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Wave Optics

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Title: Wave Optics


1
Chapter 27
  • Wave Optics

2
Wave Optics
  • 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 Huygens Principle we know the waves spread
    out from the slits
  • This divergence of light from its initial line of
    travel is called diffraction

7
Youngs Double Slit Experiment, Schematic
  • Thomas Young first demonstrated interference in
    light waves from two sources in 1801
  • The narrow slits, S1 and S2 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
Youngs Double Slit Experiment Geometry
  • The path difference, d, is found from the tan
    triangle
  • d r2 r1 d sin ?
  • This assumes the paths are parallel
  • Not exactly true, but a very good approximation
    if L gtgt 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
  • d d sin ?bright m ?
  • 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
  • d d sin ?dark (m ½) ?
  • 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 ?
  • Approximation
  • ? is small and therefore the small angle
    approximation tan ? sin ? can be used
  • y L tan ? L sin ?
  • 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 Youngs Double Slit Experiment
  • Youngs 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, w
  • The waves have a constant phase difference, f
  • The phase difference, f, depends on the angle q

20
Intensity Distribution, Phase Relationships
  • The phase difference between the two waves at P
    depends on their path difference
  • d r2 r1 d sin q
  • A path difference of l corresponds to a phase
    difference of 2 p rad
  • A path difference of d is the same fraction of l
    as the phase difference f is of 2 p
  • This gives

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

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

23
Lloyds 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 Lloyds 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 n1 toward a medium of
    index of refraction n2 undergoes a 180 phase
    change on reflection when n2 gt n1
  • There is no phase change in the reflected wave if
    n2 lt n1
  • The wavelength of light ?n in a medium with
    index of refraction n is ?n ?/n where ? 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 ) ? 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 ? 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 180o 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 (? 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 Huygens 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 q
  • 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 ?
  • 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 ?dark m? / 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, Rayleighs 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
    Rayleighs criterion

51
Resolution, Rayleighs Criterion, Equation
  • The angle of separation, qmin, is the angle
    subtended by the sources for which the images are
    just resolved
  • Since l ltlt 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

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 Rayleighs 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 ?bright m ?
  • 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 L2
  • It then reflects from mirrors M1 and M2
  • 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
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