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 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.
The standard depiction of the Inverse-Square Law at work
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.
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 8×8 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?
While , as we observed in Part 1, redistributing illuminance, also For more optically pure fixtures, his can result in “virtual origin points” that do follow the law — but from a different origin point. While other, less optically pure fixtures, like many fresnels or PARs, will have a number of variables altering the falloff rate. 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:
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.
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.
Approximation of the Inverse-Square law error, as a result of size, relative to distance. With both axis on a LOG scale, the Inverse-Square law is graphed is a constant, diagonal line. as the measurements are taken closer to the source, less of the distribution pattern is contributing to the total measurement. At a close enough point, there is no change in illuminance (radiance) when the distances change – This is represented by the constant, horizontal radiance conservation line. At a distance equal to 10x the source’s radius (5x the largest dimension), an evenly luminous, circular source will be less than 1% different than the Inverse Square law.
Falloff Graphs for an Astra Bi-Focus. Actual measurements follow the blue lines. 1 to 10 meters trend lines follow the red lines. While each has the same size dimensions, the Spot (left) has a different distribution pattern than the Flood (right). this results in very different falloff rates from 1 to 10 meters (Spot = 1.89, Flood = 1.98).
Falloff Graph for a full-spot “Junior.” The dark blue line represents actual measurements, the light blue line represents the trend line from 1-10 meters. the red line represents those measurements when distance is adjusted to the calculated “virtual point source.” At a far enough distance, the blue line will curve to meet and follow the red. However, because of the optically created distribution pattern, this distance will be much further than 5x the largest dimension.
This formula works similarly to the Inverse Square Law but with a fixed 1m Illuminance that is based off of a range of distances from 1 to 10 meters. Every fixture has an “e” number, representing a calculated approximation of energy at 1 meters, and a “regression coefficient,” representing a calculated average falloff rate that, when used with E, follows the same dynamic falloff of the fixture from 1 to 10 meters. To calculate the illuminance at any distance, use the following formula, inserting the fixture’s “e” number and regression coefficient.