1. Chapter 36
Diffraction
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2. Learning Goals for Chapter 36
Diffraction vs. Interference
Single-slit vs. Multiple-slit diffraction
Calculating intensity at points in single-slit pattern.
X-ray diffraction reveals arrangement of atoms in crystal.
Diffraction limits on smallest details
Holograms!
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3. Diffraction
Shadows created by straight edge SHOULD form a perfectly sharp line.
Nope!
Wave nature of light causes interference patterns, which blur the edge of the shadow.
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4. Diffraction
Razor blade halfway between a pinhole, illuminated by monochromatic light, and a photographic film.
Film recorded shadow cast by razor blade.
Note fringe pattern around blade outline, caused by diffraction.
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5. Chapter 35 opener. Parallel coherent light from a laser, which acts as nearly a point source, illuminates these shears. Instead of a clean shadow, there is a dramatic diffraction pattern, which is a strong confirmation of the wave theory of light. Diffraction patterns are washed out when typical extended sources of light are used, and hence are not seen, although a careful examination of shadows will reveal fuzziness. We will examine diffraction by a single slit, and how it affects the double-slit pattern. We also discuss diffraction gratings and diffraction of X-rays by crystals. We will see how diffraction affects the resolution of optical instruments, and that the ultimate resolution can never be greater than the wavelength of the radiation used. Finally we study the polarization of light.

6. Diffraction from a single slit
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7. Diffraction from a single slit
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8. Diffraction
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9. Diffraction vs. Interference?
Both use same principle! PATH LENGTH DIFFERENCES create PHASE differences in arriving light intensities
But not just how many apertures!
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10. Diffraction vs. Interference?
d (edge) a
d (center)
Path length difference d = [d(edge) – d(center)]
Phase difference Df = d /2pl = kd
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11. Fresnel vs. Fraunhofer Diffraction
d << ka2 means LARGE phase changes (Fresnel)
d>> ka2 means SMALL phase changes (Fraunhofer)
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12. Fresnel vs. Fraunhofer Diffraction
Fresnel Diffraction (near-field)
Divide aperature a into multiple point sources
Calculate path length d on near screen
You can see these!
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13. Fresnel diffraction by single slit
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14. Fresnel diffraction by a single slit
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15. Diffraction by a Single Slit or Disk
Pattern arises because different points along slit create wavelets that interfere with each other just as a double slit would.
Figure Analysis of diffraction pattern formed by light passing through a narrow slit of width D.

16. Fraunhofer diffraction by a single slit
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17. Fresnel vs. Fraunhofer Diffraction
Fresnel Diffraction (near-field)
Divide aperature a into multiple point sources
Calculate path length d on near screen
You can see these!
Fraunhofer Diffraction (far-field)
Hard to see without lens
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18. Locating dark fringes
Fraunhofer diffraction pattern from a single horizontal slit.
Central bright θ = 0, surrounded by series of dark fringes.
Central bright fringe twice as wide as other bright fringes.
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19. Intensity in single-slit pattern
Derive expression for intensity distribution for single-slit diffraction pattern using phasor-addition.
Imagine plane wave front at slit subdivided into a large number of strips.
At center point O, phasors all in phase.
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20. Intensity in single-slit pattern
Consider wavelets arriving from different strips at P.
Path length differences create phase differences between wavelets coming from adjacent strips.
Vector sum of phasors is now part of “perimeter” of a many-sided polygon.
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21. Intensity in single-slit pattern
Consider wavelets arriving from different strips at P.
Path length differences create phase differences between wavelets coming from adjacent strips.
Vector sum of phasors is now part of “perimeter” of a many-sided polygon.
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22. Intensity maxima in a single-slit pattern
Intensity versus angle in single-slit diffraction pattern.
Most of wave power goes into central intensity peak between m = 1 & m = −1 intensity minima.
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23. Width of single-slit pattern
Pattern depends on ratio of slit width a to the wavelength l.
For a ~ l can’t even see second order + minima!
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24. Width of single-slit pattern
Pattern depends on ratio of slit width a to the wavelength l
Pattern when a = 5λ (left)
Pattern when a = 8λ (right).
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25. Diffraction by a Single Slit or Disk
The minima of the single-slit diffraction pattern occur when a
Figure Intensity in the diffraction pattern of a single slit as a function of sin θ. Note that the central maximum is not only much higher than the maxima to each side, but it is also twice as wide (2λ/D wide) as any of the others (only λ/D wide each). a a a a a a

26. Width of single-slit diffraction pattern
Pattern depends on ratio of slit width a to the wavelength l.
a a a a a a
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27. What??? Wait a minute!
The minima of single-slit diffraction pattern occur when a
The maxima of double-slit interference pattern occured when d

28. Diffraction vs. Interference?
a = size of single slit
d = distance BETWEEN SLITS
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29. Single-slit diffraction vs. Double-slit Interference
Solution: a. The first minimum occurs at sin θ = λ/D = 0.75, or θ = 49°. The full width is twice this, or 98°.
b. The width is 46 cm.

