The light field

For the present purpose radiation is sufficiently described through rays, which are directed straight lines, or photons, which propagate via rectilinear orbits. These entities issue forth from primary radiators (the sun, light bulbs) and are scattered from objects that thereby become secondary radiators. Empty space (air will do) does not interact with the radiation. When rays or photons enter the eye, one “sees light” (or rather simply “sees”). Light is an aspect of consciousness; radiation is a physical entity that is never seen.

Consider empty space. It is filled with crisscrossing rays or swarms of photons. Consider some fiducial (and imaginary) volume element and add up the lengths of all rays that happen to cross it. When this total is divided by the volume, you obtain the volume (ray) density of radiation. Alternatively, you may count all photons that happen to be inside the volume within the span of 1 fiducial second of time. When this total is divided by the volume, you obtain the volume (photon) density. The two measures stand in a fixed ratio and need not be distinguished here. Volume density is important in photo kinesis of simple organisms but is rather irrelevant to the human observer.

One may also consider some fiducial (and imaginary) element of area and count all rays or photons that cross it (within 1 fiducial second, say). Add one for a crossing in one direction and subtract one for the crossing in the reverse direction (thus, you need an oriented area). The net count, divided by the area, depends upon the orientation of the area. For a small element, it is proportional to the cosine of the angle subtended by the surface normal and some direction that is characteristic of the light field. The magnitude and direction defined in this way comprise the net flux vector, which is an important descriptor of the light field (Gershun, 1939) (Fig. 72.1). It is the causal factor in the photo taxis of simple organisms. The net flux vector is the entity that photographers and interior designers (perhaps unknowingly) refer to when they discuss the “quality of the light.” Consider any closed loop in empty space. When you propagate the loop in the direction of the (local) net flux vector, you generate a tube. Area elements of the boundary of such tubes do not pass any net flux by construction (equally, many rays cross in either direction); thus, the light can be said to be transported by way of such light tubes. In contrast to light rays, light tubes are generally curved. They can even be closed. When a photographer refers to (diffuse) light “creeping around” an object, he or she is (unknowingly) referring to the tubes rather than the rays (Hunter and Fuqua, 1990) (Fig. 72.2). The human intuitive understanding of the behavior of diffuse light fields is based upon lifelong experience with net flux vector fields.

Figure 72.1..  

This is one method used to observe (or even measure) the net flux vector field. Put a grease spot on a piece of bond paper. The spot appears lighter than the paper (left) if the back of the paper receives more irradiance than the front, whereas it appears darker than the paper (right) in the opposite case. The spot disappears (center) when the net flux vector lies in the plane of the paper. This way, you can easily map out the flux tubes in a room.

Figure 72.2..  

A uniform hemispherical diffuse beam is incident from above (the top row of arrows shows the direction of the incident net flux vector). The lower part of the figure shows how the net flux vector field is influenced by a black (fully absorbing) sphere. Both direction and magnitude are indicated. The illuminance on the surface of the sphere is indicated by shading. Notice that the whole surface is illuminated, also the part that faces away from the source. The obstacle influences both the magnitude and the direction of the net flux vector field: the light tubes “bend around the object.”

Finally (after volumes and surfaces), one may consider lines. How many rays go in a certain fiducial direction through some fiducial point? Clearly, none, for if the number of rays is finite, it is infinitely unlikely to meet with any specific possibility. In order to count a finite number of rays, one needs a finite “environment” of the fiducial ray, known as the phase volume element or étendue (Gershun, 1939; Moon and Spencer, 1981). The étendue is like a slender tube, characterized by the product of its (normal) cross-sectional area and its (solid) angular spread, for neighboring rays can be shifted in space or perturbed in direction. The number of rays in the tube divided by the étendue of the tube is the radiance. The radiance is the single most important entity for the human observer; what we refer to as the light field is simply the radiance distribution. Both the volume density and the net flux vector field can be derived (through suitably averaging over space and/or direction) from the radiance. In empty space the radiance is constant along straight lines. If you pick any point and consider all lines (directions) through the point, you obtain an extreme “fish-eye-like” picture: the radiance in all directions. Thus, the radiance contains all pictures you can shoot from any point in any direction. Indeed, an excellent (because very intuitive) way to think of the radiance is as an infinite filing cabinet [like Borges's (1970) creepy library] of all possible pictures of your world. Such an intuitive grasp was, for instance, used by Gibson (1950) in his book on ecological optics. The concept of radiance has existed for centuries [Leonardo da Vinci (1927) understood it; the first to achieve some formal understanding were Lambert (1760) and Bouguer (1729) in the eighteenth century]. Very nice introductions are Gershun's (1939) paper and Moon and Spencer's (1981) book (which unfortunately suffers from arcane terminology), but the literature tends to associate the notion with Adelson and Bergen's (1991) plenoptic function.

Rays and light tubes originate from primary radiators and may end at absorbing (blackish) surfaces. They typically neither start nor end in empty space. The only exceptions are spatial sources or sinks. For instance, a volume of air may become a secondary source due to scattering of sunlight. This is important in atmospheric perspective [or air light (Koschmieder, 1924)]. Likewise, black smoke may cause the air to kill rays in midflight and thus form a sink. For the sake of simplicity, we will not consider such cases here.