Figure 8.17 depicts a two-lens system for spatial filtering (also sometimes called a “4f system,” since it involves five special planes separated by four intervals with lengths equal to the common focal length f of the two lenses). Develop a description, patterned after Abbes, of image formation with this system. Most importantly, do the following.(a) Show that the field at the filter plane is

This is like Eq. (8.31) for the one-lens system but with the spatially dependent phase factor exp[−ikx_F² /(2(v − f))] gone, so aside from a multiplicative constant, the filter-plane field is precisely the Fourier transform of the source-plane field.

(b) In the filter plane we place a filter whose transmissivity we denote K̃(−x_F/f), so it is (proportional to) the filter-plane field that would be obtained from some source-plane field K(−x_S). Using the optics conventions (8.11) for the Fourier transform and its inverse, show that the image-plane field, with the filter present, is

Here ψ_S is the convolution of the source field and the filter function

In the absence of the filter, ψ_S is equal to ψ_S, so Eq. (8.36b) is like the image plane field (8.33) for the single-lens system, but with the spatially dependent phase factor exp[−ikx²_F/(2(v − f))]gone. Thus ψ₁ is precisely the same as the inverted ψ_S, aside from an overall phase.

Fig. 8.17

Equations

VF(XF) = ik -eikf s(XF/f). 2 f (8.36a)