In this section we'll take a look at the
inverse-square law and what it tells us about light fall-off.
We'll also look at guide numbers and how they are related to this law.
If you hate math, you may want to skip this topic.
What is the
The inverse-square law for light intensity
states: The intensity of illumination is proportional to the inverse
square of the distance from the light source.
What is this Law Based
Light is just one of the many physical
phenomena that follow an inverse-square relationship. The following
example should shed some light on the subject.
Consider the simple apparatus shown
below. It consists of a very small light bulb suspended and powered by two
very fine conductors. At any given moment, the bulb will emit light with a
certain power. That power will stream out equally in all directions,
forming an ever-increasing spherical wave. While the total power of that
wave will remain the same, it will be spread over an increasing surface as
the light moves away from the source. Clearly, the light intensity, which
is the power per unit area, will diminish as the distance from the source
increases. The calculation to link intensity to distance for this example
is shown to the right of the apparatus As expected, we get an
Light bulbs and bare-bulb strobes clearly
follow the inverse square law. They are not alone. Most sources
that employ reflectors or modifiers to direct light into a conical beam will
also follow the law. The conical-beam case can be thought
of as a sub-case of our bulb example above. Consider what happens when
we replace the bulb in the example above with a source that throws a cone of
light. If we look at how this conical beam intersects the concentric
spheres of the bulb example, we will discover that the beam always cuts
through the same fraction of the total spherical surface area.
As a result the intensity will diminish at exactly the same rate as it did
for the bulb. This is illustrated, rather lamely, in the
You can use the inverse-square law to predict the rate
of illumination fall-off for bare-bulb sources, speedlights, and studio
strobes. Umbrellas and softboxes will follow the law quite
closely for distances greater than twice their diameter. So,
what are the exceptions?
Sources that use optical or physical means to
collimate light will not follow the inverse-square relationship, at
least when based on their physical location. Such devices include,
optical spots (elliptical, Fresnel) and sources whose beam has been modified
by a grid. When working very close to large, diffuse sources,
such as softboxes and scrims, the fall-off rate will be somewhat less than
predicted by the inverse-square law.
A look at
Fresnel Lenses & Grids
Lighting fixtures with Fresnel lenses are frequently
used in motion-picture work. They are also used in today's fashion
work and were the mainstay of 1930s-era Hollywood photographers.
Fresnel lenses bend the light of a source into a column of nearly parallel
rays as shown in the illustration below.
A Fresnel source will not follow the inverse-square
law when making measurements based on the position of the light source.
However the inverse-square law is not ultimately violated. The Fresnel
lens works much like a telephoto lens. It creates a source with an
increased effective "focal length". This is shown in the illustration
Grids narrow the angle of coverage of a source.
As a result, they alter the effective "focal length" of the source and will
not follow the inverse-square law for the same reason stated for Fresnels
above. The illustration below shows how a grid works.
Grids, whether made of a cloth matrix or a aluminum
honeycomb, narrow the beam by blocking light rays that fall outside a
desired coverage angle. Rays that are not within this angle will be
absorbed by the grid walls. As grids absorb energy, they reduce the
amount of light passed. The deeper and the narrower the grid channels,
the narrower the beam and the greater the light loss.
of the Inverse-Square law
As most of the mathematical relationships we encounter
are linear, many of us would guess that doubling the distance from a light
source would result in a halving of
intensity. Of course, we know otherwise. The inverse square law tells us
that the intensity would fall to square of one half or one fourth. At three times the distance,
it would fall to the square of one third or one ninth, and so on. This sharp rate of fall-off has many
An important consequence of this law is that intensity
variation with a change in position is far more pronounced close to the
source. For instance, moving your subject back one foot when they are
30 feet from the light source will result in an insubstantial change in
intensity, certainly nothing that would require an adjustment to your camera
settings. On the other hand, moving your subject back one foot when
the subject is two feet from the source will result in more than an f-stop
loss in intensity, a significant difference.
The inverse-square fall-off also explains why power
requirements can vary widely. A portrait photographer doing headshots can
get by with a few hundred watt-seconds of flash energy, but a fashion
photographer may need several thousand to light his larger and deeper set.
Numbers and the Inverse-Square Law
Guide numbers were a necessity in the days before
auto-exposure flash. If you are old enough to have used flash bulbs,
you know that. Today, they crop up primarily in marketing
materials as a relative measure of a strobe's maximum output.
While few people still make exposure calculations based on guide numbers,
guide numbers can be terribly useful when you're forced to use your flash in
manual mode or when determining if the flash has enough guts to do the job.
Common calculations using the guide number are shown below.
Here is a rather rambling derivation of the guide
number formula based on the inverse-square law and the mathematical
relationship of f-stops to intensity. I'm sure there are better
ways to present this, but this is how I reasoned it through.
At first sight, f-stops are a seemingly strange and
arbitrary sequence of numbers. As it turns out, they're based on the
powers of the square root of two (1.4142135...). The even powers of
root two give whole numbers (1,2,4,8,16,32,64) and the odd -power f-stops
are rounded (1.4, 2.8, 5.6, 11, 22, 45). Each greater f-stop results
in an aperture that reduces the light intensity on the film/sensor by half.
Another way to look at it is that a doubling in the intensity of the scene
lighting, will require moving to the next higher f-stop to maintain correct
exposure. So we have a relationship between f-stops and intensity.
The problem is that the relationship is not mathematically linear, i.e.
Intensity can't be represented by the f-stop multiplied by some constant.
But, if we square the f-stops, we will get the linear relationship we need.
When squared, each successive f-stop is equal to twice the prior, i.e.
(1.4, 2, 2.8, 4, ...) becomes (2,4,8,16, ...). The intensity can then
be represented as a constant times the square of the f-stop. The
derivation of the guide number formula is shown below.
When using guide numbers, it is important to first
determine if the guide number is based on distance in meters or feet.
Also, guide numbers are usually quoted based on a particular speedlight zoom
setting or for a strobe with a specific reflector fitted. Changing
zoom settings, or using alternate reflectors or lighting modifiers will
change the guide number.