30. Single-slit diffraction vs. Double-slit Interference
Single-slit diffraction Slit Diameter = a (or ‘s’ or ‘D’)
a is small!
a sin q = ml for minima sin q = ml / a
minima at large q
& if l ~ a, none! a
Solution: a. The first minimum occurs at sin θ = λ/D = 0.75, or θ = 49°. The full width is twice this, or 98°.
b. The width is 46 cm.

31. Single-slit diffraction
Solution: a. The first minimum occurs at sin θ = λ/D = 0.75, or θ = 49°. The full width is twice this, or 98°.
b. The width is 46 cm.

32. Single-slit diffraction vs. Double-slit Interference
Double-slit interference Slit SPACING = d
d > a typically
d sin q = ml for maxima sin q = ml/d
bright fringes at small q
Solution: a. The first minimum occurs at sin θ = λ/D = 0.75, or θ = 49°. The full width is twice this, or 98°.
b. The width is 46 cm.

33. Single-slit diffraction vs. Double-slit Interference
Solution: a. The first minimum occurs at sin θ = λ/D = 0.75, or θ = 49°. The full width is twice this, or 98°.
b. The width is 46 cm.

34. Combining Diffraction & Interference!
See Hyperphysics:
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35. Combining Diffraction & Interference!
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36. Diffraction from Two slits of finite width
Pattern from two slits with width a, separated by a distance (between centers) d = 4a.
Two-slit peaks same positions; intensities modulated by single-slit pattern.
Single-slit diffraction “envelopes” intensity function.
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37. Diffraction from Two slits of finite width
Look even more closely!!
Interference minima are present too!
But what happened to fourth interference maximum??
Oh! Interference max cancelled by diffraction minima!
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38. Diffraction AND Interference
Diffraction minima are labeled by integer md = ±1, ±2, … (“d” for “diffraction”).
Compare with interference pattern formed by two very narrow slits with distance d between slits,
Here d is four times as great as the single-slit width a (“i” is for “interference.”)
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39. Diffraction from Two slits of finite width
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40. Diffraction & Interference
Figure (a) Diffraction factor, (b) interference factor, and (c) the resultant intensity plotted as a function of θ for d = 6D = 60λ.

41. Several slits
Array of 8 narrow slits, distance d between adjacent slits.
Constructive interference occurs for rays at angle θ arriveing at P with path difference equal to integral # of l.
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42. Interference pattern of several slits
Eight slit pattern:
Large maxima, (principal maxima) @ same positions as a two-slit pattern, but much narrower.
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43. Interference pattern of several slits
16 slit pattern:
Height of principal maximum is proportional to N 2,
Energy conservation means width of each principal maximum proportional to 1/N.
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44. Diffraction grating Large # of parallel slits
GG’ = cross section of grating.
Slits perpendicular to plane.
Diagram shows six slits; actual grating may contain 1000’s.
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45. Reflection grating
Rainbow-colored reflections from surface of DVD are reflection-grating effect.
DVD grooves are tiny pits 0.12 mm deep in surface, with a uniform radial spacing of 0.74 mm = 740 nm.
Information coded on DVD by varying length of pits.
Reflection-grating aspect of disc is aesthetic feature!
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46. CDs vs. DVDs
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47. Multi-slit interference depends on l
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48. Resolution of a grating spectrograph
In spectroscopy it is often important to distinguish slightly differing wavelengths.
The minimum wavelength difference Δλ that can be distinguished by a spectrograph is described by the chromatic resolving power R.
For a grating spectrograph with a total of N slits, used in the mth order, the chromatic resolving power is:
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49. X-ray diffraction
When x rays pass through a crystal, the crystal behaves like a diffraction grating, causing x-ray diffraction.
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50. A simple model of x-ray diffraction
To better understand x-ray diffraction, we consider a two-dimensional scattering situation.
The path length from source to observer is the same for all the scatterers in a single row if θa = θr = θ.
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51. Circular apertures
The diffraction pattern formed by a circular aperture consists of a central bright spot surrounded by a series of bright and dark rings.
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52. Diffraction by a circular aperture
Airy disk = central bright spot in diffraction pattern from circular aperture.
Radius of Airy disk from angular radius θ1 of first dark ring:
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53. Diffraction & Image resolution
Diffraction limits resolution of optical equipment, such as telescopes.
Larger aperture = better resolution
Longer wavelength = worse resolution
Rayleigh’s criterion for resolution of two point objects:
Two objects are just barely resolved (distinguishable) if center of one diffraction pattern coincides with first minimum of the other.
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54. Smaller light, better resolution
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55. Bigger light, worse resolution
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56. Bigger light, worse resolution
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57. Bigger telescope, better resolution
Because of diffraction, large-diameter telescopes, such as the VLA radio telescope below, give sharper images than small ones.
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58. What is holography?
By using a beam splitter and mirrors, coherent laser light illuminates an object from different perspectives.
Interference effects provide the depth that makes a three-dimensional image from two-dimensional views.
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59. Viewing holograms
Hologram is record on film of interference pattern formed with light from a coherent source & light scattered from object.
Images formed when light is projected back through hologram.
Observer sees virtual image formed behind hologram.
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60. An example of holography
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