At focus, all of the radiators should oscillate with the same phase. Every point of a perfect aperture appears to be cycling together with every other point. A sensor at such a focus is awash in light. However, at a slight angle with the optical axis (as in Fig. 4-6), the sensor sees some of the points oscillating a little behind the other points. They aren't really doing anything different, but part of the aperture is farther away. Because time passes as the light goes the extra distance, those radiators are perceived at the offset point's location to be lagging behind the rest. Partial wave cancellation results in the total intensity of the light being less at this lateral distance from focus.
If the sensor is far enough off axis, some distant portions of the aperture are seen to be once again in phase. Actually, they are a full wavelength behind the near portions of the aperture, but since the oscillation is sinusoidal, we can't tell the difference. The sources of those waves are atomic transitions that occur quickly, but the burst of light can be a single wave packet many meters and millions of wavelengths long. In the course of a few wavelengths, we really can't see any difference from one wave crest to another.
Different delays are indicated in Fig. 4-7 by Fresnel zones,2 dark and light regions on the aperture. Zero phase occurs only at the interface between two zones. Elemental radiators that are ahead in phase are denoted as "+" regions. Points that are behind in phase are called "—" regions. The dark or light color is only an identifier of each region; it denotes a phase change of a half wavelength. It does not represent the actual appearance, nor does it represent the illumination. In reality, these regions gradually fade into each other. The actual phase varies smoothly, but these regions jerk from one sign to the other.
The detection position in Fig. 4-7 is indicated by a tiny dot, which is in the focal plane of an eyepiece. Furthermore, the pattern on the aperture indicates the Fresnel zones as seen from that dot, rather than from our outside perspective.
This pattern changes rapidly. In only 9.2 x 10-16 seconds, the pattern is reversed from positive to negative (remember, the waves are flying into the
2 Named after physicist Augustin J. Fresnel, 1788-1827.
Fig. 4-7. Fresnel zones as seen from a point offset from a star. Notice the projections of the light cone on the bottom and far wall of the telescope. This offset is 1.22fk/D above the focused image. The positive and negative areas cancel so that this offset is the first dark ring.
aperture). If we follow the motion of the bars incrementally over small time scales, they are rushing down. If the detector is on the lower side of focus, they rush upward. The actual received wave at any given location is the average over time of the summed phase values of all these tiny radiators.
The Fresnel zone description of phases on the aperture should not be confused with the appearance of the image. Each location on the image has its own set of Fresnel zones. The Fresnel zone patterns are not observable directly since they refer to the wave field and we are capable of measuring only intensity at the sensor. This Fresnel zone picture is a convenient model, nothing else. Nevertheless, it succeeds in demonstrating many of the phenomena of diffraction and is the basis of the more accurate calculations described in Appendix B.
At risk of being too simple-minded, let's think of that aperture as being a large disk of construction paper. To determine the net wave intensity, we cut out all the like-colored areas of the disk and throw them into two separate piles, one positive and the other negative. We then weigh each pile and come up with two values, say 5 grams negative and 4 grams positive. Four grams cancel each other, leaving us with 1 extra gram of negative weight—call that value the "wave sum." Thus, the average wave sum is -lg/9g = — 1/9. To calculate a number proportional to the intensity of the light, which is always a positive quantity, we need to square this sum to yield an intensity of 1/81.
The intensity is weak, so our offset must be somewhere between the rings. Figure 4-7 is actually at a balance point. It is precisely between the
rings, and so the dark area should cancel the light area, depending on the quality of the drawing and our skill with the imaginary scissors. Figure 4-8 shows the next dark ring, which occurs at a distance farther from the axis. Here the tilt is worse, so more zones show, and two more bars are visible. Between the offsets of Fig. 4-7 and Fig. 4-8 is the first diffraction ring, where the positive and negative sections do the worst job of canceling each other. Five bars show there, but the two outside bars are very thin.
The important thing to learn from Fresnel patterns is that the wave sum is inefficient everywhere except precisely at perfect focus. Think about the construction paper again. At focus the aperture everywhere has the same sign— nothing cancels. The wave sum is 9g/9g, and the intensity is 1. On even the brightest portion of the first ring, you would get an uncanceled wave sum of about 1.2g/9g and a light intensity only about 1/57 the brightest value. The farther you go sideways, the worse the situation gets. At some point you realize that all you're cutting out are almost equal narrow ribbons of construction paper.
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