Fourier Optics

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Lecture 5
Fourier Optics
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Class Test I Mark Distribution

• Mean 40
• Standard deviation 23

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Marks for Class Test I will be available from
your tutors from Wednesday.
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Class Test I

• and you were given on the front sheet

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Class Test I

• and you were given on the front sheet

You know from PC2 and the lecture notes handout
that FTtop-hat
and
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Class Test I
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Class Test I
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Common errors I

• Inability to multiply complex exponential
  expressions for sine and cosine.
• Inability to integrate exponential functions in
  Fourier analysis this is a major concern.
• Arbitrarily interchanging x, k, t, w, k0 and w0
  (and other symbols)
• Not realising that a series of sine functions is
  a Fourier series (particularly worrisome)
• Not being able to write down correct expressions
  for complex exponential form of sine and/or
  cosine.

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Common errors II

• Not being able to sketch simple filter response
  diagrams.
• Inability to write down correct form of Fourier
  integral even though it was an open book test.

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Recap.

• Diffraction and convolution double slit
  experiment

Outline of Lecture 5

• 2D Fourier transforms
• Diffraction gratings
• Fourier filtering

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Reciprocal space and spatial frequencies
Just as we can build up a complex waveform from a
variety of sinusoids of different amplitudes and
phases, so too can we generate an image from a
Fourier integral.
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2D Images and 2D Fourier Transforms
Consider an aperture
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f(x,y) in this case can be broken down into two
functions f(x) and f(y). Sketch those functions.
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2D Images and 2D Fourier Transforms
So, for a square aperture we have two sinc
functions, one along kx and one along ky
Figures taken from Optics, Hecht (Addison-Wesley,
2nd Ed. 1987)
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2D Images and 2D Fourier Transforms
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Which area of the diffraction pattern is
associated with low spatial frequencies? With
high spatial frequencies?
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2D Images and 2D Fourier Transforms
Aperture function (2 slits)
2 slit pattern
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What is the effect on the image if we only pass
the spatial frequencies within the circle shown?
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2D Images and 2D Fourier Transforms
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What is the effect on the image if we block the
spatial frequencies within the circle shown?
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Complex images Fourier transforming and spatial
filtering
Niamhs Fourier transform (modulus2)
Niamh
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Complex images Fourier transforming and spatial
filtering
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Complex images Fourier transforming and spatial
filtering
Optical computer
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The diffraction grating

• An (infinite) diffraction grating has a
  transmission function which
• looks like

• We saw earlier how the double slit transmission
  function could be represented as a convolution of
  two functions. The grating transmission function
  can be treated similarly.

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The transmission function above can be
represented as the convolution of two functions.
Sketch them.
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The diffraction grating

• The train of delta functions is known as a
  Dirac comb (or a Shah function).

whose Fourier transform is another Dirac comb
where
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The diffraction grating
has Fourier transform
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At what value of k is the first zero in G(k)
located?
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Sketch the Fourier transform (i.e. the
diffraction pattern) of the transmission function
for the infinite diffraction grating.
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The diffraction grating

• Now, what happens if we want to consider a real
  diffraction grating (i.e. one that is not
  infinite in extent)?

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The slits in the infinite grating above are
spaced by an amount L. Imagine that we want to
determine the Fourier transform of a grating
which is 50L wide. How do we convert the
transmission function for the infinite grating
into that for a real grating which is 50L wide?
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The diffraction grating