1. Chapter 8 Diffraction (1) Fraunhofer diffraction
A coherent line source
Diffraction by single slit, double slit and many slits
Rectangular aperature
Circular aperature
Diffraction grating
(2) Fresnel diffraction
Circular obstacles
Diffraction by a slit, narrow obstacle
(3) Kirchhoff’s scalar diffraction theory

2. Fraunhofer Diffraction

3. Huygens-Fresnel Principle
Every unobstructed point of a wavefront, at a given instant, serves as a source of spherical secondary wavelets (with the same frequency as that of the primary wave). The amplitude of the optical field at any point beyond is the superposition of all these wavelets (considering their amplitudes and relative phases).

4. Fraunhofer & Fresnel Diffraction
(Far-field diffraction)
Fraunhofer diffraction
Aperature size
Source (observation) distance
Wavelength
 Phase
(Near-field diffraction)
Fresnel diffraction

5. phase (SA) – phase (SB) < /4
Aspect of Phase for Fraunhofer Diffraction A B S a R R
phase (SA) – phase (SB) < /4
 R > a2 / 

6. A Coherent Line Source

7. Intensity Profile of a Coherent Line Source
(1) When D >> ,  is large  I (  0)  0
A long coherent line source (D >> ) can be treated as a single-point emitter radiating (a circular wave) predominantly in the forward,  = 0, direction.
(2) When D << ,  is small  I()  I(0)
A point source emitting sphereical waves.

8. Fraunhofer Diffraction by a Single Slitz

9. Fraunhofer Diffraction Pattern of a Single Slit

10. Fraunhofer Diffractionb
Fraunhofer Diffraction
by Double Slits a

11. Fraunhofer Diffraction Pattern of Double Slits
(Young’s experiment)
(max) kasin = m

12. Fraunhofer Diffraction by Many Slits

13. Fraunhofer Diffraction by Many Slits
(Principal maxima)

14. Experimental and Theoritical Results
Comparison between
Experimental and Theoritical Results
a=4b, N=4

15. Fraunhofer Diffraction by Rectangular Aperture
Assimption: coherent secondary point sources within the aperture.
Source point (0, y, z), Observation point (X, Y, Z)

16. Fraunhofer Pattern
of a Square Aperture

17. Matlab Demonstration

18. Fraunhofer Diffraction by Circular Aperture

19. Bessel Function of the First Kind

20. Diffracted Irradiance of a Circular Aperture

21. Airy Disk & Airy Rings
84%

22. Resolution of Imaging Systems
Just resolved when the center of one Airy disk falls on the first minimum of the Airy pattern of the other.
Angular limit of resolution
Limit of resolution

23. An Interesting Experiment on Image Resolution

24. Revision of Fraunhofer Diffraction
Fraunhofer Difraction functions as “Fourier Transform” from geomety (space) domain to wave-vector (wave-number) domain.
 Uncertainty principle

25. The Diffraction Grating
Reflection phase grating
Transmission phase grating
(Grating equation)

26. Grating Spectroscopy (I)
(Angular width of a line)
(Angular dispersion)

27. Grating Spectroscopy (II)
(limit of resolution)
(F-P spectroscopy)
(free spectral range)

28. Two- and Three-Dimensional Gratings
Why don’t use visible lights for diffraction of solid crystals?
X-ray difraction pattern for SiO2

29. Fresnel Diffraction

30. Fresnel Zones
Obliquity factor
Inclination factor
Half-period zones

31. Propagation of a Spherical Wavefront

32. Optical Disturbance from Fresnel Zones (I)
If m is odd,
If m is even,

33. Optical Disturbance from Fresnel Zones (II)
The last contributing zone occurred at  = 90o
i.e. K() = 0 for /2  | |  
and Km(/2) = 0
(Primary wave, SP)
(Secondary wavelets)

34. The Vibration Curve (I)
The resultant phasor changes in phase by  while the aperture size increases by one zone.
K() decreases rapidly only for the first few zones.

35. The Vibration Curve (II)

36. Circular Apertures Case I: m is even, E  0
Case II: m is odd, E  |E1|

37. Drffraction Patterns for Circular Apertures
of Increasing Size
The intensity at the center
The intensity at the off-axis position

38. Method for Evaluating the Intensity
at Off-Axis Positions

39. Radius of Aperture & Number of Zones
>>
Number of zones 
Examples:
 = 1 m, r0 = 1 m,  = 500 nm and R = 1 cm
 400 zones
If  and r0 are very large, such that only a small fraction of the first zone appears in the aperture,
Fraunhofer diffraction occurs.

40. Circular Obstacles
As P moves close to the disk,  increases, Kl+10, so E0

41. Fresnel Zone Plate

42. Rectangular Apertures (I)

43. Rectangular Apertures (II)
Example:
u1 = u2   and v1 = v2  
 E2 = I0

44. Diffraction Patterns for Increasing Square Apertures

45. The Cornu Spiral
A larger size of aperture corresponds to a larger arc length w.
For checking the E-field at off-axis positions, move two end points with constant w.

46. Fresnel Diffraction by a Slit

47. Fresnel Diffraction Pattern of a Slit

48. The Semi-Infinite Opaque Screen

49. Fresnel Diffraction Pattern of a Semi-Infinite Opaque Screen

50. Fresnel Diffraction by a Narrow Obstacle

51. Babinet’s Principle
When the transparent regions on one diffraction screen exactly correspond to the opaque regions of the other and vice versa, these two screen are complementary.

52. Diffraction Patterns of Complementary Screens

53. Kirchhoff’s Scalar Diffraction Theory (I)
Fresnel-Kirchhoff diffraction formula

54. Kirchhoff’s Scalar Diffraction Theory (II)

55. Homework: 11, 25, 28, 52