From a slit to a lattice

    Consider a slit of limited width: its Fourier transform (represented as its modulus) is then a very large cardinal sine function:

   
                                                                   
Object "1 slit", its Fourier transform (right) & profile along horizontal direction through Fourier space origin (image centre)

    Similarly, the same slit shifted on the image leads to a Fourier Transform having the same moduli than the first slit, but differs only by a phase term:

   
                                                                 
Object "1 slit" similar to the first slit but shifted, its Fourier transform (right) & profile along horizontal direction through Fourier space origin (image centre)

    This property is important because usually Fourier transforms are displayed as moduli, due to the fact that some physical technics realte to Fourier transforms lead to the determination of moduli (or their squares) of the Fourier coefficients, the information about the phases being lost (e.g. diffraction of a laser beam by a small aperture).

    If the object consists of two identical but shifted slits, the display of the Fourier transform is not the sum of the representations of the Fourier transform of each unique slit: one obtains oscillations superposed to an overall decreasing function. This apparent non-additivity of the Fourier Transforms is due only to the fact that these tranformations are only displayed as their moduli. Fourier coefficients are complex numbers, composed of moduli and phases: they are additive in the complex plane (in moduli AND phases):
FT{f(x)+g(x)}=FT{f(x)}+FT{g(x)}.

   
                                                               
Object "2 slits", its Fourier transform (right) & profile along horizontal direction through Fourier space origin (image centre)

    Then it is not possible to deduce the diffraction pattern of an object composed of two slits from the observation of the diffraction pattern of an object composed of one slit, since these diffraction experiments lead only to the moduli of the Fourier transform of the diffracting object.

    If one adds another slit, secondary maxima occur (small peak between two larger peaks),

   
                                                                   
Object "3 slits", its Fourier transform (right) & profile along horizontal direction through Fourier space origin (image centre)

and if one adds more & more slits, more & more secondary maxima occur (it can be seen that the number of these new peaks is the number of slits - 2) and one can observe that these maxima become more & more sharp:

   
                                                               
Objet "4 slits", its Fourier transform (right) & profile along horizontal direction through Fourier space origin (image centre)


    The limiting case consists of an image covered with vertical & equidistant slits (what one calls a lattice): the Fourier transform of this image is then a set of equidistant points, placed on the horizontal direction going through the Fourier space origin (it is then a lattice of points):

   
                                                                   
Object "lattice" formed by the recurrence of equidistant slits, its Fourier transform (right) & profile along horizontal direction through Fourier space origin (image centre)

    It is interesting to notice that if one divides by two the equidistance between the lattice slits, then the equidistance between the peaks in the Fourier space is multiplied by two: one recovers the inverse relationship between shapes in direct space (object) & Fourier space (also called reciprocal space):

   
                                                                 
Object "lattice" formed by the recurrence of slits with half equidistance compared to the first lattice, its Fourier transform (right) & profile along horizontal direction through Fourier space origin (image centre)


Remark :
    It can be seen on the Fourier transforms profiles of the two preceding lattices that the peaks in Fourier space are somewhat spread. This arises due to the fact that when one calculates the FT of an image one supposes that this object is periodically repeated in the horizontal & vertical directions. However, in the two precedent cases, the width of the image is not a multiple of the  periodicities of the lattices:



Mosaic formed by the juxtaposition of 4 images of the first lattice: the object obtained does not have the same periodicity of the original lattice itself (the images do not correspond at their edges, as can be seen along the vertical line through the image centre)

    Now, if one constructs a lattice which periodicity (e.g. 8 pixels) is a divisor of the image width (256 pixels) then the object formed by the juxtaposition of this image has also for periodicity the period of the original lattice and then the Fourier transform has peaks that are 'infinitely' sharp:



Mosaic formed by the juxtaposition of 4 images of the new lattice: this final object has the same periodicity than the lattice (no defects at the edges)

       
                                                                      
Object "lattice" which equiditance is a divisor of the image width, its Fourier transform (right) & profile along horizontal direction through the Fourier space origin (image centre) displaying 'infinitely' sharp peaks