Waves abhor boundaries, and spread to eliminate them

It's the existence of shadows that makes us believe that light travels in rays. The same shadows, when examined very closely, are incontrovertible evidence for the wave nature of light. Resolution of this paradox teaches a lot about light, rays and waves. The sharpest shadows, those produced by point sources, are outlined by delicate oscillations in intensity that are interference fringes. Although probably seen now and then for centuries, it was Grimaldi in the 17th century who first drew attention to them as something special. They were studied by Newton, and finally recognized as interference fringes by Young at the beginning of the 19th century. Fresnel's masterful synthesis of the wave theory of light was the fruit of a contest to explain these fringes.

When we punch holes in a wavefront with obstacles, or make a wavefront squeeze through apertures, the wave oozes to the sides, into the shadows, with an intensity that very rapidly decreases away from the geometric shadow edge, as would be suggested by Huyghens' Principle. Eventually, the damage to the wavefront would be completely repaired, but in the case of light we seldom go that far. Strangely, there is a wave that spreads into the geometrically lighted region as well, and it is the interference between this wave and the direct wave that makes the fringes. All of these phenomena are called *diffraction* in connection not only with light, but with other forms of waves as well.

So far as light is concerned, the wavelengths and distances involved greatly simplify the problem. Only small changes in direction are significant, and the results are very insensitive to boundary conditions or polarization. In fact, a scalar wave approximation that is a mathematical expression of Huyghens' Principle is very fruitful, and dominates the discussion in optics textbooks, to which the reader is referred for details. The formula is called the Fresnel-Kirchhoff Diffraction Integral, shown at the right. In this method, one extends the surface integral over a closed surface (which may partly coincide with a wavefront) to find the resultant intensity at any point. One should notice the factor of i, the cosine obliquity factors, and the inverse dependence on radial distances from the source and point to the area element dS. It is a less valid approximation for the diffraction of radio waves, so many alternatives have been devised that correctly include the effects of the boundaries and polarization.

A useful way of thinking about diffraction that explains its qualitative features in many practical cases is to regard the diffracted field as the superposition of the geometrically limited wave, and a cylindrical *edge wave* emanating from the boundaries of the obstacles. Sommerfeld showed in an exact solution of diffraction at a straight edge that this wave in fact existed, and was not merely an approximation. It forms an alternative description of diffraction that is sometimes useful.

For example, consider the basic case of diffraction at a straight edge. Imagine that the intensity that is diffracted into the shadow is due to the edge wave, and exactly similar wave is diffracted into the lighted region. The phase of these waves differs by an eighth of a wavelength from the phase of the direct wave, and the edge wave has a discontinuity at the shadow boundary at which the phase changes by half a wavelength. These properties can be found by a transformation of the Fresnel solution. The strange quarter wavelength comes from the fact that the edge wave is cylindrical, not spherical. If you look at such an edge from a point in the shadow, you will notice a bright line at the edge, which is the source of the edge wave. This is a frequent observation in nature, most easily observed near sunset looking in the direction of the sun, when dark obstacles will be surrounded by a bright border. One must be looking almost in the direction of the incident light, of course.

When the obstacle is a disc, the edge wave is emitted from a circular ring. At any point on the axis, all the waves will be in phase, so there will be a bright point. In fact, its intensity is the same as would be there in the absence of the disc. This deduction was noted by Poisson (but from Fresnel's theory) and observed by Arago, a famous victory of the wave theory. It is easily seen in the shadow of a small ball bearing in a laser beam. The moon's disc should also be bounded by a bright ring in a total solar eclipse.

In white light, only two or three coloured fringes can be made out at most, but in monochromatic light the pattern is amazing, with every shadow bounded by a series of fringes, and every corner with a crest of fringes. This is specially well seen with laser illumination. To observe diffraction fringes, the source must be small enough not to blur them.

An edge of a wave field is formed when a *caustic* occurs due to refraction or reflection. This happens when the deviation of the outgoing ray goes through a maximum or minimum when the angle of the incident ray changes. An easy case to observe is seen on the surface of a cup of coffee with cream when a small light is reflected from the cylindrical inner surface of the cup. A cusp-shaped bright curve will be seen. It is characteristic for the intensity to be concentrated near a caustic. On the bright side of the caustic, two rays interfere, and on the other there is only the analogue of the edge wave. The rainbow is the best natural example of this. Although a result of refraction in spherical water drops, the actual form of the rainbow is determined by diffraction and interference. Caustic interference can be seen by night when looking through those bathroom windows that are covered by evenly spaced bumps at distant lights. A pincushion of fringes is the usual result. Once you begin looking for them, it is not hard to find examples of diffraction in daily life.

Let's consider the case of a circular hole in a diaphragm, with a plane wave incident from one side. This could be the entrance pupil of a telescope observing a star, for example. We take a screen and throw the light on it, to examine the diffraction. Close to the hole, we see a bright disc surrounded by fringes, as we expect. As we move the screen farther away, the edge waves cover more and more of the bright disc, as well as spreading brightness beyond it. We have interference between the edge wave and the geometric wave in this region, but the amplitude of the geometric wave is decreasing as 1/r, while the edge wave amplitude decreases only as 1/Ör, so at great distances only the edge wave is important. The diffraction pattern changes continuously, from a fringed disc to a bull's eye surrounded by rings. The first case is called Fresnel diffraction (where the geometric ray predominates), the second Fraunhofer diffraction (where only edge rays interfere). The Fraunhofer case can be produced at a finite distance if a converging lens is put at the aperture, so that parallel rays leaving the aperture interfere at the focal plane. Fraunhofer diffraction is the one that applies to the image of a star formed in a telescope. This case was first solved by the Astronomer Royal, George Airy, and the bull's eye of the pattern is called the Airy Disc. The angular distance to the first dark fringe surrounding the Airy Disc, a quantity often needed, is 1.22λ/D radians, where D is the diameter of the aperture, and λ is the wavelength. Any physical optics text will provide full mathematical details.

