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Copyright © 2009 Pearson Education, Inc. Chapter 35 Diffraction and Polarization.

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1 Copyright © 2009 Pearson Education, Inc. Chapter 35 Diffraction and Polarization

2 Copyright © 2009 Pearson Education, Inc. Diffraction by a Single Slit or Disk Intensity in Single-Slit Diffraction Pattern Diffraction in the Double-Slit Experiment Limits of Resolution; Circular Apertures Resolution of Telescopes and Microscopes; the λ Limit Resolution of the Human Eye and Useful Magnification Diffraction Grating Units of Chapter 35

3 Copyright © 2009 Pearson Education, Inc. The Spectrometer and Spectroscopy Peak Widths and Resolving Power for a Diffraction Grating X-Rays and X-Ray Diffraction Polarization Liquid Crystal Displays (LCD) Scattering of Light by the Atmosphere Units of Chapter 35

4 Copyright © 2009 Pearson Education, Inc. If light is a wave, it will diffract around a single slit or obstacle. Diffraction by a Single Slit or Disk

5 Copyright © 2009 Pearson Education, Inc. The resulting pattern of light and dark stripes is called a diffraction pattern. Diffraction by a Single Slit or Disk

6 Copyright © 2009 Pearson Education, Inc. Pattern arises because different points along slit create wavelets that interfere with each other just as a double slit would. Diffraction by a Single Slit or Disk

7 Copyright © 2009 Pearson Education, Inc. The minima of the single-slit diffraction pattern occur when Diffraction by a Single Slit or Disk

8 Copyright © 2009 Pearson Education, Inc. What??? Wait a minute! The minima of the single-slit diffraction pattern occur when The maxima of the double-slit interference pattern occured when d

9 Copyright © 2009 Pearson Education, Inc. Single-slit diffraction vs. Double-slit Interference D

10 Copyright © 2009 Pearson Education, Inc. Single-slit diffraction vs. Double-slit Interference Single-slit diffraction Slit Diameter = D (or ‘s’) D is small! D sin  = m  for minima since sin  minima at large  D

11 Copyright © 2009 Pearson Education, Inc. Single-slit diffraction vs. Double-slit Interference Double-slit interference Slit SPACING = d d > D typically d sin  = m  for maxima sin  m  d bright fringes at small 

12 Copyright © 2009 Pearson Education, Inc. Diffraction & Interference

13 Copyright © 2009 Pearson Education, Inc. Example 35-1: Single-slit diffraction maximum Light of wavelength 750 nm passes through a slit 1.0 x mm wide. How wide is the central maximum (a) in degrees, and (b) in centimeters, on a screen 20 cm away?

14 Copyright © 2009 Pearson Education, Inc. Example 35-1: Single-slit diffraction maximum Sound of wavelength 30 cm passes through a slit 1.0 m wide. How wide is the central maximum (a) in degrees, and (b) in meters, to an ear 10 m away?

15 Copyright © 2009 Pearson Education, Inc. Conceptual Example 35-2: Diffraction spreads. Light shines through a rectangular hole that is narrower in the vertical direction than the horizontal.

16 Copyright © 2009 Pearson Education, Inc. Conceptual Example 35-2: Diffraction spreads. (a) Would you expect the diffraction pattern to be more spread out in the vertical direction or in the horizontal direction? (b) Should a rectangular loudspeaker horn at a stadium be high and narrow, or wide and flat?

17 Copyright © 2009 Pearson Education, Inc. Reducing Diffraction: Rounded Edges

18 Copyright © 2009 Pearson Education, Inc Intensity in Single-Slit Diffraction Pattern Light passing through a single slit can be divided into a series of narrower strips; each contributes the same amplitude to the total intensity on the screen, but the phases differ due to the differing path lengths:.

19 Copyright © 2009 Pearson Education, Inc Diffraction in the Double-Slit Experiment The diffraction factor (depends on β ) appears as an “envelope” modifying the more rapidly varying interference factor (depends on δ ).

20 Copyright © 2009 Pearson Education, Inc Diffraction in the Double-Slit Experiment The double-slit experiment also exhibits diffraction effects, as the slits have a finite width. This means the amplitude at an angle θ will be modified by the same factor as in the single-slit experiment: The intensity is, as usual, proportional to the square of the field.

21 Copyright © 2009 Pearson Education, Inc Diffraction in the Double-Slit Experiment Example 35-4: Diffraction plus interference. Show why the central diffraction peak shown, plotted for the case where d = 6D = 60λ, contains 11 interference fringes.

