Fresnel Zones with Defocus

The wavefront changes character when the eyepiece is defocused. Figure 4-9 shows the induced aberration at the aperture if the eyepiece is drawn back. The resulting Fresnel zones are circular, as in Fig. 4-10. With the passage of half a wavelength, the pattern reverses, producing a negative spot in the center. However, the square of the wave sum is the same, so the intensity has the same positive value. For sensor locations beyond focus, the passing of time would show a collapsing pattern with new zones appearing at the

Fig. 4-9. Defocusing aberration. The secondary sources are vibrating on the solid arc, yet the eyepiece's focal plane is at the center of the dashed arc. The aberration resembles a smoothly pushed-in deformation. The center of the aperture appears delayed relative to the outer portions.
Fig. 4-10. Fresnel zones at a defocusing aberration of 1.7 wavelengths.

edge of the aperture. These new zones move inward to disappear at the center. For locations inside focus, we'd see identical snapshot patterns, but new zones would appear exploding from the center and moving out.

Even though the pattern is collapsing and we could have picked any snapshot, the wavefront phase is chosen so that the central spot is biggest. The total amount of defocusing aberration in Fig. 4-10 is about 1.7 wavelengths. This value may be determined by counting colors from the center and moving out; the edge is about 3.4 half-wavelength zones out.

We deliberately choose the wavefront phase so that the area of the central positive zone is exactly equal to the areas of all other zones ringing it. The wave sum should be zero when the number of zones is 2, 4, 6, etc. More elaborate theories predict the on-axis image intensity goes to zero when the defocusing aberration is 1, 2, 3 ... wavelengths, just as predicted by this simple Fresnel zone model.

Moving upwards, as in Fig. 4-11, shows the effect of mixing tilt and defocusing aberrations. Figure 4-12 is the same image pattern with an arrow pointing out the sensor location on the image. The edge of geometric shadow is shown as a light ring.

The strong, outer ring seen in most defocused patterns corresponds to the location where the central Fresnel spot begins sliding off the aperture. (In fact, it is half gone at the beginning of geometric shadow.) Thus, the slightly brighter edge ring in defocused images represents the last flourish of the central Fresnel zone before it disappears and darkness closes in. Notice how the brightest parts of the disks do not fill the circle of geometric shadow and how some of the light has escaped the disk to occupy the shadow. If that pattern were described by ray optics, the disk would be an absolutely featureless circle with perfect darkness outside the radius of geometric shadow.

This image shows another intriguing behavior. Soft rings terrace the pattern, even in the shadow zone. As the sensor is offset further, fresh Fresnel zones come into view, each causing a wiggle in the intensity.

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