiral inwards
amplitude a1 = A1 : resultant of subzones in 1st half-period zone
composite amplitude at P from n half-period zones:

19. Adding up the zones individual phasors composite phasors for large N,
resultant amplitude= half that of zone 1

20. Implications of Fresnel zones
For N contributing Fresnel zones,
If N is small,
a1 ~ aN
for odd N, AN ~ a1
for even N, AN ~ 0
If N is large (i.e. huge aperture)
aN  0
for any N, AN ~ ½ a1

21. Implications of Fresnel zones
strange
. Part 1
A circular aperture is matched in size with the first Fresnel zone:
AP = a1
Now open the aperture wider to also admit zone 2:
AP ~ 0 !
Now remove aperture, allowing all zones to contribute:
AP = ½ a1 !!!
(Irradiance only ¼ !)
What is amplitude of the wavefront at P?

22. Implications of Fresnel zones
strange
. Part 2
A circular disk is matched in size with the first Fresnel zone:
all zones except the first contribute
first contributing zone is the second
AP = ½ a2
irradiance at center of shadow nearly the same as without the disk present!
What is amplitude of the wavefront at P?
How absurd!
Siméon Denis Poisson ( )

23. Poisson/Arago spot

26. The Fresnel zone plate 16 zones
If the 2nd, 4th, 6th, etc. zones are blocked, then:
Amplitude at P is 16 times the amplitude of a1 /2
Irradiance at P is (16)2 times!

27. An alternative to blocking zones
Fresnel vs. plano-convex
lens lens
phases of adjacent Fresnel zones changed by p

28. Kewaunee, Wisconsin

29. Fresnel lighthouse lens
other applications: overhead projectors
automobile headlights
solar collectors
traffic lights

30. Fresnel’s treatment of straight edges
cylindrical wavefronts diffracted by rectangular aperture:
consider source to be a slit
zones are now rectangular strips
edge view:

31. Fresnel’s treatment of straight edges
again, zone spacing = ½ l
adding the phasors gives the endpoints of a Cornu spiral

32. Phasors trace a Cornu spiral
areas of Fresnel strip zones decrease rapidly with n
successive phasor amplitudes of zones are much shorter—half circle never reached
phasors continue to spiral to limit point E (eye)
zones of lower half produce twin spiral in 3rd quadrant
2 zones:

33. Applications of the Cornu spiral
straight edge:
wire:

34. Other Cornu spirals Cornucopia Cornu spiral pendant $34.99
Roman Cornu horn
Cornu aspersum, garden snail

35. Exercises
You are encouraged to solve all problems in the textbook (Pedrotti3).
The following may be covered in the werkcollege on 13 October 2010:
Chapter 13:
1, 2, 3, 5, 6, 7, 10, 18