Single-slit diffraction

In optics and spectroscopy, resolution is often limited by diffraction. To a good approximation, the spread function may appear as a single-slit diffraction pattern (Section II). If equal-intensity objects (spectral lines) are placed close to one another so that the first zero of one sine-squared diffraction pattern is superimposed on the peak of the adjacent pattern, they are said to be separated by the Rayleigh distance (Strong, 1958). This separation gives rise to a 19% dip between the peaks of the superimposed patterns. 

Note that one of the characteristics of single-slit diffraction is that the narrower the slit width the wider is the spread of the diffraction pattern. Note also that... 

Not all orders of a given wavelength appear equally strong in the focal plane, however. The single-slit diffraction pattern corresponding to the width a of each slit forms an envelope in the focal plane that determines the positions of the orders of significant intensity. The intensity pattern in the focal plane, as derived in standard texts for Fraunhofer diffraction by multiple slits, is... 

Figure 6.15 Light intensity distribution after single slit diffraction.

Figure 18.14 The diffraction pattern of helices in fiber crystallites can be simulated by the diffraction pattern of a single slit with the shape of a sine curve (representing the projection of a helix). Two such simulations are given in (a) and (b), with the helix shown to the left of its diffraction pattern. The spacing between the layer lines is inversely related to the helix pitch, P and the angle of the cross arms in the diffraction pattern is related to the angle of climb of the helix, 6. The helix in (b) has a smaller pitch and angle of climb than the helix in (a). (Courtesy of W. Fuller.)...

Figure 13. Fraunhofer diffraction pattern of a single slit illuminate with coherent monochromatic light the intensity distribution is shown for two...

This is called a point-spread function, because it describes how what should be a point focus by geometrical optics is spread out by diffraction. The expression in the curly brackets is the one that is of interest. The other terms are phase and overall amplitude terms, as are usual with Fraunhofer diffraction expressions. The function Ji is a Bessel function of the first kind of order one, whose values can be looked up in mathematical tables. 2Ji(x)/x, the function in the curly brackets, is known as jinc(x). It is the axially symmetric equivalent of the more familiar sinc(x) = sin(x)/x (Hecht 2002), the diffraction pattern of a single slit, usually plotted in its squared form to represent intensity. Just as sinc(x) has a large central maximum, and then a series of zeros, so does jinc(x). Ji(x) = 0, but by L Hospital s rule the value of Ji(x)/x is then the ratio of the gradients, and jinc(0) = 1. The next zero in Ji(x) occurs when x = 3.832, and so that gives the first zero in jinc(x). This occurs at r = (3.832/n) x (q/2a)Xo in (3.2), which is the origin of the numerical factor in (3.1). 

For D d, the envelope function is determined by the diffraction at the single slits. The width of the interferogram B is defined by the distance of the two first order minima on each side of the maximum as... 

This is the well-known expression for the Fraunhofer diffraction by a single slit, and is plotted in Figure 1.7(b). 

Fig. 5. Diffraction profile from (a) a single slit and (c) many slits, (b) The sampling region from many slits.

Single slit (left) produces an intense band of light. Double slit (right) gives a diffraction pattern. (Courtesy of S.M. Blinder.)... 

I ig. 2.1 shows a modernized version of the famous double-slit diffraction experiment first carried out by Thomas Young in 1801. Light from a monochromatic (single wavelength) source is passed through two narrow slits and projected onto a screen. Each slit by itself would allow just a narrow band of light to illuminate the screen. But with both slits open, a beautiful interference pattern of alternating... 

FIGURE 3.4. Scattering by a single slit, (a) Diffraction by a narrow slit and (b) the diffraction pattern of a slit that is wider than that in (a). In both cases the intensity variation shown is referred to as the envelope, The zero point of the horizontal axis represents the direction of the direct beam (cf. Figure 3.5). 

So far we have only considered the diffraction pattern of a single slit and have shown that the intensity variation is bell shaped this is the envelope profile with a width inversely proportional to the width of the slit. Now we will consider what happens to the diffraction pattern when more slits are lined up parallel to the first to give the equivalent of a diffraction grating. This is a two-dimensional analogy to the buildup of a crystal... 

A three-dimensional lattice, reciprocal to the crystal lattice, is very useful in analyses of X-ray diffraction patterns it is called the reciprocal lattice. Earlier in this chapter the diffraction pattern of a series of regularly spaced slits was considered to be composed of an envelope profile, the diffraction pattern of a single slit, and sampling regions, ... 

Figure 3,5 Electron diffraction at double and single slits...

FIGURE 3-14. Diffraction of a monochromatic beam of light through a single slit. 

Chateau Lafite was a wine preferred by King Louis XV of France, and is now probably the most famous wine in the world. I offer a bottle of this great wine (vintage 1982, the best since 1959) to the first person who gives me an article or reference, old or new, that contains an experimental proof that a sequence of single particles, separated in space and in time, that has been diffracted by an ensemble of two slits gives a pattern different from the sum of the two patterns of single particles diffracted by the same slits one at a time. 

---