Wave propagation.

1
Lecture 10

• Wave propagation.
• Aims
• Fraunhofer diffraction (waves in the far
  field).
• Youngs double slits
• Three slits
• N slits and diffraction gratings
• A single broad slit
• General formula - Fourier transform.

2
Fraunhofer diffraction

• Diffraction.
• Propagation of partly obstructed waves.
• Apertures, obstructions etc...
• Diffraction régimes.
• In the immediate vicinity of the obstruction
• Large angles and no approximations
• Full solution required.
• Intermediate distances (near field)
• Small angles, spherical waves,
• Fresnel diffraction.
• Large distances (far field)
• Small angles, and plane waves,
• Fraunhofer diffraction.
• (More formal definitions will come in the Optics
  course)

3
Youngs slits

• Fraunhofer conditions
• For us this means an incident plane wave and
  observation at infinity.
• Two narrow apertures (2 point sources)
• Each slit is a source of secondary wavelets
• Full derivation (not in handout)
  is.Applying cos rule to top triangle
  gives

4
2-slit diffraction

• Similarly for bottom ray
• Resultant is a superposition of 2
  wavelets
• The term expi(kR-wt)will occur in
  allexpressions. We ignoreit - only relative
  phasesare important.Where s sinq.

5
cos-squared fringes

• We observe intensitycos-squared
  fringes.
• Spacing inversely proportional to separation of
  the slits.
• Amplitude-phase diagrams.

Spacing of maxima
Resultant
Slit 1
Slit 2
6
Three slits

• Three slits, spacing d.
• Primary maxima separated by l/d, as before.
• One secondary maximum.

7
N slits and diffraction gratings

• N slits, each separated by d.
• A geometric progression, which sums
  to
• Intensity in primary maxima a N2
• In the limit as N goes to infinity, primary
  maxima become d-functions. A diffraction grating.

Spacing, as before
N-2, secondary maxima
8
Single broad slit

• Slit of width t. Incident plane wave.
• Summation of discrete sourcesbecomes an integral.

9
Generalisation to any aperture

• Aperture function
• The amplitude distribution across an aperture can
  take any form a(y). This is the aperture
  function.
• The Fraunhofer diffraction pattern is
• putting ksK gives a Fourier integral
• The Fraunhofer diffraction pattern is the Fourier
  Transform of the aperture function.
• Diffraction from complicated apertures can often
  be simplified using the convolution theorem.
• Example 2-slits of finite widthConvolution of
  2d-functions with asingle broad slit.FT(fg)
  a FT(f).FT(g)

Cos fringes
sinc function