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More on Basic Fourier Optics

It is helpful to students who lose traction on mathematical
treatments to consider optical processes that offer much the
same effect.

1D Fourier Transform.
The idea of analysing continuous waveforms like music,
into amplitude contributions at different frequencies,
is familiar to people who use graphic equalizers in their
car stereo systems.
The intensity versus time graph is transformed to an
intensity versus frequency graph.

It is a short step to considering the information in an image -
it might be a 35mm transparency - for its spatial frequencies,
by converting the light intensity versus X and Y co ordinates
to a light intensity versus spatial frequency transform
in two axes.

2D Fourier Transform.

The optical method of generating a 2D Fourier transform of the
spatial information in a slide is ridiculously easy.
It goes like this:

On an optical bench set out a laser of a few milliwatts and a
beam expander in order to illuminate a 35 mm slide.
Set out a converging lens of convenient focal length, say
15cm, at one focal length behind the slide.
Then one focal length behind the lens, set out a white card.

The card displays a fourier transform of the spatial frequencies
present in the slide.
Consider a small hole as the image to analyze.
Its FT is a large circle of light, with two visible haloes of
reduced intensity surrounding it.
If we consider a slice through this original image, it is a square
shouldered pulse in shape, so it is no surprise that examination
of the FT at the transform plane shows a zero frequency component
at the center, surrounded by third and fifth spatial harmonics
in all radial directions. These are the haloes that I mentioned.

If we place a second similar converging lens at a focal length behind
this transform plane, we see the original image once more at a focal length
behind this second lens - the FT is symmetrical.

Optical Signal Processing

Placing a camera less lens at the FT plane enables us to take
a record of the FT. This in turn can serve as an optical filter.
If, for example, a small round aperture is placed at the transform
plane, the second lens transforms to an image with only low
spatial frequencies present.

A starting image of a silhouette of mickey mouse with a thin
belt, if passed through such a 'low pass' filter, transforms
back to a rounder image, whose thin belt has disappeared,
for example.

By contrast, a high pass optical filter (a central opaque dot)
placed at the (second) fourier transform plane results in a
mickey mouse outline only at the final focal plane.

You can see, I hope, how accessible this experimental topic is.

An atlas of optical transforms, like
Harburn, Taylor & Welberry, Bell Pub 75 ISBN 07135 1760 3
shows many more.

This was an impetus of a UNESCO pilot project of the
Commission on Crystallographic Teaching of IUC, where
X ray images were a pressing topic for visual transformation.

brian whatcott <> Altus OK