A simple example will bring out more clearly how the size of a diffraction image is determined. The diagram at the right shows a cross-section of a slit in a screen, with a plane wavefront from a distant source incident from the left. By Huyghens' Principle, the wavefront at a great distance to the right is the superposition of wavelets coming off parallel at an angle θ. When the difference in path lengths from the top and bottom of the slit are one wavelength (as shown, for m=1), then waves from the bottom of the slit cancel waves from the middle, and so on until waves from the middle cancel waves from the top. This must be the angular location of a dark fringe in the pattern. This angle is approximately λ/D, which is usually quite small, so the approximations made hold well. For the circular aperture, we found m = 1.22, which is not a great deal different.

Interesting deductions can be made from applying Huyghens' Principle to a plane (or spherical) wavefront in Fresnel's manner to find the intensity at a certain point P. Consider a ray from any point on the wavefront to the point in question. The shortest ray is that normal to the wavefront at a point Q. As we depart from this point, the ray becomes longer. We draw a circle on the wavefront contining those points a half-wavelength farther away from P. Now let us add up all the phasors for the infinitesimal circular zones of equal areas of the wavefront around Q, assuming the phase of the very first as along the +x-axis (this is the phase of the incident wave at P). The last contribution is opposite in direction, pointing towards the -x axis. Since all the infinitesimal phasors are roughly equal in length, what we have is a semicircle that can be plotted standing with its diameter normal to the x-axis at a point S. Do the same for the next zone, up to points a wavelength longer than QP. We get another semicircle, that takes us about back to where we started. However, the distances are larger, and there is an inclination with respect to the axial direction, both of which reduce the amplitude, so the semicircle is of smaller diameter. Proceeding in the same way, we see that what we get is a spiral that converges to a point T on the y-axis. The line ST then is the vector sum of all the amplitudes on the wavefront, which should add up to the amplitude of the wave at P. It seems to be advanced in phase by a quarter-wavelength, which shows us that the elementary Huyghens' wavelets are to be delayed in phase by this amount, and also gives us the amplitude that should be associated with them. All these properties also follow from a mathematical analysis based on the wave equation, but it is interesting to see evidence of them in such a simple argument.

One deduction that can be made is that blocking off a small circular area of the wave front around Q will have practically no effect on the intensity at P, since the first loops of the spiral come back to nearly the same point, so that the absence of a few will only affect the phase. Poisson, famously, arrived at this result. Another deduction, equally surprising, is that if we make a screen in which alternate zones are opaque, the contributions from the zones left open will all add in phase, so the intensity at P will be greatly increased, proportionately to the square of the number of open zones. Such *zone plates* are easily made, and have the expected effect. They resemble lenses in their effects, and can make images.

When the same thing is done using horizontal strips of the wave front, so the analysis is easily applicable to a straight-edged obstacle or a slit, the phasor diagram or *vibration curve* turns out to be expressible as a plot of the Fresnel Integrals, which is called the Cornu Spiral, a favourite of physical optics texts. The properties of the edge wave can be deduced from this representation, as well as the form of the fringes surrounding a straight edge. The predictions of Fresnel's theory are very well borne out in practice, but it should be kept in mind that the theory is an approximate one, and that there are cases in which it is not accurately valid.

Regular arrays of obstacles, apertures or scattering centres of any type yield wavelets that may interfere in interesting ways. The most common example is the *ruled grating*. If there are N equally spaced lines, then there are N wavelets with a regular phase differences between them. If a is the spacing of the lines, then the condition for a maximum in the Fraunhofer case is a (sin θ + sin θ') = mλ. m is called the *order* of the interference, and the angles are the angles of incidence and diffraction. The intensity distribution is found by adding up the phasors from each line and squaring. For large N, the maxima are very sharp and well-separated. Gratings have replaced prisms as dispersing elements, except where prisms are required for their optical properties. It is very difficult to make good gratings with a large number of lines, because any variation in the spacing has serious deleterious effects. Most gratings are *replicas*, made by using a master grating as a pattern.

Crystals are excellent examples of three-dimensional gratings. Since the spacing of the atoms is much less than a light wavelength, only a zero-order beam occurs, and there is no diffraction. In fact, when a wave enters a crystal, it is soon replaced by the beam scattered regularly forward by the atoms, that propagates with phase velocity c/n. The inverse conversion occurs when the wave exits the crystal. When the scattering centres are randomly located (as in air) there is also a wave scattered by the randomness on a scale of distance comparable with the wavelength. X-rays have wavelengths comparable to the interatomic distances in crystals. The conditions for a diffracted wave are very restrictive. The wave appears to reflect from layers of atoms with spacing d at fixed angles given by Bragg's Law: 2d sin θ = m λ, where θ is the grazing angle made with the plane, as is customary in X-ray work, not with the normal. The reflection is seen to be very dependent on wavelength, which is typical of three-dimensional arrays.

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Composed by J. B. Calvert

Created 1 May 2000

Last revised