## The Coordinates of Light

Where is a light wave? In Fig. 4-1, try to point out the location of the light wave. One might conjecture that the light wave is one of the long crests that is depicted in the figure, but this conjecture mistakes the graphical portrayal for truth. In depicting the progress of light, it is convenient to trace out the crests or troughs, but those locations have no special implication. The points halfway up the wave are no less part of the wave. In pointing out a wave, we must be careful not to allow human interpretation to color the answer.

How could we choose one crest over another? We might mark a wave crest corresponding to a feature of telescopic interest, such as the one just

Fig. 4.1. Part of a wave.

passing through the aperture. Such an assignment has no meaning, however. Light doesn't care what is planned for it; it has no memory or motivations.

The answer to the original question is easy but unsatisfying. The wave isn't at any specific location. In fact, a better way of describing a wave is to recognize it as a dynamic process.

The important thing to remember is that a wave has sources (lamps, lasers, stars, etc.) modified by boundaries. The sources are the points that can be enclosed in volumes observed to be emitting more energy than they are receiving. The boundaries are comprised of the diffracting obscurations, the edges of the aperture, the reflecting surfaces, and the transition surfaces between refracting media. The behavior of the wave at those surfaces (whether it is absorbed, reflected, or transmitted at a slower speed) must be specified. To describe diffraction properly, we must solve what is called a boundary-value problem for the wave equation in the region of interest.

Where a wave is coming from, how strong it is, and where it is going are more important than where it is. Our wave is like a choreography script, with each tiny location in space having its own dancing instructions. The wave we perceive is the totality of all of these dancers moving in concert.

The very fact that we can't point at a particle of light and say "it is there" puts some uncertainty on light's ability to be directed. This softness in light's location induces the blurring of the final image.

The quantum revolution revealed that light is packaged into discrete particles called photons. Since the quantum theory is a refinement of earlier methods, doesn't that mean that light is comprised of little rays after all?

The quantum theory is strange and non-intuitive. To understand it, we must be willing to abandon our experience of the macroscopic world. We are just too big to use what we know about the world in an effort to visualize the quantum universe.

In quantum mechanics, a way one can describe the position of a photon before it has been detected is to give its probability density, i.e., the likelihood of encountering the photon at a particular location. A particle's probability density is related to its wave function. Where the wave function is strong, the probability density is high. But the density is only measurable by running many photons through the optics.

Perhaps the best way of explaining this principle is to imagine the following experiment. Say we had the aperture of Fig. 4-2a, and a single photon was coming from the left. Photographic film is at the focus. The photon strikes the emulsion at location A, triggering a response in a single grain of the photosensitive material. If we were to stop here and develop the film, we would already have contradictory evidence. On one hand, the darkened grain isn't located precisely at the location of geometric focus, which is where the dotted line intersects the film. But the emulsion was only struck at that one point, leaving us with the impression that a particle was detected. "Maybe the alignment is off," we might say, and carefully run the experiment again. Now it hits at B.

Fig. 4-2. a) One hit at A; b) another hit at B; c) many hits with histogram.

Fig. 4-2. a) One hit at A; b) another hit at B; c) many hits with histogram.

At this point sensible photographers would recognize that they will run out of film before they get any closer to the truth. For the next trial, we leave the film in long enough to gather many photon hits. We divide the area into miniature areas called picture elements, or pixels, then count all the hits within each pixel and plot them to the right as in Fig. 4-2c. Now we see a trend—the diffraction pattern is beginning to show. But we already know that such behavior is explained by a wave.

So, we return to the original question, is light best described by particles or waves? On one hand, it is clearly observed that something impacts the photographic emulsion at individual locations. You can dim the intensity of the light until the grain darkening rate is arbitrarily slow. That seems to indicate tiny particles. On the other hand, lots of such individual particles are statistically distributed in the pattern expected for a wave.

We took it as an article of faith that light has no intelligence or motives. Here we seem to find that photons apparently can remember where other photons have already been detected. Record-keeping and communications are two more powers that are difficult to ascribe to light particles.

We have encountered so-called "wave-particle duality," one of the seeming paradoxes of quantum physics. The whole idea of particles (whether they be photons, electron, quarks, or whatever) whizzing around is an example of one of those concrete real-world ideas that doesn't apply at the quantum level. We cannot impose our kind of macroscopic reality on the microscopic situation. We would like particles to be neatly spherical, to be colored with different hues, and all to have a bright, rectangular highlight where a window reflects on the upper left side. But the quantum world doesn't present us with such comforting pictures. The important thing to remember is that we cannot measure whether a photon is a particle or a wave at the same time.

The only way we can determine the existence of a photon is to pin it against a photodetector. We can think of it as a wave as long as we don't try to locate it. When we do, its probability density collapses from an infinity of locations defined by the wave function to a single point. We can look at the tiny grain of the photograph and say "it was there" as long as we realize that the verb is in the past tense. The only way to be sure a photon exists is to destroy it.

The first physicist to put this in mathematical form was Werner Heisenberg, who expressed what is known as the uncertainty—or better, "indeterminacy"— principle. Far from being a discouraging limit to what we can know, this principle tells us new things about the dynamics of particle motion. One can even use the indeterminacy principle to derive an approximate expression for the radius of the diffraction disk.

What is unknown about this situation is where the photon (that is, the "ray") entered the aperture. We know where it ended up and where it started, but we are uncertain about the intervening path. The Heisenberg indeterminacy principle is stated roughly as follows: The uncertainty in the momentum of a particle in a certain direction times the uncertainty of its position is approximately a very small constant, or

where Apy is the fuzziness in momentum in the y direction, Ay is the fuzziness in position y, and h is that small number, called Planck's constant. We can rewrite this approximation to find the spread of the detection angle after many counts:

p d^tion

The expression for the momentum of a photon is the energy divided by the speed of light, or E/c. The photon's energy is E = hv. The constant c is the speed of light, v is the frequency of the light, and 1 is the wavelength; c = vk describes the way these quantities interrelate. The uncertainty in position Ay is just the diameter of the aperture D. Thus, the uncertainty in angle becomes

diffraction vhD D

This result is close to the 1.22k/D expression for the angular radius of the Airy disk.1