1. Chapter 27 Wave Optics

2. Wave optics is a study concerned with phenomena that cannot be adequately explained by geometric (ray) optics These phenomena include Interference Diffraction

3. Interference In constructive interference the amplitude of the resultant wave is greater than that of either individual wave In destructive interference the amplitude of the resultant wave is less than that of either individual wave All interference associated with light waves arises when the electromagnetic fields that constitute the individual waves combine

4. Conditions for Interference To observe interference in light waves, the following two conditions must be met The sources should be monochromatic Monochromatic means they have a single wavelength The sources must be coherent They must maintain a constant phase with respect to each other

5. Producing Coherent Sources Light from a monochromatic source is used to illuminate a barrier The barrier contains two narrow slits The slits are small openings The light emerging from the two slits is coherent since a single source produces the original light beam This is a commonly used method

6. Diffraction From Huygen’s Principle we know the waves spread out from the slits This divergence of light from its initial line of travel is called diffraction

7. Young’s Double Slit Experiment, Schematic Thomas Young first demonstrated interference in light waves from two sources in 1801 The narrow slits, S 1 and S 2 act as sources of waves The waves emerging from the slits originate from the same wave front and therefore are always in phase

8. Resulting Interference Pattern The light from the two slits form a visible pattern on a screen The pattern consists of a series of bright and dark parallel bands called fringes Constructive interference occurs where a bright fringe occurs Destructive interference results in a dark fringe

9. Interference Patterns Constructive interference occurs at point O The two waves travel the same distance Therefore, they arrive in phase As a result, constructive interference occurs at this point and a bright fringe is observed

10. Interference Patterns, 2 The lower wave has to travel farther than the lower wave to reach point P The lower wave travels one wavelength farther Therefore, the waves arrive in phase A second bright fringe occurs at this position

11. Interference Patterns, 3 The lower wave travels one-half of a wavelength farther than the upper wave to reach point R The trough of the bottom wave overlaps the crest of the upper wave This is destructive interference A dark fringe occurs

12. Young’s Double Slit Experiment – Geometry The path difference, δ, is found from the tan triangle δ = r 2 – r 1 = d sin θ This assumes the paths are parallel Not exactly true, but a very good approximation if L >> d

13. Interference Equations For a bright fringe, produced by constructive interference, the path difference must be either zero or some integral multiple of the wavelength δ = d sin θ bright = m λ m = 0, ±1, ±2, … m is called the order number When m = 0, it is the zeroth-order maximum When m = ±1, it is called the first-order maximum

14. Interference Equations, 2 When destructive interference occurs, a dark fringe is observed This needs a path difference of an odd half wavelength δ = d sin θ dark = (m + ½) λ m = 0, ±1, ±2, …

15. Interference Equations, 4 The positions of the fringes can be measured along the screen from the zeroth-order maximum From the blue triangle, y = L tan θ Approximation θ is small and therefore the small angle approximation tan θ ~ sin θ can be used y = L tan θ ~ L sin θ This applies to both bright and dark fringes

16. Interference Equations, final For small angles, these equations can be combined Bright fringes Note y is linear in the order number, m, so the bright fringes are equally spaced

17. Uses for Young’s Double Slit Experiment Young’s Double Slit Experiment provides a method for measuring wavelength of the light This experiment gave the wave model of light a great deal of credibility

18. Intensity Distribution – Double Slit Interference Pattern The bright fringes in the interference pattern do not have sharp edges The equations developed give the location of only the centers of the bright and dark fringes We can calculate the distribution of light intensity associated with the double-slit interference pattern

19. Intensity Distribution, Assumptions Assumptions: The two slits represent coherent sources of sinusoidal waves The waves from the slits have the same angular frequency,  The waves have a constant phase difference,  The phase difference, , depends on the angle 

20. Intensity Distribution, Phase Relationships The phase difference between the two waves at P depends on their path difference  = r 2 – r 1 = d sin  A path difference of corresponds to a phase difference of 2  rad A path difference of  is the same fraction of as the phase difference  is of 2  This gives

21. Intensity Distribution, Equation Analysis shows that the time-averaged light intensity at a given angle  is

22. Light Intensity, Graph The interference pattern consists of equally spaced fringes of equal intensity This result is valid only if L >> d and for small values of 

23. Lloyd’s Mirror An arrangement for producing an interference pattern with a single light source Waves reach point P either by a direct path or by reflection The reflected ray can be treated as a ray from the source S’ behind the mirror

24. Interference Pattern from the Lloyd’s Mirror This arrangement can be thought of as a double slit source with the distance between points S and S’ comparable to length d An interference pattern is formed The positions of the dark and bright fringes are reversed relative to pattern of two real sources This is because there is a 180° phase change produced by the reflection

25. Phase Changes Due To Reflection An electromagnetic wave undergoes a phase change of 180° upon reflection from a medium of higher index of refraction than the one in which it was traveling Analogous to a pulse on a string reflected from a rigid support

26. Phase Changes Due To Reflection, cont There is no phase change when the wave is reflected from a boundary leading to a medium of lower index of refraction Analogous to a pulse in a string reflecting from a free support

