Nodes and Antinodes

Certain locations in the volume around the brightest image point appear to be dark compared with their immediate surroundings. Figures 4-7 and 4-8 showed two such points on the first and second dark rings. In defocusing, on-axis darkness is found when the number of Fresnel zones is an even number. These locations appear to be quiet while the tumult rages around them. They are nulls, more commonly called nodes. The opposite of a node is an antinode, where the wave action is strongest and the brightness is at a local maximum. A good example of antinodes are the peaks of the diffraction rings, as well as the highest point on the central image spot itself.

Everyone has seen nodes, even though many aren't aware of it. If you look carefully at a guitar string, you can see the simplest type of vibration in Fig. 413 a. There are two nodes at the bridge and fret, and a single antinode at the center. This situation is called a standing wave. Of course, the presence of the two nodes at the edges of the string is hardly a surprise. These positions are mechanically constrained and can hardly be expected to move much.

4.4. Nodes and Antinodes

Comet Fresnel Oscillation
Fig. 4-11. An off-center bull's-eye Fresnel zone pattern showing two aberrations (defocus and tilt) mixed together.

Sensor location in Fig. 4-11 Geometric shadow outside of this radius Sensor location in Fig. 4-10

Fig. 4-12. The indicated location is where the intensity is calculated as the Fresnel zone sum of Figs. 4-10 and 4-11. The bright ring indicates the boundary of geometric shadow.

But nodes can hang freely in space, seemingly held by no physical restraints. If you pluck the guitar string as in Fig. 4-13b, you will see the pattern of Fig. 4-13c shortly after you release the string. After more time passes, it will decay to the situation in a). When you plucked the string, the stroke contained a multitude of frequencies (remember Fourier and the impulse response function). The strongest frequencies are the "fundamental" in a) and the "first harmonic" in c). No modes of vibration exist other than multiples of the fundamental frequency.

node antinode node node antinode node

Fig. 4-13. Nodes and antinodes of a guitar string.

Situations a) and c) are an octave apart. What if you want a tone somewhere in between? You can't find it with a fixed distance separating the fret and bridge. To obtain an intermediate tone you must depress the string at another fret (Fig. 4-13d), which implies a new fundamental and new harmonic sequence.

Now, if nodes and antinodes were only found in stringed instruments, they would hardly be useful for a discussion of diffraction. Standing waves are everywhere, however. For example, they are visible on the surfaces of rapidly vibrating liquids. These standing waves are sometimes seen in a coffee cup or other container shaking on the same tabletop as an appliance or tool with a fast electric motor. The surface appears almost stationary, even though you know intellectually that the speed of water waves is much faster than the almost leisurely drift of the waves. An equivalent to the bridge and fret in this case is the edge of the container, and the standing wave appears in the two-dimensional surface of the fluid. If the container is round, the most likely vibration is radially symmetric. If it is square, you see a sort of checkerboard figure.

A three-dimensional case can occasionally be found in an old microwave oven. Microwave cooking patterns are typically stirred by a hidden metal fan placed over the exit portal of the microwave transmitter. These fans can fail, and the only noticeable change is that cooking becomes very uneven. A stationary standing wave pattern is set up that overheats at the antinodes and leaves the food almost raw at the nodes.

Even with a working fan, holes and bright regions can interfere with even cooking. For this reason, the food must be moved at least once during cooking. Everyone has experienced the tiny hot spots that scorch overheated popcorn in microwave ovens. These antinodal regions are roughly 1-2 cm across.

What are the bridge and frets here? The microwave oven is a metal cavity whose walls are nodes because electrical conductors cannot have electric fields deep within them. Since any form of electromagnetic radiation has both electric and magnetic fields propagating in tandem, the metal cavity reflects microwaves efficiently. Even in the window, you are looking through a metal mesh that behaves as a solid metal wall to microwaves. The food is always elevated on a low tray because it won't heat right against a node.

Here's the connection to telescope optics. A telescope aperture is oscillating at a given unchanging frequency (like the microwave transmitter). Bright places in the image are the equivalent of hot spots in the oven. Dark places are similar to cold spots that leave food uncooked.

We can thus interpret the behavior of the diffraction structure near focus as a standing wave. A slice through the focal region of an image that suffers from a small amount of spherical aberration is depicted in Fig. 4-14. This figure shows how the light collapses to focus and expands again beyond it, with the region closer to the objective on the left above the label —2 and the region farther from the objective above the label 2. It is printed at very low contrast to emphasize the low-level structure. The nodes and antinodes are easy to identify. Because the aperture is a rigid geometrical structure, there are locations where the wave doesn't fit correctly and places where the vibration is favored.

Fig. 4-14. Longitudinal slice near focus of an aperture suffering from slight spherical aberration.

This situation is very much like a brook trickling over a rocky streambed. Over one particular pebble, the surface of the water is elevated a small amount. In other places the surface seems undisturbed. The overall look is deceptively frozen, and we forget that in only a few moments all the water has moved downstream to be replaced by fresh water occupying the same configuration.

Similarly, we interpret the standing wave of the image as some sort of fixed artifact, but energy is rushing through the instrument at the speed of light. The nodal points that can be seen as dark rings in the focused image and corrugations in the defocused image are induced by the geometry of the situation. They are more like the pebble than the water. Aberrations disturb the geometry and in doing so affect the standing waves.

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