THE INVERSE-SQUARE LAW

 

 

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 Inverse-Square Law?

 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 On?

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 inverse-square relationship.

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 following graphic. 

Some exceptions

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

Fresnel Devices

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 below.

Honeycomb Grids

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.  

Consequences 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 consequences.

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.    

Guide 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.