22 Copyright © 2009 Pearson Education, Inc. Resolution is distance at which a lens can barely distinguish two separate objects. Resolution is limited by aberrations and by diffraction. Aberrations can be minimized Diffraction is unavoidable Due to size of lens compared to wavelength of the light Limits of Resolution; Circular Apertures

23 Copyright © 2009 Pearson Education, Inc. For a circular aperture of diameter D, the central maximum has an angular width: 35-4 Limits of Resolution; Circular Apertures

24 Copyright © 2009 Pearson Education, Inc Limits of Resolution; Circular Apertures

25 Copyright © 2009 Pearson Education, Inc Limits of Resolution; Circular Apertures

26 Copyright © 2009 Pearson Education, Inc Limits of Resolution; Circular Apertures

27 Copyright © 2009 Pearson Education, Inc Limits of Resolution; Circular Apertures

28 Copyright © 2009 Pearson Education, Inc. The Rayleigh criterion states that two images are just resolvable when the center of one peak is over the first minimum of the other Limits of Resolution; Circular Apertures

29 Copyright © 2009 Pearson Education, Inc. Example 35-5: Hubble Space Telescope. The Hubble Space Telescope (HST): reflecting telescope placed in orbit above Earth’s atmosphere. Resolution not limited by turbulence in atmosphere. Its objective diameter is 2.4 m. For visible light, λ = 550 nm, estimate improvement in resolution the Hubble offers over Earth-bound telescopes, which are limited in resolution by movement of the Earth’s atmosphere to about half an arc second.

30 Copyright © 2009 Pearson Education, Inc. Example 35-5: Hubble Space Telescope. Its objective diameter is 2.4 m. For visible light, λ = 550 nm, estimate improvement in resolution the Hubble offers over Earth-bound telescopes, which are limited in resolution by movement of the Earth’s atmosphere to about half an arc second. (Each degree is divided into 60 minutes each containing 60 seconds, so 1° = 3600 arc seconds.)

31 Copyright © 2009 Pearson Education, Inc. Example 35-6: Eye resolution. You are in an airplane at an altitude of 10,000 m. If you look down at the ground, estimate the minimum separation s between objects that you could distinguish. Could you count cars in a parking lot? Consider only diffraction, and assume your pupil is about 3.0 mm in diameter and λ = 550 nm.

32 Copyright © 2009 Pearson Education, Inc. For telescopes, the resolution limit is as we have defined it: 35-5 Resolution of Telescopes and Microscopes; the λ Limit For microscopes, assuming the object is at the focal point, the resolving power is given by

33 Copyright © 2009 Pearson Education, Inc. Example 35-7: Telescope resolution (radio wave vs. visible light). What is the theoretical minimum angular separation of two stars just resolved by (a) 200-inch telescope on Palomar Mountain; & (b) Arecibo radio telescope, diameter = 300 m & radius of curvature is also 300 m.

34 Copyright © 2009 Pearson Education, Inc. Example 35-7: Telescope resolution (radio wave vs. visible light). What is the theoretical minimum angular separation of two stars just resolved by (a) 200-inch telescope on Palomar Mountain (assume λ = 550 nm for the visible-light) (b) Arecibo radio telescope, diameter = 300 m & radius of curvature is also 300 m. (assume λ = 4 cm for the radio telescope)

35 Copyright © 2009 Pearson Education, Inc. Typically, the focal length of a microscope lens is half its diameter, which shows that it is not possible to resolve details smaller than the wavelength being used: 35-5 Resolution of Telescopes and Microscopes; the λ Limit

36 Copyright © 2009 Pearson Education, Inc. The human eye can resolve objects that are about 1 cm apart at a distance of 20 m, or 0.1 mm apart at the near point. This limits the useful magnification of a light microscope to about 500x–1000x Resolution of the Human Eye and Useful Magnification

37 Copyright © 2009 Pearson Education, Inc. A diffraction grating consists of a large number of equally spaced narrow slits or lines. A transmission grating has slits, while a reflection grating has lines that reflect light. The more lines or slits there are, the narrower the peaks Diffraction Grating

38 Copyright © 2009 Pearson Education, Inc. The maxima of the diffraction pattern are defined by 35-7 Diffraction Grating

39 Copyright © 2009 Pearson Education, Inc Diffraction Grating Determine the angular positions of the first- and second-order maxima for light of wavelength 400 nm and 700 nm incident on a grating containing 10,000 lines/cm.

40 Copyright © 2009 Pearson Education, Inc. Example 35-9: Spectra overlap. White light containing wavelengths from 400 nm to 750 nm strikes a grating containing 4000 lines/cm. Show that the blue at λ = 450 nm of the third-order spectrum overlaps the red at 700 nm of the second order.

41 Copyright © 2009 Pearson Education, Inc. Conceptual Example 35-10: Compact disk. When you look at the surface of a music CD, you see the colors of a rainbow. (a) Estimate the distance between the curved lines (to be read by the laser). (b) Estimate the distance between lines, noting that a CD contains at most 80 min of music, that it rotates at speeds from 200 to 500 rev/min, and that 2/3 of its 6-cm radius contains the lines.

