Fundamentals of Diffraction and Image Formation

Propagation of the angular spectrum

Let us consider the angular spectrum of a wave U located at a distance z along the propagation axis (see figure II-2) :

Image II-2- Propagation of the angular spectrum along z.
Image II-2- Propagation of the angular spectrum along z. [zoom...]Info

To characterize the effects of wave propagation on the perturbation angular spectrum, we need to determine the relation between and . We know that U(x,y,z) can be written under the form of a TF-1 :

II-10 Info

In addition, U must follow the Helmholtz equation at any point where there is no source. By replacing (II-10) in (II-9) and after calculation, we find that must satisfy the following differential equation [] :

An elementary solution to this equation can be written as:

II-11 Info

  • This result shows that when the direction cosines satisfy the inequality , the effect of propagation on a distance z translates into a simple phase shift of the various angular spectrum components.

  • In the (rarer) case where the direction cosines verify (for example in the presence of a diopter), the square root is imaginary and the previous relation can be written: 

Since μ > 0, the spectral components are attenuated by the propagation phenomenon. Those spectral components are called “evanescent waves”.

  • The borderline case where corresponds to waves propagating perpendicularly to Oz. Consequently, they transport no energy along the z axis.

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