The Consequence of Filtering

We have already seen in Chapter 3 how the telescope has a limited spatial frequency response analogous to the 20-20,000 Hz frequency response of audio equipment. What is the implication of that filtering? Can knowledge about the limitation on spatial frequencies be used to derive interesting facts about diffraction?

The two-dimensional circular surface of an aperture injects some non-instructive mathematical complications into the problem, so it will be easier to turn once again to the simpler situation of audio electronics. In this case, we are interested only in variations in time or frequency.

The audio analogue of a star or a point optical source is an electronic pop. It could also be caused by a scratch or fault on an old vinyl LP disk. If you've ever tried to listen to a distant radio station through an intervening storm, you are probably familiar with the staccato crackle as a lightning strike takes place. One thing that you probably did not notice, however, is that every one of those pops sounds identical, varying only in amplitude.

1 Equation 4.1 is simplified. Those who are interested in this topic can find more in any elementary quantum mechanics text, e.g. Park 1974.

This fact seems innocent enough, but it gives us an insight into sound reproduction. Every little crackle sounds precisely the same through one audio system. If you listened to another audio system having a greatly different configuration—say a cheap, hand-held transistor radio compared with your stereo—it may have a different tonality. Yet each pop produced by that other radio is the same as well.

The correct interpretation of this uniform behavior is that you aren't listening to details of the electrical activity in the storm or the source. You're hearing some sort of signature of the reproduction device itself. The crackling noise is merely its best attempt at reproducing an abrupt jump.

At the beginning of the 19th century, physicist Joseph Fourier discovered that complicated, even discontinuous, functions could be simulated to arbitrarily high precision by the sum of a series of sine and cosine functions of higher and higher frequency. A "pop" could be described by this infinite series:

a(t) = 1 + 2(cos(t) + cos(2t) + cos(3t) + ...), (4.4)

where a(t) is the amplitude as a function of time. The sharp crack we actually hear is the intensity I = a2(t).

Being a filter, the audio system doesn't transfer frequencies to an arbitrarily high value; it starts failing at about 20,000 Hz. Thus, we're going to have to truncate this series somewhere. Figure 4-3 shows the effect on the intensity if we lop off the series at 10 cosine terms. The sharp spike of the crackle has been reduced to quite a different function.

This response function resembles a diffraction pattern, complete with rings. In fact, it is identical to the diffraction pattern of a slit or rectangular aperture. Diffraction is a result of the optical system's failure to transmit arbitrarily high spatial frequencies, just as the ringing of Fig. 4-3 is an effect of the limited audio bandwidth. A point-like star acts like a sharp audio impulse. A star's diffraction pattern is similar to an impulse response function. It is called a point-spread function because most stars are far away and occupy insensibly small angles. Not only do rings appear in Fig. 4-3, but also the infinitely sharp spike has broadened into a narrow peak of measurable width.

Astute readers probably realize that the symmetry of the pattern means the response curve oscillates long before the pop on the record actually happens but that such an oscillation isn't possible. The name of this limitation is causality, or the requirement that cause precede effect. Causality is a central issue of special relativity, but it even has implications here. The symmetry of Fig. 4-3 is an artifact of the simple-minded way this process was modeled. The phase shifts of the individual cosine terms in the equation above were ignored. We implicitly assumed that

IMPULSE RESPONSE 1

FUNCTION

:

Ringing

/

\A~._

Time on recording [arbitrary units]

Time on recording [arbitrary units]

Fig. 4-3. The intensity of an audio crackle is simulated as a truncation of the series in Eq. 4.4.

all signals go through the system at the same speed with the same phase. An actual system delays every frequency a little and adjusts the phases in such a way that no signal could be detected before it happened. The pattern would become asymmetrical with most of the ringing coming after the sharpest spike. Each audio system has its own characteristic slurring of this pattern.

Fortunately, optical diffraction patterns aren't calculated from time-based frequencies and don't have to obey causality. They can be nicely symmetrical. After all, spatial frequency is only a convenient analogy to true frequency. You are free to look at either side of an optical pattern, but you can't go back in time.

4.3 Waves Are Reborn

The next and most important way we will look at diffraction is a very old theory originally proposed by Christian Huygens (1629-1695) to explain why light is observed in the shadow region.

The term wave propagation is used without much attention to its origins. Propagation is a word originally associated with botany, where it meant to make new plants from cuttings. Its meanings include the multiplication, reproduction, or dissemination of plants and animals. It has come to mean the spread of ideas or even misinformation (hence, "propaganda"). But the concept of reproduction is inherent to propagation and gives a clue as to what Huygens suggested.

Huygens could explain light appearing in shadows by assuming that every point on a wavefront was emitting its own spherically diverging wavelet. The next instant of the wave would be given by the sum of all of these tiny wavelets. Only in the direction the wavefront was traveling would the wavelets not cancel each other. In other directions, a sideways-moving wavelet would cancel another point's wavelet that was out of phase with it. Similarly, if a hedge tries to grow sideways, it finds that a leaf on another branch has already taken the sunlight. The only direction that growth is free is upward and outward.

An instant later, each new point of the wavefront would emit yet another spherically diverging wavelet. Similarly, in the hedge, inner leaves wither and are replaced by outer leaves. Thus, we could describe wave progression as a rebirth at each new wavefront, a "propagation" in the old botanical sense.

The situation of Fig. 4-4 does not require a consideration of propagation, but in Fig. 4-5 we see a reason for it. When the wave propagating from a distant source encounters an aperture, a sudden disturbance occurs in the beautiful symmetry. Points inside the aperture emit wavelets up, down, or out, but no corresponding sources over in the shadow of the aperture provide canceling interference from that direction. In the center of the aperture, the wave propagates much as it did in the free-field, but off at the edge, some of the energy leaks over into the geometric shadow. The little points emitting wavelets are called elemental radiators or secondary sources. These radiators aren't envisioned as truly hanging like beads at the aperture. They are only an infinite number of mathematical points.

Fig. 4-4. Free-held divergence of a spherical wave.

Fig. 4-4. Free-held divergence of a spherical wave.

Fig. 4-5. Diffraction of a plane wave at an aperture.
Fig. 4-6. A lens, aperture, and eyepiece showing tilt aberration.

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