1. Diffraction 1 1 1

2. “Any deviation of light rays from rectilinear path which is neither
Diffraction
“Any deviation of light rays from
rectilinear path which is neither
reflection nor refraction is known
as diffraction.” (Sommerfeld)
Types or kinds of diffraction:
1. Fraunhofer ( )
2. Fresnel ( )

3. In addition to interference, waves also exhibit another property – diffraction.
It is the bending of the waves as they pass by some objects or through an aperture.
The phenomenon of diffraction can be understood using Huygen’s principle 3

4. Diffraction of ocean water waves
Ocean waves passing through slits in Tel Aviv, Israel
Diffraction occurs for all waves, whatever the phenomenon. 4

5. Huygens Principle : when a part of the wave front is cut off by an obstacle, and the rest admitted through apertures, the wave on the other side is just the result of superposition of the Huygens wavelets emanating from each point of the aperture, ignoring the portions obscured by the opaque regions.

7. 7

8. Secondary wavelets from apertures

11. μv > μR
Refraction
Deviation for blue is larger than that for red

12. Diffraction
Deviation for red is larger than that for blue

13. Fraunhofer diffraction

14. Single slit diffraction
Principal maximum

15. First minimum

16. Second minimum

17. 17

18. Young’s Two Slit Experiment 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 will be observed.
Fraunhofer diffraction patterns
Good spatial coherence
Poor spatial coherence 18

19. Diffraction from one- and two-slits
Fraunhofer diffraction patterns
One slit
Two slits 19

20. Diffraction from small and large circular apertures
Far-field intensity pattern from a small aperture
Far-field intensity pattern from a large aperture 20

21. 21

22. Diffraction from multiple slits
Slit Diffraction
Pattern Pattern 22

23. Superposition of large number of phasors
of equal amplitude a and equal successive
phase difference δ.
We have to find the resultant phasor. 23

24. GP series

25. δ 8δ 7δ 6δ 5δ 4δ 3δ 2δ
δ/2 25 25

26. 2 26

27. f Problem: Obtain intensity formula by integration
Integrate: a exp(iδ)
x sin θ is the path difference at a point x where the slit element dx is placed. 27

28. Δ = b Sin θ β = nδ/2 = πb sin θ/λ For single slit path difference
between the two ends of the slit
Δ = b Sin θ
Phase difference = 2π Δ/λ = nδ
β = nδ/2 = πb sin θ/λ 28

29. β = π b sin θ / λ tan β = β β = + m π m = 1, 2, 3… β
Intensity for single slit 2
β = π b sin θ / λ
Maxima at I
Minima at
tan β = β
β = + m π _
m = 1, 2, 3… β 29

30. 2
For Principal maxima
β = π b Sin θ / λ

31. For extrema of I(q)
tan β = β

32. Maxima
tan β = β
y= β
y=tan β β 32

33. For subsidiary maxima

34. = λ / b Diffraction envelope size Δ x0 = f λ / b Δ x0
Full width of central maximum
(First minima at sin  ~  = /b)
= λ / b
Diffraction envelope size
Δ x0 = f λ / b
Δ x0
Point source f
Smearing effect
of diffraction 34

35. Double slit 35

36. 2
Phase=(2π/) * x sin 36

37. Double slit intensity pattern for d=5b37

38. Single slit diffraction pattern X double slit interference pattern38

39. b sin θ = m λ m = 0 / d sin θ = n λ Condition for Missing orders
Diffraction Minima at
b sin θ = m λ
m = 0 /
d sin θ = n λ
Interference Maxima at
When the above two equations are satisfied at the same point in the pattern (same θ), dividing one equation by the other gives the condition for missing orders.

40. Missing orders 5, 10, 15, 20….
d/b = 5 40

41. When we use the double-source equation to find locations of bright spots, we find that there are some places where we expect to see bright spots, but we see no light. This is known as a missing order, and it happens because at that location there's a zero in the single slit pattern.

42. Remember!
If the zero in the single slit pattern, and a zero in the double slit pattern coincides, it is not called a missing order, as there is no order to be missing!
Also, if there is a local peak in the single slit pattern, and a zero in the double source pattern, there will still be a zero (remember, we multiply the functions!) - this is also not a missing order.

43. N slit grating

44. Normal
incidence
Transmission
grating

46. = Number of slits
We know that the amplitude due to each single slit:
= δ /2

47. Intensity pattern
Diffraction
Interference
for small b

48. Behavior of
sin2(x)
sin2(5*x)

49. sin2(5*x) / sin2(x)
sin2(10*x) / sin2(x)

50. When  = 0, or mπ (m = 0, ±1, ±2,…)
(m = 0, ±1, ±2,…)
The intensity is maximum when this condition is satisfied.
These are called the Primary maxima.
m gives the order of the maximum.
The intensity drops away from the primary maxima.
The intensity becomes zero (N-1) times between two successive primary maxima.
There are (N-2) secondary maxima in between.
As N increases the number of secondary maxima increases and the primary maxima becomes sharper.

51. Intensity I γ

53. The intensity becomes zero (N-1) times between two successive primary maxima.
There are (N-2) secondary maxima in between.
N=15

54. For two wavelengths

55. d/b = 4
1st Order
5th Order
6th Order
7th Order

56. Principal maxima Minima Because at minima we should have, sin   0
When  = 0, or mπ (m = 0, ±1, ±2,…)
Principal maxima
(m = 0, ±1, ±2,…)
Minima
Because at minima we should have, sin   0
If maximum is at m and Δw is the width, then the intensity should be zero at (m + Δw )

57. I
π + π/N
π - π/N γ

58. m
Oblique
incidence
Maxima

59. Let us estimate the width of the mth order Principal maximum
sin (A+B) = sin A cos B + cos A sin B
d sin θm = mλ

60. Width of principal maxima
When does the principal maxima get sharper?
Dispersive power of grating is defined as
Differentiate:

61. Chromatic resolving power of a grating
= m N

62. Chromatic resolving power of a prism-

65. Transmission grating
Reflection grating

66. Bragg’s law
X-ray diffraction from crystals:
2d Sin θ = n λ

67. X-ray diffraction from (95% Cu 5% Co) crystals:

69. Visibility of the fringes (V)
Maximum and adjacent minimum of the fringe system
Reference: Eugene Hecht, Optics, 4th Ed. Chapter 12 69