Re: Help relating 2-D Fourier Transforms to optics, diffraction, photography for math class project

It was a dark and stormy night in sci.optics, that 15-02-06, but I could still make out the words murrayatuptowngallery@xxxxxxxxx engraved on the willow's bark:

Here are a couple of the physical questions I am trying to get
comfortable with...maybe it's more accurate to say 'lay understanding'
of what seems to be abstract mathematical explanations.

I apologize for posting this after so many days.

It seems to me that you are looking for a sort of mental picture for this Fourier transforming affair. Please, bear in mind that every oversimplification carries a certain degree of incorrectness and what follows does not make exception (especially considering that I would need to draw some pictures and I am following the Winter games on tv right now : ) )

To get a *rough* picture of what's going on, try to see it in this way: consider a spherical wave originating from a point source somewhere in the first focal plane of a lens. It is completely identified by the position of the source, but has no preferential direction. The light passing through the lens will come out (approximately) as a plane wave with a well defined direction (correlated to a spatial frequency [1]) and no spatial localization. This sort of duality should remind you of the time-frequency duality of the 'usual' onedimensional Fourier transform used in signal processing.

Now, suppose the point sources in the first focal plane are generated by the light that is 'spherically' diffused from a slide illuminated by a uniform background (like a source at infinity): the weight of every point is related to how dark the slide is in that particular spatial position [2]. The light coming out of the lens will be a weighted superposition of the infinitely many plane waves due to the points that constitutes the slide in the first focal plane. This should remind you of an integral superposition in two dimensions.

To see how the FT comes out of this you have to come to terms with formulae, especially the Rayleigh Sommerfeld diffraction formula and its simplifications (which account for the 'approximately plane' above and much more).

If you already have Steward's book, I'd suggest you move to Saleh Teich's first (the chapter on Fourier Optics has many meaningful pictures), and then to Goodman's (aesthetically awful pictures but very nice formulae : ) ).

It is really hard trying to explain a concept like this in ascii, without pictures or formulae.

cheers,
Peltio

[1] You need a picture for seeing this: try have a look a Saleh Teich's "Fundamentals of Photonics" for this, or try to imagine the intersection of many parallel planes spaced by a fixed wavelength when they intersect the focal plane (or any other plane) at different angles. You will see that the traces of the planes (representing subsequent maxima or minima of the sinusoidal waveform describing the harmonic plane wave) will be spaced a distance whose minimum value is one wavelenght but will vary with the angle of incidence.

[2] An aperture can be see as a particular form of a 'slide': completely trasparent here and completely dark there.

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