Chapter 36 Diffraction © 2016 Pearson Education Inc.

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! © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

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.

Diffraction from a single slit © 2016 Pearson Education Inc.

Diffraction from a single slit © 2016 Pearson Education Inc.

Diffraction © 2016 Pearson Education Inc.

Diffraction vs. Interference? Both use same principle! PATH LENGTH DIFFERENCES create PHASE differences in arriving light intensities But not just how many apertures! © 2016 Pearson Education Inc.

Diffraction vs. Interference? d (edge) a d (center) Path length difference d = [d(edge) – d(center)] Phase difference Df = d /2pl = kd © 2016 Pearson Education Inc.

Fresnel vs. Fraunhofer Diffraction d << ka2 means LARGE phase changes (Fresnel) d>> ka2 means SMALL phase changes (Fraunhofer) © 2016 Pearson Education Inc.

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! © 2016 Pearson Education Inc.

Fresnel diffraction by single slit © 2016 Pearson Education Inc.

Fresnel diffraction by a single slit © 2016 Pearson Education Inc.

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 35-3. Analysis of diffraction pattern formed by light passing through a narrow slit of width D.

Fraunhofer diffraction by a single slit © 2016 Pearson Education Inc.

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 © 2016 Pearson Education Inc.

Locating dark fringes Fraunhofer diffraction pattern from a single horizontal slit. Central bright fringe @ θ = 0, surrounded by series of dark fringes. Central bright fringe twice as wide as other bright fringes. © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

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! © 2016 Pearson Education Inc.

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). © 2016 Pearson Education Inc.

Diffraction by a Single Slit or Disk The minima of the single-slit diffraction pattern occur when a Figure 35-4. 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

Width of single-slit diffraction pattern Pattern depends on ratio of slit width a to the wavelength l. a a a a a a © 2016 Pearson Education Inc.

What??? Wait a minute! The minima of single-slit diffraction pattern occur when a The maxima of double-slit interference pattern occured when d

Diffraction vs. Interference? a = size of single slit d = distance BETWEEN SLITS © 2016 Pearson Education Inc.

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.

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.

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.

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.

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.

Combining Diffraction & Interference! See Hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/mulslid.html#c1 © 2016 Pearson Education Inc.

Combining Diffraction & Interference! © 2016 Pearson Education Inc.

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 are @ same positions; intensities modulated by single-slit pattern. Single-slit diffraction “envelopes” intensity function. © 2016 Pearson Education Inc.

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! © 2016 Pearson Education Inc.

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.”) © 2016 Pearson Education Inc.

Diffraction from Two slits of finite width © 2016 Pearson Education Inc.

Diffraction & Interference Figure 35-10. (a) Diffraction factor, (b) interference factor, and (c) the resultant intensity plotted as a function of θ for d = 6D = 60λ.

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. © 2016 Pearson Education Inc.

Interference pattern of several slits Eight slit pattern: Large maxima, (principal maxima) @ same positions as a two-slit pattern, but much narrower. © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

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! © 2016 Pearson Education Inc.

CDs vs. DVDs © 2016 Pearson Education Inc.

Multi-slit interference depends on l © 2016 Pearson Education Inc.

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: © 2016 Pearson Education Inc.

X-ray diffraction When x rays pass through a crystal, the crystal behaves like a diffraction grating, causing x-ray diffraction. © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

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: © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

Smaller light, better resolution © 2016 Pearson Education Inc.

Bigger light, worse resolution © 2016 Pearson Education Inc.

Bigger light, worse resolution © 2016 Pearson Education Inc.

Bigger telescope, better resolution Because of diffraction, large-diameter telescopes, such as the VLA radio telescope below, give sharper images than small ones. © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

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. © 2016 Pearson Education Inc.

An example of holography © 2016 Pearson Education Inc.