1D & 2D lattices

The Fourier transform of a lattice of vertical slits is composed of equidistant peaks displayed along the horizontal direction going through the Fourier space origin:

 Object "lattice" formed by the recurrence of equidistant and vertical slits & its Fourier transform

The Fourier transform is then spread along the direction perpendicular to the slits, as evidences a lattice of horizontal slits:

 Object "lattice" formed by the recurrence of equidistant and horizontal slits & its Fourier transform

Then, one can obtain a two dimensional lattice simply by multiplying these two one dimensional vertical & horizontal lattices:

 2D lattice formed by two crossed 1D lattices & its Fourier transform

The obtained Fourier transform is a set of peaks displayed on a recantgular lattice, but contratry to the direct object the vertical equidistance is larger than the horizontal one in the Fourier space. The Fourier transform of a lattice is then a lattice.

If, starting from this last Fourier transform one only keeps the horizontal row going trough the origin (using a masking tool for example), the image obtained by inverse Fourier transform will be a lattice of vertical slits:

 Using a masking tool to hide all peaks excepted those of the horizontal direction one creates a new Fourier space object, which corresponds to a lattice of vertical slits in direct space

Similarly, if using a masking tool one only keeps the vertical column going through the origin, the image obtained by inverse Fourier transform will be a lattice of horizontal slits:

 Using a masking tool to hide all peaks excepted those of the vertical direction one creates a new Fourier space object, which corresponds to a lattice of horizontal slits in direct space

However, it should not be thought that one can select one of the original 1D vertical or horizontal lattice using the corresponding masking tool. Indeed, if the masking tool select only now a diagonal row of the FT, the reconstructed image by inverse FT is a lattice of slits perpendicular to that particular direction, and this lattice does not exists as such in the original 2D lattice! Then, applying a mask on a Fourier transform alters the corresponding image, and this is very important for example in microscopy where one wishes to form an enlarged but faithful image of a too tiny object using a apparatus (microscope) which may modify the FT of the object and thus the observed image.

 Using a masking tool to hide all peaks excepted those along a diagonal row of the FT one creates a new Fourier space object, which corresponds to a lattice of slits perpendicular to this particular direction