27. Interference in Thin Films Interference effects are commonly observed in thin films Examples include soap bubbles and oil on water The varied colors observed when white light is incident on such films result from the interference of waves reflected from the opposite surfaces of the film

28. Interference in Thin Films, 2 Facts to keep in mind An electromagnetic wave traveling from a medium of index of refraction n 1 toward a medium of index of refraction n 2 undergoes a 180° phase change on reflection when n 2 > n 1 There is no phase change in the reflected wave if n 2 < n 1 The wavelength of light λ n in a medium with index of refraction n is λ n = λ/n where λ is the wavelength of light in vacuum

29. Interference in Thin Films, 3 Assume the light rays are traveling in air nearly normal to the two surfaces of the film Ray 1 undergoes a phase change of 180° with respect to the incident ray Ray 2, which is reflected from the lower surface, undergoes no phase change with respect to the incident wave

30. Interference in Thin Films, 4 Ray 2 also travels an additional distance of 2t before the waves recombine For constructive interference 2 n t = (m + ½ ) λ m = 0, 1, 2 … This takes into account both the difference in optical path length for the two rays and the 180° phase change For destructive interference 2 n t = m λ m = 0, 1, 2 …

31. Interference in Thin Films, 5 Two factors influence interference Possible phase reversals on reflection Differences in travel distance The conditions are valid if the medium above the top surface is the same as the medium below the bottom surface If there are different media, these conditions are valid as long as the index of refraction for both is less than n

32. Interference in Thin Films, 6 If the thin film is between two different media, one of lower index than the film and one of higher index, the conditions for constructive and destructive interference are reversed With different materials on either side of the film, you may have a situation in which there is a 180 o phase change at both surfaces or at neither surface Be sure to check both the path length and the phase change

33. Interference in Thin Film, Soap Bubble Example

34. Problem Solving Strategy with Thin Films, 1 Conceptualize Identify the light source and the location of the observer Categorize Identify the thin film causing the interference Analyze The type of interference – constructive or destructive – that occurs is determined by the phase relationship between the upper and lower surfaces

35. Problem Solving with Thin Films, 2 Analyze, cont. Phase differences have two causes differences in the distances traveled phase changes occurring on reflection Both causes must be considered when determining constructive or destructive interference Finalize Be sure the answer makes sense

36. Diffraction Diffraction occurs when waves pass through small openings, around obstacles, or by sharp edges Diffraction refers to the general behavior of waves spreading out as they pass through a slit A diffraction pattern is really the result of interference

37. Diffraction Pattern A single slit placed between a distant light source and a screen produces a diffraction pattern It will have a broad, intense central band Called the central maximum The central band will be flanked by a series of narrower, less intense secondary bands Called side maxima The central band will also be flanked by a series of dark bands Called minima

38. Diffraction Pattern, Single Slit The central maximum and the series of side maxima and minima are seen The pattern is, in reality, an interference pattern

39. Diffraction Pattern, Penny The shadow of a penny displays bright and dark rings of a diffraction pattern The bright center spot is called the Arago bright spot Named for its discoverer, Dominque Arago

40. Diffraction Pattern, Penny, cont The Arago bright spot is explained by the wave theory of light Waves that diffract on the edges of the penny all travel the same distance to the center The center is a point of constructive interference and therefore a bright spot Geometric optics does not predict the presence of the bright spot The penny should screen the center of the pattern

41. Fraunhofer Diffraction Pattern Fraunhofer Diffraction Pattern occurs when the rays leave the diffracting object in parallel directions Screen very far from the slit Could be accomplished by a converging lens

42. Fraunhofer Diffraction Pattern – Photo A bright fringe is seen along the axis (θ = 0) Alternating bright and dark fringes are seen on each side

43. Single Slit Diffraction The finite width of slits is the basis for understanding Fraunhofer diffraction According to Huygen’s principle, each portion of the slit acts as a source of light waves Therefore, light from one portion of the slit can interfere with light from another portion

44. Single Slit Diffraction, 2 The resultant light intensity on a viewing screen depends on the direction  The diffraction pattern is actually an interference pattern The different sources of light are different portions of the single slit

45. Single Slit Diffraction, Analysis All the waves that originate at the slit are in phase Wave 1 travels farther than wave 3 by an amount equal to the path difference (a/2) sin θ If this path difference is exactly half of a wavelength, the two waves cancel each other and destructive interference results In general, destructive interference occurs for a single slit of width a when sin θ dark = mλ / a m =  1,  2,  3, …

46. Single Slit Diffraction, Intensity The general features of the intensity distribution are shown A broad central bright fringe is flanked by much weaker bright fringes alternating with dark fringes Each bright fringe peak lies approximately halfway between the dark fringes The central bright maximum is twice as wide as the secondary maxima

47. Resolution The ability of optical systems to distinguish between closely spaced objects is limited because of the wave nature of light If two sources are far enough apart to keep their central maxima from overlapping, their images can be distinguished The images are said to be resolved If the two sources are close together, the two central maxima overlap and the images are not resolved

