A Lens and Optical Diffraction Patterns

What does the diffraction pattern of an object look like We can visualize the diffraction pattern, the Fourier transform, of an object by making a mask about the object and then passing a collimated beam of light through the mask and onto a lens. The lens, as in 

We will further see in later chapters that it is possible to combine the two kinds of transforms illustrated here, the continuous transform of a molecule with the periodic, discrete transform of a lattice. In so doing, we will create the Fourier transform, the diffraction pattern of a crystal composed of individual molecules (sets of atoms) repeated in three-dimensional space according to a precise and periodic point lattice. 

FIGURE 1.7 In (a) the object, again exposed to a parallel beam of light, is not a continuous object or an arbitrary set of points in space, but is a two-dimensional periodic array of points. That is, the relative x, y positions of the points are not arbitrary they bear the same fixed, repetitive relationship to all others. One need only define a starting point and two translation vectors along the horizontal and vertical directions to generate the entire array. We call such an array a lattice. The periodicity of the points in the lattice is its crucial property, and as a consequence of the periodicity, its transform, or diffraction pattern in (b) is also a periodic array of discrete points (i.e., a lattice). Notice, however, that the spacings between the spots, or intensities, in the diffraction pattern are different than in the object. We will see that there is a reciprocal relationship between distances in object space (which we also call real space), and in diffraction space (which we also call Fourier space, or sometimes, reciprocal space). 

Unlike light, which, because of its refractive properties, can be focused by a properly ground glass lens, and unlike electrons, which, because of their charge, can be focused by 

FIGURE 1.8 In (a) is an arbitrary set of points that might represent the atoms in a molecule, and in (b) is the optical diffraction pattern of that set of points. It is a continuum of light and dark over the whole surface of the screen. The mask (object) in the optical diffraction experiment in (c) is the periodic arrangement of the fundamental set of points in (a) in two dimensions (i.e., the repetition of the object according to the instruction of a lattice). The diffraction pattern of (c) is shown in (d). We would find that if we superimpose the point array in (d) upon the continuous transform in ( ), the intensity at each lattice point in (d) corresponds to the value of the continuous transform beneath. That is, the diffraction pattern in (d) samples the continuous transform in (b) at specific points determined by the periodic lattice of (c). 

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