Exploring Converging Lens & Fourier Transform of Aperture

[fisico30]

Hello,

does anyone know how a converging lens forms the Fourier transform of an aperture when the obs. screen is at distance=f?
If each point emits a spherical wave, the lens should make it then parallel and the FT should be the interference resulting from that.
However, if we decompose the amperture in plane waves, each plane wave will leave be focused to a point in the focal plane.
The latter explanation (plane wave decomp.) is clear but I think the first (spherical wave) is more physically true. Any clarifications or correction to these views? How does this spherical wave interference match the plane wave result?

 

[lpfr]

I think that the easiest way to understand this is to compute de Fraunhofer (that is: at infinity) diffraction pattern of the aperture. Then, if you realize that a converging lens concentrates all rays coming from a direction to a point in the focal plane, you will see that the lens just brings the diffraction pattern from infinity to the focal plane.

If you use your cornea as converging lens, you can see the diffraction pattern at infinity with your retina. Try to see a distant point source (plane waves) through a very small hole (0.1 to 0.2 mm in diameter).

You are right; you can decompose the aperture in spherical waves but not in plane waves. A plane wave is a complete plane and not a bit of a plane.

 

[astrokreng]

I am happy to provide some clarification on this topic. When a converging lens is used to form the Fourier transform of an aperture, the observed screen will show a pattern that is the result of the interference of spherical waves. This is because, as you mentioned, each point on the aperture emits a spherical wave and the lens focuses these waves to create parallel rays. The resulting interference pattern is then the Fourier transform of the aperture.

On the other hand, if we decompose the aperture into plane waves, each plane wave will be focused to a point in the focal plane. This explanation is also correct, but it is important to note that the resulting interference pattern is not the Fourier transform of the aperture. Instead, it is the Fraunhofer diffraction pattern, which is a special case of the Fourier transform.

In terms of physical truth, both explanations are valid. The spherical wave interference is a more accurate representation of the physical process, while the plane wave decomposition is a mathematical simplification that allows us to understand the resulting pattern. In fact, the two explanations are equivalent and can be mathematically derived from each other.

I hope this clarifies your understanding of the relationship between the converging lens, Fourier transform, and aperture. Both explanations are important in understanding the behavior of light and can be used depending on the specific situation and desired outcome.