Solving James' Confusion: Abbe Theory, Spatial Filtering & Optical Computers

[JamesJames]

I am confused by this Abbe theory thing involvign the use of an optical computer.

A laser is focused onto a 10 μm screen with adjustable micrometer screws. A pinhole is provided to avoid any unnecessary dispersion of the laser beam. The first lens converts spherical wavefronts into plane waves which then pass through an object plane (Ronchi rulings). It could be thought of as a condenser that produces a parallel uniformly illuminated beam. The purpose of the second lens is to form the Fourier transform image in its focus (transform plane). Low pass filtering in the Fourier plane is acheived by making use of an aperture. The third lens, when placed at its focal distance from the Fourier plane, converts the Fourier image into the inverted image of the object.

Upon inserting a fine wire mesh into the object plane, a faithful representation of the image is obtained. A 2D periodic object like this generates a T shaped diffraction pattern with a high degree of symmetry.

Now here' s what is really bugging me...There are masks that pass only parts of the pattern. For example, the effect on the image of passing only the vertical principal maxima is horizontal lines using a narrow vertical slit at the plane of the T shapped diffraction pattern. Turning the slit to select only the horizontal maxima yields vertical lines.

What is going on here physically? I am confused about how passing vertical maxima yields a the ("opposite") horizontal lines and vice versa.
I' ve read up on Fourier transforms but don' t understand how this is happening.

James

 

[JamesJames]

Anyone? I' m sure someone can help me.

 

[sir-pinski]

, I can understand why this may be confusing to you. The Abbe theory and spatial filtering can be complex concepts to grasp, but let me try to break it down for you.

First, let's start with the Abbe theory. This theory is based on the idea that any object can be represented as a combination of different spatial frequencies, or patterns of light and dark. These spatial frequencies can be separated and analyzed using a process called spatial filtering, which involves manipulating the light waves using lenses and apertures.

In the case of the optical computer you mentioned, the first lens converts the spherical wavefronts of the laser into plane waves, which are then directed onto the object plane (in this case, the Ronchi rulings). This creates a parallel and uniformly illuminated beam, which is then passed through the object.

The second lens then forms a Fourier transform image in its focus, which can be thought of as a representation of the spatial frequencies present in the object. The low pass filtering is achieved by using an aperture to block out certain spatial frequencies, resulting in a cleaner and more focused image.

Now, let's address the issue of the vertical and horizontal lines. This has to do with the properties of the Fourier transform. When an object is rotated, the Fourier transform is also rotated in the opposite direction. So, when you pass only the vertical maxima through a narrow slit, the Fourier transform is rotated, resulting in horizontal lines in the final image. The same is true when passing only the horizontal maxima through a slit.

I hope this helps to clarify things for you. The key takeaway is that the Abbe theory and spatial filtering allow us to manipulate and analyze the spatial frequencies present in an object, resulting in different representations of the object depending on how we manipulate the light waves.