27.4 Lloyd’s Mirror

1. 27.4 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 Fig 27.6

2. 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

3. 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 Fig 27.7

4. 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 Fig 27.7

5. 27.5 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

6. Interference in Thin Films, 2 • Facts to keep in mind • An electromagnetic wave traveling from a medium of index of refraction n1 toward a medium of index of refraction n2 undergoes a 180° phase change on reflection when n2 > n1 • There is no phase change in the reflected wave if n2 < n1 • The wavelength of light λn in a medium with index of refraction n is λn = λ/n where λ is the wavelength of light in vacuum

7. 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 Fig 27.8

8. 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 …

9. 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

10. 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 180o phase change at both surfaces or at neither surface • Be sure to check both the path length and the phase change

11. Interference in Thin Film, Soap Bubble Example

19. 27.6 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

20. 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

21. 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 Fig 27.12

22. 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 Fig 27.13

23. 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

24. 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 Fig 27.14

25. Fraunhofer Diffraction Pattern – Photo • A bright fringe is seen along the axis (θ = 0) • Alternating bright and dark fringes are seen on each side Fig 27.14

26. 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

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

28. 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 θdark = mλ / a • m = ±1, ±2, ±3, …

29. 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 Fig 27.15

35. 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

36. 27.7 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 Fig 27.17

37. 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 Fig 27.17

38. 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

39. Resolution, Rayleigh’s Criterion, Equation • The angle of separation, qmin, is the angle subtended by the sources for which the images are just resolved • Since l << a in most situations, sin q is very small and sin q» q • 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 qmin

40. 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

41. 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 Fig 27.18

42. 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 Fig 27.18

43. 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 Fig 27.18

44. Resolution, Example Fig 27.19 • Pluto and its moon, Charon • Left – Earth based telescope is blurred • Right – Hubble Space Telescope clearly resolves the two objects

47. 27.8 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

48. 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

49. Diffraction Grating, cont • The condition for maxima is • d sin θbright = 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 Fig 27.20

50. 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 Fig 27.21