Wave Optics

1
Chapter 24

• Wave Optics

2
Diffraction and Polarization Sections 6 9
3
Huygens Principle

• Christian Huygens (1629 1695) assumed that
  light is a form of wave motion rather than a
  stream of particles
• Huygens Principle is a geometric construction
  for determining the position of a new wave at
  some point based on the knowledge of the wave
  front that preceded it
• All points on a given wave front are taken as
  point sources for the production of spherical
  secondary waves, called wavelets, which propagate
  in the forward direction with speeds
  characteristic of waves in that medium
• After some time has elapsed, the new position of
  the wave front is the surface tangent to the
  wavelets

4
Huygens Construction for a Plane Wave

• At t 0, the wave front is indicated by the
  plane AA
• The points are representative sources for the
  wavelets
• After the wavelets have moved a distance c?t, a
  new plane BB can be drawn tangent to the
  wavefronts

5
Huygens Construction for a Spherical Wave

• The inner arc represents part of the spherical
  wave
• The points are representative points where
  wavelets are propagated
• The new wavefront is tangent at each point to the
  wavelet

6
Diffraction

• Huygens principle requires that the waves spread
  out after they pass through narrow slits
• This spreading out of light from its initial line
  of travel is called diffraction
• In general, diffraction occurs when waves pass
  through small openings, around obstacles or by
  sharp edges

7
Single-Slit Diffraction

• A single slit placed between a distant light
  source and a screen produces a diffraction
  pattern
• It will have a broad, intense central band
  central maximum
• The central band will be flanked by a series of
  narrower, less intense secondary bands
  secondary maxima
• The central band will also be flanked by a series
  of dark bands minima
• The results of the single slit cannot be
  explained by geometric optics
• Geometric optics would say that light rays
  traveling in straight lines should cast a sharp
  image of the slit on the screen

8
Single-Slit Diffraction

• Fraunhofer Diffraction occurs when the rays leave
  the diffracting object in parallel directions
• Screen very far from the slit
• Converging lens (shown)
• A bright fringe is seen along the axis (? 0)
  with alternating bright and dark fringes on each
  side

9
Single-Slit Diffraction

• According to Huygens principle, each portion of
  the slit acts as a source of waves
• The light from one portion of the slit can
  interfere with light from another portion
• 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 d (a/2) sin ?
• Similarly, wave 3 travels farther than wave 5 by
  an amount equal to the path difference d (a/2)
  sin ?

10
Single-Slit Diffraction

• If the path difference d is exactly a half
  wavelength, the two waves cancel each other and
  destructive interference results
• d ½ ? (a/2) sin ? - sin ? ? / a
• In general, destructive interference occurs for a
  single slit of width a when
• sin ?dark m? / a m ?1, ?2, ?3,

11
Single-Slit Diffraction

• A broad central bright fringe is flanked by much
  weaker bright fringes alternating with dark
  fringes
• The points of constructive interference lie
  approximately halfway between the dark fringes

• ym L tan ?dark , where sin ?dark m? / a

Active Figure Fraunhofer Diffraction Pattern for
a Single Slit
12
Diffraction Grating

• The diffracting grating consists of many equally
  spaced parallel slits of width d
• 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

13
Diffraction Grating, 2

• 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

14
Diffraction Grating, 3

• All the wavelengths are focused 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 area
• This is in contrast to the broad, bright fringes
  characteristic of the two-slit interference
  pattern

Active Figure The Diffraction Grating
15
Diffraction Grating Spectrometer

• The emission spectrum for hydrogen contains four
  visible wavelengths
• All wavelengths are focused at m 0
• For higher orders, each wavelength deviates
  through a specific angle
• Each order contains the four wavelengths

Active Figure The Diffraction Grating
Spectrometer
16
Polarization of Light Waves

• Each atom produces a wave with its own
  orientation of
• All directions of the electric field vector are
  equally possible and lie in a plane perpendicular
  to the direction of propagation
• This is an unpolarized wave

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Polarization of Light, cont

• A wave is said to be linearly polarized if the
  resultant electric field vibrates in the same
  direction at all times at a particular point
• Polarization can be obtained from an unpolarized
  beam by
• Selective absorption
• Reflection
• Scattering

18
Polarization by Selective Absorption

• E. H. Land discovered a material that polarizes
  light through selective absorption
• He called the material Polaroid
• The molecules readily absorb light whose electric
  field vector is parallel to their lengths and
  transmit light whose electric field vector is
  perpendicular to their lengths

19
Selective Absorption, cont

• The most common technique for polarizing light
• Uses a material that transmits waves whose
  electric field vectors in the plane are parallel
  to a certain direction and absorbs waves whose
  electric field vectors are perpendicular to that
  direction

20
Selective Absorption, final

• The intensity of the polarized beam transmitted
  through the second polarizing sheet (the
  analyzer) varies as
• I Io cos2 ?
• Io is the intensity of the polarized wave
  incident on the analyzer
• This is known as Malus Law and applies to any
  two polarizing materials whose transmission axes
  are at an angle of ? to each other

Active Figure The Linear Polarizer
21
Polarization by Reflection

• When an unpolarized light beam is reflected from
  a surface, the reflected light is
• Completely polarized
• Partially polarized
• Unpolarized
• It depends on the angle of incidence
• If the angle is 0 or 90, the reflected beam is
  unpolarized
• For angles between this, there is some degree of
  polarization
• For one particular angle, the beam is completely
  polarized

22
Polarization by Reflection, cont

• The angle of incidence for which the reflected
  beam is completely polarized is called the
  polarizing angle, ?p
• Brewsters Law relates the polarizing angle to
  the index of refraction for the material
• ?p may also be called Brewsters Angle

23
Polarization by Scattering

• When light is incident on a system of particles,
  the electrons in the medium can absorb and
  reradiate part of the light
• This process is called scattering
• An example of scattering is the sunlight reaching
  an observer on the earth becoming polarized

24
Polarization of Sunlight by Scattering

• Unpolarized sunlight is incident on an air
  molecule
• The horizontal part of the electric field vector
  in the incident wave causes the charges to
  vibrate horizontally
• The vertical part of the vector simultaneously
  causes the charges to vibrate vertically
• Horizontally polarized waves are emitted downward
  toward the observer
• Vertically polarized waves are emitted parallel
  to the Earth