VII3 Introduction into Wave Optics

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VII3 Introduction into Wave Optics
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Main Topics

• Huygens Principle and Coherence.
• Interference
• Double Slit
• Thin Film
• Diffraction
• Single Slit
• Gratings
• X-Rays, Bragg Equation.
• Wave Limits of Geometrical Optics.

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Huygens Principle I

• Up to now, we have have treated situations, where
  many of wave properties could be neglected. In
  our rays model, we actually needed only their
  straight propagation.
• Now, we shall concentrate to typically wave
  properties of light which are generally valid for
  all electromagnetic (and other) waves.

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Huygens Principle II

• The basis for studying wave effects is Huygens
  (Christian 1629-1695 Dutch) principle of wave
  propagation. It states
• Every point reached by a wave can be
  considered as a new source of tiny wavelets that
  spread out (in the forward direction) at the
  speed of the wave itself. The new wave front is
  the envelope of all the wavelets that is the
  tangent to all of them.

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Huygens Principle III

• If light is traveling through homogeneous
  isotropic media without obstacles Huygens
  principle gives us the same results as
  geometrical optics including effects as
  reflection and refraction.
• However, when there is e.g. an obstacle then wave
  fronts will be not only distorted but new effects
  will appear. There will be e.g. bright or light
  or colored regions.

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Coherence I

• Typical wave properties are based on the
  principle of superposition. If several waves meet
  in one spot their common effect is the sum of all
  of them. But since waves are periodic, extremes
  may happen e.g. they are in phase and they will
  constructively interfere or they may be out of
  phase and they will interfere destructively.

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Coherence II

• Since the frequency of light is very high this
  adding of waves may have some stable result only
  if the interfering waves are coherent i.e. have
  constant phase difference. In the case of e.g.
  radio waves we can, in principle build two same
  oscillators and synchronize them. But there are
  no oscillator for visible light. It can be
  generated only by transitions in atoms.

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Coherence III

• So ideally coherent light waves should stem from
  the same transition of the same atom.
• But also a partial coherence exists under much
  less strict conditions. Diffraction can be for
  instance obtained from a Sun light when it passes
  through a very small aperture.

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Double-Slit Interference I

• This experiment was the first convincing evidence
  of wave properties of light done in 1801 by
  Englishman Thomas Young (1773-1829).
• If a plane monochromatic light wave passes
  through two thin, closely spaced slits. The
  picture on a screen behind are not two bright
  lines but rather a series of them.

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Double-Slit Interference II

• According to the Huygens principle the slits are
  sources of new wavelets but now in every point of
  screen only two of these wavelets add, instead of
  infinity, what would be the case without slits.
• Lets suppose that the distance of the slits is
  negligible to that of the screen so two rays
  entering one point of it are almost parallel.

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Double-Slit Interference III

• If two waves leave the slits under some angle ?
  their path difference is
• ?d d sin?
• Clearly if ?d is an integral multiple of the
  wavelength the waves constructively interfere.
  The exact condition for this is
• d sin? m?
• m 0, 1 order of the interference fringe.

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Double-Slit Interference IV

• If, however, ?d is odd multiple of ?/2 the waves
  will be completely out of phase and they will
  interfere destructively. The exact condition for
  this is
• d sin? (2m1)?/2 (m1/2)?
• Again m 0, 1

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Double-Slit Interference V

• We have found the positions of the maxima and
  minima but there are also apparent changes in
  intensities which we have to explain.
• The treatment of these more subtle details is
  similar to that we used in AC circuits. We can
  employ the mathematics of phasors.

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Double-Slit Interference VI

• The intensity of light is proportional to the
  square of their electric field I E2 so we
  shall find the total electric field produced by
  both waves at some angle ?, which is a sum of
  fields
• E? E1 E2
• The fields will be phase shifted
• E1 E10sin?t and E2 E20sin(?t?)

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Double-Slit Interference VII

• The phase shift ? can be easily related to the
  path difference. From
• ?/2? d sin?/? ?
• ? 2?/? d sin?
• If we expect that our point on the screen is
  equally illuminated by both slits we find
• E? 2E0 cos(?/2)sin(?t ?/2)

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Double-Slit Interference VIII

• The same result can be found from the phasor
  diagram. The phase shift of E? is clearly ?/2.
• We omit the fast changing term and relate the
  intensity to the one in the middle, where both
  waves are in phase
• I?/I0 E2?0 /(2E0)2 cos2(?/2)

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Double-Slit Interference IX

• Finally we substitute for the phase difference ?
• I? I0cos2(?/2) I0cos2(?dsin?/?)
• Again the same conditions for the positions of
  the maxima and minima are present in this formula
  but we have also obtained the information on
  intensities.

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Interference in Thin Films I

• Two waves reflected on the upper and lower
  surfaces of a thin film can interfere either
  constructively if the path difference in the film
  is equal to integer number of wavelengths or
  destructively if it is an odd number of half
  wavelengths.
• We have to consider the wavelength in the
  particular material of the film.

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Interference in Thin Films II

• It can be shown that the waves have the same
  frequency in all materials
• f c/? v/?n ? c/v n ?/?n ? ?n ?/n
• If we use white light the conditions for maximum
  at a certain angle will be valid always for some
  color color interference.