42 Copyright © 2009 Pearson Education, Inc. A spectrometer makes accurate measurements of wavelengths using a diffraction grating or prism The Spectrometer and Spectroscopy

43 Copyright © 2009 Pearson Education, Inc. The wavelength can be determined to high accuracy by measuring the angle at which the light is diffracted: 35-8 The Spectrometer and Spectroscopy

44 Copyright © 2009 Pearson Education, Inc The Spectrometer and Spectroscopy Atoms and molecules can be identified when they are in a thin gas through their characteristic emission lines.

45 Copyright © 2009 Pearson Education, Inc. Example 35-11: Hydrogen spectrum. Light emitted by hot hydrogen gas is observed with a spectroscope using a diffraction grating having 1.00 x 10 4 lines/cm. The spectral lines nearest to the center (0°) are a violet line at 24.2°, a blue line at 25.7°, a blue-green line at 29.1°, and a red line at 41.0° from the center. What are the wavelengths of these spectral lines of hydrogen?

46 Copyright © 2009 Pearson Education, Inc. The wavelengths of X-rays are very short. Diffraction experiments are impossible to do with conventional diffraction gratings. Crystals have spacing between their layers that is ideal for diffracting X-rays X-Rays and X-Ray Diffraction

47 Copyright © 2009 Pearson Education, Inc. X-ray diffraction is now used to study the internal structure of crystals; this is how the helical structure of DNA was determined X-Rays and X-Ray Diffraction

48 Copyright © 2009 Pearson Education, Inc. Light is polarized when its electric fields oscillate in a single plane, rather than in any direction perpendicular to the direction of propagation Polarization

49 Copyright © 2009 Pearson Education, Inc. Polarized light will not be transmitted through a polarized film whose axis is perpendicular to the polarization direction Polarization

50 Copyright © 2009 Pearson Education, Inc. When light passes through a polarizer, only the component parallel to the polarization axis is transmitted. If the incoming light is plane- polarized, the outgoing intensity is: Polarization

51 Copyright © 2009 Pearson Education, Inc. This means that if initially unpolarized light passes through crossed polarizers, no light will get through the second one Polarization

52 Copyright © 2009 Pearson Education, Inc Polarization Example 35-13: Two Polaroids at 60°. Unpolarized light passes through two Polaroids; the axis of one is vertical and that of the other is at 60° to the vertical. Describe the orientation and intensity of the transmitted light.

53 Copyright © 2009 Pearson Education, Inc Polarization Conceptual Example 35-14: Three Polaroids. When unpolarized light falls on two crossed Polaroids (axes at 90°), no light passes through. What happens if a third Polaroid, with axis at 45° to each of the other two, is placed between them?

54 Copyright © 2009 Pearson Education, Inc. Light is also partially polarized after reflecting from a nonmetallic surface. At a special angle, called the polarizing angle or Brewster’s angle, the polarization is 100%: Polarization.

55 Copyright © 2009 Pearson Education, Inc Polarization Example 35-15: Polarizing angle. (a) At what incident angle is sunlight reflected from a lake plane-polarized? (b) What is the refraction angle?

56 Copyright © 2009 Pearson Education, Inc. Liquid crystals are unpolarized in the absence of an external voltage, and will easily transmit light. When an external voltage is applied, the crystals become polarized and no longer transmit; they appear dark. Liquid crystals can be found in many familiar applications, such as calculators and digital watches Liquid Crystal Displays (LCD)

57 Copyright © 2009 Pearson Education, Inc Liquid Crystal Displays (LCD) This particular type of liquid crystal, called a twisted crystal, shows how the crystal passes light when the voltage is off but not when it is on.

58 Copyright © 2009 Pearson Education, Inc. Skylight is partially polarized due to scattering from molecules in the air. The amount of polarization depends on the angle that your line of sight makes with the Sun Scattering of Light by the Atmosphere

59 Copyright © 2009 Pearson Education, Inc. Light bends around obstacles and openings in its path, yielding diffraction patterns. Light passing through a narrow slit will produce a central bright maximum of width Minima occur at Summary of Chapter 35

60 Copyright © 2009 Pearson Education, Inc. Summary of Chapter 35 Diffraction limits the resolution of images. Diffraction grating has many parallel slits or lines; peaks of constructive interference are given by Polarized light has its electric field vectors in a single plane.

61 Copyright © 2009 Pearson Education, Inc. The intensity of plane-polarized light is reduced after it passes through another polarizer: Light can also be polarized by reflection; it is completely polarized when the reflection angle is the polarization angle: Summary of Chapter 35


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