Lecture 3: Introduction to wave theory (III)

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Lecture 3 Introduction to wave theory (III)
Interference and coherence (YF 15.6, 35.1)

• Principle of superposition
• When two or more waves overlap, the resultant
  displacement at any point at any instant may be
  found by adding the instantaneous displacements
  that would be produced at the point by the
  individual waves if they were alone.

t
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• Coherence
• If two overlapping waves have the same frequency
  and have a definite constant phase between them
  then they are said to be coherent waves.
• In the previous example

• Coherent source
• The source S1 is emitting coherent waves in two
  dimensions outwards.
• All crests of the wave are said to be in phase.
• For example waves on water

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• Interference
• If two sources S1 and S2 emit coherent waves,
  the resultant pattern will be due to the
  interference of the overlapping waves according
  to the principle of superposition. The amplitude
  of the wave at an arbitrary point will be the sum
  of the two amplitudes.
• Point a is equidistant from S1 and S2 (4l
  difference) sum of amplitudes
• Point b is 7l from S1 and 9l from S2 waves in
  phase so sum of amplitudes constructive
  interference
• Point c is 10l from S1 and 7.5l from S2 both
  amplitudes cancel out
• destructive interference

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• Constructive interference
• Both waves are in phase so the two waves add up.

• Destructive interference
• Both waves are exactly out of phase so the two
  waves cancel.

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• Constructive interference patterns
• The locus of all points where one obtains
  constructive interference (ie. when the crests of
  the waves align) are called the antinodal curves.
• Nodal curves are the pattern of destructive
  interference.

antinodal curves
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Interference and diffraction of light
Objectives
i) to recognise the observed phenomena of
interference and diffraction ii) to understand
Huygen's principle and its application to
both geometrical and physical optics iii) to
understand interference, in particular Young's
double slit experiment and Lloyd's mirror
experiment, and in thin films including Newton's
rings experiment iv) to appreciate the
application of the theory of interference in
such areas as non-reflective and highly
reflective coatings v) to understand the
limitation to resolving power of lenses due
to diffraction vi) to solve simple problems
involving interference and diffraction phenomena.
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Interference and diffraction of light (I)
Physical Optics wave behaviour of light (YF
35.1-2 33.7)

• Wave nature of light
• In geometric optics light considered as
  straight line rays
• Light undergoes certain phenomena that cannot be
  explained simply by light travelling in straight
  lines
• Light not only reflects on mirrors but also
  refracts in glass, water and other media
• Interference phenomena of light are observable
  every day oil spots, soap bubbles show
  multi-coloured patterns

Thin film of oil illuminated by white light
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• Wave nature of light (cont.)
• Diffraction phenomena are also common for waves
  an example is that sound bends around corners,
  due to the wave behaviour on the edges of
  objects.
• Diffraction is also visible with light on edges
  of sharp objects (for example, photograph of
  razor)
• Physical optics is the study of light, taking
  into account its wave behaviour.
• Initiated by the Dutch scientist Christian
  Huygens in 1678 who proposed the wave theory of
  light, in opposition to Newton who believed light
  was made out of corpuscles (particles). We now
  know they were both right!

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Huygens Principle (YF 33.7)

• Wave character of light the Dutch scientist
  Christian Huygens believed in the wave character
  of light and he used this to explain reflection
  and refraction
• Huygens Principle
• Every point of a wavefront may be considered as
  the source of secondary wavelets that spread out
  in all directions with a speed equal to the speed
  of propagation of the wave.
• In the figure, the new wavefront BB is
  constructed by making the surface tangent to the
  secondary wavelets (envelope of the wavelet), a
  distance rvt from the initial wavefront AA.

• The success of Huygens principle is that it
  explains reflection and refraction.

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• Reflection by Huygens Principle

Triangles OPA and OQA are equal, therefore qa qr
The reflected angle is equal to the incident
angle
qa incident angle qr reflected angle
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• Refraction by Huygens Principle

Refractive index
qa incident angle qb refracted angle
Snells law
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• Wavelength in the medium
• The wavelength in the medium will depend on the
  refractive index.
• Frequency is fixed (colour of light) but speed
  of light in the medium changes depending on the
  medium.
• In medium a

Since then
Therefore, at an interface between two media
Example Light of wavelength 550 nm in air is
incident onto a glass plate of refractive index
n 1.52. What is the wavelength of light in the
plate? na1 and nb1.52