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Interference in Thin Films III

• Experiments show important property of
  reflection If a wave reflects on a surface with
  optically denser media it changes its phase by ?.
  If the second medium is less optically dense
  there is no phase change.

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Interference in Thin Films IV

• An important application of thin-film
  interference is a non-reflective coating of
  optical elements.
• In the case of destructive interference more
  light gets through 99.
• A single layer works well for one wavelength,
  usually 550 nm.
• Actually whole field of layer reflective optics
  exists.

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Diffraction I

• Wave theory predicts that waves can be diffracted
  around edges of obstacles and interfere in the
  shadow behind them.
• Only after diffraction of light was observed the
  wave nature of it was fully accepted.
• The main ideas are again based on the Huygens
  principle.

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Diffraction II

• Lets consider a diffraction pattern produced by
  a single narrow slit of the width a.
• Every point in the slit is a source of forward
  wavelets which add on some screen behind.
• Lets find conditions for constructive and
  destructive interferences

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Diffraction III

• The condition for the first minimum is
• sin ? ?/a
• A wave from a point in the middle of the slit has
  a path difference of ?/2 from the point on the
  lower edge. These waves are out of phase and
  thereby cancel themselves. Similarly, if we
  proceed up, all waves will cancel in-pairs so we
  get a minimum.

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Diffraction IV

• The condition for the first maximum is
• sin ? 3/2 ?/a
• If we consider points in the two adjacent thirds
  of the slit, their waves will also cancel
  in-pairs but the waves from points in the last
  third will not, so we have a maximum intensity.
• Higher orders can be found similarly.

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Diffraction V

• The conditions are opposite from those for
  two-slit interference.
• Calculation of intensities can be again performed
  using phasors.
• We can divide the slit to equivalent strips ?y
  and find the phase difference of the waves from
  adjacent strips
• ?? 2?/? ?ysin? .

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Diffraction VI

• We are interested what will be the resulting
  phasor of total field built from these small
  phasors.
• In the case of minima the phasors complete a
  whole circle so the total is zero.

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Diffraction Grating I

• Its in principle multiple parallel slits. At
  present gratings can be made very precisely with
  very high densities of slits of the order of 104
  lines per centimeter. Gratings can be made both
  for transmission and reflection.
• The condition for principal maxima is the same as
  with the double slits
• sin? m?/d here d is the adjacent spacing

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Diffraction Grating II

• The main difference between the double-slit and
  multi-slit pattern is that the maxima in the
  latter case are much sharper and narrower.
• For maximum we need that waves even from far
  slits are exactly in-phase. If there is even a
  slight difference waves will cancel in pairs with
  some those from distant slits.

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Diffraction Grating III

• Gratings can be used to decompose light.
  Spectrometers using gratings are actually better
  than those using dispersive elements. The
  resolution of gratings is higher and the response
  is linear.
• This has a great impact in spectroscopy.

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X-Ray Diffraction I

• If we use X-rays which are EMW with the
  wavelengths of the order of 10-10 m then actually
  crystal planes can serve as gratings with single
  atoms as single slits.
• The refractive index for these wavelengths is
  almost 1 so the condition for maxima can be
  described by a simple Braggs equation
• 2dsin? m?

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X-Ray Diffraction II

• Beside positions of maxima, there is also much
  information in intensities which are interpreted
  by more complicated dynamical theory.
• Methods based on X-ray diffraction are important
  for characterization of structures of materials.
  Different methods exist for monocrystals, powders
  or solutions.

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Polarization I

• In unpolarized light the electric vectors have
  random position in the plane perpendicular to the
  propagation of the wave.
• It is however possible to polarize light i.e. to
  select electric vectors only in a certain plane.
  This is done by polaroids but polarization is
  produced even by reflection.

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Polarization II

• If we cross two polaroids, ideally, no light gets
  through.
• Some materials with assymetric molecules are
  capable to turn the polarization plane.
• In polarographs we can measure e.g. concentration
  of these.

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Scattering I

• Actually any atom interacts with light. Its
  electrons become roughly sources of new wavelets.
  We can compare it to Huygens principle which is
  valid even in vacuum.
• Since atoms are different there will be some
  interaction due to superposition of waves even
  for less ordered structures. But it will be seen
  only in the vicinity of the 0-order maximum.

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Scattering II

• Scattering again contains important structural
  information.
• In atmosphere scattering behaves as 1/?4.
• The sky is blue since blue light scatters most
• The sunset is red since blue waves are scattered
  and just the red ones remain.

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Wave Limits to Geometrical Optics I

• An image done by e.g. a lens, is in fact a
  superposition of diffraction patterns. We would
  find this if we display just little point. Its
  image is not a point but rather diffraction
  circles.
• This matters only when the magnification is high.

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Wave Limits to Geometrical Optics II

• A resolution of some optical device is roughly a
  distance of two points which we are still able to
  distinguish.
• If we think about the diffraction patterns the
  point can be distinguished when the main maximum
  of one falls into the first minimum of the second.

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Homework

• No more homework from physics!

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Things to read and learn

• Chapters 35, 36
• Try to understand all the details of the scalar
  and vector product of two vectors!
• Try to understand the physical background and
  ideas. Physics is not just inserting numbers into
  formulas!

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Maxwells Equations I

• .