6Q20.10 - Abbe's Theory of Imaging - Optical Fourier Transform

Image on screen without lens
Far-field image with lens in place
Code Number:
6Q20.10
Demo Title:
Abbe's Theory of Imaging - Optical Fourier Transform

Disclaimer

Procedure:

Get the "Unit" slide from the "Electron Diffraction - Optical Simulation" demo (6D20.58).  The bottom left hand pattern of small characters is the one to use.  This will give a crystal diffraction pattern when a laser beam is pointed though the slide.  Place an 18 mm lens from the Michelson Interferometer (6D40.10) demo in front of the slide and observe that the diffraction pattern is now transformed to the real image of the optical crystal showing the original "L" shaped characters.  Basically you see the diffraction pattern consisting of the body factor convolved with the form factor.  When you insert the lens you see its Fourier transform, the real image of the Optical Crystal.

This setup is that described in "Optics", 4th Ed. by Eugene Hecht, p. 524, figure 11.5 in "The Lens as a Fourier Transformer" section.  It is essentially the same as the above setup.  The difference is that the transparency slide is placed at the focal length of the lens on one side, and the screen is placed at the focal length of the lens on the other side.  The camera looks at the image on the screen without and with the lens in place.  In this case the image on the screen with the lens in place looks like the image without the lens, only smaller.  The bottom left hand pattern of the "VSEPR" slide from the 6D20.58 "Electron Diffraction - Optical Simulation" demo is used in this setup.

References:
  • Jack Higbie, "Abbe's Sine Theorem from a Thermodynamic and Fourier Transform Argument", AJP, Vol. 49, #8, Aug. 1981, p. 788.
  • Bernard M. Jaffe, "Geometrical Optics Derivation of the Fourier Transform Property of a Lens", AJP, Vol. 48, #2, Feb. 1980, p. 157.
  • D. G. Sargood, "On Abbe's Theory of Imaging: A Simple Lecture Room Demonstration", AJP, Vol. 46, #2, Feb. 1978, p. 185.
  • Dieter B. Ast, "Optical Simulation of the Origin of Contrast in the Electron Microscope", AJP, Vol. 39, #10, Oct. 1971, p. 1164.
  • Anthony Gerrard, "An Elementary Theoretical Approach to the Abbe Theory of Optical Image Formation", AJP, Vol. 31, #9, Sept. 1963, p. 723.

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