Illuminance Part 2: Falloff
In Part 1, we looked at how a fixture manipulates its distribution of light. Because the distribution describes how light spreads out from a source, for Part 2, it seems logical to explore how this spreading out directly influences the loss of illuminance measured at further distances. Conceptually, this is easy enough to understand when considering light as a quantity that thins out as it expands, like an inflating balloon. More obviously, this can be observed by noting that Illumination drops as the distance from a light source increases, and Illumination increases as the light is moved closer.
The standard depiction of the Inverse-Square Law at work
The rate of expansion, and consequently the rate at which illumination decreases is referred to as the Falloff Rate. Many of us were taught that the falloff rate for a light source is absolute and follows the Inverse-Square Law of decay. As such, understanding this Law can be a helpful tool on set. However, while the Law is scientifically sound in reference to a "point source," implementing the law on set to calculate illuminance at a distance will often produce results that are inconsistent with measurements. Nevertheless, it is important that we review the Inverse-Square Law before attempting to understand the limitations of its applications.
The easiest way to understand the Inverse-Square Law is usually to apply the rule that if you double the distance, you take one-quarter of the original intensity.
Energy at new distance
This is where the name of the law comes from -
Inverse: Taking the reciprocal of the distance
(2 feet becomes ½)
Squared: Raising that reciprocal to the power of 2
(2 feet becomes ½² = ¼)
However, while this law is absolutely true, the variables at play will often produce results that prove problematic when trying to use this formula on set.
Consider the following situation:
You are shooting a lower-light scene with an actor with a very dark skin tone.
The DP is shooting at ISO 800 with a wide open lens set to f4.
Your key light is currently shooting through an 8x8 frame of Magic Cloth™ at 5 feet away, the illuminance at the subject is measuring 3.125 footcandles.
To match the mood of the scene and due to the skin tone of the actor, the DP cannot have the actor more than 1 stop underexposed.
Proper exposure requires 25 fc meaning the least acceptable amount of light is 12.5 fc (1 stop under 25 fc).
Using the Inverse-Square Law principle, you believe that if you were to move the light and the 8x8 to the edge of the camera frame, you would be halving the distance (2.5 ft away), and multiplying the illuminance by 4x. This would give you the 12.5 fc minimum that you need. However, when you halve the distance, you’re only reading 5.4 fc: about 2⅓ stops below proper exposure! Why did this happen?
(Hover over image to view change)
The graphs above highlight one of the problems with using the Inverse-Square Law. As was said earlier, the Law applies to the expansion of light from a point source. A point source expands isotropically (equally in all directions) and therefore has a consistent rate. However, in the scenario and graphs above, the source is actually made up of billions of point sources, all with equal intensity and arranged along a plane. At a far enough distance (about 5x the largest dimension) this collection of point sources will have combined to appear as one source and will begin to falloff at the rate of the Inverse-Square Law. In the meantime, before that distance is reached, the falloff rate will be dynamically changing. The above scenarios nicely demonstrate how the distance, relative to the size/shape of a collection of point sources, all equal in intensity can skew the law's expectations. The greater number of point source's in an area, the more their combined distribution patterns will merge and influence the illumination level at any given distance.
While distance, relative to size is one way that a fixture can influence the distribution pattern, as we observed in Part 1, many fixtures employ the use of lenses or reflectors as a means of redistributing illuminance, also altering the distribution pattern. In some cases, this produces a greater concentration of light but a falloff rate that is greater than the Inverse-Square Law! In any case, this reveals an important lesson to remember:
The Distribution Pattern directly influences the Falloff Rate
A falloff rate that does not follow the Inverse-Square Law for calculation can be represented by changing the exponential power in the equation. Calculating an exponent greater than 2 means that the fixture will lose illuminance sooner over distance, whereas an exponent that is less than 2 indicates the fixture illuminance will decrease more gradually over distance.
>2 = more drastic falloff
<2 = more gradual falloff
In all cases, the exponent is actually dynamic: changing as the measured distances change. At a certain distance, this change will begin to level-out to an exponent around 2. However, with many of the fixtures that we use, that leveling-out does not occur within the common distances that the fixtures are used at. Therefore, it is best to approximate a power series trend-line to find the falloff rate across two commonly used distances. Often the deviations from the Inverse-Square Law are minor and make little difference. However, there can be situations where these deviations can make or break a lighting set up and are extremely useful to understand and be able to reference in regards to choosing fixtures. To understand the complex systems at work that transform the assumed behavior of light, we encourage you to check out our in-depth look at light behavior that will be released soon.
Below, we have included a database of each fixture we tested along with a description of some of their relative variables (listed under style). The right column gives the trend-line’s calculated exponent from 1 to 5 meters.