Fundamentals of Diffraction and Image Formation

Fraunhofer approximations

Fraunhofer made an approximation on the quadratic phase function:

This condition is even more restrictive than the Fresnel condition:

For example, if (x1²+y1²) max =1mm²=10-6m² and , then the Fraunhofer approximation corresponds to . With the same parameters, the Fresnel conditions correspond to z >>1cm. For this reason, Fraunhofer diffraction is also called far-field diffraction.

Using those conditions, the expression (III-2) can be written:

III-3 Info

Except for some multiplying factors preceding the integral, this expression is simply the FT of the field distribution at the aperture evaluated for the frequencies u=x0 /λz ; v=y0/λz.

Fraunhofer diffracting patterns can be observed at distances smaller than previously calculated if a converging lens is properly placed between the aperture and the observation plane. This is precisely what we will examine in the following section, entitled: "lens properties in relation with the Fourier transform".

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