48. Resolved Images, Example The images are far enough apart to keep their central maxima from overlapping The angle subtended by the sources at the slit is large enough for the diffraction patterns to be distinguishable The images are resolved

49. Images Not Resolved, Example The sources are so close together that their central maxima do overlap The angle subtended by the sources is so small that their diffraction patterns overlap The images are not resolved

50. Resolution, Rayleigh’s Criterion When the central maximum of one image falls on the first minimum of another image, the images are said to be just resolved This limiting condition of resolution is called Rayleigh’s criterion

51. Resolution, Rayleigh’s Criterion, Equation The angle of separation,  min, is the angle subtended by the sources for which the images are just resolved Since << a in most situations, sin  is very small and sin   Therefore, the limiting angle (in rad) of resolution for a slit of width a is To be resolved, the angle subtended by the two sources must be greater than  min

52. Circular Apertures The diffraction pattern of a circular aperture consists of a central bright disk surrounded by progressively fainter bright and dark rings The limiting angle of resolution of the circular aperture is D is the diameter of the aperture

53. Circular Apertures, Well Resolved The sources are far apart The images are well resolved The solid curves are the individual diffraction patterns The dashed lines are the resultant pattern

54. Circular Apertures, Just Resolved The sources are separated by an angle that satisfies Rayleigh’s criterion The images are just resolved The solid curves are the individual diffraction patterns The dashed lines are the resultant pattern

55. Circular Apertures, Not Resolved The sources are close together The images are unresolved The solid curves are the individual diffraction patterns The dashed lines are the resultant pattern

56. Resolution, Example Pluto and its moon, Charon Left – Earth based telescope is blurred Right – Hubble Space Telescope clearly resolves the two objects

57. Diffraction Grating The diffracting grating consists of a large number of equally spaced parallel slits A typical grating contains several thousand lines per centimeter The intensity of the pattern on the screen is the result of the combined effects of interference and diffraction Each slit produces diffraction, and the diffracted beams interfere with one another to form the final pattern

58. Diffraction Grating, Types A transmission grating can be made by cutting parallel grooves on a glass plate The spaces between the grooves are transparent to the light and so act as separate slits A reflection grating can be made by cutting parallel grooves on the surface of a reflective material

59. Diffraction Grating, cont The condition for maxima is d sin θ bright = m λ m = 0, 1, 2, … The integer m is the order number of the diffraction pattern If the incident radiation contains several wavelengths, each wavelength deviates through a specific angle

60. Diffraction Grating, Intensity All the wavelengths are seen at m = 0 This is called the zeroth order maximum The first order maximum corresponds to m = 1 Note the sharpness of the principle maxima and the broad range of the dark areas

61. Diffraction Grating, Intensity, cont Characteristics of the intensity pattern The sharp peaks are in contrast to the broad, bright fringes characteristic of the two-slit interference pattern Because the principle maxima are so sharp, they are much brighter than two-slit interference patterns

62. Diffraction Grating Spectrometer The collimated beam is incident on the grating The diffracted light leaves the gratings and the telescope is used to view the image The wavelength can be determined by measuring the precise angles at which the images of the slit appear for the various orders

63. Grating Light Valve A grating light valve consists of a silicon microchip fitted with an array of parallel silicon nitride ribbons coated with a thin layer of aluminum When a voltage is applied between a ribbon and the electrode on the silicon substrate, an electric force pulls the ribbon down The array of ribbons acts as a diffraction grating

64. Diffraction of X-Rays by Crystals X-rays are electromagnetic waves of very short wavelength Max von Laue suggested that the regular array of atoms in a crystal could act as a three-dimensional diffraction grating for x-rays The spacing is on the order of 10 -10 m

65. Diffraction of X-Rays by Crystals, Set-Up A collimated beam of monochromatic x-rays is incident on a crystal The diffracted beams are very intense in certain directions This corresponds to constructive interference from waves reflected from layers of atoms in the crystal The diffracted beams form an array of spots known as a Laue pattern

66. Laue Pattern for Beryl

67. Laue Pattern for Rubisco

68. Holography Holography is the production of three- dimensional images of objects The laser met the requirement of coherent light needed for making images

69. Hologram of Circuit Board

70. Hologram Production Light from the laser is split into two parts by the half-silvered mirror at B One part of the beam reflects off the object and strikes an ordinary photographic film

71. Hologram Production, cont. The other half of the beam is diverged by lens L 2 It then reflects from mirrors M 1 and M 2 This beam then also strikes the film The two beams overlap to form a complicated interference pattern on the film

72. Hologram Production, final The interference pattern can be formed only if the phase relationship of the two waves is constant throughout the exposure of the film This is accomplished by illuminating the scene with light coming from a pinhole or coherent laser radiation The film records the intensity of the light as well as the phase difference between the scattered and reference beams The phase difference results in the three- dimensional perspective

73. Viewing A Hologram A hologram is best viewed by allowing coherent light to pass through the developed film as you look back along the direction from which the beam comes You see a virtual image, with light coming from it exactly in the way the light came from the original image

74. Uses of Holograms Applications in display Example – Credit Cards Called a rainbow hologram It is designed to be viewed in reflected white light Precision measurements Can store visual information