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,

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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